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 , University M.Sc, McGill A THESIS SUBMITTED • . of B r i t i s h C o l u m b i a , 1978 U n i v e r s i t y , 1981 IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES Department We a c c e p t to of Physics this thesis the required as c o n f o r m i n g standard THE UNIVERSITY OF BRITISH COLUMBIA F e b r u a r y 1985 © G o r d o n L e w i s G i l e s , 1985 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of requirements f o r an advanced degree a t the the University o f B r i t i s h Columbia, I agree t h a t the L i b r a r y s h a l l make it f r e e l y a v a i l a b l e f o r reference and study. I further agree t h a t p e r m i s s i o n f o r e x t e n s i v e copying of t h i s t h e s i s f o r s c h o l a r l y purposes may department or by h i s or her be granted by the head o f representatives. my It i s understood t h a t copying or p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l gain s h a l l not be allowed without my permission. Department o f P The 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 Main Mall Vancouver, Canada 1956 V6T 1Y3 Date DE-6 (3/81) February, 1985 written Abstract The p o l a r i z e d cross-sections of the pp-»Tr precision proton + and u n p o l a r i z e d and t h e a n a l y z i n g d reaction of b e t t e r 475 MeV with unpolarized cross-sections measured a t 375, beams. A n g u l a r polarized from cross-sections differential are statistical several 20° t o 1 5 0 ° . The incident unpolarized were measured a t 350, incident beams. The and a n a l y z i n g using cross-sections with e x i s t i n g data The 425, were polarized incident and a r e expanded Legendre polynomial measurements. 375, polarized powers and t h e a°° and b ? ° e x p a n s i o n the r e s p e c t i v e compared to a 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 L e g e n d r e and A s s o c i a t e d to over distributions 350 and 500 MeV f o r 450, and 498 MeV differential respectively, angular have been measured between center-of-mass angles and power t h a n one p e r c e n t beam e n e r g i e s differential differential into series coefficients f i t resulting coefficients and r e c e n t theoretical predictions. The o b s e r v a t i o n is interpreted from t h e G « 1 498 MeV. N-N of s i g n i f i c a n t n o n - z e r o a ^ as i n d i c a t i o n of a s i g n i f i c a n t partial 0 coefficent contribution wave c h a n n e l a t e n e r g i e s a s low a s Acknowledgements I am grateful t o my c o l l e g u e s E.G. G.J. L o l o s , B . J . M c P a r l a n d , D. W.R. Falk, A u l d , G. O t t e w e l l , P.L. Jones, Walden and 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 experiment. Furthermore, I would like 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 this t o thank Jean H o l t f o r D o r o t h y Sample f o r h e r a s s i s t a n c e w i t h t h e a n a l y s i s o f t h e data. Special t h a n k s a r e 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 my Ph.D. committee, c o u r s e o f my support. for his s k i l l f u l I e x p r e s s my family, of guidance throughout s t u d i e s and t h e c o m p l e t i o n o f t h i s Above a l l , p a r e n t s and Garth Jones, the chairman sincerest the thesis. gratitude to my f o r t h e i r c o n t i n u o u s encouragement and Table of Contents 1. Introduction 1 2. Theory 5 3. and Formalism 2.1 The D i f f e r e n t i a l Power C r o s s - S e c t i o n s and A n a l y z i n g 2.2 P h e n o m e n o l o g i c a l D e s c r i p t i o n s o f t h e pp->7r d Reaction 7 2.3 Spin Amplitude A n a l y s i s 7 2.4 Orthogonal Expansion of Observables 11 2.5 D i s c u s s i o n of Theory 16 + E x p e r i m e n t a l A p p a r a t u s and Method ...19 3.1 Introduction 19 3.2 Cyclotron 20 3.3 Beam L i n e a n d 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 23 3.5 Apparatus 23 3.6 S c a t t e r i n g Chamber 25 3.7 Deuteron 27 3.8 T a r g e t s a n d Beam A l i g n m e n t 28 3.9 Particle 28 and C u r r e n t M o n i t o r Horn D e t e c t i o n System 3.10 E l e c t r o n i c L o g i c a n d S y s t e m s 31 3.11 35 Trigger Circuit Timing 3.12 D a t a A c q u i s i t i o n 4. 5 Software 37 A n a l y s i s of t h e Data. 40 4.1 Introduction 40 4.2 Experimental Evaluation Cross-Section 4.3 Event-by-Event 4.3.1 Treatment of t h e D i f f e r e n t i a l Data A n a l y s i s o f t h e Raw D a t a iv 40 43 43 4.3.2 The Primary Events 4.4 45 4.3.2.1 P u l s e - H e i g h t D i s t r i b u t i o n s 46 4.3.2.2 T i m e - o f - F l i g h t D i s t r i b u t i o n s 51 4.3.2.3 Kinematic D i s t r i b u t i o n s 56 4.3.3 The U n c o r r e l a t e d 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 M u l t i - W i r e Proportional-Chamber Efficiencies 65 4.3.6 Beam P o l a r i z a t i o n 66 4.3.7 Beam Current N o r m a l i z a t i o n 66 S o l i d Angles 68 4.4.1 Geometric S o l i d Angles 68 4.4.2 T r a n s f o r m a t i o n of the S o l i d 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 S o l i d Angle 73 4.4.5 The Muon Component of the E f f e c t i v e S o l 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 4.5 4.4.7 Comparison of the S o l i d Angle Models to Monte C a r l o E v a l u a t i o n s 80 4.4.8 Energy-Loss 82 D e t e c t o r and Geometric Calibrations 86 4.5.1 M u l t i - W i r e P r o p o r t i o n a l Chamber Calibration 86 4.5.1.1 The D e l a y - L i n e 87 4.5.1.2 The Anode Wire D i s t r i b u t i o n Image 4.5.1.3 C a l i b r a t i o n i n the V e r t i c a l Direction 87 v 88 4.5.1.4 C a l i b r a t i o n Direction i n the H o r i z o n t a l 94 4.5.1.5 S p a t i a l R e s o l u t i o n 96 4.5.2 S c i n t i l l a t o r Central Offsets 97 4.5.3 C a l i b r a t i o n of the Deuteron Arm Aperture 4.5.4 Absolute C a l i b r a t i o n of D e t e c t i o n P o l a r Angles 4.5.5 C a l i b r a t i o n of the Azimuthal Horn 99 Arm 99 Angle i n the Plane Normal to the Beam D i r e c t i o n 4.6 4.7 ..105 Carbon Background 105 4.6.1 Measurement of the Carbon Background ....108 4.6.2 Quasi-Free P a r a m e t e r i z a t i o n of the Carbon Background 4.6.2.1 F i t of the Carbon Background to the Model Experimental 4.7.1 4.7.2 4.7.3 Results 115 116 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 : U n p o l a r i z e d Beam 116 4,7.1.1 The U n c e r t a i n t y of the D i f f e r e n t i a l Cross-Sections: U n p o l a r i z e d Beam 124 The D i f f e r e n t i a l P o l a r i z e d Beam 131 Cross-Sections: 4.7.2.1 The U n c e r t a i n t y of the D i f f e r e n t i a l Cross-Section: P o l a r i z e d Beam 132 The 134 Polarized D i f f e r e n t i a l Cross-Section 4.7.3.1 The U n c e r t a i n t y of the Differential 4.7.4 111 The Polarized Cross-Section A n a l y z i n g Power 4.7.4.1 The ...139 U n c e r t a i n t y of the Analyzing power 4.8 Analyzing Powers: Kinematic 4.9 D i s c u s s i o n of U n c e r t v ia i n t i e s 134 1 39 Event D e f i n i t i o n ..143 147 5. 6. 4.10 F i t of the Unpolarized 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 t o a Sum of Legendre Polynomials 150 4.11 F i t 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 to a Sum of A s s o c i a t e d Legendre Polynomials 158 D i s c u s s i o n of the R e s u l t s 161 5.1 Introduction 161 5.2 The U n p o l a r i z e d D i f f e r e n t i a l 5.3 The P o l a r i z e d D i f f e r e n t i a l Cross-Section ....162 Cross-Section 170 Conclusion 178 APPENDIX I : THE DIFFERENTIAL CROSS SECTION FOR PROTON-PROTON ELASTIC SCATTERING AT 90°C.M. BETWEEN 300 AND 500 MEV 183 APPENDIX I I : THE MONTE CARLO • 189 11.1 Introduction 189 11.2 Apparatus Geometry and M a t e r i a l 192 11.3 Physical Interactions 192 APPENDIX 3: ANALYZING POWER OF THE p p - > 7 r d AT 37 5, 4 50, AND 500 MEV. INCIDENT PROTON ENERGIES LIST OF REFERENCES + vii 195 198 L i s t of Tables 2.1. P a r t i a l Wave Channels and Amplitude D e s i g n a t i o n . . . 1 0 2.2. 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 P a r t i a l Wave Expansion C o e f f i c i e n t s 13 2.3. The A n a l y z i n g Power P a r t i a l Wave - Expansion Coefficients 14 3.1. The Detector Geometry 33 3.2. Q u a n t i t i e s Processed by CAMAC S c a l a r s . . 38 4.1. The C o r r e c t i o n s to S o l i d Angles A s s o c i a t e d 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 P o s i t i o n a l C a l i b r a t i o n . . . . 100 4.4. The E x p e r i m e n t a l l y Determined Detector 4.5. 4.6. The 350 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 125 The 375 MeV. 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 and A n a l y z i n g Powers..126 4.7. The 425.MeV. D i f f e r e n t i a l 4.8. The 450 MeV. 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 Terms and A n a l y z i n g Powers 128 4.9. The 475 MeV. D i f f e r e n t i a l 129 4.10. The 498 MeV. 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 Terms and A n a l y z i n g Powers 130 4.11. The 375 MeV. A n a l y z i n g Powers 144 4.12. The 450 MeV. A n a l y z i n g Powers 145 4.13. The 498 MeV. A n a l y z i n g Powers.. 146 4.14. F i t s 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 s to a Sum of Legendre Polynomials..152 4.15. R a t i o 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 Expansion C o e f f i c i e n t s t o the Total Cross-Section viii Offsets...106 Cross-Sections Cross-Sections 127 154 4.16. 4.17. I I . 1. F i t s 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 to a Sum of A s s o c i a t e d Legendre Polynomials R a t i o 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 Expansion C o e f f i c i e n t s Total Cross-Section D e f i n i t i o n of a D e t e c t i o n Arm ix 159 to the by Regions 160 193 L i s t of F i g u r e s 3.1. TRIUMF F a c i l i t y 22 3.2. Beam L i n e Monitors 24 3.3. Apparatus 26 3.4. P a r t i c l e D e t e c t i o n System 29 3.5. Electronic 3.6. R e l a t i v e Timing 4.1. Pion and Deuteron S c i n t i l l a t o r P u l s e - H e i g h t s : P o l y e t h e l e n e Target 48 Pion and Deuteron S c i n t i l l a t o r P u l s e - H e i g h t s : Carbon Target 50 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 Pion S c i n t i l l a t o r Peaks and Cuts 53 4.2. 4.3. 4.4. T r i g g e r Logic and Schematic Diagram.... 34 of L i n e a r and Logic S i g n a l s Pulse-Height 36 Distribution 4.5. T i m e - o f - F l i g h t and Deuteron S c i n t i l l a t o r P u l s e - H e i g h t s : P o l y e t h y l e n e Target 54 4.6. T i m e - o f - F l i g h t and Deuteron S c i n t i l l a t o r P u l s e - H e i g h t s : Carbon Target 55 4.7. T i m e - o f - F l i g h t D i s t r i b u t i o n Peaks and Cuts 57 4.8.. A Typical 61 4.9. The E f f e c t i v e Muon S o l i d Angle F Parameters 4.10. Schematic Representation of the E f f e c t of P a r t i c l e Energy-Loss on the E f f 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: C e n t r a l 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 4.16. Pion, Deuteron, and E l a s t i c - P r o t o n Angular Correlation Distribution 79 region.90 95 x Detection Regions 1 02 4.17. The F r a c t i o n a l Carbon Background at 450 MeV 110 4.18. The E f f e c t i v 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 of the Carbon Background as a Function of cos(0) |cos(0) | ..114 4.19. The E f f e c t i v 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 of the Carbon Background 117 4.20. The 350 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 118 4.21. The 375 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 119 4.22. The 425 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 120 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 121 4.24. The 475 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 . . 122 4.25. 4.26. The 498 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 The 375 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 P o l a r i z e d Term 123 4.27. The, 450 MeV. D i f f e r e n t i a l P o l a r i z e d Term Cross-Sections: 4.28. The 498 MeV. D i f f e r e n t i a l P o l a r i z e d Term Cross-Sections: 4.29. The 375 MeV. A n a l y z i n g Powers 140 4.30. The 450 MeV. A n a l y z i n g Powers ....141 4.31. The 498 MeV. A n a l y z i n g Powers 142 5.1. The T o t a l 163 5.2. R a t i o of the C o e f f i c i e n t s Order Legendre Polynomial Cross-Sec t i o n of the Second Terms t o the T o t a l R a t i o of the C o e f f i c i e n t s Order Legendre Polynomial Cross-Section of the Fourth Terms t o the T o t a l 5.3. 5.4. 5.5. Cross-Sections 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 Polynomial Terms to the T o t a l Cross-Sec t i o n R a t i o of the C o e f f i c i e n t s of the F i r s t Order A s s o c i a t e d Legendre Polynomial Terms t o the xi ...135 ..136 137 164 165 166 5.6. 5.7. 5.8. 5.9. Total Cross-Section 171 R a t i o of the C o e f f i c i e n t s of the Second Order A s s o c i a t e d Legendre Polynomial Terms to the T o t a l C r o s s - S e c t i o n 172 R a t i o of the C o e f f i c i e n t s of the T h i r d Order A s s o c i a t e d Legendre Polynomial Terms to the Total Cross-Section 173 R a t i o of the C o e f f i c i e n t s of the Fourth Order A s s o c i a t e d Legendre Polynomial Terms to the T o t a l C r o s s - S e c t i o n 174 R a t i o of the C o e f f i c i e n t s of the F i f t h Order A s s o c i a t e d Legendre Polynomial Terms to the Total Cross-Section 175 xii 1 . INTRODUCTION The pp—>7r + d, study of the elementary pion p r o d u c t i o n r e a c t i o n , i s of fundamental s i g n i f i c a n c e . Not r e a c t i o n provide insight i n t o the fundamental process pion c r e a t i o n i t s e l f , but simultaneously i t provides i n t o the nature of the i n e l a s t i c behaviour nucleon-nucleon system. The with only does t h i s i t s r e l a t i v e l y simple understanding of of insight the of t h i s r e a c t i o n two-body i n i t i a l and final states p r o v i d e s a b a s i c element r e q u i r e d f o r the d e s c r i p t i o n of more general few-body systems. The pp—>7r + d reaction r e p r e s e n t s a s p e c i a l case of the more general r e a c t i o n , one where the f i n a l form a deuteron). As the s t a t e nucleons pp—>ir*& the pp—>7r*np are bound (to r e a c t i o n and i t s inverse (7r*d—>pp) can both be measured in the l a b o r a t o r y , reaction p r e c i s e comparison of measurements of the observables (such as the d i f f e r e n t i a l c r o s s - s e c t i o n and v a r i o u s spin-dependent q u a n t i t i e s ) provide a t e s t of fundamental symmetries such as time r e v e r s a l i n v a r i a n c e . Furthermore, these two reactions represent the simplest cases of nuclear pion p r o d u c t i o n the n u c l e a r (p,7r) r e a c t i o n f o r example) and of nuclear a b s o r p t i o n r e s p e c t i v e l y , s u b j e c t s of s i g n i f i c a n t (of pion current interest ' ' . 1 2 3 P r e c i s i o n measurements of q u a n t i t i e s such as 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 thereby the a n a l y z i n g powers) of the pp—>-n*d r e a c t i o n provide i n f o r m a t i o n regarding the nature inelastic intermediate of the h i g h l y s t a t e which c h a r a c t e r i z e s t h i s 1 2 reaction. The importance nucleon-nucleon observation of spin-dependent system h a s been of unexpected o b s e r v a b l e s of the reinforced energy by t h e d e p e n d e n c e s o f t h e Aa and Li Ao parameters T difference of the p r o t o n - p r o t o n between t o t a l anti-parallel proton direction i s either direction of the p r o t o n ' s Exotic reaction so-called to explain such such longitudinal, relative or t r a n s v e r s e , t o the motion) dependences such as those resonance", o b s e r v a t i o n s . Whether 7 precise experimental polarization , B In this thesis spin-averaged 498 MeV. require, i n performing of the s p e c i f i c however, a body o f the v a r i o u s observables. the spin-dependent for incident In a d d i t i o n , associated dependencies unpolarized d i f f e r e n t i a l pp—>-7r d r e a c t i o n + interest we d e s c r i b e t h e f i r s t measurements o f b o t h by some . data concerning dependent included a a n a l y s e s of the r e a c t i o n i n t o e x p l o r e the energy Such a n a l y s e s . r e q u i r e d h a s , however been t h e o f much c o n t r o v e r s y amplitudes. which 1 5 the i n t r o d u c t i o n of 6 partial-wave amplitude order which observables" have been p r o p o s e d Such o b s e r v a t i o n s have m o t i v a t e d full i s , the where t h e p o l a r i z a t i o n i n spin-independent mechanisms i s i n d e e d subject states, mechanisms, "dibaryon (that c r o s s - s e c t i o n s o f t h e p a r a l l e l and spin were n o t a t a l l e v i d e n t subsystem, proton precision polarized, and t h e c r o s s - s e c t i o n s of the e n e r g i e s from 350 t o we have measured and p u b l i s h e d t h e analyzing powers , 9 the spin dependent q u a n t i t y 3 more g e n e r a l l y (that i s , the most often) measured. Many p r o v i s i o n s a r e designed i n t o t h i s experiment to ensure r e l i a b l e r e s u l t s . A g e o m e t r i c a l l y - s i m p l e two-arm apparatus (devoid of c o m p l i c a t i n g magnets) was used to s i m p l i f y the d e f i n i t i o n angle of the e f f e c t i v e acceptance of the system. With t h i s apparatus, solid differential c r o s s - s e c t i o n measurements c o u l d be obtained over a l a r g e angular range i n the center-of-mass system (20° to 150°), thereby permitting accurate higher-order determination of the terms i n a s p h e r i c a l expansion 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 . The beam c u r r e n t was c a r r i e d out, i n e f f e c t , of the pp—>-pp e l a s t i c determination through simultaneous measurement r e a c t i o n (at 90° i n the centre-of-mass system) from the same p r o d u c t i o n t a r g e t as t h a t employed f o r the pp—>7r d p r o d u c t i o n . The r e q u i r e d pp—>pp e l a s t i c + 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 s s o c i a t e d s o l i d angles of the p p - e l a s t i c monitor were measured p r i o r to the pion production program. These r e s u l t s have s i n c e been p u b l i s h e d . T h i s method of beam current n o r m a l i z a t i o n has 1 0 the great advantage of being t h i c k n e s s , and of the angle independent of both the t a r g e t of the t a r g e t with respect t o the beam d i r e c t i o n . The nature of the kinematic transformation center-of-mass to l a b o r a t o r y c o o r d i n a t e a forward from the systems i s such that and a backward pion a r e both c o i n c i d e n t deuterons emitted i n t o a given apparatus was designed with l a b o r a t o r y s o l i d angle. The to permit simultaneous d e t e c t i o n of 4 these events. Because of the forward-backward the d i f f e r e n t i a l cross-section symmetry imposed symmetry of ( i n the center-of-mass), a by the f a c t that i d e n t i c a l p a r t i c l e s are i n v o l v e d , d e t e r m i n a t i o n of l a b o r a t o r y angle dependent f a c t o r s such as the system acceptance s o l i d a n g l e s , and pion-decay and e n e r g y - l o s s c o r r e c t i o n s can be The small carbon background (arising p o l y e t h y l e n e t a r g e t m a t e r i a l ) was verified. from the reduced through both the use of a p p r o p r i a t e event s e l e c t i o n and d i r e c t subtraction techniques. O v e r a l l , many steps have been taken throughout t h i s experiment to ensure the r e l i a b i l i t y measurements of the fundamental pp— >ir *d of our reaction. 2 . THEORY AND FORMALISM 2.1 THE DIFFERENTIAL CROSS-SECTIONS AND ANALYZING POWER If a p o l a r i z e d proton beam i s i n c i d e n t upon an target, the d i f f e r e n t i a l in terms of u n p o l a r i z e d cross-section unpolarized da/dfl can be w r i t t e n and p o l a r i z e d components, that i s ; do/dfl = d a / d f l + P*-n do,/dfl (01) 0 where: da /dfl 0 - Denotes the u n p o l a r i z e d differential cross-section. do^/dQ - Denotes the p o l a r i z e d differential cross-section. P - The i n c i d e n t proton beam polarization. Here n, i s a u n i t vector normal t o the s c a t t e r i n g plane i n the d i r e c t i o n k^ x k^ (the Madison Convention). C l e a r l y , i f the i n c i d e n t beam i s u n p o l a r i z e d unpolarized differential (|P|=0), then the cross-section results. If a p o l a r i z e d beam i s to be used, then both the unpolarized and p o l a r i z e d 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 a s s o c i a t e d with d i f f e r i n g of the beam p o l a r i z a t i o n v e c t o r s . of two such measurements performed 5 orientations Consider the s p e c i a l case with both of the beam 6 polarization vectors and with opposite perpendicular d i r e c t i o n s . Here, the dot products and P , the p o l a r i z a t i o n v e c t o r s are represented respectively, 2 with the u n i t vector n, where; = IP,| P} = -P -n = |P | = 2 The c o r r e s p o n d i n g dof/dO, then, between by t h e s c a l a r q u a n t i t i e s P f a n d Pf P,-n P| t o the s c a t t e r i n g plane 2 differential are given (02) c r o s s - s e c t i o n s d o f / d f l and by; d o t / d f i = d a / d f l + P| d a , / d f i (03) 0 daf/dJ2 = do /d$2 - Pf d o ^ / d f i 0 This system of l i n e a r equations p o l a r i z e d and u n p o l a r i z e d i s readily solved differential c r o s s - s e c t i o n s as a f u n c t i o n o f t h e two m e a s u r e d d i f f e r e n t i a l their f o r the c r o s s - s e c t i o n s and associated p o l a r i z a t i o n s ; that i s ; doo/dQ = i ( d a j / d f i + d o f / d f i ) - i ( dat/dfl ~ daf/dQ) P and d a , / d f l = ( da|/dJ2 - d a f / d f i )/( P| + P f ) where P = { ( P| - Pf )/( P j + Pf ) } The a n a l y z i n g power A , i s defined as the r a t i o of the (04) 7 p o l a r i z e d to unpolarized differential cross-section; that is; A no = ( d a i/ d n > / (da /dft) (05) 0 C l e a r l y , two c r o s s - s e c t i o n measurements, performed with differing beam p o l a r i z a t i o n s , are r e q u i r e d analyzing power f o r a given is t o d e f i n e the experimental c o n f i g u r a t i o n (as the case a l s o f o r do^/dA). Generally, measurement of the a n a l y z i n g powers r e q u i r e s a l e s s complex experimental procedure than t h a t the measurement of the d i f f e r e n t i a l or u n p o l a r i z e d ) . Since cross-section the a n a l y z i n g differential cross-sections, the a b s o l u t e differential (polarized power i s a r a t i o any systematic cross-sections pion-decay and e n e r g y - l o s s c o r r e c t i o n s ) of two uncertainty in (such as that due to u n c e r t a i n t i e s i n s o l i d angle, d e t e c t i o n 2.2 PHENOMENOLOGICAL DESCRIPTIONS required for e f f i c i e n c y , and simply c a n c e l out. OF THE pp-»--ir*d REACTION 2.3 SPIN AMPLITUDE ANALYSIS The p p — ^ 7 r d r e a c t i o n can be d e s c r i b e d + s t r u c t u r e of i t s i n i t i a l i n terms of the spin and f i n a l s t a t e s by a 4x3 dimensional T ( t r a n s i t i o n ) matrix. Each of these twelve complex amplitudes i s , i n t u r n , a f u n c t i o n of energy and s c a t t e r i n g angle, and i s uniquely particular transition initial, associated with a from one of the the four t o one of the three possible possible f i n a l spin states. 8 When the assumptions of p a r i t y c o n s e r v a t i o n and time r e v e r s a l i n v a r i a n c e are invoked, the number of independent T matrix amplitudes reduces to s i x , l e s s one a r b i t r a r y phase. Thus, there are i n a l l , eleven required to describe independent parameters t h i s r e a c t i o n at each kinematic configuration. When d e s c r i b e d laboratory i n terms of the usual frame spin q u a n t i z a t i o n matrix has poor r e l a t i v i s t i c spin-triplet d i r e c t i o n s , the T 1 1 transformation properties. A l t e r n a t i v e l y , formalisms c h a r a c t e r i z e d by s p i n directions either p a r a l l e l transverse (the h e l i c i t y quantization formalism) or (the t r a n s v e r s i t y formalism) to the d i r e c t i o n of the a s s o c i a t e d p a r t i c l e s ' motion, have been developed The 1 2 ' 1 3 . use of such formalisms i s j u s t i f i e d by the simpler relativistic transformation p r o p e r t i e s of the T matrix r e s u l t when the s p i n b a s i s s t a t e s are d e f i n e d that accordingly. T h i s s p i n amplitude formalism i s a l s o u s e f u l f o r p r o v i d i n g a framework i n which t o c o n c e p t u a l i z e r e a c t i o n , i n p a r t i c u l a r , to a p p r e c i a t e introduced + the complexity by the spins of the p a r t i c l e s , case, by only 6 complex amplitudes). the pp—>7r d (defined, in this Measurement of the angular s t r u c t u r e of a l l of these amplitudes as a f u n c t i o n of energy would r e q u i r e a very l a r g e number of experiments, depending, i n p a r t , on the number of angles r e q u i r e d t o d e f i n e the angular d i s t r i b u t i o n s . For beam energies i n the A(1232) isobar resonance r e g i o n , a d e s c r i p t i o n i n terms of a p a r t i a l wave expansion 9 o f f e r s 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 final s t a t e wave f u n c t i o n s i n t o 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 c e n t r i f u g a l b a r r i e r l i m i t s the number of p a r t i a l waves which can c o n t r i b u t e , can be d e s c r i b e d the system i n terms of a small number of p a r t i a l wave amplitudes. As the energy i n c r e a s e s , however, the number of amplitudes r e q u i r e d to describe markedly. The v a r i o u s associated the system p a r t i a l wave channels and the amplitude d e s i g n a t i o n s Mandl and Regge , and B l a n k l e i d e r 1 0 in table ( 2 . 1 ) . Also increases indicated (following the n o t a t i o n of and A f n a n ) are l i s t e d 1 5 i n the t a b l e (2.1) a r e some of the p o s s i b l e NA intermediate s t a t e s p e r t a i n i n g various t o the p a r t i a l wave channels. Consider, f o r example, the r e a c t i o n channel with the i n i t i a l nucleon-nucleon 'D 2 associated s t a t e and the a 2 p a r t i a l wave amplitude. Here, the two protons coupled t o a singlet spin s t a t e (S=0) and a D s t a t e (1=2) of r e l a t i v e angular momentum p r i o r t o the i n t e r a c t i o n and the subsequent formation of a NA intermediate s t a t e . The \ s p i n of the d e l t a can couple t o the i nucleon spin t o form e i t h e r a triplet (S=1) or a q u i n t u p l e t (S=2) s t a t e . Since the t o t a l angular momentum (J=2) and the p a r i t y i s conserved as the r e a c t i o n 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) f o r the q u i n t u p l e t spin state, or a D s t a t e f o r e i t h e r of these spin c o n f i g u r a t i o n s . The NA 10 Table (2.1) P a r t i a l Wave Channels and Amplitude PP Initial State 2S+1, I J parity NA Intermediate State 2S+1, 1 2 S + 1 J 'So 3 P, 3. 5 p - D5 F1 5 315 3 Pi 3. s 2 D 2 5 3 3 3 3 p i Ampli tude Designation L 1 S,p 3 3 ! TTd Final State Designation. 0 a 0 S,si a1 s,dr a 3 s,Pz a 2 s,f a 7 2 3 S,di a, 3 S,di a 3 S,d- a 6 S,gi a 9 S,gi a 1 o S,f J a s,h; a , 3 ' 5 c- - r ji • i 3 Fi 3. 5 3 , 5 ? i Fi... 3, 5p- 3 > 3 5P- 3 C 3.. . 3 Fi 3 F; 3 3 3 3 s 8 3 Here, J r e p r e s e n t s the t o t a l angular momentum of each s t a t e , 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 s t a t e , where there are three p a r t i c l e s , j and L denote the i n t e r n a l quantum numbers of the deuteron. 11 intermediate s t a t e then decays t o the f i n a l of a deuteron np state consisting ( s i m p l i s t i c a l l y d e s i g n a t e d here as a t r i p l e t system i n a S s t a t e of r e l a t i v e angular momentum) and a pion that respect i s i n a r e l a t i v e p s t a t e of angular momentum with t o the deuteron. E a r l y work ' 1 6 provided 1 7 i n d i c a t e d that the 'D 2 NN p a r t i a l wave the dominant c o n t r i b u t i o n to the s c a t t e r i n g amplitude. T h i s o b s e r v a t i o n was i n t e r p r e t e d i n terms of the formation of a NA intermediate simple c o n f i g u r a t i o n , particles s t a t e of a p a r t i c u l a r l y i n p a r t i c u l a r , a s t a t e with N and A i n a S (1=0) s t a t e of r e l a t i v e motion. 2.4 ORTHOGONAL EXPANSION OF OBSERVABLES Observables [O ), (where v simply v l a b e l s the observable) such as the d i f f e r e n t i a l c r o s s - s e c t i o n and the spin c o r r e l a t i o n parameters A — , ( f o l l o w i n g the proposal of N i s k a n e n , and using 1 8 the n o t a t i o n of B l a n k l e i d e r expanded i n terms of orthogonal f u n c t i o n s P^((6)) Associated Legendre f u n c t i o n s ) containing dependence. Here, the s u p e r s c r i p t da/dfi. In g e n e r a l , ) can be (typically the angular v denotes the A n Q and however; 4TT (doo/dQ) O where the u n p o l a r i z e d v = Z A? /»? differential f a c t o r e d out of the e x p r e s s i o n . A? are, 1 5 (06) c r o s s - s e c t i o n has been The expansion c o e f f i c i e n t s i n t u r n , l i n e a r combinations of b i l i n e a r products of the a p p r o p r i a t e p a r t i a l wave amplitudes, d e f i n e d by; 12 h?. = I C ? ( i , j ) ij where, f i n a l l y , the a. a.* (07) J coefficients are a f u n c t i o n of the a p p r o p r i a t e angular momentum c o u p l i n g 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 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 and the a n a l y z i n g powers are summarized here. 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 can be expanded i n terms of the (even o r d e r ) Legendre f u n c t i o n P.(cos(0 ) ) ; 1 7T 4rr ( d 0 / d G ) = I • a? i = 0,2,... o 1 Similarly, 0 P.(cos(0*)) 1 the a n a l y z i n g powers can be expanded i n terms of the f i r s t order A s s o c i a t e d Legendre f u n c t i o n s * o r d e r s ) , PJ(cos(9 )), that i s ; 4TT (da /dn) 0 The c o e f f i c i e n t s coefficients are (08) 17 listed A ^ = I no * o relating (of a l l b?° P. (cos(0*)) (09) 1 _1 the a ? o 0 and b?° expansion to the (sum o f ) b i l i n e a r amplitude p r o d u c t s i n table (2.2) and t a b l e 1 5 (2.3) r e s p e c t i v e l y , f o r amplitudes up to a . 8 When c o n s i d e r i n g the r e l a t i o n s h i p 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 to the p a r t i a l wave amplitudes, through the sum of a p p r o p r i a t e b i l i n e a r amplitude combinations, s e v e r a l o b s e r v a t i o n s can be made. The a°° coefficient (which i s simply the t o t a l c r o s s - s e c t i o n in this r e p r e s e n t a t i o n ) depends only on the sum of the squares of the p a r t i a l wave amplitudes. T h e r e f o r e , i t would be expected Table ( 2 . 2 ) The D i f f e r e n t i a l Bilinear Amplitude Products 2 a ai a a a< a a a a 0 2 2 2 2 3 2 2 5 2 6 2 7 2 8 Re Re Re Re Re Re Re Re Re Re Re Re Re Re Re Re a a *} a a *} a a *} a,a *} a ,a«,*} a,a *} a,a *} a a *} a a *} a a *} a a *} a a *} a»a *} C r o s s - S e c t i o n P a r t i a l Wave Expansion Coefficients. a 00 a a 0 0 a 2 a a n0 0 00 aa g 0 1 / 4 0 0 0 1 / 4 0 0 0 1 / 4 1 / 4 0 0 1 / 4 " 1 / 8 0 0 5 / 1 2 5 / 2 4 0 0 5 / 2 8 5 / 4 9 - 5 / 4 9 0 1 / 4 3 / 1 4 1 / 2 8 0 1 / 4 2 / 7 3 / 1 4 1 / 4 2 5 / 8 4 8 1 / 3 0 8 0 2 5 / 1 3 2 0 2 0 - 1 / 1 / 2 0 0 0 7 0 1 / 2 / 3 0 0 0 8 0 0 0 1 / 2 / 1 / 2 0 0 0 1 / 2 / 5 / 2 0 0 5 0 1 / 2 / 5 / 7 0 0 6 0 0 0 3 • 1 / 2 - 1 0 2 7 0 - 1 / 7 / 3 / 2 2 8 0 9 / 7 / 1 / 2 3 a 0 1 / 4 / 5 3 5 0 1 / 2 / 5 / 1 4 3 6 0 " 1 / 7 9 / 1 4 0 0 - 5 / 1 4 / 1 / 1 4 1 0 / 7 / 2 / 7 0 5 / 1 4 / 5 0 5 a a *} n 6 a a *} a a *} 0 1 / 7 / 5 - 3 / 7 / 6 0 5 / 7 / 1 / 2 0 0 0 0 0 5 6 0 1 / 7 / 1 0 / 7 5 / 7 / 5 / 1 4 7 8 0 - 1 / 7 / 1 / 3 - 1 5 / 7 7 / 3 The / symbol 0 - 2 5 / 1 i m p l i e s the square root of the q u a n t i t y to i t s right. 14 Table (2.3) The A n a l y z i n g Power P a r t i a l Wave Expansion Coefficients. Bilinear Ampli tude Products , no b, , no b , no b , no Im{a a,*} Im{a a *} Im{a a *} Im{a,a *} Im{a,a»*} Im{a,a *} Im{a,a *j Im{a,a *} Im{a a *} Im{a a„*} Im{a a *} Im{a a *} I m {a a „ *} Im{a a *} Im{a a *} Im{a a *} Im{a,a *} Im{a,a *} Im{a,a *} Im{a,a *} Im{a a *} Im{a a *} Im{a a *} Im{a a *} Im{a a *} -1/2/1/2 1/2 0 1/4 0 0 0 0 1/20/1/2 -3/4/1/10 -3/4/1/35 3/5/1/2 0 0 3/20/3 0 0 0 1/4/3/5 0 0 1/2/3/70 0 1/70/3 9/28 0 0 0 0 1/6/5/2 -1/4/5/7 0 0 0 •0 0 0 1/12/5 -1/4/5/14 0 0 1.114 -1/21/5 0 0 1/7/5/14 0 0 0 0 0 0 -1/4 0 0 0 1/2/1/6 1/4/1/2 -3/10/1/2 -1/2/1/10 -1/2/1/35 3/20/1/2 0 0 -1/5/1/3 -1/24 0 0 1/2/1/15 5/72/5 0 •1/210 5/36/5/14 1/10/1/3 -1/36 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -5/7/1/14 -1/28/5 0 0 3/28/5/14 0 0 0 0 0 0 3 0 6 2 5 7 8 2 3 2 5 2 6 2 3 3 5 3 7 3 8 5 6 7 8 5 6 5 7 5 8 6 7 6 8 2 , no Dc 3 The • symbol i m p l i e s the square root of the q u a n t i t y right. 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1/6 0 0 0 1/18/5 0 0 1/9/5/1 5/14/1/3 -11/252 to i t s 15 to be a f f e c t e d p r i m a r i l y by the most dominant amplitudes, i n a r e l a t i v e l y d i r e c t manner. The higher order terms a r e , i n g e n e r a l , composed of a sum of the r e a l p a r t s of the a p p r o p r i a t e b i l i n e a r combinations, i n a d d i t i o n 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 r e s p e c t i v e amplitudes. Although the complete d e s c r i p t i o n i s complex, the f o l l o w i n g p o i n t s emerge: 1) The e x i s t e n c e of a non-zero a°° c o e f f i c i e n t significant c o n t r i b u t i o n from amplitudes a 2 implies a or h i g h e r . 2) The e x i s t e n c e of a non-zero a°° c o e f f i c i e n t significant c o n t r i b u t i o n from amplitudes a 5 implies a or h i g h e r . 3) The e x i s t e n c e of a non-zero a l ° c o e f f i c i e n t significant c o n t r i b u t i o n from amplitudes a 8 implies a or h i g h e r . The h i g h e s t order d i f f e r e n t i a l c r o s s - s e c t i o n term observed e x p e r i m e n t a l l y , then, g i v e s i n s i g h t of p a r t i a l wave amplitudes contribute (a? ) 0 i n t o the number (and t h e i r d e s i g n a t i o n s ) which significantly. Similarly, coefficients the r e l a t i o n s h i p between the expansion of the a n a l y z i n g power (the b?°) and the sum of a p p r o p r i a t e b i l i n e a r combinations of p a r t i a l wave amplitudes (table (2.3)) i n d i c a t e a d d i t i o n a l the r e a c t i o n . In g e n e r a l , the b n o important p r o p e r t i e s of coefficients do not depend on squares of amplitudes, but depend i n s t e a d , on the sum the imaginary p a r t s of the a p p r o p r i a t e b i l i n e a r combinations. T h e r e f o r e , the b potentially n o coefficients amplitude are very s e n s i t i v e to r e l a t i v e phases of the of 16 amplitudes, and, as a consequence, are more s e n s i t i v e to the v a r i a t i o n s of smaller amplitudes. In a d d i t i o n , many of the terms i n v o l v e the product of a small amplitude with a dominant one (such as a ) , thus l e a d i n g to enhanced e f f e c t s 2 from these small amplitudes — i n some r e s p e c t s , an " i n t e r f e r e n c e " between the small and l a r g e amplitudes. I n s p e c t i o n of the b?° c o e f f i c i e n t s (table (2.3)), f o r example, i n d i c a t e s the general f e a t u r e that the b"° and b 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 c o n t a i n i n g the a 2 amplitude, whereas the b c o e f f i c i e n t s a r e , indeed, independent Thus, one may expect the bV° and b n o , b of t h i s n o coefficient 2 2 amplitude) i n the r e g i o n . A d d i t i o n a l l y , a non-zero i m p l i e s s i g n i f i c a n t c o n t r i b u t i o n s from wave amplitudes of d e s i g n a t i o n a 7 0 partial 1 A(3,3) resonance , and b ^ c o e f f i c i e n t s to n o (corresponding to the a terms amplitude. dominate as a r e s u l t of the major r o l e of the D wave channel n o by 0 partial or h i g h e r . 2.5 DISCUSSION OF THEORY To date, development of our t h e o r e t i c a l understanding of the + pp—>7r d r e a c t i o n has, roughly, kept pace along with the a v a i l a b i l i t y of experimental o b s e r v a t i o n s . A review of t h e o r e t i c a l developments given by M. Betz, B. B l a n k l e i d e r , J.A. Niskanen following and A.W. Thomas 19 serves as the b a s i s of the discussion. E a r l y attempts t o generate a f i e l d t h e o r e t i c model of the pp—>7r d r e a c t i o n p r o v i d e d some, i f l i m i t e d , + insight. 17 Because of the l a r g e momentum t r a n s f e r i n v o l v e d 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 , suggested that 21 the nature of the nucleon-nucleon short range i n t e r a c t i o n s , and the deuteron D s t a t e were important f a c t o r s in the d e s c r i p t i o n of the system. R e s c a t t e r i n g i n c o r p o r a t e d w i t h i n the context Litchtenberg 2 2 of f i e l d s h o r t l y a f t e r observation resonance. Such models, however, are n o n - r e l a t i v i s t i c and techniques t h e o r e t i c models by of the A(3,3) essentially (as a r e s u l t of the first one order u s u a l l y employed to evaluate Furthermore, they s u f f e r from the a m b i g u i t i e s with double counting was are u s u a l l y l i m i t e d t o , at most, r e s c a t t e r i n g of the pion perturbation of the pion them). associated of the pion r e s c a t t e r i n g s when attempts to i n c l u d e i n i t i a l and f i n a l s t a t e i n t e r a c t i o n s are employed. The most s u c c e s s f u l model, at l e a s t in terms of i t s q u a n t i t a t i v e , p r e d i c t i v e power, i s the coupled-channel model of Green and coupled NA Niskanen 2 3 ' 2 u ' 2 5 . I t i s based on a set of d i f f e r e n t i a l equations which i n c o r p o r a t e the NN channels on an equal f o o t i n g . The p o t e n t i a l s i n v o l v e d in t h i s n o n - r e l a t i v i s t i c model are of course, provide a framework f o r the and static and i n c l u s i o n of h e a v i e r meson exchange (exchange of the p meson f o r example). Although the three-body u n i t a r i t y of the system i s only guaranteed, e f f e c t i v e l y , approximately the summation over the pion m u l t i p l e s c a t t e r i n g s e r i e s i s complete. A reasonable the data however, does i n v o l v e s u i t a b l e c h o i c e s of f i t to 18 appropriate parameters. Recently, there has been c o n s i d e r a b l e i n t e r e s t development of 'Unitary M o d e l s ' based on the simultaneous and Trd channels 18 2 6 2 7 , models which are c o n s i d e r a t i o n of a l l of the NN, i n terms of a set of coupled d i f f e r e n t i a l e q u a t i o n s . T h i s approach and three-body unitarity i n c l u s i o n of r e l a t i v i s t i c in the NA three-body ensures exact two-body f o r a l l channels, and permits the k i n e m a t i c s . However, such equations are o f t e n e v a l u a t e d using a Tamm-Dankoff approximation 18 where i n t e r m e d i a t e s t a t e s with at most one pion are kept, thereby reducing the p r e c i s i o n a t t a i n a b l e by the technique. These models p r o v i d e l i m i t e d o p p o r t u n i t y to f i n e tune t h e i r p r e d i c t i o n s f o r a given channel, as changes to the other two channels may be e f f e c t e d as a consequence. D e s p i t e the u n i f i e d models' g e n e r a l l y poor quantitative agreement with experimental data, these models do provide a framework f o r a more complete system. understanding of the few-body 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 c r o s s - s e c t i o n of the pp-»7r d r e a c t i o n c o u l d be measured + a c c u r a t e l y , t o w i t h i n a few percent, utilizing incident proton beams of an a r b i t r a r y , but known p o l a r i z a t i o n . E i t h e r an u n p o l a r i z e d differential beam was used and the u n p o l a r i z e d c r o s s - s e c t i o n measured, or p o l a r i z e d proton beams were used so both the a n a l y z i n g unpolarized the l a t t e r extracted differential power and the c r o s s - s e c t i o n c o u l d be deduced. In case, 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 was from two sets of d i f f e r e n t i a l cross-section measurements taken with o p p o s i t e l y o r i e n t e d proton beam p o l a r i z a t i o n d i r e c t i o n s . In p r i n c i p l e , use of a p o l a r i z e d beam was adequate f o r a l l measurements d e s i r e d . Nonetheless a more accurate differential determination of the u n p o l a r i z e d c r o s s - s e c t i o n c o u l d be made with unpolarized beam, s i n c e i t s p o l a r i z a t i o n i s known to be zero To achieve a high level of confidence exactly. i n the r e s u l t s , many of the measurements were repeated a number of times using two or more independent methods. The deduction of the differential c r o s s - s e c t i o n r e q u i r e d 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 a d d i t i o n , the o v e r a l l normalization of the r e s u l t s r e q u i r e d , measurement of the i n c i d e n t beam 19 20 properties effective (beam e n e r g y , c u r r e n t , number o f t a r g e t v o l u m e . To f a c i l i t a t e and p o l a r i z a t i o n ) and t h e nuclei within the c a l c u l a t i o n of the e f f e c t i v e angle, a detector simple g e o m e t r i c c o n f i g u r a t i o n was u s e d system with each of the p a r t i c l e s The d a t a the i n t e r a c t i o n a well i n the f i n a l collected in this defined, solid relatively f o r the d e t e c t i o n of s t a t e of t h e r e a c t i o n . experiment contain redundant measurements of s e v e r a l q u a n t i t i e s , w h i c h when analyzed provide consistency. c h e c k s o f t h e s y s t e m b a s e d on i n t e r n a l These factors contributed final differential t o the o v e r a l l reliability c r o s s - s e c t i o n and a n a l y z i n g of the power results. 3.2 CYCLOTRON The TRIUMF c y c l o t r o n unpolarized current H ions 2 8 accelerates b o t h p o l a r i z e d and t o a maximum e n e r g y o f 520 MeV. The beam i s continuously v a r i a b l e up t o a maximum v a l u e d e p e n d s on b o t h t h e t y p e o f i o n s o u r c e , radius, or energy, of the c i r c u l a t i n g orbital radius current of about about a 520 MeV beam c o u l d 500 nA w i t h independently 140 ixk w i t h t h i n metal foil. radial be o b t a i n e d a t a maximum ion source, or The beam c a n be i n t o one o r more o f t h e e x t e r n a l by s t r i p p i n g e l e c t r o n s the beam. A t t h e maximum the unpolarized beam l i n e s continuously and on t h e i n t e r n a l the p o l a r i z e d ion source. extracted which The e n e r g y v a r i a b l e from p o s i t i o n of t h i s from t h e H of t h e e x t e r n a l 200 MeV ions with beam i s t o 520 MeV, stripper foil. a d e p e n d i n g on 21 During normal o p e r a t i o n the c y c l o t r o n produces beam with a 100% macroscopic duty f a c t o r . The m i c r o s t r u c t u r e c o n s i s t s of proton p u l s e s of roughly r e f e r r e d to as "beam buckets"), 5 nsec d u r a t i o n o c c u r r i n g every (also 43 nsec. The s e p a r a t i o n of the p u l s e s corresponds to the p e r i o d c h a r a c t e r i z i n g the a p p l i e d r a d i o frequency power (RF) which i s the f i f t h harmonic of the c y c l o t r o n resonance frequency. 3.3 BEAM LINE AND TARGET LOCATION The experiment was performed at t a r g e t l o c a t i o n 4BT1 on beam line 4B, represented s c h e m a t i c a l l y i n f i g u r e (3.1). The beam was e x t r a c t e d from the c y c l o t r o n and t r a n s p o r t e d through the 4B beam o p t i c system d e f i n e d by a s e r i e s of d i p o l e and quadrupole magnetic elements. At each beam energy the beam l i n e was tuned by a d j u s t i n g the strengths of the a p p r o p r i a t e s t e e r i n g and f o c u s i n g magnets i n order spots target ( 4 to 6 mm diameter t o produce small beam ) at both the 4BT1 and the 4BT2 l o c a t i o n s . T h i s process was f a c i l i t a t e d using monitors f o r i n d i c a t i n g the p o s i t i o n and p r o f i l e at v a r i o u s p o i n t s along beam c o u l d be centered the beam l i n e . A d d i t i o n a l l y , the and i t s width v e r i f i e d l o c a t i o n by remotely viewing video monitor. of the beam at the t a r g e t a s c i n t i l l a t i n g t a r g e t with a F i g u r e (3.1) TRIUMF Facility The TRIUMF C y c l o t r o n a n d t h e p r o t o n e x p e r i m e n t a l a r e a . Th e x e r i m e n t was p e r f o r m e d a t t a r g e t l o c a t i o n 4BT1 on t h e p r i m a r y p r o t o n b e a m - l i n e 4B. 23 3.4 BEAM POLARIZATION AND CURRENT MONITOR The four independent beam c u r r e n t monitors are shown schematically pp-elastic i n f i g u r e (3.2). A p o l a r i m e t e r 2 9 based on s c a t t e r i n g , l o c a t e d 2.7 m upstream of the t a r g e t , was used to measure both the beam p o l a r i z a t i o n and c u r r e n t . A p p - e l a s t i c m o n i t o r ( s e e appendix 1 0 (l) for a detailed d i s c u s s i o n of the c a l i b r a t i o n of t h i s , and other beam c u r r e n t monitors) c o n s i s t i n g of the four counters current scintillation denoted PL1, PL2, PR1, and PR2, measured the using the technique of counting elastically s c a t t e r e d a t 90° C M . choice of the s c a t t e r i n g angle, p a i r s of protons s c a t t e r i n g angle. This due to symmetry, renders the monitor i n s e n s i t i v e t o the p o l a r i z a t i o n of the beam. The rear d e t e c t o r s , at a r a d i a l d i s t a n c e of 71.9 cm from the t a r g e t , d e f i n e d the s o l i d angle of t h i s system. The beam's c u r r e n t was then measured two more times as i t passed through a secondary emission then e v e n t u a l l y monitor 21m downstream and was stopped i n a Faraday cup c u r r e n t monitor s i t u a t e d at the end of the beam l i n e . 3.5 APPARATUS The apparatus was designed with due regard f o r the kinematic p r o p e r t i e s of the r e a c t i o n , the i n t e r a c t i o n of the p a r t i c l e s with the m a t e r i a l along the t r a j e c t o r i e s , and the p r o p e r t i e s of pion decay i n t o a muon p l u s a n t i - n e u t r i n o p a i r . The apparatus was of the two-arm type, for measuring the e n e r g y - l o s s , c o n s i s t i n g of counters 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 i n the f i n a l s t a t e . In f a c t , with the a d d i t i o n of a second pion arm i t was p o s s i b l e to operate two such systems i n p a r a l l e l , f o r a given deuteron angle, as d e f i n e d since by the deuteron d e t e c t i o n arm p o s i t i o n , the a s s o c i a t e d pion was emitted one of two k i n e m a t i c a l l y p o s s i b l e angles. into The apparatus, which can be d i v i d e d i n t o s e v e r a l components, i s schematically depicted monitor was attached to a r e c t a n g u l a r were the t a r g e t holder the i n f i g u r e (3.3). The p p - e l a s t i c assembly and the deuteron horn. Both s c a t t e r i n g chamber and i t s extension, were evacuated and contained the t r a n s m i s s i o n the s c a t t e r i n g chamber, as interior the deuteron horn, windows a p p r o p r i a t e for either of p a r t i c l e s or the v i s u a l i n s p e c t i o n of r e g i o n . Three p a r t i c l e d e t e c t i o n systems, two for p i o n s and one f o r deuterons, were f i x e d t o arms which could r o t a t e independently around the t a r g e t a x i s . 3.6 SCATTERING CHAMBER In a d d i t i o n t o p r o v i d i n g an evacuated volume i n which the r e a c t i o n s occurred, the s c a t t e r i n g chamber formed the s t r u c t u r a l frame work of the whole apparatus. I t was constructed of 1/2 inch s t a i n l e s s s t e e l having the o u t s i d e dimensions o f : 91.4cm long, 61.6cm wide and 45.7cm i n depth. A t a r g e t h o l d i n g assembly was p o s i t i o n e d as shown i n f i g u r e (3.3) The 0.010 inch mylar windows mounted on t h e i r frames were attached window to the chamber on e i t h e r s i d e of the Figure (3.3) Apparatus i Scale I metre 27 beamline to allow t r a n s m i s s i o n of the pions and e l a s t i c a l l y s c a t t e r e d protons i n t o the r e s p e c t i v e d e t e c t i o n systems. Two (1/4 inch) l u c i t e windows a t t a c h e d t o the upstream end of the s c a t t e r i n g chamber p e r m i t t e d v i s u a l i n s p e c t i o n of the i n t e r i o r r e g i o n of the chamber, p a r t i c u l a r l y u s e f u l when examining the t a r g e t h o l d i n g assembly. 3.7 DEUTERON HORN The deuteron horn was a downstream e x t e n s i o n of the s c a t t e r i n g chamber r e q u i r e d f o r d e t e c t i n g the c o i n c i d e n t deuterons by e x t e r n a l counter systems at the small angles r e q u i r e d . The geometry- of the horn was d i c t a t e d by the pp—>ir*6\ r e a c t i o n k i n e m a t i c s . In p a r t i c u l a r , over the center-of-mass experiment, pion angles and e n e r g i e s e x p l o r e d i n t h i s deuterons with angles from 4° ( r e l a t i v e t o the beam d i r e c t i o n ) , up t o the maximum Jacobian angle of about 12°, had to be t r a n s m i t t e d through the horn to the e x t e r n a l d e t e c t o r s . The length of the horn depended on the minimum deuteron d e t e c t i o n angle r e q u i r e d . The minimum p o s s i b l e d e t e c t i o n angle r e s u l t e d when the d e t e c t i o n system was i n c o n t a c t with the beam p i p e . Given the 2 inch r a d i u s of the beam p i p e , simple geometry d i c t a t e d a 2.0 m deuteron arm length i n order to achieve a minimum angle of l e s s than 4 ° . 28 3.8 TARGETS AND BEAM ALIGNMENT The t a r g e t s were mounted on a t a r g e t ladder which was i n turn attached to, and c o n t r o l l e d by, an e l e c t r o - m e c h a n i c a l t a r g e t h o l d i n g d e v i c e . The ladder contained four 1.5 i n c h square t a r g e t p o s i t i o n s , t y p i c a l l y occupied by the f o l l o w i n g assortments of t a r g e t s : a t h i n CH t a r g e t , a t h i c k CH 2 (typically 45.3 mg/cm ) 2 (154.5 mg/cm ) t a r g e t , a carbon 2 2 t a r g e t (2-4.9 mg/cm ), and a z i n c s u l f i d e s c i n t i l l a t o r . The 2 remotely c o n t r o l l e d t a r g e t ladder c o u l d be p o s i t i o n e d so that any of i t s four t a r g e t s were l o c a t e d at the f o c a l p o i n t of 4BT1. The f o c a l p o i n t a t 4BT1 was known r e l a t i v e t o g r i d marked on the z i n c s u l f i d e s c i n t i l l a t o r , viewed which c o u l d be (through a l u c i t e window) by a T.V. monitor. The r e s u l t i n g video image was of great help i n tuning the 4B beam l i n e and c y c l o t r o n . 3.9 PARTICLE DETECTION SYSTEM Each p a r t i c l e d e t e c t i o n system, s c h e m a t i c a l l y represented i n figure (3.4), c o n s i s t e d of a m u l t i - w i r e p r o p o r t i o n a l chamber (MWPC) followed by a s c i n t i l l a t o r was attached t o each of the three movable arms, as d e p i c t e d in f i g u r e (3.3). The forward arm, t e l e s c o p e . One such system pion arm was designated and the backward pion arm the irB arm. deuteron arm was designated the TTF S i m i l a r l y the as e i t h e r the dF or dB arm, depending which pion arm i t was a s s o c i a t e d with, or simply as the d arm when such an a s s o c i a t i o n was i r r e l e v a n t . Figure (3.4) P a r t i c l e Detection PARTICLE Scintillator Telescope DETECTION t System SYSTEM Arm Central Axis (particle direction) 12.7 cm Multi Wire Proportional Chamber 16.5 cm 17; i Anode Plane Cathode Plane 30 With the MWPC's employed, s p a t i a l c o o r d i n a t e s of a p a r t i c l e t r a j e c t o r y c o u l d be determined b e t t e r than 1.0 mm. 15.2 x 15.2 cm with a r e s o l u t i o n of The MWPC, which had an a c t i v e area of c o n s i s t e d of three p a r a l l e l wire planes, a 2 d e l a y - l i n e read-out system, gas containment windows, and p r o v i s i o n s f o r gas c i r c u l a t i o n . The chambers were operated with a p o s i t i v e high v o l t a g e a p p l i e d to the c e n t r a l anode plane, which was separated from the adjacent cathode by 0.48 i n c h e s ) . The anode plane c o n s i s t e d of 75 cm (0.20 cm, (3/16 planes or 0.008 inch diameter) g o l d - p l a t e d tungsten wires having a s e p a r a t i o n of 2.0 mm. The two cathode planes c o n s i s t e d of 150 a c t i v e sense wires (of 0.006 cm, inch diameter) One plane was separated by 1.0 mm. e l e c t r i c a l l y connected each or 0.0025 end of each cathode to a d i s t r i b u t e d d e l a y - l i n e , with the i n d i v i d u a l cathode wires connected uniformly along the d e l a y - l i n e . Spatial i n f o r m a t i o n i s deduced from the d i f f e r e n c e i n the times i t takes s i g n a l s to t r a v e r s e the d e l a y - l i n e from the p o s i t i o n of the a c t i v a t e d sense wire, to both ends of the d e l a y - l i n e , as measured with TDC u n i t s . The spatial c a l i b r a t i o n of t h i s d i f f e r e n c e of times i s t r e a t e d i n section sum ( 4 . 5 ) . During proper o p e r a t i o n of the chambers the of the two propagation times approximately i s constant to w i t h i n 50 ns. T h i s width of a c c e p t a b l e sum results primarily from the v a r i a t i o n t r a v e l l e d by e l e c t r o n s and p o s i t i v e mixture, from the p o i n t of t h e i r times i n the d i s t a n c e s ions i n the magic gas formation to the p o i n t of 31 t h e i r d e t e c t i o n by a sense w i r e . A sum time i n t e r v a l c o u l d i n d i c a t e the d e t e c t i o n of a separated p a i r of p a r t i c l e s or i n e f f i c i e n t o p e r a t i o n of the chamber. The wire plane assembly was of time o u t s i d e of t h i s 'magic g a s ' 0.3% 3 0 immersed i n a constant composed of 70% Argon, 29.7% Freon, at a pressure only s l i g h t l y flow Butane, and exceeding atmospher i c . Two ( 5 x 5 Table thin plastic inch (3.1) 2 s c i n t i l l a t o r s with a 12.7 d e t e c t o r s from the t a r g e t , the o f f s e t s of the from the c e n t r a l t r a j e c t o r i e s , and l i g h t was guides onto RCA The 2 these scintillators the t h i c k n e s s e s of the s c i n t i l l a t i n g m a t e r i a l (see a l s o t a b l e 3.10 cm ) a c t i v e area formed the subsequent t e l e s c o p e . i n d i c a t e s the r a d i a l d i s t a n c e s of scintillation x 12.7 ( 4 . 4 ) ) . The t r a n s m i t t e d through lucite light 8575 p h o t o m u l t i p l i e r tubes. ELECTRONIC LOGIC AND SYSTEMS e l e c t r o n i c l o g i c and s i g n a l p r o c e s s i n g system, i n a s s o c i a t i o n with the o n - l i n e data a n a l y s i s system, was r e s p o n s i b l e f o r the l o g i c a l d e f i n i t i o n of a p o t e n t i a l + pp—>7r d event, and i t ' s subsequent p r o c e s s i n g p r i o r to r e c o r d i n g on magnetic tape. Furthermore, i t permitted p e r i o d i c monitoring of a l l the beam c u r r e n t and monitors, events c h a r a c t e r i s t i c s of the themselves. The event as w e l l as the important polarization e l e c t r o n i c l o g i c used to d e f i n e a p o t e n t i a l pp—^rr'd (the t r i g g e r system) i s represented s c h e m a t i c a l l y i n 32 Table (3.1) The Detector Geometry. Descr i p t ion Detector D e t e c t i o n Arm d (dF and dB) TTF TTB (d ) dF dB (dl) dF1 dBI (d2) dF2 dB2 TTF TTB 7fF1 7TB TTF2 TTB2 Desiqnat ion MWPC S c i n t i l l a t o r1 Scintillator^ 1 Radi i MWPC Scintillator*1 Scintillator#2 257.7cm 261.5cm 262.7cm 131.2cm 138.4cm 1 39.6cm 99.Ocm 107.4cm 108.6cm 3.18cm 6.25cm 1.59cm 6.35cm Thickness MWPC Scintillator*1 Scintillator#2 6.35cm 6.35cm Detector Geometry Table D e f i n i t i o n s D e s i g n a t i o n : The symbolic name a s s o c i a t e d with the v a r i o u s d e t e c t o r s . As the forward and backward branch deuteron d e t e c t o r s are the same p h y s i c a l system, the F and B d i s t i n c t i o n i s omitted in the a p p r o p r i a t e cases. R a d i i The d i s t a n c e s from the t a r g e t to the f r o n t s u r f a c e of the d e t e c t o r s . Thicknesses The width of the s c i n t i l l a t o r material. figure ( 3 . 5 ) . transmitted The six linear s c i n t i l l a t o r to the counting signals room by c o a x i a l c a b l e , were d i r e c t e d to d i s c r i m i n a t o r s modules which generated pulses ( f i r e d ) for input preset threshold (ADC) s i g n a l s whose amplitude exceeded a l e v e l . The s u i t a b l e delay) analyzed logic l i n e a r s i g n a l s were a l s o ( a f t e r by a n a l o g u e - t o - d i g i t a l i n a CAMAC system which a l s o contained converters time-to-digital converters (TDC) f o r measuring r e l a t i v e timing of associated logic s i g n a l s . The outputs from the d i s c r i m i n a t o r s which d e f i n e the forward, and the four the four which d e f i n e the backward branch of the system, were brought to a three out of four branch c o i n c i d e n c e associated 'majority' coincidence u n i t . If any scintillators i n the three out respective of the four f i r e d , these c o i n c i d e n c e units produced a l o g i c s i g n a l , thus d e f i n i n g a p o t e n t i a l pp—*-Tr d + event. A t r i g g e r s i g n a l was "OR" l o g i c module) and then formed (by the processed by a l o g i c subsequent system that i n t e r r u p t e d the data a c q u i s i t i o n computer, thus a c t i v a t i n g a "circuit busy" c o n d i t i o n , which i n h i b i t e d p r o c e s s i n g subsequent t r i g g e r s i g n a l s , u n t i l the computer had accessing finished a l l data f o r the event under c o n s i d e r a t i o n . a d d i t i o n , the s c a l e r s . The 'circuit In busy' c o n d i t i o n d i s a b l e d a l l monitor event c o i n c i d e n c e i n t e r r u p t i n g the computer was units. of s i g n a l as w e l l as used to s t a r t a l l of the TDC Figure ( 3 . 5 ) E l e c t r o n i c T r i g g e r Logic and Schematic Diagram SCINTILLATOR TTF, Tr F LINEAR dl 2 SIGNALS d 2 LEGEND OF ELECTRONIC Computer 7T B, 7TB 2 Busy COINCIDENCE S ( /4 DESIGNATES LRS • 363) GATE MODULES UNIT MAJORITY LOGIC OR 465 LRS UNIT 622 u GENERATOR LRS OR LRS DISCRIMINATOR LRS 222 621 OR 821 V CAMAC MODULE (T) CAMAC LAM STROBE. (P) Y Q © © Y I <© © © PATTERN GENERATOR EEC UNIT BIT GATE (A) ANALOGUE. (?)TDC START. (T) STOP. T 0 C © L I V E GATE. AND PATTERN UNIT C2I2 (G)ADC ADC FUNCTIONS LRS LRS (TO REGISTER. E EG C 212 2249 2228 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 s i g n a l s so that the r e l a t i v e timing of the pion and deuteron s i g n a l s at t h e i r r e s p e c t i v e d i s c r i m i n a t o r s was that shown i n 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 s i g n a l was l a s t to enter the c o i n c i d e n c e , both d e t e c t o r s recorded so d e f i n i n g the o v e r a l l timing when the same p a r t i c l e . In f i g u r e (3.6), l i n e a r s i g n a l s from the pion s c i n t i l l a t o r are shown, i n d i c a t i n g 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 s i g n a l s . The r e l a t i v e timing of the associated logic s i g n a l s p r i o r to e n t e r i n g the r e s p e c t i v e branch c o i n c i d e n c e figure unit (figure (3.6). The l o g i c (3.5)) are a l s o i n d i c a t e d i n s i g n a l s from the pion scintillators were advanced by 20ns, such that the timing of the event t r i g g e r was a l s o d e f i n e d by the d1 s c i n t i l l a t o r f o r both pp—^7r d events and in-phase random events. As a r e s u l t of + the 80ns width of the pion s c i n t i l l a t o r logic signals, t r i g g e r s i g n a l s were a l s o generated by d e t e c t i o n of e a r l y (one beam bucket) random events. These occur p r o b a b i l i t y as those with the same generated by the d e t e c t i o n of in-phase random events. Thus d i r e c t e s t i m a t i o n of the background l e v e l s a s s o c i a t e d with in-phase random events was readily obtained. The t r i g g e r s i g n a l was used to s t a r t a l l of the CAMAC TDC c l o c k s . The deuteron and pion s c i n t i l l a t o r s i g n a l s were then delayed logic a p p r o p r i a t e l y and used to stop the F i g u r e (3.6) R e l a t i v e Timing of L i n e a r and Logic S i g n a l s DEUTERON SCINTILLATOR v — LINEAR PION 43n.v (d I) SIGNAL. SCINTILLATOR LINEAR SIGNAL. 43n,s. UNCORRELATED PROTONS EARLY IN-PHASE 60 ns. LATE LINEAR SIGNALS r 80 n s. DEUTERON SCINTILLATOR LOGIC SIGNAL (ENTERING ( /<J,) COINCIDENCE UNIT) 3 PION SCINTILLATOR LOGIC SIGNAL (ENTERING ( /4) 3 I20n.s. (---) PHASES. \ COINCIDENCE UNIT) TRIGGER LOGIC SIGNAL ( /4 COINCIDENCE OUTPUT) 3 TDC U DELAY r START DEUTERON PION SIGNAL SCINTILLATOR STOP SCINTILLATOR STOP 37 TDC Units associated with them. The MWPC l o g i c s i g n a l s (four for each of the three chambers) were a l s o delayed appropriately and used t o stop the a p p r o p r i a t e TDC u n i t s . A d d i t i o n a l l y , the t r i g g e r s i g n a l was used t o generate an ADC "gate", that i s , i t defined the i n t e r v a l of time over which the CAMAC ADC u n i t s i n t e g r a t e d i n p u t s . The q u a n t i t i e s listed i n table scaled the l i n e a r s i g n a l s at i t s by the CAMAC s c a l e r s are (3.2). When the experiment was performed with u n p o l a r i z e d beam, the s c a l e r s were p e r m i t t e d t o accumulate f o r the whole d u r a t i o n of a run. When a p o l a r i z e d beam was used, the s c a l e r s were read and c l e a r e d on a p e r i o d i c b a s i s , and i n t e g r a t e d polarization over each of the beam s t a t e s by the ( a u x i l i a r y ) data a c q u i s i t i o n software. 3.12 DATA ACQUISITION SOFTWARE The data a c q u i s i t i o n system employed f o r t h i s experiment was a v e r s i o n of the TRIUMF data a c q u i s i t i o n system M U L T I , running on a PDP 11/34 computer under the 31 RSX-11M o p e r a t i n g 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 b a s i s and stored d i r e c t l y on magnetic tape. On being interrupted and by an event, a "computer busy" s i g n a l was issued the data a c q u i s i t i o n e l e c t r o n i c s i n h i b i t e d u n t i l the data h a n d l i n g task was completed. In a d d i t i o n , the MULTI system d i r e c t e d simple o n - l i n e of a subset of the d a t a . c a l c u l a t i o n s and histograming 38 Table (3.2) Quantities Quantities Accumulated Processed by CAMAC S c a l a r s . with "Live Gated" Scalers. Quantity Number of events Time i n t e r v a l s Radio frequency c y c l e s P P - E l a s t i c monitor events Faraday Cup monitor events P o l a r i m e t e r events Quantities Accumulated with "Free Running" Scalers. Quantity Time i n t e r v a l s P P - E l a s t i c monitor events P o l a r i m e t e r events S c a l e r accumulations s u b j e c t to the "Live Gate" c o n d i t i o n are c o r r e c t e d f o r the system busy time (see f i g u r e ( 3 . 5 ) ) . A l l of the above q u a n t i t i e s were s c a l e d s e p a r a t e l y f o r each of the three beam p o l a r i z a t i o n s t a t e s when a p o l a r i z e d beam was used. 39 Two on-line of additional c a l c u l a t i o n a l power, scaler were p r o g r a m s were read. quantities that and were developed t o enhance the to maintain set to zero a each running time sum they 4. ANALYSIS OF THE 4.1 DATA. INTRODUCTION. The pp— >it*d event d e f i n i t i o n together with more general p r o p e r t i e s of the data are d i s c u s s e d i n the context of a p r e c i s i o n data a n a l y s i s system with the c a p a b i l i t y of p r o c e s s i n g a l a r g e volume of data. A d e t a i l e d d i s c u s s i o n i s presented of the background c o n t r i b u t i o n from carbon nuclei (a component of the p r o d u c t i o n t a r g e t ) and of the e f f e c t s of pion-decay and e n e r g y - l o s s (and of the d e t e c t o r c a l i b r a t i o n s ) on the acceptance s o l i d a n g l e . 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 and a n a l y z i n g powers, and t h e i r a s s o c i a t e d u n c e r t a i n t i e s 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 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 angular distributions are expanded in terms of Legendre Or A s s o c i a t e d Legendre polynomials and the corresponding a ? 0 and b?° coefficients 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 c r o s s - s e c t i o n of the pp->7r*d r e a c t i o n on e x p e r i m e n t a l l y measured q u a n t i t i e s i s developed through a s e r i e s of s t e p s . In the i d e a l case where the only r e a c t i o n o c c u r r i n g was number of observed events N _ that of the pp-^-7r d, the + ^ , , would be given t pp—>TT d 40 by; 41 V-^'d • int N e d °/ d (01 ) ^ 0 where: do/an - The pp—>-7r d r e a c t i o n + differential N cross-section. - The number of p o t e n t i a l int i n t e r a c t i o n s { N(beam) N ( t a r g e t ) }. - The combined detector e f f i c ienc i e s . AO - The e f f e c t i v e acceptance solid angle. However, events a r i s i n g pp— >ir*6\ r e a c t i o n were a l s o observed. As some of these not be d i s t i n g u i s h e d during from processes other that of the from the pp—>7r*d events of the event-by-event could interest a n a l y s i s of the data, the magnitude of t h e i r c o n t r i b u t i o n t o 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 + definition included a small w e l l as random c o i n c i d e n c e s , events of i n t e r e s t . That i s , number of background events as i n a d d i t i o n t o the pp—>Tr*d 42 N = N p ++ , + N pp-*-7r d + N c (02) 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 ^ 4j a definition The number of t r u e pp—>7r d _ + pp—>TT r : r events c o n t a i n e d i n the primary event N sample. - The number of carbon c bacground events c o n t a i n e d in the primary event sample. N - The number of u n c o r r e l a t e d events (randoms) c o n t a i n e d in the primary event sample. It w i l l be shown that the number of random events can be e x t r a c t e d from a n a l y s i s of the data, and that the carbon background can be d e s c r i b e d by an e f f e c t i v e differential c r o s s - s e c t i o n d o / d f l . Thus, the number of observed events i s c given by the r e l a t i o n s h i p ; N Here N ^ n t p = N int e * do/dO 2 i d a / d f i }' tift + N c (03) r i s the product of the number of i n c i d e n t and the number of hydrogen CH + atoms i n the t a r g e t protons ( o c c u r r i n g as m o l e c u l e s ) . Thus, da/dn i s obtained by s o l v i n g the above 43 expression: dff/dn = { (N - N ) / (N. r n t e i ^ ) } - ida /dR c (04) Each component of t h i s f u n c t i o n w i l l 4.3 EVENT-BY-EVENT be d i s c u s s e d . DATA ANALYSIS The o n - l i n e data a c q u i s i t i o n system accepted a l l events which s a t i s f i e d the two-arm c o i n c i d e n c e c r i t e r i o n (backgrounds as w e l l as the pp—>ir*d events of i n t e r e s t ) and recorded these on magnetic tape. In a d d i t i o n to the problem of h a n d l i n g the background i n f o r m a t i o n , one had to contend as w e l l with the f a c t that some of the p p — z - i f d events of i n t e r e s t were l o s t due to d e t e c t o r i n e f f i c i e n c i e s . T h e r e f o r e , the o f f - l i n e data a c q u i s i t i o n system had both t o i d e n t i f y the pp—>7r d events w i t h i n a data set and c o r r e c t + the number observed f o r the i n e f f i c i e n c y of the d e t e c t i o n system. 4.3.1 TREATMENT OF THE RAW DATA There were two types of events that were w r i t t e n magnetic tape on an event-by-event onto b a s i s . The events were numbered s e q u e n t i a l l y , and the number was a t t a c h e d to each event. The two types of events, d e s i g n a t e d type A and type B, were w r i t t e n i n u n i t s r e f e r r e d to as b l o c k s . Each block c o n s i s t e d of approximately f i f t e e n one type B event. type A events f o l l o w e d by 44 Type A events d e f i n e each event represent the i n f o r m a t i o n r e q u i r e d to (ADC, TDC, and MWPC d a t a ) . Type B events represent q u a n t i t i e s i n t e g r a t e d over the type A events comprising the block, such as p o l a r i m e t e r counts and i n t e r v a l s . Due to software e r r o r s , the a c q u i s i t i o n program f a i l e d resulting time (MULTI ) data to operate as 31 specified, i n data being w r i t t e n i n an u n p r e d i c t a b l e order at t imes. It i s , however, p o s s i b l e to compensate f o r t h i s abnormality. The i d e n t i f i c a t i o n of an abnormality and c o r r e c t i v e a c t i o n taken event numbers. In a l l , can be i s based on the observed the sequence of there are three types of e r r o r s that identified. 1) D u p l i c a t e d data b l o c k s 2) M i s s i n g data b l o c k s 3) M i s s i n g type B events The d u p l i c a t e d data b l o c k s are i d e n t i f i e d by the d u p l i c a t i o n of a s e r i e s of event action numbers. The observed corrective i n t h i s case i s r e j e c t i o n of the d u p l i c a t e d events. S i m i l a r l y , a m i s s i n g data b u f f e r i s i d e n t i f i e d by a s e r i e s of missing event numbers ( a s s o c i a t e d with the a n t i c i p a t e d s e r i e s of type A and type B e v e n t s ) . In a d d i t i o n , the block of m i s s i n g events has to occur between the l a s t type B event type A event action of the p r e v i o u s block, and the of the subsequent data b l o c k . No first corrective i s r e q u i r e d (other than to renumber the subsequent events). 45 A more s e r i o u s c o n d i t i o n occurred when a type B event is ( a p p a r e n t l y ) a r b i t r a r i l y omitted. I f t h i s c o n d i t i o n i s not r e c t i f i e d , the beam c u r r e n t (and other q u a n t i t i e s summed by the CAMAC s c a l e r s ) i s d i s p r o p o r t i o n a t e l y condition number i s , however, c l e a r l y low. The i d e n t i f i e d when one event (and only one) i s m i s s i n g i n a data block, where a type B event i s expected. The c o r r e c t i v e a c t i o n requires three s t e p s . 1) A l l of the events between two complete data b l o c k s are ignored 2) A l l subsequent s c a l a r numbers are reduced by the amount i n t e g r a t e d over the ignored data b l o c k s 3) The subsequent The events are renumbered software e r r o r s r e s p o n s i b l e f o r these c o n d i t i o n s were l o c a t e d and were v e r i f i e d t o be the cause of the observed problems. 4.3.2 THE PRIMARY EVENTS Primary events were a subset of a l l recorded events satisfying the pp—?-7r d event d e f i n i t i o n . + Included i n t h i s subset, however, were events a s s o c i a t e d with the carbon impurity of the t a r g e t and events that were recorded as a result of random c o i n c i d e n c e s ( f a l s e t r i g g e r s ) between u n c o r r e l a t e d e l a s t i c a l l y s c a t t e r e d protons. The methods used to estimate the s i z e of t h i s r e l a t i v e l y small background (about three per cent) are d i s c u s s e d l a t e r i n sect ion (4.6). The primary event type was d e f i n e d by i t s a b i l i t y to s a t i s f y 46 a set of c u t s a p p r o p r i a t e l y placed on a number of experimental observables. The data were compared on an event-by-event b a s i s with the event d e f i n i t i o n , and the number of primary events determined. subset, however, were those + pp—?-7r M i s s i n g from this d events a s s o c i a t e d 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) T i m e - o f - f l i g h t q u a n t i t i e s ; a s s o c i a t e d with measurements of time intervals. 2) P u l s e - h e i g h t q u a n t i t i e s ; a s s o c i a t e d with measurements of the p u l s e - h e i g h t s of s p e c i f i e d e l e c t r o n i c d e t e c t o r s i g n a l s . 3) Kinematic q u a n t i t i e s ; a s s o c i a t e d with the kinematic c o r r e l a t i o n of the two-body f i n a l state. T i m e - o f - f l i g h t and p u l s e - h e i g h t measurements were both determined therefore from s c i n t i l l a t i o n d e t e c t o r s i g n a l s and were (weakly) c o r r e l a t e d . As the kinematic quantities were c a l c u l a t e d from the s p a t i a l c o o r d i n a t e s of the t r a j e c t o r i e s as determined by the m u l t i - w i r e p r o p o r t i o n a l chambers, they were independent of the p u l s e - h e i g h t and t i m e - o f - f l i g h t information. 4.3.2.1 Pulse-Height Distributions Charged p a r t i c l e s l o s e energy while t r a v e r s i n g such as s c i n t i l l a t o r s . Some of t h i s energy matter i s converted to l i g h t . The l i g h t p u l s e s are detected by h i g h gain p h o t o m u l t i p l i e r tubes which produce a c u r r e n t pulse f o r each 47 l i g h t p u l s e i n c i d e n t . The was converted converter proton into d i g i t a l (ADC) and t o t a l charge of each c u r r e n t pulse form by an a n a l o g u e - t o - d i g i t a l recorded. The deuteron, p u l s e - h e i g h t s were expected p i o n , muon and to vary l i n e a r l y with energy d e p o s i t e d by the p a r t i c l e of i n t e r e s t the 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 d e v i a t i o n from such a r e l a t i o n s h i p was only expected f o r the low energy pions and muons. The p u l s e - h e i g h t 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 p a r t i c l e s passing through pion and deuteron in f i g u r e the s c i n t i l l a t o r s comprising (4.1). Peaks i n the d i s t r i b u t i o n are a s s o c i a t e d + events. Three q u a l i t a t i v e with number of pp—^7r d events + (random) background f e a t u r e s of the d i s t r i b u t i o n displayed in figure than (4.1) are: greater events. c l e a n s e p a r a t i o n of the pp—=*-ir d events + background 3) The pulse-height is significantly the number of random background 2) The the arms (and t h e i r c o r r e l a t i o n ) i s i n d i c a t e d with the pp-H»-7r d r e a c t i o n , and 1) The the and the random distributions. long t a i l distributions on the high p u l s e - h e i g h t ( r e l a t e d to the Landau s i d e of the energy-loss d i s t r ibut i o n ) . Lower l i m i t c u t s imposed on both of the allowed pion deuteron p u l s e - h e i g h t v a l u e s , separate the pp—>ir*d and events from the random background. Because of the Landau shape, upper l i m i t c o n s t r a i n t s were not be a p p l i e d s i n c e some pp—>ir *6\ events would be r e j e c t e d as a result. 8fr 49 Figure (4.2) d e p i c t s 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 t a r g e t . The prominent pp—*-7r d peak of the + p u l s e - h e i g h t d i s t r i b u t i o n c o l l e c t e d using the p o l y e t h e l e n e target i s absent, while the q u a l i t a t i v e f e a t u r e s of the d i s t r i b u t i o n a s s o c i a t e d with the u n c o r r e l a t e d proton background are e s s e n t i a l l y events (about i d e n t i c a l . A small number of three percent of the pp— >it *d s i g n a l , when p r o p e r l y normalized) were d i s t r i b u t e d over deuteron events and pion p u l s e - h e i g h t s c h a r a c t e r i z i n g the pp—>rr d + arising from a CH to as carbon background The 2 t a r g e t . These events p o s i t i o n of the c e n t r o i d s of the pulse i n c i d e n t proton are r e f e r r e d events. d i s t r i b u t i o n s f o r the pp— >ir *d the the area of height r e a c t i o n were a f u n c t i o n of beam energy. As a r e s u l t , the 'cut' values of pp— >ir *d pion and deuteron detector pulse-heights v a r i e d on a run t o run b a s i s . The energy-loss dE/dx of the p a r t i c l e s has an i n v e r s e dependency on t h e i r Thus, the pion and deuteron expected scintillator energies . 0 0 p u l s e - h e i g h t s are to vary as the inverse square of the p a r t i c l e ' s veloc i t y . The c e n t r a l p o s i t i o n s of the pion and deuteron p u l s e - h e i g h t d i s t r i b u t i o n s were measured and f i t t o l i n e a r f u n c t i o n s of the i n v e r s e square of the corresponding v e l o c i t y , as determined k i n e m a t i c a l l y . The c e n t r a l of the pion and deuteron position d i s t r i b u t i o n s along with the p r e d i c t i o n of the r e s u l t i n g f i t s are i n d i c a t e d i n Figure PION AND DEUTERON CARBON (4.2) PULSE-HEIGHT DISTRIBUTIONS TARGET o 51 figure (4.3) and f i g u r e (4.4). The values of the lower limit that d e f i n e d the allowed v a l u e s of the pion and deuteron p u l s e - h e i g h t s are r e l a t e d t o the c e n t r a l values of the r e s p e c t i v e d i s t r i b u t i o n s by a constant d i f f e r e n c e indicated and are i n the f i g u r e s . 4.3.2.2 T i m e - o f - F l i g h t D i s t r i b u t i o n s Time i n t e r v a l s between the t r i g g e r deuteron s i g n a l timed to the arm s c i n t i l l a t o r s and the d e t e c t i o n 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 are l i n e a r l y related to t h e i r a c t u a l intervals value through the TDC module c a l i b r a t i o n s . A two-dimensional vs. the deuteron p l o t of a t y p i c a l pion TDC dE/dx i s d e p i c t e d i n f i g u r e spectrum (4.5). The prominent peak of the d i s t r i b u t i o n , a s s o c i a t e d with the pp—>7r d r e a c t i o n , , 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 s i n g l e background peak evident i n the p u l s e - h e i g h t d i s t r i b u t i o n ( f i g u r e now s p l i t i n t o s e v e r a l peaks centered at d i f f e r e n t pion time-of-flight Selection reaction (4.1)) i s values. of events a s s o c i a t e d with the pp—>7r d + c o u l d be obtained by t e s t i n g t h e i r pion time-of-flight values and determining whether they were c o n t a i n e d w i t h i n an a p p r o p r i a t e range of allowed v a l u e s . The s e r i e s of background peaks a r i s e from the d e t e c t i o n of u n c o r r e l a t e d protons a s s o c i a t e d with d i f f e r e n t RF beam 'buckets' (R.F. c y c l e s ) . F i g u r e (4.6) d e p i c t s the Figure Deuteron S c i n t i l l a t o r or • CD — Id §400 (4.3) Pulse-Height Cuts. Distribution Peaks and ~ PULSE HEIGHT DISTRIBUTION PEAK POSITION MODEL PREDICTIONS ---CUTS LLI < X o V Q < O 200 - tr j— Z) LU Q • 1 i 8 DEUTERON INVERSE SQUARE VELOCITY (l//3or(c/v) ) E x p e r i m e n t a l l y determined 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 probable values) are p l o t a g a i n s t the i n v e r s e square deuteron v e l o c i t y . No upper l i m i t cuts are a p p l i e d to pulse-height values. 53 Figure Pion S c i n t i l l a t o r . Pulse-Height (4.4) Distribution P e a k s and Cuts. 400 • ce £ z> z 350 - PULSE-HEIGHT DISTRIBUTION PEAK POSITIONS MODEL PREDICTIONS CUTS • 300 UJ 1< 250 ° 200 x * CJ Q < *•* 150 o CL 100 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 t h e i n v e r s e s q u a r e p i o n v e l o c i t y . No u p p e r l i m i t c u t s a r e a p p l i e d t o pulse-height values. COUNTS Figure TIME-OF-FLIGHT P, °N T OF __600 PULSE-HEIGHT D * 100, AND DEUTERON (4.6) n P r 400 DISTRIBUTIONS ojecti 2 4n~ 0r o n 0 Pulse-HeigM 200 Q <A ° CARBON TARGET m C Projection 1 100 (Bin cn cn 56 corresponding two dimensional p l o t expected, f o r a carbon t a r g e t . As the prominent peak corresponding to pp->7r d events + i s absent, while peaks r e p r e s e n t i n g the background are q u a l i t a t i v e l y unchanged (the number of counts i n both are not normalized to each o t h e r ) . Nonetheless, small number of carbon region where pp— >ir *6 background events there were a l o c a t e d i n the events would be expected p o l y e t h e l e n e t a r g e t was plots when a used. The p o s i t i o n of the pp—>-7r*d t i m e - o f - f l i g h t peak v a r i e d as a f u n c t i o n of the beam energy and pion angle values of the a s s o c i a t e d upper and lower (as d i d the l i m i t s used to d e f i n e the allowed t i m e - o f - f l i g h t values of a pp—»-7r d + e v e n t ) . Again, cut l e v e l s are d e f i n e d by l i n e a r alogarithms. C e n t r o i d s of the t i m e - o f - f l i g h t d i s t r i b u t i o n s were measured f o r a f r a c t i o n of the runs and were f i t to the corresponding c a l c u l a t e d v a l u e s , assuming a l i n e a r r e l a t i o n s h i p . The figure lower r e s u l t s of such a f i t are shown i n (4.7). A l s o i n d i c a t e d are the values of the upper and l i m i t s which d i f f e r from the value of the r e s p e c t i v e c e n t r o i d by a constant v a l u e . 4.3.2.3 Kinematic Distributions Since the c o o r d i n a t e s of both f i n a l were measured, i t was state p o s s i b l e to check on an particles event-by-event b a s i s whether the angular c o o r d i n a t e s of the two particles were c o r r e l a t e d as the r e a c t i o n kinematics p r e d i c t e d . T h i s was p o s s i b l e not only f o r the pp—>7r*d events but a l s o the pp-*-pp events, where they were d e t e c t e d . The angular 57 F i g u r e (4.7) Time-of-Fliqht D i s t r i b u t i o n TDC PEAK POSITION 7TFI • O A 77-F2 TTBI Peaks and Cuts. CUT DEFINITION CURVE O 77" B 2 600 ce u co. _J • UJ I < x (_> 500 cr o Q H 400 45 90 PION ANGLE 135 (deg. cm.) E x p e r i m e n t a l l y determined d i s t r i b u t i o n peaks are p l o t a g a i n s t the pion a n g l e . The set of curves at the lower pion angles a r e a s s o c i a t e d with the forward arm s c i n t i l l a t o r s (TTFI and TTF2) and the others with the backward pion d e t e c t i o n arm s c i n t i l l a t o r s ( T T B I and 7rB2). 58 correlation i s d e f i n e d as the c o r r e l a t i o n of the p o l a r coordinates (0) and the angular c o p l a n a r i t y i s d e f i n e d as the c o r r e l a t i o n of the azimuthal (0) c o o r d i n a t e s . As a n o t a t i o n a l a i d to s p e c i f y an otherwise i n which d e t e c t i o n arm, i n d i s t i n g u i s h a b l e proton i s d e t e c t e d , the f o l l o w i n g n o t a t i o n i s introduced; p, - Implies proton d e t e c t i o n by the pion d e t e c t o r . p _ 2 Implies proton d e t e c t i o n by the deuteron d e t e c t o r . The angular c o r r e l a t i o n A0 , = 0 Trd ,(0 ) rr rrd i s d e f i n e d by; 6 d (05) 6 PP Pi where: - The angular c o r r e l a t i o n of the pp—*-7r d reaction + products. PP - The angular c o r r e l a t i o n of the pp—*-pp r e a c t i o n products. - The deuteron determined angle kinematicalally from the (measured) pion angle and i n c i d e n t energy. proton 59 PP P2 - The proton angle (pion d e t e c t o r s i d e ) determined kinematicalally (measured) 0 P2 from the proton angle and i n c i d e n t beam energy. - The (proton) p o l a r angle measured with d e t e c t o r s mounted on the pion arm. - The (proton) p o l a r angle measured with d e t e c t o r s mounted on the deuteron arm. The angular c o p l a n a r i t y i s d e f i n e d by; 6 0 < *, - " A Kd" % P = ( * ," P 1 - *a ( 0 6 ) * » " * , P where: - The angular c o p l a n a r i t y of the pp—>-7r*d reaction products. A^pp - The angular c o p l a n a r i t y of the pp—>pp r e a c t i o n products. 0p - The (proton) azimuthal angle measured from d e t e c t o r s mounted on thepion 0 ' P2 - The arm. (proton) azimuthal angle measured from d e t e c t o r s mounted on the deuteron arm. C l e a r l y , the angular c o r r e l a t i o n s so d e f i n e d are zero if the p a r t i c l e s are p e r f e c t l y c o r r e l a t e d . In g e n e r a l , the angular d i s t r i b u t i o n a s s o c i a t e d with each r e a c t i o n i s represented by a sharp peak about a c e n t r a l v a l u e . An example of a t y p i c a l angular c o 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 f i g u r e (4.8) . 61 Figure ( 4 . 8 ) A T y p i c a l Angular C o r r e l a t i o n -10 ANGULAR Distribution 0 CORRELATION 10 (m radians) The events a s s o c i a t e d with the extreme edges of the d i s t r i b u t i o n r e s u l t from the d e t e c t i o n of random ( u n c o r r e l a t e d ) proton events and of deuteron-muon p a i r s . 20 62 4.3.3 THE UNCORRELATED EVENTS: RANDOMS. It was evident (see f i g u r e (4.5) f o r example), that the t i m e - o f - f l i g h t values a s s o c i a t e d with random events c o u l d , in a small number of cases, f a l l v a l u e s a s s o c i a t e d with the w i t h i n the range of allowed + pp->7r d r e a c t i o n . 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 The i n the number of primary events. number of such random events contained i n the sample c o u l d , however, be estimated from the t i m e - o f - f l i g h t d i s t r i b u t i o n of random events a s s o c i a t e d with particles separated by one R.F. c y c l e from the events of i n t e r e s t . Since the two complete random d i s t r i b u t i o n s accepted by the o n - l i n e data a c q u i s i t i o n system (separated by an i n t e r v a l of time a s s o c i a t e d with one R.F. c y c l e (43 nsec.)) were of s i m i l a r shape, such a s u b t r a c t i o n technique was p e r m i s s i b l e . The (to number of random events, then, were approximated w i t h i n 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 m o d i f i e d + t i m e - o f - f l i g h t c r i t e r i a . The t i m e - o f - f l i g h t values were required to f a l l w i t h i n the range allowed f o r values a s s o c i a t e d with the pp—s»7r d r e a c t i o n but s h i f t e d by an + amount corresponding to one R.F. p e r i o d . In g e n e r a l , the number of such random events represented an i n s i g n i f i c a n t fraction ( t y p i c a l l y much l e s s than one percent) of the number of primary events. 63 4.3.4 .SCINTILLATOR EFFICIENCIES I t was p o s s i b l e 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 a n a l y s i s 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 i n t o the experimental figure events, that i s events which by (3.3)). ' T r i a l ' system (see reason of the kinematics and p a r t i c l e type should have caused Trial a particular scintillator events were accepted to f i r e , were identified. i f a number of c r i t e r i a were satisfied: 1) The pp—s»-7r d angular c o r r e l a t i o n and c o p l a n a r i t y + c o n d i t i o n s were satisfied. 2) The o t h e r three s c i n t i l l a t o r s fired (the event definition c o i n c i d e n c e a i n v o l v e d 3/4 m a j o r i t y c o i n c i d e n c e ) with a p p r o p r i a t e pp—>7r d p u l s e - h e i g h t v a l u e s . + 3) A p p r o p r i a t e t i m e - o f - f l i g h t v a l u e s were o b t a i n e d , and corresponded with those of the pp—*-n*d r e a c t i o n . That i s , the t i m e - o f - f l i g h t c o n d i t i o n s were omitted f o r 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 s u c c e s s f u l event was d e f i n e d as a t r i a l event pulse-height f o r the d e t e c t o r being t e s t e d f e l l l i m i t s a s s o c i a t e d with the pp—>7r d event + Assuming binomial s t a t i s t i c s , scintillator, i n which the w i t h i n the definition. the e f f i c i e n c y of a e, and i t s u n c e r t a i n t y 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 n - The number of s u c c e s s f u l events. 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 f o r all of the runs and were observed to d e v i a t e from u n i t y 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%) m a j o r i t y of c a s e s . Somewhat l a r g e r the i n the d e v i a t i o n s occurred when average pion momentum was l e s s than 100 MeV/C, In such cases., the second p i o n s c i n t i l l a t o r appeared to have a lower efficiency real (as low as 98%). T h i s , however, d i d not r e f l e c t a inefficiency of the s c i n t i l l a t o r , but rather a breakdown of the method used to d e f i n e the e f f i c i e n c y , i n particular, 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 , c o u l d stop i n the m a t e r i a l between the f i r s t second s c i n t i l l a t o r s , and t h e r e f o r e appear and ( a r t i f i c i a l l y ) as a scintillator inefficiency. For of the r e s t of the a n a l y s i s such small i n e f f i c i e n c i e s the s c i n t i l l a t o r s were n e g l e c t e d . The apparent inefficiency of the pion arm (second s c i n t i l l a t o r ) was then taken i n t o account i n the d e f i n t i o n acceptance of the d e t e c t i o n system. of t h e . s o l i d angle 65 4.3.5 The MULTI-WIRE PROPORTIONAL-CHAMBER EFFICIENCIES e f f i c i e n c y of each MWPC was determined by a method s i m i l a r t o that employed to determine the e f f i c i e n c y of the scintillators. First, trial events, were i d e n t i f i e d , namely those events a s s o c i a t e d with a p a r t i c l e that was i n f e r r e d to have passed through a m u l t i - w i r e the m u l t i - w i r e detected chamber was t e s t e d to determine i f i t had the p a r t i c l e of these t r i a l p r o p o r t i o n a l chamber. Then, (a s u c c e s s f u l e v e n t ) . The d e f i n i t i o n events was: 1) A l l four s c i n t i l l a t o r s d e t e c t e d pulse-heights p a r t i c l e s with and t i m e - o f - f l i g h t values those of the pp->7r were smaller than the a c t i v e surface + d event d e f i n i t i o n 2) The sum time (discussed consistent (the s c i n t i l l a t o r s of the MWPC). i n sect ion(3.9)) associated the conjugate wire chamber was w i t h i n acceptable T h i s c o n d i t i o n ensured that only with with limits. single p a r t i c l e s traversed the conjugate counter. 3) The p o s i t i o n of the p a r t i c l e was w i t h i n of the center five centimeters of the conjugate wire chamber. Such a t r i a l event was deemed s u c c e s s f u l i f i t s a t i s f i e d the a d d i t i o n a l c o n d i t i o n that both the X and Y d e l a y - l i n e sum times (That i s , the sum of the t o t a l d e l a y - l i n e propagation times, d i s c u s s e d the m u l t i - w i r e i n section (3.9)) of proportional-chamber under c o n s i d e r a t i o n w i t h i n acceptable with double t r a c k s l i m i t s . Those few t r i a l events were associated i n the chamber under c o n s i d e r a t i o n were thus r e j e c t e d since the d e l a y - l i n e read-out system only 66 provides accurate p o s i t i o n information The efficiency for single tracks. e, and i t s e r r o r Ae, of the m u l t i - w i r e p r o p o r t i o n a l chamber were a l s o d e s c r i b e d by equation (07). 4.3.6 BEAM POLARIZATION The magnitude of the beam p o l a r i z a t i o n normal to the r e a c t i o n plane was monitored with the p o l a r i m e t e r . The 2 9 p o l a r i z a t i o n was determined from the measured asymmetry, e, of the l e f t - r i g h t polarimeter s c a t t e r i n g of the i n c i d e n t beam from the target: P = e / A (08) XT Where A^ i s the a n a l y z i n g power of the p o l y e t h y l e n e of the p o l a r i m e t e r , target the u n c e r t a i n t y i n the p o l a r i z a t i o n P, a r i s e s both from standard (Poisson) counting s t a t i s t i c s as w e l l as from a systematic u n c e r t a i n t y i n the a p p r o p r i a t e value of the a n a l y z i n g power, A . Although the l e f t - r i g h t P asymmetry i s dominated by the p p - e l a s t i c s c a t t e r i n g from the hydrogen component of the t a r g e t , q u a s i - f r e e s c a t t e r i n g from the protons i n carbon a l s o c o n t r i b u t e d , l e a d i n g to c o r r e c t i o n s of 5-10% from the f r e e p-p v a l u e s . The values used f o r the a n a l y z i n g power were obtained from internal TRIUMF communications. 4.3.7 BEAM CURRENT NORMALIZATION The beam f l u x i s determined from the p p - e l a s t i c s c a t t e r i n g rate at 90° C M . r e s u l t i n g from i n t e r a c t i o n of 67 the incident the pp—*-7r*d beam with the protons i n the t a r g e t reaction used f o r p r o d u c t i o n . The number of s c a t t e r e d 1 0 protons d e t e c t e d by the p p - e l a s t i c monitor are r e l a t e d to the p p - e l a s t i c d i f f e r e n t i a l cross-section d o / d f l = i{ Ns / ( N p p These terms are d e f i n e d number of p o t e n t i a l N int = N c in d e t a i l i n appendix } (1). The is identical nfc (09) f o r the r e a c t i o n , and i s given by; / * s 2 Afi) - do /dfi i n f c i n t e r a c t i o n s N^ simultaneous pp—>7r*d do^/dR by; 2 A f i [ 2 d a pp / d n + d o c / d n ]J where: Ns - Twice the number of p p - e l a s t i c events. N^ n t - The number of p o t e n t i a l interactions ( N(beam)*N(target) ) AJ2 - The p p - e l a s t i c monitor acceptance s o l i d The values of the pp—>pp e l a s t i c cross-sections angles used are l i s t e d i n appendix was subject systematic typically error. angle. and s o l i d (1). The value of N ^ t o a 0.5% random e r r o r and a 1.8% n t 68 4.4 SOLID ANGLES 4.4.1 GEOMETRIC SOLID ANGLES The geometric s o l i d angles as d e f i n e d here represent both the s o l i d angles subtended the j o i n t of geometric by i n d i v i d u a l d e t e c t o r s , and s o l i d angle subtended by a combination two d e t e c t o r s . They depend only on the apparatus geometry and the pp—>7r d r e a c t i o n k i n e m a t i c s . + The i n d i v i d u a l l a b o r a t o r y geometric s o l i d angles of the pion and deuteron d e t e c t o r s , Afl and AO,, a r e : AJ2g = J dfi and Afl d = / 6SI (11) Where the domains of the i n t e g r a t i o n v a r i a b l e s a r e : fi 0 - The set of Laboratory angles {0,(j>} subtended by the pion detector. S2, - The set of Laboratory angles {#,</>} subtended by the deuteron d e t e c t o r . In both cases the domain of the i n t e g r a t i o n v a r i a b l e was d e f i n e d by a small r e c t a n g u l a r s u r f a c e (the d e t e c t o r ) of l i n e a r dimensions Ax, and Ay, a d i s t a n c e r , from the t a r g e t . 69 Consequently these i n t e g r a l s can be approximated by; Afi = A0A0 (12) where: A0 = 2 t a n - ( Ax/2r ) 1 A 0 = 2 t a n " ( Ay/2r ) 1 4.4.2 TRANSFORMATION OF THE SOLID ANGLE TO THE CENTER-OF-MASS SYSTEM Transformation of the l a b o r a t o r y 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 r e a c t i o n . The corresponding center-of-mass s o l i d angles (designated with a * s u p e r s c r i p t ) are then: AO * = J\ dO * and * * ASK = f. dO o! (13) d Where the domains of the i n t e g r a t i o n v a r i a b l e s .are: 0 ie _ 0 The set of C M . angles subtended ic ic {0 ,</> } by the pion detector. * * * - The set of C M . angles {0 ,tf> } subtended by the deuteron detector. C a l c u l a t i o n of these q u a n t i t i e s i s s i m p l i f i e d by the f o l l o w i n g three s t e p s : 70 First, the center-of-mass s o l i d angles were obtained by i n t e g r a t i n g over the l a b o r a t o r y c o o r d i n a t e s , s o l i d angle transformations (Jacobians) j (0 ) and j , ( 0 , ) . TT Where the pion s o l i d angle u t i l i z i n g the transformation, a d 7T j (0 ), is; j ( 0 ) = dS^/dfl^ 7r and (14) 7r that of the deuteron J ( 0 d ) r i s ; j ( 0 ) = dfi*/dfi d d d d Second, these Jacobians were approximated by t h e i r values a t the c e n t r a l azimuthal integral angle and f a c t o r e d from the (such a procedure i s i n v a l i d , however, at or near the peak deuteron a n g l e ) . Thus: AO* = J j (0 )dfl = j (0 )/ dil An j it = g Tr and (15) A n d = ' W d n d • dn d T h i r d , as i n d i c a t e d , i d e n t i f i c a t i o n of the r e s u l t a n t i n t e g r a l s with the l a b o r a t o r y geometric s o l i d (equation angles (11)). The j o i n t s o l i d angle of the system i s that d e f i n e d by the c o i n c i d e n t d e t e c t i o n of both f i n a l - s t a t e p a r t i c l e s . For the apparatus d e s c r i b e d , i t was d e f i n e d by the pion detector 71 which subtended a s m a l l e r center-of-mass s o l i d angle than the deuteron d e t e c t o r . 4.4.3 THE EFFECTIVE SOLID ANGLE In a d d i t i o n to the c o n s t r a i n t s imposed by•the of the apparatus, the e f f e c t i v e acceptance dependent on the nature of the p h y s i c a l geometry of the system was interactions experienced by the p a r t i c l e s as they t r a v e r s e d the apparatus. The e f f e c t s of pion decay (TT —>n* v) , m u l t i p l e + s c a t t e r i n g , e n e r g y - l o s s , and ranging-out can be combined with the geometric c o n s t r a i n t s to d e f i n e an e f f e c t i v e angle (CM.) solid AS2^. T h i s e f f e c t i v e s o l i d angle i n c o r p o r a t e s an event d e t e c t i o n e f f i c i e n c y , e(r,fi ,fl ), i n t o the s o l i d angle def i n i t i o n : A0T = S* /* e(r,n*,J2*) dS2* dfl* (16) where: + AJ2' e(r,& - The e f f e c t i v e s o l i d ,S2 ) - The angle event d e t e c t i o n efficiency * - The i n i t i a l pion direction. (r,fl) - P o l a r c o o r d i n a t e s of the detection point. it ft„ - The set of a l l p o s s i b l e pion p r o d u c t i o n a n g l e s . As d e f i n e d here, the event d e t e c t i o n e f f i c i e n c y r e p r e s e n t s 72 the p r o b a b i l i t y of d e t e c t i n g an event with an i n i t i a l direction s p e c i f i e d by the angular p o i n t s p e c i f i e d by i t s d i s t a n c e * , with respect formalism c o o r d i n a t e s fi , at a r , and angular coordinates to the t a r g e t and beam d i r e c t i o n . In t h i s pions c r e a t e d with they would miss the'pion detected pion t r a j e c t o r i e s so d i r e c t e d that detector could, i n p r i n c i p l e , be f o l l o w i n g a change of d i r e c t i o n . I f the d e t e c t i o n of e i t h e r a pion or i t s a s s o c i a t e d muon decay product together with the c o r r e l a t e d deuteron s a t i s f i e s the event definition, then i t s d e t e c t i o n e f f i c i e n c y can be w r i t t e n i n terms of the d e t e c t i o n e f f i c i e n c i e s of the i n d i v i d u a l particles: where: R(fi ) Represents the i n i t i a l deuteron d i r e c t i o n as a f u n c t i o n of the c o r r e l a t e d pion The e (R(0*)) d direction. deuteron d e t e c t i o n efficiency. The pion detection ef f i c iency. e ( r , f i ,S2 ) The muon d e t e c t i o n ef f i c iency. 73 If t h i s form of the d e t e c t i o n e f f i c i e n c y i n t o the integrand of equation i s substituted (16), then the e f f e c t i v e * s o l i d angle separates i n t o pion and muon components, AJi^ and * An^ r e s p e c t i v e l y : A f i = Afl* + AR* (18) T U 7T where: An * = /* S* e ( r , n * , B * ) dn* dn* An* = j \ /* (r,n*,n*) dn* dn* e fi 0 ^ ft These two components have d i f f e r e n t p r o p e r t i e s , thus are evaluated separately. 4.4.4 The THE PION COMPONENT OF THE EFFECTIVE SOLID ANGLE relatively propagation through simplification (that simple nature of pion and deuteron the apparatus results in a significant of- the pion term of the e f f e c t i v e s o l i d angle i s , the pion e f f e c t i v e s o l i d a n g l e ) . I f the pions and deuterons straight are each assumed to t r a v e l lines, (on average) along (as d e f i n e d by the a p p r o p r i a t e kinematic q u a n t i t i e s ) then three approximations may be employed: F i r s t , * t h e d e t e c t o r arrangement d i c t a t e s that deuteron i s always d e t e c t e d , hence: e(R(n*)) = 1 d (19) Second, the r a d i a l dependence of the pion d e t e c t i o n efficiency i s expected to be p r o p o r t i o n a l t o the f r a c t i o n , 74 f , of pions s u r v i v i n g decay i n f l i g h t : f = f ( r ) = exp( m r / ( rp ) ) IT where p IT i s the pion Third, (20) TT 7T momentum and r i s mean l i f e at r e s t . * the angle of d e t e c t i o n 0 , becomes i d e n t i c a l to the c r e a t i o n angle 0 . T h e r e f o r e the angular d e t e c t i o n represented by a d e l t a ' f u n c t i o n , e,(R(fl*)) Substituting An* = (equation / * ;* f OQ trivial, f ir ir 6( 0*- fl* ) t h i s e f f i c i e n c y i n t o the pion angle i n t e g r a t i o n Integration and the e f f i c i e n c y becomes; e (r Q*,Q*) = f a p r o b a b i l i t y can be (21) effective solid (18)) y i e l d s : «( o*- n* ) dn* dn* (22) 04 over the i n i t i a l pion direction variable 0 i s leaving;- An* = f (r) J* dfi* * * n* The f i n a l integration i s simply the geometric s o l i d (equation (13)), and t h e r e f o r e ; AO* = f An* ir ir g (23) Furthermore, s u b s t i t u t i n g equation for angle (12) and equation (15) the geometric s o l i d angle y i e l d s ; AO* = f ir ir This representation j (6 J ir ir )A0A0 • (24) of the pion component of the e f f e c t i v e 75 s o l i d angle was verified (to w i t h i n a one Monte C a r l o s i m u l a t i o n s of the experiment percent) through (appendix (2)) for runs of average pion momenta g r e a t e r than 100 MeV/c (greater than approximately 4.4.5 THE 35 MeV.). MUON COMPONENT OF THE EFFECTIVE SOLID ANGLE E v a l u a t i o n of the muon component of the e f f e c t i v e angle (equation (18)) solid i s not as s t r a i g h t f o r w a r d as i t i s in the case of the pion component. P r i m a r i l y , t h i s i s a consequence of the g e n e r a l l y n o n - c o l i n e a r pion-muon t r a j e c t o r i e s . T h i s p o i n t i s r e f l e c t e d by non-zero values the event d e t e c t i o n e f f i c i e n c y of e^(r,R ,fl ), i n cases where ~* the i n i t i a l pion d i r e c t i o n $2 , and * detection point c o o r d i n a t e s Q , d i f f e r . Consequently, the pion angular production s o l i d angle, as d e f i n e d by the pion d e t e c t o r alone, i s l a r g e r f o r d e t e c t i o n of muons than i t i s i f pions detected. detector are In a d d i t i o n , the acceptance of the deuteron i s not l a r g e enough to d e t e c t a l l the deuterons a s s o c i a t e d with parent pion t r a j e c t o r i e s d i r e c t e d i n t o i n c r e a s e d s o l i d angle; t h e r e f o r e the angle was ( j o i n t ) muon s o l i d no longer determined by the pion detector acceptance alone. T h i s can be shown by decomposing the angle the solid i n t o terms that d i s p l a y the e x p l i c i t dependence on the 76 deuteron arm geometry. An* = s* S* e (r,o*,o*) dn* dn* = / * { ; * eM n n 0 + (25) (r,n*,n*)dn* 2 e (r,n*,n*)dn* } dn* n M 3 where the i n t e g r a t i o n v a r i a b l e s domains ( s e t s ) s a t i s f y : n* n* - {n*} : R(n*) e {n*} - {n*} : R(n*) \ = n* u si* Q* {n^} - The s e t of angular coordinates subtended by the deuteron detector. If the deuteron straight i s assumed to t r a v e l (on average) i n a l i n e , then the d e t e c t o r geometry d e f i n e s the following detection e f f i c i e n c y ; 1; e (R(ji*)) d if R(n*) e {R* ) d = (26) 0; if R(n*) v {n*} d C l e a r l y , the second term i n the muon e f f e c t i v e s o l i d angle 77 v a n i s h e s , l e a v i n g the double integral * An (27) i n t e g r a t i o n over both of the pion and deuteron d e t e c t o r angular c o o r d i n a t e s r e s u l t s . 4.4.6 SEMI-PHENOMENOLOGICAL MODEL OF THE MUON COMPONENT OF THE EFFECTIVE SOLID ANGLE E v a l u a t i o n of the muon component of the e f f e c t i v e * solid angle A$2^ was of s u f f i c i e n t complexity that n o n - a n a l y t i c methods were employed. I t s e v a l u a t i o n , t h e r e f o r e , was c a r r i e d out i n two s t e p s . F i r s t , a semi-phenomenological model of the s o l i d angle was developed. Then, d e t e r m i n a t i o n of the f r e e parameter of the model was c a r r i e d out u s i n g the r e s u l t s of Monte-Carlos s i m u l a t i o n s of the experiment number of s e l e c t e d experimental The daughter s o l i d angle subtended for a configurations. by the parent pions (whose muons are detected) i s again much l a r g e r than that of the a s s o c i a t e d deuteron Afl^, and i s (approximatly) bound by a maximum muon s o l i d angle AS2 , d e f i n e d by the Jacobian peak angle 9 c h a r a c t e r i z i n g the pion decay. That i s ; AO* = 2TT{ 1 - cos( 9 ) } (28) As a r e s u l t of the g r e a t e r s i z e of t h i s maximum muon s o l i d * angle r e l a t i v e t o that of the a s s o c i a t e d deuteron Afl^, the joint s o l i d angle of the two d e t e c t i o n systems i s no longer determined by the s i z e of the pion d e t e c t o r alone (as i t i s 78 * for AO ). The on initial investigation the e f f e c t i v e fraction of the e f f e c t s o l i d angle i n v o l v e d comparison of the of the t o t a l e f f e c t i v e * of pion decay s o l i d angle c o n t r i b u t e d by t the muon (Afl^/ASr ) to the r a t i o of the "maximum" muon t o deuteron s o l i d angles, (Afi^/AO^). C l e a r l y , this ratio depends on the f r a c t i o n of muons present, f •. That i s ; AO*/AQ^ = f { F( Afi*/Afi* ) } M M y d (29) where: Interestingly, as shown i n f i g u r e (4.9), the Monte C a r l o s i m u l a t i o n of the experiment f o r a s e l e c t configurations indicated set of a simple e x p o n e n t i a l relationship for F as a f u n c t i o n of the argument d i s p l a y e d i n equation (29). By i n t e r p o l a t i n g the r e s u l t s of t h i s figure to other values of the argument, (AO^/AO^), the t o t a l effective equation s o l i d angle c o u l d be determined using (18) r e w r i t t e n as; AJT* = Afi*/( 1 - AO^/AO " ) (30) 1 Again, r e w r i t t e n as a f u n c t i o n of the parameter F, t h i s yields; AQ? = Afi*/( 1 - Ff Substituting the e x i s t i n g ) e x p r e s s i o n f o r the pion (31) effective 79 Figure (4.9) The E f f e c t i v e Muon S o l i d Angle F Parameters, N \ • \ • 0.8 t a b • \ \ N \ \ n \ 0.6 \ model S — 0.4 0.2 011 • i I I I I Ij l(f i i i i i,i A%i/AilTd I I I 10' The F parameters determined from Monte C a r l o s i m u l a t i o n s of the experiment f o r s e l e c t e d c o n f i g u r a t i o n s . The s o l i d l i n e i n d i c a t e s the p r e d i c t i o n s 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 t h i s data. 80 s o l i d angles Afi The (equation (23)) i n t o t h i s equation T = AO* { f / ( 1 - F f ^ ) } (32) i r e f f e c t i v e s o l i d angle Afi^ was determined the f i r s t yields; i n t h i s way, to order, f o r a l l the experimental c o n f i g u r a t i o n s + employed. F i n a l v a l u e s of AO f o r a small number of cases 1 involved a d d i t i o n a l correction described i n section f o r e n e r g y - l o s s e f f e c t s as (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 e v a l u a t e d i n a Monte C a r l o s i m u l a t i o n which i n c o r p o r a t e d pion-decay m u l t i p l e - s c a t t e r i n g and e n e r g y - l o s s f o r both pions and muons. As the p a r t i c l e e n e r g y - l o s s c o n t r i b u t i o n t o the e f f e c t i v e s o l i d angles was found t o be i n s i g n i f i c a n t i n the m a j o r i t y of cases, these e n e r g y - l o s s e f f e c t s are n e g l e c t e d in the f o l l o w i n g d i s c u s s i o n and t r e a t e d as a small c o r r e c t i o n at a l a t e r p o i n t . Assumptions used to d e r i v e the pion e f f e c t i v e s o l i d angle e x p r e s s i o n (equation (24)) were v e r i f i e d , as were a s e l e c t number of the a s s o c i a t e d s o l i d angle p r e d i c t i o n s , to w i t h i n a one percent (statistical) accuracy. Monte C a r l o e v a l u a t i o n s of the complete effective s o l i d angle AQ', were then combined with v a l u e s c a l c u l a t e d * for the geometric and c r o s s s e c t i o n s Afi^, the pion f r a c t i o n s f the muon f r a c t i o n s f ^ , to determine parameters a c c o r d i n g t o the formula; f f , the aforementioned F 81 F = { 1 - £ As d e p i c t e d in figure reasonably linear (Afig/Afi ) } / f (33) 1 v (4.9), M t h e y were f o u n d to exhibit a d e p e n d e n c e on t h e l o g a r i t h m of the r a t i o (An*/AJ2*) ; F = { a log 1 0 ( Afi*/Afi* ) + b } ± A (34) where; a This, within reasonable and A b B 0.84 the i n d i c a t e d u n c e r t a i n t y , = 0.05 provided a 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 obtained = -0.39 uncertainty of the e f f e c t i v e by d i f f e r e n t i a t i n g c a l c u l a t i n g the root appropriate equation solid (33) w i t h mean s q u a r e d e v i a t i o n s angles i s respect t o F, of the variables. d(An ')/An = { f / ( 1 - F f 1 t ~ f ) } dF (35) dF where: d(Afl^) - The u n c e r t a i n t y of the t effective dF solid - The u n c e r t a i n t y a n g l e AJ2 . of the F parameter. Given the uncertainty uncertainty than of the e f f e c t i v e two p e r c e n t , fraction. o f F ( dF = A = 0.05 ), t h e depending solid angle i s typically less ( a p p r o x i m a t e l y ) on t h e muon 82 4.4.8 The ENERGY-LOSS Monte C a r l o s i m u l a t i o n s enerqy-loss of the p a r t i c l e s was i n d i c a t e d that i f neglected, m u l t i p l e s c a t t e r i n g e f f e c t s c a n c e l l e d out figure then small-angle (refer to (4.10)). For low values of the pion energy, however, such a c a n c e l l a t i o n ceases to be exact. The effect is p r i m a r i l y due that d e f i n e s to the f a c t that the aperture geometric s o l i d angle (the MWPC), and that f o r the the particle 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 p h y s i c a l l y separated. The p a r t i c l e s which are s c a t t e r e d i n t o the before the f i r s t aperture have f u r t h e r to t r a v e l system and t h e r e f o r e more m a t e r i a l to t r a v e r s e than those which s c a t t e r out. As the pion (and muon) energies decrease, the particles that t r a v e r s e l a r g e r d i s t a n c e s s u f f e r an i n c r e a s i n g p r o b a b i l i t y of e i t h e r ranging-out s c a t t e r i n g out. F i g u r e (stopping) or of (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 apparatus. t r a v e r s i n g the These e f f e c t s l e a d to a r e d u c t i o n of e f f e c t i v e s o l i d angle as the pion the l a b o r a t o r y energy decreases beyond some t h r e s h o l d v a l u e . The associated correction i s negligible magnitude of the (much l e s s 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 c o r r e c t e d f o r e n e r g y - l o s s , s i z e of the c o r r e c t i o n are t a b u l a t e d in table and (4.1). the Figure Schematic Representation MWPC (4.10) of t h e E f f e c t o f P a r t i c l e S o l i d Angle. Energy-loss on t h e E f f e c t i v e SCINTILLATORS APERTURE 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 o f p a r t i c l e s a r e i n d i c a t e d s u p e r i m p o s e d on t h e a p p a r a t u s . The t r a j e c t o r i e s above t h e c e n t r e l i n e r e p r e s e n t t h o s e r e s p o n s i b l e f o r t h e c a n c e l l a t i o n of s m a l l - a n g l e m u l t i p l e - s c a t t e r i n g s . T h o s e below t h e l i n e indicate t h e e f f e c t of r a n g i n g - o u t and l a r g e a n g l e s c a t t e r i n g s on t h e l o n g e r t r a j e c t o r y , and hence a mechanizm f o r t h e b r e a k down o f s u c h cancellations. 84 Figure Low (4.11) Energy Pion Energy Distributions, 2400h co lo o LL. o tr LU LTJ 0 10 20 KINETIC ENERGY (MeV) 30 The energy d i s t r i b u t i o n of pions i s shown at the t a r g e t (the higher energy d i s t r i b u t i o n ) and upon e n t e r i n g the f i n a l scintillator (sintillator # 2 ) . Table (4.1) The C o r r e c t i o n s to S o l i d Angles A s s o c i a t e d with Low Energy P i o n s . Inc ident Proton Energy Pion Energy Pion Angle' (CM. ) Target Thickness (MeV) (MeV) (degrees) (cm) 350 350 350 350 350 375 375 375 375 375 375 375 375 375 425 425 450 450 12.3 14.0 16.0 17.0 28. 1 13.8 21 .3 28.5 35. 1 14.1 18.6 19.6 23.6 33.3 26.2 32.7 26. 1 31 .3 138.6 134.9 131.0 128.9 110.2 146.1 132.6 121.9 113.0 145.4 136.9 135.2 128.8 115.3 142.7 134.3 150.5 143.2 0.340 0.300 0.270 0.260 0.330 0.071 0.110 0.083 0.070 0.250 0.320 0.340 0.350 0.240 0.069 0.089 0.058 0.067 Solid Angle correction Factor ( ± 2%) 0.91 0.95 0.96 0.98 0.89 0.98 0.99 1 .00 0.94 0.95 0.96 1.01 0.99 1 .00 0.96 1 .00 86 4.5 DETECTOR AND GEOMETRIC CALIBRATIONS M u l t i - w i r e p r o p o r t i o n a l chambers d e l a y - l i n e read-out systems provide i n f o r m a t i o n on p a r t i c l e p o s i t i o n s and t r a j e c t o r i e s as a f u n c t i o n of d e l a y - l i n e t i m i n g d i f f e r e n c e s measured with TDC's. In order to be able to i n f e r s p a t i a l i n f o r m a t i o n , c a l i b r a t i o n of the system was necessary. The a b s o l u t e p o s i t i o n s of the MWPC's c o u l d then be determined through study of the r e s u l t s of simultaneous measurements of 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 + pp—*-7r d angular c o r r e l a t i o n s . D e t a i l e d d i s c u s s i o n of these c a l i b r a t i o n s , i n a d d i t i o n t o those of the s c i n t i l l a t o r p o s i t i o n s i s presented in the f o l l o w i n g sections. 4.5.1 MULTI-WIRE PROPORTIONAL CHAMBER D e t e c t i o n of an event spatial i n i t i a t e d the r e a d i n g of the i n f o r m a t i o n from the cathode d e l a y - l i n e read-out system CALIBRATION planes of the MWPC's. A such as that employed here i n v o l v e s the e l e c t r i c a l c o n n e c t i o n of the v a r i o u s cathode wires at r e g u l a r l y spaced (discussed in section times of a cathode i n t e r v a l s along a d e l a y - l i n e (3.9)). A comparison of the a r r i v a l s i g n a l at the opposite ends of the d e l a y - l i n e thus p r o v i d e s q u a n t i t i e s that must be c a l i b r a t e d to y i e l d s p a t i a l c o o r d i n a t e s . When a MWPC was i l l u m i n a t e d with r a d i a t i o n , data read from the cathode plane whose sense wires were o r i e n t e d p a r a l l e l t o the anode plane wires contained i n f o r m a t i o n r e l a t e d to the p o s i t i o n of the anode wires. An image of the 8 7 anode wires c o u l d be observed by histograming the TDC channel number d i f f e r e n c e 6. T h i s image, when combined with the known anode wire p o s i t i o n s provided a s t r a i g h t f o r w a r d means f o r i n t e r n a l l y c a l i b r a t i n g t h i s cathode plane. C a l i b r a t i o n of the d e l a y - l i n e read-out a s s o c i a t e d with the o p p o s i t e cathode plane was more complex as no comparable i n t e r v a l technique c o u l d 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 e f f e c t e d through the comparison apparent of t h e i r dimensions with those expected by geometry. 4.5.1.1 The Delay-Line The p r i n t e d c i r c u i t d e l a y - l i n e s used are f a r from i d e a l . E l e c t r i c a l i n such chambers s i g n a l s were both attenuated and d i s p e r s e d when propagated along the d e l a y - l i n e . The overall effect concerned) (so f a r as the f o l l o w i n g a n a l y s i s was was that the apparent group v e l o c i t y of the s i g n a l v a r i e d along the d e l a y - l i n e . The form of the v e l o c i t y dependence, however, was c o n s t r a i n e d t o be symmetric about the c e n t e r of the d e l a y - l i n e . For t h i s reason, a small n o n - l i n e a r component was i n c o r p o r a t e d i n t o the c a l i b r a t i o n r e l a t i o n s h i p f o r the system (see s e c t i o n 4.5.1.3). , 4.5.1.2 The Anode Wire D i 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 f u n c t i o n was denoted T ( 8 ) . I t represented the p r o b a b i l i t y of a d e l a y - l i n e being recorded with a (TDC) full signal channel number d i f f e r e n c e 5, f o r i l l u m i n a t i o n of the MWPC s u r f a c e . 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 a s s o c i a t e d with i n d i v i d u a l anode wires were e a s i l y i d e n t i f i e d . In the envelope Figures of the peaks was (4.13) and symmetric about the c e n t e r . (4.14) i n d i c a t e the v a r i a t i o n shape of the peaks a s s o c i a t e d function could be approximated that by a sum normalized gaussian d i s t r i b u t i o n s of v a r y i n g (resolution) i n the with the c e n t r a l and edge regions r e s p e c t i v e l y . These diagrams i n d i c a t e d distribution addition, the of width c e n t e r e d at each anode wire. Let: i = The sequential number of an anode wire. 6. i = The channel d i f f e r e n c e number corresponding to the i * " * al = The i f c standard d e v i a t i o n 1 wire. of the ^ Gaussian d i s t r i b u t i o n . Then, TU) = I i { expU-6^ The parameters 5^, and 2 / 2oi } / y/2^h~ , were dependent on both (36) the spacing of the anode wires and the e l e c t r i c a l p r o p e r t i e s the of delay-line. 4.5.1.3 C a l i b r a t i o n i n the V e r t i c a l D i r e c t i o n After the d i s c r e t e r e l a t i o n s h i p 6^(x^) between the channel number d i f f e r e n c e 6^, of the i k anode wire x., was fc and the corresponding determined, inversion position 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 D i s t r i b u t i o n Image; C e n t r a l region 500 c o h - O o Lu O OC UJ CD 0 TDC -200 CHANNEL NUMBER 0 DIFFERENCE ( 8 ) 91 Figure The (4.14) Anode Wire D i s t r i b u t i o n Image: Edge Region 500 -600 TDC CHANNEL NUMBER - 800 DIFFERENCE (S) 92 yielded the spatial position form of the signal the if delay-line the $ constrains c S(x = c x(6). velocity the number d i f f e r e n c e = c propagation x , channel function about form of 6 The 6. symmetric the In of particular, is defined c center by; ) (37) 6'(0) where: 6'(x) Then, g i v e n center of the constrained differing (5'(0)), two at to x-x positions, delay-line, direction Therefore, constant 6( t o c h a n g e by ) c each a d i s t a n c e the an (sign) function Ax) 6'(x) be = AX from 6'(±AX) r e l a t i v e to the the is e q u a l m a g n i t u d e , but each extreme p o i n t 6'( o required = by central respectively, that a point is; -6'(-Ax) (38) i s anti-symmetric, anti-symmetric consequently, (neglecting ( i n s t r u m e n t a l ) ) about a central an 6(x) is additive position x . c Furthermore, a higher account for the non-linear position-dependendent delay-line. The order term effect signal functional (cubic) of was introduced to the propagation relationship v e l o c i t y within used was: the 93 6(x)/o - p = ( x-x c ){ 1 + ( x-x 7 c ) 2 (39) } where a - s e t s the o v e r a l l s c a l e p - i s an i n s t r u m e n t a l o f f s e t x - d e f i n e s the center c of 7 (the p o i n t anti-symmetry) - d e f i n e s the extent of non-linearity The v a l u e s of these parameters are obtained by a l e a s t squares f i t of t h i s f u n c t i o n t o the data p o i n t s ( x ^ , 6 ^ ) . As d e f i n e d 5(x) i s a c u b i c f u n c t i o n which was i n v e r t e d . By analogy equation with standard (39) was expressed 0 = z 3 + 3qz techniques , 3 3 i n standard - 2r z = x - x„ c 1 / 7 -2r = ( p - 5/a ) / form; (40) where: 3q'= readily 7 94 As the d i s c r i m i n a n t d, i s p o s i t i v e , and real, then the r e a l root of equation z = ( r - /d ) 1 / 3 Finally, figure + r the x c o o r d i n a t e x(6) The 3 = z + x are (40) i s ; + ( r + /d where the d e f i n i t i o n of the descriminant d = q a l l coefficients ) 1 / (41) 3 d, is ; 2 i s then; (42) c r e s u l t s of such a c a l i b r a t i o n are d e p i c t e d i n (4.15) where the q u a n t i t y A6^ i s p l o t t e d versus the wire number f o r a t y p i c a l run, where; M i This quantity = 6. + 1 (43) - 6. i s shown s i n c e i t i s g r a p h i c a l l y more s e n s i t i v e to the n o n - l i n e a r the v i s i b l e peak spacing s e p a r a t i o n . The wire (7) term then i s 6^(x). Here, represents the (0.2cm) anode wire p a r a b o l i c shape, symmetric about the (as opposed to a constant center f u n c t i o n ) r e s u l t e d from the n o n - l i n e a r i t y of the p o s i t i o n f u n c t i o n , x ( 5 ^ ) . 4.5.1.4 C a l i b r a t i o n i n the H o r i z o n t a l D i r e c t i o n The by read-out system of the cathode plane d i s t i n g u i s h e d sense wires p e r p e n d i c u l a r to those of the anode plane c a l i b r a t e d with a d i f f e r e n t method. The s c i n t i l l a t o r was s i z e of each measured with a MWPC. Comparison of i t s 'shadow' s i z e to i t s known (projected) s i z e provided the was Figure The (4.15) Anode Wire Spacing The i n t e r v a l (A5.) of the TDC Channel number d i f f e r e n c e 8, between anode wires as a f u n c t i o n of the anode wire number. Each i n t e r v a l i s a s s o c i a t e d with the 2.0 mm p h y s i c a l s e p a r a t i o n of the anode wires. The n o n - l i n e a r shape d i s p l a y e d i n d i c a t e s the n o n - l i n e a r i t y of the d e l a y - l i n e spatial calibration. 97 .ff. = a + b{ 1 - exp[( i - i c ) / 2a ] } w (44) where; i a = The center wire w = The Gaussian T h i s form of the r e s o l u t i o n o^ of T(6) shown i n f i g u r e channel equation T(5), r number. (envelope) width. r e q u i r e d f o r the d e s c r i p t i o n (4.12) and the p r e v i o u s l y determined number d i f f e r e n c e 5 ( x ^ ) , were s u b s t i t u t e d i n t o the (36) of the anode wire d i s t r i b i b u t i o n f u n c t i o n and the f r e e parameters a, and b, were f i t (by l e a s t squares) t o the data. The r e s u l t i n g a and b c o e f f i c i e n t s are used t o c a l c u l a t e the r e s o l u t i o n at the c e n t e r , and a t the edges of the d e t e c t o r . The r e s u l t s a r e : C e n t r a l R e s o l u t i o n : 0.05cm R e s o l u t i o n more than 3cm from the c e n t e r : 0.08cm 4.5.2 SCINTILLATOR CENTRAL OFFSETS As d e s c r i b e d i n the p r e v i o u s s e c t i o n , an image a s s o c i a t e d with each s c i n t i l l a t o r was p r o j e c t e d 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 measured and i t s dimensions and i t s p o s i t i o n image was ( i n the C a r t e s i a n c o o r d i n a t e system a p p r o p r i a t e t o the MWPC) were deduced. The c o o r d i n a t e s of the center of each are t a b u l a t e d i n t a b l e (4.2). scintillator Table Relative Scintillator Arm x (c.m.) D F B (4.2) 0.57(16) 0.08(16) 0.00(16) Central Centres Offsets y (Degrees) -0.13(4) 0.04(7) 0.00(9) Centres (cm. ) -0.04(20) 0.42(20) 0.00(20) The measured s e p a r a t i o n o f t h e 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 s y s t e m ( p e r p e n d i c u l a r t o t h e c e n t r a l a x i s ) . The q u a n t i t i e s i n b r a c k e t s r e p r e s e n t t h e u n c e r t a i n t y of t h e l a s t digits. 99 4.5.3 CALIBRATION OF THE DEUTERON ARM HORN APERTURE An image of the deuteron the deuteron horn aperture was formed on MWPC. The v e r t i c a l dimension and center of the aperture were deduced and the r e s u l t s a l s o t a b u l a t e d i n (4.3). table I t s known p r o j e c t e d 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 e r r o r s i n the measured p o s i t i o n s of the two arms, i t was p o s s i b l e f o r the angular c o o r d i n a t e s of p a r t i c l e s c a l c u l a t e d as a f u n c t i o n of t h e i r s p a t i a l coordinates (me-asured by a MWPC) t o d i f f e r from the ' a c t u a l ' v a l u e s . The term 'absolute' used i m p l i e s the a c t u a l values of the angular absolute polar coordinates somewhat here, c o o r d i n a t e s . The (with respect t o the beam d i r e c t i o n ) of a p a i r of c o r r e l a t e d p a r t i c l e s are a b s o l u t e l y s p e c i f i e d by the two body kinematics of the r e a c t i o n . The measurement of t h e i r a s s o c i a t e d azimuthal coordinates (measured i n the plane normal t o 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 a r b i t r a r y o r i g i n . T h i s i s due t o the c y l i n d r i c a l symmetry of the r e a c t i o n kinematics about the a x i s of the beam d i r e c t i o n . Nonetheless, simply r e l a t i v e c o o r d i n a t e s of the two p a r t i c l e s were r e l a t e d by the c o p l a n a r i t y of the two-body final state. The p o l a r angle of a p a r t i c l e deduced from a MWPC s p a t i a l measurement (that i s with no c o r r e c t i o n s a p p l i e d ) 100 Table (4.3) Deuteron-Horn P r o j e c t e d width: Measured width: Measured c e n t r e : Aperture P o s i t i o n a l Calibration. 10.5cm 10.5±0. 02cm -1,0±0. 02cm 101 was designated, by way of the s u p e r s c r i p t s i n d i c a t e d , 6, when deduced from the pion MWPC measurements, or 6, when deduced from the deuteron MWPC measurements. In each case, the measured angle was r e l a t e d t o the a b s o l u t e angles, 0^ or the a d d i t i v e p o l a r o f f s e t s rj^, or T?^; (? , through d 6 = 0it - it 7? ; Pion arm. 8 = # Deuteron arm. D - T}^; (45) Absolute c a l i b r a t i o n of the p o l a r o f f s e t s of both of the d e t e c t i o n arms was based two on the kinematic p r o p e r t i e s of r e a c t i o n s that were measured s i m u l t a n e o u s l y . At p a r t i c u l a r values of the i n c i d e n t beam energy s e t t i n g s of the d e t e c t i o n arms, both + pp—*-7r and angular d events and pp—>pp e l a s t i c events c o u l d be s i m u l t a n e o u s l y d e t e c t e d . The d i f f e r i n g kinematic p r o p e r t i e s of the two r e a c t i o n s c o n s t r a i n e d the i n t e r s e c t i o n ( d e t e c t i o n ) of the t r a j e c t o r i e s of the a s s o c i a t e d r e a c t i o n products to d i f f e r i n g areal regions of the MWPC's a c t i v e s u r f a c e s . The four r e g i o n s , one f o r each of the r e a c t i o n products, a r e i n d i c a t e d i n figure (4.16). Since the pion and deuteron acceptance MWPC's d e f i n e the s o l i d angle f o r d e t e c t i o n of the pp—>it*d and pp—>-pp r e a c t i o n s r e s p e c t i v e l y ; the p i o n and deuteron MWPC's are f u l l y i l l u m i n a t e d with pions and protons As a n o t a t i o n a l a i d to s p e c i f y an otherwise i n which d e t e c t i o n arm, i n d i s t i n g u i s h a b l e proton following notation i s introduced; respectively. i s d e t e c t e d , the 1 02 Figure (4.16) Pion, Deuteron, and E l a s t i c - P r o t o n D e t e c t i o n PION MWPC DEUTERON — 77" d + —• pp Regions MWPC events events The shaded regions of each MWPC s h e m a t i c a l l y i n d i c a t e the a r e a l r e g i o n s of d e t e c t i o n of p a r t i c l e s a s s o c i a t e d with e i t h e r of the two simultaneous r e a c t i o n s . The axes represent the r e c t a n g u l a r coordinate system of the MWPC d e t e c t o r . The l i n e a r s e p a r a t i o n of two such regions on the MWPC s u r f a c e s X, and X , are r e l a t e d t o the angular q u a n t i t i e s A, and A , d i s c u s s e d i n the t e x t . 2 2 103 - Implies proton d e t e c t i o n by the pion detector. (46) p 2 - Implies proton d e t e c t i o n by the deuteron Although the regions depicted i n the C a r t e s i a n c o o r d i n a t e appropriate i n f i g u r e (4.16) a r e s p e c i f i e d system a p p r o p r i a t e to the MWPC, the a s s o c i a t e d p o l a r 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 approximation The detector. s i m i l a r ( w i t h i n the small angle framework). opening angles A ^ 3 indicated reactions „, and A pp—>ir d 3 pp—5-pp' , of the i s then d e f i n e d by the c e n t r a l values of the p o l a r angle d i s t r i b u t i o n s a s s o c i a t e d with the four regions i n d i c a t e d i n f i g u r e (16), that i s ; A _ + , = 0 Pp—^-7T*d A ' = TT 6 PP-^PP - r? + 0 + 6, = 6 d TT +6 Pi = P 2 0 'TT p d 2 - 7 7 + 0 Pi ( 47 ) 77, - 7 7 , p 2 'd where the s u p e r s c r i p t e d q u a n t i t i e s take on the c e n t r a l value of the a s s o c i a t e d p o l a r angle d i s t r i b u t i o n s . The unknown polar offsets 7j f f and 7?^, w i l l cancel out when the d i f f e r e n c e of these opening angles i s formed; that i s ; A^^., pp—>i:*a T h i s expression designated - A^^^ = 0 + 0, - ( 0„ + 0 ^ PP->PP a d PT p can be r e w r i t t e n ) (48) 2 i n terms of q u a n t i t i e s A1 and A2, which a r e d e f i n e d i n terms of the d i f f e r e n c e s between the c e n t r a l p o s i t i o n s of the two p o l a r 1 04 angle d i s t r i b u t i o n s observed on each MWPC r e s p e c t i v e l y ( r e f e r t o f i g u r e (4.16)). That i s i f : * Pi d p Pi 1 1 d 2 p 2 then; A_ - A pp—>-7r d ^ = A, + A (50) 2 pp—=>-pp These A's then, are each d e f i n e d within a s p e c i f i c MWPC, and are thus independent of the p o l a r angle o f f s e t s 7?^ and T J ^ . These A's could be deduced from the ( u n c a l i b r a t e d ) arm positions (which d e f i n e 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 a s s o c i a t e d MWPC's) together with the measured angular representing c o r r e l a t i o n s ( s e c t i o n 4.3.2.3.) the d e v i a t i o n s of d i s t r i b u t i o n s from t h e i r p o s i t i o n s ; that i s ; A ' A But = 2 6 «~ W - { ={ e^Cej - A ^ D M P P } ( 5 , ) }-8 p2 the A ' s c o u l d a l s o be c a s t as a f u n c t i o n of the absolute unknown angles 8 ir A, and 8 ; p 2 = e ,-'(e A 2 = e id + A n p 7rd f 2 2 ) - e^ffi ) pp p 2 - e (e - A , ) pp 7T Where these two equations a r e dependent of course. (52) 105 Once the values of the A's were determined from the experimental values (equation (51)) , they were s u b s t i t u t e d i n t o equation (52); which was then s o l v e d n u m e r i c a l l y u s i n g the r e q u i r e d kinematic f u n c t i o n s , to y i e l d the absolute p o l a r v a l u e s of the angles of the arms. The arm o f f s e t s , were then simply obtained from equation o f f s e t s were not expected the experiment, (45). As these to change s i g n i f i c a n t l y they were c a l c u l a t e d throughout i n d e t a i l only f o r one run. The r e s u l t s are t a b u l a t e d i n t a b l e (4.4). 4.5.5 CALIBRATION OF THE AZIMUTHAL ANGLE IN THE PLANE NORMAL TO THE BEAM DIRECTION The angular o f f s e t s i n t h i s c o o r d i n a t e r e s u l t from v e r t i c a l o f f s e t s of the d e t e c t i o n systems. The v e r t i c a l o f f s e t with respect t o the surveyed p o s i t i o n of the forward pion d e t e c t o r was a r b i t r a r i l y origin taken to be zero (as the f o r t h i s c o o r d i n a t e 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 o f f s e t of the other d e t e c t o r s were then deduced on the b a s i s of the measured c o p l a n a r i t y d i s t r i b u t i o n 4.3.2.3.) of the two-body f i n a l (section s t a t e s . The r e s u l t s of these c a l i b r a t i o n s are t a b u l a t e d i n t a b l e (4.4). 4.6 CARBON BACKGROUND Carbon background events arose from i n t e r a c t i o n of the i n c i d e n t proton beam with n u c l e i of carbon i n the t a r g e t . P o l y e t h e l e n e , the t a r g e t m a t e r i a l , i s a polymer consisting of hydrogen and carbon atoms i n a two-to-one r a t i o . The 106 Table (4.4) The E x p e r i m e n t a l l y Determined Detector Arm d Axis X Survey MWPC -11.91(2)° -11.878(3)° -11.878(3)° 0.91(1)cm 0.91(1)cm 0.87(2)cm -0.14(1 )° -0.14(1)° -0.10(7)° 0.00cm 0.42(2)cm Y TTF X 0.26(4)° Y TTB X Y Offsets. 0.00cm 0.29(6)° Scint.#1 Scint.#2 -12.01(4)° -0.05( 1 )° -0.05(1 ) ° -0.05(9)° 0.06(1)cm 0.06(1)cm 0.06(2)cm The Surveyed angle of the arm i s mesured with respect t o the p h y s i c a l centre of The MWPC. The c e n t e r of the f i r s t s c i n t i l l a t o r i s taken here as the MWPC c e n t r e , which i s the reason f o r the magnitude of the d i f f e r e n c e between the survey and MWPC o f f s e t s . 107 f r a c t i o n of events w i t h i n 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 ; i m p o s i t i o n of s u i t a b l e constraints q u a n t i t i e s such as; the e n e r g y - l o s s e s , the t i m e - o f - f l i g h t s , and ( i n the case of the a n a l y z i n g power data) the angular correlations, r e q u i r e d to d e f i n e an event. 2) Background S u b t r a c t i o n ; d i r e c t s u b t r a c t i o n of the number of carbon background events as determined from data c o l l e c t e d with a carbon The target. f r a c t i o n of carbon background events i n a sample c o u l d not be reduced to l e s s than approximately three percent by method ( 1 ) . Examination of data c o l l e c t e d with a carbon t a r g e t i n d i c a t e d that the events which s u r v i v e d the p u l s e - h e i g h t and e n e r g y - l o s s c o n s t r a i n t s had i n t e r e s t i n g p r o p e r t i e s . In p a r t i c u l a r , t h e i r angular c o r r e l a t i o n and c o p l a n a r i t y d i s t r i b u t i o n s were s i m i l a r to those of the + pp—>-Tr d r e a c t i o n . Although the d i s t r i b u t i o n s were c o n s i d e r a b l y 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 s h o r t , + the observed p a r t i c l e s which had the same e n e r g y - l o s s and t i m e - o f - 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 f r e e pp—*-jr d + r e a c t i o n , were a l s o d i s t r i b u t e d , on average, a c c o r d i n g to the same two-body k i n e m a t i c s . Thus, the apparent pp->7r + d c h a r a c t e r of these carbon background events suggested a q u a s i - f r e e p p — ^ " d origin w i t h i n the carbon n u c l e u s " . That i s , the i n c i d e n t proton 3 i n t e r a c t e d with one of the nucleons, (a proton) bound w i t h i n 108 the carbon nucleus, v i a a two-body r e a c t i o n with the r e s t of the carbon nucleons p a r t i c i p a t i n g only as ' s p e c t a t o r s . ' The momenta (and thus angular c o r r e l a t i o n s ) 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 p r o d u c t i o n r e a c t i o n because of the fermi momentum ( c h a r a c t e r i s t i c of bound nucleons) of the s t r u c k nucleon. 4.6.1 MEASUREMENT OF THE CARBON BACKGROUND Carbon background measurements were taken with a carbon t a r g e t , at s e v e r a l proton beam e n e r g i e s and angular s e t t i n g s of the d e t e c t i o n arms. The beam c u r r e n t was monitored by the p o l a r i m e t e r since the use of the p p - e l a s t i c monitor was i n a p p r o p r i a t e without a hydrogen bearing t a r g e t . The p r e c i s e c a l i b r a t i o n of the p o l a r i m e t e r was, however, unknown. Thus, in each case the data were c r o s s normalized to a s i m i l a r run taken with a p o l y e t h e l e n e t a r g e t where the beam c u r r e n t was measured with both p p - e l a s t i c and p o l a r i m e t e r monitors s i m u l t a n e o u s l y . The number of carbon background events as a f r a c t i o n of the number of pp—*-7r*d events was thereby determined. The r e s u l t s f o r a t y p i c a l proton energy are illustrated in figure (4.17). The d e t e c t o r e f f i c i e n c i e s were not taken i n t o account d u r i n g the f o l l o w i n g a n a l y s i s due to the ambiguties a s s o c i a t e d with t h e i r d e f i n i t i o n when a carbon t a r g e t was employed. Nonetheless, s i n c e the d e t e c t o r e f f i c i e n c i e s were expected, i n g e n e r a l , t o vary s l o w l y , and s i n c e the background i s determined from a r a t i o of two ( u s u a l l y ) c o n s e c u t i v e runs, the d e t e c t o r e f f i c i e n c i e s were 109 expected to c a n c e l l . A q u a n t i t y analogous for the carbon background based on two First, t o the d i f f e r e n t i a l was formed. cross-section I t s d e f i n i t i o n was assumptions: the r e a c t i o n was a two-body process having the same kinematic d e s c r i p t i o n as that of the f r e e pp—^7r"d r e a c t i o n . Second, the acceptance detection ( e f f e c t i v e s o l i d angle) of the apparatus was i d e n t i c a l f o r the q u a s i - f r e e and the pp->7r d r e a c t i o n s . The l a t t e r + assumption, i t will be shown, has l i m i t e d regions of a p p l i c a t i o n . As a r e s u l t of these two assumptions an e f f e c t i v e carbon background cross-section differential i s d e f i n e d by; do /d£2 = 2 f ( 0 * ) do/dR c (53) c where: da /dfl c - The carbon background differential f c (6 it ) cross-section. - The f r a c t i o n of carbon events to pp— >ir + d background events. do/dJ2 The pp—>n*d differential cross-section (estimated, see text) The f a c t o r of two r e s u l t s from the r a t i o of hydrogen to carbon atoms i n the t a r g e t . As p r e c i s i o n v a l u e s of the carbon background pp— >ir *d were not r e q u i r e d , the v a l u e s of the differential cross-section were o b t a i n e d from 1 10 Figure The (4.17) F r a c t i o n a l Carbon Background at 450 MeV. model data spin: up down off 0.08 — • o A 2 O 0O.O6 < or u. 2 m 0.04 cc < 0.02 120 SCATTERING ANGLE 160 {6\) The number of d e t e c t e d carbon background events as a f r a c t i o n of the number of d e t e c t e d pp-^-rr*d events. The s o l i d l i n e r e p r e s e n t s the p r e d i c t i o n s of the q u a s i - f r e e pp—*-7r*d model of the carbon background. The e r r o r bars represent s t a t i s t i c a l uncertainties only. 111 published 4.6.2 data 3 5 . QUASI-FREE PARAMETERIZATION OF THE CARBON BACKGROUND The carbon background d i f f e r e n t i a l c r o s s - s e c t i o n was parameterized on the b a s i s of the q u a s i - f r e e r e a c t i o n model d i s c u s s e d above. I t was assumed that the angular d i s t r i b u t i o n of the carbon background differential c r o s s - s e c t i o n would have the same shape, (but d i f f e r e n t magnitude) as that of the f r e e d r e a c t i o n . Thus, + pp-»7r (54) d a / d f l = X da/d£2 c = X agVUrr) { i ( i=0,2,... + p-n I i=1 Where the c o e f f i c i e n t a o o / a o o ) 1 P .(cos(0*)) * 1 (b?°/ag ) p j ( c o s ( 0 * ) ) } 0 2 X, s c a l e d 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 f r e e + pp—>-Tr d reaction. When presented i n t h i s form the terms that d e f i n e the shape of the angular d i s t r i b u t i o n are i n s i d e the c u r l y b r a c k e t s . Since the carbon background t y p i c a l l y represented a three percent c o r r e c t i o n t o the p p — d i f f e r e n t i a l cross-sections, i t s form c o u l d by reduced i n complexity at the expense of only a small l o s s of p r e c i s i o n cent) by the f o l l o w i n g 1) The r a t i o a ^ V a , 0 0 (about ten per approximations: i s approximatly constant over beam e n e r g i e s from 3 5 0 MeV t o 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 e n e r g i e s used t o c o l l e c t the data i s t h e r e f o r e denoted k; k = 1.08 = a ^ / a g 2) 0 The h i g e r order terms a?°/ag°, are n e g l e c t e d s i n c e t h e i r magnitudes are c o n s t r a i n e d by; aSVag 0 < 0.1 ag°/ag° = 0.0 3) A l l polarization terms b?°/ag°, are n e g l e c t e d s i n c e t h e i r magnitudes are c o n s t r a i n e d by; | b / a g ° | < 0.1 n o b / a g ° = 0.0 n o b ° / a g ° < 0.05 n b /ag° n o =0.0 T h e r e f o r e , to t h i s l i m i t e d - p r e c i s i o n , only the f i r s t two terms 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 sum are r e q u i r e d . That i s ; d a / d n = X a g V U r r ) { P (cos ( 0* )) c (55) 0 + (a§°/ag°) P ( c o s ( 0 * ) ) } 2 E v a l u a t i n g the Legendre f u n c t i o n s and s u b s t i t u t i n g the average value k f o r the a 2 / a g 0 da /dO = X ag°/(47r) C 0 ratio, yields; { 1 + k cos (0*) 2 } 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 (56) 1 13 magnitude i s p r o p o r t i o n a l to the t o t a l c r o s s - s e c t i o n ag°, the pp—>ir*d r e a c t i o n . In t h i s way be c o n s i d e r e d expression simultaneously. 0 Therefore, a l l of the carbon data D i v i d i n g both s i d e s of by the t o t a l c r o s s - s e c t i o n ag°, (ag )- 1 do /dS2 = X / U r r ) can this yields; { 1 + k cos (0*) } (57) 2 c a l l of the carbon background data c o u l d , in p r i n c i p l e , be d e s c r i b e d by a simple model c o n t a i n i n g only one The of quasi-free reaction f r e e parameter, observed carbon background X. differential c r o s s - s e c t i o n , however, appears to f a l l below this * prediction depicted in-t-he forward hemisphere i n f i g u r e (4.18) where the c r o s s - s e c t i o n normalized plot against < 90°). T h i s i s differential to the t o t a l c r o s s - s e c t i o n ag°, i s the q u a n t i t y cos( d were s a t i s f i e d , (6 )|cos(6 )|. I f equation the p l o t would e x h i b i t a m i r r o r (57) symmetry * about the p o i n t cos(0 An explanation )=0. of t h i s asymmetry was based on acceptance of the apparatus for each of the two differing (quasi-free vs. f r e e ) r e a c t i o n types. T h i s r e s u l t e d from the weak angular c o r r e l a t i o n of the q u a s i - f r e e r e a c t i o n f i n a l p a r t i c l e s . The q u a s i - f r e e r e a c t i o n e f f e c t i v e acceptance s o l i d angle c o u l d not be e v a l u a t e d (with the e x i s t i n g Monte C a r l o s s i m u l a t i o n procedure) s i n c e the angular of the final state s t a t e p a r t i c l e s was distribution unknown. Nonetheless r e l a t i v e decrease of the q u a s i - f r e e r e a c t i o n the (product) d e t e c t i o n acceptance c o u l d be q u a n t i t i v l y e x p l a i n e d by the 11 4 Figure (4.18) The E f f e c t i v e Di f f prpnt- i a 1 Cross-Sect ion of the Carbon Background as a Function of cos(6) c o s ( 0 ) The carbon background 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 normalized to the t o t a l pp-^iTd c r o s s - s e c t i o n i s p l o t as a f u n c t i o n of cos(65) |cos(65) | . Carbon data of a l l energies i s i n c l u d e d . The l i n e , a g a i n , r e p r e s e n t s the p r e d i c t i o n s of the model d i s c u s s e d i n the t e x t . 115 d e t e c t o r geometery and the (pp—>ir*&) r e a c t i o n k i n e m a t i c s . 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 c r o s s - s e c t i o n s broke down i n the forward hemisphere section ( i n p a r t i c u l a r , assumption #2; 4.6.1). Nonetheless, the shape of the carbon background angle c o u l d be f i t to the f o l l o w i n g solid semi-phenomenological model; X/(4TT) if 6* { 1 + k cos (0*) 2 }; > 90°. TT da / d f l / ag° = c (58) X/(4TT) if 6* { 1 + k cos (90°)}; 2 < 90°. 7T Where the shape of the carbon background hemisphere i n the forward has been approximated with a constant f u n c t i o n . 4.6.2.1 F i t of the Carbon Background to the Model The two parameters X , and k, were f i t t o the carbon d a t a . The r e s u l t i n g c o e f f i c i e n t average value of the r a t i o k, was c o n s i s t e n t with the a° /ao°. 0 T h e r e f o r e , the carbon background was found to be d e s c r i b e d to s u f f i c i e n t accuracy by the r e l a t i o n ; 116 d a / d f i = X { da/dQ ± ag°A } (59) c where: The X = 0.07 A = 0.02 carbon data and t h i s d e s c r i p t i o n of i t are p l o t t e d i n 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 c r o s s - s e c t i o n s presented here were c a l c u l a t e d as d i s c u s s e d (04) in section (4.2.). Here, equation • i s r e w r i t t e n as a f u n c t i o n of $; da/dfl. = S/Afi " - i ( da /dR ) (60) 5 = ( N (61) 1 c where, p - N r ) / ( N i n t 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 s evaluated by t h i s means f o r the four data s e t s a s s o c i a t e d with the u n p o l a r i z e d energies o f : 350 MeV, 375 MeV, 425 MeV,and 475 MeV, and are shown as a f u n c t i o n of c o s ( 0 2 figures (4.20)-, (4.21 ), ( 4 . 2 2 ) , * ) in IT and (4.24) r e s p e c t i v e l y . The l i n e s i n d i c a t e d on the f i g u r e s represent using Legendre polynomials. values i n c i d e n t beam a f i t to the data In a d d i t i o n , the numerical f o r the c r o s s - s e c t i o n are t a b u l a t e d i n 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 C r o s s - S e c t i o n Background. SCATTERING ANGLE of the Carbon (0^) The carbon background 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 normalized to the t o t a l pp-s-TTd c r o s s - s e c t i o n i s p l o t as a f u n c t i o n of the C M . s c a t t e r i n g angle. Carbon data of a l l energies i s i n c l u d e d . The l i n e , again, represents the p r e d i c t i o n s of the model d i s c u s s e d i n the t e x t . 118 Figure The 350 MeV. 0.2 (4.20) D i f f e r e n t i a l Cross-Sections. 0.4 0.6 0.8 COS (0*) 2 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 an u n 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 . 119 Figure The (4.21) 375 MeV. D i f f e r e n t i a l Cross-Sections I20 0.2 0.4 0.6 COS (0") 0.8 2 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 u n p o l a r i z e d and p o l a r i z e d i n c i d e n t proton beams, represented on the f i g u r e by c i r c l e s and squares r e s p e c t i v e l y . 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 represents the r e s u l t s of f i t s of f o u r t h order Legendre polynomials to these r e s u l t s . 120 Figure The 425 MeV. (4.22) D i f f e r e n t i a l Cross-Sections. 200 C0S (#*) 2 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 an u n 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 represents 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 . 121 Figure The 450 MeV. (4.23) Differential Cross-Sections. 300 11 0 i_! 0.2 i 0.4 1 1— 0.6 0.8 COS (0*) 2 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 h e r e a r e o b t a i n e d from d a t a c o l l e c t e d w i t h a p o l a r i z e d i n c i d e n t p r o t o n 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 d e d u c e d from measurements w i t h the backward p i o n 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 t h e r e s u l t s of a f i t of a f o u r t h o r d e r L e g e n d r e p o l y n o m i a l t o t h e s e results. 1 22 Figure The 475 MeV. (4.24) D i f f e r e n t i a l Cross-Sections. 300 COS (6?*) 2 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 an u n 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 represents 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 23 Figure The 498 MeV. (4.25) D i f f e r e n t i a l Cross-Sections. 400 —" . 0.4 i_ 0.6 COS (i9*) 2 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 results. 124 tables (4.5),(4.6),(4.7), and (4.9) r e s p e c t i v e l y . 4.7.1.1 The U n c e r t a i n t y Cross-Sections: The of the D i f f e r e n t i a l Unpolarized Beam u n c e r t a i n t y of the d i f f e r e n t i a l c o n t a i n s both random and systematic q u a n t i t i e s are expected to vary c o n t r i b u t i o n s . Random randomly about a mean value on a run to run b a s i s . Systematic uniform cross-sections e r r o r s , however, have a e f f e c t on a l l r e s u l t s . These e f f e c t s a r e d i s c u s s e d in d e t a i l The in section (4.9). u n c e r t a i n t y 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 as a r e s u l t of random f l u c t u a t i o n s of the independent v a r i a b l e s d i s p l a y e d by equation (60) above, i s given by; { Mda/dfl] } 2 = ( S/AR *) 1 + ( A£/S ) 2 2 { [ A(AR )/AR T } + { iA[do /dJ2] } 2 = S 2 { (N + ( AN A significant precision i n t + N )/(N r /N i n t simplification i s achieved the above equation 2 c where the u n c e r t a i n t y of the q u a n t i t y AS T ) 2 $, A$, - N ) ] 2 (62) is; 2 r + ( Ae/e ) 2 } with an i n s i g n i f i c a n t (63) l o s s of by approximating the l e a d i n g f a c t o r of by 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 , that is; S/AR" = da/dR 1 Then, the random u n c e r t a i n t y of the d i f f e r e n t i a l (64) 125 Table (4.5) The 350 MeV. D i f f e r e n t i a l Pion Angle e * (degrees) it 90.5 90.6 103.5 108.9 110.2 63.3 58.2 56.5 53.2 128.9 131.0 134.9 40.2 35. 1 33.3 Cross-Sections. D i f f e r e n t i a l Cross-Sect ions * Cos (0 ) 2 7T 0.000 0.000 0.054 0. 105 0.119 0.202 0.278 0. 305 0.359 0.394 0.430 0.498 0.583 0.669 0.699 da /dfl 0 Ub/sr. ) 15.7( 15.9( 19.2( 20.7( 22.0( 25.4( 28.3( 30.4( 33.2( 34.1( 35.8( 40.3( 42.5( • 48.3( 49.8( 0.5) 0.4) 0.5) 0.7) 0.7) 0.6) 0.7) 0.7) 1.0) 1.2) 1.2) 1.4) 1.3) 1.0) 1.1) do,/dJi (nb/sr.) - -- - - - Analyzing Powers A no _ — - — — — -- - 126 Table (4.6) The 375 MeV. 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 and A n a l y z i n g Powers. Pion Angle 6 Differential Cos (0 ) 2 It it (degrees) Cross-Sections do /dft da,/dfi (/xb/sr.) (Mb/sr.) 0 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 0.000 0.000 0.035 0.082 0. 183 0, 210 0, 281 0, 382 0,,393 0.503 0.516 0.626 0.656 0.686 0.774 0.768 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 0 0 0 1 1 0 1 1 1 2 2, 2 2, 2, 2 2, 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 0.001 0.010 0.009 0.041 0. 153 0.258 0.278 0.364 0.456 0.648 0.689 0.820 23.7 23.0 24.^ 25.3 36.8 44, 45, 56, 60, 81 , 83 88, 0 0, 0, 0, 1 1 , 1 1 2.0 2.0 3.0 1.9 •1 1 .5( 0.3) •10.8( 0.3) •1 1 .8( 0.4) -9.9( 0.3) -9.4( 0.8) -6.0( 0.5) -8.2( 0.5) -3.6( 0.5) -6.0( 0.6) 1 .7( 0.8) -2.6 ( 0.8) 3.2( 0.7) Cross-Section Analyz ing Powers no -0.48( .01 ) -0.47( .01 ) -0.48( .01 ) -0.39( .01 ) -0.26( .02) -0. 14( .01 ) -0 . 18(.01 ) -0.06( .01 ) -0. 10( .01 ) 0.02(. 01 ) -0.03( .01 ) 0.04(. 01) 127 Table (4.7) The 425 MeV. D i f f e r e n t i a l Pion Angle * e Cos (e*) 2 (degrees) 89.7 89.8 97.5 104.7 108. 1 112.5 61 .2 56.3 125. 1 53. 1 50.7 1 34.3 38. 1 142.7 35.0 28. 1 19.4 Di f ferent i a l C r o s s - S e c t i o n s da /dfl 0 (/xb/sr. ) 0.000 0.000 0.017 0.064 0.097 0. 146 0.232 0.308 0.331 0.361 0.401 0.488 0.619 0.633 0.671 0.778 0.890 Cross-Sections. 42.1( 1.2) 42.2( 1.2) 45.1( 1.2) 53.5( 1.3) 58.7( 1.5) 64.5( 1.6) 73.0(2.1) 90.6( 2.0) 92.7( 2.9) 99.9( 2.2) 111.8( 2.7) 117.2( 3.6) 144.0( 4.9) 140.5( 4.3) 158.6( 4.5) 168.7( 4.8) 178.9( 5.2) do,/dfi (/ib/sr.) - -- - -- Analyzing Powers A no - ---- -- - 128 Table (4.8) The 450 MeV. 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 Terms and A n a l y z i n g Powers. Pion Angle 6 * ir Differential Cos (6*) 2 ir (degrees) 93. 1 83.9 100.4 78.4 100.4 65.3 57.6 52.8 1 28.2 134.1 143.2 35.3 31.3 1 49.9 26. 1 20.7 0,003 0.011 0.033 0.040 0.033 0. 175 0.287 0.366 0.382 0.484 0.641 0.666 0.730 0.748 0.806 0.875 Cross-Sections d0 /dO da,/dn Ub/sr.) (yb/sr.) o 62. 1( 1.7) 61.1( 1.7) 6.8 . 7 ( 1.8) 64.8( 1.7) 68.8( 1.8) 96.0( 2.2) 1 18.7( 2.6) 139.8( 3.0) 149.8( 3.5) 174.1( 4.0) 208. 1( 6.2) 219.3( 5.2) 228.7( 4.8) 242.8( 9.3) 241.9( 4.8) 251.4( 6.5) -15.7( 0.8) -12.6( 0.7) -13.7( 0.7) -10.6( 0.8) -14.0( 0.9) 0.9( 0.9) 7.7( 1 1 ) 17.4( 1.3) 2.3( 1.8) 8.3( 2.1) 17.7( 2.0) 31.5( 2.0) 32.3( 2.3) 22.3( 2.9) 28.7( 1.9) 20.6( 2.8) Cross-Section Analyz ing Powers A no -0.25(.01 ) -0.21(.01 ) -0.20(.01 ) -0.16( .01 ) -0.20(.01 ) 0.01(.01 ) 0.07(.01) 0.12(.01) 0.02(.01 ) 0.05(.01 ) 0.09(.01) 0.14(.01 ) 0.14(.01 ) 0.09(.01 ) 0.12(.01) 0.08(.01 ) 129 Table (4.9) The 475 MeV. D i f f e r e n t i a l Pion * Angle Cos (0*) 0 (degrees) 90.1 90.3 95.3 102.4 1 12.3 62. 1 55.9 51.2 131.8. 135. r 141.1 34.8 31.3 24.6 20.9 2 0.000 0.000 0.009 0.046 0 . 144 0.219 0.314 0.393 0.444 0.502 0.606 0.674 0.730 0.827 0.873 Differential Cross-Sections. Cross-Sections da /dfi da,/dn (Mb/sr.) (nb/sr.) 0 68.6( 68.6( 71.6( 82.2( 103.4( 120.4( 147.0( 173.0( 181.7( 202.4( 228.8( 248.5( 252.5( •274.9( 286.1( 2.0) 2.0) 2.0) 2.2) 2.6) 2.8) 3.3) 6.1) 4.2) 4.7) 5.1) 7.1) 5.2) 7.1) 5.8) - -- - - Analyzing Powers A no - 130 Table (4.10) The 498 MeV. 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 Terms and A n a l y z i n g Powers. Pion Angle * Cos (0*) 2 (degrees) 90.0 83.5 97.5 107.8 65. 1 115.0 115.1 60.6 126.4 51 .2 134.7 141.4 36.4 148.6 31.3 26.2 19.2 0.000 0.013 0.017 0.093 0.177 0. 179 0. 180 0.241 0.352 0.393 0.495 0.611 0.648 ' 0.729 0.730 0.805 0.892 Differential Cross-Sections da /dfl da,/dfl Ub/sr.) Ub/sr. ) 80.8( 2.2) 83.5( 2.3) 89.6( 2.3) 1 13.2( 2.8) 132.6( 3.1) 141 . 1 ( 3.3) 138. 3( 3.2) 154.3( 3.4) 190.9( 4.3) 216.5( 4.5) 237.9( 5.5) 273.8( 6.0) 289.0( 8.2) 316.8( 9.1 ) 299.1( 6.1) 320.9( 6.5) 338.2( 6.6) -3.8( 0.6) -0.8( 0.6) -1,8( 0.7) 3.7( 1.3) 20.8( 1.3) 14.1( 1.4) 14.6( 1.6) 29.9( 1.4) 27.9( 2.3) 51.0( 2.1) 45.2( 3.3) 42.4( 2.6) 67.4( 3.4) 47.8( 3.3) 67.9( 3.0) 67.5( 3.5) 55.4( 2.5) 0 Cross-Section Analyzing Powers A no -0.05(.01) -0.0K.01) -0.02(.01) 0.03(.01) 0.16(.01) 0.10(.01 ) 0.11(.01 ) 0.19( .01 ) 0.15( .01 ) 0.24(.01 ) 0.19( .01 ) 0.16( .01 ) 0.23( . 01 ) 0.15( .01) 0.23(.01 ) 0.21( .01 ) 0.16(.01 ) 131 cross-section i s given by; { A[da/dfi] + ( } 2 = ( do/dfi ) ) 2 2 { [ A(Afl )/AB ] T } + { iA[do /dfi] } T 2 (65) 2 c 4.7.2 THE DIFFERENTIAL CROSS-SECTIONS; POLARIZED BEAM The u n p o l a r i z e d according differential cross-section i s evaluated to the equation: do /dR = i ( da|/dR + daf/dR ) (66) 0 - i ( doj/dfi - daf/dR) P where: P = ( P| - P| )/( Pj + P| ) | - Indicates a quantity measured with the spin ( d i r e c t i o n ) up. { - Indicates a quantity measured with the spin ( d i r e c t i o n ) down. P|,P| - The magnitude (a p o s i t v e quantity) of the beam polarizations. S u b s t i t u t i n g the spin dependent v a l u e s of the determined q u a n t i t i e s i n t o the above experimentally differential 132 cross-section expression dao/dfi = - i( ST yields; >/ 5t + " ( i< H - U Ant i ( d a c T / d f l + dff |/dfl ) H c " i ( d a c T / d f l - do \/dQ ) P } c The d i f f e r e n t i a l cross-sections data s e t s a s s o c i a t e d energies: figures 375, 450 with the and 498 are (67) e v a l u a t e d f o r the three i n c i d e n t p o l a r i z e d beam MeV,'and are (4.21),(4.23), and i n d i c a t e d on } P shown in (4.25) r e s p e c t i v e l y . The the p l o t s represent the line r e s u l t s of a f i t of Legendre polynomials to the data. The associated numeric values are (4.8), (4.10). tabulated in tables (4.6), and f o l l o w i n g values were used for the p o l a r i m e t e r analysing power: 0.409 at 375 0.432 at 498 MeV. See MeV, section 0.422 at 450 (4.9) MeV, and for a d i s c u s s i o n of The this quantity. 4.7.2.1 The Uncertainty Cross-Section: As a basis simplified 1) The P o l a r i z e d Beam the following magnitude of the are approximately equal, 2) The PT Differential for e r r o r c a l c u l a t i o n s , equation using ( of the - Pf )/( (67) was assumptions: s p i n up and s p i n down p o l a r i z a t i o n s then; PT + P| ) = P = 0 (68) s p i n averaged value of the carbon background d i f f e r e n t i a l cross-section i s approximately its unpolarized 133 value, that i s ; da /dS2 = i ( da f/dO + da }/dfi ) C C (69) O Then 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 expression approximated is by; da /dJ2 = i( St 0 U + )/AO It f o l l o w s that the u n c e r t a i n t y c r o s s - s e c t i o n i s then given { A[da /dR] } of the d i f f e r e n t i a l by; {[A(Afi )/Aa ] T (70) c = { i( H 2 0 " ida /dR T T S\ )/AO + + ( A$t 2 + { iA[da /dR] } 2 + t ) 2 AS} )/( H + U 2 simplification i( Finally, U + - U i s obtained ) / AR the u n c e r t a i n t y using the approximation = da /dfi T 2 (71) 2 c A Further )} (72) 0 of the d i f f e r e n t i a l cross-section due to random f l u c t u a t i o n s of the independent q u a n t i t i e s on which i t depends, i s ; {A[da /dfl]} 0 {[ 2 = { da /dfi 0 A(AJ2 )/An ] t t 2 + { iA[da /dfi] } c ( 2 } 2 A$f 2 + A*} 2 )/( 5! + U )) 2 (73) 134 4.7.3 The THE POLARIZED DIFFERENTIAL CROSS-SECTION 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 c a l c u l a t e d a c c o r d i n g to the expression, da,/dO = ( dof/dO - daf/dJ2 )/( P| + Pf ) (74) Upon s u b s t i t u t i o n of the s p i n dependent measured q u a n t i t i e s , the expression i s : do,/dn = [ ( H " U )/^ ] / ( Pj + + Pf ) " i ( ( da |/dJ2 - d a f / d f i ) } / ( P| + Pf ) c c (75) The p o l a r i z e d p o r 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 s are evaluated f o r the three data unpolarized s e t s a s s o c i a t e d with i n c i d e n t beam e n e r g i e s of; 375, and are shown i n f i g u r e s (4.26) , (4 . 27), and lines i n d i c a t e d on the p l o t s represent of A s s o c i a t e d Legendre polynomials A d d i t i o n a l l y , the numerical tables (4.6),(4.8), and the 450,and 498 (4.28). The the r e s u l t s of a f i t to the data. r e s u l t s are t a b u l a t e d i n (4.10). The f o l l o w i n g values were used f o r the p o l a r i m e t e r a n a l y s i n g power: 0.409 at 375 0.422 at 450 MeV, and MeV, 0.432 at 498 MeV. See section MeV, (4.9) for a d i s c u s s i o n of t h i s q u a n t i t y . 4.7.3.1 The U n c e r t a i n t y of the P o l a r i z e d D i f f e r e n t i a l Cross-Sect ion As a b a s i s f o r c a l c u l a t i o n of the random u n c e r t a i n t i e s , equation (75) can be approximated by assuming that the 135 Figure The 375 MeV. 30 (4.26) D i f f e r e n t i a l Cross-Section 60 90 120 SCATTERING ANGLE (69*) P o l a r i z e d Term, 150 180 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 measured with the backward pion d e t e c t i o n arm. The l i n e represents the r e s u l t s of a f i t of a f i f t h order A s s o c i a t e d Legendre polynomial to these r e s u l t s . 136 Figure The 450 MeV. (4.27) 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 : P o l a r i z e d Term. 60 40h -20' 0 1 30 — • 1 i 60 90 I20 SCATTERING ANGLE (69") I50 I I80 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 measured 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 i f t h order A s s o c i a t e d Legendre polynomial to these r e s u l t s . 1 37 Figure The 498 MeV. (4.28) 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 : P o l a r i z e d Term, 120 30 60 90 120 SCATTERING ANGLE (6*) 150 180 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 measured with the backward pion d e t e c t i o n arm. The l i n e represents the r e s u l t s of a f i t of a f i f t h order A s s o c i a t e d Legendre polynomial to these r e s u l t s . 138 contribution uncertainty of the carbon background term to the o v e r a l l is insignificant. and i t s a s s o c i a t e d That i s the f o l l o w i n g contribution term, towards the u n c e r t a i n t y can be n e g l e c t e d ; ii ( d a | / d f i - da }/dfi c c ) } / ( Pf + Pf ) = 0 (76) thus; da,/dfi = [( U " U >/A8 T ] / ( P| + Pf ) (77) Then, on the b a s i s of t h i s approximation of the d i f f e r e n t i a l cross-section, the a s s o c i a t e d { A.[d0,/dfi] } = 2 + ( APt T Approximating the l e a d i n g the u n c e r t a i n t y becomes; ) / A n ] / ( P| + P{ f + A${ )/( Pt + P{ 2 )/( 2 ) 2 $T - )} 2 $})2 } (78) f a c t o r by the p o l a r i z e d d i f f e r e n t i a l cross-section,leads for SI + ( AH2 2 + AP{ 2 5f - {[( {[A(AR )/AR ] T uncertainty to the f o l l o w i n g expression i n the p o l a r i z e d d i f f e r e n t i a l cross-section. { A[do,/dn] } 2 = { da,/dfi {[A(Afi )/An ] + + ( APt 2 T + ( A$t 2 + APf } 2 2 2 + A${ )/( Pt + P| )/( H 2 ) 2 } " St )2 (79) 139 4.7.4 THE The ANALYZING POWER analyzing power i s simply the polarized d i f f e r e n t i a l cross-section d i f f e r e n t i a l crosssection, A no = ( d ( i / 7 d to the the unpolari'zed that i s ; do /dn >/ ( n r a t i o of 0 ) (80) N The analyzing data are figure powers of the (4.31) r e s p e c t i v e l y . The 450 MeV, (4.6),(4.8), 0.422 at 450 MeV, and and n o 498 MeV and a l s o be found MeV. See power. be approximated i n the - M ) / ( 5T + u form; ) ) • ) } i f do /d£2 ]/[ do/dfi ] + ...} (81) c r i g h t hand sides) of equations (77) term of the denominator has e x p r e s s i o n ) such that the following Which r e s u l t s (with some manipulation) from the denominator expanded MeV, sect ion.(4.9) of the A n a l y z i n g { 2 / ( Pj + Pf + power: 0.409 at 375 the a n a l y s i s of u n c e r t a i n t i e s , = { ( n { 1 f o l l o w i n g values were quantity. Uncertainty powers can A analysing 0.432 at 498 the b a s i s - o f analyzing (4.30), data can (4.10). The for a d i s c u s s i o n of t h i s 4.7.4.1 The and encoded i n t o used f o r the p o l a r i m e t e r As MeV, shown in f i g u r e (4.29), f i g u r e alphanumerically tables 375 (70). The been f a c t o r e d out (the f i n a l the to ratio and (of leading the f a c t o r in the above s o l i d angles c a n c e l out of the 140 Figure The 375 MeV. (4.29) Analyzing Powers. 0.8r0.6o < ;l 0 i 30 ' i i 60 90 120 SCATTERING ANGLE (0*) 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 backward pion d e t e c t i o n arm. The l i n e a n a l y s i n g power deduced from the f i t s polarized d i f f e r e n t i a l cross-sections I 150 1 180 from measured with the r e p r e s e n t s the to the u n p o l a r i z e d and . 141 Figure The 450 MeV. (4.30) Analyzing Powers 0.8 30 60 90 120 SCATTERING ANGLE (#") 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 backward pion d e t e c t i o n arm. The l i n e a n a l y s i n g power deduced from the f i t s polarized d i f f e r e n t i a l cross-sections 150 180 from measured with the r e p r e s e n t s the to the u n p o l a r i z e d and . Figure The 498 MeV. 0 30 (4.31) Analyzing Powers. 60 90 I20 SCATTERING ANGLE (67*) 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 backward pion d e t e c t i o n arm. The l i n e a n a l y s i n g power deduced from the f i t s polarized d i f f e r e n t i a l cross-sections I50 I80 from measured with the represents the to the u n p o l a r i z e d and . 143 r a t i o . The term r e p r e s e n t i n g the denominator i s then approximated by u n i t y since the r e l a t i v e carbon background contribution i s taken to be i n s i g n i f i c a n t and the a n a l y z i n g power i s approximated by; A N { ( n = O - n ) / sT ( ) u + J { 2 / ( Pj + P{ ) } { 1 } The u n c e r t a i n t y (82) (random) of the a n a l y z i n g powers i s then given by; (AA ) n Q 2 = A { ( { ( AH 2 n o A$T { ( AP| 2 + A$f + AU ) / ( 2 + APf 2 2 2 ) / 2 n + ( 5T ~ U M ) ) 2 2 ) / ( P| + P| ) 2 } (83) 4.8 ANALYZING POWERS; KINEMATIC EVENT DEFINITION The a n a l y z i n g powers of the pp—»-7r d r e a c t i o n were d e r i v e d + from the p o l a r i z e d beam data correlation of the f i n a l utilizing the kinematic s t a t e p a r t i c l e s as a c o n s t r a i n t t o reduce the r e l a t i v e background l e v e l to the p o i n t where a background s u b t r a c t i o n was unnecessary. The r e s u l t s , which are p u b l i s h e d reproduced i n Appendix (Giles ( 3 ) . The numerical a n a l y z i n g powers were not p u b l i s h e d , t a b u l a t e d here i n Tables et a l . ) , are 9 values of the thus, (4.11),(4.12), and they are (4.13). 1 44 Table The (4.11) 375 MeV. A n a l y z i n g Powers. Pion Angle A n a l y z i n g Powers Target Material * e (degrees) Polyethylene CH Carbon C (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 TT 2 pp—>ii * d Table (4.12) The 450 MeV. A n a l y z i n g Powers. Pion Angle A n a l y z i n g Powers Target Material * e (degrees) Polyethylene CH Carbon C (Hydrogen) 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 2 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 PP—>7T* d 146 Table (4.13) The 498 MeV. A n a l y z i n g P o w e r s , Pion Angle Analyzing Target (degrees) Polyethylene CH. 2 19.5 26.4 31 . 36. 51 . 60. 65. 78. 83. 90. 97. 1 07 8 11 5 1 1 20 0 1 26 4 1 34 7 141 5 1 49 6 0.162±0 .004 0.20610 .008 0.229+0 .007 0.24010 .006 0.23210 .006 0.19210 .006 0.15910 .006 0.03610 .008 - 0 . 0 0 8 1 0.005 - 0 . 0 4 7 1 0.005 - 0 . 0 2 3 1 0.005 0.04310 .007 0.10510 .008 0 1 5410.009 0 1 53 + 0 .009 0.18410 .006 0.16310 .006 0.15610 .005 Powers Material Carbon C (Hydrogen) 0.01 0.0 0.010 .001 .0 + 0 .001 .010 .001 .010 .001 .010 .001 0011 0.001 0011 001 001 + 001 001 0011 001 0021 001 0021 001 0021 001 0021 002 + 001 001 0011 001 + 001 001 0011 0. 16210. 004 0. 20610. 008 0. 229+0. 007 0. 24010. 006 0. 23210. 006 0. 19210. 006 0. 16010. 006 0. 03710. 008 -0 .007+0 .005 -0.04610 .005 -0.02110 .005 0.04510. 007 0, 1 07 + 0. 008 0, 15610. 009 0, 155+0. 009 0, 18510. 006 0, 16410. 006 0. 15710. 005 pp—>77* d 147 D i f f e r e n t i a l c r o s s - s e c t i o n r e s u l t s c o u l d not be obtained with t h i s technique, as the kinematic constraints used t o elimimate the background a l s o e l i m i n a t e d from the data s e t , an unknown f r a c t i o n of pp—>n*d events ( i n p a r t i c u l a r , of those events f o r which the pion decayed and the subsequent muon was d e t e c t e d ) . Thus, f o r 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 , a background technique as d e s c r i b e d in section subtraction (4.3) had to be employed. 4.9 DISCUSSION OF UNCERTAINTIES Systematic u n c e r t a i n t i e s and u n c e r t a i n t i e s other a s s o c i a t e d 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 d i s c u s s e d in this There i s an o v e r a l l u n c e r t a i n t y values section. of 1.8% i n the a b s o l u t e 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 s due to the uncertainty elastic than those of the e f f e c t i v e s o l i d angle of the pp-»-pp beam current monitor. T h i s u n c e r t a i n t y as that d e s c r i b e d i n our p u b l i s h e d pp—>-pp d i f f e r e n t i a l c r o s s - s e c t i o n r e s u l t s . I t , of course, c a n c e l s r a t i o of the pion production cross-sections the a?°/aB out when the to pp—>pp d i f f e r e n t i a l (at 90°cm) i s c o n s i d e r e d . when c o n s i d e r i n g i s the same 0 I t a l s o c a n c e l s out or b"°/ao° r a t i o s that the angular shapes 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 cross-sections respectively. A d d i t i o n a l l y , there i s an u n c e r t a i n t y a s s o c i a t e d with the i n c i d e n t proton energy. of ±1 MeV define 1 48 The a n a l y z i n g powers 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 subject to a systematic u n c e r t a i n t y that i s a s s o c i a t e d with the p o l a r i z a t i o n of the i n c i d e n t proton beam. T h i s u n c e r t a i n t y , estimated result of c a l i b r a t i o n at 5 percent, a r i s e s as a ( u n c e r t a i n t i e s ) of the beam energy dependent a n a l y z i n g power ( p) of the beam-line p o l a r i m e t e r . A If c a l i b r a t i o n s to higher p r e c i s i o n are ever a t t a i n e d , the systematic u n c e r t a i n t i e s of the a n a l y z i n g powers and polarization-dependent differential the cross-sections could be determined more a c c u r a t l y . Systematic and carbon u n c e r t a i n t i e s a s s o c i a t e d with s o l i d background s u b r a c t i o n s are, i n g e n e r a l , angles angle dependent. Because of the forward-backward symmetry of pp—>7r + d r e a c t i o n , such u n c e r t a i n t i e s can e r r o r s where both superimposed forward and the simulate random backward angle data are (as happens, f o r example, when the cross-section i s p l o t t e d as a f u n c t i o n of c o s ( 0 ^ ) example, F i g u r e (see, f o r 2 (4.20)). Consider, f o r example, the systematic u n c e r t a i n t i e s a s s o c i a t e d with the measurement of the MWPC dimensions, the pion-decay and c o r r e c t i o n s to the s o l i d a n g l e s , and energy-loss the carbon s u b t r a c t i o n s ; a l l of which are expected smooth f u n c t i o n of the proton l a b o r a t o r y a n g l e . As such, characterizing to be beam energy and background reasonably pion the systematic u n c e r t a i n t i e s the d i f f e r e n t i a l c r o s s - s e c t i o n s for a c l o s e l y spaced pion l a b angles may not be apparent. not the case when p o i n t s of s i m i l a r c o s ( 6 2 ) but few This i s very 149 different l a b o r a t o r y angles are compared (take as an extreme case, the pion l a b o r a t o r y angles a s s o c i a t e d with * c o s ( 0 )<1). 2 7T Such p o i n t s of s i m i l a r c o s ( 0 * 2 ) were measured with d i f f e r e n t d e t e c t i o n systems at d i f f e r e n t pion l a b o r a t o r y e n e r g i e s and a n g l e s . Furthermore, the pion-decay, energy-loss and carbon different background c o r r e c t i o n s w i l l be very f o r these p o i n t s as w i l l t h e i r a s s o c i a t e d systematic u n c e r t a i n t i e s . T h e r e f o r e , some of the d e v i a t i o n * between two p o i n t s of s i m i l a r c o s ( 0 ) (but d i f f e r e n t 2 7T l a b o r a t o r y angle) can be due, i n p a r t , to systematic uncertainties. If the e r r o r s a s c r i b e d f o r the data p o i n t s are not 'normally' d i s t r i b u t e d , but a r e , nonetheless, usual minimum x 2 of used i n the c r i t e r i o n to e s t a b l i s h a f i t , then the use common s t a t i s t i c a l tests (such as the F t e s t ) to evaluate the goodness of the f i t so obtained are not r i g o r o u s l y justified. Notwithstanding, the estimated a s s o c i a t e d with the s o l i d angles systematic e r r o r s (that- i s , of the d e t e c t o r dimensions and of the pion-decay and e n e r g y - l o s s c o r r e c t i o n s ) and with the carbon background s u b t r a c t i o n s were combined with the random e r r o r s and t r e a t e d as incoherent e r r o r s on a p o i n t - b y - p o i n t b a s i s . Although leads t o reasonable table (4.14), values of x /v 2 this f o r the f i t s , (see f o r example) due c a u t i o n must be e x e r c i s e d i n the i n t e r p r e t a t i o n of the e r r o r s assigned t o the e x t r a c t e d 1 50 c o e f f i c i e n t s , and the goodness of the f i t s as i n d i c a t e d by the (x /v and F) s t a t i s t i c a l 2 tests. 4.10 FIT OF THE UNPOLARIZED DIFFERENTIAL CROSS-SECTIONS TO A SUM OF LEGENDRE POLYNOMIALS 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 expanded i n terms of even-order coefficients Legendre polynomials, and the expansion (the a ? ) were determined 0 l e a s t squares, using general-purpose by the method of fitting r o u t i n e s . For 3 6 each set of 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 ( f o r example, at each proton energy) a number of such each with the expansion f i t s were c a r r i e d out, s e r i e s t r u n c a t e d at a d i f f e r e n t order of Legendre polynomial (second, f o u r t h , s i x t h , and e i g h t h order t r u n c a t i o n s were examined). The r e s u l t s of these f i t s are t a b u l a t e d i n t a b l e f o l l o w i n g we f i r s t adding (4.14) and (4.15). In the d i s c u s s the s t a t i s t i c a l f o u r t h order terms to second s i g n i f i c a n c e of order f i t s , and then d i s c u s s the e f f e c t of the a d d i t i o n of s i x t h and e i g h t h order terms to the expansion f u n c t i o n s e r i e s . The higher order terms ( i n p a r t i c u l a r , those a s s o c i a t e d with the a°° and a c o e f f i c i e n t s ) a r e , i n the intermediate energy expected to be i n s i g n i f i c a n t 0 0 region, (near zero) f o r energies below some "turn-on t h r e s h o l d " , above which they might be expected to d i s p l a y an a p p r o p r i a t e energy G l o b a l l y , when averaged e n e r g i e s , the reduced (from an average x2 dependence. over a l l data s e t s f o r a l l (x2/^) changes insignificantly value of 1.4) when the f o u r t h order terms 151 Table (4.14) F i t s 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 s to a Sum of Legendre Polynomials. = a 0 0 0 399(3) 401(4) 407(7) 398(20) a 0 0 a 2 397(8) 405(13) 430(26) 392(80) ,00 e a 350 MeV data; 15 p o i n t s a (, a a 0 0 8 V X 2 6.16 1 2 5.60 11 4.49 10 4.24 1.: 9(12) 44(35) 6(103) 26(24) 16(87) -20(40) X A 2 0.47 0.47 0.41 0.41 375 MeV data; 28 p o i n t s 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) 425 MeV data; 1200(10) 1200(10) 1200(10) 1190(10) 1340(20) 1350(30) 1330(40) 1310(40) 20(30) -30(50) -80(50) 1700(10) 1700(10) 1680(20) 1680(20) 1910(30) 1940(40) 50(40) 1880(40) -100(60) 1870(50) -120(80) 1930(20) 1930(20) 1920(20) 1920(20) 2130(30) 0(50) 2130(40) 2100(40) -90(60) 2090(50) -1 10(70) 1 .92 2.00 1 .74 1 .80 15 14 13 12 22.4 21.9 19.7 17.3 1 .49 1 . 56 1 .52 1 .44 -70(50) 14 13 12 1 1 25.7 23.9 12.5 12.3 1 .84 1 .84 1 .04 1.12 13 12 1 1 10 9.67 9.67 4.72 4.49 0.74 0.81 0.43 0.45 16 p o i n t s -210(60) -240(90) 475 MeV data; 49.9 49.9 41.7 41.4 17 p o i n t s -60(40) 130(60) 450 MeV data; -15(27) 26 25 24 23 30(70) 17 p o i n t s -130(60) -160(90) -40(70) 152 a 0 0 ao a 0 0 2 3 a 0 0 a it a 0 0 6 498 MeV data; 2320(20) 2310(20) 2310(20) 2310(20) 2570(40) 2500(40) -130(50) 2470(40) -230(70) 2460(50) -240(70) The c o e f f i c i e n t s a 0 0 a a V X 2 X /» 2 8 17 p o i n t s -140(60) -150(90) -20(70) are measured in 15 14 13 12 Mb/sr. 29.7 21.2 15.7 15.7 1 .98 1 .51 1.21 1.31 153 Table (4.15) R a t i o 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 Expansion C o e f f i c i e n t s t o the T o t a l C r o s s - S e c t i o n . aSVag 0 - 0 0 /_ 0 0 a« /ao a 0 0 /a 0 0 a6 /ao 350 MeV 0.99(2) 1.01(3) 1.06(7) 0.02(3) 0.11(9) 0.06(6) 375 MeV 1.10(2) 1.10(2) 1.06(3) 0.00(2) -0.10(4) -0.10(3) 425 MeV 1.12(2) 1.13(3) 1.11(3) 0.02(3) -0.03(4) -0.05(3) 450 MeV 1.12(2) 1.14(2) 1.12(2) 0.03(2) -0.06(3) -0.13(4) 475 MeV 1.10(2) 1 .10(2) 0.00(3) 1.09(2) -0.05(3) -0.07(3) X A 2 Fx P r o b a b i l i t y of Exceeding Fx Randomly 1.19 2.7 10%-^25% 10%->25% 0 4.7 2.5%-»5% 0.3 1 .5 >50% 25%-^50% 1 .0 1 1 ~40% .5%->1% 0 . 12 .5%->1% results; 0.47 0.47 0.41 results; 1 .92 2.00 1 .74 results; 1 .49 1 .56 1 .52 results; 1 .84 1 .84 1 .04 results; 0.74 0.81 0.43 1 54 a 0 0 /_ 0 0 a2 / o a aa 0 0 /_ 0 0 ft / a 0 a a 0 0 /a 0 0 6 /o a x A 2 Fx P r o b a b i l i t y of Exceeding Fx Randomly 498 MeV r e s u l t s ; 1.11(2) 1 .08(2) 1 .07(2) -0.06(2) -0.10(3) -0.06(3) 1 .98 1.51 1.21 5.6 4.6 2.5%->5% 5%->10% 155 are i n c o r p o r a t e d i n t o the f i t s . I t i s q u e s t i o n a b l e whether a more d e t a i l e d a n a l y s i s of the ( i n d i v i d u a l ) x 2 distributions would be a p p r o p r i a t e i n t h i s case. Nonetheless, i n s p e c t i o n of the s t a t i s t i c a l t e s t s of a°° c o e f f i c i e n t s i n d i c a t e s that only f o r the case of the 498 MeV data i s the term significantly different = 2.00) (x /v 2 lowest x 2 i s a s s o c i a t e d with the 375 MeV data, and the 2 ( x / f = 0.47) The from z e r o . The l a r g e s t reduced with the 350 MeV data. 375 MeV data set c o n s i s t s of 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 e x t r a c t e d from runs with both p o l a r i z e d and u n p o l a r i z e d i n c i d e n t beams. T h i s data set has the l a r g e s t number of p o i n t s that d i f f e r more than two standard d e v i a t i o n s (4/28 e x p e c t a t i o n of .046 based from the f i t by compared to an on pure random Gaussian errors). The poorer q u a l i t y of t h i s data may be the r e s u l t of u n c e r t a i n t i e s a s s o c i a t e d with the r e s t r i c t i o n s t h i s data set than f o r any of the others) a p p l i e d to the d e t e c t o r s i z e s r e q u i r e d to c o r r e c t Determination (more f o r f o r t h e i r misplacement. of the adequacy of these f i t s was supplemented using standard s t a t i s t i c a l a n a l y s i s based on the F d i s t r i b u t i o n . This test 3 7 a p p r o p r i a t e r a t i o s of x functional 2 i s based on e v a l u a t i o n of values a s s o c i a t e d with d i f f e r e n t forms f i t to the d a t a . The r a t i o s are d e f i n e d i n such a way that systematic m u l t i p l i c a t i v e these x 2 factors affecting values w i l l c a n c e l . The Fx q u a n t i t y i s d e f i n e d as: 156 Fx = { ( n - 1 ) 2 X 2 X ( n ) }/{ x (n)/(N-n-1) } 2 2 (84) = Ax /(x A) 2 Where N - The number of data p o i n t s n - The number of c o e f f i c i e n t s ( l e s s one f o r the constant term) being f i t to the data. The value of Fx i s as an i n d i c a t i o n of the q u a l i t y of the f i t on a term-by-term b a s i s . I t t e s t s the s i g n i f i c a n c e of the highest order term i n c o r p o r a t e d i n t o the f i t . I t does not give an i n d i c a t i o n of the a b s o l u t e v a l i d i t y of the f i t in q u e s t i o n . On the b a s i s of the Fx t e s t above, the aj° term i s most s i g n i f i c a n t i n the case of the 498 MeV data (Fx=5.6). T h i s value of Fx has l e s s than a 5% p r o b a b i l t y of being exceeded by that of a randomly d i s t r i b u t e d data s e t . In general, the a d d i t i o n of s i x t h order terms, u n l i k e that of f o u r t h order, a c c o r d i n g to the Fx t e s t , has s t a t i s t i c a l s i g n i f i c a n c e . G l o b a l l y , the energy reduced x 2 decreases averaged from the p r e v i o u s value of 1.4 to 1.1. Furthermore, a l l of the Fx values i n d i c a t e that t h i s term i s s i g n i f i c a n t , the r e s u l t s of the f i t s , the forementioned largest two (with the exception of 375 MeV r e s u l t s , which s t i l l x / v v a l u e ) , suggest 2 groups. The f i r s t has the that the data can be s p l i t into group c o n s i s t s of the two low energy (350 and 425 MeV) r e s u l t s , and the second c o n s i s t s of the 157 three h i g h e s t energy (450, 475, and 498 MeV results. The r e l a t i v e s i z e s of the Fx values a s s o c i a t e d with these two groups suggests the s i g n i f i c a n c e of the s i x t h order term i s i n c r e a s i n g with energy. In g e n e r a l , i n c l u s i o n of the a°° terms i n t o the r e s u l t s i n a decreased correlation value of the a ? 0 fits terms. The i s such that the a°° terms a l l change sign become negative, with the e x c e p t i o n s of the 350 MeV and 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 which was already negative. O v e r a l l , the 375 MeV and the 450 MeV data) term (with the exception of the changes i n a°° are w i t h i n the e r r o r s a s s o c i a t e d with t h i s q u a n t i t y as determined by the f i t t i n g procedure. a s s o c i a t e d with the 498 MeV The value of a°° data e x h i b i t s the s m a l l e s t change. I n t e r e s t i n g l y , the magnitudes of both the a°° a°° c o e f f i c i e n t s are s i m i l a r at a given The expansion energy. i n c o r p o r a t i o n of e i g h t h order terms i n t o the s e r i e s r e s u l t s in g e n e r a l l y i n s i g n i f i c a n t c o e f f i c i e n t s . G l o b a l l y , the energy remains unchanged data does the x /v 2 e n e r g i e s the x2/'v (at a value of decrease averaged reduced a%° x2 1.1). For only the 425 values i n c r e a s e ( s l i g h t l y ) . Ideally, the would be greater i n 10% to 25% of randomly d i s t r i b u t e d data sets, suggesting a moderate s i g n i f i c a n c e f o r t h i s term. Nonetheless, MeV ( s l i g h t l y ) whereas f o r a l l other Fx value a s s o c i a t e d with the 425 MeV only and given the none i d e a l d i s t r i b u t i o n of the u n c e r t a i n t i e s , a l l a§° c o e f f i c i e n t s are c o n s i d e r e d 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 i n the intermediate energy r e g i o n , that they i n s i g n i f i c a n t p r o v i d e s an i n d i c a t i o n of a lack of are systematic c o n t r i b u t i o n s to 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 , to the e i g h t h order at l e a s t . 4.11 FIT OF THE SUM The of POLARIZED DIFFERENTIAL CROSS-SECTION TO A OF ASSOCIATED LEGENDRE POLYNOMIALS expansion c o e f f i c i e n t s b"° c h a r a c t e r i z i n g the 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 i n terms of A s s o c i a t e d 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, set, expansion f o r each data f i t s were done f o r a v a r y i n g number of terms. r e s u l t s are l i s t e d the b ^ 0 i n t a b l e s (4.16) and term i s s t a t i s t i c a l l y The (4.17). A d d i t i o n of significant (as d e f i n e d by the F t e s t ) f o r a l l data s e t s . I t i s by f a r most s i g n i f i c a n t i n the case of the 498 MeV data. A d d i t i o n of a b g 0 f i t s does not s i g n i f i c a n t l y change the values of i n d i c a t i n g a very small i n t e r - c o r r e l a t i o n of c o e f f i c i e n t s . However, there i s very l i t t l e reason f o r adding by adding i t , as the x /v 2 t h i s term. The case of the 450 MeV b^ 0 b^ , 0 these statistical are a f f e c t e d only slightly term i s most s i g n i f i c a n t data, although j u s t over one e r r o r bar. term to the i n the i t d e v i a t e s from zero by 159 Table (4.16) F i t s 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 to a Sum of A s s o c i a t e d Legendre Polynomials. , no .no , no b , no , no 3 6 37 5 MeV. -108(3) -109(2) -109(2) 17(2) 17(2) 17(2) 24(2) 26(2) 25(2) data; 3(2) 2(2) 3(2) 48(5) 49(5) 51 (5) 133(4) 139(4) 143(4) 9(3) 3(4) 4(5) 498 MeV. 316(6) 315(6) 315(6) 78(6) 72(6) 72(6) 245(5) 259(5 )• 259(6) 22(4) 19(4) 17(5) X 2 x A 2 1(2) 8 7 6 8.47 3.32 2.21 1 .06 0.47 0.37 1 2 33.7 1 1 20.4 1 0 13.1 2.81 1 3 34.9 1 2 10.3 1 1 10.2 2.68 0.85 0.93 16 p o i n t s 12(4) 17(4) data; V 12 p o i n t s 3(2) 2(2) 4 50 Mev. data; 6(5) 2(5) -1(6) , no b -8(5) 1 .85 1.31 17 p o i n t s 16(3) 16(4) -1(4) The c o e f f i c i e n t s are measured i n Mb/sr. Table (4.17) R a t i o 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 Expansion C o e f f i c i e n t s to the T o t a l C r o s s - S e c t i o n . b?°/a8° b n b?%8 % 8 ° 375 MeV. -.167(5) 0.026(3) -.169(3) 0.026(3) -.169(3) 0.026(3) results; by%8° b results; n 0 /ag° results; Fx a g ° = 645;ib. 0.012(3) 0.002(3) 1 1 3.0 ag° = 1700/ib. 0.078(2) 0.005(2) 0.082(2) 0.002(2) 0.007(2) 0.084(2)- 0.002(3) 0.010(2) • •0.005(2) 498 MeV. 0.137(3) 0.034(3) 0. 136(3) 0.031(3) 0.136(3) 0.031(3) b?°/a8° 0.037(3) 0.006(3) 0.040(3) 0.003(3) 0.006(3) 0.039(3) 0.006(3) 0.003(3) 450 MeV. 0.004(3) 0.028(3) 0.001(3) 0.029(3) -.001(4) 0.030(3) 0 7.5 5.6 ag° = 23lO.Mb. 0. 106(2) 0.010(2) 0.112(2) 0.008(2) 0 . 007(1 ) 0,112(2) 0.007(2) 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 u n p o l a r i z e d 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 are p l o t t e d compared with e x i s t i n g r e s u l t s in f i g u r e s through (5.9). Model, and the other two are U n i t a r y d i f f e r e n t i a l cross-sections of pion m^/c. shown, one several i s a Coupled Channel Model p r e d i c t i o n s . are c o n s i d e r e d here as near-threshold pion momentum was convenient v a r i a b l e to use reaction (and c o n s i d e r e d to be a when comparing the i t s inverse, the 7r*d->pp r e a c t i o n ) np—^7r°d polarized d i f f e r e n t i a l cross-sections i s o t r o p i c part of the a°°) unpolarized are to those unpolarized (other than total gross energy dependence of the c o e f f i c i e n t s (which, in general, s i m i l a r to that of the the shown here normalized to the 0 and differential c r o s s - s e c t ion a°> , in order to remove the 1 pp— reaction expansion c o e f f i c i e n t s f o r both the cross-section, differential r e s u l t i n g from measurements of the deduced form measurements of the All of importance of phase-space in t h i s region, cross-sections The functions center-of-mass momentum TJ, expressed in u n i t s Because of the are t o t a l c r o s s - s e c t i o n ) . T h i s method of d i s p l a y i n g the c o e f f i c i e n t s a l s o e l i m i n a t e s of the and (5.1) In a d d i t i o n , the p r e d i c t i o n s of t h e o r e t i c a l approaches are and systematic u n c e r t a i n t i e s i n d i v i d u a l data s e t s . The e f f e c t s of some characterizing s i g n i f i c a n c e of the 161 the s i x t h order 162 expansion c o e f f i c i e n t of the u n p o l a r i z e d cross-section, a ? , which was found to be g e n e r a l l y 0 s i g n i f i c a n t at higher e n e r g i e s section differential (4.10)), i s also more (discussed i n discussed. 5.2 THE UNPOLARIZED DIFFERENTIAL CROSS-SECTION The total cross-section the remaining a ? % o ° ag 0 i s plotted in figure ratios describing unpolarized d i f f e r e n t i a l cross-section are p l o t t e d in figures (5.2), (5.3), measurements (surveyed by G. J o n e s predictions Blankleider The of N i s k a n e n 3 3 3 5 the shape of the angular d i s t r i b u t i o n s and (5.4). i n d i c a t e d on these p l o t s a r e r e l e v a n t ' (5.1) and Also existing precision 3 8 ) and the t h e o r e t i c a l (the Coupled Channel Model), 25 and Lyon g r o u p " 0 (both using Unitary Models). t h e 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 t h e o r i e s are able to describe t h i s fundamental r e a c t i o n . On each p l o t 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 r e s u l t s from f i t s of the data to Legendre s e r i e s terminated a t the f o u r t h order terms, and the second set r e s u l t s from f i t s expansion s e r i e s truncated of a ? 0 of the data to the a t the s i x t h order terms. The s e t c o e f f i c i e n t s c o n s i d e r e d t o most reasonable ( s i g n i f i c a n t ) are i n d i c a t e d by s o l i d symbols on the respective plots. Consider f i r s t a ) , depicted 0 0 the t o t a l d i f f e r e n t i a l in figure cross-section, (5.1). T h i s c o e f f i c i e n t i s r e l a t i v e l y l a r g e and i s , as expected, q u i t e i n s e n s i t i v e to 163 Figure (5.1) The Total PION Cross-Sections MOMENTUM (77) The c o e f f i c i e n t s of the z e r o t h order (the i s o t r o p i c ) term of the Legendre polynomial expansion 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 as a f u n c t i o n of the pion centre-of-mass momentum 77. Here, the c o e f f i c i e n t a s s o c i a t e d with the recommended order 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 polynomial 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. 164 Figure (5.2) R a t i o of the C o e f f i c i e n t s of the Second Order Legendre Polynomial Terms to the T o t a l C r o s s - S e c t i o n . 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 § o <£ O CM O I.O 0.5 0 I.O PION 2.0 M O M E N T U M (77) 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 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 normalized to the t o t a l c r o s s - s e c t i o n a g i s shown as a f u n c t i o n of the pion centre-of-mass momentum 77. Here, the c o e f f i c i e n t a s s o c i a t e d with the recommended order 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 polynomial 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. 0 1 65 Figure (5.3) Ratio of the C o e f f i c i e n t s of the Fourth Order Legendre Polynomial Terms to the T o t a l C r o s s - S e c t i o n . PION MOMENTUM (77) The c o e f f i c i e n t s of the f o u r t h order term of the Legendre polynomial_expansion 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 normalized to 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 pion centre-of-mass 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 with the recommended order 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 polynomial 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. 1 66 F i g u r e (5.4) Ratio of t h e 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 t h e T o t a l Cross-Section. 0.1 o o O o <£ O -0.1 o 0.2 o o PP-7Td THIS WORK (6th ORDER FIT) PP-7Td NANN ET AL NP-7Td ROSSLE ET AL NISKANEN BLANKLEIDER LYON 1.0 0 PION 2.0 MOMENTUM (17) The c o e f f i c i e n t s o f t h e s i x t h o r d e r t e r m o f t h e L e g e n d r e polynomial expansion of the unpolarized 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 t h e t o t a l c r o s s - s e c t i o n a°° i s shown a s a f u n c t i o n o f t h e p i o n c e n t r e - o f - m a s s momentum 17. H e r e , t h e 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 t h e 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 o r s i x t h ) o f t h e L e g e n d r e 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 s y m b o l . 167 the number of terms i n the f i t . Our t o t a l c r o s s - s e c t i o n s are in good agreement with the p r e c i s i o n measurements of H o f t i e z e r et a l . * 1 at higher values of TJ. They are i n s i g n i f i c a n t disagreement however, (that i s , by typically many standard d e v i a t i o n s , depending on the p o i n t ) with those of R i t c h i e et a l . " over the lower v a l u e s of r\ where the two 2 data s e t s o v e r l a p . The o r i g i n of t h i s large discrepancy i s probably the r e s u l t of a l a r g e systematic u n c e r t a i n t y a s s o c i a t e d with t h e ' n o r m a l i z a t i o n of the i n c i d e n t pion beam current f o r the 7r*d—*-pp measurements of R i t c h i e et a l . " As 2 the method of n o r m a l i z a t i o n of the i n c i d e n t proton beam c u r r e n t used i n our experiment the w e l l known p p - e l a s t i c l a r g e systematic e r r o r u n c e r t a i n t i e s . The i s based on measurements of r e a c t i o n c r o s s - s e c t i o n s , no 1 0 i s expected Coupled to c o n t i b u t e to our Channel M o d e l t r e n d of the t o t a l c r o s s - s e c t i o n but not whereas the U n i t a r y M o d e l s 3 9 ' 2 5 reproduce the i t s magnitude, are i n r e l a t i v e l y 4 0 such good agreement with the data. The c o e f f i c i e n t governing the second the r e l a t i v e c o n t r i b u t i o n of order Legendre term a 0 0 /a 0 0 , i s the dominant term d e s c r i b i n g the shape 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 angular d i s t r i b u t i o n energy i n the intermediate region. It i s depicted in figure the f i g u r e , the value of t h i s r a t i o was (5.2). As seen i n found to be q u i t e i n s e n s i t i v e to the number of terms i n c l u d e d i n the Legendre polynomials f i t to the data. The agreement between the v a r i o u s data s e t s i s , with the e x c e p t i o n of the o l d datum of 168 Dolnick et a l . " (renormalized as suggested 3 by J o n e s ) , 3 5 q u i t e s a t i s f a c t o r y . Reasonable agreement should be expected, however, s i n c e both a ^ 0 and a g 0 are l a r g e 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 u n c e r t a i n t y a s s o c i a t e d with a p a r t i c u l a r experiment will c a n c e l when such a r a t i o i s formed. T h e o r e t i c a l l y , t h e Coupled for Channel M o d e l 2 5 under estimates the a ° / a o ° 0 ratio rj < 0.65(350 MeV) and over estimates i t f o r l a r g e r values of 77. The t h e o r e t i c a l p r e d i c t i o n s shown i n the f i g u r e do, however, c o r r e c t l y reproduce data with B l a n k l e i d e r ' s 3 9 aggreement i n t h i s energy the o v e r a l l trend of the u n i t a r y theory g i v i n g the best region. - The magnitudes of the higher order terms (aj° and a ? ) 0 are an order of magnitude s m a l l e r than those of the l e a d i n g terms. In f a c t , the combined c o n t r i b u t i o n to 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 these terms at a t y p i c a l point i s similar data i n magnitude (a few percent) to that of the u n c e r t a i n t y a s s o c i a t e d with that p o i n t . As such, some degree of c o r r e l a t i o n between the aj° and a ? expected 0 coefficients i s to be p r e s e n t . Such a c o r r e l a t i o n i s manifested by the o b s e r v a t i o n of a dependence of the value f o r the a°° coefficient fit on the order assumed f o r the Legendre polynomial to the d a t a . The r a t i o s of the f o u r t h to z e r o t h order c o e f f i c i e n t s , a°°/ao°, are d e p i c t e d i n f i g u r e as d i s c u s s e d i n S e c t i o n (4.10), there appears statistical expansion (5.3). S i n c e , to be s i g n i f i c a n c e to the s i x t h order terms at the 169 three h i g h e s t energies (450, 475, and 498 MeV), the recomended values f o r the a2°/a§-° are thus obtained from f i t s t o the s i x t h order Legendre f u n c t i o n s . For the three lower energy p o i n t s , the a2°/a§° r a t i o s recomended are those d e r i v e d from the r e s u l t s of f i t s of the data to f o u r t h order Legendre f u n c t i o n s . These "recommended" values are designated as s o l i d symbols on the f i g u r e s . As such, our aS%o° r a t i o s are c o n s i s t e n t with zero f o r energies from 350 to 425 MeV (0.65 < r) < 1.00). In t h i s energy r e g i o n , our data are not i n c o n s i s t e n t with those of R i t c h i e et a l . " 2 (7r + d->pp) or Rossle et a l . " ( n p - » 7 r ° d ) . I f anything, our r e s u l t s i n t h i s region are somewhat c l o s e r to zero than the o v e r a l l p o s i t i v e trend c h a r a t e r i z i n g the other data. For energies g r e a t e r than 425 MeV (TJ>1) our data d i s p l a y s a negative t r e n d c o n s i s t e n t with the data of Rossle et a l . ( n p - > T r ° d ) , R i t c h i e et a l . " datum of Aebischer et a l . " (pp—>-7r d), 5 + 2 (7rd->pp) and the but disagree i n magnitude with the p r e c i s i o n r e s u l t s of H o f t i e z e r et a l . " . 1 In f a c t , the weight of the evidence r e s u l t s of H o f t i e z e r et a l . " o v e r a l l systematic 1 suggests that the a r e i n c o r r e c t , perhaps by an factor. For the h i g e r order terms, the t h e o r e t i c a l p r e d i c t i o n s are much l e s s s a t i s f a c t o r y , with only the Coupled Model p r e d i c t i n g the c o r r e c t Channel sign of the measured r e s u l t s i n t h i s energy r e g i o n . I n t e r e s t i n g l y , booth U n i t a r y Models p r e d i c t a small p o s i t i v e value of a S % o ° f o r T? < 1 . 170 The r a t i o of t h e s i x t h expansion c o e f f i c i e n t s Of the v a l u e s three highest statistically energy higher the They a r e n e g a t i v e the Rossle energies. O v e r a l l , there trend i s not c l e a r l y for this ratio although i t s t h e o r i e s are negligable b a s e d on in this CROSS-SECTION direct and ( 5 . 9 ) . They a r e d e r i v e d p r e c i s i o n measurements differential cross-sections in this c o m p l i m e n t t h o s e of H o f t i e z e r e t a l . " energies. Previous (Mathie et a l . " 6 results in this differential measured a n a l y z i n g presented h e r e were o b t a i n e d from 1 at of t h e energy n o fits region higher region of e s t i m a t e d cross-sections p o w e r s . The b our p o l a r i z e d d i f f e r e n t i a l published energy were b a s e d on t h e p r o d u c t measured) u n p o l a r i z e d deduced at a p p e a r s t o be determined. Expectations (5.5),(5.6),(5.7),(5.8) polarized to i n the region r e s u l t s are depicted i n from t h e f i r s t with only the region. The b"°/a8° and plot, r e s u l t s are negative 5.3 THE POLARIZED DIFFERENTIAL figures on t h i s 0 formentioned current energy presented (5.4). et a l . ( n p - » - 7 r d ) r e s u l t s are e s s e n t i a l l y e v i d e n c e of a n e g a t i v e magnitude order r e s u l t s a r e b e l i e v e d t o be significant. Nonetheless, slightly to the z e r o t h ag°/a8°, a r e shown i n f i g u r e from our f i t s over which R o s s l e zero. order (or together coefficients (see t a b l e (4.16) ) c r o s s - s e c t i o n s , wheras our r e s u l t s ( s e e f i g u r e (2) i n a p p e n d i x from t h e measured a n a l y z i n g ( 3 ) ) were powers ( s e e 171 Figure (5.5) R a t i o of the C o e f f i c i e n t s of the F i r s t Order A s s o r i a r ^ Legendre Polynomial Terms to the T o t a l C r o s s - S e c t i o n 400 • o A o 200 8 o THIS WORK HOFTIEZER ET AL NANN ET AL MATHIE ET AL NISKANEN BLANKLEIDER LYON o 200 • -400 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 A s s o c i a t e d Legendre polynomial expansion 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 normalized to 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 pion centre-of-mass momentum T J . 172 Figure (5.6) R a t i o of the C o e f f i c i e n t s of the Second Order A s s o c i a t e d Legendre Polynomial Terms to the T o t a l 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 A s s o c i a t e d Legendre polynomial expansion 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 normalized to 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 pion centre-of-mass momentum 77. 173 F i g u r e (5.7) R a t i o of the C o e f f i c i e n t s of the T h i r d Order A s s o c i a t e d Legendre Polynomial Terms to the T o t a l C r o s s - S e c t i o n — a o A o 400 200 1 THIS WORK HOFTIEZER ET AL NANN ET AL MATHIE ET AL NISKANEN BLANKLEIDER LYON - • ~ o A ^*r~—"aT-^ 0 -200. 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 A s s o c i a t e d Legendre polynomial expansion 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 normalized to 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 pion centre-of-mass momentum T J . 174 Figure (5.8) Ratio of the C o e f f i c i e n t s of the Fourth Order A s s o c i a t e d Legendre Polynomial Terms to the T o t a l C r o s s - S e c t i o n , 60 • o A 40 8 o o THIS WORK HOFTIEZER ET AL NANN E T A L NISKANEN BLANKLEIDER LYON 20 00 4 0 -20 0 PION MOMENTUM (77) The c o e f f i c i e n t s of the f o u r t h order term of the A s s o c i a t e d Legendre polynomial expansion 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 normalized to 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 pion centre-of-mass momentum 77. 0 0 175 Figure (5.9) R a t i o of the C o e f f i c i e n t s of the F i f t h Order A s s o c i a t e d Legendre Polynomial Terms to the T o t a l C r o s s - S e c t i o n . D O A THIS WORK HOFTIEZER ET AL NANN ET AL NISKANEN BLANKLEIDER LYON 20 o v o 0 •20 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 A s s o c i a t e d Legendre polynomial expansion 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 normalized to 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 pion centre-of-mass momentum TJ. 176 figures (4.29), (4.30), and (4.31)) together with estimates of the shape 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 s obtained from p u b l i s h 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 data. Only minor changes from our p u b l i s h e d values c a h a r a c t e r i z e d the more exact The b " 0 analysis. coefficient results, significant i s , a c c o r d i n g to the F t e s t i n a l l cases significance i s reflected v a l u e s . T h i s term (see t a b l e (4.17)). T h i s i n the drop of the a s s o c i a t e d x / v 2 i s most s i g n i f i c a n t (according to the F t e s t ) and thus the s m a l l e s t u n c e t a i n t y at 498 MeV. 375 MeV the b n o term, although s t a t i s t i c a l l y according to the F t e s t , At significant i s not i n c o n s i s t e n t with zero when the magnitude of the e r r o r bars i s c o n s i d e r e d . A d d i t i o n of a s i x t h order term to the expansion y i e l d s bg° values c o n s i s t e n t with zero f o r the 375 498 MeV data even though t h i s term the F t e s t and the a s s o c i a t e d drop and i s deemed s i g n i f i c a n t in x /v 2 of the f i t . c o r r e l a t i o n s of the b?° c o e f f i c i e n t s , evident through v a r i a t i o n s i n value of the lower f u n c t i o n of the order Legendre polynomial 450 MeV series order b n 0 by The the c o e f f i c i e n t s as a (number of terms) of the A s s o c i a t e d f i t to the data, are g r e a t e s t w i t h i n the data s e t . O v e r a l l , however, such v a r i a t i o n s are w i t h i n the u n c e r t a i n t y l i m i t s d e r i v e d from the e r r o r The values of the b" /a°, 0 0 f i f t h order expansion matrix. of these r e s u l t s are c o n s i s t e n t with our p u b l i s h e d r e s u l t s , r e s u l t s obtained from a s i g n i f i c a n t l y a n a l y s i s of our data. l e s s rigourous 1 7 7 Values of the comparison c o e f f i c i e n t s together with a to other data and p r e d i c t i o n s Channel Model are presented i n d e t a i l publication . 9 Blankleider here 2 5 ' 3 9 reproduce Predictions 0 Coupled i n our p r e v i o u s of the U n i f i e d Models of and Lyon are i n d i c a t e d '* . of the on the f i g u r e s presented In g e n e r a l , the U n i f i e d Models q u a l i t a t i v e l y the t r e n d of the energy r a t i o s but, again, inadequate dependence of the b"°/ao° quantitativly. 6. In t h i s t h e s i s the CONCLUSION first 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 and p r e c i s i o n measurements of the u n p o l a r i z e d for proton energies differential l e s s than 498 MeV cross-sections are presented. A two-arm apparatus c o n s i s t i n g of s c i n t i l l a t i o n counters multi-wire p r o p o r t i o n a l chambers was constructed of and simple geometric p r o p e r t i e s , capable of measuring pp—*-ir*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 over an angular range of 20° 150° CM., for both p o l a r i z e d and unpolarized proton beams. T r a j e c t o r y r e c o n s t r u c t i o n using from the p r o p o r t i o n a l chambers, together redundant counter systems which enabled determination incident information with employment of on-line 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 d e f i n i t i o n to an accuracy r e q u i r e d to event f o r the p r e c i s i o n desired. In a d d i t i o n , the normalization, i n c i d e n t proton beam current a critical such as t h i s , was based on the simultaneous measurement of the pp->pp e l a s t i c r e a c t i o n and the same production of the + pp-»7r t a r g e t . T h i s development knowledge of the 90° CM. higher element of a p r e c i s i o n experiment d r e a c t i o n from required d i f f e r e n t i a l c r o s s - s e c t i o n to a accuracy than e x i s t e d . P r i o r to t h i s experiment, such measurements were made and method e l i m i n a t e s target thickness the r e s u l t s p u b l i s h e d . 1 0 uncertainties associated or the angle of the This with e i t h e r t a r g e t r e l a t i v e to the the beam d i r e c t i o n . In a d d i t i o n , u n c e r t a i n t i e s r e s u l t i n g from 178 1 79 beam l o s s that can r e s u l t when the p r o d u c t i o n t a r g e t and the beam c u r r e n t monitoring d e v i c e are p h y s i c a l l y separated were also eliminated. The r e l a t i v i s t i c forward-backward t r a n s f o r m a t i o n p r o p e r t i e s of the symmetry of the r e a c t i o n center-of-mass system i n t o the l a b o r a t o r y kinematics i n the system were e x p l o i t e d to estimate and reduce systematic uncertainties a s s o c i a t e d with the apparatus acceptance s o l i d a n g l e s , and pion-decay and e n e r g y - l o s s c o r r e c t i o n s . Carbon background c o n t r i b u t i o n s , although small initially, were c l e a r l y identified through measurements c a r r i e d out with a pure carbon t a r g e t . A model f o r the carbon background was c o n s t r u c t e d and used as a b a s i s f o r a background s u b t r a c t i o n t e c h n i q u e . Furthermore, i n the case of the a n a l y z i n g power r e s u l t s ( r e s u l t s that have a l r e a d y been p u b l i s h e d , G i l e s et a l . ) the background was 9 an i n s i g n i f i c a n t reduced to l e v e l by a method based on the kinematic r e c o n s t r u c t i o n of each event. The r e l i a b i l i t y of our background handling t e c h i q u e s i s demonstrated by the c o n s i s t e n c y of the r e s u l t s o b t a i n e d by the two methods. P r i o r to t h i s experiment, knowledge of the t o t a l c r o s s - s e c t i o n of t h i s fundamental r e a c t i o n was surprisingly poorly known i n t h i s energy r e g i o n . The work of H o f t e i z e r et a l . " 1 d e f i n e d the magnitude of the c r o s s - s e c t i o n 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 i n v e s t i g a t i o n 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 , their + internal i n c o n s i s t e n c i e s of t h e o r d e r results suffered from 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 t h e t o t a l cross-section (ag°) a r e now a v a i l a b l e f r o m 350 t o 498 MeV a s a r e s u l t o f t h e work p r e s e n t e d here. t h e two t e r m s a s s o c i a t e d w i t h t h e a°° and a°° Since c o e f f i c i e n t s dominate the angular dependence of t h e r e a c t i o n , a n d s i n c e common s y s t e m a t i c calculating their ratio, the a ^ / a o 0 e r r o r s c a n c e l when ratio i s experimentally t h e most s t r a i g h t f o r w a r d t o m e a s u r e 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 evident i n published the trends r e s u l t s . Nonetheless, t h e much s m a l l e r a ° / a ° ° r a t i o , 0 already when c o n s i d e r i n g the r e s u l t s of previous w o r k e r s a r e much l e s s c o n s i s t e n t w i t h e a c h o t h e r . case, our r e s u l t s a r e reasonably Rossle et al.*'" (obtained c o n s i s t e n t w i t h those of f r o m m e a s u r e m e n t s o f t h e np—>-7r d 0 r e a c t i o n ) and R i t c h i e e t a l . " 2 ( 7 r d—^pp) , n e i t h e r o f w h i c h + were d e d u c e d f r o m d i r e c t m e a s u r e m e n t s o f t h e However, our r e s u l t s d i s a g r e e Hofteizer et a l . " 1 In t h i s (which pp-»7r + d system. w i t h those of may s u f f e r an o v e r a l l systematic 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 , m e a s u r e d 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 o f t h e pp->;r d r e a c t i o n + Our a j % o ° to support by Rossle r e s u l t s at the highest the negative et al."' 1 energy measured trend established at higher tend energies (np-»7r°d). T h e r e i s no- s t a t i s t i c a l term directly. requirement f o r an e i g h t h order ( a s s o c i a t e d w i t h t h e a2° c o e f f i c i e n t ) t o d e s c r i b e o u r 181 data. I f one assumes that the a£° c o e f f i c i e n t (as p r e d i c t e d by, f o r example, the Coupled i s indeed zero Channel Model of N i s k a n e n ) then the o b s e r v a t i o n that i t i s i n s i g n i f i c a n t 25 suggests the absence of an angular dependent systematic u n c e r t a i n t y , to the e i g h t h order at l e a s t . The first ever d i r e c t p r e c i s i o n measurement 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 below 498 MeV are presented i n t h i s t h e s i s . The b n o expansion coefficients d e r i v e d from these r e s u l t s are i n agreement, w i t h i n the s t a t e d u n c e r t a i n t i e s , with our p r e v i o u s l y p u b l i s h e d r e s u l t s ( G i l e s et a l . ) . 9 The energy b and b " n 0 0 c o e f f i c i e n t s are dominant i n t h i s region and our r e s u l t s i n t h i s case, a g a i n , v e r i f y a trend i n d i c a t e d by p u b l i s h e d work. T h i s 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 c o n s i d e r e d . Of these c o e f f i c i e n t s only the b 2 ° term has been p u b l i s h e d f o r energies below 498 MeV, and the e r r o r s a s s o c i a t e d with these data are l a r g e . Thus, our r e s u l t s provide the only p r e c i s i o n determination of the spin dependent b 498 n o , b n o and of b n o c o e f f i c i e n t s at e n e r g i e s below MeV. I n t e r e s t i n g l y , the only ( i f l i m i t e d ) evidence of a non-zero b ^ same energy rat i o . 0 coefficient i s present at 450 MeV, which i s the as our l a r g e s t ( i n magnitude) determined agVa , 0 0 182 A non-zero a°° c o e f f i c i e n t contribution a 8 requires a s i g n i f i c a n t from the p a r t i a l wave amplitude of d e s i g n a t i o n or h i g h e r , which i n turn i s a s s o c i a t e d with a ' G i , higher r e l a t i v e angular momentum c o n f i g u r a t i o n ) NN (or initial s t a t e . When compared t o the t h e o r e t i c a l d e s c r i p t i o n s of t h i s r e a c t i o n , the Coupled Channel M o d e l 2 5 which p r o v i d e s the best q u a l i t a t i v e p r e d i c t i o n s of our r e s u l t s , f a i l s to take i n t o account c o n t r i b u t i o n s from such channels, the G « 1 p a r t i c u l a r , and thus cannot be expected t o y i e l d results i n the 498 MeV energy in realistic region. As high p r e c i s i o n r e s u l t s such as ours become a v a i l a b l e it i s i n c r e a s i n g l y c l e a r that the present t h e o r e t i c a l d e s c r i p t i o n of t h i s fundamental p r o c e s s , even i n the near t h r e s h o l d r e g i o n , r e q u i r e s s u b s t a n t i a l refinement, a development availability that w i l l of such undoubtedly be guided by the results. APPENDIX I; ELASTIC Nuclear Physics A412 © THE DIFFERENTIAL SCATTERING AT CROSS 90°C.M. SECTION BETWEEN FOR 300 PROTON-PROTON AND 500 (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. J O N E S , G.J. L O L O S , B.J. M c P A R L A N D and W. ZIEGLER Physics Department, University of British Columbia, Vancouver, BC, Canada V6T 2A6 and W. FALK 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° c m . 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; Comparison with phase-shift analyses. measured cr( 6 = 90°). 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 protoninduced 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 1 0 % ) 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 M e V to a precision of approximately 1.8%. The experiment 189 Januir) 1984 183 MEV. 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 T l target position on the 4B external proton beam at T R I U M F . The experimental set-up is shown in fig. 1. The protons resulting from the pp elastic scattering were detected in coincidence 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 cm 3 at 71.9 cm) defined the solid angle. The logic for each event was (PL1 • PL2) • PR 1 + (PR1 • PR2) • P L 1 , or left-arm events plus right-arm events. The percentage of events counted twice by this logic never exceeded 1 0 % . 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 cm 3 ) 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 n A to 2.5 n A 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. A l l 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 S E M 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 S E M , 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. A l l 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 differential 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 C H : target is da 60 (1) pp where da/dO\c is a measure of events from proton-carbon scattering (discussed below), N k 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 (CH 2 ) 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\ was defined by the equation c (2) 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 thefluctuationsin 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 C H : target runs the fluctuations in the ratio dominated the error whereas for the C-target runs the counting statistics dominated the error. 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 C H : target Carbon £ p (MeV) (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 pp elastic da/dfi90°c.m. (mb/sr) 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 infig.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 infig.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 infig.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 B A S Q U E phase-shift predictions4) are also plotted. It is remarkable how similar the two analyses are considering that the B A S Q U E 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 infig.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. 188 194 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 B A S Q U E 4 ) (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. References 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. G A . Ludgate. J.R. Richardson and N.M. Stewart, Nucl. Phys. A 3 7 7 (1982) 5 5 4 APPENDIX I I : THE II.1 MONTE CARLO INTRODUCTION Monte C a r l o techniques were used to evaluate angle i n t e g r a l s d e f i n e d in the numerical i n t e g r a t i o n was the solid t e x t . T h i s method of more capable of e v a l u a t i n g e f f e c t i v e s o l i d angles c h a r a c t e r i z i n g the system the (solid angles depending on complex p h y s i c a l p r o p e r t i e s ) 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, on s i m p l i f y i n g assumptions c o u l d be v e r i f i e d . Aft^) based Furthermore, the muon component of the e f f e c t i v e s o l i d angle could be evaluated The using a Monte C a r l o technique. event d e t e c t i o n e f f i c i e n c y was explicitly; therefore the event d e t e c t i o n i t was the apparatus geometry and i n t e g r a l c o u l d be evaluated not known integrated i m p l i c i t l y . efficiency i s an implicit simulating angle events, t r a c k i n g the p a r t i c l e s through the apparatus to detection point, subject i f any. In-flight, to the geometrical example; w a l l s and Since f u n c t i o n of m a t e r i a l , the s o l i d by only and their the p a r t i c l e s were c o n s t r a i n t s of the apparatus ( f o r a p e r t u r e s ) i n a d d i t i o n to the simulated i n f l u e n c e of pion-decay, m u l t i p l e - s c a t t e r i n g , and energy-loss i n t e r a c t i o n s . Since be removed from the simulation, any of these processes i t was could p o s s i b l e to determine which processes or c o n s t r a i n t s were most s i g n i f i c a n t . In Monte C a r l o system used, randomly d i s t r i b u t e d p a r t i c l e 189 the 190 d i r e c t i o n s were generated over a given s o l i d angle i n the center-of-mass system. The p a r t i c l e s were then t r a c k e d and the e f f e c t i v e s o l i d angle determined from the f r a c t i o n of p a r t i c l e s d e t e c t e d . Two such systems (computer programs) d e s i g n a t e d PEPI, and REVMOC* , each with d i f f e r e n t 7 c a p a b i l i t i e s were u t i l i z e d : 1) PEPI: Designed f o r a two arm d e t e c t o r . T h i s system was capable of s i m u l a t i n g : - A two-arm d e t e c t i o n system; both the pion and deuteron were t r a c k e d . - Energy-loss e f f e c t s not i n c l u d e d . - Small-angle m u l t i p l e s c a t t e r i n g ('optional) - Pion decay (optional) - A f i n i t e s i z e beam spot - A f i n i t e beam energy d i s t r i b u t i o n 2) REVMOC" : A g e n e r a l purpose 7 system width. beam ( p a r t i c l e ) transport supported and maintained a t TRIUMF. With supplementary r o u t i n e s developed where necessary, i t could simulate: - A quasi-two arm system; Events with deuterons would escape d e t e c t i o n on the b a s i s of t h e i r d i r e c t i o n only were r e j e c t e d . Otherwise that initial the deuteron was assumed d e t e c t e d , and only the pion t r a c k e d i n detail. - Energy-loss e f f e c t s (optional) - Small angle m u l t i p l e s c a t t e r i n g - Pion decay (optional) (optional) 191 - A f i n i t e s i z e beam spot - A monochromatic proton beam energy distribution was r e q u i r e d . REVMOC* 7 in i t s original s i m u l a t i n g the- experiment. correct form was not capable of I t was unable t o d u p l i c a t e the random pion momentum and angular c o o r d i n a t e d i s t r i b u t i o n s . Furthermore, one-arm system; that i t was i n h e r e n t l y o r i e n t e d t o a i s , i t c o u l d only t r a c k one of the two p a r t i c l e s r e q u i r e d . The f o l l o w i n g improvements were thus implemented. The angular c o o r d i n a t e s of c o r r e l a t e d pions and deuterons were evenly d i s t r i b u t e d over a given s o l i d in the center-of-mass angle system. These angular- c o o r d i n a t e s and the a s s o c i a t e d p a r t i c l e momenta were then transformed the l a b o r a t o r y system. into The r e s u l t i n g deuteron c o o r d i n a t e s were then examined and a t e s t performed t o determine whether the deuteron would h i t the deuteron d e t e c t o r . I f i t d i d not, the event was r e j e c t e d . Thus, the assumption deuteron travelled REVMOC* was not r e q u i r e d t o t r a c k the second 7 deuteron) i n a s t r a i g h t l i n e was enforced, and in d e t a i l . c o o r d i n a t e system, that the p a r t i c l e (the I f the deuteron was d e t e c t e d , the initially d i r e c t i o n , was r o t a t e d about with the Z-axis i n the beam the v e r t i c a l (Y-axis) such that the Z-axis d i r e c t i o n was along the c e n t r a l a x i s of the pion d e t e c t o r system. F i n a l l y , the momenta and r e s u l t a n t angular c o o r d i n a t e s a s s o c i a t e d with the pions were t r a n s f e r r e d t o REVMOC" 7 which c a r r i e d out the t r a c k i n g of the pion the remaining arm. through 192 II.2 APPARATUS GEOMETRY AND MATERIAL The apparatus was d i v i d e d i n t o elements or regions i n the format r e q u i r e d by the Monte C a r l o systems. Each region of a d e t e c t i o n arm was d e f i n e d by a s e c t i o n of uniform m a t e r i a l . In general, the m a t e r i a l contained w i t h i n each region was d i f f e r e n t Table from that of the region on e i t h e r s i d e . (1) shows an example. The depth of a region (Z) corresponds t o the l e n g t h of the m a t e r i a l along the c e n t r a l a x i s of the arm. The other two dimensions d e f i n e a r e c t a n g u l a r aperture a s s o c i a t e d with each r e g i o n . P a r t i c l e s passing o u t s i d e of an aperture were c o n s i d e r e d The Table stopped. p h y s i c a l p r o p e r t i e s of the m a t e r i a l s are l i s t e d i n (1b). REVMOC' 7 only c o n s i d e r s a m a t e r i a l s p e c i f i e d by three or l e s s atomic s p e c i e s (elements). Thus, the composition of some m a t e r i a l s (eg. magic gas) were approximated by the three dominant s p e c i e s i n d i c a t e d i n Table (1b). II.3 PHYSICAL The INTERACTIONS three p h y s i c a l i n t e r a c t i o n s invoked were pion decay, small-angle m u l t i p l e - s c a t t e r i n g , and e n e r g y - l o s s . A d e s c r i p t i o n of these processes the REVMOC* 7 i s given i n the appendix of documentation which i s reproduced i n Table ( 2 ) . When both the energy-loss and pion decay i n t e r a c t i o n s were invoked (within REVMOC" ) subsequent energy-loss of the 7 muons subsequent to the pion decay was d i s r e g a r d e d . T h i s omission was c o r r e c t e d with the f o l l o w i n g method. Since most 193 Table 1 la) DEFINITION OF A DETECTION ARM BY REGIONS REGION DIMENSION description 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 TARGET VACUUM MYLAR #1 AIR n MYLAR #2 MAGIC GAS #1 CATHODE / / l MAGIC GAS #2 ANODE MAGIC GAS //3 CATHODE 02 MAGIC GAS #4 MYLAR //3 AIR #2 WRAPPING #1 SCINTILLATOR / / l WRAPPING //2 AIR //3 WRAPPING #3 SCINTILLATOR / / l The lb) Z (cm) 0.088 0.507 0.025 8.468 0.025 0.925 0.006 0.472 0.002 0.472 0.006 0.925 0.025 5.476 0.066 0.159 0.066 1.539 0.066 0.683 X (cm) to < from 1.0 30.0 40.7 100.0 100.0 100.0 100.0 100.0 5.0 100.0 100.0 100.0 100.0 100.0 100.0 6.35 100.0 100.0 6.35 6.35 -1.0 -30.0 -40.7 -100.0 -100.0 -100.0 -100.0 -100.0 -5.0 -100.0 -100.0 -100.0 -100.0 -100.0 -100.0 -6.35 -100.0 -100.0 -6.35 -6.35 to Y (cm) < from 1.0 30.0 6.4 100.0 100.0 100.0 100.0 100.0 5.0 100.0 100.0 100.0 100.0 100.0 100.0 6.35 100.0 100.0 6.35 6.35 -1.0 -30.0 -6.4 -100.0 -100.0 -100.0 -100.0 -100.0 -5.0 -100.0 -100.0 -100.0 -100.0 -100.0 -100.0 -6.35 -100.0 -100.0 -6.35 -6.35 geometry of a t y p i c a l p i o n arm i s d e f i n e d by the above r e g i o n s . TABLE OF ASSUMED PHYSICAL PROPERTIES OF THE MATERIALS MATERIAL ATOMIC COMPOSITION DENSITY g/cm COMMENTS 3 Polyethylene Mylar Air Magic Gas Cathode wires Anode wires Scintillators The composition approximated. (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 f o r wrapping R a t i o s by volume of the m a t e r i a l s above has, i n some c a s e s , been 194 of the pions decay p r i o r to the f i r s t s c i n t i l l a t o r , the i n t e g r a t e d a r e a l d e n s i t y of the system from t h i s point on was c a l c u l a t e d . A c u t - o f f muon energy was d e f i n e d , below which muons could not be expected to t r a v e r s e the d e t e c t o r . The f i n a l number of s u c c e s s f u l events was then reduced by the number of muons with energies resulting angle. below the c u t - o f f value i n a p r o p o r t i o n a l drop of the muon e f f e c t i v e solid APPENDIX 3: ANALYZING POWER OF THE pp->ir»d AT 375, 450, AND 500 MEV INCIDENT PROTON ENERGIES. RAPID COMMUNICATIONS PHYSICAL REVIEW C V O L U M E 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. A u l d , 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 The 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). p p — 7 r d reaction is the simplest pion production + beam was continuously monitored during the experimental process that can be studied. Because the inverse reaction runs using an upstream polarimeter which monitored represents process, asymmetry of pp elastic scattering. The beam intensity was essential in- measured by a number of devices, the most important of the elementary knowledge of the reaction gredient to understanding pion absorption is therefore an the absorption of low energy the which involved the detection of the 90° [center-of-mass pions in nuclei.' Much recent interest in the reaction has (cm.)] elastically scattered protons from the target itself.8 been the The time of flight, energy-loss, and angular coordinates of channel provides a major source of information coincident deuterons and pions were measured with a two- associated with pp — w + d the fact that the study of towards the understanding of the complete nucleon-nucleon arm system. The between 20° and 150°. A single 38.3 mg/cm2 polyethylene importance of spin-dependent observables of detection system for pions with center-of-mass angles the nucleon-nucleon system has been enhanced by the ob- [(CH;),] target was used for all the pion production mea- servation of unexpected energy dependence of the Acr and surements. Data were also obtained from a 24.9 mg/cm2 L Exotic carbon target in order to delineate the contribution of the reaction mechanisms, such as those which include a highly carbon background. Each of the arms used for detecting the inelastic intermediate state that contains a so-called " d i - pion and deuterons consisted of a pair of thin scintillation baryon resonance," have been proposed to explain this type counters together with a multiwire proportional chamber Ao-r parameters of the proton-proton subsystem. of observation. be expected 4 2,3 If such a mechanism should exist, it could to manifest itself in the inelastic pp—' ir +<i nucleon-nucleon channel. In fact, spin-dependent observ- used for determining the angular coordinates of the trajectories. The hardware event definition consisted of (any) threefold coincidence of the four scintillators. Thus the ef- ables (such as the analyzing power) provide particularly ficiencies of all detectors could be extracted from the data. stringent constraints on the theoretical models constructed The data were recorded on magnetic tape for subsequent Existing theoretical off-line analysis. Only time-of-flight and energy-loss con- models fail to provide an adequate description of the pre- straints were required for the off-line event definition for + to describe the p p — j r d reaction. cision data from 517-578 MeV. 6 5 At lower energies, nearer threshold, where a theoretical description should be simpler the 375 M e V data. Only a small (typically 0.01) correction to the analyzing power resulted from the carbon subtraction. because of the reduced number of angular momentum com- For the 450 and 500 MeV ponents, no tion and angular coplanarity constraints were applied with precision analyzing power data exist over a data, additional angular correla- range of angles sufficient to permit a definitive comparison the result that no carbon background subtractions were re- with existing theories.7 quired. In all cases, the error in the analyzing powers asso- 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 together with extensive measurements of the unpolarized differential cross section, a body of results which is currently is less than ± 0 . 0 1 . Jn addition, an overall systematic un- certainty of 2% the 500 M e V for the 375 and 450 MeV The experiment was mounted on an external proton beam The polarization of the 28 data and 4% for 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. being analyzed. line at the T R I U M F cyclotron. ciated with both carbon background and counting statistics MeV. R. Falk et a i 1 0 at 450 The agreement of the two 450 MeV data sets is ex- cellent. Although the data of Ref. 10 are also from TRI2551 ©1983 The American Physical Society 195 196 RAPID COMMUNICATIONS 2552 G . L . a • • • o • • o ° o O • • o • o • Q a• • 0 0 0 0 o 28 a o n 4 O etal o o o * G I L E S 0 o 0 oo o PION FIG. 1. Analyzing power responding symbol A- oo ANGLE for the tion of the pion angle ( c m . ) . 500 MeV o - 450 MeV Rel. 9 0 - 375 MeV • - 0 (cm ) pp — reaction as a n*d T h e e r r o r b a r is s m a l l e r than unless otherwise func- the cor- indicated. T h e data of Ref. 10 different b e a m a at 4 5 0 M e V a r e i n c l u d e d . U M F , they were single-arm obtained experimental o n a configuration line employing a with magnetic spectrometer. T h e analyzing s h o w n in section tion) Eq. (i.e., Legendre with values obtained These powers (1) as b each aj/o-, to where a X -4,vo(fl) 17 > tions in of cients where T|, of are those traction and of of M a t h i e era/. error to J. the bars associated with counting bk c o e f f i c i e n t s the series, terms, the was whereas indicated twice that found for the pion error of the bars presented here trends established by higher dependence dicated cient for the 1.25, than 1.3. case of erwise for even for k O u r as well 7) g r e a t e r 500 M e V . completely data clarify 1. have this the data 0.75. are an for the v0 /<r, The the s o m e - which bars is overall of 1 (a) i ( v for the The TJ l e s s the uncertainties pion ln (the remain- than line dashed lOxjf'o/o-. the o u r results solid the % 7 . functions of 1 and depicts curve (b) la a n d t h e d a s h e d c u r v e 0 only the the f o r b^ associated a for solid 0 with la. the subtraction. the of 7)-1.5 for experimental of bi c o e f f i c i e n t s for clear that the oth- even in 0.75 a n d surements. in the bi!" c o e f f i c i e n t effort, m o d e l of various 1 3 coupled-channel intermediate the energy section, are generally as data Niskanen. improves, as well p p — • 77 indicates as further + d it is in the region experimental reaction parameters. than quality b e c o m i n g a clear need require for the values o b value neighborhood the pertinent good of theoretical negative the a experimental A s theoretical models near-threshold This the to cross zero by for provides dependence the m o r e addition, fails predicted the present the In formalism state, cross experimentally. of N i s k a n e n , a Ni. bk c o e f f i c i e n t s values TJ l e s s o n the description magnitude o n include represent Ref. for Legendre function is s e l to z e r o l . In for curves polarization-dependent precise example, prediction dotted based treatment served TJ b e t w e e n the of 1.3. a to M e V of For 13) thaan c o u n t i n g statistics a n d the b a c k g r o u n d coeffi- reported and error the i n - o f (Ref. symbols the results rj g r e a t e r as a, (ij-0.774) is t h e p r e d i c t i o n f o r i f As a shoulder increase A l t h o u g h N o situation. coefficient over Niskanen the associated section T h e solid TJ. represent for of h"", cross of m o m e n t u m the order been indicate in up with at 6 f>l o d d total (cm.) symbols Ref. curve 5 0 0 T h e ing the the the A j v 0 c o e f f i c i e n t at 3 7 5 M e V of term Coefficients to momentum the 6* 2. relative of a n d T h e sub- order obtained 6 increase than k terms, as a noticeable than M e V , coefficients is s m o o t h m a r k e d which A?°/o- for the consistent era/. TJ g r e a t e r b^" la, increasing a n d 450 data t 0.01 at precision a ( c m . ) coeffi- k limits were Hoftiezer terms, the o d d than they J. with b an additional the by of the o d d A region, resulting the smaller than T) energy are func- sensitivity 375 bars o u r reasonable less as FIG. a n d (for 6 background T h e terms 2(a) era/. TJ^I) for the carbon be at error results what to this (1) m o m e n t u m s h o w n within even in . in Figs. (for 1 2 statistics only. variations to noted: Hoftiezer 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 in c o e f f i c i e n t s . ' " /VtcosO) results T h e sec- associated a the TJ r e p r e s e n t s m „ c . as cross cross using referred bk c o e f f i c i e n t s a r e p l o t t e d those c o m b i n e d differential fit are h k with a n d units bSfla the fycostO-S— — resulting along 1.3) a n d 7 coefficients even J °" T h e were the cr i s t h e - t o t a l data, yield bf/cr of coefficients, unless otherwise k 2(b), energy estimate published functions normalized paper of from at an to m o r e of the increasingly refinement, these m e a - theoretical measurement of the ANALYZING POWER OF THE pp - ir d REACTION AT 375, 2553 + 28 T h e ported extensive assistance i n part the by of Natural D. S a m p l e Sciences a n d a n d C. G r e i n Engineering in the data Research analysis C o u n c i l of is gratefully acknowledged. This w o r k was sup- Canada. 'G. Jones, in Ref. 5, p. 15. 'A. W. Thomas and R. H. Landau, Phys. Rep. 58, 121 (1980). I. P. Auet, E. Colton, H. Halpern, D. Hill, H. Spinka, G. Theodo- D. 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Kost and P Reeve, TRIUMF Design Note, 1982, (unpublished); and r e f e r e n c e s c o n t a i n e d w i t h i n . 461 552 (1981). (1970). (l981). 214 (1976). Gordon G i l e s REFEREED PAPERS IN SCIENTIFIC JOURNALS THE ANALYZING POWER OF THE pp ->• i x d REACTION AT 375, 450, AND 500 MeV INCIDENT PROTON ENERGIES G.L. G i l e s , E.G. A u l d , G. Jones, G.J. L o l o s , B . J . McParland, W. Z i e g l e r , D. O t t e w e l l , P. Walden, and W. F a l k . Phys. Rev. C28 (1983) 2551 + THE DIFFERENTIAL CROSS-SECTION FOR PROTON-PROTON ELASTIC SCATTERING AT 90° c m . BETWEEN 300 AND 500 MeV D. O t t e w e l l , P. Walden, E.G. Auld, G.L. G i l e s , G. Jones, G.J. L o l o s , B . J . McParland, W. Z i e g l e r . a n d W. F a l k . N u c l . Phys. A412 (1984) 189 ANGULAR DEPENDENCE OF THE L i (ix , He) He REACTION AT 60 AND 80 MeV B.J. McParland, E.G. A u l d , P. Couvert, G.L. G i l e s , G. Jones, X. A s l a n o g l o u , G.M. Huber, G.J. L o l o s , S.I.H. Naqvi, Z. Papandreou, P.R. G i l l , D.F. O t t e w e l l , and P.L. Walden. Manuscript submitted f o r p u b l i c a t i o n t o P h y s i c s L e t t e r s . 6 + 3 3 POLARIZED-PROTON-INDUCED EXCLUSIVE PION PRODUCTION IN C 225, 237 AND 250 MeV INCIDENT ENERGIES G.J. L o l o s , E.G. A u l d , W.R. F a l k , G.L. G i l e s , G. Jones, B.J. McParland, R.B. T a y l o r , and W. Z i e g l e r . Phys. Rev. C30 (1984) 574 1 2 AT 200, 216, ANALYSING POWER OF THE pp + it d REACTION AT 400 AND 450 MeV W.R. F a l k , E.G. A u l d , G.L. G i l e s , G. Jones, G.J. L o l o s , 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. L o l o s , E.L. M a t h i e , G. Jones, E.G. A u l d , G.L. G i l e s , B.J. McParland, P.L. Walden, W. Z i e g l e r , and W. F a l k . N u c l . Phys. A386 (1982) 477 + PION PRODUCTION FROM DEUTERIUM BOMBARDED WITH POLARIZED PROTONS OF 277 and 500 MeV G.J. L o l o s , E.G. A u l d , G.L. G i l e s , G. Jones, B . J . McParland, D. O t t e w e l l , P.L. Walden, and W. Z i e g l e r . N u c l . Phys. A422 (1984) 582 SPECTROSCOPY OF DOUBLY RESONANT THIRD HARMONIC GENERATION L. T a i , F.W. Dalby, and Gordon L. G i l e s . Phys. Rev. A20, (1978) 233 IN \
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A study of the differential cross-section and analyzing powers of the pp-->[pi]+d reaction at intermediate… Giles, Gordon Lewis 1985
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