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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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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. 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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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