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Nuclear proton radii from low energy pion scattering Barnett, Bruce M. 1985

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NUCLEAR PROTON RADII FROM LOW ENERGY PION SCATTERING  by  BRUCE M. BARNETT B.Sc,  McGill University, 1978  A THESIS SUBMITTED IN PARTIAL FULFILLMENT THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY  in THE FACULTY OF GRADUATE STUDIES Department of Physics  We accept this thesis as conforming to the required  standard  November 1985  © Bruce MacLeod Bamett 1985  In  presenting  degree freely  at  the  available  copying  this  of  department publication  of  in  partial  fulfilment  of  the  University  of  British  Columbia,  I  agree  for  this or  thesis  reference  thesis by  this  for  his thesis  and  study.  scholarly  or  her  for  of  financial  Physics  The University of British Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 D  a  t  e  DF.fin/ft-n  December 3, 1985  purposes  gain  shall  that  agree  may  representatives.  permission.  Department  I further  requirements  be  It not  that  the  be  Library  permission  granted  is  for  by  understood allowed  the  an  advanced  shall for  make  extensive  head  that  without  it  of  copying my  my or  written  ii Abstract  The subject of this thesis i s the study of the use of e l a s t i c scattering d i f f e r e n t i a l cross section ratios of positive pions on l i g h t nuclei i n the determination of nuclear proton matter d i s t r i b u t i o n differences and their moments.  These are compared with results from  electron scattering and muonic X-ray measurements of the charge density differences. The measurements are r e l a t i v e to nuclei whose matter d i s t r i b u t i o n s and absolute cross sections are considered as references.  The use of the r a t i o  of cross sections, rather than the absolute cross sections themselves, minimises the effects of uncertainties i n the understanding  of the pion  nuclear interaction i n our extraction of density difference information; these effects are investigated.  A furthur advantage of the technique i s  that the measurement of cross section ratios i s insensitive to many systematic experimental  effects encountered i n measurement of absolute cross  sections. The ratios of e l a s t i c scattering of positive pions at 38.6 MeV and 47.7 MeV on the isotones ^ B ,  1 2  C are presented.  at 48.3 MeV and 62.8 MeV on the nuclei C , 1 2  Also, cross section ratios 1 6  0 and  1 8  0 are presented.  The measured cross sections do not r i v a l the quality of the cross section r a t i o s , but are also  presented.  The RMS radius differences extracted from the pion e l a s t i c scattering cross section ratios at low energy are consistent (within a standard deviation) with the results of other methods.  The proton matter radius  (N+Z ) differences which we obtain are as follows: r+  iii  (U  r +  _12  r + )  (lt  r +  _12  r +  )  r +  )  =  r +  )  =  (  16  r +  (18  _14  r +  _16  =  _.Q64  (28)  fm  =  .098  (44)  fm  #  .  The e r r o r s r e f l e c t uncertainties  1  5  0  (44)  9  8  4 (  7  f  )  (llB,12 ) C  (12  m  G >  N >  16 ) 0  m  f  m  {-.032(33)  [ALK83] }  {  .074(21)  [SCH75]  }  {  .180(22)  [SCH75]  }  {  .077  [MIS79] }  s t a t i s t i c a l a n d , to a l a r g e e x t e n t ,  i n the q u a n t i t i e s .  (5)  the  systematic  The best e l e c t r o n s c a t t e r i n g r e s u l t s  are  shown w i t h i n b r a c e s . A n a l y s i s of the  1  2  C , ^ B experiment  i n d i c a t e s systematic u n c e r t a i n t i e s statistical uncertainties.  1 8  0,  1 6  0  c h o i c e of o p t i c a l  s e t i s of l i m i t e d  potential,  importance.  experiments i n terms of F o u r i e r B e s s e l and  n o v e l F o u r i e r Laguerre p r o t o n matter ( r a d i a l ) d e n s i t y (App(r))agree  differences  of about the same o r d e r as the  In p a r t i c u l a r ,  d e n s i t y form and o p t i c a l parameter A n a l y s e s of the  i n terms of RMS r a d i u s  w i t h p r e c i s i o n "model independent"  differences  [NOR82]  electron  s c a t t e r i n g r e s u l t s , i n the r e g i o n i n which the p i o n can be s e n s i t i v e to n u c l e a r p r o t o n matter d i s t r i b u t i o n s . fm.)  (The p h y s i c a l l i m i t s  The e f f e c t s of o p t i c a l parameter u n c e r t a i n t i e s S i m i l a r a n a l y s e s of the  1  2  C,  1 4  N,  1 6  0  the p r o t o n matter d e n s i t y d i s t r i b u t i o n of  are  enforce r ? discussed.  N  towards the c e n t e r of  n u c l e u s , r e l a t i v e to t h a t suggested by e l e c t r o n s c a t t e r i n g [SCH75]  1.5  experiments i n d i c a t e a s h i f t 1 4  the  in  that  experiments  and at about the same r a d i u s suggested by S _ e l f - C o n s i s t e n t S i n g l e  Particle Potential  (SCSPP) c a l c u l a t i o n s [H0D85].  The a n a l y s e s of  e l e c t r o n s c a t t e r i n g experiments were not model i n d e p e n d e n t , w e l l generate d e n s i t i e s which a r e i n c o r r e c t at transfer:  r < 2  these  though, and may  s m a l l r a d i u s ( h i g h momentum  fm.)  We p r e s e n t these r e s u l t s as testimony to the a b i l i t y  of the T f  +  to  iv  probe App(r) r e l i a b l y . especially  The o p t i c a l  parameter  s e n s i t i v i t y i s minimal,  when c r o s s s e c t i o n f i t t i n g i s used to determine the r e f e r e n c e  nucleus o p t i c a l  parameters.  T h i s work p r o v i d e s  c o r r o b o r a t i o n f o r the  analogous tr" n e u t r o n d e n s i t y measurements of [JOH79, GYL85].  Table of Contents i i  Abstract  v  Table of Contents  x  List of Figures  xxi  List of Tables A c k n o w l e d g e m e n t s Chapter I  xxvi  1  Introduction  1.1  Introduction  1  1.2  T r a d i t i o n a l Matter Probes  1  1.3  Low Energy P i o n S c a t t e r i n g : The I s o r a t i o Method  3  1.4  The P i o n Nuclear I n t e r a c t i o n  5  1.4.1  The P i o n Nucleon I n t e r a c t i o n :  1.4.2  The P i o n N u c l e a r I n t e r a c t i o n  1.5  Matter D i s t r i b u t i o n S e n s i t i v i t y  1.6  P r e v i o u s I s o r a t i o Experiments  14  1.7  Current Objectives  16  1.8  Summary  16  Chapter II  Currents  5 9  of the Ratio  12  18  Experimental Details  2.1  Introduction  18  2.1.1  Machine D e t a i l s  18  2.2  1 2  2.2.1  E x p e r i m e n t a l Setup  25  2.2.2  S c a t t e r i n g Telescopes  25  2.2.2.1  Operating C h a r a c t e r i s t i c s  28  2.2.3  Electronics  33  2.2.4  Targets  36  2.3 2.3.1  1  Carbon,  2  C ,  n  1  B o r o n (CB) Experiment  6  0,  1  8  0 (CNO Experiments  The QQD Spectrometer  2  20  37 41  vi 2.3.2  E l e c t r o n i c Logic  42  2.3.3  Targets  44  2.4  Summary  49  Chapter I I I  3.1 3.2  50  Analysis  Introduction 1  2  C ,  n  50  B Experiment  3.2.1  Preliminary  3.2.2  E v a l u a t i o n of A b s o l u t e Cross S e c t i o n s  51  3.2.3  E v a l u a t i o n of R a t i o s  66  3.2.4  Results  68  3.3  CNO  81  3.3.1  Momentum Spectrum O p t i m i s a t i o n  81  3.3.2  Software  82  3.3.2.1  NMR Cuts  82  3.3.2.2  T a r g e t Traceback Cuts  83  3.3.2.3  TOF Cuts  86  3.3.2.4  Muon Cuts  86  3.3.2.5  Energy S p e c t r a  89  3.3.3  Peak F i t t i n g  94  3.3.4  A b s o l u t e Cross S e c t i o n Measurement  96  3.3.4.1  BM1»BM2 Rate Loss  96  3.3.4.2  Spectrometer O f f s e t Angle  103  3.3.4.3  Average S c a t t e r i n g Angles  103  3.3.4.4  S o l i d Angle D e t e r m i n a t i o n  104  3.3.5  R a t i o E v a l u a t i o n s and C o r r e c t i o n s  107  3.3.6  Results  108  3.4  Summary  108  2  A n a l y s i s : Software  50 Cuts  Experiments  Cuts  51  Interpretations  134  4.1  Introduction  134  4.2  Optical Potential Analysis  135  Chapter IV  vii  4.2.1  SMC Potentials  137  4.2.1.1  The Potential  137  4.2.1.2  The Calculation  139  4.3  CB Experiment  140  4.3.1  Cross Section Calculations with the SMC Potential  140  4.3.2  Generic Low Energy Pion Scattering  143  4.3.3  Ratio Calculations  146  4.3.4  S e n s i t i v i t i e s to Higher Moments  153  4.3.4.1  Modified Density Results  155  4.3.5  S e n s i t i v i t i e s to Optical Parameters  160  4.3.6  Miscellaneous S e n s i t i v i t i e s  160  4.3.7  Optical Model Dependence of Results  165  4.4  CNO Experiments  165  4.4.1  Cross Section F i t t i n g  165  4.4.2  Density D i s t r i b u t i o n Difference Analysis  170  4.4.3  2  1 8  0 , 0 Experiments 1 6  176  4.4.3.1  Matter D i s t r i b u t i o n Determinations  176  4.4.3.2  Optical Parameter S e n s i t i v i t i e s  180  4.4.4  ^C, ^, ^  181  4.4.4.1  Matter D i s t r i b u t i o n Difference Analysis  181  4.5  Summary  187  Chapter V  1  1  (CNO) Experiments  Discussion and Summary  192  5.1  Overview  192  5.2  O p t i c a l Potential F i t t i n g  193  5.3 5.4 5.4.1  1 2  C , B Experiment n  CNO Experiments 2  1 8  0 , 0 Experiments 1 6  194 197 197  5.4.2  CNO Experiments  199  5.5  Summary  202  5.6  Epilogue  203  viii  205  List of References Appendix I  Subscripts and Superscripts  Appendix II  P o w d e r Boron Target M a s s M e a s u r e m e n t s  Appendix III  D e c a y Kinematics  215 216 219  A3.1  Muon Cone Angle  219  A3.2  Beam A t t r i t i o n  221  Appendix IV  Core + Valence Matter Distributions  225  A4.1  The D i s t r i b u t i o n s  225  A4.1.1  The MGCV D i s t r i b u t i o n  225  A4.1.2  The CF D i s t r i b u t i o n  229  A4.1.3  The CFCV D i s t r i b u t i o n  231  Appendix V  A Replacement for Viewfit  Appendix VI  Spectrometer Transfer Coefficient Optimisation  Appendix VII  P e a k Fitting in the C N O Experiments 2  234 236 242  A7.1  Introduction  242  A7.2  The Peak Shape  242  A7.3  Mechanics of F i t t i n g  245  A7.4  Correlation Errors  245  A7.5  Propagation of E r r o r s  247  A7.5.1  R a t i o Measurements  249  Appendix VIII A8.1  Fourier Expansions of the Nuclear Density Introduction  255 255  ix  A8.2  Fourier Bessel Analysis  255  A8.3  F o u r i e r Laguerre Expansions  256  A8.3.1 A8.3.2 A8.3.3 A8.3.4 A8.3.5 A8.3.6  The B a s i s  Appendix IX  256 258  Zero Sum C o n s t r a i n t D e r i v a t i v e s of  PFLU)  2 5 8  259  R a d i a l Moments Uncertainties i n ppL( ) r  2  6  0  261  The D e n s i t i e s  Of Charge Densities and Matter Densities  263  A9.1  Convolution  o f the Proton Form F a c t o r  263  A9.2  F o l d i n g Ppp i n t o a F o u r i e r L a g u e r r e Sum  264  A9.3  Folding p  p p  i n t o a F o u r i e r B e s s e l Sum  266  A9.4  Folding p  p p  i n t o the S t a r t i n g D e n s i t y  A9.5  Evaluations  A9.5.1 A9.5.2 A9.5.3 A9.5.4 A9.5.5  «(» f o r the Folded  <|> f o r the Folded  A9.6  Unfolding  Pj(r)  of $  266  Q  0  MG Charge D e n s i t y  <(> f o r the Unfolded FB Charge D e n s i t y 0  0  FB Charge D e n s i t y  <j> f o r the Unfolded FL Charge D e n s i t y 0  <|> f o r the Folded 0  FL Charge D e n s i t y  o f the Proton Form F a c t o r  267 267 267 268 268  from  E l e c t r o n S c a t t e r i n g Data  Appendix X  266  Moments and the Folding Process  268  274  A10.1  The F o l d i n g and the RMS I n t e g r a l s  274  A10.2  Other Moments v i a the F o u r i e r T r a n s f o r m  275  L i s t of Figures 1.1  A low energy n  scattering experiment (The 15 MeV  +  experiment: [GIL82]) similar i n configuration  to the  carbon-boron T T r a t i o experiment described i n the text. +  1.2  4  Kinematic observables f o r pions scattering from nuclear material.  The pion, of mass m, charge e^, isospin x,  momentum nk and angular momentum I scatters from a nucleus with Z protons, N neutrons, t o t a l i s o s p i n t, t o t a l spin J . The protons and neutrons have d i s t r i b u t i o n s p+(r) and p _ ( r ) , respectively.  1.3  6  The r e s u l t s of ir~ scattering experiments (29 MeV) on isotopes y i e l d r e l a t i v e neutron r a d i i .  1 2  a _ ( a_) 13  i s the e l a s t i c d i f f e r e n t i a l scattering cross section for pions on C ( C ) [GYL79a]. 1 2  2.1  13  a) Channel configuration and b) momentum resolution of M13 [0RA81].  2.2  15  19  Comparison of fluxes available from meson channels Mil and M13.  21  2.3  M13 TOF spectra at 40 and 50 MeV.  23  2.4  Typical beam p r o f i l e s at 40 (a and b) and 50 (c and d) MeV, f o r i r  on M13.  a) and c) are horizontal p r o f i l e s ,  b) and d) v e r t i c a l .  Structure on the v e r t i c a l p r o f i l e s  +  i s an a r t i f a c t . 2.5  Scattering  24  table arrangement, showing TT scattering  telescopes and beam monitoring apparatus.  26  6  Peak, h e i g h t and r e s o l u t i o n f o r S i ( L i )  detectors.  7  P a s s i n g r e s o l u t i o n of the S i ( L i ) d e t e c t o r s to p i o n s .  8  Energy spectrum f o r the Nal d e t e c t o r .  9  L i n e a r i t y of Nal d e t e c t o r s .  10 Contour p l o t of E vs AE f o r 128 MeV/c p a r t i c l e s emerging  from Ml3.  11 C i r c u i t diagram f o r CB experiment.  12 E x p e r i m e n t a l setup showing c o n f i g u r a t i o n on M13  (a) s p e c t r o m e t e r - c h a n n e l  and (b) enlargement of  spectrometer.  13 P i o n TOF  s p e c t r a of beams from (a) M13  (b) M i l a t 65  14 QQD  at 50 MeV  and  MeV.  spectrometer e l e c t r o n i c  logic.  15 Beamspot p r o f i l e at the t a r g e t frame at 65 MeV 120 degrees (RUN1116).  and  Contours show i n c r e a s e s i n  i n t e n s i t y of 10% of the maximum.  1  Uncut  scatterplot  f o r (run28, ARMl).  on  at 47.7  as s c a t t e r e d though a l a b a n g l e  1 2  C  MeV  Pions i n c i d e n t  of 9 0 ° .  2  Uncut AE h i s t o g r a m f o r (run28, ARMl). to E h i s t o g r a m i n f i g u r e 3.3  are shown.  AE c u t s  applie  xii  3.3  Energy h i s t o g r a m f o r  (run28, ARM1) a f t e r cut on A E .  P i o n s i n s i d e the c u t s are e l a s t i c a l l y  3.4  AE h i s t o g r a m as i n f i g u r e f i n a l E spectrum. figure  3.5  3.2,  scattered.  55  but cut o n l y on the  T h i s shows the AE cut of  that  to have adequate width and c e n t e r i n g .  56  C a l c u l a t e d p i o n l o s s e s w h i l e s t o p p i n g i n a Nal detector:  s o l i d curve shows l o s s e s from r e a c t i o n s ;  broken curve shows those from p i o n d e c a y .  63  3.6  Muon decay c o n t r i b u t i o n  65  3.7  Excited states  3.8  1 2  to t e l e s c o p e e f f i c i e n c y .  of s t a b l e boron and carbon i s o t o p e s .  C 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 at  Curves are from o p t i c a l model (SMC79) w i t h parameter  3.9  1 2  C elastic  +  +  Diagram of the QQD definitions.  MeV.  calculations  C and  U  1 2  1 1  set  1 1  at  38.6  2. 79  C and  v a l u e of  spectrometer,  B  71  r+.  O p t i c a l c a l c u l a t i o n s use parameter  C e n t r a l curve uses best f i t  3.12  1 2  v a l u e of  R a t i o of c r o s s s e c t i o n s of TT on MeV.  47.7  O p t i c a l c a l c u l a t i o n s use parameter  C e n t r a l curve uses best f i t  3.11  70  s e t s 1 (broken) and l b ( s o l i d ) .  R a t i o of c r o s s s e c t i o n s of TT on MeV.  calculations  d i f f e r e n t i a l c r o s s s e c t i o n at  w i t h parameter  MeV.  s e t s 1 (broken) and l a ( s o l i d ) .  Curves a r e from o p t i c a l model (SMC79)  3.10  38.6  67  at set  47.7  2.  r+.  showing  80  coordinate 84  3.13 T y p i c a l d e l t a d i f f e r e n c e , DDIF, spectrum [RUN220, 50 MeV;  1 8  0 , 70°].  3.14 T y p i c a l , a n g l e c o r r e l a t i o n spectrum, ANGL, [RUN220, 50 MeV;  1 8  0 , 70°].  3.15 D e n s i t y p l o t i l l u s t r a t i n g t h e c o r r e l a t i o n between the DDIF and ANGL s p e c t r a o f f i g u r e s 3.13 and 3.14. 3.16 T y p i c a l energy spectrum 48.3 MeV.  w  +  ir  +  on C a t 80°. 1 2  [RUN007]  3.17 T y p i c a l energy spectrum 62.8 MeV.  on C a t 120°. 1 2  [RUN1110]  3.18 N u c l e a r l e v e l s observed i n C , ^ N , 1 2  3.19 Peak f i t t i n g r e s u l t s from T T 110°.  +1 8  1 6  0 , and  1 8  0.  0 spectrum a t 62.8 MeV,  [RUN1109]  3.20 Rate c o r r e c t i o n due t o m u l t i p l e p i o n s p e r beam b u r s t . The range o f t h e c o r r e c t i o n found i n t h e s e e x p e r i m e n t s i s shown as a shaded a r e a .  3.21 T y p i c a l observed s o l i d a n g l e of s p e c t r o m e t e r as a f u n c t i o n of T . Q  [RUN1228 , 62.8 MeV;  1 6  0 , 50°]  3.22 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 o f T T i n c i d e n t +  on  1 2  C and  1 6  0 a t 48.3 MeV.  densities derived  from e l e c t r o n s c a t t e r i n g ( F L ) and  the SMC81 p o t e n t i a l . Set  Ef50f.  C a l c u l a t i o n s use  S o l i d , S e t E50;  broken,  3.23 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 of T C i n c i d e n t +  on  C and  1 2  1 6  0  a t 62.8 MeV.  C a l c u l a t i o n s use d e n s i t i e s  d e r i v e d from e l e c t r o n s c a t t e r i n g  ( F L ) and the SMC81  potential.  long d a s h e s , Set  EC65f  Solid,  Set EBLE65f;  (Set E065f f o r  Set E65;  1 6  0  calculation);  short dashes,  and d o t s , Set EBLE65co.  3.24 R a t i o s of e l a s t i c c r o s s s e c t i o n s f o r TT at 48.3 MeV: +  c/ a.  lk  12  C a l c u l a t i o n uses Set E f 5 0 f i n an SMC81  p o t e n t i a l and a FL p a r a m e t e r i s a t i o n independent e l e c t r o n s c a t t e r i n g The  l t f  (cf.  of a model  density for  N p r o t o n d e n s i t y i s a best f i t figure  1  2  C .  FL form  4.15).  3.25 R a t i o s of e l a s t i c c r o s s s e c t i o n s f o r T T at 48.3 MeV: +  a/ a.  16  lk  C a l c u l a t i o n uses Set Ef50f  p o t e n t i a l and FL p a r a m e t e r i s a t i o n s independent e l e c t r o n s c a t t e r i n g 1 2  C.  i n an SMC81  of the model  d e n s i t i e s of  The **N p r o t o n d e n s i t y i s a best f i t 1  (cf.  figure  o7  1 6  a.  The (cf.  0  FL form  at 48.3 MeV:  +  C a l c u l a t i o n uses Set EIM50 i n an SMC81  p o t e n t i a l and a MG form 1 8  0 and  4.16).  3.26 R a t i o s of e l a s t i c c r o s s s e c t i o n s f o r it 1 8  1 6  f o r the  1 6  0 matter d e n s i t i e s .  p r o t o n d e n s i t y i s a best f i t  figure  FL form  4.13).  3.27 R a t i o s of e l a s t i c c r o s s s e c t i o n s f o r TT at 62.8 MeV: +  lk  a/ o. l2  C a l c u l a t i o n uses Set EC65f i n an SMC81  p o t e n t i a l and a FL p a r a m e t e r i s a t i o n independent e l e c t r o n s c a t t e r i n g The (cf.  ll  density for  * N p r o t o n d e n s i t y i s a best f i t figure  4.17).  of a model  FL form  1  2  C.  3.28  R a t i o s of e l a s t i c c r o s s s e c t i o n s f o r T T at 62.8 1 6  a/  1 4 t  a.  C a l c u l a t i o n uses Set E065f i n an SMC81  p o t e n t i a l and FL  parameterisations  of the model  independent e l e c t r o n s c a t t e r i n g d e n s i t i e s of 1 2  C.  The  (cf.  3.29  11+  figure  N p r o t o n d e n s i t y i s a best f i t  a/  a.  The (cf.  0  The 40 MeV Los Alamos [BLE79].  The 49.9 data  0  T T  +  ,  1  2  FL form  e l a s t i c cross  C  and 2 ( s o l i d  MeV Los Alamos  [MOI78].  T T  +  ,  1  2  C  e l a s t i c cross  O p t i c a l c a l c u l a t i o n s use and 2 ( s o l i d  MeV.  section  parameter  curve).  from  C h a r a c t e r of Coulomb-nuclear  d i s t i n g u i s h e s p i o n charge  contrast 1 2  C  at  interference  states.  O p t i c a l model c a l c u l a t i o n s (parameter set 2)  of  s c a t t e r i n g of p o s i t i v e  curve)  and  U  B  at  43.1  and the s h i f t  4.5  parameter  curve).  the natures of charged p i o n s c a t t e r i n g  4.4  section  O p t i c a l model c a l c u l a t i o n s (parameter set 2)  43.1  MeV:  matter d e n s i t i e s .  O p t i c a l c a l c u l a t i o n s use  set 1 (dashed c u r v e )  4.3  62.8  4.14).  set 1 (dashed c u r v e )  4.2  1 6  p r o t o n d e n s i t y i s a best f i t  figure  data  and  C a l c u l a t i o n uses Set EIM65 i n an SMC81  p o t e n t i a l and a MG form f o r the 1 8  0  4.18).  +  1 6  1 6  FL form  R a t i o s of e l a s t i c c r o s s s e c t i o n s f o r T T at 1 8  4.1  MeV:  +  MeV.  p i o n s on  1 2  C (dashed  N o t i c e the r e l a t i v e  i n the minimum.  R a t i o of c r o s s s e c t i o n s i n f i g u r e  4.4.  magnitudes  4.6  x  values generated from equation 4.24.  are f i t s to equation 4.25. increases i n x 4.7  from values at the respective minima.  Contour plot f o r MGCV density at 47.7 Mev. x l  RAD  and RCOP i s the core radius.  ll  4.9  Dashed lines indicate unity  Contour plot for MGCV density at 38.6 MeV. i s r+  4.8  2  Solid curves  RAD i s  r + and RCOP i s the core radius.  Contour plot for Re(c ) at 38.6 MeV. Q  r+  RAD i s  ll  and RECS i s R e ( c ) . Q  4.10 Contour plot f o r Re(c ) at 47.7 MeV. Q  RAD i s  n  r  +  and RECS i s R e ( c ) . 0  4.11 E l a s t i c d i f f e r e n t i a l cross sections of TT on 0 at +  1 6  62.8 MeV and 48.3 MeV showing SMC81 potential calculations with parameter sets E (E50 and E65; broken curves) and the f i t t e d sets EIM (EIM50 and EIM65).  Matter densities are of the MG form.  4.12 E l a s t i c d i f f e r e n t i a l cross sections at 65 MeV for Tf  +  on  1 2  c  as measured by Blecher et a l .  with various parameter sets are shown: EBLE65F; E65;  Long dashed, Set EC65F;  Dotted, Set EBLE65co.  of the FL form.  Calculations Solid, Set  Short dashed, Set  Matter densities are  xvii 4.13  P r o t o n matter d e n s i t y d i f f e r e n c e s d e r i v e d from 48.3 (a)  Variation  (except  MeV TT r a t i o s  as i n d i c a t e d )  Set EIM50 v a l u e s : curve, ii),  ImB ;  p (r)-  1 6  p  p  figure  w i t h +10%  p (r))  3.26).  variations  i n o p t i c a l parameters  from  Broken c u r v e (+5%), Imt> ;  Dotted  0  i),  0  1 8  (cf.  +  i n d e r i v e d Ap(r)  (  Reb ;  iii),  Q  those r e m a i n i n g .  (b)  Imc ;  Best f i t  Set EIM50 o p t i c a l p a r a m e t e r s . completeness e r r o r .  iv),  Q  ImC ; 0  FL d e n s i t y w i t h  E r r o r envelope i n c l u d e s  E l e c t r o n s c a t t e r i n g matter  d e n s i t i e s are shown f o r c o m p a r i s o n .  4.14  Same as f i g u r e (a)  Variation  4.13  at  fit  Imc ; Q  MeV ( c f .  i n d e r i v e d Ap(r)  i n o p t i c a l parameters ii),  62.8  iv),  X;  177  figure  w i t h +10%  variations  from Set E65 v a l u e s : iii),  3.29).  i),  those r e m a i n i n g ,  ImB ; Q  (b)  Best  FB d e n s i t y w i t h Set E65 o p t i c a l parameters.  envelope i n c l u d e s completeness e r r o r .  Error  Electron  s c a t t e r i n g matter d e n s i t i e s are shown f o r c o m p a r i s o n .  4.15  P r o t o n matter d e n s i t y d i f f e r e n c e s ( * P p ( r ) - P p ( r ) ) l l  MeV i r , ' N /  d e r i v e d from 48.3 figure  3.24).  +  Best f i t  o p t i c a l parameters.  1  T  1 2  C  1 2  ratios  (cf.  FL d e n s i t y w i t h Set  E r r o r envelope i s  Dashed c u r v e i s d i f f e r e n c e  of model  in  1 2  independent density  C and the b e s t a v a i l a b l e MG d e n s i t y f o r  A l s o shown a r e SCSPP c a l c u l a t i o n s w i t h a nuclear p o t e n t i a l  developed i n the  Same as f i g u r e 4.15 from 48.3 figure  MeV T T ,  3.25).  +  1 6  for 0/  1 u  ( *p (r)l l  p  N  1 2  Ef50f  statistical.  e l e c t r o n s c a t t e r i n g d e r i v e d p r o t o n matter  4.16  178  Ca  k0  N.  'standard'  region  p (r) ) p  1 I +  [HOD85].  183  derived  r a t i o s w i t h Set EF50f  (cf. 184  xviii  4.17  Same as f i g u r e 4.15 from 62.8 figure  4.18  MeV T T , +  for  1 1 +  (  l l t  pp(r)-  1 2  Pp(r)  derived  N / C r a t i o s w i t h Set EC65f  (cf.  1 2  3.27).  185  Same as f i g u r e 4.15 from 48.3  MeV T f , +  1 6  for 0/  l l +  ( **P (r)1  1 2  p  N  p (r)) p  derived  r a t i o s w i t h Set E065f  (cf.  figure 3.28).  4.19  186  R a t i o c a l c u l a t i o n (SMC81, Set E f 5 0 f ) w i t h the fit  density ( c f .  f i g u r e 4.15)  w i t h the same RMS r a d i u s . In  and a MG  best  density  t h i s case use of  (solid)  solely  the MG form would l e a d to a s e r i o u s o v e r e s t i m a t i o n the  1 1 +  N RMS r a d i u s .  189  A2.1 Apparatus f o r d e t e r m i n i n g the  ^B  t a r g e t mass t h i c k n e s s  by measurement of e l e c t r o n m u l t i p l e  A3.1 D e f i n i t i o n of the k i n e m a t i c LABo ratory (LAB) f r a m e , frame  of  scattering effects.  observables in  217  the  and the Center of Momentum  (CM).  220  A3.2 A p e n c i l beam of p i o n s i n c i d e n t upon a p l a n e c i r c u l a r d e t e c t o r p e r p e n d i c u l a r to the beam.  222  A3.3 CM angle c o s i n e s c o r r e s p o n d i n g to p i o n s of LAB energy 43 MeV d e c a y i n g i n t o to one h a l f  a cone w i t h a v e r t e x  of the muon cone a n g l e .  the shaded r e g i o n s e n t e r  2  equal  Decays i n s i d e  t h i s cone.  A4.1 V a r i o u s d e n s i t y d i s t r i b u t i o n s f i v e nucleons with < r  angle  >  1 / 2  P+(r) = 2.25  223  evaluated fm.  for 227  xix A4.2 QD (dashed)  and < r > 2  as c a l c u l a t e d f o r the  1 / 2  Fermi  d i s t r i b u t i o n as a f u n c t i o n of c / t .  A6.1 Spectrometer c o e f f i c i e n t  A6.2 Improvement Spectrum,  232  determination  of QQD r e s o l u t i o n : 48.3 a)  With t r a n s p o r t  algorithm.  MeV Energy  c o e f f i c i e n t s and no  muon c u t s ,  b) With o p t i m i s e d c o e f f i c i e n t s and no  muon c u t s ,  c) With o p t i m i s e d c o e f f i c i e n t s and muon  c u t s on DDIF and ANGL. A7.1 a)  241  Covariance supermatrix  c o n s t r u c t e d i n peak, f i t t i n g  to the energy s p e c t r a from the QQD. between GROUP 1 v a r i a b l e s k and I, written ' k £ '  i n the m a t r i x .  error matrices with free  A7.2 R e d u c t i o n of s u p e r m a t r i x 'subgroups'  240  b) The  cr^,  correlation  is  The E ' s are the u s u a l  variables u,h,  and  f.  250  f o r a ' g r o u p ' w i t h two  c o r r e s p o n d i n g to s c a t t e r i n g from the  same angle and t a r g e t .  A8.1 a) and b) The f i r s t  252  s i x (zero  norm) FL d e n s i t i e s .  The t o t a l number of l o c a l minima and maxima on a g i v e n curve i s the o r d e r of the d e n s i t y , D e n s i t i e s of a) factor.  A9.1  1 6  0  and  [a  1 2  0  and b)  c) and d)  f o l d e d w i t h the p r o t o n  form 262  normalisation]  C charge d e n s i t i e s ,  a)  and b) The  solid  curves a r e f o l d e d m o d i f i e d G a u s s i a n d e n s i t i e s and the broken c u r v e s are the model independent of  [NOR82] and [CAR80].  c)  and d) FL components  r e p r e s e n t i n g the d i f f e r e n c e s between the i n a)  and b ) .  [a  0  densities  normalisation]  distributions 271  XX  A9.2 P r o t o n matter ( s o l i d ) d i f f e r e n c e s between  and charge ( s h o r t  >>o.  18  1(  dash)  density  The u n f o l d i n g was  c a r r i e d out as d e s c r i b e d i n s e c t i o n A 9 . 6 .  Also  shown i s the charge d e n s i t y d i f f e r e n c e w i t h MG forms w i t h the RMS charge r a d i i  of  1 8  0  and  1 6  0  (long  dash).  The d o t t e d curve i s the charge d e n s i t y from a c a l c u l a t i o n of Brown et a l .  [BR079a,  BR079b].  272  A9.3 M a t t e r d e n s i t i e s (a and b) and charge d e n s i t i e s and d)  c o r r e s p o n d i n g to the f i r s t  c o n s t r a i n e d to z e r o RMS i n t e g r a l [a  0  (c  6 FL d e n s i t i e s , and zero norm.  normalisation]  273  <  xx i  L i s t of Tables  2.1  D e t e r m i n a t i o n of Beam E n e r g i e s at  2.2  Isotopic  2.3  CNO  2.4  Isotopic  3.1  Meaning  C o m p o s i t i o n of S c a t t e r i n g  Experiment  2  Target  Targets  Composition of Target  MeV.  Angles Weighted  are Denoted  38  Material  48  of Symbols used i n the C a l c u l a t i o n Cross  of  Elastic  Sections  by TT  57  for  Pions  Cross  +  at  Sections  by CM"" ^"^  59  1  3.3  Symbols used i n P i o n Decay  3.4  S o l i d Angle Data f o r  3.5  Quantities  Correction  Scattering  60  Telescopes  Involved i n C a l c u l a t i o n  of R a t i o  62  Correction  Factors  3.6  22  46  Angle A v e r a g i n g and CM T r a n s f o r m a t i o n s 43.1  Center  Summary  D i f f e r e n t i a l Scattering  3.2  the Target  69  E l a s t i c D i f f e r e n t i a l Cross S e c t i o n s f o r TT on +  Lab Energy 38.6  1 2  C  for  MeV 72  3.7  E l a s t i c D i f f e r e n t i a l Cross S e c t i o n s f o r u Lab Energy 47.7  3.8  Ratio on  n  B  +  on  1 2  C  for  MeV  73  of TT E l a s t i c Cross S e c t i o n f o r 7 t on +  at Lab Energy 38.6  +  MeV  1 2  C to  that 75  XXII  3.9  Ratio of n on B  E l a s t i c Cross Section for T T on  +  +  at Lab Energy 47.7  n  12  C  to that  MeV  76  3.10  Correction Factors for Ratios at Lab Energy 38.6 MeV  77  3.11  Correction Factors for Ratios at Lab Energy 47.7  78  3.12  MeV  Values of Target Traceback Coefficients Defined i n Equation 3.18  3.13  Ratio of I n e l a s t i c (4.44 MeV) for  3.14  Tf+  on  12  C  Cross Sections to E l a s t i c  at 48.3 MeV  98  Ratio of I n e l a s t i c (4.44 MeV) for TT+ on  3.15  85  12  C  Cross Sections to E l a s t i c  at 62.8 MeV  99  Meaning of Symbols used i n the Calculation of E l a s t i c D i f f e r e n t i a l Scattering of T T on +  12  C  at 48.3 MeV  and  62.8 MeV 3.16  D i f f e r e n t i a l Cross Sections for E l a s t i c Scattering of Tf  3.17  +  on  at 48.3 MeV  110  11+  N  at  48.3 MeV  111  +  on  1 6  0  at  48.3 MeV  112  D i f f e r e n t i a l Cross Sections for E l a s t i c Scattering of Tf  3.20  C  D i f f e r e n t i a l Cross Sections for E l a s t i c Scattering of Tf  3.19  1 2  D i f f e r e n t i a l Cross Sections for E l a s t i c Scattering of TT+ on  3.18  100  +  on  1 8  0  at  48.3 MeV  113  D i f f e r e n t i a l Cross Sections for E l a s t i c Scattering of Tf  +  on  12  C  at  62.8 MeV  114  3.21  Differential TT  3.22  on  +  N  1 6  0  +  on  1 8  3.31  0  1 6  0  1 8  0  Differential +  on  1 H  N  Differential of TT+ on  3.30  1 6  Differential  of T T  3.29  N  Differential  of TT+ on  3.28  14f  Differential  of Tr+ on  3.27  at  Differential  of TT+ on  3.26  at  0  of TT+ on  3.25  at  Differential TT  3.24  on  11+  Differential ir  3.23  +  1 6  0  Cross S e c t i o n s 62.8  f o r E l a s t i c S c a t t e r i n g of  MeV  Cross S e c t i o n s 62.8  for Elastic  S c a t t e r i n g of  MeV  Cross S e c t i o n s 62.8  f o r E l a s t i c S c a t t e r i n g of  MeV  Cross S e c t i o n R a t i o s of E l a s t i c and  C  1 2  at 48.3  MeV  Cross S e c t i o n R a t i o s of E l a s t i c and  C  1 2  at 48.3  N  1 4  a t 48.3  and  0  a t 48.3  1 2  C  at 62.8  1 2  C  a t 62.8  Cross S e c t i o n R a t i o s of E l a s t i c  of TT+ on  and  0  Differential of TT+ on  1 8  0  ll  * N a t 62.8  1 6  0  at 62.8  Scattering  MeV  Cross S e c t i o n R a t i o s of E l a s t i c and  Scattering  MeV  Differential 1 6  Scattering  MeV  Cross S e c t i o n R a t i o s of E l a s t i c and  Scattering  MeV  Cross S e c t i o n R a t i o s of E l a s t i c and  Scattering  MeV  Cross S e c t i o n R a t i o s of E l a s t i c 1 6  Scattering  MeV  Cross S e c t i o n R a t i o s of E l a s t i c and  Scattering  MeV  Scattering  X X I V  4.1  V a r i o u s O p t i c a l Parameter  Sets.  C,  1 2  X 1  B  Experiment.  SMC79 P o t e n t i a l  4.2  RMS M a t t e r R a d i i of Calculations. ( U n i t s are  4.3  142  n  B  from O p t i c a l Model  Uncertainties  are Shown i n  Parenthesis.  fm)  154  R e s u l t s of T e s t s f o r S e n s i t i v i t y Matter D i s t r i b u t i o n .  1 2  C,  X 1  to D e t a i l s  B Experiment.  of  the  (Units  are  fm)  4.4  Dependence of Measured Radius.  4.5  156  f  on the Assumed Core  ( U n i t s are am per percent  Sensitivities  of  l l  are am per p e r c e n t  4.6  1 1  i  change)  159  to O p t i c a l Parameters.  (Units  change)  M i s c e l l a n e o u s Parameter  161  Dependencies.  1 2  C,  1 1  B  Experiment  4.7  D e v i a t i o n of  164  l  x  f from the P o t e n t i a l  C o l o r a d o , LT and SMC79 A n a l y s e s . over O p t i c a l Parameter  n  Averaged Means f i s the  Sets 0 and 2.  4.8  O p t i c a l Parameter  Sets (SMC81).  1 8  4.9  O p t i c a l Parameter  Sets (SMC81).  1 2  0,  C,  Average  ( U n i t s are am)  1 6  1 4  0  N,  Experiments  1 6  F o u r i e r Laguerre P a r a m e t e r i s a t i o n s Densities  166  169  0  Experiments  4.10  in  171  of  Reference 175  XXV  4.11  Proton Matter D i s t r i b u t i o n Differences:  1 8  4.12  Proton Matter D i s t r i b u t i o n Differences:  12  0- 0  179  1 6  C,  ll+  N, 0 16  Experiment  188  5.1  Measured Values of  5.2  Proton Matter D i s t r i b u t i o n Differences:  1 8  5.3  Proton Matter D i s t r i b u t i o n Differences:  12  1 1  r +  1 2  r  for B  196  n  +  0- 0  198  1 6  C,  ll+  N, 0 16  Experiment A4.1  200  Summary of Characteristics of Matter Density Distributions  226  A4.2  Solutions to Equation A4.16 for 2 through 8 Nucleons  233  A6.1  P r o l i f e r a t i o n of Spectometer Transfer Coefficients  237  A7.1  D e f i n i t i o n of Peak. F i t t i n g Parameters  244  A9.1  Nucleon Form Factor Parameterisation  265  xxvi  Acknowledgement s  It  is difficult  t o put pen to paper  to compose these l i n e s .  During  the  course of t h i s p r o j e c t many c o l l a b o r a t o r s and f r i e n d s have taken time t o c o n t r i b u t e e x p e r t i s e or encouragement towards i t s c o m p l e t i o n ; somewhat r e l u c t a n t  to thank those who  I  am  most r e a d i l y come t o mind l e s t  I  n e g l e c t o t h e r s d e s e r v i n g of mention. The  project i t s e l f ,  of course, would not have been p o s s i b l e without  support of TRIUMF and NSERC. environment  TRIUMF p r o v i d e s an i n t e l l e c t u a l l y n o u r i s h i n g  i n which to work and  study.  NSERC makes t h i s p r a c t i c a l .  s t a f f a t TRIUMF I thank f o r t h i s environment r e s o u r c e s (such as the AES  word p r o c e s s i n g and VAX  would have been more t e d i o u s by f a r .  My  thanks  helped to p r o v i d e a smooth i n t e r f a c e to the UBC  11/780 computer  p l e a s u r e to work.  thesis  to c o l l a b o r a t e  To s e v e r a l of these I owe  work on the QQD  spectrometer.  g r a t i t u d e f o r h i s humour, c r e a t i v i t y and David G i l l .  fellows,  summer students w i t h whom I have had particular  thanks.  To B i l l  Dave i s a dynamic i n d i v i d u a l who  i n t e r e s t e d i n and a v a i l a b l e f o r c o n s u l t a t i o n .  their  Gyles I express  invaluable insight.  the  Randy Sobie  Roman T a c i k , S i g M a r t i n and C h r i s Wiedner c o n t r i b u t e d much through developmental  who  p h y s i c s department.  I thank the numerous r e s e a r c h e r s , post d o c t o r a l  s t u d e n t s , t e c h n i c i a n s and  systems,  a l s o to those people  D u r i n g the course of the work, I have been p r i v i l e g e d  graduate  The  and f o r the p r o v i s i o n of  and much t e c h n i c a l s u p p o r t ) without which the p r o d u c t i o n of t h i s  w i t h many p e o p l e .  the  my  I thank a l s o  has always been both He c o n t i n u e s to p l a y a key  r o l e i n the P i s c a t Group's e x p e r i m e n t a l endeavours.  Finally,  l e t me  mention  Byron Jennings i n g r a t i t u d e f o r h i s r e a d i n e s s to d i s c u s s p h y s i c s ( n u c l e a r or otherwise).  xxv i i  I have not f o r g o t t e n my f r i e n d and r e s e a r c h He i s d e s e r v i n g  o f s p e c i a l thanks.  supervisor  D i c k Johnson.  I have enjoyed the p r i v i l e g e of working  w i t h him and am g r a t e f u l f o r having had the b e n e f i t o f h i s guidance and experience.  Here a l s o my thanks t o the members of my t h e s i s  advisory  committee f o r t h e i r time, c o n t r i b u t i o n s and i n t e r e s t i n t h i s  project.  Many p e r s o n a l Brosing, halves:  f r i e n d s deserve mention.  People such as B i l l  W a l l y F r i e s e n and Roland P i e r r o t (and t h e i r r e s p e c t i v e  Gyles,  Juli  better  L i z Hewetson, K e i t h Lecomte, Irma F r i e s e n and Sharon P i e r r o t ) h e l p e d  to m a i n t a i n my p e r s p e c t i v e  on l i f e .  Pat B e l l i s a dear and much l o v e d  f r i e n d who a l o n g w i t h h e r f a m i l y ( e s p e c i a l l y Jean B e l l ) has p r o v i d e d the l o c a l s t a b i l i t y afforded  by f a m i l y r e l a t i o n s h i p s .  At l a s t , l e t me express my s i n c e r e and l o v i n g thanks t o my own f a m i l y : Don,  Linda  provided  and, i n p a r t i c u l a r , my mother and f a t h e r .  encouragement and a s u p p o r t i v e  love, respect  and thanks.  atmosphere.  They have always To them, my h e a r t f e l t  xxviii  Through f a i t h we understand that the worlds were framed by the word of God, so t h a t t h i n g s which a r e seen were not made of t h i n g s which do appear.  Hebrews  11:3  1  Chapter I Introduction  1.1  Introduction One  nuclear was  of the t r a d i t i o n a l i n t e r e s t s of n u c l e a r  s i z e and  Rutherford's  a landmark c o n t r i b u t i o n .  innovation He  shape.  that explained  The  alpha  physics  i s the study of  s c a t t e r i n g paper [RUTH] of  p o i n t - l i k e nucleus t h a t he  1911  proposed was  an  the s t a t i s t i c s of l a r g e a n g l e a l p h a s c a t t e r i n g .  remarked t h a t :  Considering  the evidence as a whole, i t seems s i m p l e s t  t h a t the atom c o n t a i n s  to suppose  a c e n t r a l charge d i s t r i b u t e d through a  very  s m a l l volume... ...it  should  be p o s s i b l e from a c l o s e study of the n a t u r e of  the  d e f l e c t i o n to form some i d e a of the c o n s t i t u t i o n of the atom to produce the e f f e c t s observed.  This l a s t observation  has  formed the b a s i s of many of the attempts at  q u a n t i f y i n g the term ' d i s t r i b u t e d ' .  1.2 T r a d i t i o n a l Matter Probes There are two first  t r a d i t i o n a l methods of measuring n u c l e a r  of t h e s e uses the e l e c t r o m a g n e t i c  p a r t i c l e s such as muons and i s a r e f i n e d and  electrons.  size.  probes; t h i s c l a s s i f i c a t i o n  The includes  E l e c t r o n s c a t t e r i n g [DON75, FRI75]  t e s t e d t e c h n i q u e which u t i l i s e s w e l l understood dynamics.  E l a s t i c s c a t t e r i n g of e l e c t r o n s y i e l d s p r e c i s e i n f o r m a t i o n  about charged  2  n u c l e a r matter  through measurement of the e l e c t r o m a g n e t i c form f a c t o r ,  i s c l e a r l y i n s e n s i t i v e to n e u t r a l m a t t e r .  Attempts  a t measuring  v a l e n c e neutron d e n s i t i e s from i n f o r m a t i o n gleaned from backscattering  [LI70, DON73] have met  but  nuclear  magnetic  w i t h some s u c c e s s [SIC77,  PLA79].  A major disadvantage of e l e c t r o n s c a t t e r i n g as a technique i s the mass of the p a r t i c l e .  T h i s n e c e s s i t a t e s the i n c l u s i o n of numerous t e d i o u s  c o r r e c t i o n s such as those f o r r a d i a t i v e e f f e c t s . redeems i t ,  somewhat, on t h i s p o i n t .  The  muonic atoms p r o v i d e s i n f o r m a t i o n about the c o r r e s p o n d i n g n u c l e i . these probes .0129  The muon's h e a v i e r mass  study of e m i s s i o n s p e c t r a  the RMS  (Root M_ean Square) r a d i i of  1 2  C:  (<r > 2  ± .0053) [RUC82] ) are o f t e n a t t r i b u t e d to QED  The  from  S m a l l d i f f e r e n c e s i n the r a d i i determined  ( f o r example, i n the case of  p r o p e r l y accounted  1 / 2  ^ - <r > 2  1 / 2 e  c o r r e c t i o n s not  f o r i n the muonic atom a n a l y s e s .  s t r o n g l y w i t h n u c l e a r matter  [TH081].  p r o t o n s and the a l p h a p a r t i c l e s .  scatterer.  with  ) =  second broad c l a s s of n u c l e a r probes c o n t a i n s those which  are s e n s i t i v e ,  low  interact  I n c l u d e d i n t h i s c l a s s are the  Protons have the disadvantage that  they  i n essence, o n l y t o the t o t a l matter d e n s i t y (p++p_) of t h e i r  (See appendix  I f o r a d i s c u s s i o n of l a b e l l i n g  conventions.)  A l p h a p a r t i c l e s , on the o t h e r hand, absorb s t r o n g l y a t the n u c l e a r s u r f a c e so t h a t they tend to be i n s e n s i t i v e to the d e t a i l of the n u c l e a r i n t e r i o r . Furthermore,  the composite  n a t u r e of a l p h a s makes them somewhat d i f f i c u l t  handle from t h e o r e t i c a l f i r s t  to  principles.  In r e c e n t y e a r s , the copious q u a n t i t i e s of pions a v a i l a b l e  from  h i g h - f l u x meson f a c t o r i e s have made p o s s i b l e the e x t e n s i v e use of pions (and  3  the muons r e f e r r e d to e a r l i e r ) i n n u c l e a r s i z e measurements. near the A-resonance  (1^=180 MeV)  comparisons  of T T and i r ~  At e n e r g i e s  elastic  +  s c a t t e r i n g have been used to e x t r a c t p r o t o n and n e u t r o n matter r a d i i by u s i n g the s t r o n g l y d i f f r a c t i v e n a t u r e of the c r o s s s e c t i o n s .  [JAN78]  The s t r o n g  a b s o r p t i o n of p i o n s a t these e n e r g i e s , however, p r o h i b i t s d i r e c t p r o b i n g of the n u c l e a r i n t e r i o r . moments of the matter  These p i o n s cannot d i r e c t l y measure the lowest distributions.  1 . 3 Low Energy Pion Scattering: The Isoratio Method. A major stumbling b l o c k i n p i o n - n u c l e a r p h y s i c s i s found i n the f a c t t h a t the i n t e r a c t i o n p o t e n t i a l i s not w e l l  'understood'.  One  can, a t b e s t ,  use knowledge of p i o n i c atoms [BAC70] and of f r e e p i o n - n u c l e o n (TTN) scattering  [ROW78] i n a semi-phenomenological  o p t i c a l p o t e n t i a l i n the hope  of d e s c r i b i n g p i o n - n u c l e a r s c a t t e r i n g p r o c e s s e s .  A problem  i n the f a c t t h a t a m b i g u i t i e s and u n c e r t a i n t i e s i n parameters obscure  exists, may  though,  tend to  the e f f e c t s of bona f i d e n u c l e a r s t r u c t u r e .  The P I S C A T t e r i n g group  a t TRIUMF has been u t i l i z i n g a method which  i s o l a t e s n u c l e a r s i z e e f f e c t s , w h i l e a v o i d i n g many of the e f f e c t s t h a t a r i s e from e x p e r i m e n t a l and t h e o r e t i c a l s y s t e m a t i c s .  can  T h i s i s done by  c o n s i d e r i n g the r a t i o s of c r o s s s e c t i o n s , r a t h e r than the c r o s s s e c t i o n s themselves  ( f i g u r e 1.1).  The method makes use of the low energy  dependence of the TTN e l a s t i c s c a t t e r i n g amplitude  [ERI70].  isospin  As w i l l be  seen,  near c a n c e l l a t i o n s i n the p i o n - n u c l e o n p-wave amplitude r e s u l t i n a Tr n -  ( T r p ) amplitude +  s i g n i f i c a n t l y l a r g e r than the Tr'p ( i r n ) amplitude.  be understood by examining [ERI70]:  +  the f r e e p i o n n u c l e o n s c a t t e r i n g  This  amplitude  may  4  1.  Alignment  Location  7 .  Passing  2 .  AlIgnment  Marks  8.  Ion  3.  Bending  9.  Stopping  4 .  Cable  10.  5.  Cabling  €.  Cyclotron  Magnet  B2  Tray  Shielding  Scintillator  C1  13.  Scattering  14 .  Passing  15.  Beam C o u n t e r  NIM B i n S t a t i o n  16.  Target  11 .  Quadrupole  07  17 .  Beam V e t o  12 .  Scattering  Arm ARM0  18.  To  Chamber C o u n t e r NaI1  Counter  +  (The 15 MeV  r a t i o experiment d e s c r i b e d i n the t e x t .  S1»S2  ( 1 5 MeV)  C o u n t i n g Room  experiment:  [GIL82]) s i m i l a r i n c o n f i g u r a t i o n to the carbon-boron T T  S1(L1)1  Location  F i g u r e 1.1  A low energy i r s c a t t e r i n g experiment  Table  +  5  f(9) = b  0  + biCTfr) + ( c  0  + C j a t T ) ) ^ ' )  where one n e g l e c t s a s m a l l s p i n dependent term.  (1.1)  irN s c a t t e r i n g l e n g t h s  determine the parameters: b  Q  = -0.005 fm  b  :  = -0.13  fm  c  0  =  0.64  fm  3  c  =  0.43  fm  3  :  T»T  = +1  (TT+p.rr-n)  (1.6)  -1  (TT-p.Tr+n)  (1.7)  (isoscalar  s-wave)  (1.2)  ( i s o v e c t o r s-wave)  (1«3)  ( i s o s c a l a r p-wave)  .(1.4)  ( i s o v e c t o r p-wave)  (1«5)  where:  Some o f t h e r e l e v a n t dynamical q u a n t i t i e s a r e shown i n f i g u r e  1.2.  1.4. The Pion Nuclear Interaction L e t us c o n s i d e r now, i n a h e u r i s t i c way, the o r i g i n of the form of t h e s c a t t e r i n g amplitudes of n u c l e a r s i z e s .  1.2 through 1.5 and t h e i r i n c o r p o r a t i o n i n t o our study  We w i l l  see the manner i n which the n u c l e a r s t r u c t u r e and  nuclear i n t e r a c t i o n are convoluted  to a r r i v e at a pion-nuclear  optical  potential. The  s t a r t i n g p o i n t i n our d i s c u s s i o n i s of n e c e s s i t y the TT n u c l e o n  (TTN) i n t e r a c t i o n , as we assume the n u c l e o n which our i n t e r a c t i o n  t o be the b a s i c b u i l d i n g b l o c k on  occurs.  1.4.1 The Pion Nucleon Interaction: Currents R e c a l l t h a t i n t h e f o r m u l a t i o n of c l a s s i c a l p e r t u r b a t i o n t h e o r y f o r a charged p a r t i c l e , e, w i t h an e l e c t r o m a g n e t i c  field  c h a r a c t e r i s e d by a v e c t o r  p o t e n t i a l A, the i n t e r a c t i o n i s i n t r o d u c e d t o a f r e e system by making  Figure  1.2  K i n e m a t i c o b s e r v a b l e s f o r p i o n s s c a t t e r i n g from n u c l e a r  material.  The p i o n , of mass m, charge e^, i s o s p i n T , momentum nk and a n g u l a r momentum I s c a t t e r s from a n u c l e u s w i t h Z p r o t o n s , N neutrons, t o t a l i s o s p i n t, have d i s t r i b u t i o n s  t o t a l s p i n J . The p r o t o n s and neutrons p+(r)  and p _ ( r ) ,  respectively.  7  minimal s u b s t i t u t i o n : k i n e t i c energy and statement of how  V£ =L-L n t  0  the e l e c t r o m a g n e t i c f i e l d  Since £  has  i n the L a g r a n g i a n L=T-V, where T i s  V the p o t e n t i a l energy of the system.  Note t h a t p_ f ° current j_.  _p_ -> £ - eA  r  appears i n T q u a d r a t i c a l l y , the  terms p r o p o r t i o n a l  i s just a  a f f e c t s a p a r t i c l e ' s motion.  charged p a r t i c l e i s p r o p o r t i o n a l  a  This  to e(j»A) and  to i t s charge  interaction potential  e (A«A). 2  E l a s t i c s c a t t e r i n g of an e l e c t r o n from a n u c l e o n , n e g l e c t i n g c o m p l i c a t i o n s of s p i n , i s e s s e n t i a l l y the particle  ( j ) w i t h the  particle.  s t a t i c vector  the  the  i n t e r a c t i o n of a non-massive  p o t e n t i a l A created  by a massive  T h i s A i s a c t u a l l y t h a t of a v i r t u a l photon whose exchange  mediates the e l e c t r o s t a t i c f o r c e . Note the emphasis h e r e , t h a t  the  exchanged quantum i s not  the  probe.  I n t e r a c t i o n of a f r e e photon w i t h a charged s p i n l e s s p a r t i c l e i s known as 'Compton S c a t t e r i n g w i t h the (2i _f #A  1  , and  i s described  by  the  c l a s s i c a l second o r d e r p e r t u r b a t i o n  HJi*Ai)»  w  i  t  h A  t  n  a  t  0 1  t n e  Minimal C o u p l i n g i n QED might expect the  has  term e (Af»A-f) combined 2  t h e o r y term of the  f r e e photon. proven to be  i n t e r a c t i o n between p i o n s and  current-field interaction.  Yukawa d e s c r i b e d  i s a case analogous to the  a s u c c e s s f u l premise, so  one  nucleons to a l s o appear as  a  s p i n l e s s nucleon-nucleon  s c a t t e r i n g as the exchange of v i r t u a l p i o n s : an a "j]»A" and  form  elastic  i n t e r a c t i o n that looks s c a t t e r i n g of an  like  electron  from a n u c l e o n . P i o n n u c l e o n s c a t t e r i n g i s a p r o c e s s analogous to Compton s c a t t e r i n g , then, and  i s a second o r d e r p r o c e s s .  the e f f e c t s of i s o s p i n ) . of c r e a t i o n and  We  write  the  S i n c e <J> i s to be w r i t t e n  a n n i h i l a t i o n o p e r a t o r s and  pion  field  as an  appropriate  as <j> ( i g n o r i n g  expansion i n terms  plane waves,  the  8  o p e r a t o r form of _j s h o u l d i n c l u d e a g r a d i e n t ( V e ~ ^ ^ * current).  We  acknowledge now  r  « velocity «  t h a t the n u c l e o n has s p i n ; the s i m p l e s t  a p p r o p r i a t e c u r r e n t o p e r a t o r i s then:  (£ V) #  [EIS80].  i n t e r a c t i o n , we w i l l a r r i v e a t a s c a t t e r i n g amplitude  In a second  like:  (o»k)(£»k')=k«k'+!£•(kxk') where k and k' are the i n i t i a l momenta.  order  and f i n a l  N o t i c e t h a t k»k'is p r o p o r t i o n a l to the c o s i n e of the  pion  scattering  angle and i s hence, i n the u s u a l t e r m i n o l o g y , a p-wave i n t e r a c t i o n . The  s c a t t e r i n g amplitude i s , t o f i r s t  perturbation theory matrix  the  classical  element:  <«i' |v(r)|'i' > f  order, just  i  = / V ( r ) exp(i£«r) dr = V ( )  where V ( r ) i s the i n t e r a c t i o n p o t e n t i a l , ^ ± f are the i n i t i a l y  final just  s t a t e s , and aj=k_-k'.  where we  and  Hence the p i o n n u c l e o n s c a t t e r i n g amplitude i s  the momentum space p o t e n t i a l and has the f(9)  (1.8)  a  = b(E) + c(E)k«k' +  form:  i d(E) (o»kxk'),  (1.9)  have added an s-wave term b ( E ) , the i n c l u s i o n of which i s r e q u i r e d  by a proper r e l a t i v i s t i c  treatment  [BJ065].  To a r r i v e a t a f i n a l form f o r the TTN s c a t t e r i n g amplitude, we must d e a l w i t h the e f f e c t s of i s o s p i n . neutron-neutron  The  now  symmetry between p r o t o n - p r o t o n and  s t r o n g i n t e r a c t i o n i s i n c o r p o r a t e d i n t o n u c l e a r p h y s i c s by  the i n t r o d u c t i o n of the concept of the i s o t o p i c s p i n of a p a r t i c l e .  Protons  and neutrons a r e c o n s i d e r e d ( a p a r t from e l e c t r o m a g n e t i c n i c e t i e s ) to be i d e n t i c a l p a r t i c l e s , both w i t h an i s o t o p i c s p i n 1/2 the p r o j e c t i o n ( T = ± l / 2 ) of t h a t s p i n . 3  i s o s p i n 1, w i t h t h r e e s t a t e s :  and d i f f e r i n g o n l y i n  S i m i l a r l y , the p i o n i s a p a r t i c l e o f  9  Tf+ : T =+1  (1.10)  TT° : T =  0  (1.11)  IT" : x = - l  (1.12)  3  3  3  We know t h a t under the s t r o n g i n t e r a c t i o n , i s o s p i n and i t s p r o j e c t i o n are conserved q u a n t i t i e s .  Our s c a t t e r i n g a m p l i t u d e must r e f l e c t t h i s f a c t .  U s i n g s t a n d a r d a n g u l a r momentum c o u p l i n g r u l e s have t o t a l i s o s p i n of 3/2  or 1/2;  [EDM74], the TTN system  i t i s easy t o show t h a t f o r t o t a l  may  isospin  I_==T+j_/2 where j_ i s t h e i s o s p i n o p e r a t o r f o r a meson and J_ t h a t f o r t h e nucleon:  project  Ql/2  =  (1-T»T)/3  (1.13)  Q3/2  =  (2+T«T)/3  (1.14)  out s t a t e s w i t h t o t a l i s o s p i n 1/2  and 3/2  respectively.  We may  then  w r i t e the i s o s p i n dependent a m p l i t u d e a s : f - f° - f ^ T « r ) where f° i s a symmetric, isovector.  i s o s c a l a r p a r t and f  1  i s antisymmetric  and  Combining t h i s w i t h e q u a t i o n 1.9 we a r r i v e a t :  f(q)=  b ( E ) + c (E)k»k* + ( b ^ E ) + c j (E) ( k f k / ) ) T * r Q  (1-16)  0  + i (d  1.4.2  (1.15)  0  +  djT^^kxk'  Pion Nuclear Interaction The d i s c u s s i o n  u n t i l now  has c o n s i d e r e d o n l y the i n t e r a c t i o n of a  s i n g l e p i o n on a s i n g l e n u c l e o n .  I n a n u c l e u s ( f i g u r e 1.2)  one might  the p i o n t o i n t e r a c t t h r o u g h a p o t e n t i a l which i s j u s t the sum of due t o i n d i v i d u a l n u c l e o n s .  expect  potentials  The n u c l e u s i s t y p i c a l l y s p i n s a t u r a t e d ( t h e  i n d i v i d u a l n u c l e o n s p i n s c o u p l e t o g i v e a p p r o x i m a t e l y n e t z e r o s p i n ) so we  i g n o r e t h e presence o f t h e n u c l e o n s p i n f u n c t i o n . the i n i t i a l  For e l a s t i c  scattering  and f i n a l n u c l e a r s t a t e s must be i d e n t i c a l .  Including pion-nucleon t o t a l s p i n - i s o s p i n functions, X J » F  i»f  e q u a t i o n 1.8 i s w r i t t e n : f ( 6 ) = < ¥ | x t V ( r ) X |l' >, f  f  i  (1.17)  i  so we w r i t e t h e p i o n n u c l e a r p o t e n t i a l as a f o l d i n g o f t h e c o o r d i n a t e space d e n s i t y w i t h t h e c o o r d i n a t e space n u c l e a r p o t e n t i a l :  U(r) = /  I  t V ( | r - r ' | ) p (r«)  x  x  n  summed over t h e nucleon's a l l o w a b l e e i g e n v a l u e s o f T  3  ±  dr'  (1.18)  ( i e : n = ± 1).  t h a t V ( r ) and p ( r ) a r e w e l l behaved [MES58] t h i s means t h a t  Given  the momentum  space Tv-nuclear p o t e n t i a l i s j u s t : U(q) = E  X  f  t V(q) p (q)  X  (1.19)  ±  Tl  where V(q) and p ( q ) a r e the F o u r i e r t r a n s f o r m s o f V ( r ) and p ( r ) , respectively. Now, f o r a charged p i o n of charge  i n c i d e n t upon a n u c l e o n o f  i s o s p i n p r o j e c t i o n n/2 (n = +1 f o r p r o t o n s and -1 f o r n e u t r o n s ) the TTN i s o s p i n f u n c t i o n i s :  | e  1 n > - - { |e +n | | 3/2,3e „/2  ' T t  2  Tt  >  _ + |e -n | |3/2,e /2 > / /3 1  Tl  ' '  + /2" (e -n) I 1/2, e 12 > / / I Tf  1  TT  }  (1-20)  TT  T h i s i s o b t a i n e d from t h e s t a n d a r d C l e b s c h Gordan d e c o m p o s i t i o n o f t h e mixed i s o s p i n TTN s t a t e s  | TT N > i n t o i s o s p i n 3/2 and 1/2 components. 1  11  Performing  the sum i n e q u a t i o n 1.19, we a r r i v e a t : 1  U(q) = - ( ( + l ) £i  + ( e - 1)2/3 + 2 ^ - l ) / 3 ) p U 2  2  f f  +  Q  4 +  +  +  1 - ( ( e - l ) 2 + Uv+ 4 1 - ( {e + l ) + ( 4 1 - ( (e^- l) + U+ f f  2  £it  2  v  [  U  0  D /3 + 2U 2  1)2/3 - 4 (  D /3  - 4(  2  (p+(q)+p-(q)) +  U  + D /3  )  - D /3  ) p U  2  2  E i i  E 7 r  0  +  x  + 1)2/3 ) p-Uj  (p+(q)-p-(q))  L  p-U  ]  (1.21)  where: u  o  U To a r r i v e a t e q u a t i o n  x  =  C o oii•ii b  + c  ,  ^»  a  n  (1.22)  d  = (bi+Cikfk')  (1.23)  1.21, t h e v a l u e s : 1  (1=3/2)  (1.24)  T»T_ = -2  (1=1/2)  (1.25)  T«T -  have been u s e d . It  i s customary t o make the d e f i n i t i o n s : (1-26)  p(q) = p+(q) + P - ( q ) ,  (1.27)  6p(q) = p_(q) - p+(q) which g i v e s : U(q) = ( b + c k « k ' ) p ( q ) - ( b + Cjkpk')6p(q)e Q  Consider  0  x  f o r a moment the i s o s c a l a r p a r t of t h i s U (q) = ( b + c k«k )p(q). ,  0  Then  Q  0  if  (1.28)  expression: (1.29)  U (q)=(b 0  0  + c k«k')/p(r)exp(iq»r) d r  (1-30)  0  = (b /exp(-ik'«£)p(r)exp(ik«r) dr; 0  c J e x p ( - i k » £ ) ( V « p ( r ) V ) e x p ( i k « r ) d_r  (1.31)  ,  In the second term, the r e s u l t o f a p a r t i a l i n t e g r a t i o n , each g r a d i e n t i s u n d e r s t o o d t o a c t on a l l f u n c t i o n s o f £ t o the r i g h t o f i t . The c o n f i g u r a t i o n space p o t e n t i a l i s then o b t a i n e d f a c t o r s i n the i n t e g r a n d , V f o r the i s o s c a l a r term.  by removing the p l a n e wave  so t h a t :  Q p t  (r)  = b p(r) - c Vep(r)V Q  ,  0  (1.32)  Hence i n c l u d i n g the e x p l i c i t i s o v e c t o r terms o f  e q u a t i o n 1.23 we have a r r i v e d a t : V  opt  =  b  0  p  (  r  )  " 0 °P( ) c  V  r  V  " b  l E 7 r  6p(r) + c  ie7T  V«6p(r)v,  which i s the K i s s l i n g e r form [KIS55] f o r the p i o n n u c l e a r I t i s t h i s p o t e n t i a l which, when m o d i f i e d allows  the s e p a r a t i o n  optical potential.  by some second order  o f the i n t e r a c t i o n ( c h a r a c t e r i s e d by  the d e n s i t y i n f o r m a t i o n  (1.33)  corrections,  b g j C Q . b j j C j )  from  ( p ( r ) , 6 p ( r ) ) i n the n u c l e u s .  1.5 Matter D i s t r i b u t i o n S e n s i t i v i t y of the Ratio The  Born a p p r o x i m a t i o n c r o s s s e c t i o n i s p r o p o r t i o n a l t o the square o f  the momentum space p o t e n t i a l ( e q u a t i o n s  1.17 and 1.19).  I g n o r i n g the  i s o s p i n s e n s i t i v i t y o f t h a t p o t e n t i a l , f o r the moment, we can see t h a t t h e r a t i o o f p i o n e l a s t i c s c a t t e r i n g c r o s s s e c t i o n s on d i f f e r e n t n u c l e i , one o f which i s c o n s i d e r e d  as a r e f e r e n c e , i s : (V(q)p(q))  p (q)  2  2  R =  = (V(q)p(q))  2  (1.34) P (q) 2  r e f  r e f  From t h i s , one expects a r a t i o o f c r o s s s e c t i o n s t o be i n s e n s i t i v e t o the  s i n g l e nucleon p o t e n t i a l d e t a i l s .  The i s o s p i n d e t a i l s , though, mix p+ p_  and V, so t h a t t h e f a c t o r i s a t i o n o f t h e p o t e n t i a l i s n o t t o t a l . We choose ^ k - k ' , so t h e F o u r i e r t r a n s f o r m p(q) = p ( k - k ' ) <* /" s i n ( q r ) q 0 - /" 0  r  {  2  I ((") n=0  p(r) r dr  - 1  / (2n+l)!)p(r)  n  <* 1 - q <r >/6 + q'*<r »>/120 2  =  2  1  1 - q <r >/6 2  of the nuclear density i s :  } dr  • • •  (dimensionless)  2  (1.35) (1.36)  E q u a t i o n 1.36 assumes, o f c o u r s e , t h a t t h e c o r r e c t i o n terms o f equation  1.35 a r e s m a l l : q  <r' >  2  t  «  20 <r > 2  (1.37)  To o b t a i n some i d e a o f t h e c o n s t r a i n t , we use a hard sphere d i s t r i b u t i o n w i t h moments: <r > = ^  (1.38)  n  and  radius R = 1.45 A  1 / 3  (1.39)  The r e s u l t i s : k = A  _ 1 / 3  (1.40)  or: T  = 25 MeV  (1.41)  u  E q u a t i o n 1.18 g i v e s t h e r e s u l t i n g c o n s t r a i n t : t h e s e n s i t i v i t y o f t h e r a t i o  i s e x c l u s i v e t o the RMS r a d i u s o n l y a t low energy ( a r e s u l t which I s n o t surprising). If positive  one c o n s i d e r s the r a t i o of the low energy e l a s t i c s c a t t e r i n g of ( n e g a t i v e ) p i o n s from i s o t o n e s ( i s o t o p e s ) , then i t i s apparent t h a t  the r a t i o i s p r i m a r i l y dependent upon the RMS r a d i i of the p r o t o n (neutron) distributions.  I n p a r t i c u l a r , t h e r e l a t i v e RMS r a d i i o f t h e d i s t r i b u t i o n s  determine the r a t i o n o r m a l i s a t i o n . relative  Measuring  the r a t i o then determines the  size.  The q u e s t i o n a r i s e s as t o whether the s e n s i t i v i t y i n i t s e l f a l l o w s a b s o l u t e measure of the n u c l e a r matter r a d i i .  I t i s t h e l a r g e degree t o  which the ir n u c l e a r p o t e n t i a l f a c t o r s i n e q u a t i o n 1.34 t h a t insensitivity  to the p r e c i s e n a t u r e of t h a t p o t e n t i a l .  implies  Furthermore, g i v e n  n u c l e a r s i z e i n f o r m a t i o n f o r the r e f e r e n c e from some independent s o u r c e , the p o t e n t i a l V ( q ) may be o p t i m i s e d t o d e s c r i b e the r e f e r e n c e n u c l e u s and hence n u c l e i w i t h n e a r l y the same s t r u c t u r e .  Attempts  to extract absolute nuclear  s i z e d i r e c t l y from the p i o n s c a t t e r i n g e l a s t i c c r o s s s e c t i o n s would  suffer  on both c o u n t s .  1.6 Previous Isoratio Experiments In "Neutron R a d i i D e t e r m i n a t i o n s from the R a t i o o f TT" E l a s t i c S c a t t e r i n g from  1 2  C>  1 3  C and 16,18 0  [JOH79,GYL79] the i s o r a t i o method  has been used t o determine the s i z e s of respectively.  1 3  C and  1 8  0 r e l a t i v e to  1 2  C and  1 6  0  The n e u t r o n r a d i u s was found t o be e s s e n t i a l l y independent o f  p o t e n t i a l model and i n s e n s i t i v e t o u n c e r t a i n t i e s i n the a s s o c i a t e d p r o t o n distributions.  In f i g u r e 1.3 the r e s u l t s o f t h i s experiment f o r the carbon  case a t 29 MeV a r e i l l u s t r a t e d ,  a l o n g w i t h an o p t i c a l model c a l c u l a t i o n  15  1.9 1.8 1.7 -  b'  -  1.6  CM  b' 1.5 1.4 1.3 O  1.2 I.I  -  1.0  -  20  40  60  80  100  120  140  160  eC M . Figure  The  1.3  r e s u l t s of TT~ s c a t t e r i n g experiments (29 MeV)  yield  r e l a t i v e neutron  differential  radii.  12,a-  s c a t t e r i n g cross section [GYL79a].  (  1 3  on  o _ ) i s the  f o r pions on  1 2  isotopes elastic C  ( C) 1 3  [GYL79a].  The v a l u e of A r _ =  1 3  r_-  1 2  r _ was found to be 0 . 0 4 ± 0 . 0 3 f m .  the c h a r a c t e r i s t i c s t r u c t u r e o c c u r i n g near 9 0 ° ; between i s o t o p e s of an s - p p a r t i a l - w a v e  this  Notice  r e s u l t s from a  shift  i n t e r f e r e n c e minimum.  More r e c e n t experiments w i t h n e g a t i v e p i o n s on i s o t o p e s of magnesium and  sulfur  [GYL85] have extended the technique i n t o a r e g i o n where  diffractive  e f f e c t s must be c o n s i d e r e d c a r e f u l l y i n the e x t r a c t i o n  relative size information,  of  but where comparison w i t h the r e s u l t s of n u c l e a r  s t r u c t u r e c a l c u l a t i o n s such as those u s i n g m o d i f i e d H a r t r e e Fock t e c h n i q u e s become more  1.7  tenable.  Current Objectives In o r d e r to v e r i f y  the r e l i a b i l i t y  determining nuclear s i z e ,  0  with p o s i t i v e pion r a t i o s  1 1  B, ' N, 1  +  1 8  [BAR80, BAR85].  method, of c o u r s e , had o r i g i n a l l y been i n i t s radii,  but i t  as w e l l .  soon became apparent  Furthermore,  immediately  0 r e l a t i v e to those of The main i n t e r e s t ability  in  I 2  C and  the  to measure neutron  that p r o t o n r a d i u s measurement was v i a b l e  r e s u l t s f o r the p r o t o n d i s t r i b u t i o n s of n u c l e i are  comparable to the more common e l e c t r o n s c a t t e r i n g  p r e v i o u s l y mentioned.  in  a s e r i e s of experiments were proposed to measure  the r e l a t i v e p r o t o n matter s i z e s of i 6  of the i s o r a t i o method  The remainder of t h i s  d e s c r i p t i o n of these experiments and t h e i r  results  t h e s i s i s devoted to a  detailed  analysis.  1.8 Summary This,  the i n t r o d u c t o r y c h a p t e r ,  r a t i o experiment,  in light  s t a r t e d w i t h an overview of the T T  of the h i s t o r y of n u c l e a r s i z e measurement.  pion-nuclear o p t i c a l p o t e n t i a l ,  +  The  which i s the b a s i s of e x t r a c t i o n of matter  17  distribution difference differential  i n f o r m a t i o n from the r a t i o  c r o s s s e c t i o n s , was examined.  of T T e l a s t i c +  Background f o r  the v a l i d i t y  the method was d i s c u s s e d i n terms of the i s o s p i n dependence of the and the enhanced s e n s i t i v i t y  of  potential  of p o s i t i v e p i o n s to n u c l e a r p r o t o n m a t t e r  distributions. In Chapter II,  a d i s c u s s i o n of the e x p e r i m e n t a l  setups used to  elastic  s c a t t e r i n g data on f i v e n u c l e i at a s e l e c t i o n of t h r e e  given.  Chapter III  then d e t a i l s  subsequently the e x t r a c t i o n  collect  energies  the a n a l y s i s of the data s e t s and  and c o r r e c t i o n of r a t i o s  and c r o s s s e c t i o n s .  Chapter IV f o l l o w s w i t h d e s c r i p t i o n of the o p t i c a l p o t e n t i a l  a n a l y s i s of  ratio  Discussion  data to e x t r a c t  relative  nuclear density information.  about p o s s i b l e weaknesses i n the a n a l y s e s induced by the l i m i t a t i o n s optical potential these e x p e r i m e n t s .  is included.  is  of  the  the  Chapter V p r o v i d e s overview and summary of  Comparison of the r e s u l t s of these experiments  obtained through o t h e r methods i s made.  to  those  Chapter II Experimental Details  2.1  Introduction The experiments i n c l u d e d i n t h i s work f a l l n a t u r a l l y i n t o two  categories:  1 2  C , B and C , N, 0 , n  1 2  11+  1 6  1 8  0 experiments.  This d i v i s i o n i s  suggested by d i f f e r e n c e s i n t h e e x p e r i m e n t a l t e c h n i q u e s and i n t h e times a t w h i c h t h e e x p e r i m e n t s were performed. In  t h i s c h a p t e r , we w i l l d i s c u s s t h e e x p e r i m e n t a l d e t a i l s : a p p a r a t u s ,  hardware and f a c i l i t i e s , w h i c h were used t o c o l l e c t t h e d a t a .  After dealing  w i t h some g e n e r a l d e t a i l s common t o a l l o f the e x p e r i m e n t s , we w i l l d e s c r i b e t h e ^ C , ^ B experiment which was t h e f i r s t g e n e r a t i o n T T r a t i o 2  experiment.  +  The r e s u l t s o f t h i s experiment have been summarised  F o l l o w i n g t h i s , we w i l l proceed t o d e s c r i b e t h e C , 1 2  as summarised  1 4  N,  1 6  0,  1 8  i n [BAR80].  0 experiments,  i n [BAR85].  2.1.1 Machine Details A l l o f t h e measurements were performed u s i n g t h e low energy i r beams +  a v a i l a b l e from the TRIUMF 500 MeV i s o c h r o n o u s c y c l o t r o n [RIC63] l o c a t e d on the  campus o f The U n i v e r s i t y o f B r i t i s h Columbia.  P r e s e n t l y , two c h a n n e l s ,  d e s i g n a t e d M i l [STI80] and M13 [0RA81], p r o v i d e s u i t a b l e beams.  These  c h a n n e l s b o t h a r e l o c a t e d a t t h e T l p r o d u c t i o n t a r g e t on t h e 1A beamline i n the  meson h a l l a t TRIUMF. M13 p r o v i d e s p i o n s o f energy up t o 50 MeV.  T h i s c h a n n e l ( f i g u r e 2.1a)  i s 9.5 meters i n l e n g t h and p r o v i d e s f l u x e s o f about 5 x 1 0  5  p a r t i c l e s per  second ( a c h r o m a t i c a l l y tuned) f o r a momentum r e s o l u t i o n o f 1% ( s e l e c t e d by  t >86  5  * CALCULATED VALUE o MEASUREMENT WITH 2mm CARBON o MEASUREMENT WITH Icm CARBON 5 Fl  HORIZONTAL  10 SLIT  15 WIDTH  cm  F i g u r e 2.1  a) Channel c o n f i g u r a t i o n and b) Momentum r e s o l u t i o n o f M13  [0RA81].  the s e t t i n g of momentum l i m i t i n g primary beam c u r r e n t .  per 10 uA  M i l , on the o t h e r hand, p r o v i d e s p i o n s of energy  g r e a t e r than about 65 MeV, decreased.  s l i t s a t F l and F2, f i g u r e 2.1b)  w i t h f l u x e s dropping r a p i d l y as energy i s  M i l has a forward t a k e o f f  beam) w h i l e the lower energy M13  ( a t an acute a n g l e to the primary  has a backward t a k e o f f .  Figure  2.2  compares the a v a i l a b l e f l u x e s on the two channels as a f u n c t i o n of energy. In a d d i t i o n to p i o n s , muons and e l e c t r o n s ( o f both charges) are produced at T l .  The secondary channels s e l e c t p a r t i c l e s of g i v e n momentum  and charge, so that muons and e l e c t r o n s are a l s o t r a n s p o r t e d to the secondary t a r g e t l o c a t i o n .  Such p a r t i c l e s , however, have d i f f e r i n g  masses,  hence v e l o c i t i e s , and so r e q u i r e d i f f e r e n t amounts of time to t r a v e l the p r o d u c t i o n t a r g e t  2.2  l2  Carbon, Boron 11  The channel.  12  C_, B_ 11  to the secondary t a r g e t  from  location.  (CB) Experiment  (CB) experiments were performed e x c l u s i v e l y on the  Data were taken at two e n e r g i e s .  M13  The f i n a l quoted energy i s the  energy of p i o n s momentum s e l e c t e d a t the second bending element, B2, a l l o w s f o r the energy l o s t  i n a r r i v i n g at the t a r g e t c e n t e r ( t a b l e  but  2.1).  The d i f f e r e n c e i n B l and B2 momenta i s caused by the presence of ~0.2  g/cm  2  p o l y e t h y l e n e i n s e r t e d a t F l to e l i m i n a t e p r o t o n c o n t a m i n a t i o n from the beam. Time Of F l i g h t  (TOF) s p e c t r a are shown i n f i g u r e 2.3.  r e c o r d e d on a L e c r o y 4001  These  qvt w i t h a p l a s t i c s c i n t i l l a t o r  s p e c t r a were  l o c a t e d at the  t a r g e t p o s i t i o n and show p a r t i c l e c o n t a m i n a t i o n s of Tf: y : e=348:15:1 at MeV  and 2500:62:4 a t 47.7  MeV.  38.6  Beam p r o f i l e s f o r each channel tune,  r e c o r d e d w i t h a s i n g l e M u l t i Wire P r o p o r t i o n a l Chamber (MWPC) at the t a r g e t p o s i t i o n , are shown i n f i g u r e 2.4.  These chambers c o n s i s t  of anode and  10  J  -i  r-  CM  E o /  M  i  l  Q.  < <  M13 10*  w  50  J  100  1  i  150  • i  200  MOMENTUM (MeV /  Figure  Comparison of f l u x e s a v a i l a b l e  250  300  c)  2.2  from meson channels M i l and  M13.  J  Table 2.1 Determination of B e a m Energies at the Target Center  Data Set  50 MeV  B l Momentum  40 MeV  126.90  112.91  126.12  112.26  48.54  39.54  47.7  38.6  ± 0.23  ± 0.15  (MeV/c)  B2 Momentum (MeV/c)  Pion  KE, p o s t B2  (MeV)  P i o n KE, t a r g e t c e n t e r (MeV)  Uncertainty  i n pion  KE ( t a r g e t  center)  (MeV)  23  38.6 MeV  J  u. 47.7 MeV  I  1 = 10 n 8  Figure 2.3 M13 TOF s p e c t r a at 40 and 50 MeV.  24  o.  b.  c.  d.  •—• •  Figure  Typical beam p r o f i l e s at 40 M13.  a) and  (a and  I cm  2.4  b) and  50  (c and  c) are horizontal p r o f i l e s , b) and  Structure on the v e r t i c a l p r o f i l e s i s an  d) MeV,  for TT  d) v e r t i c a l . artifact.  +  on  cathode p l a n e s of w i r e s w i t h a p o t e n t i a l of s e v e r a l k i l o v o l t s p l a c e d between them.  When a p a r t i c l e passes between the p l a n e s , a gas avalanche  i s created  and i n d u c e s s i g n a l s on a d e l a y l i n e a t t a c h e d to p l a n e s of sense w i r e s i n each d i r e c t i o n .  The d i f f e r e n c e of the times of a r r i v a l of the r e s u l t i n g  p u l s e s a t the ends of these l i n e s i s p r o p o r t i o n a l t o the d i s t a n c e a l o n g t h a t l i n e a t which the event o c c u r r e d . was  e s t i m a t e d t o be ~2°  Beam d i v e r g e n c e through these chambers  [GYL80, ORA79].  2.2.1 Experimental Setup A s c a t t e r i n g t a b l e (see f i g u r e s 1.1 c e n t e r l o c a t e d 1.35  and 2.5)  was  meters from the l a s t quadrupole  placed with i t s  (Q7).  The  t a b l e was  a l i g n e d beneath the s c a t t e r i n g plane w i t h a t h e o d o l i t e p l a c e d ~5 meters downstream, the t h e o d o l i t e b e i n g p l a c e d w i t h the use of benchmarks on Beam f l u x e s were monitored  Q7.  upstream of the t a r g e t by an i o n i z a t i o n chamber  [SH079, SH079a, SH081] and downstream by two 1.5  mm  t h i c k by 10 cm square  s c i n t i l l a t i o n counters set i n coincidence.  2.2.2  Scattering Telescopes S c a t t e r e d p i o n s were d e t e c t e d i n two three-element  stopping  t e l e s c o p e s ( d e s i g n a t e d ARMO and ARM1), i d e n t i c a l i n c o n f i g u r a t i o n to one a n o t h e r , but s l i g h t l y d i f f e r e n t i n s o l i d a n g l e . t e l e s c o p e was  a 1 mm  The  f i r s t element of each  t h i c k s c i n t i l l a t i o n p a s s i n g counter (NE102) [NUC80], 5  cm I n d i a m e t e r , l o c a t e d 15 cm from the t a r g e t a x i s .  These c o u n t e r s ,  d e s i g n e d t o l i m i t the f i e l d of v i e w of the s c a t t e r i n g t e l e s c o p e s t o the immediate v i c i n i t y of t h e t a r g e t , d i d not a f f e c t the t e l e s c o p e s o l i d angles.  26  B E A M  \ 2 5  L I N E  S C A T T E R I N G  M I 3  T A B L E  C M  Figure 2.5  S c a t t e r i n g t a b l e arrangement, showing IT s c a t t e r i n g t e l e s c o p e s and beam m o n i t o r i n g  apparatus.  The  second  lithium-drifted  element o f each t e l e s c o p e was a 2 mm t h i c k Kevex [KEV80] silicon  ( S i ( L i ) ) d e t e c t o r w i t h an a c t i v e a r e a of 1250 mm . 2  T h i s d e t e c t o r a c t e d as t h e s o l i d a n g l e d e f i n i t i o n f o r the t e l e s c o p e and p r o v i d e d a AE s i g n a l f o r t h e s c a t t e r e d p a r t i c l e s . 0.5%  I t was c a p a b l e o f about  r e s o l u t i o n when used as a s t o p p i n g counter f o r low energy  but e x h i b i t e d ~30% The  FWHM i n t h i s  particles,  configuration.  f i n a l s t o p p i n g element was a 3 - i n c h diameter by 4 - i n c h deep  Harshaw sodium i o d i d e ( N a l ( T Z ) ) c r y s t a l  [HAR80].  T h i s c r y s t a l was capable  of s t o p p i n g up t o 90 MeV p i o n s o r 175 MeV p r o t o n s . connected  I t s phototube was  t o a p r e a m p l i f i e r which p r o v i d e d an anode s i g n a l as w e l l as a  s i g n a l from t h e t h i r d dynode. to p r o v i d e an energy  The dynode s i g n a l ( r e f e r r e d t o as E) was used  s i g n a l w i t h b e t t e r l i n e a r i t y than t h a t a v a i l a b l e  from  the anode. The  s t o p p i n g t e l e s c o p e s were a l i g n e d i n t h e i r mountings u s i n g the  theodolite.  The S i ( L i ) s were mounted on aluminium, p r o v i d i n g some l a t e r a l  s h i e l d i n g from s t r a y r a d i a t i o n .  P l a s t i c , 20 mg/cm  2  t h i c k on e i t h e r s i d e o f  the c r y s t a l mount, m a i n t a i n e d a l i g h t - f r e e environment.  The N a l  c r y s t a l - p h o t o t u b e a s s e m b l i e s were p r o v i d e d by the manufacturers mu-metal magnetic aluminium  cans.  s h i e l d i n g and .048 cm t h i c k (.025 a t t h e entrance window) A d d i t i o n a l l e a d s h i e l d i n g was p l a c e d around  p r o v i d e some p r o t e c t i o n from s t r a y charged o b s t r u c t i n g t h e e n t r a n c e window. 20 mg/cm  2  with  plastic.  and n e u t r a l p a r t i c l e s  without  The f r o n t p a s s i n g c o u n t e r s were wrapped i n  They were coupled to the standard TRIUMF  a s s e m b l i e s , which p r o v i d e magnetic  the c r y s t a l t o  and l i g h t  shielding.  phototube  Operating Characteristics  2 . 2 . 2 . 1  Some d i s c u s s i o n o f the o p e r a t i n g c h a r a c t e r i s t i c s o f the TT s t o p p i n g +  telescopes i s u s e f u l .  As noted, t h e S i ( L i ) p a s s i n g r e s o l u t i o n i s decreased  markedly over i t s s t o p p i n g r e s o l u t i o n .  F i g u r e 2.6 shows peak h e i g h t and  r e s o l u t i o n f o r these counters i n the s t o p p i n g mode f o r ARMO as o b t a i n e d f o r an  2<+1  Am alpha source.  I t can be seen  w e l l b e f o r e the o p e r a t i n g p o i n t . elastically The  C o n t r a s t t h i s r e s o l u t i o n with t h a t o f  s c a t t e r e d 47.7 MeV p i o n s from  shape o f the AE spectrum  energy  t h a t the d e t e c t o r reaches d e p l e t i o n  1 2  C , about 30% FWHM ( f i g u r e 2 . 7 ) .  h i n t s t h a t Landau s t r a g g l i n g  [LAN44] i n p i o n  l o s s t o t h e S i ( L i ) i s the cause o f t h i s poor r e s o l u t i o n .  are w e l l w i t h i n t h e Landau regime [RIT61];  the expected  In f a c t , we  FWHM i s about 33%,  i n good agreement w i t h our r e s u l t . F i g u r e 2.8 shows a t y p i c a l energy MeV. set  f o r a N a l element a t 47.7  R e s o l u t i o n i s 1.5 MeV FWHM w i t h the channel's momentum s e l e c t i o n a t 10.7 mm.  The s t a t e s suggested  the ground (g.s.) and f i r s t presence  e x c i t e d s t a t e s o f carbon.  F i g u r e 2.9 shows an attempt  l i n e a r i t y I n f o r m a t i o n f o r the N a l s . (i.e.:  the e f f e c t  We n o t i c e t h e poorer  t o e x t r a c t from the 38.6 MeV data  With the e x c e p t i o n o f the p o i n t s marked  r u n 56), a c l e a r t r e n d w i t h a n g l e can be seen. of r e s i d u a l magnetic f i e l d s on the phototubes  (Subsequent experiments  slits  assume the 4.44 MeV s e p a r a t i o n between  o f s c a t t e r i n g from h i g h e r e x c i t a t i o n s t a t e s , a l b e i t with  statistics.  56  spectrum  T h i s I s probably [ENG52].  w i t h these t e l e s c o p e s such as the one shown i n  f i g u r e 1.1 [GIL82] i n c o r p o r a t e d a .64 cm t h i c k s o f t i r o n c y l i n d e r around the Nals t o a v o i d g a i n changes r e s u l t i n g elements.  from f r i n g e f i e l d s o f the beam l i n e  These c y l i n d e r s a l s o p r o v i d e d a d d i t i o n a l r a d i a t i o n s h i e l d i n g f o r  those d e t e c t o r s . )  The e x c i t e d s t a t e g a i n s a r e about 0.8% per MeV l a r g e r  Figure  Peak height and  2.6  resolution for Si(Li)  detectors.  30  200  A  in  o o ZD  200  400 600 CHANNEL NUMBER  Figure  800  2.7  P a s s i n g r e s o l u t i o n of the S i ( L i ) d e t e c t o r s to p i o n s .  gg ID ID O O O )  I + I w a n> n  I  OQ  200  to •a n o rt  l-t  g o II  rt ET (D  25  a.  o  — om  •n +  o  CM  O  III  A  A  OQ  c  ARM I  150 A  n> 00  E  c/>  i-  I  o u  100 4 50 J  •1  420  440  CHANNEL NUMBER  460  480  32  :  *  t •  run 56  • A  30  ARM  0  (G.S.)  ARM  0  (4.44  ARM  I  ( G.S.)  ARM  I  (4.44  1  60  1  MeV) MeV)  1  90  120 0LAB  F i g u r e 2.9  L i n e a r i t y of N a l d e t e c t o r s .  1  150  180  than the g . s . g a i n s , i n d i c a t i n g a n o n l i n e a r i t y o f t h a t o r d e r .  Scattering  t e l e s c o p e e f f i c i e n c i e s a r e d i s c u s s e d i n c h a p t e r 3. Composite i n f o r m a t i o n from the s t o p p i n g t e l e s c o p e s may be d i s p l a y e d i n the form o f E-AE s c a t t e r p l o t s . s c a t t e r p l o t seen i n f i g u r e 3.1.  F i g u r e 2.10 shows a contour  p l o t of the  A l t h o u g h the o v e r a l l s t a t i s t i c s a r e n o t  good, s e v e r a l f e a t u r e s a r e apparent.  These i n c l u d e p r o t o n  and d e u t e r o n  bands, the TT e l a s t i c peak, TT r e a c t i o n s i n the Nal c r y s t a l , a decay muon +  +  cone and p o s i t r o n s r e s u l t i n g from muon decay ( t h e r e i s no TOF c u t h e r e ) .  2.2.3 E l e c t r o n i c s P r e a m p l i f i c a t i o n f o r s i g n a l s from the N a l was p r o v i d e d phototube bases.  The S i ( L i ) s were used i n c o n j u n c t i o n w i t h Canberra 970D  p r e a m p l i f i e r s , which a l s o p r o v i d e d u n i t s provided was d i g i t i s e d adjacent  b i a s through 110  i s o l a t e d energy and t i m i n g o u t p u t s . i n the e x p e r i m e n t a l  counting  i n their  area.  room v i a standard  resistors. Ion Chamber  These  (IC) current  A l l s i g n a l s were then passed t o an  50ft (RG-58/U) BNC c a b l e s .  F i g u r e 2.11 i s a c i r c u i t diagram f o r the TT r a t i o experiment. +  f r o n t end c o n s i s t s of t h r e e p a r t s : ARMO, ARMl, and BEAM MONITORS. monitor stream i n c l u d e s i n p u t from the SI and S2 c o u n t e r s , c a p a c i t i v e probe l o c a t e d i n the primary beam l i n e . b u r s t s of protons  scaler.  The  t h e IC, and a  T h i s probe d e t e c t s  as they move t o the p i o n p r o d u c t i o n  waveform r e f e r r e d to here as RF.  The  t a r g e t and produces a  IC counts a r e s c a l e d on a CAMAC  [BAR69]  The SI and S2 s i g n a l s , p l a c e d i n d i s c r i m i n a t o r s , d e f i n e a  coincidence b i t  ( r e f e r r e d to as 1«2 o r S1«S2).  For each t e l e s c o p e , s i g n a l s f o l l o w two paths. a m p l i f y and a t t e n u a t e  energy s i g n a l s b e f o r e r e a c h i n g  The analogue paths i n t e g r a t i n g ADCs.  Time  D D  OR GATE AND GATE  ( o . INVERSION )  ACTIVE ELEMENT TYPE C  (INHIBIT) C SCALERS"") VISUAL SCALERS (ENABLE)  CONST. FRACT. DISC.  0  -  DELAY AMP  E  -  FANOUT. ANALOGUE  F  -  FANOUT, LOGIC  G  -  GATE GENERATOR  L  -  LOWER LEVEL DISC  P  -  PREAMP  S  -  SPECTROSOPY AMP  T  -  TIMING FILTER AMP  U  -  UPDATING DISC  CAMAC MODULE O > 0  t»  LATCH) (8T0>) (•ENJD  -  CZI2 PATTERN UNIT 2249W ADC TDC SCALER  ATTENUATOR  s  i  VISUAL SCALER INPUT CURRENT INTEGRATOR 50 a TERMINATOR  CAMAC INPUT CHANNELS  36  constants  a r e s e l e c t e d to p r o v i d e moderate r i s e times and  t i m i n g paths proceed Constant  The  F i l t e r A m p l i f i e r s (TFA)  F r a c t i o n D i s c r i m i n a t o r s (CFD)  p a s s i n g counter Timing  through Timing  l o g i c l e v e l s are f a s t  and  then  f o r the c r y s t a l d e t e c t o r s ;  s i g n a l s are passed d i r e c t l y  s i g n a l s then d e f i n e one  short t a i l s .  t h r e e f o l d c o i n c i d e n c e b i t f o r each t e l e s c o p e .  U ' S , and  w i l l a c t i v a t e a Master Strobe  e's.  (MS)  A 1*2  b i t i s time sampled at a  time s i g n a t u r e .  (Look At Me)  nS gate  of these  any level  three  bits  and  to the ADCs, and s i g n a l s  to i n i t i a t e CAMAC r e a d s .  energy i n f o r m a t i o n , c o l l e c t e d by a Data General [FEE79] system, i s p l a c e d on 800  Any  which i n h i b i t s f u r t h e r s t r o b e s  s c a l i n g , s t a r t s CAMAC TDCs, sends a 700  2.2.4  i n an  T h i s c o i n c i d e n c e i s wide enough to accept  i s a l s o g i v e n the RF  the computer w i t h a LAM  the  NIM.  a d d i t i o n a l coincidence unit.  of ~.005% and  through  into level discriminators.  Both t e l e s c o p e b i t s a r e g i v e n the time s i g n a t u r e of the RF  of the beam r e l a t e d I T ' S ,  The  Nova based data  Timing  and  acquisition  BPI magnetic tape f o r l a t e r a n a l y s i s .  Targets T a r g e t s were mounted at the t a b l e c e n t e r w i t h t h e i r normals d i s p l a c e d  i n the s c a t t e r i n g plane from the i n c i d e n t beam d i r e c t i o n . a n g u l a r displacement at d i f f e r e n t a n g l e s . experiments  was  s e l e c t e d to o p t i m i s e the e f f e c t i v e  A technique  [GYL79a] was  The  used.  amount of  target thickness  s i m i l a r to t h a t used i n e a r l i e r  Three t a r g e t s , B , n  1 2  C,  and EMPTY  ratio (MT),  were p l a c e d i n the p i o n beam d u r i n g s u c c e s s i v e runs f o r each p a r t i c u l a r angle.  In t h i s way,  no p o i n t i n the r a t i o s u f f e r e d from the e f f e c t s of l o n g  term v a r i a t i o n s i n the machine performance.  The  i d e n t i c a l low mass s t y r o f o a m - p l a s t i c c o n t a i n e r s .  t a r g e t s were c o n t a i n e d The MT  target provided  in  i n f o r m a t i o n f o r a background s u b t r a c t i o n . The boron t a r g e t was composed o f powdered n a t u r a l boron. verified  by an assay performed by t h e Chalk R i v e r N u c l e a r  Atomic Energy o f Canada L i m i t e d rather large i n area, method was d e v i s e d  [0LI80].  As t h i s  t h i c k n e s s measurement  T h i s was  L a b o r a t o r i e s of  t a r g e t was powder and  required s p e c i a l a t t e n t i o n .  A  [GYL80] t o use the m u l t i p l e s c a t t e r i n g p r o p e r t i e s of  e l e c t r o n s t o measure the boron t a r g e t t h i c k n e s s as a f u n c t i o n o f p o s i t i o n . T h i s method i s d e s c r i b e d was i n c o r p o r a t e d w i t h target  i n some d e t a i l  beam p r o f i l e  i n appendix I I . The r e s u l t a n t data  information  t h i c k n e s s , of 377.6±3.0 mg/cm . 2  The carbon t a r g e t m a t e r i a l was a  p l a t e o f r e a c t o r grade g r a p h i t e w i t h a u n i f o r m This target data  2.3  1 2  C,  1 H  N,  1 6  0,  1 8  t h i c k n e s s 332.6±1.0 mg/cm . 2  i s summarised i n t a b l e 2.2.  0  (CNO ) Experiments 2  The experiments on referred  t o a r r i v e a t a weighted, mean  t o as t h e CNO  Data was c o l l e c t e d  2  1 2  £,  14t  N, 0_, 0_, which w i l l 16  18  f o r t h w i t h be  experiments, were performed a t each o f two e n e r g i e s .  on Ml3 w i t h a 50 MeV tune.  The 65 MeV measurements were  made on M i l , as M13 was n o t a t t h e time a b l e t o p r o v i d e the h i g h e r  energy p i o n s  due t o l i m i t a t i o n s  adequate f l u x e s o f  i n the q u a d r u p o l e s .  The quoted  e n e r g i e s , 48.3 MeV and 62.8 MeV, a r e d e r i v e d from NMR measurements of the bending f i e l d energy l o s t  o f c h a n n e l d i p o l e s , M13B1 and M11B1, but a l l o w f o r the mean  i n travelling  t o the s c a t t e r i n g t a r g e t  Beam c o n s t i t u e n t s were sampled c o n t i n u o u s l y with  centers. during  these  experiments  beam m o n i t o r s BMl and BM2 (see f i g u r e 2.12) s t r a d d l i n g the t a r g e t  position.  The c o n t a m i n a t i o n s  were t y p i c a l l y Tr:u:e = (261:13:4) on M13 a t  48.3 MeV and Tr:y:e = (154:24:17) on M i l a t 62.8 MeV ( f i g u r e 2.13).  Proton  Table  2.2  Isotopic Composition of Scattering Targets  Target  Isotopes  %  Type  Mass (mg/cm ) 2  boron  carbon  UB  80.8  10  B  19.2  12  C  13  C  98.89 1.11  377.6±3.0  powder  332.6±1.0  plate  b) Figure  2.12  E x p e r i m e n t a l setup showing (a) s p e c t r o m e t e r - c h a n n e l on M13  and  (b) enlargement  of  configuration  spectrometer.  o  9  4  to > LU  Z  0  4  25 50 75 BEAM T O F (channels) [50 MeV. RUN  100 7]  I — i — i — i — i — | — i — i — i — i — | — i — i — i — i — | — i — i — i — r  BEAM T O F  (channels)  [65 MeV, RUN 11]  Figure 2.13 P i o n TOF s p e c t r a  of beams from (a) M13 (b) M i l a t 65 MeV.  a t 50 MeV and  c o n t a m i n a t i o n i n the beam was e l i m i n a t e d w i t h p o l y e t h y l e n e absorber p l a c e d a t t h e momentum d i s p e r s e d f o c i o f t h e beamlines.  2.3.1  The QQD Spectrometer The  s c a t t e r e d pions were d e t e c t e d w i t h the use o f the QQD  spectrometer  p l a c e d a t t h e end o f t h e secondary  QQD spectrometer  [SOB84] subtends  c o n s t r u c t e d w i t h t h r e e magnetic  ( M i l o r M13) beamline.  a s o l i d a n g l e o f about  elements:  magnetic The  17 msr, and i s  two quadrupoles, p r o v i d i n g  v e r t i c a l and h o r i z o n t a l f o c u s s i n g , and a d i p o l e to p r o v i d e momentum a n a l y s i s ( f i g u r e 2.12). the a n g u l a r range between  The spectrometer 90° and - 1 3 7 ° .  i s p h y s i c a l l y capable o f spanning A rate l i m i t a t i o n i n the f r o n t  end of t h e spectrometer c u r r e n t l y l i m i t s u s e f u l angles t o those g r e a t e r than about  2 5 ° . At a n g l e s l e s s than about  30° d i r e c t beam and beam r e l a t e d muons  s c a t t e r i n t o the e n t r a n c e to the spectrometer, c a u s i n g v o l t a g e breakdown and l o s s o f e f f i c i e n c y i n the MWPCs l o c a t e d t h e r e .  These MWPCs, d e s i g n a t e d WC1  and WC3, measure the t r a j e c t o r y o f p a r t i c l e s en r o u t e , and a l l o w r a y traceback to the target plane.  A t t h e back end o f the s p e c t r o m e t e r ,  larger  chambers WC4 and WC5 measure the e x i t t r a j e c t o r y o f p a r t i c l e s e x i t i n g the d e v i c e , thereby d e f i n i n g the p a r t i c l e s ' momenta. E3, when f i r i n g t o g e t h e r , s i g n a l a s c a t t e r i n g  Counters BM1, E l , E2 and  event.  Beam f l u x e s were measured by s c i n t i l l a t o r c o u n t e r s BM1 and BM2, p l a c e d i n coincidence. two  In a d d i t i o n , r e l a t i v e f l u x m o n i t o r i n g was p r o v i d e d by a  element muon counter  [WAD76], yl»u.2, p l a c e d upstream o f the t a r g e t .  T h i s t e l e s c o p e was a l i g n e d above and a t a n g l e o f about  9° t o the beam l i n e  to d e t e c t muons d e c a y i n g from beam p i o n s , and hence i n d i r e c t l y measure t h e incoming p i o n f l u x .  The a n g l e chosen  i s w e l l l e s s than the maximum  k i n e m a t i c a l l y allowed  f o r n->u  decay (17.9° at 50 MeV  and  15.3°  see appendix I I I f o r a d i s c u s s i o n of muon decay k i n e m a t i c s ) . f l u x was  measured by a Rate R e d u c t i o n  the t a r g e t .  T h i s monitor was  M o n i t o r (RRM)  s e n s i t i v e to the passage of charged  2.3.2  MeV;  L a s t l y , the  l o c a t e d downstream of  c o n s t r u c t e d by p l a c i n g d i s k s of  i n t o a p l e x i g l a s l i g h t g u i d e so t h a t o n l y 10%  at 65  scintillator  of the d e t e c t o r a r e a  was  particles.  E l e c t r o n i c Logic F i g u r e 2.14  an e v e n t .  summarises the e l e c t r o n i c l o g i c used to d e f i n e and  record  There were i n p u t s i g n a l s from each of the s c i n t i l l a t o r s i n f i g u r e  2.12b.  The  three E counters,  event.  E l and E2 were l a r g e r e c t a n g u l a r counters  phototubes at e i t h e r end;  along w i t h a BM1  event, d e f i n e d a spectrometer with  l i g h t - g u i d e s and  t h i s construction increased detection e f f i c i e n c y  and minimised the e f f e c t s of l i g h t a t t e n u a t i o n on energy measurement. S i g n a l s from these counters  passed through mean timers  time s i g n a t u r e f o r each e v e n t . 65 MeV  experiment had  T h i s was  The LAM,  output  a l s o the case w i t h E3, which i n the  the c o n f i g u r a t i o n shown.  s i n g l e p h o t o t u b e - s c i n t i l l a t o r was  open ADC  gates.  In the 50 MeV  experiment, a  used i n p l a c e of the E3 shown here.  of the El«E2«E3 c o i n c i d e n c e was  s t a r t TDCs and  to g i v e a p r e c i s e  LAM  anded w i t h BM1  s i g n a l l e d the data  computer to i n i t i a t e an i n t e r r u p t sequence.  to c r e a t e a  acquisition  Subsequently, " s t o p s " a r r i v e d  at the CAMAC TDCs, g i v i n g event time i n f o r m a t i o n and MWPC p o s i t i o n a l information. and  The  computer then read the CAMAC ADCs and TDCs.  event r e l a t e d q u a n t i t i e s such as BM1»BM2 and  s c a l e r s were gated There was  V a r i o u s beam  yl«u2 were s c a l e d ;  those  o f f d u r i n g the i n t e r r u p t s e r v i c i n g p r o c e s s .  a second source  of LAMs.  At r e g u l a r i n t e r v a l s BM1»BM2  was  E1R  -00  E1L -00  E2R  -co  r  E2L  -00  E3R< E3R-  ^_  Enable (Monuol)  -00  12  LAM  -00  E3L*  ADCAOC Stort  -00  E3L-00  MWPC _ xjxxiy xx iv) <  o D E> • o Discriminator pi L J  OR  AND  Mean Timer  Gate Generator  Scalers (Visual and CAMAC) 00 CAMAC ADC  V CZ ? Pattern Unit O t T1 Capocitive Probe 1  Logical not  Linear Output  CAMAC TDC Stop  F i g u r e 2.14  QQD  spectrometer e l e c t r o n i c  logics  44  a l l o w e d t o generate  a "beam sample" event  strobe.  The t i m i n g of such  e v e n t s , r e l a t i v e t o protons a r r i v i n g a t t a r g e t 1AT1, was r e c o r d e d t o a l l o w m o n i t o r i n g o f t h e beam contaminants  (u,e) i n the p i o n beam.  The sample  i n t e r v a l was s e l e c t e d t o g i v e adequate (not overwhelming) s t a t i s t i c s . type o f event g e n e r a t i n g the s t r o b e was recorded as a b i t - f l a g  The  i n a C212 b i t  pattern unit. The  d a t a a c q u i s i t i o n system DA, [MIL84] r a n on a PDP11/34 o p e r a t i n g  under RSX-llM.  T h i s system s e r v i c e d t h e CAMAC i n t e r r u p t s and managed  b u f f e r of approximate l e n g t h 8K b y t e s . event  When t h i s b u f f e r was f i l l e d  a data  with  i n f o r m a t i o n a tape w r i t e f o l l o w e d by a CAMAC s c a l e r read was  initiated.  The data was w r i t t e n a t 1600 BPI i n a n t i c i p a t i o n of o f f - l i n e  analysis. [FER79],  O n - l i n e d a t a m o n i t o r i n g and a n a l y s i s was performed  w i t h MULTI  running c o n c u r r e n t l y .  2.3.3 T a r g e t s T a r g e t s were mounted on a t a r g e t l a d d e r [GYL84] p l a c e d w i t h i n a s c a t t e r i n g chamber a t t h e f r o n t end o f the spectrometer. were chosen t o minimise in particular, spectrometer The  energy  The t a r g e t angles  s t r a g g l i n g e f f e c t s on t h e energy  resolution;  t h e t a r g e t normal angles were chosen t o b i s e c t the  measurement a n g l e s .  t a r g e t s were p l a c e d i n the beam d u r i n g s u c c e s s i v e r u n s .  Measurement times were kept s h o r t and m u l t i p l e runs performed, to minimise  as necessary,  the e f f e c t s of long term s y s t e m a t i c s on the r e s u l t s .  Between  runs a t a g i v e n a n g l e s e t t i n g , o n l y t h e v e r t i c a l t a r g e t l a d d e r p o s i t i o n was changed, and t h i s without  access t o the e x p e r i m e n t a l a r e a , a g a i n a v o i d i n g  the p o s s i b i l i t y of e f f e c t s due t o s y s t e m a t i c changes ( e g . r e s u l t i n g  from  magnet h y s t e r e s i s ) i n the channel The  tune.  t a r g e t c h a r a c t e r i s t i c s a r e summarised i n t a b l e 2.3.  c o n t a i n e d i n an i d e n t i c a l frame.  The t a r g e t support  frames were aluminum.  The windows were composed of 50 urn Kapton glued under t e n s i o n ; p r o v i d e d the r i g i d i t y plane.  Each was  this  r e q u i r e d to m a i n t a i n u n i f o r m i t y a c r o s s the t a r g e t  Aluminium f o i l of t h i c k n e s s 12.5 ym was p l a c e d over the Kapton t o  prevent e v a p o r a t i o n from the water t a r g e t s and i s o t o p i c exchange i n the case of  the 0 t a r g e t . 1 8  F i g u r e 2.15 shows the dimensions  r e l a t i o n t o the beamspot The  1 6  w i t h agar; important The  0 and  1 8  of the t a r g e t frames i n  profile.  0 t a r g e t s were made from water combined 1.5% by weight  the r e s u l t i n g g e l I s s e l f s u p p o r t i n g .  The  1 8  0 composition  i s an  q u a n t i t y , the v a l u e a f f e c t s the c r o s s s e c t i o n r a t i o n o r m a l i s a t i o n .  c o m p o s i t i o n was measured i n d e p e n d e n t l y  consistent determinations. comparison.  The  [OBE66], a white  1Lf  by t h r e e l a b o r a t o r i e s t o a s s u r e  Those compositions  a r e g i v e n i n t a b l e 2.4 f o r  N t a r g e t was c o n s t r u c t e d from ammonium a z i d e (NH N ) tf  c r y s t a l l i n e powder.  The  1 2  3  C was r e a c t o r grade p l a t e  graphite. The  a c t u a l c o n s t r u c t i o n techniques used  v a r i e d from t a r g e t to t a r g e t .  Agar i s a white powder when purchased and  w i l l d i s s o l v e i n water which has been heated mixture  i n the t a r g e t manufacture  to near b o i l i n g .  i s a l l o w e d t o c o o l , the g e l becomes more v i s c o u s .  As the  T h i s g e l must be  p l a c e d i n t o the t a r g e t frame b e f o r e i t s o l i d i f i e s and w h i l e i t i s s t i l l warm enough t o f l o w .  Care must be taken, on the o t h e r hand, to not i n s e r t  m a t e r i a l i n t o a frame w h i l e t o o hot, l e s t window d i s t o r t i o n should The water t a r g e t s were manufactured by f i r s t frame and subsequently  filling  occur.  p l a c i n g windows on the  through a s m a l l h o l e i n the frame.  This hole  46  Table 2.3 CNO Experiment Target Summary 2  Target  Mass  Type  (mg/cm ) 2  18  361±3  0  Enriched  317±3  0  (306)  1<*  N  249±3 (248)  1 8 2  0  1.5% agar g e l  (355)t  16  H  Natural,  demineralised  distilled  H 0. 1.5% agar g e l 2  NH^Nj, ammonium a z i d e . Natural isotopic  content.  White c r y s t a l l i n e powder.  1 2  C  333±1  Reactor  grade, g r a p h i t e  Natural isotopic  MT  = 17  plate.  composition.  I d e n t i c a l frame and windows to those of above  t Numbers i n p a r e n t h e s i s p e r t a i n  to 62.8 MeV experiment.  o  I  1  I  E  (i:g*l) 31IJ0dd A F i g u r e 2.15  Beamspot p r o f i l e a t the t a r g e t frame a t 65 MeV and 120 degrees (RUN1116).  Contours show i n c r e a s e s of the maximum.  i n i n t e n s i t y of 10%  Table 2.4 Isotopic Composition of 0 1 8  Target Material  H  2  H  2  H  2  AECL  LASL  95.0  95.3  MPI  1 8  0  1 7  0  2.57  2.8  —  1 6  0  2.52  1.9  6.2  +  Methods:  93.8  .1  1.2  AECL: D i r e c t  reduction  with chlorine  f l u o r i d e . Gas mass  tripento-  spectroscopy.[CR083]  LASL: Chemical E q u i l i b r a t i o n w i t h C 0 . 2  Gas  MPI:  mass s p e c t r o s c o p y . [CAP83]  High r e s o l u t i o n mass s p e c t r o s c o p y of water vapour. [WIE83]  was  plugged w i t h epoxy a f t e r f i l l i n g .  was  completed  The  powder t a r g e t , on the o t h e r hand,  window by window, the f i l l i n g  t h a t of a sandwich.  P r e s s u r e was  b e i n g p l a c e d i n t o the frame l i k e  a p p l i e d to the m a t e r i a l b e f o r e mounting  the second window to ensure u n i f o r m i t y i n the t a r g e t d e n s i t y and n e a r - c l o s e s t packing. assemble; mounted  The  the m a t e r i a l was  p l a t e g r a p h i t e t a r g e t was  the e a s i e s t  target  to  g l u e d by i t s edges t o a frame and windows  subsequently.  2.4 Summary The  e x p e r i m e n t a l techniques used  described.  These experiments  h i g h q u a l i t y and The  Two  s i g n i f i c a n t range Finally,  have been  of the meson channels  The channels p r o v i d e d low energy  p i o n beams  structure.  and setups have been d e s c r i b e d , a l o n g w i t h  types of d e t e c t o r were used:  3 element magnetic  on two  f l u x , and of a c o n v e n i e n t time  d e t e c t i o n apparatus  t h e i r use.  +  were performed  l o c a t e d at the TRIUMF f a c i l i t y . of  i n the TT r a t i o experiments  spectrometer.  The  the TT s t o p p i n g t e l e s c o p e and +  experiments  thereby covered a  i n measurement t e c h n i q u e .  the d e s i g n and manufacture of the s c a t t e r i n g t a r g e t s has been  discussed.  The  experiment,  as a c r o s s s e c t i o n measurement v a r i e s i n v e r s e l y w i t h the  thickness.  a  t a r g e t q u a l i t y i s an important  f e a t u r e of any  scattering target  50  Chapter I I I Analysis 3.1 Introduction The  t h e o r e t i c a l and e x p e r i m e n t a l background t o t h e TT r a t i o +  experiments  which a r e t h e focus o f t h i s work have been p r e s e n t e d .  c h a p t e r , f a c e t s o f t h e a n a l y s i s o t h e r than those concerned i n t e r p r e t a t i o n o f t h e experiment  w i t h the  a r e d e s c r i b e d and q u e s t i o n s  concerning  c o r r e c t i o n s t o t h e data and measurement e f f i c i e n c i e s a r e a l s o d e a l t The  In t h i s  with.  e v a l u a t i o n o f s c a t t e r i n g c r o s s s e c t i o n s i s d e s c r i b e d as i s t h a t o f t h e i r  ratios.  P a r t i c u l a r c a r e I s taken i n t h e e v a l u a t i o n o f t h e c r o s s s e c t i o n  ratios.  These a r e used  matter  i n chapter IV to p r o v i d e a measure o f t h e p r o t o n  d e n s i t y d i f f e r e n c e s between n u c l e i . As i n t h e l a s t c h a p t e r , t h e CB and CNO  2  experiments  are dealt  with  individually.  3.2 C , B Experiment 1 2  1 1  The a n a l y s i s o f t h e e x p e r i m e n t a l d a t a proceeded  i n several steps.  The  d a t a tapes were t r a n s p o r t e d from t h e e x p e r i m e n t a l s i t e t o t h e U n i v e r s i t y o f B r i t i s h Columbia Computing Center.  A f t e r t r a n s f e r r i n g t h e data t o h i g h e r  d e n s i t y t a p e s , i t was t r a n s l a t e d t o a format based  system and p l a c e d i n f i l e s  compatible w i t h t h e Amdahl  s t o r e d on a d d i t i o n a l t a p e s .  the a n a l y s i s was subsequently performed  The b u l k o f  w i t h FIOWA [HAY79] ( F o r t r a n Input  Output Working A r r a y s ) , a package which a l l o w s f l e x i b l e one and two d i m e n s i o n a l b i n n i n g of d a t a .  3.2.1  Preliminary Analysis: Software Cuts V a r i o u s s o f t w a r e c u t s were a p p l i e d t o e l i m i n a t e i n e l a s t i c events and  background cut;  from the a n a l y s i s .  The TOF s i g n a t u r e o f an event i s one such  i t a l l o w s t h e e l i m i n a t i o n o f s c a t t e r i n g events a s s o c i a t e d w i t h beam  p a r t i c l e s of p i o n momentum but non-pion mass. valuable with fast s c i n t i l l a t o r s .  T h i s technique i s very  The o v e r a l l beam c o m p o s i t i o n f o r TT  +  ( f i g u r e 2.3) was such t h a t TOF c u t s were n o n - e s s e n t i a l . TOF c u t s were used i n the ARMO a n a l y s i s o n l y . More important i n the a n a l y s i s were P a r t i c l e I D e n t i f i c a t i o n (PID) c u t s utilising loss.  t h e Bethe-Bloch  [BET53, NOR79, HEC69] formula f o r s p e c i f i c  Such c u t s e l i m i n a t e p a r t i c l e s o f mass  energy  Ml<M <M2 and a r e o f the 7r  f u n c t i o n a l form f(E,AE,£)=0 , where SL i s the p a s s i n g counter t h i c k n e s s . F o r example, i n the regime  i n which: dE -cE°  dx  (3.1)  ( t h a t i s , p a r t i c l e e n e r g y « minimum i o n i z a t i o n e n e r g y ) , i t f o l l o w s (E+AE) " 1  In  01  the p r e s e n t experiment  - E " 1  0 1  =  c(l-a)£  that: (3.2)  AE d i d n o t change r a d i c a l l y a c r o s s the w i d t h  i n E o f the e l a s t i c p i o n peak, so t h a t a cut on the LHS o f e q u a t i o n 3.2 would have o f f e r e d no advantages  over c u t t i n g on AE a l o n e .  In practice,  then, the PID c u t was i n the form o f a AE cut a p p l i e d through the use o f computer s o f t w a r e . r e s u l t i n g energy for  A second, o r E c u t , was then a p p l i e d by hand t o t h e  (E) h i s t o g r a m s .  Care was taken t h a t the c u t s were the same  each o f t h e t a r g e t s and t h a t t h e r e was no v a r i a t i o n i n t h e i r  efficiency  with angle.  In p a r t i c u l a r , the l o c a t i o n o f the c u t allowed  differences i n particles  s c a t t e r e d from d i f f e r e n t  f o r kinematic  targets at different  angles. F i g u r e s 3.1 through 3.4 show a p r o g r e s s i o n o f computer binned  outputs.  In f i g u r e 3.1 we see an uncut E-AE s c a t t e r p l o t f o r t h e  47.7 MeV TT , C data a t a l a b angle o f 9 0 ° . +  1 2  AE h i s t o g r a m .  generated  F i g u r e 3.2 i s the accompanying  A f t e r a AE c u t , f i g u r e 3.3 r e s u l t s , t h i s l a s t  having  been c u t by hand t o o b t a i n the number of events  peak.  A AE h i s t o g r a m  histogram  i n the e l a s t i c  pion  w i t h no c u t s but the f i n a l E c u t s i s shown i n f i g u r e  3.4 and v e r i f i e s t h a t the AE c u t shown i n f i g u r e 3.2 e l i m i n a t e s few elastically  scattered pions.  3.2.2 Evaluation of Absolute Cross Sections The  TT ,  low energy  +  1 2  C  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 have been  measured by v a r i o u s groups, so an e x t e n s i v e data s e t e x i s t s a t 40 MeV and 50 MeV  [AJZ80, AJZ82].  The data have disagreements, p a r t i c u l a r l y a t  backward a n g l e s , but the g e n e r a l f e a t u r e s a r e understood.  The f o l l o w i n g  e x p r e s s i o n was used t o c a l c u l a t e a b s o l u t e c r o s s s e c t i o n s from the experimental  q u a n t i t i e s o f t h i s experiment: ,  da a(e)=  N  r  d a" (  .  N  mt-i/-  c,b T— 3  ~  c,b Table  cos* J  TT M mt  ,£A  ,  c,b c,b  NQ u  •  s  AQ  T . )  /-> i\ ( 3  -  3 )  ° c , b  3.1 g i v e s the meanings o f the symbols used, but some e x p l a n a t i o n i s  necessary. An  e f f e c t i v e J a c o b i a n allowed  f o r the d e v i a t i o n o f the s c a t t e r i n g  experiment from a p o i n t geometry, and f o l d e d such i n f o r m a t i o n i n v i a an  a o o c  rt O  cu rt  •  oooooooooooo5ooooooooo  rr  O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O  (TJ  < CO  H XI  :8  O rt  I -*  Ml  St  O  CO  o  H  Co rr  rr (0 r|  n>  a.  Co H Co  >  l-t  c  3 N> 00  rt  =r o c 0Q rr  - fa M fa m  95 c H n to  I-*  •  i  fat U 1 • U ss;  I  MtlU  3  •  z §  I  ui in i tfl -* W a w •  CO  0 0Q i— CD 1  CO H*  3  O  O  (X  VO  rt  Hi  Oo •  (T> 3 O 3 ho  O  kJ  M — fa  - -. —  I  MUM  58— M -* M  -  fat M - it fa Ul - 9l U * -._ U U A U » » a o - 91 » tt « e w «t fa _ _ J- o M -JQ«I f » n u i < 9> B> IB J ot -• - - ui -• — .> M - » - » - • M — PO - -» u u u u o ^ » • ft >l I 9> 01 -J 91 91 M ft -* M — — M • Ul — I. — — M — U U U WW M IO at M hi (, -* * J -— - . k i > o t t - e . w * i -J <J" J w -. -» Mu->>outjkuucr>w w u -• -« M -» -—— •* «J w » w M " U u u t a ' t i falM-tflM • -* -• M —ff»U » - » "J l» * Utfa)«t * -» M »J UftM ~ — U U K> M A U U 1 -» u -» MM M t lfatM — - - M •» UUMUIU^.k • — M M — — — -* W U - kj a k> M - - - - WO fal * n * - M ft -* M M Mfat— W — .> — a u M u M M U U ' i i ft -*MW->fal-»Mfal - . . . M M —  -»fal' — A » M Is?  i »fal-*  -. — 'J m M- m • —Mm —  Si 5 n  -< at o —ftm  8* M -» -.WW i l l s * * * *falfal • • at J ^ k • u u i • ft  - - w w M w -  IMM  A -  S:  ss: 8s  M M  * ft  • -» MfatUt M M K • — • • f a t - W - — MM-> -> M W JkM^^MMM M — M —  H-  O 3  --  Si  1  cr  •» — -  * M ft Mtilat uM U y M » -• M -> — — M -» — *ftftw u MU>-~Ma~MMWftMft M -«.«-. ^ fthl9JM«IM«>MM M fal M W — -» W * « U >J* -J • W * M M W — - M — i M V M -» •"!•«» lf««flM4MUUN"U^ M mft-i »IIIIUW'<IMMU*--• M — M ft • • • t u g MWU ^ ' M U UM M WftW) » U A « ' M » . hl<*-* -> M •> — 0tM i • atfat— -»M MfalftM U M M —fatM - » - • - • 0»M — UD-iaAIIV'JI i * k fa> -hiuM —- ^—w — -t-uia hi hifta> •J - fat Ot Mfatfa)— M M *• (M — ~> k U M  *  *ftI  w  -» M » -> u " w  — ->  M -> ^  ~ —  ^ w  M Utftm A fl W i • W SM  usuumvoiK  - fat atft-j _ fflUII»-J—OIM-> ft M at fa> WOfalftftfal-h ^ M M M 41 M ft JWfa|UI9l9iai(B fa)faiM a M *i « • MfalVftftftft *h)UI<|tfa|9ia>M ^ * M lthl«9l*l»UI9l j m k at « ' m u kft»>UIUI>4fal-*  oo~SSS:°!SS:iS3J!!S58l3?2SSS3XtS28S;Si8S;8 = 2 S S S 5 S 8 S ; S 8 S S S 5 3 S ; ; 3 8 J S S S S S 2 S 3  I 1  M« H CB Ul  I (H 91  CM  X  IP  CO *> 0 03  —  n  X  X  x CM x  (£ X CD x X  X  x x x  Oft x 0} * -  X X X  X  10  X  X  X  eo  x  x  W  X X  X X  X  X  X  X  - x x x  - x x x : x x x ) X } ) ( I  If X X  X X X X C X ) X  x • x 10 V X X I X  X X X X X X X X X X X X  x x x x  X X X X X X X X  X X « X 0) X X X X X X X X X 01 X X X X X X X X X X X X X XX X X X X X X X X X X X X X X X X X  • f» X X X 0) X rX X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X  X X X X X X X x X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X  X X X X X X X x X X X X X X X X X X X X' X X X X X X X X X X' X X X X X X X X X X' X X X X X X X X X X' X X X X X X X X  tgggggg-L m x  (  X  X X X X X X X X X CO X — X X X X X X x x X X X X X CM X X X X CO X X X «0 X X X X CD X X X X X X X X 0) X X X 10 X X X CO X X X X x x x X X X X X X X x x x X X x x x X X x x x X X x x x X X X X X X X x x x X X x x x X X x x x X X x x x X X x x x X X x x x X X x x x X X x x x X X x x x X X x x x X X x x x X X X X X X X  r*  X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X  X X X X X X X X X X ' X X X X X X X X X X ' X X X X X X X X X X X X X X X X  X X X X X X X  X X X X X X X  X X ' X X X X X  X X V) X X X  X X X  X X X  x X  m  }  PX X  X  0) X X X x x x X — x x x x x x x x x X n x x x x x x x x x x X x x x x x x x x x x x x X i-  a  r  i  C  O  x  C  X  x  D  X  x  X  X  x  —  X  X  X  x  X  x  X  x  X  x  X  x  X  x  X  x  X  x  a  (  X  t  x  X  x  P  X  x  x  X  x  X  X  x  x  X  x  X  X  x  x  X  x  X  x  X  x  X  t X  x  X  x x  X  x X  x  X  x x  X  x  X  x  X  x x  X  x  X  x  X  x  x  X  x  X  x  X  x  x  X  x  X  x  X  x  x  X  x  X  x  X  x  x  X  x  K  x  X  X  x  x  M  x  T  X  x  X  x  X  x  x  X  X  x  X  x  X  x  x  X  X  x  X  x  X  x  x  X  X  x  X  x  X  x  X  X  X  X  X  X  x  x X  x  x  x x x x x x X  X  X  X X X X X X X X  CM X  (0  PTST  nx CO X  X X X X X X X X X X X X X X X X  X X X X X X X X X X X X X X X X X  X X X X X X X X X X X X X X X X X  X X  X X  X X  x  x  x  X X X  X X X  X X X X X x  X X  X X  r> x  ggggggggggg  X X  x X *  CD  x  x x x  x x x  x x x  X X X X X  (0 I*-  x x x x x x  in x  * CM - CM  0  (0 (  m ' pt n( • Ci « i • Ci ii '  CM CM —  «  IP •» -5!  - 8 : - 8 8 ) CM  t — r»  i  _> o ! CD 0) ( OB CO I at 0) 0> 0> 0)  : =§s = R8 —ftCM  i (0 ( kf) V < r> C  I  0) CM  n  — — — —  i  O — CM — *  CO CO CO CM CM  — r» ( 0 — MT CM — CM O — m »  a>  CO  — — — —  nf x ti  (0 * r> *  — m  10 (0 CM  (0  CM  — (0 *  co — i  m co «r  — (0  — m CM * • MT O — mv — o — —  CM V) O * CO  —tf>o  i t  IP I , 10 CD  > co  « m < (0 IP (0 (0 *»  m m cn tn  CO n <  m r> t m CM t to — (  - o  • W W  — r» ip — O*  Mr < n t f l ( — ( O < ot (  co ( r- < u> < cn <  i-  = r* 88w  to (0 (0 < ID  — r> w  > — — *- « u?  —  CM  *  — to — n CM  — - tp —  <  *> CM  CM »- Mf CM I P O - C M r» CD CM 0 ) V M O « Mf CM ( P  c» eo  CO f - CM  m  co  UJ <P  n  <o  K M P - I P CD CM Mf CO O  p» I o f- I r> i p (  n m « n p» o  CD  n t- U>  ID  CM O MT — 10 U? CM M O  co (  M> <  o <  c* <  — < (  i O co  CM ( CM CO < CM f - ( CM IP I  • CD CO  CM m (  CM ^ < CM n < CM CM < CM — <  x 01  t- o —  CO CO  CM O  -8  o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o  >ooooooooo6o6o6ooooooooooQd6ooo6oooobbooo ' n n n n n n n n n t  O O 3  Z »-  V)  F i g u r e 3.2  Uncut AE histogram f o r (run28, ARM1). histogram  AE c u t s a p p l i e d  i n f i g u r e 3.3 a r e shown.  to E  ft  won to • tt  X  n x tt x n x n x • C» X  * t tt 10 Mt • «  <r •  Mt  0  et x x » x ft X  -8  MT to PS MtttCM Mt • -Mt MJ t> CA Mt »* • M» r- r-  t M  X  Q  M l r« W f- —  *  * 10 tt MT IP Mt tfn Ml IP CM  t  MT LP - M T <P MT  —  Q  tta  CM X X CM X X r» x x x x x x X n x x x x x x x x x x x X X ttXXXXXXXXXXXXXXXX X ttxxxxxxxxxxxxxxxxxxxxxxxxxx X X X X X X X X X X X X X X X X X X X X X X X X X X X - X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X K X X X K X X X X X X X X X X X X X X X X X X X X X X X X X X X r - x x x x x x x x x x x x x x x x x x x x x x x x x x x x X X X X X X X X X X X X X X X X X X X X X X K X X X X X X t » X X X X X X X K X X X X X X X X X X X n x x x x x x x x x x x x x x C M X X X X X X X X X X X X f- X X X X X X X X X X X O X X X CM  x X X X X X X X X X X X X X X X X X X ft  M) Mt Mt MT •MJ  x x X X X X tt co X X MT X X X X MT tt CM X X Mrtt— X x « - M T tt O X X MT * ft X X X X w MT r Mf MT <0 X X X X X X MT MT Mt X X MT Mt o X X Mt MT CM X X Mt Mr — X x « - M t Mt O X X n X X n co ^ r> em x r- x O IC Mt ft tt * ft Mt Mt P ) P ) MT PI CM  tt  IP * 0) CM — CM P ) —ttp)  M m PI  MM «  tt  MT M  -Eg  rt a -8  Mt CM r Mt CM IP Mt CM tt Mt CM Mt fl D Mt CM CM  tt  x CM X CM X CM X tt  tt CO tt I tt IP  X X P) X x x X X X X X X • X X CO X  M  — CO — 01  — o> — 0» — tt  — o Six!'  > CM  -M8S P) P) P) P)  i P)P>  0) CO 0) f 0) IP 01 tt cn Mr 0) n 0» CM 0> —  -P) n P) PJ P> PI PI n P) -ft ft pj p> P> P) ft ft ft  ft co co « pCO IP CO tt CO Mt CO P) co CM CO CO O r* ft r- to r~ f~ r- IP r» i n f» Mt f". « f* CM  r» Pi  i  -  o  ooooooooooooooooooooooooooooooooooooooooo  * O M t t t r t u > O M r e r M t p O M r a r M U 9 0 M t B C M t t O M t t t r « t t O M t t t C M t t O M t t t C M t t O M t O ^ ft CM CM — »- o c n f t c o c o p - i p i p t t t t ^ n n c M C M * ' 00ftftttP-^-iPtPttMtMrnmcM«-*-  -"CMCMCMCMCMCMCM—  —  —  —  —  Figure 3.3 Energy h i s t o g r a m f o r (run28, ARM1) the  a f t e r c u t on AE.  cuts are e l a s t i c a l l y  scattered.  Pions i n s i d e  to x n x x cn x x x  r* x x (O X X X P> X fP* P>  X X X X X X  ai  ID  (  OJ o c ee cn CD CD (  f  m<  CD CD V  (  co n (  p- O ( ID 0) < ID CO ( ID r» < ID ID < ID ID *r ( ID P> ( ID W ID '  m(  n  X X X X P - X X X X X X X tfixxxxxxxxx x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x c o x x x x x x x x x x x x x x x x x x x x x x x x r - x x x x x x x x x x x x x x x x x x x x x x x n x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x W X X X X  x x x x x x X x  x x x x x x X x  x x x x x x X x  x x x x  x x x x  x x x x  P» X X X X X X X X X X X X X X X f»  X X X X X X X X X X X X X X X X X  P> Ct P> P) X X X X X X X X X X X X X X X X X  P3 X CO X P* X X rt x X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X P» X X r- x n x cn x  i  m co ( m r- 1 tn ID < B Y in < If) * < m PI i cn r t ( — \ O i * T 0) (  i  P> n  O)  P) p> PI p» P) ct rt «  V p> Pt — o cn co r  , i < ( < j i  rt rt rt rt rt  ' p> rt — O 0)  ( ( ( < <  CD <  « - CD (  — r»- < — IP <  ooooooooooooooooooooooooooooooooooooooooo  < • s o « c D N ( D O « c D r t u > 0 « c o r « i D O * c o r i i p o « c o r i i D O M C D N i D O M r c D C M i P O v c o r i ( D O O M . r ) N c < - - o c ^ 0 ) « « ^ » ^ p l n l l l « ^ ^ ^ N w • • o o » 0 > » ^ ^ u » « 0 D t t n B N • • »  3  —Z  U  O3 Z^  I  F i g u r e 3.4  AE h i s t o g r a m as i n f i g u r e 3.2, but c u t o n l y on the f i n a l E spectrum.  T h i s shows the AE c u t o f t h a t f i g u r e t o have adequa w i d t h and c e n t e r i n g .  Table 3.1 Meaning of S y m b o l s U s e d In the Calculation of Elastic Differential Scattering Cross Sections  SYMBOL  MEANING  N  ()  Counts  i n () e l a s t i c  peak  l  O  IC counts f o r ( ) r u n  4>  Angle o f t a r g e t normal  J()(6)  E f f e c t i v e Jacobian  5  Number of IC counts per p i o n a t t a r g e t  t o beam d i r e c t i o n  dft/dn* f o r ()  F r a c t i o n o f p i o n s r e a c h i n g ARM w i t h i n c u t s  *  F r a c t i o n o f s c a t t e r e d p i o n s s u r v i v i n g t o ARM P o i n t s o l i d a n g l e p r e s e n t e d by ARM  da  Center o f Mass (CM) d i f f e r e n t i a l s o l i d  dn*  LABoratory (LAB) d i f f e r e n t i a l s o l i d  9  E f f e c t i v e CM s c a t t e r i n g a n g l e o f ARM  a(8)  CM d i f f e r e n t i a l s c a t t e r i n g c r o s s s e c t i o n  T  ()  A  ()  N  0  ()  T a r g e t t h i c k n e s s i n g/cm Atomic weight  2  for ()  for ()  Avogadro's number One o f (carbon, _boron, o r empty)  angle  angle  averaging process. weighted  by do  The  e f f e c t i v e s c a t t e r i n g angle, i n a d d i t i o n ,  was  (SMC79 [STR79, STR80] p o t e n t i a l , parameter s e t 0 ) .  were computed w i t h a Monte C a r l o i n t e g r a t i o n r o u t i n e and from the p o i n t v a l u e s of g e n e r a l l y l e s s than 0.1° f o r the J a c o b i a n .  These  r e s u l t i n changes  f o r the angle and  0.001  Some sample c a l c u l a t i o n s a r e shown i n t a b l e 3.2.  each s c a t t e r i n g angle the optimum t a r g e t angle was  assumed.)  (At  These  c o r r e c t i o n s a r e s m a l l enough t o be i g n o r e d ; p o i n t geometry v a l u e s were used f o r J ( 9 ) and  9.  The  f a c t o r £ was  important  n o r m a l i s a t i o n f o r the c r o s s s e c t i o n . d e t e r m i n i n g the number of IC counts counters  ( f i g u r e 2.5)  and  The  i n d e t e r m i n i n g an a b s o l u t e  n o r m a l i s a t i o n was  obtained  by  per p a r t i c l e p a s s i n g through the S1»S2  then making a p p r o p r i a t e decay c o r r e c t i o n s to  o b t a i n the number of p i o n s i n c i d e n t on the t a r g e t per IC count. We ° r i C= — [ (1-PF)+PF[A(TGT,S1«S2)]J PF  have:  5  (3.4)  where: 1 _ P F  PF=(  and  1+  (  0  )  A(IC,TGT)exp(x /x ) ) 1  the o t h e r symbols are e x p l a i n e d i n t a b l e PF  0  (see f o r example f i g u r e 2.3)  e n e r g i e s of i n t e r e s t .  The  the p i o n f r a c t i o n i l l e g i t i m a t e l y .  (3.5)  3.3.  i s g i v e n i n c h a p t e r I I f o r both  f a c t o r A allowed f o r the f a c t  i n t o muons a f t e r the l a s t quadrupole  -1  0  had p i o n TOF  t h a t pions  decaying  and hence c o n t r i b u t e d to  I t I s shown i n s e c t i o n A3.2  that, for a  p e n c i l beam i n c i d e n t on a c i r c u l a r beam counter at e n e r g i e s such t h a t a cone exists:  Table 3.2 Angle Averaging and CM Transformations for Pions at 43.1  MeV.  Angles Weighted by * • + < ) Cross Sections Are D e n o t e d by C M * * ) -  e  (degrees)  dQ/dQ* ( p o i n t : J ) Q  or  Lab  CM  CM+  30.00  30.49  30.45  30.25  1.0298  1.0289  40.00  40.63  40.55  40.37  1.0263  1.0255  50.00  50.75  50.65  50.44  1.0220  1.0213  60.00  60.85  60.83  60.64  1.0170  1.0164  70.00  70.92  70.97  71.01  1.0115  1.0111  80.00  80.96  81.01  81.07  1.0057  1.0055  90.00  90.98  90.94  90.97  0.9998  0.9999  100.00  100.96  101.04  101.05  0.9939  0.9936  110.00  110.92  110.96  110.96  0.9882  0.9878  120.00  120.85  120.86  120.85  0.9829  0.9823  130.00  130.75  130.73  130.72  0.9781  0.9775  140.00  140.63  140.59  140.58  0.9740  0.9733  150.00  150.49  150.44  150.42  0.9707  0.9700  J (e) 0  J(e)  Table 3.3 Symbols Used i n Pion Decay Correction  SYMBOL  P F  0  PF  MEANING  Observed p i o n f r a c t i o n a t t a r g e t Corrected  pion f r a c t i o n at target  x  0  Decay l e n g t h o f p i o n a t beam energy  x  l  IC t o t a r g e t d i s t a n c e  x  2  Target  to S1»S2 d i s t a n c e  ^0  IC counts per S1»S2 p a r t i c l e  A(a,b)  F r a c t i o n of pions a t 'a' g i v i n g pion signals at  'b'  H(l-e)(l-x ) : 2  A = 1 - A / I exp(C/x)  (  1 / 2  ) dx  (3.6)  x (x +B) 2  where A , B , C , E unit  are kinematics-geometry dependent and H ( l - e ) i s the  step f u n c t i o n . The  2  T h i s was  Heaviside  numerically integrated.  s o l i d a n g l e s of the t e l e s c o p e s were determined from the  usual  expression: fi =2TT[l-(l+E )-l ] 2  (3.7)  /2  0  w i t h 5 being  the r a t i o of the S i ( L i ) a c t i v e r a d i u s t o i t s d i s t a n c e from  target.  C counters  The  do not e n t e r i n t o  t h i s s o l i d angle  C a l c u l a t i o n s have i n d i c a t e d t h a t fewer than 0.05 would have s c a t t e r e d from the t a r g e t and having  passed through the r e s p e c t i v e C The  r e l e v a n t s o l i d angle  a c t i v e area of the S i ( L i ) s was the a c t i v e areas  data  definition.  % of the p i o n s i n c i d e n t  proceeded to a S i ( L i )  without  counter.  i s shown i n t a b l e 3.4.  taken  the  to be 1250  mm [KEV80]. 2  The  mean  The  of the d e t e c t o r s a t t h e i r o p e r a t i n g p o i n t s was  r a t i o of then assumed  to be p r o p o r t i o n a l t o the r a t i o of the squares of t h e i r p h y s i c a l r a d i i . Some f i n a l comments a r e n e c e s s a r y efficiency decay the  of the s c a t t e r i n g two  t e l e s c o p e s to p i o n s .  We  observed  r e c a l l that f o r pion  y ->  e  (+v's)  (3.8)  fraction:  l-exp(-/ of  to the  s t e p c h a i n predominates: TI ->  A  with regard  the p i o n s decay w h i l e  r a n 8 e 0  stopping  I n c o r r e c t Energy S i g n a t u r e  (IES).  (ecyx)- dx)  (3.9)  1  [SAL72] and F i g u r e 3.5  many of these w i l l have an gives t h i s f r a c t i o n  as  a  Table 3.4 S o l i d Angle Data f o r Scattering Telescopes  ARM  Active  Target  Radius  Distance  Angle  (cm)  (cm)  (msr)  Point  Solid  0  2.07  45.0  6.64  1  1.92  45.5  5.59  0  10  20 INITIAL  30  40  50  PION ENERGY  (MeV)  60  F i g u r e 3.5  Calculated pion losses while stopping curve  i n a Nal detector:  shows l o s s e s from r e a c t i o n s ; broken curve p i o n decay.  solid  shows those  from  f u n c t i o n o f p i o n energy,  assuming the form of e q u a t i o n 3.1 f o r the s t o p p i n g  power a l o n g w i t h a p p r o p r i a t e v a l u e s o f a [MEA72].  T h i s was a s m a l l  c o n t r i b u t i o n t o the o v e r a l l i n e f f i c i e n c y , of the order o f 1 %. A more important pion.  contribution resulted  A 700 ns time gate width  to the muon s t a g e .  from the decay of the stopped  t o the ADCs allowed most pions t o decay o n l y  As a r e s u l t , the e l a s t i c peak a c t u a l l y had an energy  4.1 MeV g r e a t e r than t h a t k i n e m a t i c a l l y a l l o t t e d  to the p i o n .  p i o n - r e l a t e d muons which d i d decay, however, l e f t  behind 52.3 MeV p o s i t r o n s ,  most of which would not have come t o r e s t i n the N a l . spread i n energy  Any  The r e s u l t was the  above the p i o n peak seen i n f i g u r e 3.1.  The p r o b a b i l i t y  t h a t a p i o n decays o n l y as f a r as a muon d u r i n g a time gate, t , i s : X [ exp(-X t)-exp(-X t)] ii  V  where ^ y  - 1  and X ^  respectively.  -1  (3.10)  ij  TT  a r e the mean l i v e s of the muon and p i o n  E q u a t i o n 3.10, p o r t r a y e d g r a p h i c a l l y as a f u n c t i o n of t i n  f i g u r e 3.6, i s the r e s u l t i n g e f f i c i e n c y .  The maximum occurs a t t=117 n s . A  gate width o f 700 ns r e s u l t s i n an e f f i c i e n c y of 74%. F i n a l l y , n u c l e a r e l a s t i c s c a t t e r i n g s caused  some of the pions to  s c a t t e r out o f the N a l b e f o r e coming t o r e s t , g i v i n g IESs f o r these The  t e l e s c o p e geometries  were designed  t o minimise  this effect.  events.  The  i n e l a s t i c events which o c c u r r e d i n the N a l d i d , however, e f f e c t i v e l y remove some events  from the e l a s t i c s c a t t e r i n g peak.  I f , following Richard-Serre  et a l . [RIC68], we u s e : o.  = (70/32) A  2 / 3  [E -12E +36E] mb, 3  2  (3.11)  Figure  3.6  Muon decay c o n t r i b u t i o n to t e l e s c o p e  efficiency.  we  find  before  t h a t some 5%  (see f i g u r e 3.5)  inelastically  stopping. From the above c o n s i d e r a t i o n s  c u t t i n g p r o c e d u r e s , we  alone,  expect t e l e s c o p e  i s a d i s a d v a n t a g e of these n  even b e f o r e  inefficiencies in  e f f i c i e n c i e s of l e s s than 70%.  our This  telescopes.  +  3.2.3  of the pions i n t e r a c t e d  Evaluation of Ratios Given the c r o s s  the r a t i o s . constants  s e c t i o n data,  In t h i s p r o c e s s ,  i t i s conceptually  simple to c a l c u l a t e  decay c o r r e c t i o n f a c t o r s and  normalisation  tend to f a c t o r , so t h a t t h e i r u n c e r t a i n t i e s do not a f f e c t the  u n c e r t a i n t i e s i n the r a t i o . the experiment as one  T h i s , i n f a c t , i s a major advantage of doing  r a t i o experiment i n s t e a d of as two  separate  cross  s e c t i o n measurements. E q u a t i o n 3.3  i s c o r r e c t f o r the case of  a l l o w e d easy s e p a r a t i o n at  a t most a n g l e s (and  a l l a n g l e s ) of the f i r s t  the boron i s o t o p e s  1 0  11  completely r e s o l v a b l e . required  F i g u r e 3.7  net  q u a l i t y estimates  shows, though, t h a t  l y i n g e x c i t e d s t a t e s which were not the  to the c r o s s s e c t i o n s of these n u c l e i .  Furthermore, the n a t u r a l boron t a r g e t ' s h i g h The  where system r e s o l u t i o n  O p t i c a l model c a l c u l a t i o n s were used to e s t i m a t e  i n e l a s t i c contributions  allowed f o r .  C,  at l e a s t high  excited states.  J * } have low  1 2  1 0  B  c o n t a m i n a t i o n had  c o r r e c t i o n f a c t o r s by which the e x p e r i m e n t a l  to  be  cross  s e c t i o n s were m u l t i p l i e d a r e : ((l-f  1  0  ) f +  L = b  0  *j  1  0  ) —  [(l+g )(l-f n  and:  1  1 0  )+f  1 0  *k(8)*(l+g  (3.12) 1 0  )]  67  5H  2*  3/2"  2  >  5/2 3/2"  +  CD  1/2*  >Q:  2«  -  I  .  |  0  UJ  1/2"  +  +  UJ  H  |  +  OH  3 10  B  +  11  B  Figure  Excited  0  3/2"  l 2  C  +  1/2"  l3  C  3.7  s t a t e s of s t a b l e boron and carbon  isotopes.  *  68  L  with  t a b l e 3.5 The  c  =  td-fnJ+f^^^]  e x p l a i n i n g the meaning of the  q u a n t i t y k ( 6 ) was  model e s t i m a t e was 1 2 D  _  symbols.  not e x p e r i m e n t a l l y known and  used.  The  o  ((N / i )-(N J x J ) c c mt mt  _  (3.13)  3.14  7  J  /y  c  assumes t h a t the normal angles  r i g o u r o u s l y the same.  T, A b c  ,  s  b  of the t a r g e t s are a l l  D i f f e r e n c e s of order e° would have l e a d to  u n c e r t a i n t i e s of the o r d e r  tan(0)*e*Tr/18O° i n the r a t i o s ; a s m a l l e f f e c t  comparison to the s t a t i s t i c a l  expression f o r propagation  in  errors involved.  S t a t i s t i c a l e r r o r s i n R and  Xj.  optical  f i n a l e x p r e s s i o n f o r the r a t i o i s :  b e Equation  hence an  o(9) were c a l c u l a t e d u s i n g the  standard  of e r r o r s [BEV69] w i t h u n c o r r e l a t e d v a r i a b l e s  For: y=f(x ,x ,...,x 1  2  ),  n  (3.15)  n (6y) = I 2  (6x . ) ( 3 y / 8 x . )  j-l where 6XJ  3.2.4  (3.16)  2  1  i s the u n c e r t a i n t y i n X j .  Results The  p i o n s on and  2  3.7.  experimental 1 2  C The  relativistic  e l a s t i c d i f f e r e n t i a l cross sections f o r p o s i t i v e  a r e shown i n f i g u r e s 3.8 kinematic formulae  and  transformations  3.9  and  to CM  [BAL61] averaged over  tabulated i n tables  were performed u s i n g  t a r g e t n u c l e i mass.  The  3.6  standard errors  Table 3.5 Quantities involved in calculation of Ratio Correction Factors  SYMBOL  f  13  MEANING  I s o t o p i c abundance o f C 1 3  *10  I s o t o p i c abundance o f  8ll  Ratio of d i f f e r e n t i a l cross sections, 11 *  «10  R a t i o o f d i f f e r e n t i a l c r o s s s e c t i o n s , i o * /IOB  k(9)  Ratio of d i f f e r e n t i a l cross sections,  J10  R a t i o o f atomic masses,  1  0  B B  B  1°B/HB  J13  R a t i o o f atomic masses,  Lb  C o r r e c t i o n f a c t o r f o r boron  Lc  C o r r e c t i o n f a c t o r f o r carbon  h(6)  Ratio of correction factors  1 3  C/  1 2  C  (Lc/Lb)  1 0  B/ U B  70  Figure  *C e l a s t i c  differential  from o p t i c a l  cross  3.8  s e c t i o n at 38.6 MeV.  Curves are  model (SMC79) c a l c u l a t i o n s w i t h parameter (broken) and l a ( s o l i d ) .  sets 1  71  Figure  2  C  elastic  differential  from o p t i c a l  cross  3.9  s e c t i o n a t 47.7  MeV.  Curves a r e  model (SMC79) c a l c u l a t i o n s w i t h parameter (broken) and l b ( s o l i d ) .  sets 1  Table 3.6 E l a s t i c D i f f e r e n t i a l Cross Section for w  on  +  1 2  C  e (deg  f o r Lab Energy 38.6 MeV  0(6) cm)  6  (mb/sr)  (mb/sr)  30.5  4.88  0.45  45.6  3.49  0.14  60.8  2.62  0.24  75.9  3.29  0.09  90.9  5.17  0.12  105.9  6.73  0.19  120.8  7.75  0.28  135.6  8.42  0.21  150.4  7.66  0.26  Table 3.7 E l a s t i c D i f f e r e n t i a l Cross Section for i r on +  1 2  C at Lab Energy 47.7 MeV  o-(e)  e  6  (mb/sr)  (mb/sr)  30.5  5.53  0.24  40.6  5.13  0.24  50.7  3.21  0.11  60.8  2.78  0.10  70.9  3.23  0.19  80.9  4.22  0.13  91.0  5.91  0.35  100.9  7.22  0.36  110.9  7.95  0.27  120.8  8.34  0.29  130.7  7.28  0.41  140.6  7.44  0.24  150.4  7.45  0.67  (deg  cm)  are p u r e l y s t a t i s t i c a l except u n c e r t a i n t i e s i n the The  1 2  C*  a t backward a n g l e s , where they  (4.44  MeV)  contribution.  c r o s s s e c t i o n s d i p below the t h e o r e t i c a l c a l c u l a t i o n s at  a n g l e s , a problem p o s s i b l y r e l a t e d cone.  to h i g h e r c o u n t i n g r a t e s near  T h i s cone i s at 20.8° f o r 38.6  pions.  The  MeV  pions and  c o r r e c t to w i t h i n about T a b l e 3.8  i s o t o p i c a l l y pure MeV.  1 1  The  B),  and g  1 Q  [KAR69, ASP74] f o r B . n  [BEE79, 0LI81] f o r the the SMC  1 0  that f o r f i t t i n g  fitting  C/  l i  B  parameter,  1 1  l x  MeV  are t a b u l a t e d i n  (see t a b l e s 3.10 and  2.14  and  B  charge  3.11)  MeV  taken as 0.67  [SWI76] f o r  f a c t o r k(6) i s a s t r o n g f u n c t i o n 1 0  B  RMS  matter  radius (  1 0  r+).  the best a v a i l a b l e e x p e r i m e n t a l v a l u e of 2.43 radius,  1  0  r  0  ; h(9) i s c a l c u l a t e d here and v a l u e s of  fm with 1 1  to the r a t i o data at the r e s p e c t i v e e n e r g i e s .  purposes,  r+.  previous  e l a s t i c cross  p o t e n t i a l , u s i n g parameter s e t 2 (see t a b l e 4.1)  r e s u l t i n g from best f i t s note  The  i s dependent upon the v a l u e of the  c o r r e c t e d r a t i o s use  1 2  MeV  was  2  The  MeV  [EIS74, EIS76] code CHOPIN [R0S79].  The v a l u e of the d e f o r m a t i o n parameter, B ,  of 9 and  f o r 47.7  absolute normalisation i s  r e s u l t s at 47.7  inelastic contributions g  r e s p e c t i v e l y w i t h the coupled channel  and 0.25  the muon  the c o r r e c t i o n f a c t o r h ( 9 ) , and  were c a l c u l a t e d f o r the e x c i t e d s t a t e s at 0.72  B  forward  10%.  the c o r r e c t e d r a t i o a t 38.6 The  The  g i v e s the e x p e r i m e n t a l r a t i o of  s e c t i o n (assuming  1 0  18.4°  backward angle measurements are much l a r g e r than  measurements have I n d i c a t e d to be the c a s e .  t a b l e 3.9  reflect  The  h(9) must be allowed t o v a r y w i t h  r a t i o s are p l o t t e d i n f i g u r e s 3.10  along with s e v e r a l t h e o r e t i c a l  calculations.  the  and  3.11  r+ We  Table 3.8 Ratio of T T E l a s t i c Cross Section on +  to that on  6 (degrees)  1 1  B  at Lab Energy 38.6  Uncorrected Uncorrected  h(6)  1 2  C  MeV  Corrected  Corrected  Ratio  Error  Ratio  Error  45.6  1.353  0.068  0.983  1.329  0.067  60.8  1.596  0.077  1.000  1.595  0.077  75.9  1.648  0.066  1.034  1.705  0.068  90.9  1.479  0.062  1.036  1.533  0.064  105.9  1.398  0.108  1.032  1.443  0.111  120.8  1.096  0.080  1.030  1.129  0.082  135.6  1.138  0.073  1.029  1.171  0.075  150.5  1.038  0.071  1.028  1.068  0.073  Table 3.9 Ratio of * Elastic Cross Section on +  1 2  C  to that on B at Lab Energy 47.7 M e V n  e  Uncorrected Uncorrected  h(9)  Corrected  Corrected  Ratio  Error  Ratio  Error  30.5  1.083  0.062  0.985  1.067  0.061  40.6  1.134  0.049  0.986  1.118  0.048  50.7  1.115  0.060  0.982  1.094  0.059  60.8  1.299  0.068  0.985  1.279  0.067  70.9  1.583  0.076  1.009  1.597  0.076  80.9  1.426  0.068  1.022  1.458  0.069  91.0  1.432  0.039  1.022  1.464  0.040  100.9  1.400  0.063  1.021  1.429  0.064  120.8  1.232  0.034  1.017  1.253  0.034  130.7  1.024  0.042  1.016  1.040  0.042  (degrees)  Table 3.10 Correction Factors f o r Ratios at Lab Energy 38.6  e  k(6)  MeV  (1+8 )  (l+g >  10  n  (degrees)  45.6  0.817  1.0060  1.0015  60.8  0.902  1.0090  1.0024  75.9  1.078  1.0080  1.0024  90.9  1.089  1.0060  1.0018  105.9  1.070  1.0060  1.0019  120.8  1.058  1.0070  1.0023  135.6  1.051  1.0090  1.0029  150.5  1.045  1.0110  1.0035  Table 3.11 Correction Factors f o r Ratios at Lab Energy 47.7 MeV  e  k(6)  (1+810>  d+g ) n  (degrees)  30.5  0.831  1.0030  1.0005  40.6  0.836  1.0040  1.0006  50.7  0.811  1.0060  1.0008  60.8  0.826  1.0080  1.0012  70.9  0.948  1.0080  1.0015  80.9  1.015  1.0070  1.0016  91.0  1.019  1.0060  1.0015  100.9  1.010  1.0070  1.0018  120.8  0.988  1.0100  1.0030  130.7  0.981  1.0130  1.0038  150.5  0.966  1.0180  1.0056  T  1  r  80  100  120  B  (CM.)  Figure  3.10  R a t i o of c r o s s s e c t i o n s of i r on +  c a l c u l a t i o n s use  1 2  C  and  1 X  parameter s e t 2.  best f i t v a l u e of  B  r 140  at 38.6  MeV.  C e n t r a l curve 1 1  r+.  T  160  Optical uses  1.8  T-^47.7 MeV  Cr > 4  2288  2.238 2 188 20  i 40  i 60  T  80 6  — i —  —I  100 (CM.)  120  - I —  160  140  F i g u r e 3.11  R a t i o o f c r o s s s e c t i o n s o f TT on +  1 2  C and -B a t 47.7 MeV. 13  c a l c u l a t i o n s use parameter s e t 2. best f i t v a l u e o f  1 1  C e n t r a l curve r+.  Optical  uses  3.3 CNO  2  Experiments  The d a t a a n a l y s i s was performed under VAX/VMS.  a t TRIUMF w i t h a VAX 11/780 o p e r a t i n g  The data tapes were scanned  w i t h MOLLI ( M u l t i £ffline  I n t e r a c t i v e A n a l y s i s , an i n t e r a c t i v e v e r s i o n o f FIOWA) [BEN83] t o remove random time background. all  Only events w i t h the C212 spectrometer f l a g s e t and  f o u r MWPC's f i r i n g were a c c e p t e d .  then w r i t t e n t o summary ARRAY f i l e s .  I n f o r m a t i o n from these events was The i n e f f i c i e n c i e s o f the spectrometer  MWPC's were c a l c u l a t e d f o r events p a s s i n g through the c e n t r a l r e g i o n o f the s p e c t r o m e t e r ' s e x i t p o r t from t h e number o f events w i t h o n l y t h r e e of f o u r MWPC's f i r i n g .  As an example, the i n e f f i c i e n c y , EF2, i n WC2, was : N(WClFeWC2F»WC3FeWC4F) N(WC1F«WC3F«WC4F) '  where WC(i)F  1  ?  U - l / ;  i s t r u e i f WC(i) f i r e d and N(x) i s t h e number o f events w i t h  c o n d i t i o n "x" t r u e .  The u n c e r t a i n t y i n t h i s i n e f f i c i e n c y i s j u s t  r o o t o f t h e numerator d i v i d e d by t h e denominator  the square  i n e q u a t i o n 3.17.  i n e f f i c i e n c i e s , a l o n g w i t h a beam c o n t a m i n a t i o n spectrum  These  and the accumulated  s c a l e r v a l u e s f o r t h e r u n were w r i t t e n a t the end o f each ARRAY  file.  3.3.1 Momentum Spectrum Optimisation The combined e f f e c t s o f f i n i t e s i z e o f the T l p r o d u c t i o n t a r g e t , secondary c h a n n e l a c h r o m a t i c i t i e s , energy  s t r a g g l i n g i n the s c a t t e r i n g  t a r g e t s and o p t i c a l i m p e r f e c t i o n s i n t h e QQD spectrometer l i m i t e d t h e r e s o l u t i o n a t t a i n e d i n these experiments  t o about  1.6 MeV FWHM.  This  r e s o l u t i o n was o b t a i n e d a f t e r e x t e n s i v e e f f o r t was a p p l i e d a t o p t i m i s i n g a s e t o f " t r a n s f e r c o e f f i c i e n t s " which d e s c r i b e t h e t r a n s p o r t o f charged p a r t i c l e s through the s p e c t r o m e t e r .  A computer code was developed (QQD  Optimisation of Parameters or QQDMP) to perform t h i s o p t i m i s a t i o n much c o n c e p t u a l l y code has  to a l e s s s o p h i s t i c a t e d  s i n c e become a de  f a c t o standard  code due  to W.  Gyles  [GYL84].  D e t a i l s of the procedure are  appendix VI;  that  from the  that  spectrometer depends upon t h e i r  3.3.2  the momentum d e v i a t i o n  spectrometer c e n t r a l momentum i s d e r i v e d  t r a n s f e r c o e f f i c i e n t s , and  the  full-acceptance  Analysis)  of  via  r e s o l u t i o n of  an  the the  optimisation.  ARRAY f i l e s were a n a l y s e d w i t h i n t e r a c t i v e r o u t i n e which i s a m o d i f i e d v e r s i o n  complete a r r a y fitting  and  files.  placed  3.3.2.1 NMR  of  quantities  ( h i g h and  A number of cuts  low were  in this analysis.  Cuts  Dipole  f i e l d s were ensured to be  and  BT  (see  the  s t a b l e by  f i g u r e 2.12a) f i e l d s  most c a s e s , t h i s c o n s t r a i n t stability,  (QQD  QQDANA c o n t a i n s code f o r d i s p l a y i n g data, peak  i n t e g r a t i n g , as w e l l as f o r s e t t i n g software cuts  on r e l e v a n t  B l , B2  QQDANA  of QQDMP s u i t a b l e f o r a n a l y s i s  l i m i t s of a v a r i a b l e between which data Is a c c e p t e d ) .  the  found i n  Software Cuts The  the  The  scheme f o r d e t e r m i n i n g  spectrometer t r a n s f e r c o e f f i c i e n t s .  event, <5p,  owes  [TAC84,R0Z84,WIE84] f o r o t h e r s  w i s h i n g a convenient p a r a m e t r i s a t i o n / o p t i m i s a t i o n  here i t s u f f i c e s to say  and  d i d not  r e s o l u t i o n of the  r e q u i r i n g NMR  readings  to v a r y by l e s s than .25%.  come i n t o p l a y .  apparatus was  not  By  of In  ensuring t h i s  a f f e c t e d by  the  e x i t chamber c o o r d i n a t e s that would have o t h e r w i s e o c c u r r e d .  spread i n  3.3.2.2 T a r g e t Traceback  Cuts  T a r g e t c u t s e l i m i n a t e d p i o n s seen t o o r i g i n a t e from the v i c i n i t y o f the t a r g e t frame. angles 6  0  The t a r g e t p r o j e c t i o n c o o r d i n a t e s X  and Y  Q  and cj> ( p r o j e c t i o n s of s c a t t e r i n g angle on the X -Z 0  Q  planes r e s p e c t i v e l y , f i g u r e 3.12; a x i s ) a r e d e r i v e d from the f i r s t  Z i s the d i r e c t i o n of the  Q  and the and Y -Z Q  spectrometer  o r d e r c o e f f i c i e n t s r e l a t i n g them t o the  v a l u e s o f XI, Y l , X3 and Y3:  XI X3 Yl Y3  1  0 0 1  SL  00  A  A  0 0  01  0 0  B  0 0 £  00  B  X  0  6  0 0  Y  •o  01  here £=211/1000. S p a t i a l c o o r d i n a t e s are i n mm  (3.18)  and a n g u l a r c o o r d i n a t e s a r e  i n mrad. V a l u e s of a ^ j and bj_j a r e g i v e n i n t a b l e 3.12 The  p r o j e c t i o n c o o r d i n a t e s a r e r e l a t e d to the l o c a t i o n on the t a r g e t  (^tar> t a r ) Y  a  t  which the s c a t t e r i n g took p l a c e . Y  tar  X  tar  " =  Y  X  o/  To f i r s t  order, (3.19)  0  c o s  (3.20)  (*>  where i)r i s the acute angle between the t a r g e t normal and the axis.  spectrometer  More p r e c i s e l y , however: XQ  X  = i  tar  s  COS0Q  , and  cos(6 +e ) 0  XQ  Y  = Y  n  +  tancfig  i  -  i  COS9Q  sine  = cos(e +e ) 0  e  (3.21)  9  , where  (3.22)  i  tgt  ) + e  s ec - f P  ]  (3.23)  Figure 3.12 Diagram o f the QQD  spectrometer,  showing c o o r d i n a t e d e f i n i t i o n s .  Table  3.12  Values of Target Traceback C o e f f i c i e n t s defined i n Equation 3.18  Coefficient  Energy  Value  Units  (MeV)  a  00  a  01  b  00  b  01  a  00  a  01  b  00  b  01  50  1.4  mm/mm  50  1.0  mm/mrad  50  .79  mm/ram  50  .61  mm/mrad  65  1.6  mm/mm  65  1.2  mm/mrad  65  .60  mm/  65  .60  mm/mrad  mm  8 p S  i s the a n g l e between the spectrometer a x i s and the beam ( p o s i t i v e  e c  a n g l e s are taken t o be to the r i g h t as the beam sees them). ®tgt *  s  t  n  a c u t e a n g l e between the t a r g e t normal and the beam  e  direction. ijj, i s +1  (-1)  when the spectrometer i s on the l e f t  ( r i g h t ) s i d e of the  i n c i d e n t beam. i  t  i s +1  (-1)  f o r t r a n s m i s s i o n ( r e f l e c t i o n ) geometry.  i  s  i s +1  (-1)  to s e t X  t a r  =+X  L i m i t s were s e t on X  t a r  originating  from i n s i d e of 2.5  F i g u r e 2.15  shows the u t i l i t y  (9  s p e c  ).  The  (-X  t a r  and Y mm  t a r  t a r  ). so t h a t o n l y those  events  from the t a r g e t frame edge were a c c e p t e d .  of these c u t s f o r the 62.8  MeV  data at  t a r g e t o u t l i n e s are shown over a contour diagram  t r a c e b a c k of p i o n s c a t t e r i n g events t o t h e i r o r i g i n on the  120°  of the  target.  3.3.2.3 TOF Cuts The spectrometer events were r e q u i r e d t o have the time s i g n a t u r e o f p i o n s i n c i d e n t on the t a r g e t .  The  time s i g n a t u r e i s g i v e n by the d i f f e r e n c e  i n times of the spectrometer event s t r o b e ( t i m i n g s e t on s c i n t i l l a t o r and the c a p a c i t i v e probe Tl.  E2)  t i m i n g p u l s e s i g n a l l i n g the a r r i v a l of protons a t  T h i s cut removes few e v e n t s , s i n c e the n u c l e a r s c a t t e r i n g a t a l l  measured a n g l e s i s much s t r o n g e r than the Coulomb s c a t t e r i n g which i s r e s p o n s i b l e f o r beam l e p t o n s s c a t t e r i n g i n t o the s p e c t r o m e t e r .  3.3.2.4 Muon Cuts As p i o n s decay  in flight,  some 30% of the p i o n s s c a t t e r e d from a  t a r g e t i n t o the s p e c t r o m e t e r w i l l decay b e f o r e r e a c h i n g the e x i t  chambers.  Those t h a t decay b e f o r e the d i p o l e w i l l have a momentum which p r e c l u d e s t h e i r passage through t h e c e n t r a l r e g i o n o f t h e e x i t chambers. which pass t h e bender s u c c e s s f u l l y b e f o r e decay, however, w i l l as l e g i t i m a t e s c a t t e r i n g e v e n t s . energy spectrum. to  Many p i o n s be observed  These events harm t h e r e s o l u t i o n o f t h e  Furthermore, as t h e p i o n decays p r e f e r e n t i a l l y a t an a n g l e  the p i o n d i r e c t i o n , t h e energy c a l c u l a t e d f o r a g i v e n decay event w i l l  not  be t h a t o f t h e d e c a y i n g p i o n , so t h a t e l a s t i c and i n e l a s t i c events i n  the  energy spectrum w i l l  be i n t e r m i x e d .  Two types of c u t s were used t o remove decay muon events from the f i n a l energy spectrum.  Both o f these were c u t s which demanded c o n s i s t e n c y i n t h e  observed WC4 and WC5 c o o r d i n a t e s . cut,  The f i r s t  c u t , DDIF, o r D e l t a  Difference  demanded t h a t t h e p e r c e n t v a r i a t i o n o f momentum from t h e spectrometer  c e n t r a l momentum (100*6p/p="6") as determined from WC4 s h o u l d be e q u a l to t h a t from WC5, w i t h i n some c u t l i m i t s .  I n the p r e s e n t case,  -1.5<6 -6 <1.5 lf  (3.24)  5  A t y p i c a l DDIF spectrum i s shown i n F i g u r e 3.13. The second muon c u t a p p l i e d was an Angle c o n s i s t e n c y c u t [GYL84]. T h i s c u t used the spectrometer t r a n s p o r t c o e f f i c i e n t s t o determine the angle at  which a g i v e n event's r a y was expected t o pass through the spectrometer  exit region. relation  R e c a l l t h a t X4=X4(Xl,Yl,X3,Y3,6 ). £f  6^ i s determined from t h i s  , so we cannot use i t t o determine the expected v a l u e o f X4.  have, however a s i m i l a r r e l a t i o n f o r X5: solved to give  We  X5=X5(XI,Y1,X3,Y3,6 ) which i s 5  6 . 5  The expected c o o r d i n a t e s a t t h e e x i t may be c a l c u l a t e d  from:  88  o  CM CM  3  CL <  H V  ( 001 x )  S1N3A3  Figure 3.13 T y p i c a l d e l t a d i f f e r e n c e , DDIF, spectrum [RUN220, 50 MeV;  1 8  0 , 70°].  ^  89  X4*=X4(X1,Y1,X3,Y3,6 )  (3.25)  Y4'=Y4(X1,Y1,X3,Y3,6 )  (3.26)  XS'-XSCXI.YI.XS.YS^)  (3.27)  Y5'=Y5(X1,Y1,X3,Y3,6 )  (3.28)  5  5  1+  These g i v e an expected p o l a r a n g l e o f e x i t  from the s p e c t r o m e t e r .  The  d i f f e r e n c e between t h i s a n g l e and the e x p e r i m e n t a l l y observed a n g l e as c a l c u l a t e d from X4, Y4, X5 and Y5 i s the c o n s i s t e n c y a n g l e , ANGL. a n g l e i s n o r m a l l y expected t o be about  This  1.2° (50 MeV) which i s the RMS p o l a r  m u l t i p l e s c a t t e r i n g angle due t o m a t e r i a l s (windows, a i r , e t c . ) w i t h i n the spectrometer; muon decays  c o n t r i b u t e a l a r g e angle t a i l  t o the observed  spectrum. F i g u r e 3.14 i s a h i s t o g r a m o f ANGL t y p i c a l t o these experiments, which a t y p i c a l c u t has been shown.  F i g u r e 3.15 shows a d e n s i t y  d e m o n s t r a t i n g t h a t the DDIF and ANGL s p e c t r a a r e s t r o n g l y Together  upon  plot  correlated.  they c o n t r i b u t e to an improvement i n the spectrometer  resolution.  (See f i g u r e A6.2 i n appendix VI.)  3.3.2.5 Energy Spectra F i g u r e s 3.16 and 3.17 show energy s p e c t r a a t 48.3 MeV and 62.8 MeV respectively.  F i g u r e 3.16, w i t h t h e spectrometer a t 80° t o the i n c i d e n t  beam shows s m a l l i n e l a s t i c c o n t r i b u t i o n s t o the s c a t t e r i n g , t y p i c a l of l i g h t elements  a t low incoming p i o n energy.  F i g u r e 3.16 i s taken w i t h the  spectrometer a t 120° f o r 62.8 MeV p i o n s on  1 2  C.  Here t h e r e i s a l a r g e  c o n t r i b u t i o n from i n e l a s t i c s c a t t e r i n g s , i n p a r t i c u l a r the 4.44 MeV s t a t e .  90  Figure  3.14  T y p i c a l , a n g l e c o r r e l a t i o n spectrum, ANGL, [RUN220, 50 MeV;  1 8  0,  70°].  91  LO  CM  O CM O CM CM 3  w  O  V  < - 1  m  ro  ( d/d  V  % ) Figure  Density plot i l l u s t r a t i n g  the  s  o  -  m  m  +Q  3.15  c o r r e l a t i o n between the DDIF  ANGL s p e c t r a of f i g u r e s  3.13  and  3.14.  and  600  X)  o  ' I  1  1  •  '  |  '  '  '  '  |  i  i  II  |  i  i  II  |  i  II  i  CB  ID 3  n  JI  CO ID  ft) o ft  400  1 5  e  -p-  oo 2  n <  o 3  00  e n to  co  co  200  N>  n  Co  rt  0 0 O -im.  S3  O  o  25  30  35 ENERGY  40 45 50 ( MeV ) [RUN 7]  55  NJ  o  o  LO CM  O CM  o LO *~  o  o  O  LO  o  SING A3  Figure  3.17  T y p i c a l energy spectrum 62.8 MeV.  T T on +  1 2  C  at 120°.  [RUN1110]  3.3.3 Peak F i t t i n g As  t h e combined c h a n n e l - s p e c t r o m e t e r r e s o l u t i o n i n the CNO  experiments was 1.6 MeV, peak f i t t i n g i n e l a s t i c scattering contributions Figure 1 8  0  2  t e c h n i q u e s were employed t o remove the  from t h e e l a s t i c  scattering  3.18 shows t h e low l y i n g n u c l e a r energy l e v e l s i n C , 1 2  li+  spectra. N,  1 6  0 and  [LED78]. The  separation  o f t h e lowest l y i n g e x c i t e d  ground s t a t e i s not d i f f i c u l t .  The c r o s s  e v a l u a t e d d i r e c t l y from the energy s p e c t r a e l a s t i c region.  The 2  +  s t a t e i n 0 from the  sections  1 6  i n t h i s case were  number o f binned counts i n the  s t a t e i n 0 , however, i s expected to e x h i b i t a 1 8  l a r g e e x c i t a t i o n i n our energy regime which, combined w i t h t h e e l a s t i c cross  section's  decrease a t l a r g e a n g l e s , r e q u i r e d  a careful  subtraction  ( e s p e c i a l l y a t 65 MeV where the i n e l a s t i c s c a t t e r i n g a t 120° i s 35% of the e l a s t i c i n strength). resolve,  The  analysis  ( f i g u r e 3.18).  use o f peak f i t t i n g  consistency  to a v o i d  i n these r a t i o experiments  1 6  and the C c r o s s 1 2  u t i l i s e d a peak f i t t i n g  s e c t i o n was determined  s e c t i o n d e t e r m i n a t i o n only used peak  as a c o r r e c t i v e t e c h n i q u e , t h e e v a l u a t i o n  The  due to peak shape  F o r t h i s reason, a l t h o u g h the 0 c r o s s  without peak f i t t i n g  requires  s y s t e m a t i c e f f e c t s i n the r a t i o s , and c a r e f u l e r r o r  t o a l l o w f o r c o r r e l a t i o n s i n the f i t t i n g  systematics.  routines  s t a t e a t 2.31 MeV i s a l s o d i f f i c u l t t o  but i t s e x c i t a t i o n i s suppressed because o f t h e p a r i t y and a n g u l a r  momentum o f t h e s t a t e  fitting  The n i t r o g e n  o f t h e r a t i o s i n a l l cases  protocol.  d e t a i l s o f t h e peak f i t t i n g  a n a l y s i s , as implemented i n f o r t r a n  QQDANA and GSRATIO a r e g i v e n i n appendix V I I .  r e s u l t s w i t h t h e 62.8 MeV data s e t f o r t h e 0 t a r g e t 1 8  An example of t h e i s shown i n f i g u r e  8  6  \ -  4 h  o  2  h  0  Figure  N u c l e a r l e v e l s observed  3.18  in  1 2  C,  *N,  14  1 6  0,  and  1 8  0.  96  3.19.  There are two states contributing to the peak at around 4  MeV  e x c i t a t i o n energy, but the exact r e l a t i v e amplitudes of the two i s not important as t h e i r e f f e c t on the area correction to the e l a s t i c peak i s small.  The  f i r s t excited state has a s i g n i f i c a n t amplitude.  The sum  of the  contributions from the various scatterings i s also shown. Tables 3.13  and 3.14  l i s t the f i t t e d strengths, of the f i r s t  state r e l a t i v e to the e l a s t i c scattering for TT on +  12  C  at the two  These, combined with the cross section data of tables 3.16  In each of the other cases, the peak f i t t i n g was  +  +  energies.  and 3.20  give  .  i n e l a s t i c cross sections for the scattering C(ir ,ir ') C*C* 12  excited  12  used only i n the correction  of the e l a s t i c scattering data.  3.3.4  Absolute Cross Section Measurement The peak f i t t i n g used to extract the number of e l a s t i c a l l y scattered  events from the raw data has been discussed  i n section 3 . 3 . 3 , and  the  relevant error analysis i n appendix VII. The  expression  that of equation 3.3  for the absolute cross section, then, i s the same as where the symbols used have the meanings i n table  A number of corrections are referred to i n that table; we now  3.15.  discuss t h e i r  implementation.  3.3.4.1 BM1«BM2 Rate Loss This loss was  due  to a number of e f f e c t s .  a beamspdt exceeding the target i n extent. pions through the target which may the BM1*BM2 monitor i n d i c a t e s .  The f i r s t i s the e f f e c t of  In t h i s case, the f l u x rate of  cause scattering events i s smaller than  This correction i s dependent upon angle, as  Figure  Peak f i t t i n g  r e s u l t s from T T  3.19  +1 8  0  spectrum a t 62.8  [RUN1109]  MeV,  110°.  Table 3.13 Ratio of Inelastic (4.44 MeV) Cross Sections to Elastic for v on C +  12  at 48.3 M e V  da^Vdfi 6  (%)  ±(%)  (deg cm)  da/dn  50.2  5.06  0.56  60.4  5.88  0.43  70.4  4.02  0.47  81.0  6.79  0.49  91.1  7.59  0.75  100.7  9.24  0.54  110.7  12.26  0.76  120.3  15.87  0.98  130.2  24.8  1.3  Table  3.14  Ratio of I n e l a s t i c (4.44 MeV) to E l a s t i c f o r ir at 62.8  +  on  Cross Sections 1 2  C  MeV  da^Vcm e  (%)  ±(%)  (deg cm)  da/dn  50.0  2.58  0.67  51.1  2.97  0.96  60.1  4.44  0.81  71.6  3.64  0.63  81.6  5.70  0.74  91.5  11.6  1.4  101.5  24.0  1.6  111.1  42.3  1.8  120.9  78.5  2.9  Table 3.15 Meaning of S y m b o l s U s e d In the Calculation of Elastic Differential Scattering Cross Sections  SYMBOL  N  (  )  ()  l  MEANING  Counts i n () e l a s t i c  peak  BM1S-BM2 counts f o r ( ) run  4>  Angle o f t a r g e t normal  to beam d i r e c t i o n  J()(6)  J a c o b i a n d^/d^* f o r ( )  I  BM1«BM2 Rate l o s s MWPC e f f i c i e n c y product F r a c t i o n o f pions s a t i s f y i n g cuts  4>  «o  F r a c t i o n of scattered Solid  pions s u r v i v i n g  to Dipole  a n g l e p r e s e n t e d by spectrometer  Center o f Mass (CM)  differential solid  angle  dn*  LABoratory  9  E f f e c t i v e CM s c a t t e r i n g a n g l e of Spectrometer  0(8)  CM d i f f e r e n t i a l  (LAB) d i f f e r e n t i a l  ()  T a r g e t t h i c k n e s s i n g/cm  A  ()  Atomic weight  o  ()  2  for ()  for ()  Avogadro's number One  of ( C, 1 2  1 4  angle  s c a t t e r i n g cross section  T  N  solid  N,  1 6  0,  1 8  Q o r eMpTy)  the s i z e o f t a r g e t t h a t the i n c i d e n t beam sees i s dependent upon a n g l e . c o r r e c t i o n was runs.  The  made by n o t i n g the average beamspot shape over a number of  A s i m p l e Monte C a r l o s i m u l a t i o n then gave a c o r r e c t i o n f a c t o r as a  f u n c t i o n of a n g l e .  The  and 30% f o r the 48.3  s i z e of t h i s c o r r e c t i o n was  and 62.8 MeV  The next e f f e c t  data sets  found t o be about  10%  respectively.  t o c o n s i d e r i s t h a t of m u l t i p l e p i o n s a r r i v i n g a t the  s c a t t e r i n g t a r g e t per beam b u r s t a t the p r o d u c t i o n t a r g e t . time s t r u c t u r e i s such t h a t ( a t 100%  The  cyclotron  duty f a c t o r ) , a b u r s t of p r o t o n s  reaches the p r o d u c t i o n t a r g e t every 43 ns.  As the p r o t o n beam f l u x  i n c r e a s e s , the p r o b a b i l i t y of h a v i n g more than one p i o n produced a t the same time ( t h e FWHM of the p r o t o n beam b u r s t i s 2.5  ns) and hence a r r i v e at the  beam c o u n t e r s (which, w i t h o n l y lower energy l e v e l d i s c r i m i n a t i o n  interprets  a double p i o n event to be a s i n g l e more e n e r g e t i c p a r t i c l e ) i n c r e a s e s . is  u  0  the average number of BM1«BM2 p i o n events observed per 43 ns time  interval.  p  0  *  s  p r o p o r t i o n a l to I P ( i ) , where P ( i ) i s the p r o b a b i l i t y of  observing exactly i pions.  u, the P o i s s o n average number o f events per time  i n t e r v a l i s then u = - l o g ( l - u ) , so t h a t the m u l t i p l i c a t i v e e  factor i s l o g ( l - u ) e  U  0  was  Q  - 1  correction  0  /u . Q  taken t o be the number of BM1»BM2 events per run d i v i d e d by the  number of beam buckets which a r r i v e d a t the p r o d u c t i o n t a r g e t (1AT1) d u r i n g the run's d u r a t i o n ( l i v e  time o n l y ) .  F i g u r e 3.20  shows u / l o g ( l - u ) 0  e  - 1  0  f o r v a r i o u s v a l u e s of u , a l o n g w i t h the range of lip's observed i n these 0  experiments. The  The c o r r e c t i o n was  f l u x l o s s due  typically  l e s s than  10%.  to beam decay muons w i t h p i o n TOF  s c a t t e r i n g t a r g e t i s c a l c u l a t e d from e q u a t i o n s 3.5  i n c i d e n t upon the  and 3.6.  This  effect  Rate c o r r e c t i o n due t o m u l t i p l e p i o n s p e r beam b u r s t .  The  range o f the c o r r e c t i o n found i n these experiments i s shown as a shaded  area.  a l t e r s the e f f e c t i v e beam p i o n f r a c t i o n a t the t a r g e t , a f r a c t i o n which c r e a t e s an apparent BM1»BM2 r a t e i n c r e a s e a t the  target.  3.3.4.2 Spectrometer Offset Angle The  spectrometer o f f s e t was  s e c t i o n measurements a t 50° f o r  1 2  e s t i m a t e d by p e r f o r m i n g s e p a r a t e c r o s s C  o p p o s i t e s i d e s o f the M i l beam.  and  1 6  0  t a r g e t s w i t h the spectrometer  on  As the c r o s s s e c t i o n drops r a p i d l y w i t h  i n c r e a s i n g s c a t t e r i n g a n g l e i n t h i s a n g u l a r r e g i o n , the d i f f e r e n c e i n the c r o s s s e c t i o n s between measured v a l u e s on the d i f f e r e n t  s i d e s i s a measure  of the a n g u l a r o f f s e t of the beam w i t h r e s p e c t to the s p e c t r o m e t e r . The c r o s s s e c t i o n s were indeed observed to be d i f f e r e n t , but c o n s i s t e n t w i t h the d i f f e r e n c e i n the average asymmetries i n the spectrometer a c c e p t a n c e . s c a t t e r i n g a n g l e s i s d i s c u s s e d i n the next  s c a t t e r i n g a n g l e s caused The measurement of the  by  average  section.  3.3.4.3 Average Scattering Angles As the spectrometer t a r g e t t r a c e b a c k p r o v i d e s t r a j e c t o r y i n f o r m a t i o n i n terms of the a n g l e s 6Q and <j>, i t i s p o s s i b l e to average 0  a n g l e s on an event by event b a s i s .  the  scattering  W r i t i n g the p o l a r a n g l e of the  p a r t i c l e ' s t r a j e c t o r y w i t h r e s p e c t t o the spectrometer a x i s as tan The  2  ¥  = tan  2  9  0  + tan  <J».  2  (3.29)  0  s c a t t e r i n g a n g l e to f o r t h i s event i s g i v e n by: cos 9 - i  (1 - c o s  2  cos to =  6)  1 / 2  tan  6  _____—_— (tan  2  T + 1)  1 / 2  0  (3.30)  104  where 6 i s the spectrometer a n g l e and i ^  = +1 (-1) f o r the spectrometer  p l a c e d on t h e l e f t  For N events,  ( r i g h t ) o f t h e beam.  = (E arccos(o) .))/N,  (3.31)  where the i n d e x j ranges from 1 t o N. Now 50 = 9 - S i s the o f f s e t generated by t h i s a n g l e a v e r a g i n g process.  The s c a t t e r i n g a n g l e s g i v e n f o r the measured a b s o l u t e c r o s s  s e c t i o n s have been transformed t o the CM frame by a f u n c t i o n F o f the masses and e n e r g i e s i n v o l v e d .  P r o p e r l y , t h i s s h o u l d be:  p r a c t i c e , the a p p r o x i m a t i o n 0  9  c m  = F(iij).  In  = F ( 0 ) - 59 has been used.  c m  3.3.4.4 S o l i d Angle Determination Transport c a l c u l a t i o n s  [S0B84, SOB84a] have shown the spectrometer  acceptance t o be around 17 msr. geometry o f the s p e c t r o m e t e r .  T h i s number i s dependent upon the exact  A more p r e c i s e d e t e r m i n a t i o n o f s o l i d  angle  c o u l d be made w i t h e x t e n s i v e Monte C a r l o s i m u l a t i o n s [ALT85], but here we have used the e x p e r i m e n t a l data s e t to a r r i v e a t a v a l u e . The p o l a r a n g l e ¥ o f an event r a y w i t h r e s p e c t to the spectrometer a x i s i s d e f i n e d by t a n Y o f e q u a t i o n 3.29. 2  Use o f the r e l a t i o n :  ¥ = arcsin((l+tan" ¥r 2  1 / 2  then y i e l d s a v a l u e o f ¥ i n t h e range 0 < ¥ < TT.  ).  For a s e r i e s of rays  o r i g i n a t i n g a t t a r g e t c o o r d i n a t e s ( X , Y ) = (0,0) the s o l i d a n g l e Q  (3.32)  0  subtended  by the s p e c t r o m e t e r i s :  An =  ( T o t a l Number o f e v e n t s ) x 2TT (1-cos Y Q ) (Number o f events w i t h * < ¥ ) 0  (3.33)  105  where *  0  *  s  w i t h V < "P  0  chosen such t h a t the spectrometer s o l i d a n g l e p r e s e n t e d to r a y s is still  g e o m e t r i c a l and a s u f f i c i e n t number of events o c c u r s i n  this region. For expected  a point o f f - a x i s ,  (X ,Y ) # (0,0), Q  0  the subtended  solid  angle i s  to be g e o m e t r i c a l i n some d i r e c t i o n o t h e r than t h a t of the  spectrometer a x i s . the bending  A r a y c o n n e c t i n g ( X , Y Q ) w i t h the g e o m e t r i c a l c e n t e r of Q  d i p o l e i s d e f i n e d by: 6  = -tan-^Xg/dg)  0  (3.34)  4> = - t a n "  1  0  (Y /d ), 0  0  The d i s t a n c e from the t a r g e t c e n t e r to the c e n t e r of the s p e c t r o m e t e r being d  Q  = 1 3 8 0 mm.  acceptance.  dipole  T h i s r a y i s assumed to d e f i n e the d i r e c t i o n of maximum  Aft i s then c a l c u l a t e d from e q u a t i o n 3 . 3 3 w i t h : t a n Y = t a n ( 9 - 9 ) + t a n C<p—* ) 2  2  F i g u r e 3 . 2 1 shows a p l o t of Aft vs t a r g e t a t 6 2 . 8 MeV. •PQ  (3.35)  2  0  0  f o r T T s c a t t e r e d from the +  1  6  Note t h a t the curve r i s e s s h a r p l y and s t a b i l i s e s  0 around  = 4 0 mrad, b e f o r e r i s i n g as the number of events w i t h ¥ < ¥ Q becomes l e s s  than would be the case i f the s o l i d a n g l e were s t i l l  geometrical, ( i e . a  l o s s i n s o l i d a n g l e o c c u r s a t the "edges" of the spectrometer  entrance  port.) The v a l u e s of Aft determined were averaged each energy.  The v a l u e of Aft f o r the 6 2 . 8 MeV  1 6 . 8 ± 0 . 8 and a t 4 8 . 3 MeV, due  to the f a c t  18.0±0.4.  over  1 2  C  and  experiment  1 6  was  0  runs a t  found  to be  The decrease observed a t 6 2 . 8 MeV  t h a t , i n t h a t experiment,  the d a t a was  is  c o l l e c t e d w i t h the  106  Figure  T y p i c a l observed  3.21  s o l i d angle of spectrometer as a f u n c t i o n of [RUN1228 , 62.8  MeV;  1 6  0,  50°]  W. Q  107  h o r i z o n t a l l y f o c u s s i n g element QT1  s e t to z e r o .  The  decrease of about  10%  i s i n agreement w i t h REVMOC c a l c u l a t i o n s [S0B84b].  3.3.5  Ratio Evaluations and The  Corrections  form of the r a t i o c a l c u l a t i o n i s the same as t h a t g i v e n  3.2.3, e q u a t i o n 3.14.  The  error evaluation i s discussed  where some e f f o r t i s a p p l i e d to a l l o w i n g the peak shape which was  used.  areas i n a l l cases (except The involving  The  f o r MT  isotopic correction for 1 6  0,  11+  N  and  1 2  C  r a t i o analyses  0,  1 2  i n appendix V I I ,  f o r the e f f e c t s of u n c e r t a i n t i e s i n  target 1 6  i n section  used the f i t t e d  peak  subtractions).  C  cross  s e c t i o n s and  ratios  were c a r r i e d out as i n e q u a t i o n 3.13;  w i t h simple adjustments to the number of t a r g e t n u c l e i due  that i s  to i s o t o p i c  v a r i a t i o n s i n atomic number. In the case of the oxygen i s o t o p e  ratio,  the c o r r e c t i o n can  more p r e c i s e l y because a major contaminant i n the H  1 8 2  0 is H  1 6 2  be made  0.  If  8-exp i s the e x p e r i m e n t a l r a t i o of c r o s s s e c t i o n s i n the l a b of t a r g e t to t a r g e t Y, and  and  (l-FR(X))  the e x p e r i m e n t a l t a r g e t X c o n t a i n s  of Y,  a f r a c t i o n FR(X)  X of X  by weight, then o ( X ) F R ( X ) J ( X) q(Y)(l-FR(X))J(Y) A(X) MY) g(Y)T(X)J(Y)  R  exp  A(Y)T(Y) where the a's w e i g h t s , J's thicknesses FR(Y)  = 1,  are the CM  d i f f e r e n t i a l cross  the J a c o b i a n i n g/cm  closely.  2  r a t i o s d ^ / d ^ a b , and c m  of the t a r g e t s . If R  s e c t i o n s , A's  0  t a r g e t s , then e q u a t i o n 3.37  i s the CM gives R  Q  T's  R e c a l l that H 0 2  the  atomic  the i s 99.8%  H  1 6 2  0 so  r a t i o of c r o s s s e c t i o n s f o r pure from R p « e X  that  108  FR(X)J(X)A(Y) T(Y) +(1-FR(X))) J(Y)A(X) T(X)  3.3.6  Results The  14  N,  1 6  0  d i f f e r e n t i a l e l a s t i c c r o s s s e c t i o n s f o r p o s i t i v e pions on and  1 8  statistical. fit  (3.37)  and  0  are g i v e n i n t a b l e s 3.16  The  1 2  C  and  1 6  0  data  through 3.23  .  The  i s shown i n f i g u r e s 3.22  parameters i n the 62.8  MeV  C,  e r r o r s are  and  " G l o b a l parameter" c a l c u l a t i o n s are shown f o r each.  1 2  3.23.  The  Best  Global  c a l c u l a t i o n s do not reproduce the back  angle  data. The  1 2  C  c r o s s s e c t i o n s agree w i t h  those  PRE81, BLE83, TAC84, SOB84a] as does the 48.3 previous  The  1 6  0  [PRE81].  at 62.8  Tables 1 6  3.24  through 3.31  o / * a and l l  1 8  o/  1 6  o  (LAB)  (or  This uncertainty  with  and muon  f o r TF+ s c a t t e r i n g on  1 2  C,  *N,  1J  through 3.29  These c a l c u l a t i o n s are d i s c u s s e d  by  t h a t of SAID [ARN82].  g i v e r a t i o s of c r o s s s e c t i o n s  r e s u l t s a r e d i s p l a y e d i n f i g u r e s 3.24  calculations.  MeV  n o r m a l i s a t i o n i s c o n s i s t e n t with that obtained  +  3.4  data  to the u n c e r t a i n t y i n the spectrometer s o l i d angle  comparing the i r p c r o s s s e c t i o n at 50°  The  0  exists.  decay c o r r e c t i o n s .  a,  No  1 G  a b s o l u t e n o r m a l i s a t i o n i s c o r r e c t to about 10%.  i s l a r g e l y due  1 2  MeV +  The  o/  [DYT79,  e l a s t i c c r o s s s e c t i o n measurement f o r t r on  w i t h i n a few MeV)  1 6  of other experimenters  1 6  1 1 +  0,  a/ o,  and  along w i t h  i n chapter  1 2  1 8  0.  fitted  IV.  Summary In chapter  I I I , the e x t r a c t i o n of the e l a s t i c c r o s s s e c t i o n s and  their  109  r a t i o s from t h e raw d a t a has been d e s c r i b e d .  A number of t e c h n i q u e s common  to n u c l e a r p h y s i c s have been d i s c u s s e d , such as the use of PID ( p a r t i c l e i d e n t i f i c a t i o n ) c u t s , peak f i t t i n g , s e c t i o n s , and compensation  i s o t o p i c c o r r e c t i o n t o the c r o s s  f o r p a r t i c l e decay.  s p e c i f i c t o the p a r t i c u l a r apparatus  Some techniques more  o f t h i s experiment  c o n s i d e r e d , such as t h e QQD s o l i d angle and average  have a l s o been  angle e v a l u a t i o n s .  R e s u l t s have been p r e s e n t e d f o r the 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 t h e i r r a t i o s f o r C , B a t 38.6 and 47.7 MeV and C, '*N, 0 / 1 2  1 8  0,  1 6  0 a t 48.3 and 62.8 MeV.  n  12  1  16  The e x p e r i m e n t a l c r o s s s e c t i o n r a t i o s a r e  f r e e from many o f the s y s t e m a t i c e f f e c t s which need t o be a l l o w e d f o r i n a r r i v i n g at the absolute cross s e c t i o n s . the s t r e n g t h o f t h i s work.  The e x p e r i m e n t a l r a t i o s a r e thus  I n c h a p t e r IV, an a n a l y s i s w i l l be made t o  e x t r a c t from them p r o t o n matter d i f f e r e n c e s f o r the c o r r e s p o n d i n g  nuclei.  Table 3.16 Differential Cross Sections for Elastic Scattering of ir on C +  12  at 48.3 MeV  6 (deg cm)  do/dQ. (mb/sr)  +  (mb/sr)  50.2  3.67  0.23  60.4  2.52  0.09  70.4  2.44  0.18  81.0  3.52  0.16  91.1  4.28  0.29  100.7  5.64  0.25  110.7  6.44  0.38  120.3  6.49  0.42  130.2  6.32  0.41  Table 3.17 Differential Cross Sections for Elastic Scattering of i r on +  1<f  N  at 48.3 MeV  e (deg cm)  dcr/dft  +  (mb/sr)  (mb/sr)  50.1  4.76  0.31  60.3  3.47  0.23  70.4  3.54  0.26  80.8  4.95  0.41  90.8  5.75  0.45  101.1  6.97  0.39  110.9  8.09  0.53  120.2  7.12  0.61  130.0  6.83  0.68  Table 3.18 D i f f e r e n t i a l Cross Sections f o r E l a s t i c Scattering of *  +  on  1 6  0  at 48.3 MeV  6 (deg  cm)  da/dft  +  (mb/sr)  (mb/sr)  50.0  5.24  0.26  60.2  3.90  0.19  70.3  4.24  0.23  80.7  5.86  0.40  90.7  6.42  0.42  101.0  7.34  0.30  110.8  7.94  0.47  120.1  6.95  0.50  129.9  5.99  0.47  Table 3.19 D i f f e r e n t i a l Cross Sections f o r E l a s t i c Scattering of n  +  on * 0 8  at 48.3 MeV  do/dtt  0 (deg  cm)  (mb/sr)  +  (mb/sr)  49.9  6.39  0.36  60.1  3.78  0.23  70.2  3.63  0.22  80.6  4.99  0.43  90.6  5.35  0.40  100.9  6.03  0.28  110.7  6.72  0.47  120.0  5.23  0.44  129.8  4.19  0.45  Table 3.20 D i f f e r e n t i a l Cross Sections f o r E l a s t i c Scattering of *  +  on  1 2  C  at 62.8 MeV  0 (deg  cm)  do/dti  +  (mb/sr)  (mb/sr)  50.0  6.72  0.48  51.1  5.83  0.51  60.1  2.53  0.16  71.6  2.26  0.13  81.6  2.90  0.16  91.5  3.49  0.21  101.5  3.96  0.19  111.1  3.23  0.16  120.9  2.72  0.11  Table 3.21 D i f f e r e n t i a l Cross Sections f o r E l a s t i c Scattering of n  +  on  1<f  N  at 62.8 MeV  6 (deg  cm)  do/dQ.  +  (mb/sr)  (mb/sr)  61.4  3.33  0.27  71.5  2.74  0.19  81.3  3.79  0.21  91.1  4.73  0.47  101.3  4.93  0.35  111.0  3.54  0.18  120.6  3.09  0.18  Table 3.22 Differential Cross Sections for Elastic Scattering of if on 0 +  16  at 62.8 MeV  9 (deg cm)  do7d°,  +  (mb/sr)  (mb/sr)  49.7  8.77  0.77  51.0  7.06  0.62  61.3  3.61  0.22  71.4  3.16  0.16  81.2  4.09  0.18  91.0  4.74  0.32  101.2  4.18  0.24  110.9  2.81  0.12  120.5  1.89  0.09  Table 3.23 D i f f e r e n t i a l Cross Sections f o r E l a s t i c Scattering of i r on +  1 8  0  at 62.8 MeV  e (deg  cm)  da/dft  +  (mb/sr)  (mb/sr)  61.2  3.14  0.32  71.3  2.56  0.18  81.1  3.44  0.21  90.9  3.59  0.30  101.1  3.13  0.24  110.8  1.96  0.11  120.4  1.20  0.10  118  10  CO _Q  E 10  7  "O  b  TD  10'  30 50 70 90 110 130 150 6 (degrees, CM) Figure  Elastic  d i f f e r e n t i a l cross  a t 48.3 MeV.  3.22  sections  Calculations  of T T i n c i d e n t on +  use d e n s i t i e s d e r i v e d  1 2  C  from  e l e c t r o n s c a t t e r i n g (FL) and the SMC81 p o t e n t i a l . S o l i d , Set E50;  broken, Set E f 5 0 f .  and  1 6  0  Figure  Elastic at  3.23  d i f f e r e n t i a l c r o s s s e c t i o n s of I T  62.8 MeV.  scattering  +  i n c i d e n t on  C a l c u l a t i o n s use d e n s i t i e s d e r i v e d  ( F L ) and the SMC81 p o t e n t i a l .  l o n g dashes, Set EC65f  (Set E065f f o r  s h o r t dashes, Set E65;  1 2  and  1 6  0  from e l e c t r o n  S o l i d , Set 1 6  C  EBLE65f;  0 calculation);  and dots Set EBLE65co.  Table 3.24 Differential Cross Section Ratios of Elastic Scattering of n on N and C +  1H  12  at 48.3 MeV  9  l^o/lZa  +  (deg cm)  50.1  1.300  0.043  60.3  1.374  0.040  70.4  1.440  0.063  80.8  1.419  0.062  91.0  1.349  0.065  100.8  1.241  0.039  110.8  1.260  0.041  120.2  1.101  0.046  130.0  1.082  0.047  Table 3.25 Differential Cross Section Ratios of Elastic Scattering of * on +  1 6  0 and C 12  at 48.3 MeV  e  16 /12 a  a  +  (deg cm)  50.1  1.432  0.045  60.3  1.564  0.045  70.4  1.739  0.080  80.8  1.678  0.069  90.9  1.506  0.069  100.8  1.311  0.039  110.8  1.236  0.036  120.2  1.075  0.042  130.0  1.950  0.038  Table 3.26 D i f f e r e n t i a l Cross Section Ratios of E l a s t i c Scattering of n  +  on 0 and 1 6  H  lk  at 48.3 MeV  +  6 (deg  cm)  50.0  1.101  0.035  60.2  1.124  0.035  70.3  1.197  0.045  80.7  1.183  0.058  90.8  1.116  0.050  100.7  1.053  0.033  110.7  0.981  0.030  120.1  0.976  0.043  129.9  0.877  0.044  Table 3.27 Differential Cross Section Ratios of Elastic Scattering of w on 0 and 0 +  18  16  at 48.3 MeV  9  18 /16 0  a  +  (deg cm)  50.0  1.220  0.029  60.2  0.970  0.024  70.2  0.855  0.021  80.7  0.851  0.045  90.7  0.833  0.035  101.0  0.821  0.018  110.8  0.846  0.029  120.0  0.752  0.032  129.9  0.699  0.040  Table 3.28 D i f f e r e n t i a l Cross Section Ratios of E l a s t i c Scattering of *  +  on N and 1 W  1 2  C  at 62.8 MeV  +  0 (deg  cm)  60.8  1.318  0.092  71.6  1.214  0.077  81.4  1.309  0.067  91.3  1.343  0.075  101.5  1.247  0.066  111.1  1.228  0.050  120.8  1.139  0.050  Table 3.29 Differential Cross Section Ratios of Elastic Scattering of v on 0 and C +  16  12  at 62.8 MeV  9  16 /12 a  a  +  (deg cm)  49.8  1.306  0.108  51.0  1.211  0.120  60.7  1.428  0.099  71.5  1.400  0.084  81.4  1.412  0.069  91.2  1.358  0.072  101.4  1.057  0.054  111.0  0.872  0.039  120.7  0.695  0.034  Table 3.30 D i f f e r e n t i a l Cross Section Ratios of E l a s t i c Scattering of w  +  on 0 and 1 6  llf  N  at 62.8 MeV  +  6 (deg  cm)  60.6  1.084  0.078  71.4  1.153  0.075  81.3  1.079  0.050  91.1  1.002  0.062  101.3  0.848  0.045  110.9  0.710  0.032  120.6  0.611  0.031  Table 3.31 D i f f e r e n t i a l Cross Section Ratios of E l a s t i c Scattering of i r on 0 and +  1 8  1 6  0  at 62.8 MeV  18 /16  e (deg  a  CT  +  cm)  61.2  0.870  0.081  71.3  0.811  0.053  81.2  0.841  0.043  90.9  0.757  0.036  101.1  0.748  0.045  110.9  0.779  0.039  120.4  0.633  0.051  0.9  I 30  •  1  •  50  •  1  70 6  Figure  •  1  •  1  90  —  110  130  (degrees, CM)  3.24  R a t i o s of e l a s t i c c r o s s s e c t i o n s f o r T T at 48.3 MeV: +  C a l c u l a t i o n uses Set E f 5 0 f  1  o/ o.  lh  12  i n an SMC81 p o t e n t i a l and a FL  p a r a m e t e r i s a t i o n of a model independent e l e c t r o n scattering density f o r  1 2  C.  The  1I+  N  proton  d e n s i t y i s a best f i t FL form ( c f . f i g u r e 4.15).  1.4  T  1  1  r  i  1  r  I  .  L  1.3  1.2  1.1  3  1.0  0.9  0.8  -  48.3 MeV  +  TT  0.7  -i  30  50  70 6  90  110  130  (degrees, CM)  F i g u r e 3.25  R a t i o s of e l a s t i c c r o s s s e c t i o n s f o r T T a t 48.3 MeV: +  C a l c u l a t i o n uses Set E f 5 0 f parameterisations  1 6  a/  i n an SMC81 p o t e n t i a l and FL  of the model independent e l e c t r o n  s c a t t e r i n g d e n s i t i e s of  N  proton  d e n s i t y i s a best f i t FL form ( c f . f i g u r e  4.16).  1 6  l l +  0 and  1 2  C.  The  li+  o.  Figure  R a t i o s of e l a s t i c c r o s s C a l c u l a t i o n uses Set form f o r the density  1 6  0  3.26  sections  f o r T T at 48.3  MeV:  +  1 8  o7  EIM50 i n an SMC81 p o t e n t i a l and matter d e n s i t i e s .  The  1 8  0  proton  i s a best f i t FL form ( c f . f i g u r e 4.13).  a  1 6  a.  MG  F i g u r e 3.27  Ratios  of e l a s t i c c r o s s s e c t i o n s f o r i r a t 62.8 MeV: +  1 1 +  o7 a. 1 2  C a l c u l a t i o n uses Set EC65f i n an SMC81 p o t e n t i a l and a FL parameterisation  o f a model independent e l e c t r o n  scattering density f o r C . 1 2  i s a best  The *N p r o t o n ll  density  f i t FL form ( c f . f i g u r e 4.17).  F i g u r e 3.28  R a t i o s of e l a s t i c c r o s s  sections  f o r T T a t 62.8 MeV: +  16  a/ '*o. 1  C a l c u l a t i o n uses Set E065f i n an SMC81 p o t e n t i a l and FL p a r a m e t e r i s a t i o n s o f the model independent scattering densities density  of  1 6  0 and  1 2  C.  electron  The *N p r o t o n ll  i s a best f i t FL form ( c f . f i g u r e  4.18).  o  O J  O _ ci O  CP  CO  - g  O _| CO O  o o  (0 ) UP/-op / (0 8 1 ) U P/-°P 9l  ftl  F i g u r e 3.29  R a t i o s o f e l a s t i c c r o s s s e c t i o n s f o r i r a t 62.8 MeV: +  1 8  o7  1 6  a.  C a l c u l a t i o n uses Set EIM65 i n an SMC81 p o t e n t i a l and a MG form f o r the  1 6  0 matter d e n s i t i e s .  The  1 8  0 proton  d e n s i t y i s a best f i t FL form ( c f . f i g u r e  4.14).  Chapter IV Interpretations  4.1  Introduction To  t h i s p o i n t , we  pertaining  to d a t a a c q u i s i t i o n and  s e c t i o n and are  have d i s c u s s e d  ratio results.  f r e e from many of the  e x t r a c t i n g absolute cross  The  a s p e c t s of the  the e x t r a c t i o n and  cross  n  +  r a t i o experiments  c o r r e c t i o n of  section ratios, i t w i l l  be r e c a l l e d ,  s y s t e m a t i c e f f e c t s which were c o n s i d e r e d sections.  We  have p r e v i o u s l y  cross  indicated  in that  o p t i c a l model c a l c u l a t i o n s of these r a t i o s are  s e n s i t i v e to p r o t o n matter  differences  dependence upon the  between the n u c l e i , without s t r o n g  of the p o t e n t i a l i t s e l f . model c a l c u l a t i o n s and information  and  section  extract  p r e s e n t c h a p t e r , we  perform such o p t i c a l  p r o t o n matter d i s t r i b u t i o n d i f f e r e n c e  from the r a t i o s .  There s t i l l (see  In the  1.5),  e x i s t s some p o s s i b i l i t y of ambiguity i n t h i s because the n u c l e a r p h y s i c s  the n u c l e a r s t r u c t u r e  Our  optical potential i s  from a n a l y s e s of many independent d a t a s e t s to the c u r r e n t  a n a l y s e s by  of the ir-nucleus  fitting  to r e f e r e n c e  cross  then determined by  section ratios.  to the c r o s s  s e p a r a t e the a n a l y s i s here i n t o two  folded  pararaeterised  sections  The  tailored  to determine  n u c l e a r s t r u c t u r e parameters  parts.  experiment i n terms of simple RMS  The  first  a n a l y s e s of the  CB  distributions.  In t h i s experiment the unknown n u c l e u s i s an  reference.  second p a r t d e a l s w i t h a n a l y s i s of the CNO  The  interaction  [STR79,STR80,CAR82] and  a s e l e c t i o n of the o p t i c a l parameters. fitting  extraction  (matter d i s t r i b u t i o n s ) of the nucleus are  t o g e t h e r i n the o p t i c a l p o t e n t i a l .  We  details  radii  are  deals with  of p r o t o n matter  2  isotone  of  the  experiments w i t h  135  the use of F o u r i e r expansions  of the d e n s i t y d i f f e r e n c e s .  The  measurement I s c o n s i d e r e d as a p r e c i s i o n t e s t , s i n c e e l e c t r o n  1 8  0/  1 6  0  scatterers  have measured A ( p ( r ) ) between the i s o t o p e s over the e x t e n t of the n u c l e a r volume.  The  1 2  C,  l l +  N , 0 (CNO) 1 6  d a t a s e t i s a t e s t of the T T r a t i o  technique  +  i n measuring A ( p ( r ) ) = A ( p _ ( r ) ) i n n u c l e i w i t h p r o t o n number equal to  neutron  +  number, but where the n u c l e o n c o n t e n t v a r i e s s i g n i f i c a n t l y w i t h n u c l e u s .  4.2 O p t i c a l Potential Analysis In the opening  shown t h a t the r a t i o of T T  c h a p t e r , i t was  d i f f e r e n t i a l s c a t t e r i n g c r o s s s e c t i o n s i s extremely matter  d i f f e r e n c e s between n u c l e i .  +  elastic  s e n s i t i v e to p r o t o n  To perform a q u a n t i t a t i v e  interpretation  of any e x p e r i m e n t a l d a t a , one must use a more s o p h i s t i c a t e d model.  This  model should i n c l u d e i n f o r m a t i o n about n u c l e a r s t r u c t u r e , as w e l l as k i n e m a t i c , c o r r e l a t i v e , and a b s o r p t i v e p r o c e s s e s . In the low energy (3,3)  resonance  regime ( E « A ,  peaks: T ^ I S O MeV),  where A i s the energy  a t which the  some of the most p r o m i s i n g models  d e s c r i b i n g p i o n s c a t t e r i n g p r o c e s s e s are the o p t i c a l models.  So named  because of t h e i r r e m i n i s c e n c e of c l a s s i c a l photon s c a t t e r i n g , these models develop  i n t e r a c t i o n p o t e n t i a l s ( o p t i c a l p o t e n t i a l s ) which do not depend upon  i n d i v i d u a l n u c l e o n c o o r d i n a t e s ( e q u a t i o n 1.33). S e v e r a l e x c e l l e n t o p t i c a l p o t e n t i a l s have been developed the s y s t e m a t i c e f f e c t s observed  i n pion-nucleus  Landau-Thomas t h e o r y (LT) [LAN78, LAN82] extends al.  (Kerman, McManus and T h a l e r o r KMT)  in its first  The  the f o r m a l i s m of Kerman e t  [KER59] to p i o n - n u c l e u s  p e r f o r m i n g c a l c u l a t i o n s i n a momentum space s e m i - f a c t o r e d approximation  scattering.  to e x p l a i n  framework.  scattering,  I t uses a  order o p t i c a l p o t e n t i a l  (thereby  neglecting  any E r i c s o n - E r i c s o n  L o r e n t z - L o r e n z ( E L ) e f f e c t s ) and chooses a 2  2  s e p a r a b l e p i o n - n u c l e o n t - m a t r i x which i s simply r e l a t e d shifts.  I t includes  to observed phase  a n g l e t r a n s f o r m a t i o n and P a u l i c o r r e c t i o n s ,  as w e l l as  true pion absorption i n the nucleus. Other p o t e n t i a l s ,  such as the Colorado [DIG77] p o t e n t i a l and that of  S t r i e k e r , McManus and £arr (SMC) [STR79, STR80, CAR82], have t h e i r i n the E r i c s o n - K i s s l i n g e r  potential.  This  Ericson  origins  [ERI66] form takes the  scattering  amplitude of e q u a t i o n 1.1, w r i t e s the a p p r o p r i a t e  scattering  e q u a t i o n s , and a r r i v e s a t t h e v e l o c i t y dependent K i s s l i n g e r  potential  [KIS55] ( e q u a t i o n 1.33).  multiple  The v e l o c i t y dependence a r i s e s  from the  p-wave (k _k') term i n the s c a t t e r i n g amplitude and i s important i n #  characterising  the i n t e r a c t i o n .  Short range c o r r e l a t i o n s between nucleons  i n n u c l e a r matter r e s u l t i n an e f f e c t s i m i l a r to that the  s c a t t e r i n g of d i p o l e  ( o p t i c a l ) r a d i a t i o n : the c e l e b r a t e d  weakens the p-wave a t t r a c t i o n o f pions t o a n u c l e u s . for Pauli effects  (increases  s-wave r e p u l s i o n )  (two-nucleon) a r e i n c l u d e d  2  2  effect  Generalisations  and k i n e m a t i c e f f e c t s .  allow  Absorptive  order o p t i c a l p o t e n t i a l .  SMC theory a l t e r s the E r i c s o n  order c o r r e c t i o n s  E L  as complex 'rho-squared terms' to be  added t o those of the otherwise f i r s t The  induced by a medium on  p o t e n t i a l by removing the h i g h e r  to the p-wave a b s o r p t i o n  [STR79] and i n c l u d i n g  e f f e c t s due  to the t r a n s f o r m a t i o n between p i o n - n u c l e u s and p i o n - n u c l e o n c e n t e r of momentum c o o r d i n a t e systems (angle t r a n s f o r m a t i o n s ) ( V p , V p ) [ T H I 7 6 ] . 2  2  2  The  C o l o r a d o p o t e n t i a l i s a l s o of the K i s s l i n g e r form, but a l l o w s f o r a b s o r p t i v e e f f e c t s by the i n c l u s i o n o f a semi-phenomenological s-wave term.  S i c i l i a n o and Johnson effort  i n describing  [J0H80, J0H83, JOH83a, JOH83b] have spent some  the i s o s p i n dependence of the o p t i c a l p o t e n t i a l .  g e n e r a l form f o r a TT2N type of i n t e r a c t i o n generates i s o t e n s o r ir-nucleus  i n t e r a c t i o n i n much the same way  i s o s c a l a r and i s o v e c t o r terms l i k e  (T«n)  them.  (The i s o t e n s o r  i n the i s o s p i n v a r i a b l e s . )  2  c o n s i d e r e d by the SMC include  terms.  potentials.  ( E L , p exchange, 2  2  [J0H83b,  isovector  and i s o t e n s o r  SMC  terms a r e c h a r a c t e r i s e d Such i s o t e n s o r  by  terms are not  examining the e f f e c t s of p-wave  and exchange  IR083, C0084, ALT85a].  graphs) on charge  exchange  The r e l a t i v e importance of  terms i n the o p t i c a l p o t e n t i a l w i l l be most  s e n s i t i v e l y probed by the c u r r e n t  4.2.1  the TTN i n t e r a c t i o n g e n e r a t e s  F u r t h e r work by S i c i l i a n o , i n p a r t i c u l a r , has extended the  reactions  exchange  terms i n the  S i c i l i a n o and Johnson, however, do  work t o low energy, where he i s c u r r e n t l y correlations  that  The  g e n e r a t i o n of s i n g l e and double charge  reactions.  Potentials  To demonstrate t h a t matter r a d i i o b t a i n e d through a n a l y s e s of p i o n cross  section  r a t i o data a r e independent of p o t e n t i a l model,  of a v a i l a b l e o p t i c a l model approaches s h o u l d be made.  This  some sampling has i n f a c t  the p r a c t i c e of the 'PISCATtering' group a t TRIUMF [JOH79, BAR80]. r e s u l t s of such a n a l y s e s w i l l be d i s c u s s e d l a t e r . a n a l y s i s , however, we  4.2.1.1 The  The SMC  p o t e n t i a l , U, i s w r i t t e n  as:  The  For the purposes of t h i s  concern o u r s e l v e s p r i m a r i l y w i t h the SMC  Potential  been  potential.  2_U = -4TT [ b ( r ) + B ( r ) +  V c(r)+  V C(r)l  2  2 C  2  0  4TT [ v « L ( r ) c ( r ) V + — p  +  2  V»p (r)V + 2 w V ( r ) ]  (4.1)  2  c  2  with: b(r)-P [b p(r)-e b 6p(r)] 1  0  i r  (4.2)  1  bo-bo-SCbo^b! )^^  (4.3)  2  c(r)=[c p(r)-e c 6p(r)]/P 0  7 r  1  (4.4)  1  B(r)=P B p (r)  (4.5)  2  2  0  C(r)=C p (r)/P  (4.6)  2  0  2  6p(r)=p-(r)-p (r)  (4.7)  +  L(r) = [l  +  - ^ i A l _ _  c  (  r  p  i-rrfe7A>  P  2=rTT 7A)  )  ] - l  (4.8)  ( 4  Here k ^ b <* l a b an  w  a  r  e  a  t  n  e  =  10  1 +"(e/A)  (=1.4  fm  - 1  Q  )  t h e n u c l e a r mass per n u c l e o n and  e is  (  4  *  U  )  momentum and t o t a l energy i n the l a b  and co i s t h e t o t a l energy of t h e p i o n i n the ACM system, charge, A  9 )  (*- >  e  5  -  IO/AQ.  (=931 MeV) , kf  i s the p i o n the f e r m i momentum  The above, so c a l l e d SMC79 p o t e n t i a l , was the form o r i g i n a l l y introduced  [STR79].  I s o v e c t o r a b s o r p t i o n terms may be i n c l u d e d : B(r)=P [B p (r)2  2  0  C(r)=[c p (r)2  0  e i r  e i r  B (r)6p(r)]  C (r)6p(r)]/P , l P  treatment o f the E L 2  2  (4.13)  2  but the e f f e c t s of Bj and C j a r e s m a l l i n e l a s t i c The e x a c t  (4.12)  l P  scattering.  e f f e c t i s not w e l l d e f i n e d , but i n  l a t e r papers the a b s o r p t i o n i s i n c l u d e d i n L ( r ) [STR80, CAR82] so t h a t t h e term: V»L(r)c(r)V + V«C(r)V = 7»[L(r)c(r)+C(r)]v  (4.14)  V«[L(r)[c(r)+C(r)]]v,  (4.15)  becomes:  where: L(r) =[l +4TrX(c(r)+C(r))/3]  _ 1  The f a c t o r ( A - l ) / A p r e v i o u s l y accompanying X, due to T h i e s  (4.16) [THI76], has been  dropped. T h i s form of the SMC p o t e n t i a l , w i t h B^CjMD, we r e f e r t o here as the SMC81 p o t e n t i a l .  4.2.1.2 The C a l c u l a t i o n The e l a s t i c s c a t t e r i n g c r o s s s e c t i o n s a r e c a l c u l a t e d i n c e n t e r o f momentum c o o r d i n a t e s w i t h use o f the Krell-Thomas code [KRE68, TH081]. code matches  This  the Coulomb wave f u n c t i o n s of p a r t i c l e s a p p r o a c h i n g a n u c l e u s  w i t h s o l u t i o n s t o the wave e q u a t i o n : (V +k -2uU) ¥ ( r ) = 2  2  0  (4.17)  a t about 15 fm from the n u c l e a r c e n t e r . thereby  Partial-wave p h a s e - s h i f t s are  obtained.  The  o p t i c a l parameters a r e somewhat e l u s i v e , but may  [KRE69, MCM79, STR80] from f i t t i n g  lengths).  The E L 2  obtained  to p i o n i c atom l e v e l s h i f t s and  w i d t h s , a l o n g w i t h data o b t a i n e d i n p i o n - n u c l e o n (scattering  be  scattering  line  experiments  parameter, A, i s o f t e n assumed to have the  2  l o n g wavelength l i m i t i n g v a l u e of 1, even though t h e r e i s some e v i d e n c e 1.2  i s a better value  [THI76].  that  A n u l l v a l u e i n t h i s parameter would  i n d i c a t e an absence of the e f f e c t .  Values as h i g h as 2 have been  suggested  [JEN83]. P r o t o n and n e u t r o n matter have a  M o d i f i e d Gaussian  (MG)  p (r) = T The  q u a n t i t y T can be r e l a t e d  d e n s i t i e s i n l i g h t n u c l e i a r e assumed to form:  4.3 CB 4.3.1  (4.18)  ( r / a ) 2  to the number of nucleons, T, i n a g i v e n  i s o s p i n s t a t e ( i . e . : N or Z) and the s h e l l model  (1 + T ( r / a ) ) e " 2  P f J 0  to the r e l e v a n t RMS  matter  radius  through  [ELT61].  Experiment Cross Section Calculations with the SMC  Potential  I f the o p t i c a l parameters are t r e a t e d as f r e e ( o r c o n s t r a i n e d ) parameters,  the o p t i c a l codes may  be u t i l i s e d  experimental nuclear cross s e c t i o n data.  i n the f i t t i n g  T h i s can a d m i t t e d l y be a somewhat  r i s k y p r a c t i c e , as the parameters a r e numerous and correlated.  C e r t a i n semi-ad hoc assumptions  providing constraints.  of  i n some cases s t r o n g l y  [KRE69] prove u s e f u l i n  For example, the a b s o r p t i o n parameters may  assumed to be o f the form i c ( l - i ) where K has a r e a l v a l u e .  be  141  In o r d e r  to a v o i d m i s l e a d i n g  r o u t i n e VIEWFIT was  written.  fits  2  by p r e s e n t i n g x  and  This routine allows  fact,  sense 'best f i t s ' ,  F i t s generated  well yield  p a r t s of the  Set 0 i s the  scattering  by S t r i e k e r et a l . f o r 50 MeV  X  2  little l a and  minimisation  fitting 2  approach.  The  l b use 1 2  C  to set  These  coordinates  0,  two  1  The  f i t t i n g was  carried  K , 2  and  In  this  made: B Q = K ( 1 - I ) and 1  X (The E L 2  2  parameter,  out w i t h a m o d i f i e d v e r s i o n of  the  a FORTRAN r o u t i n e FLETCH [PAT78] i n the  F i g u r e s 3.8  the s c a t t e r i n g  i n a true  c r o s s s e c t i o n data of t h i s experiment.  f r e e parameters were < ,  s c a t t e r i n g processes  and  3.9  show the r e s u l t s of the  calculations:  r e q u i r e s a good knowledge of the h i g h e r  (absorption,  order  E L ). 2  2  Parameter s e t 2 i s a 'best f i t ' ( i e : VIEWFIT) s e t o b t a i n e d MeV  suggested  Set 1 i s i d e n t i c a l  f o r pions of energy 30 MeV.  s e t 1 as the i n i t i a l  Krell-Thomas code, u t i l i s i n g parameter s e a r c h .  The  i n t h e i r predicted cross s e c t i o n s .  on the  4.16).  reproducing  Appendix V  l e n g t h s and volumes are those  an assumption f o r the t r u e a b s o r p t i o n terms was  CQ=K (1-1). equation  fitting  (In  the  [STR79] G l o b a l s e t 1.  pion s c a t t e r i n g .  uses the parameters suggested  Sets  i n the  shows s e v e r a l s e t s of parameters r e f e r r e d to i n the  a n a l y s i s of t h i s experiment.  sets d i f f e r  experiment f o r  say,  the b e t t e r ' f i t ' . )  d i s c u s s e s an a l t e r n a t i v e n o r m a l i s a t i o n - i n d e p e n d e n t  but  reliable  but are h e r e a f t e r r e f e r r e d to as such.  n o r m a l i s a t i o n , t h i s method may  imaginary  of  i n t h i s manner are not  i f the t r e n d of the data i s of more importance than,  T a b l e 4.1  fitting  the g e n e r a t i o n  a v i s u a l d i s p l a y of c a l c u l a t i o n and  user a l t e r e d parameter s e t s . strictest  convergence, an i n t e r a c t i v e  d a t a of B l e c h e r et a l . [BLE79].  T h i s data and  the f i t t e d  f o r the  curve  are  40  Table  4.1  Various O p t i c a l Parameter Sets C, B  Experiment: SMC79 P o t e n t i a l  11  Para-  Parameter  meter  Set ( R e a l and Imaginary P a r t s )  X b  B  0  o  0  1  1.000  1.000  -.028  .002  -.028  .002  -.080  .000  -.078-.001  -.043 .043  -.043 .043  la  0.765 -.028  2  lb  .002  1.420 .002  -.026 .001  -.078-.001  -.078-.001  -.080 .000  -.032  -.047  -.026 .038  .032  -.028  0.875  .047  .270 .002  .265 .0025  .265 .0025  .265 .0025  .267 .0025  c  l  .220 .001  .219 .001  .219 .001  .219 .001  .220 .001  c  o  -.100 .100  -.099 .099  -.056 .056  -.009 .009  -.065 .065  143  shown i n f i g u r e 4.1.  F i g u r e 4.2  r e a s o n a b l e f i t t o the 50 MeV  demonstrates  that set 2 a l s o gives a  d a t a o b t a i n e d by the same group [MOI78].  These  d a t a w i l l be r e f e r r e d to h e r e a f t e r as the 'Los Alamos' d a t a . I t must be emphasised a t t h i s p o i n t t h a t d i s c r e p a n c i e s between the Los Alamos d a t a s e t and t h a t of the p r e s e n t experiment The 40 MeV  d a t a s e t s agree r e a s o n a b l y , the s l i g h t l y h i g h e r back a n g l e d a t a  of t h i s experiment 40 MeV  a r e not worrisome.  I s p a r t i a l l y t h e r e s u l t of the d i f f e r e n c e between  s c a t t e r i n g and t h a t a t 38.6 MeV.  There a r e no d i s c r e p a n c i e s i f one  assumes a n o r m a l i s a t i o n d i f f e r e n c e of the o r d e r of 8% between the d a t a s e t s . (The Los Alamos d a t a c l a i m s 4% t o 7% u n c e r t a i n t y i n t h i s q u a n t i t y . ) 47.7  MeV  The  c r o s s s e c t i o n d a t a d i s a g r e e s a t back a n g l e s by an amount g r e a t e r  t h a n can be a t t r i b u t e d t o the energy d i f f e r e n c e between 49.9  and 47.7  MeV.  The d a t a a t f o r w a r d a n g l e s seems t o slump w i t h r e s p e c t to o t h e r measurements, as d i s c u s s e d i n s e c t i o n 2.2.2.1.  I n any c a s e , i f t h i s c r o s s  s e c t i o n d a t a s u f f e r s from u n c o r r e c t e d s y s t e m a t i c e f f e c t s such as might accompany the a n g l e dependent g a i n i l l u s t r a t e d i n f i g u r e 2.9,  no e f f e c t upon  the r a t i o d a t a s h o u l d be p r e s e n t .  4.3.2  Generic Low  Energy Pion Scattering  The c r o s s s e c t i o n s i n f i g u r e s 3.8,  3.9, 4.1, and 4.2  a l l exhibit  f e a t u r e s c h a r a c t e r i s t i c of low energy p i o n s c a t t e r i n g ' on l i g h t n u c l e i . d i p a t 60°  A  I s due t o an i n t e r f e r e n c e between the r e p u l s i v e s-wave and  a t t r a c t i v e p-wave p a r t s of the p o t e n t i a l .  I t i s f i l l e d i n by t h e e f f e c t s o f  i m a g i n a r y terms i n the s c a t t e r i n g l e n g t h s and volumes. p i o n s of b o t h p o s i t i v e and n e g a t i v e charge.  T h i s d i p i s seen f o r  For p o s i t i v e pions a  d e s t r u c t i v e i n t e r f e r e n c e a l s o o c c u r s between the p i o n - n u c l e a r p-wave and  Figure  The 40 MeV  Los Alamos  T T  +  ,  1  2  C  4.1  e l a s t i c cross  O p t i c a l c a l c u l a t i o n s use parameter and 2 ( s o l i d  s e c t i o n data  [BLE79].  s e t 1 (dashed curve)  curve).  o  00  O  ^  to  10  {e)**zi  ( J S / quu )  Figure  The 49.9 MeV  Los Alamos  T T  +  ,  CJ  1  2  C  4.2  e l a s t i c cross  s e c t i o n data  [MOI78].  O p t i c a l c a l c u l a t i o n s use parameter s e t 1 (dashed c u r v e ) and 2 ( s o l i d  curve).  Coulomb a m p l i t u d e s . The  For n e g a t i v e p i o n s , t h i s i n t e r f e r e n c e i s c o n s t r u c t i v e .  t r e n d a t l a r g e a n g l e s i s a d i p toward the f i r s t d i f f r a c t i v e minimum.  T h i s t r e n d becomes more apparent as i n c i d e n t p i o n energy i s i n c r e a s e d (compare f i g u r e s 4.1  and 4.2).  F i g u r e 4.3  of p o s i t i v e and n e g a t i v e p i o n s a t 43.1 The  MeV  c o n t r a s t s the e l a s t i c s c a t t e r i n g on  12  C.  form of the d i f f e r e n t i a l e l a s t i c s c a t t e r i n g c r o s s s e c t i o n s i n t u r n  p r e d i c t s the form of t h e i r r a t i o s .  F o r s m a l l a n g l e s c a t t e r i n g the  w i l l go r o u g h l y as the r a t i o of the f o u r t h power of the charges of respective nuclei.  At l a r g e r a n g l e s  ratio the  (9>30°) the n u c l e a r s c a t t e r i n g begins  to predominate. As one chooses the c o n v e n t i o n o f always u s i n g the h e a v i e r  nucleus'  c r o s s s e c t i o n as the numerator i n the r a t i o , the r a t i o w i l l r i s e above unity.  The  relative  p r e c i s e manner i n w h i c h t h i s o c c u r s i s dependent upon the  s i z e s of the s and p p a r t i a l - w a v e a m p l i t u d e s  i n the n u c l e i .  After  r e a c h i n g a maximum, the r a t i o a g a i n drops as the c r o s s s e c t i o n s d i p toward t h e i r d i f f r a c t i v e minima. F i g u r e 4.4 isotones Figure  4.3.3  1 2  C and  shows c a l c u l a t e d c r o s s s e c t i o n s f o r s c a t t e r i n g from the 1 1  B a t 43.1  MeV.  The  r a t i o of these i s shown i n  4.5.  Ratio Calculations We have seen t h a t o p t i c a l p o t e n t i a l models use a l a r g e number o f  parameters i n t h e i r r e p r e s e n t a t i o n o f the p h y s i c s o f p r o j e c t i l e - n u c l e u s interactions.  I f such models a r e r e a s o n a b l e ,  be independent o f n u c l e u s  one expects  or, at worst, slowly varying.  r a t i o of such v a r i a t i o n s would be s m a l l .  the parameters to The  e f f e c t on a  F u r t h e r m o r e , a r a t i o w i l l tend  to  Figure 4.3 Optical model calculations (parameter set 2) contrast the natures of charged pion scattering from  1 2  C  at 43.1 MeV.  Character of  Coulomb-nuclear interference distinguishes pion charge states.  Figure  O p t i c a l model c a l c u l a t i o n s p i o n s on  1 2  C  4.4  (parameter s e t 2) of s c a t t e r i n g of p o s i t i v e  (dashed curve) and  n  B  a t 43.1 MeV.  N o t i c e the  r e l a t i v e magnitudes and the s h i f t i n the minimum.  20  40  60  80 6  Figure  100  120  (CM.)  4.5  R a t i o of c r o s s s e c t i o n s i n f i g u r e  4.4.  140  160  150  e l i m i n a t e the e f f e c t s o f u n c e r t a i n t i e s i n the p r e c i s e v a l u e s o f the o p t i c a l parameters. I f we assume f o r the moment t h a t the parameters  themselves  are well  known, then the d i f f e r e n c e between n u c l e i i s c h a r a c t e r i s e d by t h e matter d i s t r i b u t i o n and the i s o s p i n o f nucleons  i n the n u c l e u s .  We p a r a m e t e r i s e the p r o t o n (neutron) matter d e n s i t y a c c o r d i n g t o e q u a t i o n 4.18, where: POO = [ ^ - { l + ^ } a 3 ] - l .  (4.19)  For n u c l e i w i t h T protons ( n e u t r o n s ) : T = (T-T )/3  (4.20)  0  In the s h e l l model, T  Q  f o r p - s h e l l n u c l e i (T = {2,3  8}) i s g i v e n the  v a l u e 2; t h i s v a l u e I s used here u n l e s s s p e c i f i c a l l y s t a t e d t o the contrary. The ( 2 m )  t n  moment i s :  < r > = *3/2  i L ^ l 2m+l)!! { l  2 m  P  q  o  (  T ( 2 m + 3 > +  }  (4.21)  so t h a t :  The RMS r a d i i , r ( ) (where ( ) e {+,protons; - , n e u t r o n s } ) , the matter d i s t r i b u t i o n s and the r a t i o . RMS charge r a d i u s (appendix X ) , r r  2 +  T  where r AND77]).  p  p  0  characterise  F o r p r o t o n s , r+ i s r e l a t e d t o t h e  , by:  = r 2 0  2 r  (4.23)  pp  i s t h e RMS charge r a d i u s o f a s i n g l e p r o t o n (0.8 fm [COL67,  151  The  e x p e r i m e n t a l r a t i o d a t a were f i t t e d  the SMC79 p o t e n t i a l u s i n g parameter s e t 2 w i t h  t o the r a t i o s p r e d i c t e d by r + a f r e e parameter.  n  v a l u e f o r r + (r+ f o r C ) was assumed to be e q u a l t o 1 2  1 2  from t h e e x p e r i m e n t a l l y determined [KLI73] f o r t h a t n u c l e u s . r_.  1 2  The v a l u e f o r  1  0  r  0  charge  The v a l u e o f  1  1  radius,  1  2  r  1  Q  2  The  r _ and was o b t a i n e d  , o f 2.44±0.02 fm  r _ was a l s o taken t o be equal to  of 2.43±0.06 fm has been measured [BEE79, 0LI81]  and y i e l d s the assumed v a l u e f o r r + .  I t was assumed t h a t  1 0  F i t t i n g was performed  1 0  r+=  1 0  r_.  i n the u s u a l manner, s e a r c h i n g f o r a minimum  in:  X < r+> = E [ i 2  Rj[ and R  a r e the e x p e r i m e n t a l  Q i  respectively.  ]  ll  6  (4.24)  2  i  ( c o r r e c t e d ) and c a l c u l a t e d  The 6^ a r e the ( c o r r e c t e d ) e x p e r i m e n t a l e r r o r s .  q u a n t i t y h(6) ( s e e t a b l e 3.5 and e q u a t i o n s 3.12 through vary with  1 J  ratios  r+.  x  2  The  3.14) was a l l o w e d t o  was e v a l u a t e d a t i t s approximate minimum i n ^ r ^ . and a t  6 o t h e r p o i n t s l o c a t e d s y m m e t r i c a l l y about t h a t p o i n t . were s e p a r a t e d by about 1/3 of the s t a t i s t i c a l  These v a l u e s o f * v+  f i t t i n g error.  l  A parabola of  the form: 2 X  was f i t t e d of  ( r + ) = a + b( r+) + c ( r ) n  n  n  (4.25)  2  +  to t h i s d i s t r i b u t i o n i n order t o determine  the p r e c i s e l o c a t i o n  the minimum a t f , d e f i n e d by: r = (-b/2c) ± l / / c The u n c e r t a i n t y i s the average  i n c r e a s e the x X  2  2  displacement  (4.26) i n **r+ r e q u i r e d t o  by u n i t y from i t s v a l u e a t the minimum [CLI70].  A typical  p l o t f o r each d a t a s e t i s shown i n f i g u r e 4.6, a l o n g w i t h the r e s u l t s of  21  Figure 4.6 X  2  v a l u e s generated to e q u a t i o n 4.25. in x  2  from e q u a t i o n 4.24.  S o l i d curves a r e f i t s  Dashed l i n e s i n d i c a t e u n i t y i n c r e a s e s  from v a l u e s a t the r e s p e c t i v e minima.  153  f i t s t o e q u a t i o n 4.25.  The goodness o f t h e p a r a b o l i c f i t s i n d i c a t e  The x 's f o r t h e 38.6  r a t i o s are l i n e a r i n r + i n the regions considered. 1 1  2  MeV r a t i o a r e s m a l l e r than f o r t h e 47.7 MeV d a t a ; t h i s suggests potential  parameter s e t b e t t e r d e s c r i b e s t h e former.  of f a r e g i v e n i n t a b l e 4.2 a l o n g w i t h t h e r e s u l t s  that the  that the  The r e s u l t a n t v a l u e s  of s i m i l a r  calculations  u s i n g o p t i c a l parameter s e t 0.  4.3.4  S e n s i t i v i t i e s to Higher Moments As was seen i n c h a p t e r I , low energy s c a t t e r i n g  sensitive  i s expected  o n l y t o t h e s m a l l e s t even moment o f t h e n u c l e a r  distribution. distributions  matter  To t h e e x t e n t t h a t t h i s i s t r u e , a l l r e a s o n a b l e should y i e l d i d e n t i c a l r e s u l t s  t h i s , four different  density distributions  1) M o d i f i e d G a u s s i a n  t o be  matter  i n our r a t i o a n a l y s i s .  To t e s t  were used:  (MG)  2) M o d i f i e d £aussian Core + V a l e n c e Nucleons (MGCV) 3) C o n s t r a i n e d Fermi (CF) 4) C o n s t r a i n e d F e r m i Core + V a l e n c e Nucleons (CFCV)  Each d i s t r i b u t i o n c o n s i s t s o f a c o r e d i s t r i b u t i o n ( C ) t o which t h e 1 2  d i s t r i b u t i o n of s i n g l e nucleons  i n a harmonic p o t e n t i a l  i s added.  The  c o n s t r a i n e d Fermi d i s t r i b u t i o n i s an o r d i n a r y Fermi d i s t r i b u t i o n w i t h quadrupole r+.  m a t t e r moment e q u a l t o t h a t o f t h e MG d i s t r i b u t i o n w i t h t h e same  These d i s t r i b u t i o n s  summarised i n t a b l e A4.1.  a r e d i s c u s s e d i n more d e t a i l i n Appendix IV and  Table 4.2 RMS  Proton Matter Radius Differences: H  r +  _12  r +  Derived from O p t i c a l Model Calculations  Parameter  Set  n  38.6  r+-  MeV  1  2  r  +  (fm)  47.7  MeV  0  -.022  (29)  -.078  (15)  2  -.031  (23)  -.071  (11)  4.3.4.1 Modified Density Results T a b l e 4.3 shows the b e s t v a l u e s o f f f o r the v a r i o u s forms o f density.  A l l o f these c a l c u l a t i o n s  potential.  u t i l i s e parameter s e t 2 i n the o p t i c a l  ( F i g u r e A4.1 i l l u s t r a t e s the d e n s i t y d i s t r i b u t i o n s  f o r r+=2.25 11  fm.) The U  c o r e + v a l e n c e d i s t r i b u t i o n s use a  C  core f o r the c a l c u l a t i o n o f  B  and  1 0  B matter  C  and  1 0  B RMS r a d i i a r e assumed t o be those p r e v i o u s l y quoted.  1 2  distributions  1 2  (1 and 2 v a l e n c e h o l e s r e s p e c t i v e l y ) .  There i s some s e n s i t i v i t y t o the exact form of t h e d i s t r i b u t i o n a l t h o u g h each f i s l e s s than i t s f i t t i n g e r r o r  from the mean.  The  used,  I t may w e l l  be t h a t the Fermi d i s t r i b u t i o n i s not a reasonable form t o use; the MG is closer totally  to r e a l i t y f o r l i g h t n u c l e i .  independent  distributions The  I n any case, the r e s u l t s a r e not  o f t h e h i g h e r r a d i a l moments o f the assumed mass  (including  those h i g h e r than <r**>).  s e n s i t i v i t y t o the p r e c i s e form o f the mass d i s t r i b u t i o n can  be made more q u a n t i t a t i v e plots  a r e generated  c e n t e r e d near ?.  by the p r e p a r a t i o n o f x  by i n t e r p o l a t i n g  x  2  2  contour p l o t s .  These  v a l u e s e v a l u a t e d on a 7 by 7 g r i d  F i g u r e s 4.7 and 4.8 show some r e s u l t s f o r the MGCV  d e n s i t y ; RCOP i s t h e core r a d i u s f o r the p r o t o n d i s t r i b u t i o n and RAD i s ^r^..  form  Assuming r (  c o r e  )  v a r i a t i o n i n ? with r ( calculation  t o be the i n d e p e n d e n t l y known q u a n t i t y , the c o r e  )  can be found.  are given i n table  4.4.  The r e s u l t s o f t h a t  Table 4.3 Results of Tests for Sensitivity of Derived Proton Matter Density Difference to Details of the Matter Distribution C, B Experiment  12  Distribution  n  (  38.6  ll  MeV  r  +  _12  r  +  )  (  f  m  )  47.7  MeV  MG  -.031  (23)  -.071  (ID  MGCV  -.042  (20)  -.077  (13)  CF  -.050  (21)  -.083  (9)  CFCV  -.039  (22)  -.088  (14)  MEAN  -.041  (11)  -.080  (6)  157  Figure 4.7  Contour p l o t f o r MGCV d e n s i t y at 38.6 RCOP i s the c o r e  MeV.  radius.  RAD  Is  n  r  +  and  Figure 4 . 8 Co«o„r  p  l  o  t  f  o  r  M  G  C  V  d  e  n  s  U  y  >  t  M  e  y  _  RCOP i s the c o r e r a d i u s .  ^  i  s  ^  ^  Table 4 . 4 D e p e n d e n c e of M e a s u r e d t 11  o n the  A s s u m e d Core Radius (Units are a m per percent change)  Distribution  Data Set  38.6 MeV  MGCV  2.9  47.7 MeV  -3.8  160  4.3.5  S e n s i t i v i t i e s to O p t i c a l Parameters For  the i s o r a t i o method of determining  to be c o n s i d e r e d  v i a b l e , i t should  the matter r a d i u s of a n u c l e u s  be demonstrated that the r e s u l t s are  s t r o n g l y a f f e c t e d by u n c e r t a i n t i e s i n the o p t i c a l T h i s has plotting  calculations.  Table  4.9  and  4.10  of the best  though, t h a t parameter s e t s 0 and An  contour  shows the r e s u l t s of  f i t r a d i u s i s made.  e f f e c t upon the RMS  Re(c ), Q  radius.  2 d i f f e r by o n l y a percent  Notice,  in this  o v e r a l l u n c e r t a i n t y of 5 % i n our knowledge of the  optical  parameters would c o n t r i b u t e u n c e r t a i n t i e s i n f of somewhat l e s s than statistical  4.3.6  4.6  Sensitivities  demonstrates the s e n s i t i v i t y of the d e t e r m i n a t i o n s  various miscellaneous  parameters.  (~2.5%) y i e l d s a 13 am  uncertainty  The  2  The am  shown  1 0  B  uncertainty  boron t a r g e t  in  of f to 1 0  r+  thickness  u n c e r t a i n t y i n ?.  i n the mean energy at the t a r g e t c e n t e r  %, g i v i n g an u n c e r t a i n t y i n r of ~1 The  experimental  in f.  u n c e r t a i n t y , 3 mg/cm , r e s u l t s i n an ~8 uncertainty  the  f i t t i n g errors.  Miscellaneous Table  The  i s of the order  of  i n chapter  1/2  am.  model dependency r e s u l t s i f the Colorado p o t e n t i a l  i n s t e a d of SMC79 i s used i n the c a l c u l a t i o n of the c o r r e c t i o n f a c t o r discussed  these  In set 2 f o r both  show the accompanying contour p l o t s f o r  which demonstrates by f a r the s t r o n g e s t  parameter.  4.5  of the  Each parameter i s a l t e r e d from i t s v a l u e  n u c l e i , then a r e d e t e r m i n a t i o n Figures  parameters.  been accomplished here through the use  r o u t i n e p r e v i o u s l y mentioned.  not  I I I ( t a b l e 3.5  and  equations 3.12  through 3.14).  k(9) Using  Table S e n s i t i v i t i e s of  1  1  4.5  r to O p t i c a l Parameters.  (Units are am per percent change)  Parameter  38.6 MeV  Re(b )  0.12  Im(b )  0.054  Q  0  -0.12 0.028  Re(b )  -0.68  InKb^t  -0.011  0.002  Re(B )  0.013  -0.056  Im(B )  0.47  0.26  Re(c )  3.48  2.02  Im(c )  0.015  0.002  1  Q  0  Q  Q  -0.15  Re(c )  -1.08  -0.73  Im(c )  0.18  -0.20  Re(C )  -0.29  -0.11  Im(C )  0.21  0.13  -0.22  -0.34  3.77  2.19  x  1  0  0  X Quadrature  t  47.7 MeV  c e n t r a l v a l u e of 0.001 assumed when s e t 2 v a l u e i s 0  Figure  Contour p l o t  4.9  f o r R e ( c ) at 38.6  MeV.  Q  RECS i s R e ( c ) . 0  RAD  is  1  1  r  +  and  Figure  4.10  Contour p l o t f o r R e ( c ) a t 47.7  MeV.  Q  RECS i s R e ( c ) . 0  RAD  is  l x  r+  and  Table 4.6 Miscellaneous Parameter Dependencies 1 2  C,  n  B  Experiment  Dependence Quantity  Units (38.6 MeV)  10  1o _  (47.7  MeV)  /  -0.23  -0.22  -18  -10  Normalisation  11.8  8.9  am  /  (% change)  Energy  1.2  2.5  am  /  (% change)  0.97  1.20  am  /  (am change)  r + =  1 0  B  r  am  (am change)  correction  model dependency  am  Ratio  12 -12 _«ll _ R +  r  r  t h i s k ( 6 ) i n the SMC79 r a t i o c a l c u l a t i o n r e s u l t s i n the i n d i c a t e d change i n f. The  e f f e c t of varying  the assumed v a l u e of  am change i n ^ r + i n d i c a t i n g t h a t differences  4.3.7  r + i s close  t o 1 am per  the r a t i o technique a c t u a l l y measures  i n the r a d i i of n u c l e i and not the a b s o l u t e r a d i i .  t h i s dependence, then, t a b l e 4.6 i m p l i e s 38.6  1 2  uncertainties  Excluding  of about 23 am a t  MeV and 18 am a t 47.7 MeV.  O p t i c a l Model Dependence of Results I t has been noted t h a t t h e r e e x i s t , a t p r e s e n t , s e v e r a l o p t i c a l  p o t e n t i a l s which s u c c e s s f u l l y d e s c r i b e The  low energy p i o n - n u c l e u s  scattering.  a n a l y s i s here has used the SMC79 approach almost e x c l u s i v e l y .  analysis  of [BAR80], i t has i n f a c t been shown that  c h o i c e of o p t i c a l model i s not s t r o n g .  I n the  the dependence upon the  T a b l e 4.7 summarises the r e s u l t s .  4.4 CNO Experiments 2  4.4.1  Cross Section F i t t i n g In the s e c t i o n s  The  protocol  differed  that  follow,  observed i n c r o s s  from t h a t d e s c r i b e d  the SMC81 form of the p o t e n t i a l i s used.  s e c t i o n f i t t i n g s on the CNO  i n s e c t i o n 4.3.1.  In general,  a g l o b a l s e t o f parameters t o r e p r e s e n t a p a r t i c u l a r c r o s s w i t h the need f o r f i n e t u n i n g parameters a r e h e a v i l y  2  data s e t s  one might expect section  of some subset of the parameters.  i n t e r c o r r e l a t e d , though, so t h a t  also  well,  The o p t i c a l  trends i n a c r o s s  s e c t i o n can be f i t t e d by changes i n a number of d i f f e r e n t parameters. Given a cross  s e c t i o n and a r a t i o , however, one may f i n d  parameters whose v a r i a t i o n s produce e q u a l l y  good f i t s  that  to the cross  section  Table Deviation of  1 1  4.7  f from the Potential Averaged Means,  i n Colorado, LT and SMC79 Analyses.  1  1  1 i s the Average over  Optical Parameter Sets 0 and 2. (Units are  Potential  Data Set  38.6  SMC Colorado LT  am)  MeV  47.7  0.  4.  5.  -2.  -5.  -2.  MeV  may, how  when used i n a r a t i o c a l c u l a t i o n , have markedly w e l l the r a t i o c a l c u l a t i o n f i t s It  in  was  the r a t i o d a t a .  found by s t u d y i n g the i r 0 +  the imaginary p a r t s of b , Q  the r a t i o markedly,  B  Q  and C  Q  lists  Q  1 8  /0  data a t 48.3  1 6  MeV  Q  and ImC  Q  these o p t i c a l parameter  as r e g u l a r x  2  Sets E50  to  1 2  C  the a n a l y s i s o f the CNO  and E65  and  1 6  throughout t h i s was  0  d a t a , i t was  found t o be v e r y d i f f i c u l t  valid  For the 48.3  MeV  data,  In the case of the 62.8  to  MeV  to a r r i v e a t such a parameter  set.  exacerbated by the l a c k of f a r forward angle d a t a i n the c r o s s  s e c t i o n measurement; such data appear number of the parameters to  experiments.  by v a r y i n g a l a r g e number of the o p t i c a l parameters  a r r i v e a t a s u i t a b l e p o t e n t i a l (Set E f 5 0 f ) .  T h i s was  4.8  i n t e n d e d to f i t c r o s s s e c t i o n s  the range of atomic number encountered.  d a t a , i t was  Table  a r e g l o b a l Set E of  s i m u l t a n e o u s l y t o a r r i v e a t an o p t i c a l p o t e n t i a l  accomplished  Hence the  minimisations searching  ( S e t s EIM50, EIM65; f i g u r e 4.11). sets.  changes  w i t h n u c l e u s a f f e c t e d the form of  r e f e r e n c e SCM80 s c a l e d q u a d r a t i c a l l y t o the e n e r g i e s of these In  that  e s p e c i a l l y a t the t h r e e forward a n g l e p o i n t s .  c r o s s s e c t i o n f i t t i n g s were performed o n l y upon Imb , ImB  d i f f e r e n t e f f e c t s upon  to be very important i n d e t e r m i n i n g a  ( e s p e c i a l l y the E L 2  parameter,  2  the n u c l e a r amplitude's i n t e r f e r e n c e w i t h the Coulomb.  X) which c o n t r i b u t e The  exact  s y s t e m a t i c change i n t h i s i n t e r f e r e n c e w i t h nucleus does not appear  to be  c o r r e c t l y reproduced by the c u r r e n t form of the o p t i c a l p o t e n t i a l .  The  e f f e c t s of a l a c k of s p i n s a t u r a t i o n i n the [BR078] but found to produce  1 2  C  n u c l e u s were c o n s i d e r e d  e f f e c t s of a t l e a s t an o r d e r o f magnitude too  s m a l l to reproduce the observed d i f f e r e n c e s between c a l c u l a t i o n experiment.  and  62.8  8  MeV  2(/>  E  <3  8  b XJ  0  40  60  80  100  120  140  ANGLE (deg. cm.) Figure  E l a s t i c d i f f e r e n t i a l cross 48.3  MeV  4.11  sections  of i r  +  on  1 6  0  a t 62.8 MeV  and  showing SMC81 p o t e n t i a l c a l c u l a t i o n s w i t h parameter  s e t s E (E50 and E65; (EIM50 and  EIM65).  broken c u r v e s ) and the f i t t e d s e t s M a t t e r d e n s i t i e s a r e of the MG  EIM  form.  Table 4.8 Optical Parameter Sets (SMC81) 1 8  Optical  0,  Parametert  X  1 6  0 Experiments  E50  1.4  Reb  0  EIM50  E65  EIM65  1.4  1.4  1.4  -.0427  -.0463  Imbg  .0040  .0829  Rebj  -.0920  -.0920  Imbj  -.0013  -.0018  ReB  0  -.0048  -.0069  ImB  0  .0285  Rec  0  .2472  Imc  0  .0092  -.1821  .0062  .0192  .0490  -.0930  .2513 .0156 C  Recj  .1622  .1678  Imc^  .0043  .0069  ReC  .0447  .0481  0  ImC  0  .0694  -.2230  .0543  t expressed i n u n i t s o f m ^ / C R c ) . E n t r i e s  -.0058  left  blank are unchanged from the p r e c e d i n g column.  The 62.8  d i f f i c u l t i e s i n o b t a i n i n g simultaneous  fits  t o the  1 2  C and  1 6  0  MeV d a t a a r e d o u b t l e s s c o m p l i c a t e d by the i n c r e a s e d importance o f  a b s o r p t i o n and of the i n c r e a s i n g l y d i f f r a c t i v e shape o f the e l a s t i c  cross  s e c t i o n s a t the h i g h e r e n e r g i e s . The  r a t i o a n a l y s i s i n the 62.8 MeV CNO d a t a w i l l r e f e r t o f o u r o p t i c a l  parameter s e t s , which are summarised i n t a b l e 4.9:  i)  Set EC65f: r e s u l t i n g from a f i t t o the 62.8 MeV c r o s s s e c t i o n i n v a r y i n g Imb , B Q  Q  1 2  C  and C . Q  ii)  Set E065f: as above but f o r t h e 62.8 MeV  iii)  Set EBLE65f: r e s u l t i n g from a f i t t o the d a t a o f B l e c h e r e t a l . [BLE83] a t 65 MeV on  iv)  1 2  C  1 6  0  ( i n c l u d i n g p o i n t s as f a r  forward as 25° i n the angular  distribution).  Set EBLE65co:  f i t t o the  a simultaneous  data o f t h i s experiment,  F i g u r e s 3.22 and 3.23 showed the  1 2  0  C and  1 6  c a l c u l a t i o n s u s i n g the above parameter s e t s . d a t a of B l e c h e r e t a l . , a l o n g w i t h s i m i l a r  1 2  C and  1 6  0  s t a r t i n g a t s e t EBLE65f and  thence v a r y i n g Imb , Imc , B Q  data.  Q  and C . Q  0 cross s e c t i o n s along with  F i g u r e 4.12 i l l u s t r a t e s the  calculations.  4.4.2 Density D i s t r i b u t i o n Difference Analysis E a r l y n u c l e a r s i z e s t u d i e s w i t h charged  p a r t i c l e s a l l used  a n a l y t i c forms t o d e s c r i b e the r a d i a l d e n s i t y of the nucleus  simple  [BAR81].  The  most common o f these were the MG and the Fermi d i s t r i b u t i o n s , d i s c u s s e d i n s e c t i o n 4.3.3 and appendix IV.  These forms have been used  i n our s t u d i e s  Table 4.9 Optical Parameter Sets (SMC81) 12  t  X  E50  1.4  Reb  0  Imbg Reb  C,  1,f  N , 0 Experiments  EF50f  E65  1.072  1.4  -.0427  -.0231  -.0463  .0040  -.0272  .0062  16  EC65f  E065f  EBLE65f EBLE65co  1.229 -.0553 .0267  .0323  .0466  -.0544  -.0920  -.0920  Imbj  -.0013  -.0018  ReB  -.0048  -.0219  -.0069  .0405  .0197  .0592  .0982  ImBg  .0285  .1228  .0192  -.0324  -.0556  -.1192  .1896  Recg  .2472  .1682  .2513  .2860  Imc  .0092  -.0923  .0156  -.0064  .0680  x  Q  0  .1622  .1678  Imcj  .0043  .0069  ReCg  .0447  .1878  .0481  -.0366  -.0175  -.1022  -.1673  ImCg  .0694  .3621  .0543  .0728  .0930  .1942  .1388  Rec  x  t o p t i c a l parameters a r e expressed i n u n i t s o f m ^ / C n c ) . E n t r i e s l e f t blank a r e unchanged column.  from t h e p r e c e d i n g  1 0  °  I — I — I — I — I — I — I  10 3 0 5 0 7 0 9 0 110 130 9 (degrees, CM) Figure  Elastic measured  differential by B l e c h e r  s e t s a r e shown: Short  cross  4.12  s e c t i o n s a t 65 MeV  f o r i r on +  et a l . C a l c u l a t i o n s w i t h v a r i o u s  S o l i d , Set EBLE65F;  dashed, Set E65;  1 2  C  as  parameter  Long dashed, Set EC65F;  D o t t e d , Set EBLE65co.  d e n s i t i e s a r e of the FL form.  Matter  of the p r o t o n d i s t r i b u t i o n i n  r e l a t i v e to  1 2  C.  The use of these  a n a l y t i c forms l i m i t s the a n a l y s i s t o the gross f e a t u r e s of a r a d i a l d i s t r i b u t i o n ( f o r example the RMS f o r more d e t a i l e d r a d i a l study. dependent  r a d i u s ) and i g n o r e s a probe's More s e r i o u s l y , i t can r e s u l t  potential i n results  upon the model used f o r the d e n s i t y , and i n u n d e r e s t i m a t e s i n the  u n c e r t a i n t i e s i n the d e r i v e d moments.  In r e c e n t y e a r s , e l e c t r o n  scattering  d a t a have been a n a l y s e d 'model i n d e p e n d e n t l y ' u s i n g n e a r l y complete  s e t s of  o r t h o g o n a l f u n c t i o n s to r e p r e s e n t the d e n s i t y d i s t r i b u t i o n s o f nucleons within nuclei  [FRI75,  FRI78].  F o u r i e r B e s s e l (FB) expansion i s e s p e c i a l l y w e l l s u i t e d to e x t r a c t i o n of charge d i s t r i b u t i o n s from e l a s t i c e l e c t r o n s c a t t e r i n g d a t a because  the  Coulomb charge form f a c t o r i s e s s e n t i a l l y the 3-dimensional F o u r i e r t r a n s f o r m of the charge d i s t r i b u t i o n .  where R  c  p(r) =  P l  (r) + I a  =  P l  ( ,r )\  i s a cutoff radius.  n  The expansion i s w r i t t e n :  sin[ § 1 ] r " l c  r <  R  c  > c  r  R  The o r t h o g o n a l i t y of the expansion a l l o w s  the e x t r a c t i o n of the g r e a t e s t amount of i n f o r m a t i o n per added degree of freedom.  The s t a r t i n g d e n s i t y , p j , i s o f t e n taken to be a Fermi of MG  form.  The FB and a l s o a F o u r l e r - L a g u e r r e (FL) expansion: P(r) =  P l  (r) + Z a  were used i n these experiments.  n  Ll  / 2  (2(r/a) )  The parameter  2  _ e  a may  ^  r  /  a  )  (4.28)  2  be r e l a t e d  to the  s t r e n g t h of the n u c l e a r p o t e n t i a l , and L i s the Laguerre p o l y n o m i a l . The mathematical p r o p e r t i e s of and s t a t i s t i c a l a n a l y s i s t e c h n i q u e s a p p r o p r i a t e to these F o u r i e r expansions a r e d i s c u s s e d i n appendix V I I I .  In  174  implementing them i n the o p t i c a l model a n a l y s i s , the n o r m a l i s a t i o n s were chosen such t h a t : /  (r)r dr 2  P l  = T  / Ll/2 (2(r/a) ) m  ;  L^ n  2  2  / p (r)r dr = 0  and  2  F  (2(r/a) ) -2(r/a) 2  2  e  r  2  where T i s the number of neutrons o r protons and p P  component of the d e n s i t y .  The v a l u e of a  second i n t e g r a l c o n d i t i o n , and hence was  Q  was  d  r  =  "(nx) mn  fi  chosen to implement  never a f r e e parameter.  4.27  and 4.28, The  The s c a l e l e n g t h s a=1.5  were t r e a t e d as f r e e  ' r e f e r e n c e ' and  i n F o u r i e r forms.  2 9 )  the In  was  imposed  fm and R =6 fm, of e q u a t i o n s c  parameters.  'unknown' n u c l e i had t h e i r d e n s i t i e s p a r a m e t e r i s e d  The r e f e r e n c e n u c l e i (  w e l l d e s c r i b e d by the MG  '  i s the F o u r i e r  a d d i t i o n , a c o n s t r a i n t r e q u i r i n g p o s i t i v e d e n s i t i e s everywhere on the c a l c u l a t i o n s .  ( 4  a  forms; i n the  1 8  1 2  0/  C, 1 6  1 6  0 ) however are r e a s o n a b l y  0 a n a l y s i s the F o u r i e r  c o e f f i c i e n t s a j of the r e f e r e n c e were s e t to z e r o .  The CNO  a n a l y s i s used  FL p a r a m e t e r i s a t i o n s f o r the r e f e r e n c e n u c l e i u n f o l d e d from the model independent charge d e n s i t y d i s t r i b u t i o n s tabulated i n table  [CAR80, REU82, NOR82].  These a r e  4.10.  The a c t u a l r a t i o f i t t i n g s were accomplished w i t h the SMC81 o p t i c a l p o t e n t i a l a c c e s s e d through the program PIFIT  [GYL84].  PIFIT was  modified to  a l l o w F o u r i e r r e p r e s e n t a t i o n s of both p r o t o n and n e u t r o n matter d i s t r i b u t i o n s s i m u l t a n e o u s l y and to d e a l p r o p e r l y w i t h charged n u c l e o n s . P a r a l l e l code was w r i t t e n to implement fit  ( e q u a t i o n 4.24)  was  the FL d e n s i t y o p t i o n .  a g a i n used as the goodness  of f i t  The x  criterion.  2  of  the  Table 4.10 Fourier Laguerre Parameterisations of Reference Densities  Parameter  1 2  P r o t o n MG RMS  (fm)  Neutron MG RMS (fm)  FL (a  16  C  Q  2.3098  2.3985  2.5735  2.2725  2.3542  2.5203  parameters - 3 7  "  2  normalisation)  a  p  (fm)  1.48781  1.54542  a  n  (fm)  1.46380  1.51348  -.11254  -1  -.96695  -.69905  -2  -.16652"  .15982  -2  a t 0  l  a  a  2  a  3  a  5  .19844" -.27029  o t 6  t a  Q  and a  g  2  - 1 2  .92090  -2  2  -2  •23373"  2  -.22989~  2  a r e determined here by c o n s t r a i n t s on  volume and RMS i n t e g r a l s o f the F o u r i e r  component.  4.4.3  1 8  0/  I 6  0 Ratio Experiments  4.4.3.1 Matter D i s t r i b u t i o n Determinations The d e n s i t y of the p r o t o n matter a F o u r i e r form ( e q u a t i o n 4.27) d i s t r i b u t i o n of the measured The  1 8  0  1 8  0-  1 6  was  P l  0 neutron radius d i f f e r e n c e ,  p r o t o n d i s t r i b u t i o n of 1 6  0  was  r a t i o s were f i t t e d  1 6  0  was  1 8  0  was  taken t o have  i s MG w i t h r+=2.67 fm.  assumed, a l s o , t o be MG  n e u t r o n d i s t r i b u t i o n of The  where  d i s t r i b u t i o n of  The  neutron  with radius obtained .21 fm  from  [JOH79].  assumed MG w i t h r+=2.59 fm.  taken e q u a l to i t s p r o t o n  by a l l o w i n g a  n  The  distribution.  (1 < n < nmax) and R  c  for  the h e a v i e r n u c l e u s t o v a r y , s u b j e c t to c o n s t r a i n t s which r e q u i r e d t h a t the F o u r i e r sum's volume i n t e g r a l was  zero and a l l d e n s i t i e s u n i f o r m l y p o s i t i v e .  E r r o r bands were p l o t t e d by c a l c u l a t i o n of the p r o p o g a t i o n of e r r o r s f o r the d e n s i t y f u n c t i o n and error matrix.  i n c l u d e the e f f e c t s of the o f f d i a g o n a l terms of the  To t h i s e r r o r was  added a completeness e r r o r  [FRI75],  r e s u l t i n g from the t r u n c a t i o n of the F o u r i e r s e r i e s a t nmax terms and reflecting densities. realistic 1 8  0  fit  the i n a b i l i t y The  of the d e n s i t y expansion  completeness e r r o r was  the r e a l i s t i c  The the two  by f i t t i n g d i r e c t l y t o a  d e n s i t y ( i n t h i s case the p r o t o n matter d e n s i t y d i s t r i b u t i o n f o r  from e l e c t r o n s c a t t e r i n g and  determined  t o model a complete s e t of  [MIS79, NOR82]).  d e n s i t y was  The d i f f e r e n c e between the  taken t o be the completeness  r e s u l t s a r e p r e s e n t e d g r a p h i c a l l y i n f i g u r e s 4.13b  energies.  T a b l e 4.11  shows the RMS  probes.  and 4.14b  for  r a d i u s d i f f e r e n c e as d e r i v e d  under v a r i o u s assumptions about the d e n s i t y d i f f e r e n c e s . w i l l be seen t h a t these r a d i i  error.  In chapter V i t  agree w e l l w i t h those o b t a i n e d w i t h o t h e r  5.0  1  i  1  1  (a) 2.5  I  0  IB -2.5  •  -5.0  -7.5h  48.3  ^W/  ii  -10.0  MeV  •  i  i  1  1  1  I 1  1  5.0  O O O  (b)  2.5  0 *  -2.5  -5.0  i  /IF  7  •  i  -7.5  10.0  |  | ir* Ratio  3HLDQE i  *  2  ••Scatl.ring i 4  3  -  i 5  RADIUS (fm) F i g u r e 4.13  P r o t o n matter d e n s i t y d i f f e r e n c e s (  1 8  P (r)p  48.3 MeV ir+ r a t i o s ( c f . f i g u r e 3.26).  1 6  p ( r ) ) derived p  from  (a) V a r i a t i o n i n d e r i v e d  Ap(r) w i t h +10% v a r i a t i o n s (except as i n d i c a t e d ) i n o p t i c a l parameters from Set EIM50 v a l u e s : D o t t e d c u r v e , ImBo; i ) , those r e m a i n i n g , o p t i c a l parameters.  Broken curve (+5%),  Rebo; i i i ) ,  Imco; i v ) , ImCg;  Imb ; Q  ii),  (b) Best f i t FL d e n s i t y w i t h Set EIM50 E r r o r envelope i n c l u d e s completeness  error.  E l e c t r o n s c a t t e r i n g matter d e n s i t i e s a r e shown f o r comparison.  F i g u r e 4.14  Same as f i g u r e 4.13 a t 62.8 MeV ( c f . f i g u r e 3.29). derived  (a) V a r i a t i o n i n  A p ( r ) w i t h +10% v a r i a t i o n s i n o p t i c a l parameters from Set  E65 v a l u e s : i ) , ImB ; i i ) , 0  Imc ; i v ) , X; i i i ) ,  (b) Best f i t FB d e n s i t y Error  envelope i n c l u d e s  0  those r e m a i n i n g ,  w i t h Set E65 o p t i c a l parameters. completeness e r r o r .  Electron  s c a t t e r i n g matter d e n s i t i e s a r e shown f o r comparison.  Table 4.11 Proton Matter D i s t r i b u t i o n Differences:  Energy  O p t i c a l Parameter Set  Density  1 8  nmaxt, nx  0-  1 6  0  6 < r  2>l/2  62.8  E65 EIM65 E65 E65  FB FL FLttt FL  4 3 3 3  0 0 0 0  .091(26) .079(12) .100(37) .097(28)  48.3  EIM50 EIM50  FL FB  3 3  0 0  .084(11) .087(15)  t T y p i c a l v a l u e s : R =6.0 fm ( F B ) ; a=1.5 fm ( F L ) tt E r r o r s shown i n p a r e n t h e s i s a r e s t a t i s t i c a l , t t t I n t h i s case o n l y , t h e 0 n e u t r o n matter RMS r a d i u s was s e t e q u a l t o i t s p r o t o n matter RMS r a d i u s . c  1 8  t t  180  Analysis electrons  on  1 8  of e l a s t i c d i f f e r e n t i a l s c a t t e r i n g c r o s s 0  and  1 6  0  [MIS79], has  s e c t i o n r a t i o s of  p r o v i d e d a p r e c i s i o n FB  the charge d i s t r i b u t i o n d i f f e r e n c e between i s o t o p e s  expansion f o r  [N0R82].  p r o t o n matter d i s t r i b u t i o n d i f f e r e n c e d i s t r i b u t i o n , the  To  a r r i v e at  e f f e c t s of  the  p r o t o n charge form f a c t o r have been u n f o l d e d from the charge  density  d i f f e r e n c e by assuming a 3 G a u s s i a n p a r a m e t e r i s a t i o n of t h a t  factor, after  Chandra and  Sauer [CHA76].  appendices IX and  X.  The  unfolding  In f i g u r e s 4.13b  matter d i s t r i b u t i o n s a r e seen to be pion  i s discussed  and  4.14b, the  a t more l e n g t h electron  a  in  scattering  i n agreement w i t h those d e r i v e d  from  the  r a t i o data.  4.4.3.2 O p t i c a l Parameter S e n s i t i v i t i e s The  gradient  p o t e n t i a l c h a r a c t e r i s i n g pion  s c a t t e r i n g from  the  n u c l e u s obscures u n d e r s t a n d i n g of the c o r r e l a t i o n s between the o p t i c a l p o t e n t i a l parameters and  the matter d e n s i t y  i n which a change i n Ap(r)  i s mimicked by  parameters.  and  various  Figures  4.13a  4.14a  difference;  t h a t i s , the manner  changes i n the  optical potential  show the e f f e c t s of +10%  changes i n  the  o p t i c a l parameters upon the d e t e r m i n a t i o n of the matter d i s t r i b u t i o n  differences.  At 48.3  parameters, where the  MeV  ( f i g u r e 4.13a), the  imaginary terms Imb , ImB , and Q  d r a m a t i c changes i n _ p ( r ) .  In f a c t , Imb  0  0  MeV)  a l l o w a r e a s o n a b l e f i t to Ap(r)  f o r the 48.3  parameter, X,  62.8  MeV  Ap(r)  i n addition  ImC  from Set EIM50 0  produce the most  i s so i n f l u e n t i a l (a +5%  shown f o r t h i s v a r i a b l e a t 48.3  The  changes are  t h a t the  Set E parameters w i l l MeV  data  E L 2  2  imaginary terms r e f e r r e d to above.  f i g u r e shows v a r i a t i o n s from Set E65  not  set.  ( f i g u r e 4.14a) i s s e n s i t i v e to the  to the  change i s  This  parameters, parameters which r e p r e s e n t  the  r a t i o s much b e t t e r  than they do the c r o s s s e c t i o n s (see  figure  The  precise  the  d e t a i l s o f i t s i n t e r a c t i o n w i t h t h e n u c l e u s , i n p a r t i c u l a r the  4.11).  r a d i a l s e n s i t i v i t y o f t h e probe [FRI82, FRI83] i s determined by  dependence o f the i n t e r a c t i o n upon t h e n u c l e a r d e n s i t i e s  and t h e i r  derivatives.  4.4.4  l2  C,  ll|  H  16 l  0 (CNO)  Experiments  4.4.4.1 Matter D i s t r i b u t i o n Difference In t h i s a n a l y s i s , assumed to be s t r o n g l y  Determinations  t h e p r o t o n and n e u t r o n matter d i s t r i b u t i o n s correlated.  were  T h i s i s expected to be the case i n *N, ll  e s p e c i a l l y , where t h e n u c l e a r i n t r i n s i c and i s o t o p i c  s p i n s (T=0, J = l )  suggest t h e presence o f a q u a s i - d e u t e r o n o u t s i d e of a ( r e l a t i v e l y i n e r t ) core.  The c o r r e l a t i o n was assumed t o be of the form  1 2  C  [LAW80]:  1 / 2  a  = a (1-6 ) p n p  2  2  (Z-l)e M 2  =  6  R  p  3  n  2  where:  (4.30)  (Z-l) - .0229 (fm)  a*  Z  (4.31)  3  T h i s r e l a t i o n was taken t o a p p l y t o the RMS r a d i i of t h e s t a r t i n g distributions  Pj and t h e FL s c a l e  parameters a f o r a l l d i s t r i b u t i o n s ; i t  p r o v i d e s a simple s c a l i n g of p r o t o n matter s i z e r e l a t i v e t o t h e neutron matter s i z e t o a l l o w f o r Coulomb r e p u l s i o n The the  e f f e c t s i n the n u c l e u s .  FL p a r a m e t e r i s a t i o n s o f the r e f e r e n c e n u c l e i d i s t r i b u t i o n s (and  starting density for  llf  N) a r e l i s t e d i n t a b l e  4.10.  , ( r ) were MG i n a l l c a s e s , as was t h e case i n s e c t i o n  The forms used f o r 4.4.3.  The r a t i o s were f i t t e d by a l l o w i n g  the  proton d i s t r i b u t i o n to vary, again subject  and a f o r the ^N  nuclear  X  to c o n s t r a i n t s on the FL  component to e n f o r c e z e r o o v e r a l l mass c o n t r i b u t i o n and u n i f o r m l y densities. respect  Each o f the  1 4  N:  to the a p p r o p r i a t e  1 2  C and  1 6  0:  reference  1 4  N r a t i o s was a n a l y s e d , w i t h  nucleus.  d i s t r i b u t i o n s , however, a r e shown wrt  1 2  C.  A l l density  most u n c e r t a i n probe i n any Figures  1 2  C  and  1 6  difference  T h i s a l l o w s easy  of the r a t i o r e s u l t s , and adds n e g l i g i b l e u n c e r t a i n t y (<1 fm) o f the r a d i u s , where the  positive  0  intercomparison  except a t s m a l l  values  electron s c a t t e r i n g r e s u l t s are  (and the p i o n r a t i o has i n s u f f i c i e n t momentum t r a n s f e r to  case). 4.15 through 4.18 p r e s e n t the r e s u l t s f o r * * p + ( r ) - p + ( r ) . 1  12  Above about 1.8 fm, t h e d e n s i t y d i f f e r e n c e s a r e i n t e r n a l l y c o n s i s t e n t . shown a r e the r e s u l t s of a c a l c u l a t i o n w i t h a S e l f C o n s i s t e n t P a r t i c l e P o t e n t i a l (SCSPP) o p t i m i s e d  i n the *°Ca r e g i o n l  Single  [HOD85].  The  e l e c t r o n s c a t t e r i n g matter d e n s i t y assumes a simple form f o r the density  Also  [SCH75] and a FB model independent form [CAR80] f o r the  1 2  1<+  N  charge  C  density.  No model independent e l e c t r o n s c a t t e r i n g charge d e n s i t y measurements f o r 1**N, o r of the presently  1 2  C,  l l f  N charge d e n s i t y  available.  d i f f e r e n c e f o r t h a t matter, i s  The p i o n r a t i o c a l c u l a t i o n i n d i c a t e s a s h i f t  of the  p r o t o n matter d i s t r i b u t i o n d i f f e r e n c e r e l a t i v e to t h a t i n d i c a t e d by the e l e c t r o n s c a t t e r i n g experiment.  The l o c a t i o n of the maximum at about 2 fm  i s the same as suggested by the SCSPP c a l c u l a t i o n . There i s a s e r i o u s d e f i c i e n c y i n the e l e c t r o n s c a t t e r i n g data i n the present case.  The a v a i l a b l e e l e c t r o n s c a t t e r i n g a n a l y s e s were not d e s i g n e d  to minimise s y s t e m a t i c  problems i n the d a t a , and u n c e r t a i n t i e s i n the  e l e c t r o m a g n e t i c form f a c t o r s are p r e s e n t i n d i f f e r e n t ways f o r the two  40  i  —  1 r  48.3 MeV 20  o  0  a.  CM  'tf-20 h .Q.  -40  0  1 2 3 4 5 RADIUS (fm)  F i g u r e 4.15  P r o t o n matter d e n s i t y 48.3 MeV i r , +  1 4 |  N/  w i t h Set E f 5 0 f  1 2  C  differences (  l l f  pp(r)-  1 2  pp(r))  r a t i o s ( c f . f i g u r e 3.24).  o p t i c a l parameters.  derived  from  Best f i t FL d e n s i t y  E r r o r envelope i s s t a t i s t i c a l .  Dashed curve i s d i f f e r e n c e of model independent e l e c t r o n s c a t t e r i n g derived density  p r o t o n matter d e n s i t y for  11+  N.  'standard' n u c l e a r  Also  i n C and the best 1 2  a v a i l a b l e MG  shown a r e SCSPP c a l c u l a t i o n s w i t h a  p o t e n t i a l developed i n the C a 4 0  region  [H0D85].  184  40  1 1 r  T  48.3 MeV 20  ro  0  •»  A  A  Q. CN  Q .  '^-20  —40  0  1 2 3 4 5 RADIUS (fm)  Figure  Same as f i g u r e 4.15 MeV  ir+, 0/ 16  1Jt  for (  l l t  4.16  p p ( r ) - P p ( r ) ) d e r i v e d from 1 2  N r a t i o s w i t h Set EF50f ( c f . f i g u r e  48.3 3.25).  185  40  1 1 1 r  T  62.8 MeV 20  to I  0 CM  7T  '_-20  Ratio  - - (e.e)  Q .  ' 40  SCSPP J  0  1  Same as f i g u r e 4.15  for (  L  2 3 4 5 RADIUS (fm)  Figure  MeV  +  1 4  4.17  p (r)p  1 2  p ( r ) d e r i v e d from p  i r + . ^ N / ^ C r a t i o s w i t h Set EC65f  62.8  ( c f . f i g u r e 3.27).  40 62.8 MeV 20  0 Q. CM  7T  L-20  Ratio  -- (e,e)  Q .  * -40  j  0  Same as f i g u r e 4.15  SCSPP  i  i  i  i  1 2 3 4 5 RADIUS (fm)  Figure  MeV  +  f o r ( *p 1J  4.18  (r)-  1 2  p ( r ) ) d e r i v e d from D  i r + , 0 / * N r a t i o s w i t h Set E065f ( c f . f i g u r e 16  ll  48.3  3.28).  nuclei. the 1J  *N  T h i s a l o n e w i l l produce l a r g e model e r r o r s .  simple 2 parameter MG data may  Furthermore, use  form i n the a n a l y s i s of the e l e c t r o n s c a t t e r i n g  give misleading  r e s u l t s , e s p e c i a l l y at small nuclear  through l a c k of completeness of the d e n s i t y  form.  In s h o r t ,  radii,  though  u n c e r t a i n t i e s are shown i n the e l e c t r o n s c a t t e r i n g matter d e n s i t i e s are p u b l i s h e d ) ,  of  of f i g u r e s 4.15  through 4.18,  the  r e l a t i v e errors  no (none  are  d o u b t l e s s f a r l a r g e r than those of the e l e c t r o n s c a t t e r i n g matter d e n s i t i e s shown i n f i g u r e s 4.13 a n a l y s i s was  4.14  f o r the oxygen i s o t o p e s , where a p r e c i s i o n  undertaken.  T a b l e 4.12 ratios.  and  shows d e r i v e d RMS  r a d i i f o r various  These r a d i i are s e l f c o n s i s t e n t ;  parameter s e t s  i n c h a p t e r V i t w i l l be  they are a l s o c o n s i s t e n t w i t h the r e s u l t s from other probes.  No  has  been shown w i t h set EBLE65co; those c a l c u l a t i o n s would not  the  shape of the e x p e r i m e n t a l r a t i o s , thereby s u g g e s t i n g an  o p t i c a l parameter s e t f o r Figure at 48.3 the  MeV  same RMS  4.19  w i t h the radius.  (best  through 4.18)  shape of the c r o s s  f i t ) FL matter d e n s i t y  It i s evident  i s to s h i f t  calculation  converge near  inappropriate  section ratio and  (Set  a simple MG  EF50f)  density  of  that a s i z e determination f o r t h i s  form would y i e l d an  I n c i d e n t a l l y , the  seen t h a t  N.  compares the  n u c l e u s w i t h a simple MG radius.  11+  and  i n c o r r e c t v a l u e f o r the  e f f e c t of the added F o u r i e r the n u c l e o n d e n s i t y  RMS  terms ( f i g u r e s  towards the c e n t e r  of  4.15  the  n u c l e u s , r e l a t i v e to what might o t h e r w i s e have been expected.  4.5  Summary Chapter IV has  cross  s e c t i o n and  seen d i s c u s s i o n of the o p t i c a l model a n a l y s i s of  r a t i o d a t a p r e s e n t e d i n chapter I I I .  the  Some overview of  the  Table 4.12 Proton Matter D i s t r i b u t i o n Differences: C , N , 0 Experiment 12  Energy  Optical Parameter Set  48.3 62.8 62.8  EF50f EC65f EBLE65f  48.3 62.8 62.8  EF50f E065f EBLE65f  llf  16  Ratio  Density  FL  1 6 . 1 4 0  N  nmaxt,nx  6<r > 2  1 / 2  5 3 3  -3 -3 -3  .106(11) .002(94) .017(75)  4 4 3  -3 -3 -3  .162(11) .198(51) .157(12)  T y p i c a l v a l u e s : R =6.0 fm (FB); a=1.5 fm (FL) E r r o r s shown i n p a r e n t h e s i s are s t a t i s t i c a l . Difference i s wrt the r e f e r e n c e : RMS r a d i u s o f h e a v i e r n u c l e i l e s s t h a t o f the l i g h t e r . c  tt  30  50  70 6  90  110  130  150  (degrees, CM)  Figure  4.19  R a t i o c a l c u l a t i o n (SMC81, Set E f 5 0 f ) w i t h  the best  ( c f . f i g u r e 4.15) and a MG ( s o l i d ) d e n s i t y w i t h r a d i u s . I n t h i s case use of s o l e l y to a s e r i o u s o v e r e s t i m a t i o n  f i t density  the same RMS  the MG form would l e a d  of the N 1 4  RMS  radius.  190  a v a i l a b l e o p t i c a l models has coordinate The  been made, w i t h p a r t i c u l a r emphasis on the  SMC  space p o t e n t i a l . 1 2  C, B  experiments used the SMC79 form of the p o t e n t i a l to  n  e x t r a c t d i f f e r e n c e s i n p r o t o n matter r a d i i between those n u c l e i , assuming a modified  G a u s s i a n (MG)  density  form.  The  systematic  u n c e r t a i n t i e s i n these  e x t r a c t i o n s were examined: i n p a r t i c u l a r , s e n s i t i v i t i e s to d e n s i t y ( t a b l e s 4.3  and  4.4)  and  o p t i c a l p o t e n t i a l parameter v a l u e s ( t a b l e 4.5)  found to be of s i z e comparable to the a n a l y s i s d i d not  incorporate  s t a t i s t i c a l e r r o r s (=.020 fm).  F o u r i e r d e s c r i p t i o n s of the d e n s i t i e s :  p r o t o n matter d i f f e r e n c e d i s t r i b u t i o n a n a l y s i s (as a f u n c t i o n of radius)  was  attempted.  form  T h i s was  deemed the w i s e s t  were The  no  nuclear  course as the 20%  1 0  B  content i n the boron t a r g e t would have made i n t e r p r e t a t i o n of such a n a l y s e s difficult. The  1 8  0,  1 6  0 r a t i o a n a l y s e s used the SMC81 form of the o p t i c a l  p o t e n t i a l , s i m p l e assumptions about the neutron matter d e n s i t y between the i s o t o p e s ,  and  p r o t o n matter d e n s i t y  d i f f e r e n c e between the i s o t o p e s .  differences  a F o u r i e r L a g u e r r e (FL) p a r a m e t e r i s a t i o n  o p t i c a l parameter u n c e r t a i n t y  was  discussed  and  The  of  the  e f f e c t s of  noted to be of the same  o r d e r as the e r r o r bands ( s t a t i s t i c a l + completeness) f o r the  density  differences. The derive  1 2  C,  1 1 +  N,  the d e n s i t y  1 6  0 r a t i o s were a n a l y s e d w i t h the SMC81 p o t e n t i a l to d i s t r i b u t i o n d i f f e r e n c e s between the n u c l e i under  the  assumption of c o r r e l a t e d n e u t r o n - p r o t o n matter d e n s i t y d i s t r i b u t i o n differences. reference No  ( C 1 2  The  density  f o r a/ a 1H  12  d i f f e r e n c e s were evaluated and  1 6  0  f o r o/ a), 16  lk  but  p r e c i s i o n e l e c t r o n s c a t t e r i n g measurement of the  wrt  the  appropriate  i l l u s t r a t e d wrt 1I+  N  charge  1 2  C.  density  d i s t r i b u t i o n , w i t h which to compare, i s c u r r e n t l y a v a i l a b l e .  Hence  comparison of the p i o n r a t i o d e r i v e d p r o t o n matter d e n s i t y d i f f e r e n c e s w i t h those  from e l e c t r o n s c a t t e r i n g i s d i f f i c u l t  c u r r e n t r e s u l t s suggest to  and  perhaps m i s l e a d i n g .  t h a t the p r o t o n matter i n *N 1J  i s concentrated  The closer  the n u c l e a r c e n t e r than i n d i c a t e d by the a v a i l a b l e ( n o n - p r e c i s i o n )  e l e c t r o n s c a t t e r i n g data. r a d i u s , but  The  c o n c e n t r a t i o n i s at about 2 fm:  c l o s e to  the  to a g r e a t e r extent than t h a t , p r e d i c t e d by a SCSPP  calculation. In each of the a n a l y s e s , f i t t i n g  of the o p t i c a l p o t e n t i a l parameters  (but i n t h i s case not the d e n s i t y d i s t r i b u t i o n parameters) to the nucleus'  d i s t r i b u t i o n of 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 was  reference made.  192  Chapter V Discussion and Summary  5.1 Overview The  r a t i o s of p o s i t i v e p i o n e l a s t i c  s e c t i o n s have been used nuclei  n  B,  1I+  N  and  1 8  0  to e x t r a c t p r o t o n matter  r e l a t i v e to the  These measurements have been performed and 65 MeV. used  Two  on  s e c t i o n d a t a at 48.3 The  1 2  C , B ' a t 38.6 1 1  and 62.8  MeV  experimental  cross  d e n s i t y i n f o r m a t i o n on  'reference n u c l e i '  1 2  C  and  1 6  the  0.  a t a v a r i e t y of e n e r g i e s between 35  types of apparatus were used.  i n experiments  spectrometer.  scattering d i f f e r e n t i a l  MeV  A IT  and 47.7  +  stopping telescope  MeV.  were taken w i t h the QQD  techniques were designed  R a t i o and  was  cross  magnetic to  minimise  s y s t e m a t i c u n c e r t a i n t y i n the r a t i o d a t a ( s y s t e m a t i c u n c e r t a i n t i e s which i n v a r i a b l y occur i n a b s o l u t e c r o s s s e c t i o n measurement). were performed  i n the meson h a l l of the TRIUMF  In Chapter  I, the expected  d i s t r i b u t i o n excess was regime, the p r o t o n RMS  facility.  s e n s i t i v i t y of i r s c a t t e r i n g to p r o t o n +  outlined. matter  I t was  shown t h a t , i n the low  The  energy  r a d i u s , r+, a f f e c t s the n o r m a l i s a t i o n of a  c r o s s s e c t i o n r a t i o c a l c u l a t i o n i n a simple way. o f , a t l e a s t , r+.  A l l measurements  T h i s a l l o w s measurement  r a t i o s were a n a l y s e d w i t h a v a r i e t y of  p o t e n t i a l s to d e r i v e ( r e l a t i v e ) RMS d i f f e r e n c e s of the p r o t o n matter  r a d i i and,  i n some c a s e s ,  density distributions.  p o t e n t i a l s were o p t i m i s e d by f i t t i n g  The  optical radial optical  to the e l a s t i c c r o s s s e c t i o n d a t a .  In t h a t which f o l l o w s , the a n a l y s i s which has been d e s c r i b e d i n some d e t a i l i n e a r l i e r c h a p t e r s w i l l be summarised, and  the r e s u l t s of the  p r e s e n t experiments compared t o those a r r i v e d a t v i a o t h e r n u c l e a r measurement  density  techniques.  5.2 O p t i c a l P o t e n t i a l Implicit  Fitting  i n the a n a l y s i s  d i s c u s s e d i n t h i s t h e s i s has been t h e  e x i s t e n c e o f an o p t i c a l p o t e n t i a l which produces e l a s t i c s c a t t e r i n g  cross  sections  There  are  i n r e a s o n a b l e agreement w i t h the measured c r o s s  sections.  a number o f such p o t e n t i a l s , but none o f them reproduces w i t h complete  s u c c e s s the e l a s t i c c r o s s principles.  sections  o f t h i s energy regime e n t i r e l y from  As a r e s u l t , one must o f t e n  parameters to e x p e r i m e n t a l c r o s s  r e s o r t to f i t t i n g  first  the p o t e n t i a l  s e c t i o n data to a r r i v e a t r e a s o n a b l e  agreement between t h a t data and the c a l c u l a t i o n .  The s i t u a t i o n i s somewhat  a m e l i o r a t e d i n the c o n s i d e r a t i o n  r a t i o s , since  model b e t t e r  predicts  of cross  trends i n , r a t h e r  In g e n e r a l , the c l o s e r  that  section  than a b s o l u t e , c r o s s  the o p t i c a l  sections.  an o p t i c a l parameter f i n d s i t s o r i g i n i n a  first  p r i n c i p l e , the l e s s i t should v a r y i n an o p t i c a l p o t e n t i a l f i t t o  cross  section data.  scattering fitting An  length  described  F o r example, the r e a l p a r t s  s h o u l d be c l o s e  to t h e i r free nucleon values.  i n t h i s t h e s i s , t h i s has f o r the most p a r t  i n t e r e s t i n g r e s u l t common t o the c r o s s  i s an i n c r e a s e length,  r e ( b ) , within  the n u c l e u s .  This  been the c a s e .  Q  o f the s wave s c a t t e r i n g  Only s l i g h t l y  b  Q  repulsive  ir n u c l e o n s c a t t e r i n g , t h e ( i s o s c a l a r ) s wave s c a t t e r i n g  becomes much more r e p u l s i v e when the e f f e c t s of n u c l e a r s h o r t correlations,  however,  i s not s u r p r i s i n g , s i n c e  ( i e : B ) i s an e f f e c t i v e s c a t t e r i n g l e n g t h . when c o n s i d e r i n g  In the  section f i t t i n g herein,  i n the magnitude o f the r e a l p a r t  Q  of the s and p wave  i n particular Pauli repulsion,  are considered  range  [ERI70].  The  t r u e a b s o r p t i o n parameters, i n most c a s e s , have been g r e a t l y i n c r e a s e d i n magnitude, i n d i c a t i n g t h a t a d d i t i o n a l s t u d i e s of the e f f e c t s of t r u e a b s o r p t i o n i n the n u c l e u s [M0I84] c o n t i n u e to be warranted. The s e n s i t i v i t y of the c r o s s s e c t i o n s i n the and s-p i n t e r f e r e n c e r e g i o n s t o the E L 2  2  Coulomb-nuclear  parameter,  number of f i g u r e s , i n p a r t i c u l a r f i g u r e 4.12  X, can be seen i n a  ( t a b l e 4.9).  In o r d e r to  determine t h i s parameter w e l l , and hence a v o i d the e f f e c t s of some of i t s correlations  [BR081, SEK83, SEK83a] t o the o t h e r s ( I n p a r t i c u l a r the p wave  s c a t t e r i n g volumes), v e r y s m a l l a n g l e data on r e f e r e n c e c r o s s should be accumulated.  sections  In p a r t i c u l a r , the t r e n d w i t h n u c l e u s i n t h i s  a n g u l a r r e g i o n s h o u l d be s t u d i e d a t e n e r g i e s such as 65 and 80 MeV  w i t h the  aim of improving the o p t i c a l model d e s c r i p t i o n . The o p t i c a l model a t 65 MeV Experiments  appears to be e n c o u n t e r i n g some problems.  [JEN83, SOB84c] have i n d i c a t e d t h a t A-hole p r o c e s s e s s h o u l d not  c o n t r i b u t e s i g n i f i c a n t l y , a t t h i s energy, t o the e l a s t i c s c a t t e r i n g . o t h e r p r o c e s s e s may  need  t o be c o n s i d e r e d to p a r a m e t e r i s e the o p t i c a l  potential satisfactorily i n this  5.3  1 2  C, B  There was,  1 2  regime.  Experiment  n  The  Some  C,  1 1  B experiment was  the f i r s t  i n f a c t , one e a r l i e r experiment  poor s t a t i s t i c s and was  made to e x t r a c t the RMS  r e l a t i v e to t h a t i n m a t t e r d e n s i t i e s was  1 2  C  +  [DYT78],  but i t s u f f e r e d  from  not c o n c e i v e d , nor d i d i t attempt, to e x t r a c t p r o t o n  matter d e n s i t y d i s t r i b u t i o n i n f o r m a t i o n . a n a l y s i s was  of the i r r a t i o experiments.  ( r+). 1 2  assumed.  In the c u r r e n t  experiment,  r a d i u s of p r o t o n matter i n  A m o d i f i e d Gaussian (MG)  1 X  B  ( r ) n  +  form f o r the  A r i g o u r o u s a n a l y s i s r e q u i r e d the use of a  r e l i a b l e pion o p t i c a l potential; parameter s e t s were used  the SMC79 p o t e n t i a l was  Two  i n the a n a l y s i s ; the r e s u l t s were p r e s e n t e d i n  t a b l e 4.2.  The  -.064  (8).  The unweighted v a l u e i s -.050  -.064  (28) we  r e s u l t s y i e l d e d a weighted  average (28).  f o r 0 r+- r.(.) of 1  12  The v a l u e ( r n  1  +  then take as being the best v a l u e .  some degree,  chosen.  2  r ) = +  The e r r o r r e f l e c t s ,  to  both s t a t i s t i c a l and s y s t e m a t i c u n c e r t a i n t i e s .  A simple A '' 1  3  behaviour < r  Q  >  [COL67a] of the RMS  1 / 2  = 0.82  (N+Z)  1 / 3  charge r a d i u s g i v e s :  + 0.58  (5.1)  which p r e d i c t s a v a l u e f o r the matter r a d i u s change of Ar+=0.058 fm between the i s o t o n e s .  T h i s agrees, w i t h i n the e x p e r i m e n t a l u n c e r t a i n t y , w i t h the  v a l u e measured i n t h i s  experiment.  A number of t e s t s were performed r a t i o method.  to assure the i n t e g r i t y of the  These I n c l u d e d **r+ e v a l u a t i o n s u s i n g v a r i o u s d e n s i t y  d i s t r i b u t i o n s and o p t i c a l parameter s e t s , as w e l l as v a r i o u s v a l u e s f o r a few m i s c e l l a n e o u s parameters. fit  v a l u e of  statistical  1 1  r+,  (i.e.:  In each of these i t was  * * ? ) , was  f i t t i n g errors.  found t h a t the best  not changed by more than the  quoted  T h i s i s not to say t h a t n u l l e f f e c t s were  seen. There e x i s t a t p r e s e n t s e v e r a l o t h e r v a l u e s of the charge 1 1  B  to which the r e s u l t s of t h i s experiment  radius, r e c a l l ,  may  be compared.  r a d i u s of  (The  charge  i s o b t a i n e d by the a d d i t i o n of the s i n g l e p r o t o n r a d i u s to  the n u c l e a r p r o t o n matter  r a d i u s i n quadrature.)  T a b l e 5.1  shows t h a t the  v a r i o u s v a l u e s a r e w i t h i n r e a s o n a b l e agreement, although t h a t g i v e n by R i s k a l l a quotes p r e c i s e to date.  no e r r o r .  The r e s u l t of the present experiment  i s the most  196  Table 5.1 Measured Values of for  U  1 1  r+- r+ 1 2  B  Experiment  Method  1 1  1 2  r+  (fm)  Present  [BAR80]  IT  S t o v a l l et a l . t  [ST066]  (e",e-)  -.05(13)  Riskalat  [RIS71]  (e",e-)  -.107  A l k a z o v and Domchenkovt [ALK83]  (e",e-)  -.032(33)  Auerbach  [AUE80]  O l i n et a l . *  t Assuming  [0LI81]  1 2  C  1 2  C  -.064(28)  HF I  -.048  HF I I  -.077  -  atoms  -.11(4)  charge r a d i u s of 2.470(5) fm [RUC82]  and p r o t o n RMS * Assuming  u  +  ratio  r+-  r a d i u s of .7754 fm  [CHA76].  charge r a d i u s of 2.4832(18)  fm [RUC82]  197  The p r e s e n t 1 2  C,  r e s u l t f o r the matter r a d i u s of ^ B ,  r e l a t i v e to t h a t of  l e a d s to no d i s c r e p a n c i e s w i t h e x i s t i n g charge r a d i u s i n f o r m a t i o n .  This  i s t r u e a t l e a s t to the p r e c i s i o n quoted. Note t h a t the TRIUMF r e s u l t a l l o w s i n parameter s e t s and the model used. l a r g e l y a f u n c t i o n of a spread (38.6, 47.7 MeV). given  f o r , i n some measure, u n c e r t a i n t i e s  The f i n a l quoted u n c e r t a i n t y i s  of the r e s u l t s over the data  T h i s dependence i s a p p a r e n t l y  sets with  energy  a random e f f e c t , e s p e c i a l l y  the r e s u l t s of the f o l l o w i n g s e c t i o n s .  5.4 CNO 5.4.1  1 8  2  Experiments  0,  1 6  0 Experiments  In these experiments, the proton matter d e n s i t y of r e l a t i v e to t h a t of  1 6  0.  The a n a l y s i s i n c o r p o r a t e d  1 8  0  was measured  a F o u r i e r expansion f o r  the d e n s i t y d i f f e r e n c e , a p p l i e d to the proton matter d i s t r i b u t i o n of the 1 8  0.  The r e f e r e n c e  simple with  proton  MG form w i t h  and the n e u t r o n d e n s i t i e s were assumed to be of  p r e v i o u s l y measured r a d i i .  the SMC81 p o t e n t i a l .  The best  ( r+- r+)=.084±.008, with 18  16  seen i n t a b l e Figures  N0R82J  (average) value was found to be  the e r r o r r e f l e c t i n g  4.13 and 4.14 p r e s e n t e d Ap (r)=  1 8  p  p (r)p  1 6  +  d e r i v e d proton  p (r).  The T T r e s u l t s agree w i t h  p  Table  i n the v a l u e s  energy o r assumed  1 8  0  neutron density.  [CHA76] has been u n f o l d e d  density difference  5.2 shows the RMS  those of e l e c t r o n s c a t t e r i n g [MIS79,  and muonic atom [BAC80] experiments.  distributions.  the spread  5.2  distributions: radii.  C a l c u l a t i o n s were performed  There i s l i t t l e  The proton  s e n s i t i v i t y to  charge form f a c t o r  from the e l e c t r o n s c a t t e r i n g charge  Table 5.2 Proton Matter D i s t r i b u t i o n Differences: 0- 0 1 8  Energy  O p t i c a l Parameter Set  Density  nmaxt,nx  1 6  6<r2>l/2  62.8  E65 EIM65 E65 E65  FB FL FLttt FL  4 3 3 3  0 0 0 0  .091(26) .079(12) .100(37) .097(28)  48.3  EIM50 EIM50  FL FB  3 3  0 0  .084(11) .087(15)  Weighted Average Unweighted Average Precision Electron Scattering  .084(7) .089(5) [MIS79, NOR82] [SCH75]  .077(5) .075(8)  [BAC80]  .079(5)  BR079b] BR079b]  .011 .077  [WES84]  .107  ( e q u a t i o n 5.1) [C0L67a]  .088  Muonic Atom S t u d i e s SCSPP ( (  1 6  1 6  0 + (sd) ) 0 + (psd))  [BR079a [BR079a  2  Semi E m p i r i c a l P a r a m e t e r i s a t i o n Simple P a r a m e t e r i s a t i o n  t T y p i c a l v a l u e s : R =6.0 fm (FB); a=1.5 fm ( F L ) tt E r r o r s shown i n p a r e n t h e s i s a r e s t a t i s t i c a l , t t t I n t h i s case o n l y , the 0 n e u t r o n matter RMS r a d i u s was s e t equal to i t s proton matter RMS r a d i u s . c  1 8  tt  It charge  i s i n t e r e s t i n g to note  [BR079a, BR079b], t h a t the shape o f the  d e n s i t y d i s t r i b u t i o n d i f f e r e n c e s between n u c l e i appears  generated  by the e f f e c t s o f i n t e r a c t i o n s i n v o l v i n g up t o 6 - p a r t i c l e  c o n f i g u r a t i o n s i n the 0 wave f u n c t i o n .  ( i e : The  1 8  The T T  +  data e r r o r envelopes  1 6  The  encompass those r e s u l t i n g from a ±10% poorly  terms i n t h e p o t e n t i a l a r e d e r i v e d from c r o s s s e c t i o n  p i o n probes  the n u c l e u s a t r a d i i g r e a t e r than 1.5 fm, l i m i t e d by  c h a r a c t e r i s e d by X-1.5 fm. FRI83] i s determined  CNO  resolution  The r a d i a l s e n s i t i v i t y o f the probe by d e t a i l s o f the i n t e r a c t i o n ' s dependence on  the n u c l e a r d e n s i t i e s and t h e i r  5.4.2  determined  fitting.  momentum t r a n s f e r , and samples d e n s i t y d i f f e r e n c e s w i t h a r a d i a l  [FRI75,  4-hole  0 c o r e i s not i n e r t . )  o p t i c a l parameter s e t u n c e r t a i n t y , p r o v i d e d t h a t otherwise imaginary  t o be  derivatives.  Experiments  In these experiments  t h e r e f e r e n c e n u c l e i were taken as  1 2  C and  1 6  0.  the p r o t o n matter d e n s i t y d i s t r i b u t i o n .of ^ N was found w i t h r e s p e c t t o these.  The r e f e r e n c e p r o t o n matter  d e n s i t i e s were of FL forms d e r i v e d from  p r e c i s i o n e l e c t r o n s c a t t e r i n g experiments.  Neutron matter  d e r i v e d from  about Coulomb r e p u l s i o n e f f e c t s  these w i t h a simple assumption  d e n s i t i e s were  which I n c r e a s e the p r o t o n d i s t r i b u t i o n r a d i u s ( t y p i c a l l y by 1.5 %) r e l a t i v e to t h e neutron r a d i u s . The  C a l c u l a t i o n s were made w i t h the SMC81 p o t e n t i a l .  r e s u l t s f o r the RMS r a d i i a r e r e t a b u l a t e d i n t a b l e 5.3, a l o n g w i t h  r e s u l t s o b t a i n e d w i t h other  probes.  At 48.3 MeV, the i r r a t i o r e s u l t s were e x t r a c t e d w i t h a p o t e n t i a l +  parameter s e t f i t t e d distributions.  s i m u l t a n e o u s l y to  1 2  C and  The r e s u l t i n g v a l u e s f o r (  l l +  1 6  r +  0 e l a s t i c cross section 1 2  r + ) and (  1 6  r - ' * r ) are 1  +  +  Table 5.3 Proton Matter Distribution Differences: C, N, 0 Experiment 1 2  Energy  Optical Parameter Set  48.3 62.8 62.8  EF50f EC65f EBLE65f  48.3 62.8 62.8  EF50f E065f EBLE65f  1 , f  1 6  Ratio  Densityt (nmax,nx)  *N: C  FL(5,-3) (3,-3) (3,-3)  ll  12  16  0 :  U  N  (4,-3) (4,-3) (3,-3)  Weighted Average Unweighted Average Muonic Atom  U  r  r (fm)tt 1  +  .106(11) .002(94) .017(75)  r - *r (fm)ft 1 J  +  .159(7) .192(44)  .082(11) .159(24)  Electron Scattering  [SCH75] .074(21) [ALK83]** .139(33)  .180(22) .116(34)  Semi E m p i r i c a l P a r a m e t e r i s a t i o n  [WES84]  .124  .099  [COL67a]  .106  .096  Simple  t tt * **  Parameterisation  T y p i c a l v a l u e s : R =6.0 fm ( F B ) ; a=1.5 fm ( F L ) E r r o r s shown i n p a r e n t h e s i s a r e s t a t i s t i c a l . Value assumes t h e b e s t v a l u e o f r + - r + o f .254(10) fm [KIM78, SIC80, RUC82]. Assumes C charge r a d i u s o f 2.470(5) fm [RUC72]. 0 charge r a d i u s o f 2.709(9) fm [KIM78, SIC80] c  1 6  1 2  1 6  1 2  +  .148(15)* .252(95)* .237(75)*  .092(15)* .162(11) .056(52)* .198(51) .097(17)* .157(12) .098(8) .062(44)  [RUC82, SCH80] [SCH80, DUB74]  1 6  2  +  201  i n t e r n a l l y c o n s i s t e n t , and c o n s i s t e n t w i t h the r e s u l t s o f t h e o t h e r The  probes.  c a l c u l a t e d r a t i o s f i t t h e data w e l l . The  fitted  r e s u l t s a t 62.8 MeV were d e r i v e d w i t h o p t i c a l parameter s e t s  separately to  1 2  C and  1 6  0 e l a s t i c cross section d i s t r i b u t i o n s .  The  d e r i v e d RMS r a d i i have l a r g e r e r r o r s than do those o b t a i n e d a t 48.3 MeV: t h i s i s merely  a statistical effect.  The form o f t h e c a l c u l a t e d r a t i o s do  not f i t the r a t i o d a t a as w e l l a t t h i s energy, inadequacies The .159  i n the o p t i c a l  p r o b a b l y because of  potential.  best v a l u e f o r ( r - » r + ) i s .098 ± .044 fm and f o r ( 1 6  l l  1 1 +  +  r+-  1 2  r ), +  ± .044 fm. The d e r i v e d d e n s i t y d i f f e r e n c e d i s t r i b u t i o n s  i n f i g u r e s 4.15 through errors.  4.18.  l t  * p ( r ) - p ( r ) a r e shown 1 2  The e r r o r s t h e r e do not r e f l e c t  completeness  The s a l i e n t f e a t u r e common t o the d e n s i t y d i s t r i b u t i o n  is a shift  differences  i n t h e p r o t o n matter d e n s i t y towards the i n t e r i o r o f t h e n u c l e u s ,  r e l a t i v e t o t h a t suggested  by t h e a v a i l a b l e e l e c t r o m a g n e t i c measurements.  Such an e f f e c t c o u l d be c r e a t e d by d e t a i l s  (such as might be produced by  phenomena such as "core p o l a r i s a t i o n " ) i n the o r t h o g o n a l t o t h e d e n s i t y form used s c a t t e r i n g from ^ N .  1I+  N n u c l e a r d e n s i t y which a r e  i n the a n a l y s e s o f e l e c t r o n  An a n a l y s i s o f  11+  N ( e , e ) N experiments 1I+  w i t h a FB  d e n s i t y form f o r the p r o t o n charge d e n s i t y (model independent might h e l p t o r e s o l v e the apparent  discrepancy.  elastic  analysis)  Such an a n a l y s i s would,  however, be c o m p l i c a t e d by the s p i n and l a c k of s p h e r i c i t y o f the  1I+  N  nucleus. The T T  +  r a t i o d e r i v e d d e n s i t i e s agree w i t h one another  i n the r a d i a l  r e g i o n where momentum t r a n s f e r o f t h e p i o n e l a s t i c s c a t t e r i n g p r o c e s s a l l o w s s e n s i t i v i t y t o be e x h i b i t e d : r ? 1.5 t o 2 fm.  I n s i d e of t h i s , t h e  202  s t a t i s t i c a l e r r o r bands do not encompass the v a r i a t i o n s i n the d e r i v e d d e n s i t i e s , as might be expected.  The completeness e r r o r i s a s i g n i f i c a n t  c o n t r i b u t i o n t o the o v e r a l l e r r o r band e s p e c i a l l y a t s m a l l n u c l e a r  radii.  Furthermore, dependence o f the o p t i c a l p o t e n t i a l upon the d e n s i t y d e r i v a t i v e s may appear to enhance the s e n s i t i v i t y o f the p i o n i n the c e n t e r of t h e n u c l e u s  over what I t p h y s i c a l l y can be, as t h e d e r i v a t i v e s o f t h e  F o u r i e r components a r e t y p i c a l l y l a r g e i n t h i s  5.5  region.  Summary The  s u b j e c t o f t h i s t h e s i s has been the study  of t h e use of p o s i t i v e  p i o n e l a s t i c s c a t t e r i n g d i f f e r e n t i a l c r o s s s e c t i o n r a t i o s i n the determination moments.  o f n u c l e a r p r o t o n matter d i s t r i b u t i o n d i f f e r e n c e s and t h e i r  The measurements have been r e l a t i v e t o n u c l e i whose matter  d i s t r i b u t i o n s and a b s o l u t e c r o s s s e c t i o n s have been used as r e f e r e n c e s . use o f the r a t i o o f c r o s s s e c t i o n s , r a t h e r than the a b s o l u t e  cross  The  sections  themselves, minimises t h e e f f e c t s of u n c e r t a i n t i e s i n the understanding  of  the p i o n n u c l e a r i n t e r a c t i o n i n our e x t r a c t i o n o f d e n s i t y d i f f e r e n c e information.  Furthermore, the measurement o f c r o s s s e c t i o n r a t i o s i s  i n s e n s i t i v e t o many s y s t e m a t i c  experimental  e f f e c t s encountered i n the  measurement of the a b s o l u t e c r o s s s e c t i o n s . We have seen t h a t RMS r a d i i e x t r a c t e d from the p i o n r a t i o s a t low energy a r e c o n s i s t e n t ( w i t h i n 1 standard o t h e r methods. as f o l l o w s :  d e v i a t i o n ) w i t h the r e s u l t s of  The p r o t o n matter r a d i u s d i f f e r e n c e s which we o b t a i n are.  ( r+n  1 2  r ) = -.064 (28) fm  {-.032(33) [ALK83] }  +  (  lt  r +  _12  r + )  =  >  (  16  r +  _lt  r + )  =  a  (  18  r +  _16  r + )  =  >  The e r r o r s r e f l e c t  0  9  5  0  8  8  (  4  4  )  f  m  9  (  4  4  )  f  m  4  (  7  )  f  { .074(21) [SCH75] } { .180(22) [SCH75] } ( 0,16 ) 1 8  m  { .077 (5) [MIS79] }  0  s t a t i s t i c a l and, t o a l a r g e e x t e n t , s y s t e m a t i c  u n c e r t a i n t i e s i n the q u a n t i t i e s .  The b e s t e l e c t r o n s c a t t e r i n g r e s u l t s a r e  shown i n b r a c e s . Analyses of the  1 8  0,  1 6  0 experiments i n terms o f F o u r i e r p r o t o n matter  d e n s i t i e s agree w e l l w i t h p r e c i s i o n "model independent" e l e c t r o n  scattering  r e s u l t s , i n the r e g i o n i n which t h e p i o n can be s e n s i t i v e to the n u c l e a r p r o t o n matter  distributions.  S i m i l a r a n a l y s e s of the C , 1 2  11+  N,  1 6  0 experiments i n d i c a t e a s h i f t i n  the p r o t o n matter d e n s i t y towards the n u c l e a r c e n t e r , r e l a t i v e t o t h a t suggested by e l e c t r o n s c a t t e r i n g experiments.  We r e i t e r a t e , however, t h a t  a n a l y s e s o f t h e s e e l e c t r o n s c a t t e r i n g experiments were n o t model independent, and may w e l l generate d e n s i t i e s which a r e i n c o r r e c t a t s m a l l r a d i u s ( h i g h momentum t r a n s f e r : r K. 2 fm.)  5.6 E p i l o g u e We p r e s e n t these r e s u l t s as testimony to the a b i l i t y o f the i r probe App(r) r e l i a b l y .  We observe some o p t i c a l parameter  +  to  sensitivity.  T h i s i s minimal, e s p e c i a l l y when c r o s s s e c t i o n f i t t i n g i s used t o determine the r e f e r e n c e n u c l e u s o p t i c a l parameters.  S t a r t i n g w i t h the e l e c t r o n  s c a t t e r i n g d e n s i t i e s one might have proceeded t o study i s o t o p i c dependence i n the o p t i c a l p o t e n t i a l [DYT78, BLE83].  Without  measurements such s t u d i e s a r e o f l i m i t e d scope.  r e l i a b l e neutron density T h i s s t r e n g t h e n s the  204  o r i g i n a l m o t i v a t i o n of t h i s work found i n the p r o v i s i o n of c o r r o b o r a t i o n f o r the I T " n e u t r o n d e n s i t y measurements [J0H79, GYL85].  In f a c t , the  indication  of the p r e s e n t work i s t h a t i t i s c o m p l i c a t i o n s i n the o p t i c a l p o t e n t i a l  due  to changes i n the Coulomb i n t e r f e r e n c e s which generate much of the observed o p t i c a l model s e n s i t i v i t y , c o m p l i c a t i o n s whose importance  i s greatly  d i m i n i s h e d when n e g a t i v e p i o n s are used i n the study of neutron differences.  distribution  T h i s i s an e x c i t i n g p r o s p e c t , as few o t h e r t e c h n i q u e s access  the moments of n e u t r o n d i s t r i b u t i o n s i n such an e l e g a n t manner. N e v e r t h e l e s s , i t i s c l e a r t h a t our understanding of the p i o n nucleus interaction i s s t i l l  not f a r advanced  from i t s i n f a n c y .  h i g h e r o r d e r a b s o r p t i v e and c o r r e l a t i v e e f f e c t s must be quantified.  Experiments  In p a r t i c u l a r , rigorously  c u r r e n t l y e x p l o r i n g more e x o t i c r e a c t i o n channels,  such as those of double and s i n g l e charge exchange [ALT85, C0084], scattering may  [TAC84],  inelastic  and t o t a l a b s o r p t i o n c r o s s s e c t i o n measurements  w e l l p r o v i d e e s s e n t i a l i n s i g h t i n these d i r e c t i o n s .  [NAV83]  Together w i t h the  h i g h p r e c i s i o n e l e c t r o n s c a t t e r i n g charge d e n s i t y measurements now  becoming  a v a i l a b l e , r e l a t i v e neutron and p r o t o n d e n s i t y d i f f e r e n c e i n f o r m a t i o n d e r i v e d from experiments g e n e r a t i o n of experiments nuclear  interaction.  such as d e s c r i b e d h e r e i n s h o u l d a i d the  new  to advance the study of many a s p e c t s of the  205  L i s t of R e f e r e n c e s  AJZ80  F. A j z e n b e r g - S e l o v e , N u c l . Phys. A336 (1980) 1  AJZ82  F. A j z e n b e r g - S e l o v e , N u c l . Phys. A375 (1982) 1  ALK83  G.D. 46  ALT85  A. Airman, p r i v a t e communication.  ALT85a  A. Altman, R.R. Johnson, U. Wienands, N. Hessey, B.M. B a r n e t t , B.M. F o r s t e r , N. G r i o n , D. M i l l s , F.M. Rozon, G.R. Smith, R.P. T r e l l e , D.R. G i l l , G. S h e f f e r and T. A n d e r l , Phys. Rev. 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Wienands, p r i v a t e communication  Press,  215  Appendix I Subscripts and Superscripts  Throughout  this thesis t r a i l i n g subscripts  '+',  p r o t o n , n e u t r o n and charge d i s t r i b u t i o n s r e s p e c t i v e l y .  and '0' r e f e r t o Leading s u p e r s c r i p t s  i n d i c a t e atomic number and l e a d i n g s u b s c r i p t s are atomic charge ( Z ) . ambiguities  result,  r  r e f e r t o Root Mean Square  =<r > 2  r )  1 / 2 n  appropriate charge}.  leading subscripts are omitted.  abbreviations  (RMS) r a d i i weighted by the  d e n s i t y : n e {+ o r 'p', p r o t o n s ; - o r 'n' ,neutrons; 0 o r 'ch',  T h i s n o t a t i o n s e r v e s t o remind one o f the predominant  of TT t o p r o t o n s and TT t o n e u t r o n s . +  The  Unless  -  sensitivity  Appendix II Powder Boron Target Mass Measurements  The measurement was  performed by making use of the c o r r e l a t i o n between the  range of the e l e c t r o n s e m i t t e d by a 8 boron t a r g e t . 1 0 6  Ru  A ^^Ru  decays t o  energy g  - ,  s  energy: 3.53  1 0 6  Rh  s o u r c e was (t  of i n t e r e s t MeV)  1 / 2  -  source and the mass t h i c k n e s s of the  chosen as a p p r o p r i a t e f o r the purpose.  = 3 6 7 d a y s ) , which subsequently y i e l d s the h i g h  (predominantly with t  1 / 2  ~ 3 0 seconds; end  point  [WEA73].  The apparatus used i s shown i n f i g u r e A2.1.  The source was  fixed  r e l a t i v e to the G e i g e r - M u l l e r (G.M.) tube (bias=500 V o l t s ) and p a r a l l e l to the t a r g e t normal. was  To perform the i n i t i a l c a l i b r a t i o n ,  p l a c e d on a v e r t i c a l p l a n e .  the source-tube a x i s  S e v e r a l mock t a r g e t s were c o n s t r u c t e d of  the same m a t e r i a l s used f o r the o r i g i n a l t a r g e t .  These  targets  differed  from the o r i g i n a l o n l y i n t h e i r a r e a (somewhat l e s s i n the mock t a r g e t s i n an e f f o r t was  to a v o i d weight d e f o r m a t i o n s ) and t h e i r mass t h i c k n e s s .  of the same e l e m e n t a l c o m p o s i t i o n as t h a t In the o r i g i n a l  the range of mass t h i c k n e s s e s was original  The  target,  boron and  chosen to span t h a t expected i n the  target.  A number of runs were performed and r e s u l t s averaged to determine, f o r each mock t a r g e t , the number of counts s c a l e d from the G.M. runs were made a t d i f f e r e n t  tube.  These  p o i n t s on the mock t a r g e t , so the s t a n d a r d e r r o r  i n the average r e f l e c t e d w e l l both the s t a t i s t i c a l e r r o r s and those induced by l o c a l a n i s o t r o p i e s i n the mock t a r g e t The mock t a r g e t d a t a was  thickness.  then used to r e l a t e e l e c t r o n count r a t e to  P~  SOURCE  MOCK  G.M.  TARGET  TUBE  TIMER-SCALER  Figure A2.1 Apparatus for determining the B U  target mass thickness  measurement of electron multiple scattering e f f e c t s .  t a r g e t mass t h i c k n e s s . intercept  The r e l a t i o n was l i n e a r , so t h a t a s l o p e  (as w e l l as u n c e r t a i n t i e s i n those q u a n t i t i e s ) c o u l d  and  be e a s i l y  determined. To measure the mass t h i c k n e s s apparatus was r o t a t e d enabling  a measurement  of the o r i g i n a l  1 1  B  t a r g e t , the  so t h a t i t s a x i s was i n a h o r i z o n t a l p l a n e , thus i n the in-beam o r i e n t a t i o n .  Measurements were  performed on a symmetrical g r i d of 9 l o c a t i o n s on the t a r g e t . A q u a n t i t a t i v e i n d i c a t i o n of t a r g e t n o n - u n i f o r m i t i e s  was o b t a i n e d from  v a r i a t i o n s i n count r a t e over the t a r g e t . The boron t a r g e t mass t h i c k n e s s  was determined to be 377.6 ± 3.0 mg by  p e r f o r m i n g a weighted average of the t h i c k n e s s e s  measured on t h a t  The w e i g h t i n g f a c t o r s were o b t a i n e d from the beam p r o f i l e s  ( f i g u r e 2.4).  (Here, o n l y , was an average v a l u e f o r the t a r g e t normal angle T h i s r e s u l t was averaged f o r the two e n e r g i e s , profiles  r e s u l t e d i n , a t most, ~0.5 mg/cm  2  target.  assumed.)  as the d i f f e r e n c e i n beam  d i f f e r e n c e i n the t a r g e t masses.  Appendix III Decay Kinematics  Two  body k i n e m a t i c s have been c a l c u l a t e d throughout  the r e l a t i v i s t i c  formulae  of B a l d i n e t a l . [BAL61].  t h i s t h e s i s with  Their conventions  e s s e n t i a l l y the same as i n the f o l l o w i n g , and are shown i n f i g u r e ( N o t i c e , here, t h a t CM q u a n t i t i e s are denoted  by the t i l d e  are  A3.1.  overscore.)  A3.1 Muon Cone Angle With M i l i n i t i a l l y a t r e s t , we  V  have:  ElSlI  =  l* ' ) 3  1  and: V  PI E l  pi =  =  (A3.2)  61 The k i n e m a t i c s r e q u i r e s  P(EI+EII)  t h a t the c o n d i t i o n : 1 -  tan  2  V  2  91 <  (A3.3) pi  - 1  2  must be f u l f i l l e d , and i t i s t h i s which determines 1 9 = tan  - 1  V  [  the cone a n g l e :  2  ]  1  /  (A3.4)  2  pl -l 2  The maximum cone has It  9=90° ( i . e . :  pl=l).  i s e a s i l y shown t h a t , f o r a simple p a r t i c l e decay where one of the  decay p r o d u c t s i s m a s s l e s s ,  p l = l i s s a t i s f i e d when: MI -Ml 2  Tl  . m  i  n  =  2  { [ 1 - (  ) ] MI +M1 2  2  _  1  /  2  - 1 } MI  (A3.5)  LAB:  ff*m, pi ''01  O—>  MI, PI  <5-  Mil  \  0  2  \MZ,  <  P2  CM J*A\,  /  P1  e  /  J.  MI, PI  /  MII.PII  P2 Figure  A3.1  D e f i n i t i o n of the k i n e m a t i c o b s e r v a b l e s i n the L A B o r a t o r y frame, and the C e n t e r of Momentum frame  (CM).  (LAB)  For p i o n decay t h i s g i v e s T.-=5.445 MeV, below which energy no cone i s observed.  A3.2 B e a m Attrition One may c a l c u l a t e the f r a c t i o n , f , of pions which decay i n t o muons and subsequently miss a plane c i r c u l a r d e t e c t o r c e n t e r e d on and p e r p e n d i c u l a r t o the beam a x i s .  C o n s i d e r t h e geometry i n f i g u r e A3.2 w i t h a p e n c i l beam o f  pions o r i g i n a t i n g a t p o i n t 0. p=cos6.  Now d e f i n e s=tan9, d = ( l - V ) 2  1 / 2  , and  Then k i n e m a t i c s r e q u i r e s t h a t : (1 s  2  u )d 2  2  =  (A3.6) (pl+u)  2  hence: -pi  s  d (d -s (pl -l))  2  2  V =  ±  2  .  2  [  ]  (s +d ) 2  2  (s +d )  2  2  2  1  /  (A3.7)  2  2  F i g u r e A3.3 i l l u s t r a t e s the v a l u e s of u c o r r e s p o n d i n g t o v a l u e s o f s i n the lab. The f r a c t i o n o f decaying p a r t i c l e s e n t e r i n g a ( l a b ) cone o f known v e r t e x a n g l e , 9 , i s p r o p o r t i o n a l t o the s o l i d angle subtended In the CM Q  frame by t h a t cone. 6  0 (cone)» < e  We assume t h a t T > T ( i ) , so t h a t f o r i r  /d(cos9) i s j u s t : d f« = (s +d )  T f  2  2  I f 9o- (cone)»  then f=0.  >9  f  l  =  m  s  2  d  2  [ 1  n  (Pl -D 2  2  ]  1  /  2  -  (A3.8)  In s h o r t :  [  (s*+d ) £  1  " Tl d*  (  P  1  2  ~  L  )  H ( S  (cone)-  s )  ( A 3  '  9 )  Figure A3.2 A p e n c i l beam o f p i o n s i n c i d e n t perpendicular  upon a plane c i r c u l a r t o the beam.  detector  Mi = - A + B H-z  = - A- B  A  = 0.1146  B  = V0.6926  Figure A3.3 CM a n g l e c o s i n e s c o r r e s p o n d i n g t o p i o n s of LAB energy 43 MeV d e c a y i n g i n t o a cone w i t h a v e r t e x angle e q u a l t o one h a l f of t h e muon cone a n g l e .  Decays i n s i d e the shaded r e g i o n s  e n t e r t h i s cone.  224  At a p o i n t x, a f r a c t i o n :  f  2  -  E X  P  [ cl7t7 ]  6 X  P  t clT?  of the p i o n beam remains, (x i s the p i o n mean l i f e . )  ]  (  A  3  a  I n t e g r a t i n g along  o  )  the  beam, the f r a c t i o n of pions which decay i n t o muons which miss the d e t e c t o r at D i s : s, (  J s  c  o  n  e  )  f  l  f  -±  2  d  CYSTS  0  s  ( A 3  .11)  2  Writing  " 0 R  A - CB exp [ —  _d_ = [ __2 5'(A cone)  B  ]  (A3.12)  ]2  C = -z  CBYTS,  (A3.13)  (A3.14)  .  (cone)  and e = it  follows  = (cone)  (A3.15)  that ! f - A /  C/x  (l-x ) 2  1 / 2  e £  H ( l - e ) dx  (A3.16)  x (x +B) 2  2  The f r a c t i o n of pions which decay and miss a d e t e c t o r of the geometry i s f . either  Equation  3.6,  1-f, i s then the f r a c t i o n of pions  which  do not decay or decay i n t o muons which subsequently h i t the  detector.  specified  Appendix IV Core + Valence Matter Distributions  A4.1 The Distributions The v a r i o u s d i s t r i b u t i o n s used i n s e c t i o n 4.3.4 t o t e s t f o r s e n s i t i v i t i e s t o moments g r e a t e r t h a t < r > 2  1 / 2  were:  1) M o d i f i e d G a u s s i a n (MG) 2) M o d i f i e d Gaussian Core + V a l e n c e Nucleons  (MGCV)  3) C o n s t r a i n e d Fermi (CF) 4) C o n s t r a i n e d Fermi Core + V a l e n c e Nucleons  (CFCV)  These d i s t r i b u t i o n s a r e summarised i n t a b l e A4.1 and i l l u s t r a t e d A4.1.  The f i r s t  i n figure  o f these (MG) was d i s c u s s e d i n s e c t i o n 4.3.3 and w i l l not  be d e a l t w i t h h e r e .  A 4 . 1 . 1 The M G C V Distribution The MGCV d i s t r i b u t i o n i s s i m i l a r t o t h a t used by W. G y l e s [GYL79a] h i s i n v e s t i g a t i o n s o f the n e u t r o n r a d i u s o f C . 1 3  in  Here i t i s g e n e r a l i s e d t o  a l l o w m u l t i p l e v a l e n c e nucleons ( o r h o l e s ) i n e i t h e r i s o s p i n s t a t e .  A  c l o s e d s h e l l c o r e w i t h known r a d i u s i s assumed t o have the MG form. The i t h v a l e n c e n u c l e o n i s assumed t o occupy a s i n g l e o r b i t a l o f a harmonic w e l l , so t h a t i t c o n t r i b u t e s an angle-averaged matter d e n s i t y o f :  Table A4.1 S u m m a r y of Characteristics of Matter Density Distributions  Matter  Parameters  Constraints  Degrees of Freedom  Distribution  MG MGCV  <r > 2  • • •  1  <r >,  • • •  2  2  <r >(core) 2  CF CFCV  c/t  c,t  fixed  c(core),  c(core),t(core)  t(core),  f i x e d by QD;  <r > 2  <r >(core) fixed 2  1 1  10  0  1  2  3 r  Figure  Various density d i s t r i b u t i o n s with < r  4  (FERMIS)  2  A4.1  p+(r) >  1 / 2  evaluated f o r f i v e  - 2.25  fm.  nucleons  w i t h the R .  g i v e n by  n£  [BER72]  Kn£ o = " n, £ l  =  "( v'r * ) ~ e t  N  v >  /  2  w  2  /  2  L» "£+1/2  (A4.2)  / 0  where:  L  £+l/2  v! (2£+l)!!  N  n£ r  _  °kn£ We  =  f  j  k  n  i ;  V  r  2  )  < ' >  k  A4  [ 2 - ' (2n+2£+l)!! nl 7 i £  L  _ k ^  W  Q  z  lH  ,1/2  2  n!  J  l  M  ,  *  ;  (2£+l)!!  W-*)  k ! ( n - k ) ! (2£+2k+l)!!  r e c a l l t h a t the angular momentum and p r i n c i p a l quantum numbers, £  and n, a r e non-negative  integers.  The w e l l s t r e n g t h , v, i s j u s t mco/n  where m i s the reduced mass and a> i s the o s c i l l a t o r frequency. j v a l e n c e n u c l e o n s , then v i s g i v e n  vt = 1  matter  i s a reminder  radii  j Z (2n.+£.+ 3/2) i=l — 2 total total core r core  ( t o t a l and  ( t o t a l and c o r e ) and the r ' s  The v a l u e s of £ and n are determined  p a r t i c l e model  The  subscript t  of d i f f e r e n t i s o s p i n are to be d e a l t  n u c l e o n from the o r d e r of (2J+l)-degenerate independent  (A4.6)  core) f o r a g i v e n i s o s p i n .  t h a t v a l e n c e nucleons  with separately.  I f t h e r e are  by:  where the T's a r e numbers (N or Z) of nucleons are RMS  3  f o r each v a l e n c e  l e v e l s p r e d i c t e d by  an  [FEL53].  F o r a v a l e n c e p r o t o n , one must a l l o w f o r the e l e c t r o s t a t i c e f f e c t s of that nucleon.  The  i t h v a l e n c e p r o t o n c o n t r i b u t e s a p o t e n t i a l , as a  f u n c t i o n of d i s t a n c e r from the c e n t e r of the n u c l e u s , o f :  * -  e 2 ( f 0  —  0  The f i r s t  (  r  >  i  1  =  e  r  °  L_-_.'l  I  2  .2  d r  ,  +  d  -  jr _ _ L _ t  0  r  ( A 4 , 7 )  _ _  r  i  ] d r  ,  (  M  >  8  )  r  term i n e q u a t i o n A4.8 i s d e a l t w i t h a n a l y t i c a l l y and i s  just:  e  The  2 0  —  v  *  I C m, k=0  k n £  C  m n £  (m+kfl) !.  (A4.9)  second term i s i n t e g r a t e d w i t h Simpson's Rule i n a manner i d e n t i c a l t o  t h a t o f the o r i g i n a l Krell-Thomas  code.  T h i s i s a two parameter d i s t r i b u t i o n . r a d i u s , the RMS r a d i u s o f the MG c o r e .  The f i r s t  parameter  The second parameter  i s a core  i s the a c t u a l  RMS r a d i u s of the matter d i s t r i b u t i o n o f nucleons of the a p p r o p r i a t e isospin. altered  By v a r y i n g t h e c o r e r a d i u s , the o v e r a l l matter d i s t r i b u t i o n may be w h i l e k e e p i n g the RMS r a d i u s c o n s t a n t .  A4.1.2 The CF  Distribution  The Fermi d i s t r i b u t i o n i s o f the u s u a l form: p(r) = T p  I  ———--  r _£ l  0 0  1 + e  L  a  (A4.10)  1  w i t h the 90% t o 10% s k i n t h i c k n e s s o f : t = 4 (ln3) a and has the ( 2 m )  t n  moment:  (A4.ll)  230  . 2m. . <r > = 4TT  2m+3 r k ^ " {-^+  r , , r . n! .n-r , ~ r.. z [[!+(-) ] _ _ _ [ i - ] r=0  n+  Poo  a  n  -  r i l  r  k  2  + C(r+1)] + C }  (A4.12)  n  with  5  I C - ) ^  -  J-l  n  where k=c/a,<=Tr/k, and n=2(m+l). (p  0 0  - ^  1  j " *  2  t h a t <r°> = 1.)  distribution  i s a two parameter d i s t r i b u t i o n .  QD=<r >  2  1/2  Hence the Fermi  We note t h a t t h e r a t i o  / < r > d e f i n e s t h e f o u r t h moment o f the d i s t r i b u t i o n  the second. A4.12  (A4.13)  k  T h i s d e f i n e s <r > i n terms o f c and a.  i s d e f i n e d by t h e requirement  4  e" J 1  I f we wish  r e l a t i v e to  t o match t h i s t o the MG v a l u e of QD, then  equations  and A4.13 w i t h v a l u e s o f t h e Riemann z e t a f u n c t i o n [HAY65] imply  that: 1 [ _U+K H  3  2  21  k  QD, , = fermi  5 }{I  7  k  2  h  49 31 i» 6 K  H  K  3  3  7 H  g }]i/2 6  3  k  7  (A4.14) 1  10  7  [l+—K H  —  3  5 <-+ — ? J  2  5 so t h a t w i t h  +  3  k  5  (see e q u a t i o n 4.21 of s e c t i o n 4.3.3): [15(2+3a)(2+7cO] QD  , = gaussian  = 3  QD may be s a t i s f i e d .  [5T(7T-8)]  1/2  (  2  +  5  . gaussian  a  )  1 / 2  , (  QD, fermi  5  T  _  4  (A4.15)  )  (A4.16)  231  E q u a t i o n A4.14 i s p l o t t e d as a f u n c t i o n o f c / t i n f i g u r e A4.2 a l o n g with < r > 2  1 / 2  /c.  S o l u t i o n s t o e q u a t i o n s A4.14 through A4.16 a r e g i v e n i n  t a b l e A4.2. Matching  moments i n t h i s way c o n s t r a i n s c / t .  The r e s u l t i s a one  parameter d e n s i t y d i s t r i b u t i o n w i t h <r > u n i q u e l y d e f i n e d : 2  distribution.  An MG d i s t r i b u t i o n w i t h the same <r > d i f f e r s i n even moments 2  o n l y i n those g r e a t e r than the f o u r t h . t e s t i n g f o r s e n s i t i v i t y t o such  A4.1.3 The CFCV Finally,  1 2  moments.  the CFCV d e n s i t y uses a CF core and adds harmonic  oscillator  As Implemented here, i t i s a one parameter d e n s i t y ; the  core radius i s f i x e d (ie:  The CF d i s t r i b u t i o n then a l l o w s the  Distribution  valence nucleons.  nucleus  the CF  t o correspond  t o t h e measured v a l u e f o r the c o r e  C ) . Without c o n s t r a i n t s , i t i s a t h r e e parameter d e n s i t y .  232  0-6  H  ,  ,  ,  ,  1  0  I  2  3  4  5  C/  Figure  QD  (dashed) and  <r > 2  1 / 2  T  A4.2  as c a l c u l a t e d f o r the Ferini d i s t r i b u t i o n  as a f u n c t i o n of c / t .  Table  A4.2  Solutions to Equation  A4.16  f o r 2 through 8 Nucleons  QD  N  c/t  2  1.01688  1.29099  3  1.12626  1.26948  4  1.23846  1.25000  5  1.23214  1.23718  6  1.38366  1.22836  7  1.43190  1.22198  8  1.47030  1.21716  Appendix V A Replacement f o r VIEWFIT  V i e w f i t was o r i g i n a l l y w r i t t e n w i t h the i n t e n t of e l i m i n a t i n g normalisation displayed x  e f f e c t s from the f i t t i n g t o a c r o s s s e c t i o n .  This  routine  and a v i s u a l p l o t of d a t a and c a l c u l a t i o n to h e l p i n  2  d e t e r m i n i n g a "good" f i t .  Here we d i s c u s s a more s o p h i s t i c a t e d method of  a r r i v i n g a t the same r e s u l t . Normally, one d e f i n e s : X  = I [ i(exp)- K c a l c )  2  X  X  Suppose t h a t , i n s t e a d , we 2  =  A  5  <  1  )  define:  [ i(exp)~ X  s  (  °i  1  x  ] 2  5  X  a  i  i(calc)  ] 2  (  A  5  >  3  )  >  2  )  ±  Then:  x  2  .  X E  ^exp)  i f f  _  x u  s  1  2  This i s just a quadratic for  x> 2  x  i(calc) cr  +  ?  2  ^  x  i(calc) 1  2  i n £; c l e a r l y there  (  A  5  ff  2  i s e x a c t l y one minimum  occuring f o r : — — — 9 5  or:  i(exp)  = 0 =  2£ £ 1  X 2 l  < a  c a l c  i  >  - 2 E i(exp) i(calc) X  1  X  a  i  (  A  5  >  4  )  £  X  i(exp) i(calc) x  1  5 = E  i Defining x  2  of the  1  (A5.5)  2  i(calc) 0  <  i n t h i s manner removes a data s e t ' s s e n s i t i v i t y to i t s  overall normalisation explicitly  x  r e l a t i v e to a t h e o r e t i c a l  c a l c u l a t i o n , while  i n t r o d u c i n g an e x t r a parameter i n t o the  process.  theoretical  not  description  236  Appendix VI Spectrometer Transfer Coefficient Optimisation  In i t s c u r r e n t implementation, spectrometer.  the QQD spectrometer i s a s o f t w a r e  T h i s means t h a t , i n s t e a d of measuring the d i s t r i b u t i o n o f 6p  (the f r a c t i o n a l v a r i a t i o n o f a p a r t i c l e ' s momentum from the s p e c t r o m e t e r ' s ' c e n t r a l momentum' as d e f i n e d by t h e spectrometer's geometry and bending field),  a t t h e f o c a l p l a n e of t h e s p e c t r o m e t e r , the 6p a r e d e r i v e d from  p a r t i c l e t r a j e c t o r y i n f o r m a t i o n measured a t the e n t r a n c e t o and e x i t the spectrometer  from  [BAN66].  The p a r t i c l e t r a j e c t o r i e s a r e c h a r a c t e r i s e d by t h e q u a n t i t i e s X , Y , Q  Q  9Q , C|)Q, and 6pHfi a t the t a r g e t p l a n e , as i l l u s t r a t e d by f i g u r e 3.12. The e x i t c o o r d i n a t e s , X4,Y4,X5,Y5, a r e r e l a t e d t o these q u a n t i t i e s . The measured e n t r a n c e parameters,  however, a r e not the c o o r d i n a t e s a t t h e t a r g e t  p l a n e , but r a t h e r the c o o r d i n a t e s , X1,Y1,X3,Y3, a t the spectrometer's  front  end. The  e x i t c o o r d i n a t e s may be w r i t t e n i n terms o f the f r o n t end  c o o r d i n a t e s and 6p.  F o r example: X  = P(m ) + Q(m )6 + R ( m ) 5 ,  (A6.1)  2  u  Q  1  2  where P, Q, and R a r e p o l y n o m i a l s o f o r d e r s m , m-^, and m , r e s p e c t i v e l y , i n Q  the e n t r a n c e c o o r d i n a t e s .  2  N o t i c e t h a t as the o r d e r s m^ i n c r e a s e , t h e  p o s s i b l e number o f t r a n s f e r c o e f f i c i e n t s r i s e s r a t h e r d r a m a t i c a l l y , as illustrated  i n t a b l e A6.1.  F o r example, i f m =2, then: Q  Table A6.1 Proliferation of Spectrometer Transfer Coefficients POSSIBLE DISTRIBUTION OF EXPONENTS AMONG (X1,Y1,X3,Y3) ORDER In  D e c r e a s i n g Order of exponent  (0  0  3  0 0  ON  1  0 0  ON  4  v » > > I  1 2  Number o f Unique Permutations  \ (  \  » > » i  (2 0 0 O N \  »  ,3  /i*  »  >  /  >  \ 1 0 ON  (  \  *  »  »  /  2 1 0 ON (1 1 1 O N  0 0 ON  (  /3 1 0 O N / » v . » » » / ON Z-2 1 1 O N i 1 MN' ' ' » » IJ  0 0 ON  \ ) » » (2 2 0 ^ ' ' , v  4 9 6 4  4 6  4 12 12  /5 1 0 O N » » » / /•<+ 1 1 O N  4 12 6 4  0 0 ON  /6 0 0 O N \ » » » / » h 2 0 ON (  sVo'o  /3  1 1 I N V  \  ' hVl'ON /2 2 2 O N  > > » I  > »  > > >  > > > >  6  10  12 20 12 12  1  ;  ,-3  »  4  /-^ 1 0 O N 2 0 ON Z-3 1 1 O N 2Vl'0 ' (Vl'l'l  (5  TOTAL NUMBER OF COEFFICIENTS  25 12 12 4 12 12 24 4  56  84  P(m =2) = c ^ X l  2  0  + a Yl  + a X3  2  2  2  3  + a^Y3  2  + a X l Y l + a XlX3 + a XlY3 + a X3Yl 5  6  + a X3Y3 + a g  + a for  1 3  Yl + a  The e x p r e s s i o n A6.1 the  coefficients  1 Q  11+  a t o t a l of 15 p o s s i b l e  ?  YlY3 + a X l u  Y3 +  a  (A6.2)  8  + a  1 2  X3  1 5  coefficients. may  be m e a n i n g f u l l y i n v e r t e d  to determine 6,  a r e known (a s i m i l a r e x p r e s s i o n e x i s t s f o r X 5 ) .  once  The  c o r r e s p o n d i n g e x p r e s s i o n s f o r Y4 and Y5, however, have a t most weak dependence upon 6 and may  not be used to determine that  N o t i c e t h a t e q u a t i o n A6.2 c o e f f i c i e n t s may [BEV69] was  files  i s l i n e a r i n the parameters a^, so those  be determined v i a a l i n e a r r e g r e s s i o n t e c h n i q u e .  used to remove s t a t i s t i c a l l y  expansions.  insignificant  The i n p u t d a t a to the r e g r e s s i o n was  containing  quantity.  1 2  C  An F - t e s t  terms i n the  c a l c u l a t e d from a r r a y  data i n the mid angle range (90° t o 1 1 0 ° ) .  With the  two d e l t a degrees of freedom a v a i l a b l e from t h a t d a t a (ground s t a t e and first  e x c i t e d , 4.44  polynomials.  R was  MeV  s t a t e of c a r b o n ) , one may  s e t to zero; t h i s was  determine o n l y the P and Q  a r e a s o n a b l e c h o i c e f o r the  momentum spread encountered i n the experiments.  C o e f f i c i e n t s were  determined s e p a r a t e l y f o r the two d a t a s e t s at d i f f e r e n t  energies.  A p r e l i m i n a r y a n a l y s i s f o r e v a l u a t i o n of 6 used TRANSPORT [BR080] coefficients.  Each of about 2000 s c a t t e r i n g events was  s t a t e event ( t a g G ) , 4.44 (tagU).  MeV  tagged as ground  i n e l a s t i c event ( t a g X ) , or unknown event  With t h i s t a g g i n g , e q u a l numbers of tagX and tagG events were  s e l e c t e d t o a l l o w even s t a t i s t i c a l w e i g h t i n g ( i n momentum).  The  c o e f f i c i e n t s determined i n t h i s manner a r e , a p p r o p r i a t e l y , weighted by the  beamspot ( f l u x d i s t r i b u t i o n on t a r g e t ) . technique The  tend t o conform the momentum d i s t r i b u t i o n t o a G a u s s i a n r e g r e s s i o n a l g o r i t h m was performed  IMSL r o u t i n e [IMS82] t o a c c o m p l i s h A6.1  The s t a t i s t i c s o f the r e g r e s s i o n shape.  by program QQDMP, which  the a c t u a l r e g r e s s i o n and F - t e s t .  calls Figure  i l l u s t r a t e s t h e sequence which QQDMP f o l l o w e d . F i g u r e A6.2b i l l u s t r a t e s the improvement i n spectrometer  resolution  w i t h o p t i m i s e d c o e f f i c i e n t s over t h a t o b t a i n e d w i t h TRANSPORT c o e f f i c i e n t s ( f i g u r e A6.2a).  F i g u r e A6.2c shows the a d d i t i o n a l Improvement a c h i e v e d  the i m p o s i t i o n o f muon e l i m i n a t i o n c u t s .  with  240  (  LOGON  *)  No  Run MOLLI. Coll PRINTER(): Croote RUNnnnARR  Found Yes  T  t >  »  (Run QQDMP) Choose: Energy, Mode  Figure  Spectrometer c o e f f i c i e n t  A6.1  determination  algorithm.  F i g u r e A6.2  Improvement of QQD r e s o l u t i o n : a) With t r a n s p o r t c o e f f i c i e n t s coefficients  48.3 MeV Energy  and no muon c u t s ,  and no muon c u t s ,  b) With o p t i m i s e d  c) With o p t i m i s e d  and muon c u t s on DDIF and ANGL.  Spectrum,  coefficients  242  Appendix VII Peak F i t t i n g i n the CNO Experiments 2  A7.1  Introduction In S e c t i o n  required  3.3.3  we  of peak f i t t i n g  spectrometer.  discussed  the m o t i v a t i o n  f o r and  i n the a n a l y s i s of energy s p e c t r a  In t h i s appendix we  discuss  specifications from the  the peak shape and  error  produced from the  QQD  QQD analysis  accompaning t h i s p r o c e s s .  A7.2  The Peak Shape The  peak shape of the energy s p e c t r a  spectrometer w i t h o p t i m i s e d  t r a n s f e r c o e f f i c i e n t s i s a complicated  function  dependent upon many d e t a i l s of the channel-spectrometer c o n f i g u r a t i o n . assume the b a s i c shape to be of the  form of two  here as a component) on a common c e n t e r . of the  r e l a t i v e heights  shape a t a g i v e n one be  The  Gaussians (each r e f e r r e d  r a t i o s of the h a l f w i d t h s  of the components are assumed to d e f i n e  s c a t t e r i n g angle, regardless  We  the  of t a r g e t m a t e r i a l .  to  and  peak There i s  of these b a s i c peaks f o r each s c a t t e r i n g s t a t e i n the energy spectrum to analysed.  be a d j u s t e d  The  energy s e p a r a t i o n s  f o r the v a r y i n g  d i f f e r upon l e a v i n g the (biggest)  of the  s t a t e s are known [LED78] and  s p e c i f i c energy l o s s of the p i o n s whose e n e r g i e s  target n u c l e i .  The  halfwidth  component i s assumed to be a c o n s t a n t , v a l i d  states at a given  angle f o r a g i v e n  target.  The  d i f f e r i n g s c a t t e r i n g s t a t e s , of c o u r s e , d i f f e r The  can  peak shape i s , a n a l y t i c a l l y :  of the  fundamental  for a l l scattering  r e l a t i v e i n t e n s i t i e s of  f o r each t a r g e t and  angle.  the  + 6 e  where N, i n p a r t i c u l a r , i s a f u n c t i o n of E . v a r i a b l e s i n e q u a t i o n A7.1  2  }  (A7.1)  The d e f i n i t i o n s of the  are given i n table  The parameters 3, n, and  (N/ia)  A7.1  i d e f i n e the peak shape and were determined  from carbon s p e c t r a a t each a n g l e .  The e f f e c t i v e d i s p e r s i o n , n, was  allowed  to v a r y f o r each group of runs a t a g i v e n angle s e t t i n g to a l l o w f o r p o s s i b l e l i m i t a t i o n s i n the s p e c t r o m e t e r c o e f f i c i e n t s due to n e g l e c t e d h i g h e r o r d e r terms i n 6p. of  E , a, a^, and N Q  0  The v a l u e s of E^ were not v a r i e d .  were f i t t e d  The v a l u e s  i n each r u n .  I t s h o u l d be noted t h a t the two-Gaussian  p r e s c r i p t i o n f o r the peak  shape a t g i v e n e x c i t a t i o n energy E^ i s not s t r i c t l y n e c e s s a r y f o r a r e a s o n a b l e f i t to the observed s p e c t r a , as the second Gaussian i s meant to r e p r e s e n t the muon decay background  s u r r o u n d i n g the main peaks.  The  i n c l u s i o n of the second G a u s s i a n p r i m a r i l y a l l o w s degrees of freedom fit  the decay contaminants not e l i m i n a t e d by the c u t s and which  otherwise r e s u l t  i n an o v e r e s t i m a t e of i n e l a s t i c c o n t r i b u t i o n s .  use of the e r r o r m a t r i c e s from the peak shape d e t e r m i n a t i o n and  which  may Judicious spectra  f i t t i n g s a l l o w the i n c l u s i o n of e f f e c t s i n the o v e r a l l u n c e r t a i n t y due to c o r r e l a t i o n s i n the e x t r a c t e d number of counts due t o the peakshape ( c o r r e l a t i o n s i n the parameters f i x e d from c o r r e l a t i o n s i n the parameters energy s p e c t r a ) themselves.  1 2  C ) , as w e l l as due  ( f r e e parameters, f i t t e d  to the  to the individual  Table A7.1 D e f i n i t i o n of Peak F i t t i n g  VARIABLE  Parameters  DEFINITION  - r\{\Z -Z )< l-e )  N  { E - E e  imax  Number of e x c i t e d s t a t e s i n c l u d e d  0  (i=0  i 0  Q  ±  k  }  oi  corresponds to the ground  6  Relative strength  T)  Effective dispersion (n^l)  in fitting state)  of the second component  E  0  Energy c e n t r o i d of the ground s t a t e peak  E  i  E x c i t a t i o n energy of the i  e  0i  Kronecker f u n c t i o n : e  a  t  n  peak ( i * 0 )  =l, egi ^  ^  =  0 0  o r  Width of the fundamental ( l a r g e s t ) component of the two Gaussian p r e s c r i p t i o n Ratio  1  of w i d t h s , a /a 2  of the  2  n d  component Gaussian to the fundamental Energy a t which G ( E ) , i s e v a l u a t e d  E N  0  a  i  S t r e n g t h of the ground s t a t e peak Ratio  of the s t r e n g t h  of the i  t  h  inelastic  peak to that of the ground s t a t e peak  A7.3 Mechanics of F i t t i n g The a c t u a l f i t t i n g was c a r r i e d out w i t h the r o u t i n e QQDANA a l r e a d y referred  t o i n s e c t i o n 3.3.1 and appendix V I .  and parameter the  T h i s r o u t i n e handled the data  i n t e r f a c e to the IMSL r o u t i n e ZXMIN, which was used to perform  n o n l i n e a r l e a s t squares parameter s e a r c h e s .  i n determining f i n a l  fitting  IMSL r o u t i n e s LINV1P a i d e d  c o r r e l a t i o n m a t r i c e s f o r the f r e e parameters.  G a u s s i a n i n t e g r a t i o n was performed w i t h the r o u t i n e MERF [IMS82],  which  e v a l u a t e s the sum of ERROR f u n c t i o n s ( e r f ) a p p r o p r i a t e t o t h e d e f i n e d peakshape over s p e c i f i e d energy  regions.  The f i t t i n g was accompanied of  by g r a p h i c s p l o t s , generated w i t h the a i d  CALCOMP type FORTRAN p l o t t i n g r o u t i n e s  fitting  [KOS84], which a i d e d i n t h e  and h e l p e d t o ensure the i n t e g r i t y o f the f i n a l  fits.  A7.4 Correlation Errors We wish to f i t an observed d i s t r i b u t i o n w i t h a f u n c t i o n o f m f r e e parameters and n f i x e d parameters. e i t h e r ab i n i t i o  The f i x e d parameters a r e determined  ( f o r example, the p o s s i b l e e x c i t e d s t a t e e n e r g i e s f o r the  element under c o n s i d e r a t i o n ) or from f i t t i n g  to an e a r l i e r  run.  F o r the  f i x e d parameters b^ we have a c o r r e l a t i o n m a t r i x N^£; the d i a g o n a l element, N ^ , i s the v a r i a n c e o f b^ and Nj-^ i s the c o v a r i a n c e between b^ and b j .  R e c a l l t h a t N i s symmetric, and  \£  - < < V  V  (  V  V  >•  G i v e n the f i x e d parameters, a s o l u t i o n f o r the m f r e e  ( A 7  parameters  (denoted by a j ) e x i s t s , a l o n g w i t h the a s s o c i a t e d c o r r e l a t i o n m a t r i x Mjk  [MAT65].  We r e c a l l  that  - > 2  246  M  j k  = < (a - a ) ( a - a ) > j  k  (A7.3)  k  where I j i s t h e best v a l u e o f a j which would r e s u l t from f i t t i n g s t o a l a r g e number o f s t a t i s t i c a l l y  independent data  sets.  The c o r r e l a t i o n m a t r i x , N, a f f e c t s our c e r t a i n t y i n the a j determinations.  We d e f i n e a m a t r i x A j ^ = 9aj/9bi , which t e l l s how c  much the d e r i v e d v a l u e of a j v a r i e s w i t h changes i n the v a l u e o f b^. Suppose t h a t a j i s the v a l u e o f ( a j - a j ) when b^Ebj^.  Then f o r b ^ b ^ : _ (a - a ) - a  9a . J +  ( V V >  9 b. k  (  A  7  '  4  )  so t h a t : < (a.-a.Xa^)  > s ^ . a ^+ A  5  j i  M  = M  +  A  j k  A  i £  N  jk i£ k£ A  (A7.5)  k £  N  .. Ji  (  A  7  '  6  )  (A7.7)  and: 9a < ( a - a . ) ( b -b.) >  -  J  _ _ <(b -b. ) ( b -b )>  (A7.8)  k  A  Jk ki N  (  A  ?  -  9  )  The  The parameters a±  m a t r i x A j ^ may be determined as f o l l o w s .  x •  are determined from a r e g r e s s i o n a n a l y s i s m i n i m i s i n g hypersurface  dependent upon the a-j. 3X 3a  where Xo. * 2  s  t  n  e  3X  2  3 X  2  2  0  2  defines  a  0,  (A7.10)  The dependence of the second  o f b^ w i l l  2  term i s l a r g e r , when b 3 X  2  2  0  3 a, i  s  2  a t the minimum.  8X  *  0  6a. -  d e r i v a t i v e 3 X o / ( S a ^ S a j ) upon the c h o i c e change i n the g r a d i e n t  a  Sa^a..  ±  of x  value  2  At a minimum,  + I %a j  i  The x ( i )  2  k  k  be s m a l l , but the  varies.  So:  2 0  6b  (A7.ll)  k  3 a, 3 b, i k  the minimum w i t h a d i f f e r e n t v a l u e  of b^.  Combining these  gives:  6a . A., s jk  = - (M-l ),, , ~ jk'  (A7.12)  0  k when w r i t i n g : 3 X 2  3 X  2  2  0  and l  33,3b, i k  k  2 0  M, . = 1  3  (A7.13) 3a,3a. i j  A7.5 P r o p a g a t i o n o f E r r o r s Given energy s p e c t r a and f i t t e d may e s t a b l i s h the c r o s s f i t t i n g allows necessitates  curves f o r each of s e v e r a l t a r g e t s , we  s e c t i o n s and r a t i o s of c r o s s s e c t i o n s .  Although the  easy e x t r a c t i o n o f the number of e l a s t i c events, i t a l s o  considerable  care  i n error analysis.  248  The differ  measured d a t a i n c l u d e s e v e r a l groups o f measurements; these groups  i n spectrometer c o n f i g u r a t i o n  time by more than s e v e r a l hours. from each t a r g e t .  N :  w i t h each o f i t s runs:  The raw number o f counts i n a s t a t e o f i n t e r e s t w i t h i n a  f  given ii)  Each group c o n s i s t s o f a t l e a s t one r u n  Each group i s s t a t i s t i c a l l y independent, and has the  following quantities associated  i)  ( i e : angle s e t t i n g ) or a r e s e p a r a t e d i n  N:  energy r e g i o n  s p e c i f i e d by a low and a h i g h  The peak f i t t e d number o f counts i n t h a t s t a t e over a l l E, r e g a r d l e s s  i i i ) Nf  The peak f i t t e d  t  the  The  region  integrated  o f the r e g i o n s p e c i f i e d .  t o t a l number of counts i n t e g r a t e d  i n which N  number of e l a s t i c  limit.  over  was determined.  r  events c o r r e s p o n d i n g t o the s t a t e i s then: N N  = N  that a l l three  uncertainties a N  r  and N  f t  r  differ  specified limits, that and  f o r N.  An  o f these q u a n t i t i e s , and i n p a r t i c u l a r t h e i r  a, and Of > t  are strongly correlated.  o n l y by the x  2  We note  that  o f the f i t [BEV69] w i t h i n the  so the a p p r o p r i a t e  c o r r e l a t i o n matrix f o r N i i s just e  That i s , we may i g n o r e  concern o u r s e l v e s  (A7.14)  1  6 1  Notice  .  the u n c e r t a i n t i e s i n N  o n l y w i t h those i n N  r  and N^  t  explicitly.  a p p r o x i m a t i o n t o e q u a t i o n A7.14 i s : N N  -  + x (within 2  f  [  limits) ] * N.  N  ft  (A7.15)  This expression The  full  was used t o perform a s m a l l c o r r e c t i o n t o the r a t i o  expression  carbon cross separation  values.  i n e q u a t i o n A7.14 was used i n the e x t r a c t i o n of the  s e c t o n s , whereas the ground s t a t e t o f i r s t  i n the case of  be used i n that c r o s s  1 5  excited  state  0 was l a r g e enough t h a t d i r e c t i n t e g r a t i o n  section  could  extraction.  A7.5.1 R a t i o Measurements We now d i s c u s s  the p r o p a g a t i o n of e r r o r s i n f u n c t i o n s  of N, i n  p a r t i c u l a r i n t h a t i t concerns r a t i o measurements. C o n s i d e r , f o r s i m p l i c i t y a case where each energy spectrum of two peaks.  consists  We choose f i v e v a r i a b l e s :  h : Height of the e l a s t i c peak f  : Ratio  of the h e i g h t  o f the i n e l a s t i c peak  to t h a t of the e l a s t i c peak W : Width common t o both peaks u  : Energy c e n t r o i d of the e l a s t i c peak  6E:  Energy s e p a r a t i o n  o f the i n e l a s t i c peak  from the e x c i t e d s t a t e peak We suppose t h a t the data s e t c o n s i s t s of a number o f groups of d a t a : each having runs f o r C , 1 2  1 6  0  and MT t a r g e t s .  The carbon and oxygen  are assumed t o generate beam energy l o s s e s and m u l t i p l e same e x t e n t s ,  so t h a t W i s the same f o r both t a r g e t s .  c o v a r i a n c e s u p e r m a t r i x , as shown i n f i g u r e A7.1.  s c a t t e r i n g t o the We c o n s t r u c t  The  w i t h a modest  due t o the energy l o s s d i f f e r e n c e s o f d i f f e r e n t energy  W i s determined i n a s e p a r a t e f i t t i n g  a  There a r e two types o f  f i x e d v a r i a b l e here, 6E's a r e f i x e d to a p r i o r i known v a l u e s , uncertainty  targets  pions.  to the carbon data, and i t s e r r o r  12  12  c  16  0  T 3 O -L_  Z3  O or o  16r  1 12,  3 O or o  16  0 a)  16  12,  V  JE,  V  11*  «E,W  «1  6EjV  «E,'  »1 "l  ».»  0  »1«  0  u,  0  h,  f,  !>,«  <1»  0  0  «  I!, SEjV  >,  f  b,  yW  f  f  2  2  2  H  <E,V  n I  0  <1  «E,H  U SEjW  «' 2  »J  V,»  »2«  0  "j  h,»  h,«  0  '»  h,w 0  0  0  i  0  b)  F i g u r e A7.1  a) C o v a r i a n c e s u p e r m a t r i x c o n s t r u c t e d i n peak f i t t i n g s p e c t r a from the QQD.  b) The c o r r e l a t i o n between GROUP 1 v a r i a b l e s  k and I, a £ , i s w r i t t e n 'k£' i n the m a t r i x . k  to the energy  The E's a r e the  u s u a l e r r o r m a t r i c e s w i t h f r e e v a r i a b l e s u,h, and f .  251  i s determined from that The MT  fit.  t a r g e t runs have no e n t r i e s i n t h i s m a t r i x .  l a c k of s t a t i s t i c s p r o h i b i t s m e a n i n g f u l peak f i t t i n g c o r r e c t i o n i s s m a l l , and i s taken d i r e c t l y the energy spectrum related target).  (ie N  f i x e d parameter  f o r the MT  r  f o r those r u n s .  target).  By c o n s i d e r i n g the v a r i a b l e s W^, ww w»  CT  =0  1  2  t  n  The  MT  from the raw number of counts i n A l s o , the peak shape  e n t r i e s occur twice i n each group  with perfect c o r r e l a t i o n s : correct  T h i s i s because  (once f o r each  f o r example, as independent counting w i l l  e  2  i n the p r o p a g a t i o n of e r r o r s t h a t  be  follows.  To a r r i v e a t the c o r r e l a t i o n between the numbers of i n t e g r a t e d counts, N, f o r the group  ( w r i t t e n here as:  c o r r e l a t i o n s i n the peak f i t t e d  12  N,  16  N)  that occurs through the  parameters, we p r e - and p o s t - m u l t i p l y  the  group s u p e r m a t r i x by the d i a g o n a l m a t r i x D, where D^i=di and d^ i s the v e c t o r : 3 N  3 N  12  d = [  3 N  12  , 3W  , 36E  The r e s u l t we  3 N  12  1  call  , 3y  x  3 N  12  3 N  12  , 3h  x  , 3f  3 N  16  1  3W  3 N  16  , 36E  3 N  16  , 2  15  , 3u  2  , 3h  2  ] 3f  (A7.16)  2  N.  At t h i s p o i n t , a group which has two subgroups be combined  3 N  16  i n t o a s i n g l e subgroup,  a d d i t i o n Is element by element.  as i l l u s t r a t e d  f o r the same t a r g e t  i n f i g u r e A7.2.  may  The  The c o r r e s p o n d i n g v a l u e s of the i n t e g r a t e d  f l u x e s , f , f o r the runs a r e then added. What i s the u n c e r t a i n t y i n q u a n t i t i e s d e r i v e d from the N's? t h a t a l l parameters may  be c o n s i d e r e d , now,  Recall  as s e p a r a t e and c o r r e l a t e d ,  one need not worry about double c o u n t i n g i n the p r o p a g a t i o n of e r r o r s .  and  252  c  CO  C0  O  Q  COb  a  O Ob  a  a  O Ob  C Ob  Ob  a  C O  c  a  +  C Ob C O  a  Oa+Ob  C Ob  2 OaOb  F i g u r e A7.2  Reduction  of supermatrix  corresponding  f o r a 'group' w i t h two 'subgroups'  to s c a t t e r i n g from  the same angle and t a r g e t .  253  Furthermore, (albeit v(  1 2  N,  1 6  1 6  N i s dependent upon a c o m p l e t e l y d i f f e r e n t s e t o f v a r i a b l e s  some p e r f e c t l y c o r r e l a t e d ) N,  0 0  than i s  1  2  N .  Then, f o r a f u n c t i o n  N)=v(n ,n ,n )=v(n): 1  3  2  9v  °  9v  )+ Z l a  2 = 0  [  i j  3n,  0  3  9v 9x.  i  J  (A7.17)  ]  ][ 3x,  3  The second term i s :  E  [  9n  1  9v  +  9 n 9x_  i,j So,  9v  1  9n  2  9n 9x  L  2  9v  ][  3n  x  9v  +  9 n 9X_.  i  9n  2  ]  (A7.18)  9TI 9X^ 2  1  that: 9v  9v  9v  (A7.19) 9n  i, j  c  3T1  g(i) V i ) a  where g ( k ) i s the subgroup (1 o r 2) whose peak the k describes.  (Subscripts  and s u p e r s c r i p t s  t n  variable  '00' r e f e r t o MT  target  quantities). In the case v( the  1 2  following  N,  1 6  that:  N,  0 0  N ) = (16 /16f - 00 /00 )/(12 /12 N  N  f  N  f  _ 00 /00 N  f ) >  (  A 7  . 0) 2  relations result: 9v  = [16f(12 /12 N  9  1 6  f  _  00 /00 ]-l N  f )  (A7.21)  N  9v  = 9  1 2  N  [-l/l f]  16  N  00  16  f  00f  12  N  00  N  f  00  f  N  (A7.22)  2  iJ 12  254  and: 12,  3v g00  N  00  f  12  00  f  N  00<  ]-M  v -1  ]  (A7.23)  255  Appendix VIII Fourier Expansions of The Nuclear Density  A8.1 Introduction I t i s c o n v e n i e n t f o r the purpose of n u m e r i c a l c a l c u l a t i o n to choose one o f our a n a l y t i c forms, e q u a t i o n 4.18 o r e q u a t i o n A4.10, t o p a r a m e t e r i s e the n u c l e a r p r o t o n and n e u t r o n d e n s i t i e s , p+(r) and p _ ( r ) , r e s p e c t i v e l y . Such a c h o i c e ,  however, r e s t r i c t s an a n a l y s i s  s i z e information serious  to the g r o s s f e a t u r e s  i n the e x t r a c t i o n of n u c l e a r  of a d i s t r i b u t i o n .  T h i s may be a  l i m i t a t i o n , e s p e c i a l l y i f the momentum t r a n s f e r a v a i l a b l e i n e l a s t i c  s c a t t e r i n g experiments a l l o w s the p r o b i n g of the n u c l e a r i n t e r i o r . serious  l i m i t a t i o n , however, i s t h a t  the c o n s t r a i n t  A more  imposed by an a n a l y t i c  form may r e s u l t i n u n d e r e s t i m a t e s of the u n c e r t a i n t i e s  i n the d e r i v e d  moments.  A8.2 Fourier Bessel Analyses The a n a l y s i s of e l e c t r o n s c a t t e r i n g d a t a expansion of the n u c l e a r charge d e n s i t y  [FRI75] f r e q u e n t l y  assumes an  of the form:  sin(mrr/R ) c p(r) = =  P l  P l  (r) + Ea n (r) +  - Pi(r) with p i ( r ) a s t a r t i n g density, and  R  c  a cutoff  radius.  P p  » < r  n  (r)  R c  •  r ,r<R  (A8.1)  c  ,r >R  c  o f t e n of the Fermi or M o d i f i e d  G a u s s i a n Form,  The d i s t r i b u t i o n of e q u a t i o n A8.1  has been used r e c e n t l y i n the  a n a l y s i s of n e u t r o n d i s t r i b u t i o n s from TT~ r a t i o experiments on i s o t o p e s of s u l f u r and magnesium.  V a r i a t i o n s have been used by o t h e r s [BAR81] i n t h e i r  a n a l y s e s of n u c l e a r matter d i s t r i b u t i o n s u s i n g h a d r o n i c probes. Many of the p r o p e r t i e s of the expansion found i n e q u a t i o n A8.1 a r e r e l e v e n t to the p r e s e n t a n a l y s i s have been e x t e n s i v e l y elsewhere  [GYL84, FRI78], and w i l l not be r e p e a t e d h e r e .  that  discussed We r e c a l l t h a t , as  w i t h the MGCV d i s t r i b u t i o n d i s c u s s e d e a r l i e r , we must d e a l p r o p e r l y w i t h the e l e c t r o s t a t i c e f f e c t s of the added n u c l e o n m a t t e r . i n appendix IX, as i t was  T h i s w i l l be d i s c u s s e d  not a matter of importance In the a n a l y s i s of  [GYL84].  A8.3 Fourier Laguerre Expansions The c h o i c e of the FB expansion f o r our s t u d i e s i s perhaps not the wisest choice. utilises  A more l o g i c a l and c a l c u l a t i o n a l l y convenient c h o i c e  the harmonic o s c i l l a t o r s o l u t i o n s seen In appendix IV.  s q u a r i n g , these g i v e r i s e  With  to d e n s i t i e s of the form:  P(r) = a"  2n - ( r / a ) 2 Z B (r/a) e  3  n  (A8.2)  A 8 . 3 . 1 The Basis An orthonormal b a s i s s e t of such d e n s i t i e s on the i n t e r v a l  N  m  e  -(r/a)2 Li'2[ (r/«)2] 2  f  (0,°°) i s :  (A8.3)  257  1/2 i s a w e l l depth parameter and the L^'  where 'a', c o r r e s p o n d i n g t o R , c  are Laguerre polynomials. a  n x  /2. h w  e  r  e  n  x  i  s  N  i s chosen t o n o r m a l i s e the J t  m  to  m  chosen f o r c a l c u l a t i o n a l convenience: a N  2  m  * [  [r(m+3/2)]  3  3  ] = a"  • 4/2  (A8.4)  n X  r(m+l)  from which: r(m+l)  4/2 i  B  . ]  1  a a 6  -(  3 _ n X  r / a  [r(m+3/2)] >  , ,, e" >  / 0  = [  /  2  2  ( r / a  L /2[2(r/a)2] , m  2k k E L(m,k) ( r / a ) 2 , k=0 m  =  (A8.6)  K  a  (A8.5)  3  (3-nx)/2  where L(m,k)=0 f o r k > m.  The v a l u e s of L(m,k) a r e then u n i q u e l y d e f i n e d by  the r e c u r s i o n r e l a t i o n s [MOR53]: Lj  / 2  L}  / 2  ( z ) = T(3/2)  (A8.7)  ( z ) = r(5/2)(3/2-z)  (A8.8)  2n+l L  l/2 n  (  z  I (  ) n  2 n  -l/2) L  2n  (z) - z L ( z ) - (n-1/2) n-i n—i 1 / 2  1 / 2  2  LJ/ (Z) } n-z 2  (A8.9)  Now, -(r/a) Pt  , (r) = E a I T  F  L  n  n  nmax n = E o £ n=0 k=0 n  , L(n,k)2 (r/a) f c  2  (A8.10)  Z l c  (3-nx)/2  258  nmax E k=0  "(r/a)2  nmax 2k I a L(n,k) 2* (r/a) n=k = 0,.  (A8.ll) (3-nx)/2  n  "(r/a)  nmax £ 0,, k=0  2k (r/a)  fc  2  (A8.12) (3-nx)/2  A 8 . 3 . 2 Zero S u m Constraint We now evaluate: nmax _ 1 p ( r ) r dr = E 8,<3+nx)/2 _  m  J  0  *  2k+3  w  2  F L  a  k=0  L  {  f r  *  ]  2  }  2  which must give zero contribution to the t o t a l density, as the density p j ( r ) has the normalisation T, the relevant number of nucleons.  Hence the  constraint: 0  -1  nmax  T(3/2)  k=l  s  O  0  k  r[  2k+3  ],  (A8.14)  2  K  from which: nmax cc = [ e - E a L(n,0) ] / L(0,0). k-1 0  (A8.15)  0  This allows us to rewrite pp^  a  s  ;  2k+3 nmax 2k P ( r ) - £ 0. { (r/a) K T  F  L  k-1  k  F [  T  "  e^  ]  r(3/2)  }  r / a  >  2  (A8.16) (3-nx)/2  A 8 . 3 . 3 Derivatives of PpL() r  In order to evaluate uncertainties i n the density due to uncertainties  i n the f i t t e d 3p,, (r) FL  , 2  T  3a  density coefficients a  we w r i t e  n >  the d e r i v a t i v e s  -(r/a) e '  2  nmax Z kg ( r / a ) } (3-nx)/2 k-1 v  { p(r) [(r/a) -(3-nx)/4] 2  a  (A8.17)  2 k  k  a  and: 9 p  FL  nmax 9p ( r ) = £ [ — — k=l 36, k  ( r )  3a. i  8 f 5  k ]  (A8.18)  3a i J  2k+3 i = I L(i,k) 2 k=l  T[  { (r/a)  k  2 k  ~T  e"  ]  -  ( r / a  >  2  } T(3/2)  (A8.19) (3-nx)/2 3.  A8.3.4 R a d i a l Moments The r a d i a l moments f o r the F o u r i e r L a g u e r r e d i s t r i b u t i o n a r e calculated  from: to J o " [ ( P i < 0 + <  r  p  ( >] r  F L  r  J  +  2  d  r  J > =  (A8.20) to Jo"[(Pi<r) + P  Given t h a t p j ( r ) i s normalised  F L  (r)]  r  2  dr  t o T, we have: 4ir  < r  where < r J >j i s the j  t  J  n  > - < r  3  >. + —  < r  3  >_.  (A8.21)  moment o f the s t a r t i n g d i s t r i b u t i o n ,  P l  (r).  Now,  < r  J  > - Jo" FL FT  Pp, ( r ) r FL  3  +  2  dr = - *-<3+nx>/2 T\ a  2  k  =  1  k  TT{j,k)  (A8.22)  260  where the I T ( j , k . )  i s defined as:  r(  j+2k+3 rr(j,k)  ={ r(  2k+3 2  )  j+3  )  r(  2  r(3/2)  (AS.23)  ) }  2  The d e r i v a t i v e s of the r a d i a l moments are as f o l l o w s :  <  9  V  r J  1 =-  3 0  8  <  f  J  a*-  rr(j.k)  ( 3 + n x ) / 2  (A8.24)  k  >FL  (j+(3+nx)/2)  M  l  a  3a  +  m  0  /  nmax  2  J+U+nx;/^  2  k  E q u a t i o n A8.24, when combined 3 <r  J  rr(j,k)  z  =  (A8.25)  1  with:  > FL  3 <r nmax = Z [  FL  k=l  3a. i  30. k  j  38,  3a, i  k  ]  (A8.26)  gives: 3 < r  J  >,  1  .. , J ( /  +  a  3a. l  2  n  3  +  W n  x  i  n  )/  2  (rr(j,k)L(i,k)2 ) k  r  (A8.27)  k=l  A8.3.5 Uncertainties in ppi,(r) and the Radial M o m e n t s We wish t o e v a l u a t e the u n c e r t a i n t i e s i n p p L ( r ) and i t s d e r i v e d moments. p =a , 2  2  The nmax parameters p±, i e {0,1,2...,nmax) a r e P g a , P j ^ i i =  Pnmax nmax« =a  T  h  e  covariance matrix a  2  ^  which  r e p r e s e n t s the u n c e r t a i n t i e s i n t h i s parameter s e t i s then O j the  v a l u e s of j and k r u n over the same range as does i .  k  , where  261  Then: 3 p  °* p  (  r  FL^  )  =  ol  2 i,j=0  r ;  1  FL  ( r )  8 p  FL  ( r )  (A8.28) 3p. i  J  3 . j P  w i t h the d e r i v a t i v e s d e f i n e d i n e q u a t i o n s A8.17 and A8.19. S i m i l a r l y , u s i n g the d e r i v a t i v e s i n equations A8.25 and A8.27, the u n c e r t a i n t i e s i n the r a d i a l moments a r e :  a  -  2  < r  nmax I i,k=0  3  >  F L  3 < r  j  > 3 < r FL  CJ2 1  K  j  >„ FL (A8.29)  3  P i  3p  k  A8.3.6 The D e n s i t i e s F i g u r e A8.1 shows the f i r s t zero net i n t e g r a l requirement a± were s e t to z e r o , except function i l l u s t r a t e d ,  6 F o u r i e r Laguerre  defining  f o r the  the v a l u e of a . Q  corresponding  A l l other v a l u e s of  to the order of the  i n which case i t was s e t to 0.01. The v a l u e of 'a' was  1.5 fm, a v a l u e near t o t h a t suggested potential.  d e n s i t i e s , w i t h the  by the s t r e n g t h of the n u c l e a r  262  CO C o  o D  0  1 2  3  4  5  6  0  1 2  3  4  5  6  1  3  4  5  6  0  1 2  3  4  5  6  C  O O O X  2  r  (fm)  F i g u r e A8.1  a) and b) The f i r s t of  s i x ( z e r o norm) FL d e n s i t i e s .  The t o t a l number  l o c a l minima and maxima on a g i v e n curve i s the o r d e r o f the density.  c) and d) D e n s i t i e s o f a) and b) f o l d e d w i t h the p r o t o n form f a c t o r .  [a  0  normalisation]  263  Appendix IX Of Charge Densities and Matter Densities  A9.1 Convolution of the Proton F o r m Factor By v i r t u e of the form of i t s i n t e r a c t i o n , the p i o n i s a b l e to probe matter  densities  directly.  d e n s i t y as w e l l , potential,  There i s a dependence upon the n u c l e a r  but f o r the purposes  i t i s usually  The de f a c t o  of i n c l u d i n g  s u f f i c i e n t to assume the two i d e n t i c a l .  density.  though, p r o v i d e s us  To m e a n i n g f u l l y compare our r e s u l t s  the s t a n d a r d , then, we must p r o v i d e a p r e s c r i p t i o n charge  Coulomb e f f e c t s i n our  s t a n d a r d f o r measuring n u c l e a r s i z e s  w i t h the n u c l e a r charge  charge  with  f o r r e l a t i n g matter  and  densities. There a r e a number of e f f e c t s which may be c o n v o l u t e d w i t h a matter  d e n s i t y to a r r i v e a t a charge i m p l i c i t l y folded are not u s u a l l y  density.  i n t o our matter  seen  The CM motion of a nucleus i s  d e n s i t y measurement.  Magnetic  i n the l i g h t e r n u c l e i , but r a t h e r i n h e a v i e r  effects nuclei  such as ** Ca [BER72], where h i g h a n g u l a r momentum o r b i t a l s a r e occupied and 8  c o n t r i b u t e s i g n i f i c a n t m a g n e t i c a l l y induced c u r r e n t s .  The e f f e c t of the  f i n i t e p r o t o n s i z e i s the l a r g e s t , and i s the one upon which our e f f o r t s here are c o n c e n t r a t e d . We w r i t e a c o n v o l u t i o n i n t e g r a l  [BAR77] r e l a t i n g the charge and matter  d e n s i t i e s as:  p  ch  ( r )  =  '° m C l - ' l ] P  r  r  < V ' > £' R  D  <  A9A  >  264  where the charge d e n s i t y here i s Pch> the n u c l e a r matter p  m  ( i e : p+(r)  ppp.  or p _ ( r ) ), and  Chandra and  values  the s i n g l e p r o t o n matter d e n s i t y  Sauer [CHA76] suggest:  p  where £^^=1  p p  3 I ^  (r) =  -(r/r.)2 [ITr^]  2  —  and  r j  guarantees t h a t <r  A9.2 Folding P  p  p  2 c n  2  >  those g i v e n  (A9.2)  1  e  as a v i a b l e p r e s c r i p t i o n f o r the proton  of  density  i n t a b l e A9.1.  form f a c t o r , w i t h  The  folding  the  process  = < r > + <r pp>. 2  2  m  into a Fourier Laguerre S u m  For a F o u r i e r L a g u e r r e d i s t r i b u t i o n , we may  p. (r) = a ^ T  3  >  /  T % . ( r / a ) j-0  2  2  write:  3  e - ^  /  a  >  (A9.3)  2  3  P e r f o r m i n g the c o n v o l u t i o n and  some a l g e b r a i c m a n i p u l a t i o n  -(r/ P  F F L  (nx+l)/2 (r) = ± ± r  3 ^ i E i-1 r / A  )  (2j+l)!  2  (B2/A e  6  i  2j+l E k=0  r i  i  ) nmax j E j-0 J + l / 2 6  *  A  B  i  EVEN(k) [ k!(2j+l-k)!  yields:  . J * 2  •A  (k-1)!! (A9.4)  1 - 1 - 1 1  ±  2  k/2+l  where: EVEN(k) = 1 = 0  f o r k even f o r k odd  (A9.5) ,  Table A9.1 Nucleon F o r m Factor Parameterlsatlon  i  +1 PROTON  NEUTRON  *i  A  i  2  3  < r >  .506373  .327922  .165705  .775425  .431566  .139140  1.525540  1  -1  .444136  .521464  -  2  2x.7242/3  ar A  A9.3 Folding p  pp  The  i  -  ]  and  .  B  FB f u n c t i o n a l form f o r p ( r ) does not i n c l o s e d form.  f o l d i n g numerically  CPU-intensive.  .  (A9.6)  A  1  into a Fourier Bessel S u m  f o l d e d charge d e n s i t y the  = [  1 +  In the  has  allow  e v a l u a t i o n of  the  A n u m e r i c a l technique to accomplish  been developed, but  i s cumbersome  and  t e x t o f t h i s t h e s i s where the FB matter d e n s i t y i s  used i n o p t i c a l model a n a l y s e s ,  the charge d e n s i t y of the nucleus was taken  e q u a l t o the p r o t o n matter d e n s i t y , as the  s m a l l d i f f e r e n c e between the two  d i d not warrant s a t i s f y i n g the a d d i t i o n a l n u m e r i c a l demands.  A9.4 Folding  P  p  p  into the Starting Density P(r) 1  In the F o u r i e r a n a l y s e s  discussed  i n the  were always assumed to be of the MG form. p r e s c r i p t i o n f o r the p charge d e n s i t y  p  p  allows  to be w r i t t e n  P  c h  ( >l  c n  "  T  starting densities  I t i s w e l l known t h a t a G a u s s i a n  an a n a l y t i c form f o r the  corresponding  [ELT61]:  3 a 3(b2.-a ) a r -(r/b ) POO + i - T [ 1 + T { — 5 — + - — } ] e 1=1 b. 2b b b. l i i i 3  r  t e x t , the  2  2  2  2  S  3  2  2  (A9.7)  2  where b^ = a + 2 ( r | ) / 3 . 2  A9.5  Evaluations of < f r 0  In our  d i s c u s s i o n of core+valence n u c l e o n d i s t r i b u t i o n s , we noted  e v a l u a t i o n of the n u c l e a r know:  e l e c t r o s t a t i c p o t e n t i a l , <|>(r), r e q u i r e s  that  t h a t we  *o We  =  4 i r e  o /o  r  2  dr.  2  (A9.8)  r  assumed the matter and charge d i s t r i b u t i o n s to be e q u a l , i n t h a t  discussion.  We w i l l now  d i s c u s s the v a l u e s of 4>  f o r the f o l d e d  0  charge  d i s t r i b u t i o n s c o r r e s p o n d i n g t o the d e n s i t y forms r e f e r r e d to i n t h i s thesis.  A 9 . 5 . 1 < t > for the Folded M G Charge density 0  For the f o l d e d MG charge d i s t r i b u t i o n d i s c u s s e d immediately above, we have: 4-rreg a' 2  3  a  3T  3  a  a  5  (A9.9)  0  2  A 9 . 5 . 2 4> for the Unfolded FB Charge Density 0  In the case of p g ( r ) , P  i f we assume the P h Pm> =  w  e  c  have:  $0 to =  Jo e  c =  2 0  4Tre  2 0  r  [  ]  TT  nmax n I [— ]• n=l (odd)  (A9.10)  n  A 9 . 5 . 3 $ for the Folded FB Charge Density 0  In the case of a F B d i s t r i b u t i o n , <|>Q may  not be expressed i n a c l o s e d  a n a l y t i c form, and must be e v a l u a t e d n u m e r i c a l l y .  A 9 . 5 . 4 $ for the Unfolded FL Charge Density 0  When p p L ( r ) i s c o n s i d e r e d , then i n the case t h a t charge and matter d e n s i t i e s a r e c o n s i d e r e d i d e n t i c a l  •  0  = 27re  2  (  0  n  x  +  1  )  /  2  a  ( i e , no f o l d i n g i s performed):  T * k=0  k! 8,  (A9.ll) k  A 9 . 5 . 5 4> for the Folded FL Charge Density 0  I f we choose  to use the proper f o l d e d charge d e n s i t y to accompany a F L  matter d e n s i t y , then:  ,  2  (  n x +  3)/2  n m a x  h  3  1-1  (a + r ) 2  1  /  2  j  j-0  1  (k-1)!! k/2  y ,j A  1  ( 2 3 + 1 ) 1  k=0 k ! ( 2 j + l - k )  ±  a [{ — } r  EVEN(k) 2  2  6  2 j _ k  Kj-k/2+1) ]  (A9.12)  t  A9.6 Unfolding of the Proton F o r m Factor from Electron Scattering Data To compare the matter d e n s i t i e s o b t a i n e d i n these experiments w i t h the charge d e n s i t i e s measured i n e l e c t r o n s c a t t e r i n g , the charge d e n s i t i e s o f Norum e t a l . [NOR82] f o r  1  6  0 and  1  8  0 and Cardman  [CAR80] f o r  1  2  C were  fitted  w i t h f o l d e d FL charge d e n s i t i e s , where the f i t t i n g parameters were a and of  e q u a t i o n A8.10. The f a c t t h a t the F o u r i e r component must g i v e a net zero c o n t r i b u t i o n  to  the matter d e n s i t y of the n u c l e u s means t h a t the FL d e n s i t y and i t s  f o l d e d c o u n t e r p a r t must have the same RMS i n t e g r a l .  We choose:  B a  r, nmax  nmax . '  =  m a v  3 and:  a  Q  un  R(nmax,nmax)  - a  (A9.13)  nmax-1 l. a R(n,0) n=l n '  R(nmax,0) nmax '  =  (A9.14) R(0,0)  where: nmax-1 " 2 k-l 8  IIIQ3 X  2k+3 W  T[  k  ] 2  =  (A9.15) 2nmax+3 nmax-1 R(nmax,k) 2k+3 (nmax r [ ] + Z [ k T ( )]) 2 k=l R(nmax,nmax) 2 nmax-1 k 2k+3 Z B ( 1) r [ ] k=l nmax 2 e = T(3/2) k  (A9.16)  Q  R(j,k) = L ( j , k ) 2  and  B, k  =  6, + k  a  nmax  (A9.17)  k  R(nmax,k)  which guarantees the FL d e n s i t i e s t o have z e r o RMS  (A9.18)  integral.  The f i t t i n g was performed w i t h the n o n - l i n e a r l e a s t squares r o u t i n e ZXMIN from the IMSL [IMS82] l i b r a r y . and a to  n m a x  ,  this f i t t i n g  fitting  With the above c h o i c e s f o r ct  was f o r c e d , by c h o o s i n g a p p r o p r i a t e  P l  (r)'s,  g i v e r e s u l t a n t matter d e n s i t i e s and charge d e n s i t i e s c o n s i s t e n t w i t h the  0  experimental e l e c t r o n s c a t t e r i n g r e s u l t s . b e s t determined  As the RMS  q u a n t i t y from such experiments,  charge  r a d i i a r e the  i t i s important  to be  c o n s i s t e n t w i t h r e s p e c t to them. F i g u r e A9.1 shows the e x p e r i m e n t a l e l e c t r o n s c a t t e r i n g r e s u l t s f o r the charge  d e n s i t i e s of  corresponding to  P l  1 2  C  (r).  and  1 6  0 , a l o n g w i t h the s t a r t i n g  The FL Components r e p r e s e n t i n g the d i f f e r e n c e s  between the two a r e a l s o i l l u s t r a t e d . form f a c t o r from the p r o t o n matter i n F i g u r e A9.2. functions.  (folded) densities  The e f f e c t of u n f o l d i n g the p r o t o n  difference,  ( p + ( r ) - p + ( r ) ) , i s shown 1 8  1 6  F i g u r e A9.3 shows some o f the z e r o - R M S - i n t e g r a l  FL d e n s i t y  100  T  1  1  1  r  100  0  1 2  3 4  5 6  1 2  3 4  5 6  -5 -  -10  1 2  3 4  5 6  -10  0  (fm) Figure A9.1 1 6  0  and  1 2  C  charge d e n s i t i e s ,  a) and b) The s o l i d c u r v e s a r e  f o l d e d m o d i f i e d G a u s s i a n d e n s i t i e s and the broken curves a r e the model independent d e n s i t i e s of [NOR82] and [CAR80]. c) and d) FL components r e p r e s e n t i n g the d i f f e r e n c e between the d i s t r i b u t i o n s i n a) and b ) .  [ a normalisation] 0  5.0 2.5 ro  -  0.0 -  I  E  x  2-2.5 Q^5.0  CO  •%7.5 -10.0  0  1 2 3 4 5 RADIUS (fm)  6  Figure A9.2 P r o t o n matter between  1 8  »  1 6  0.  s e c t i o n A9.6. MG  ( s o l i d ) and charge ( s h o r t dash) d e n s i t y  differences  The u n f o l d i n g was c a r r i e d out as d e s c r i b e d i n A l s o shown i s the charge d e n s i t y d i f f e r e n c e w i t h  forms w i t h the RMS  charge r a d i i of  1 8  0  and  1 6  0  (long dash).  The d o t t e d curve i s the charge d e n s i t y from a c a l c u l a t i o n of Brown e t a l . [BR079a, BR079b].  -2  12  3  4  5  -2  6  J  0  L  1 2  J  3  L  4  5  6  (fm) Figure  Matter d e n s i t i e s  (a and  c o r r e s p o n d i n g to the f i r s t RMS  i n t e g r a l and  A9.3  b) and  charge d e n s i t i e s  (c and  d)  6 FL d e n s i t i e s , c o n s t r a i n e d to zero  zero norm.  [a  0  normalisation]  274  Appendix X M o m e n t s and the Folding Process  A10.1 The Folding and the RMS Integrals The  c o n v o l u t i o n i n t e g r a l used i n f o l d i n g  the p r o t o n form f a c t o r w i t h a  matter d e n s i t y t o a r r i v e a t a charge d e n s i t y has been  ch  p  where p  m  ( r )  / PmUr-jL'l)  =  p  ( P  r  ,  )  d  P  written:  i '  i s a matter d i s t r i b u t i o n and Ppp i s a form  (  A  l  o  a  )  factor.  C o n s i d e r , now, the RMS i n t e g r a l : I = /  P  c h  (r) r  2  dr =  Changing i n t e g r a t i o n noting that the  / [/ P ( | r - r ' | ) r  variables  angular i n t e g r a t i o n ,  P p p  (r')  dr'  V  J  (A10.2)  (rj-r_' ) -> _r' i n the i n n e r i n t e g r a l and vanish i n  we a r r i v e a t :  h  f 4TT / °° p ( r ) r m 0  unit  dr ]  terms c o n t a i n i n g odd powers o f cos6'= _r»£'/(|r| |r'|)  I = ( 4TT f °° p ( r ) v  For  2  m  normalised d i s t r i b u t i o n s ,  d r )( 4ir / °°p  2  (r') r ' dr' ) + 2  d r ) ( 4ir f "p 0 PP this  ( r ' ) r' * d r ' ) 1  gives:  < r > = < r > + < r > ch m pp 2  (A10.3)  2  2  (A10.4)  275  A10.2 Other M o m e n t s via the Fourier Transform The s u b s t i t u t i o n above may be used to c a l c u l a t e the o t h e r  even moments  of the d i s t r i b u t i o n P h ( ) » but the c a l c u l a t i o n q u i c k l y becomes r  c  tedious.  A more e l e g a n t way of a r r i v i n g a t a g e n e r a l  t h a t the c o n v o l u t i o n  i n t e g r a l A10.1 i m p l i e s  P  c h  (q) = p j q )  expression  i s to note  that:  P p p  (q),  (A10.5)  where the p(q) a r e the F o u r i e r transformed p ( r ) ' s :  p(q) =  / q  r s i n ( q r ) p(r) dr  0  , n (-)  . 2n <r >  x  v  = 4rr I [ n=0 ( 2 n ) ! (2n+l) The product i n e q u a t i o n  < r  ch c  n ? T V  > "  n  Note t h a t i f P  p p  n  ]  (A10.6)  A10.5 then g i v e s : 2i <r > y  ,„  . q"  ^ ^ j=0  . i  2n Z  J  . 2(n-i) <r >  v  v  PP  p i e c e s t h a t a r e u n i t n o r m a l i s e d ) then p  (2^1-1)  c n  <  A 1 0  ' > 7  ( o r c o n s i s t s o f a sum of  w i l l have the same  n o r m a l i s a t i o n as ( o r be the sum o f p i e c e s w i t h m  v  m  1  n  ( q ) i s u n i t normalised  sum of p i e c e s c o m p r i s i n g ) p «  J /  the same n o r m a l i s a t i o n as the  

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