UBC Theses and Dissertations

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UBC Theses and Dissertations

Low energy π± interactions with S-D shell nuclei Tacik, Roman 1984

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LOW ENERGY i r * INTERACTIONS WITH S-D SHELL NUCLEI by ROMAN TACIK B.Sc, McGill University,  1978  M.Sc, McGill University,  1980  A THESIS SUBMITTED  IN PARTIAL FULFILLMENT OF  THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY  in THE FACULTY OF GRADUATE STUDIES Department of P h y s i c s  We a c c e p t t h i s t h e s i s as conforming to the r e q u i r e d  standard  THE UNIVERSITY OF BRITISH COLUMBIA J u l y 1984 © Roman T a c i k ,  1984  In presenting t h i s thesis  in partial fulfilment  r e q u i r e m e n t s f o r an a d v a n c e d d e g r e e a t of B r i t i s h Columbia, it  the  and s t u d y .  agree t h a t p e r m i s s i o n f o r extensive f o r s c h o l a r l y purposes  further thesis  It  is thesis  g a i n s h a l l n o t be a l l o w e d w i t h o u t my w r i t t e n  permission.  Department o f  PHYSICS  The U n i v e r s i t y o f B r i t i s h 1956 Main M a l l V a n c o u v e r , Canada V6T 1Y3 J u l  make  may be g r a n t e d b y t h e h e a d o f my  understood that copying or p u b l i c a t i o n of t h i s  Date  I  copying of t h i s  d e p a r t m e n t o r by h i s o r h e r r e p r e s e n t a t i v e s . financial  the  University  I agree that the L i b r a r y s h a l l  f r e e l y available for reference  for  of  Y 26, 1984  Columbia  Abstract  A magnetic spectrometer low  energy  pion scattering  has been commissioned f o r use i n p e r f o r m i n g  experiments.  I t has been used  d i f f e r e n t i a l c r o s s s e c t i o n s f o r the e l a s t i c 1 2  C,  1 8  0 , and M g  s t a t e s o f these The  scattering  from  to the 2 ^  nuclei.  analysis  of t h e e l a s t i c  an o p t i c a l p o t e n t i a l model. tions  s c a t t e r i n g o f 50 MeV ir*  t a r g e t s , as w e l l as the i n e l a s t i c  2 6  to measure the  f o r C and 1 2  1 8  s c a t t e r i n g d a t a has been performed  I t i s found  with  t h a t the ir* e l a s t i c c r o s s sec-  0 a r e w e l l d e s c r i b e d by c a l c u l a t i o n s employing  a glo-  b a l parameter s e t (CMS 8 2 ) , which has been p r e v i o u s l y shown to reproduce ir  +  e l a s t i c s c a t t e r i n g d a t a f o r a range o f n u c l e i .  The TT* e l a s t i c  f o r ^Mg r e q u i r e a v a r i a t i o n of the p-wave p o t e n t i a l parameters. 2  lar  requirement  i s seen w i t h ir* -  3 2  S e l a s t i c data  D i s t o r t e d wave c a l c u l a t i o n s , i n p u t t i n g t r a n s i t i o n d e n s i t i e s , have been performed tic  data.  data A simi-  (Sob 8 4 a ) .  s e p a r a t e neutron and p r o t o n  f o r the a n a l y s i s o f the i n e l a s -  These c a l c u l a t i o n s demonstrate a s t r o n g s e n s i t i v i t y o f i r  p r o t o n s , and i r " t o n e u t r o n s ,  to  +  present i n the n u c l e u s .  By v a r y i n g t h e s t r e n g t h s of the neutron and p r o t o n t r a n s i t i o n d e n s i ties,  i n o r d e r t o reproduce  sible  to determine  for  the g i v e n 2  +  the r a t i o o f neutron  states.  t h i s way i s independent calculation. and  the TT* i n e l a s t i c  to p r o t o n m a t r i x elements  n  p  n  of t h e o p t i c a l p o t e n t i a l parameters used n  1 2  (M /M )  I t i s shown t h a t t h e v a l u e of M /Mp o b t a i n e d i n  The e x t r a c t e d v a l u e s o f M /M  0.90 ± 0.07 f o r C ,  c r o s s s e c t i o n s , i t i s pos-  1 8  p  i n the  a r e : 0.97 ± 0.08, 1.81 ± 0.15,  0 , and M g r e s p e c t i v e l y . 2 6  agreement w i t h those o b t a i n e d by other means.  These v a l u e s a r e i n  -  iii  -  Table of Contents  Abstract  i i  T a b l e o f Contents List  of F i g u r e s  List  of T a b l e s  i i i v viii  Acknowledgements  x  Chapter I  1  INTRODUCTION  1.1  N u c l e a r P r o p e r t i e s and the S h e l l Model  3  1.2  S c a t t e r i n g Experiments  7  1.3  Pion-Nucleon S c a t t e r i n g  9  1.4  Experimental Considerations  Chapter I I  14  EXPERIMENTAL DETAILS  18  2.1  M13 Beamline  19  2.2  QQD  Spectrometer  24  2.2.1  Detectors  24  2.2.2  E l e c t r o n i c s and Data A c q u i s i t i o n  30  2.2.3  Momentum D e t e r m i n a t i o n  33  2.2.4  Resolution  36  2.3  Targets  Chapter I I I 3.1  40  EXPERIMENTAL ANALYSIS  .  ... .42  C a l c u l a t i o n o f Cross S e c t i o n s  3.1.1  Beam N o r m a l i z a t i o n  44  3.1.2  Target Thickness.  49  3.1.3  Efficiency  50  3.1.4  Peak F i t t i n g  53  3.1.5  Spectrometer Acceptence  54  3.2  Results  60  - iv -  C h a p t e r IV 4.1  THEORETICAL DETAILS  Elastic  67  Scattering  4.1.1  The MSU O p t i c a l P o t e n t i a l  68  4.1.2  Further Discussion  71  4.2  Inelastic  Scattering  4.2.1  C a l c u l a t i o n of Cross S e c t i o n s  76  4.2.2  Transition Densities  78  4.2.3  Neutron and Proton M a t r i x Elements  81  4.2.4  Isovector  S e n s i t i v i t y o f I T * I n e l a s t i c S c a t t e r i n g . . .83  Chapter V  RESULTS AMD CONCLUSIONS  92  5.1  Elastic  Scattering  95  5.2  Inelastic  Scattering  5.2.1  Results  103  5.2.2  Model Dependence  109  5.2.3  Comparison w i t h Other E x p e r i m e n t s  113  References  •  117  Appendix A  D e t a i l s of E l a s t i c C a l c u l a t i o n  123  Appendix B  D e t a i l s of I n e l a s t i c C a l c u l a t i o n  126  Appendix C  N u c l e a r D e n s i t y Models..  Appendix D  TT •*• uv K i n e m a t i c s  ...129 130  - v -  L i s t of Figures  Fig.  1.1  S h e l l Model S t r u c t u r e o f  1 8  0 and M g  Fig.  1.2  Energy L e v e l Schemes f o r  1 2  C,  Fig.  1.3  ir—p T o t a l C r o s s S e c t i o n s  Fig.  1.4  i r p D i f f e r e n t i a l Cross S e c t i o n s  Fig.  1.5  R a t i o o f T r p / i r ~ p D i f f e r e n t i a l Cross  1 8  0 , and M g  ±  2 6  6  10  a t 50 and 160 Mev  +  at  4  2 6  12  Sections  Several Energies  13  Fig.  2.1  M13 Beamline  and QQD Spectrometer  Fig.  2.2  QQD Spectrometer with A s s o c i a t e d  Fig.  2.3  C o n f i g u r a t i o n of E l e c t r o n i c s  Fig.  3.1  T y p i c a l T T and TT~ Time of F l i g h t  Fig.  3.2  Pion Flux Correction Factors  Fig.  3.3  Typical  1 2  C Energy Spectrum  55  Fig.  3.4  Typical  1 8  0 Energy  56  Fig.  3.5  Typical  Fig.  3.6  REVMOC C a l c u l a t i o n of V a r i a t i o n o f QQD  +  Solid  2 6  Mg  20  Detectors  25  31  Spectra  Spectrum...  Energy Spectrum  Angle o v e r S c a t t e r i n g T a r g e t . .  46  48  57  59  -  Fig.  4.1  50 MeV T f D i f f e r e n t i a l Cross M g ( 2 + ) with 8 1  4.2  Sections f o r  1  2 6  Fig.  vi -  = Bp - 0.50  n  84  R a t i o o f I T ~ / T T D i f f e r e n t i a l Cross S e c t i o n s f o r M g ( 2 ) f o r S e v e r a l Values o f 6 +  2 6  85  +  1  Fig.  4.3  50 MeV T T * D i f f e r e n t i a l Cross  2 6  Mg(2  + 1  )  Sections f o r  f o r S e v e r a l Values  of 6  Fig.  4.4  /c+ vs 8 f o r 50 MeV I T S c a t t e r i n g  Fig.  4.5  a /a~  Fig.  5.1  Fig.  Fig.  Fig.  5.2  5.3  5.4  1  +  vs B / B p  87  to M g ( 2 2 6  f o r 50 MeV T T S c a t t e r i n g 1  n  + 1  )  to M g ( 2  89  2 6  + 1  )  91  C and 0 E l a s t i c Cross S e c t i o n s C a l c u l a t i o n w i t h S e t E Parameters  96  M g and S E l a s t i c Cross S e c t i o n s C a l c u l a t i o n w i t h Set E Parameters  97  1 2  1 8  2 6  3 2  M g and S E l a s t i c Cross S e c t i o n s C a l c u l a t i o n with 'Best F i t ' Parameters  2 6  3 2  Typical x to  Fig.  5.5  1 2  Fig.  5.6  1 8  Fig.  5.7  2 6  2 6  Mg  Plot  2  f o r F i t of 8  I n e l a s t i c Cross  C(TT,TT')  1 2  0(TT,TT')  1 8  C*(2  1  0*(2  1  and B  p  Sections  104  +  )  I n e l a s t i c Cross  Sections  106  +  )  I n e l a s t i c Cross  Sections  107  Mg(ir,TT ) Mg*(2 ) l  n  101  2 6  +  1  I n e l a s t i c Cross  Sections  108  - vii -  Fig.  5.8  Variation  Fig.  5.9  x 2 Contour P l o t U s i n g 'Best F i t ' O p t i c a l P o t e n t i a l Parameters  Fig.  5.10  of  'Best F i t ' B's  Comparison of M /M as O b t a i n e d Using D i f f e r e n t n  with Changes i n R e c  Probes  TT  F i g . D.2  E  F i g . D.3  Decay Muon A n g u l a r D i s t r i b u t i o n  uv Decay i n Center of Mass and  vs <|> f o r 50 MeV u  110  Ill  p  F i g . D.l  u  Q  IT'S  ....115  L a b o r a t o r y Frames  131  133  134  - viii  -  L i s t of Tables  Table  2.1  T a b l e 2.2  T a b l e 2.3  Specifications  f o r QQD  MWPCs  27  T y p i c a l Set of QQD T r a n s f e r C o e f f i c i e n t s f o r WC4 P o s i t i o n  Typical  Set of QQD  37  Transfer  Coefficients  f o r WC5  T a b l e 3.1  Experimental  Target Thicknesses  Table  3.2  Measured  Cross S e c t i o n s f o r  T a b l e 3.3  Measured  Cross  Sections f o r  T a b l e 3.4  Measured  Cross  Sections f o r  1  Table  3.5  Measured  Cross  Sections f o r  1 8  T a b l e 3.6  Measured  Cross S e c t i o n s f o r M g ( T r , T r ) M g  65  Table  Measured  Cross S e c t i o n s f o r M g ( T r , T r ' ) M g * ( 2 + , 1.81)  66  3.7  Position  1 2  8  C  C(TT,TT)  1 2  Table  Nuclear  61  C ( i r , T r ' ) C*(2+,4.44) 12  0(ir,Tr)  0(Tr,Tr'  1 8  0  18  2 6  2 6  Parameters  for Optical Potential  64  72  D e n s i t y D i s t r i b u t i o n Parameters  Best F i t V a l u e s  62  63  ) 0 * ( 2 + , 1.98)  2 6  Set E O p t i c a l P o t e n t i a l  T a b l e 5.2  2  51  2 6  T a b l e 4.1  5.1  I  38  Parameters  93  99  - ix -  T a b l e 5.3  C a l c u l a t e d T o t a l R e a c t i o n Cross S e c t i o n s  102  T a b l e 5.4  F i n a l R e s u l t s f o r M /M D e r i v e d from A n a l y s i s of I n e l a s t i c P i o n S c a t t e r i n g Data  105  Comparison of P r e s e n t R e s u l t s w i t h Those Obtained Using Other  114  T a b l e 5.5  n  p  Techniques  - x -  Acknowledgements  First haircut  and  and  foremost,  I suppose I should thank the man  y e l l o w framed g l a s s e s , my  w i t h the  r e s e a r c h s u p e r v i s o r , D i c k Johnson.  A l s o , both f o r the i n c r e d i b l e amount of time he spent on the and  t h a t he d i d p h y s i c s because he enjoyed G y l e s and  before I joined i t .  the graduate  Drake, our man  s t u d e n t s t h a t do a l l the hard  to get done.  Of c o u r s e , t h i s i s not meant  our Swiss r e s e a r c h a s s o c i a t e ; and  chamber our  Tom  i n Toronto.  I musn't n e g l e c t to thank our t e c h n i c i a n s .  Robert Oppenshaw f o r h i s  of the a r t o f wire chamber maintenence; Doug Maas f o r h i s know-  l e d g e o f e l e c t r o n i c s ; and Grant  S c h e f f e r f o r a whole host of t h i n g s .  I wish I c o u l d thank the engineer who  worked w i t h us f o r a w h i l e , but  the memory o f the u n n e c e s s a r i l y heavy s t a i n l e s s back w i r e chambers i s s t i l l Unfortunately, myself,  Every-  C h r i s Wiedner, beam o p t i c i a n s e x t r a o r d i n a i r e s ,  German c o n t i n g e n t ; Hans Roser,  mastery  s t u d e n t s w i t h the group  the v a l u a b l e c o n t r i b u t i o n s of K a r l Erdman, our d r i f t  e x p e r t ; S i g M a r t i n and  shifts,  Gill.  Randy Sobie got i n v o l v e d not long a f t e r w a r d .  work which e n a b l e s experiments underplay  i t , I thank Dave  Bruce B a r n e t t were graduate  one knows t h a t i t i s r e a l l y  to  spectrometer,  f o r s a y i n g , a t f o u r o ' c l o c k one morning, a f t e r a week of n i g h t  Bill  funny  too s t r o n g .  f o r the  Maybe some day.  I d i d a l l the t y p i n g and  diagrams f o r the  so I c a n ' t thank anyone f o r h e l p i n g me  l e d g e the support p r o v i d e d by f r i e n d s who  s t e e l housings  thesis  t h e r e , but I must acknow-  would have h e l p e d , had  I asked.  - 1 -  CHAPTER  I  INTRODUCTION  In t h i s  thesis,  the r e s u l t s of experiments  involving  the s c a t t e r i n g  o f 50 MeV p i o n s from s e v e r a l n u c l e i a r e p r e s e n t e d and d i s c u s s e d . of these experiments  was to p r o v i d e data from which i n s i g h t s  d e t a i l s o f the p i o n - n u c l e u s  The aim  i n t o the  i n t e r a c t i o n c o u l d be g a i n e d , and n u c l e a r  structure information extracted. The one  pion-nucleus  definitive  system i s s u f f i c i e n t l y complex, t h a t t h e r e i s no  experiment  answer to a l l q u e s t i o n s .  which can be performed,  which w i l l  Knowledge i n t h i s  i s gained  field  examination  of large q u a n t i t i e s of d i v e r s e data.  sed i n t h i s  t h e s i s represent a s i g n i f i c a n t  in several respects: p i o n s o f both  charge  states,  through the  The experiments  contribution  We have measured the e l a s t i c  p r o v i d e the  i n this  scattering  from a range o f n u c l e i ,  and found  discusdirection  of 50 MeV that a l l  the data can be d e s c r i b e d c o n s i s t e n t l y w i t h i n the framework of a s i n g l e model. We have measured, f o r the f i r s t 50 MeV p i o n s of both charges nuclei,  specifically  1 8  same model used the neutrons  2 6  extracted  from  a c t u a l model  s t a t e s of s e v e r a l  The shapes o f the measured  scattering.  angular  The r e l a t i v e c o n t r i b u t i o n s of  i n these n u c l e i t o the e x c i t e d  the d a t a , i n a manner which shows l i t t l e  s t a t e s can be dependence on the  used.  In an e f f o r t present  s c a t t e r i n g of  r e a s o n a b l y w e l l w i t h i n the c o n t e x t of the  f o r the e l a s t i c  and p r o t o n s  the i n e l a s t i c  to low-lying e x c i t e d  0 and M g .  d i s t r i b u t i o n s a r e reproduced  time,  t o put these r e s u l t s i n t o a more g l o b a l c o n t e x t , the  i n t r o d u c t o r y chapter of t h i s  t h e s i s begins w i t h a s h o r t , g e n e r a l  - 2 -  discussion  of nuclear properties,  model, which p r o v i d e s a p i c t u r e states. ation  In s e c t i o n  p a r t i c l e s from n u c l e i .  protons(neutrons). noticed  possible  1.3, a d i s c u s s i o n  I t i s p o i n t e d out that  This  excited  consider-  by s c a t t e r i n g  various  of the p i o n -  nucleon  i n low energy  scat-  t o n e u t r o n s ( p r o t o n s ) than t o  low energy s e n s i t i v i t y , l a r g e l y i g n o r e d or un-  by p h y s i c i s t s i n the f i e l d ,  vestigating  continues with a general  i s much more s e n s i t i v e  +  the s h e l l  i n both t h e i r ground and  can be i n v e s t i g a t e d  In s e c t i o n  i n t e r a c t i o n i s undertaken. the T T ~ ( T T )  of n u c l e i  1.2, the d i s c u s s i o n  of how n u c l e a r p r o p e r t i e s  tering  and t h e i r d e s c r i p t i o n w i t h i n  provides a strong motivation f o r i n -  pion-nucleus s c a t t e r i n g .  In s e c t i o n  1.4 then, the d i f f e r e n t  experimental techniques f o r performing pion s c a t t e r i n g  experi-  ments a r e e x p l o r e d , and reasons g i v e n f o r the use o f the a p p a r a t u s which has ject  i n f a c t been employed i n the s e r i e s o f experiments which a r e the subof t h i s  thesis.  In Chapter  I I , a more d e t a i l e d d e s c r i p t i o n  methods used i n p e r f o r m i n g the p r e s e n t contains a quantitative  discussion  o f t h e equipment and  experiments i s g i v e n .  o f how the raw d a t a r e s u l t i n g  these measurements i s a n a l y s e d , and concludes w i t h a t a b u l a t e d t i o n o f the d i f f e r e n t i a l  cross sections.  o p t i c a l model c a l c u l a t i o n s performed a r e o u t l i n e d . dictions  from  presenta-  I n Chapter IV, the t h e o r e t i c a l  for differential  cross sections  which have been  Some time i s spent i n the d i s c u s s i o n  of the i n e l a s t i c c a l c u l a t i o n s ,  reported previously.  Chapter I I I  since  o f the p r e -  s i m i l a r ones have n o t been  C o n n e c t i o n i s made w i t h the e x p e r i m e n t a l  results i n  Chapter V, which c o n t a i n s an examination of the c o n c l u s i o n s which may be drawn from the p r e s e n t the  r e s u l t s ; the i m p l i c a t i o n s  f o r , and l i m i t a t i o n s o f  t h e o r e t i c a l model; and a comparison w i t h o t h e r e x p e r i m e n t a l r e s u l t s .  - 3 -  1.1  Nuclear Properties The  simplest  and t h e S h e l l Model  conceptual picture  of atomic n u c l e i  i s one i n which  t h e y a r e viewed as c o n g l o m e r a t i o n s o f i n d i v i d u a l p r o t o n s and n e u t r o n s , i n r e l a t i v e motion w i t h r e s p e c t fact  to each o t h e r .  Although t h i s p i c t u r e  is in  t o o simple i n many i n s t a n c e s , i t does p r o v i d e a q u a l i t a t i v e  description  o f many b a s i c  A more q u a n t i t a t i v e m a t h e m a t i c a l model.  nuclear properties, description  The b a s i c  such as charge and mass.  can o n l y be p r o v i d e d by a  one Is known as the s h e l l model, i n which  each i n d i v i d u a l n u c l e o n i s viewed as moving i n a p o t e n t i a l g e n e r a t e d by all  the o t h e r s , i n much the same way as an e l e c t r o n  field  o f the p r o t o n i n a hydrogen atom.  atom, the s h e l l model p o t e n t i a l energy l e v e l s . and  1 8  0  and  2 6  As i n the case o f the hydrogen  permits o n l y c e r t a i n allowed o r b i t s o r  These a r e i l l u s t r a t e d  neutrons i n  moves i n the Coulomb  i n Fig.  1.1, as o c c u p i e d by protons  Mg.  U n l i k e the case o f the hydrogen atom, however, where the form o f t h e Coulomb p o t e n t i a l potential  i s not.  i s well  known from f i r s t  I n some I n s t a n c e s , the shape o f the n u c l e a r  may be approximated, o r c a l c u l a t e d  i n a phenomenological,  way.  But the most s a t i s f a c t o r y d e r i v a t i o n s  basic  nucleon-nucleon p o t e n t i a l .  Paris  potential  nucleons.  p r i n c i p l e s , the n u c l e a r  self-consistent  a r e those b u i l t  up from the  The best modern N-N p o t e n t i a l s  (e.g. the  ( V i n 7 8 ) ) are based on the exchange o f p i o n s between  Thus, t h e r e i s some m o t i v a t i o n f o r the i n t e r e s t i n p i o n - n u c l e u s  interactions.  Also,  i t has been p o i n t e d out by Gyles ( G y l 84) that low  energy T T ~ - n u c l e u s e l a s t i c  scattering  the  o f n u c l e a r ground s t a t e s .  of  potential  neutron d i s t r i b u t i o n s the n u c l e a r p r o p e r t i e s  i s perhaps the best way o f s t u d y i n g  one would hope to c a l c u l a t e  T h i s i s c e r t a i n l y one w i t h the use of some  A  E  Fig.  1.1  S h e l l Model S t r u c t u r e of  1 8  0 and  2  6  Mg  - 5 -  m a t h e m a t i c a l model. W i t h i n the framework o f the s h e l l model, n u c l e a r formed by promoting nucleons occupying l o w - l y i n g t h o s e h i g h e r i n energy. which w i l l  In g e n e r a l ,  there  excited  s h e l l model l e v e l s t o  a r e many p o s s i b l e  r e s u l t i n excited n u c l e i with i d e n t i c a l spins  s i m i l a r energies.  Thus, e x c i t e d n u c l e a r  i n low energy ir-nucleus i n e l a s t i c  more s e n s i t i v e to n e u t r o n s , and i r s + ,  combinations  and p a r i t i e s , and  wavefunctions are l i n e a r  combinations o f the many p o s s i b l e n u c l e o n e x c i t a t i o n s . i n Chapter IV, that  states are  It will  be shown  scattering, T T  to p r o t o n s then v i c e - v e r s a .  one  o f the d i r e c t aims o f the experiments d i s c u s s e d  use  a c o m b i n a t i o n o f low energy ir*-nucleus  - ,  S are  Thus,  i n t h i s t h e s i s was to  Inelastic scattering to  s e p a r a t e the n e u t r o n and p r o t o n c o n t r i b u t i o n s  to s e v e r a l nuclear  excited  states. It multiple  i s worth n o t i n g  that when the number o f d i f f e r e n t s i n g l e o r  nucleon e x c i t a t i o n s c o n t r i b u t i n g  to a n u c l e a r  excited  become l a r g e , i t i s sometimes more c o n v e n i e n t t o d e s c r i b e e x c i t a t i o n In terms of a c o l l e c t i v e model, r a t h e r  state  the n u c l e a r  than the s h e l l model.  In c o l l e c t i v e models, e x c i t a t i o n s are assumed to a r i s e as a r e s u l t o f rotations  or v i b r a t i o n s o f the nucleus as a whole.  Note a l s o , t h a t w i t h i n  the s h e l l model, t h e number o f l e v e l s t o  which an i n d i v i d u a l n u c l e o n may be promoted i s q u i t e h i g h . facilitate  c a l c u l a t i o n s , the s h e l l model space i s u s u a l l y  compensated f o r by a s s i g n i n g different The  In order t o truncated,  and  neutrons and p r o t o n s e f f e c t i v e c h a r g e s ,  from t h e i r r e a l ones. energy l e v e l s of  energy, a r e g i v e n  1 2  C,  1 8  0 , and M g , up to =7.5 MeV e x c i t a t i o n  i n F i g . 1.2 (LS 7 8 ) .  26  7.65 7.12 s. ee  (o-i . ,3" r  « * 6.40  2"  5.53 ^ 5 . 2 6 5.?^  —*  4,44  5^  34.46  2*  3.92 3.63,  _&2i  -(2r  6.20  1-  IL2L  4* ar  5.47  E-4«,.-  4.97 4.90  ML  4.351 n / •»•? 3*  _2Ji 359  c c  2.94  1.98  1.81  26 M g  16O  F i g . 1.2  Energy L e v e l Schemes f o r  1  2  C,  1  8  0, and  2  6  Mg  -  1.2  Scattering Perhaps  7  -  Experiments  the most common way o f measuring  through s c a t t e r i n g e x p e r i m e n t s .  An example t h a t s p r i n g s to mind  away i s that of t h e s c a t t e r i n g of l i g h t the b a s i s of v i s i o n .  by f o i l s  into  the human eye, which  of d i f f e r e n t  involving  forms  the s c a t t e r i n g  of a l p h a  materials.  M a t h e m a t i c a l l y , the d i f f e r e n t i a l c r o s s particle  right  The e x i s t e n c e of atomic n u c l e i was suggested by  R u t h e r f o r d as a r e s u l t o f experiments particles  p h y s i c a l observables i s  section f o r scattering a  from a p o t e n t i a l U ( r ) can be o b t a i n e d from Fermi's Golden  Rule.  That i s , doVdt. = |<? |u(r) | Y > |  ,  2  f  where ¥^ and f f a r e t h e i n i t i a l  ±  and f i n a l p a r t i c l e w a v e f u n c t i o n s .  If  these can be taken t o be plane waves, i . e . „, ik-rr 4*i = e  and T f = e  1  then  <¥ |U(r)|f > = / f  where q = k-j-kf.  i  i e  q  *  ikf r  ,  1  U ( r ) d r = U(q)  r  3  ,  And so da/dfi = |U(q) |  For the case of p a r t i c l e  2  scattering  . from a n u c l e u s , i f the  i n t e r a c t i o n between t h e p a r t i c l e and a p o i n t n u c l e o n i s g i v e n by a p o t e n t i a l V ( r ) , then U(r) = / V(r-r') p(r') d r ' 3  ,  where p ( r ' ) i s the d e n s i t y d i s t r i b u t i o n o f n u c l e o n s i n the n u c l e u s . <Tf|U(r)|?i> = / e  i  q  #  r  d r / V(r-r') p(r') d r ' =V(q)p(q) 3  3  Then ,  - 8 -  da/dQ.  and  «  |V(q)p(q) |  From the above, i t can be seen nuclei will nucleon  2  .  that i n g e n e r a l , p a r t i c l e s c a t t e r i n g  depend both on the p a r t i c l e - n u c l e o n i n t e r a c t i o n , and  d i s t r i b u t i o n s w i t h i n the  from  the  nuclei.  In the case of e l e c t r o n s c a t t e r i n g , where the i n t e r a c t i o n i s the f a m i l i a r e l e c t r o m a g n e t i c one, e x t r a c t e d from  n u c l e a r s t r u c t u r e i n f o r m a t i o n can  the d a t a q u i t e e a s i l y .  However, because the i n t e r a c t i o n i s  e l e c t r o m a g n e t i c , the n u c l e a r p r o p e r t y one distribution.  From t h i s , proton matter  i s sensitive  to i s the  charge  d i s t r i b u t i o n s can be o b t a i n e d .  i n f o r m a t i o n i s o b t a i n e d about the neutrons order to study t h e s e , i t i s necessary  be  p r e s e n t i n the n u c l e u s .  to employ h a d r o n i c probes,  No  In  such  as  n u c l e o n s , a l p h a p a r t i c l e s , or p i o n s . U s u a l l y , the hadron-nucleus of an o p t i c a l  p o t e n t i a l , which i s independent  i n d i v i d u a l nucleons  w i t h i n the n u c l e u s .  and  imaginary  The  o p t i c a l p o t e n t i a l used  scattering  use  the  o p t i c a l p o t e n t i a l s have r e a l  p a r t s , the l a t t e r a c c o u n t i n g f o r v a r i o u s i n e l a s t i c i n the present a n a l y s i s of low energy  be p o i n t e d out i n s e c t i o n 1.3,  low energy  be used,  The  of the c o o r d i n a t e s of  the  channels. pion  i s d i s c u s s e d i n s e c t i o n 4.1.1.  It w i l l that  I n t e r a c t i o n i s d e s c r i b e d through  tr s - l  and  then i n s e c t i o n 4.2.4,  are e s p e c i a l l y s e n s i t i v e to n e u t r o n s .  i n combination  This fact  w i t h t e c h n i q u e s such as s c a t t e r i n g from  i s o t o p e s , o r simultaneous  fits  to T T * d a t a , t o reduce  adjacent  the dependence of  the n u c l e a r s t r u c t u r e I n f o r m a t i o n e x t r a c t e d from p i o n s c a t t e r i n g data the d e t a i l s of the p i o n - n u c l e u s same i s t r u e of o t h e r h a d r o n i c  interaction. probes.  can  I t i s not  on  e v i d e n t t h a t the  - 9 -  1.3  Pion-Nucleon Fig.  sections,  1.3 i l l u s t r a t e s  this  the ir* - p r o t o n t o t a l  as c a l c u l a t e d with the phase s h i f t s  These phase s h i f t s u^p  Scattering cross  o f Arndt and Roper (AR 8 2 ) .  a r e the r e s u l t o f a g l o b a l f i t t o a l l the a v a i l a b l e  s c a t t e r i n g data, at a l l energies.  The most o u t s t a n d i n g f e a t u r e o f  p l o t i s the peak a t a p p r o x i m a t e l y 180 MeV incoming p i o n  r e s o n a n c e i n the c r o s s  section i s interpreted  f o r m a t i o n of a new p a r t i c l e , are  scattering  the A.  energy.  This  as being due t o the  I t s c h a r a c t e r i s t i c quantum numbers  s p i n = 3/2 and i s o s p i n = 3/2, thus i t i s f r e q u e n t l y  referred  to as the  3-3 r e s o n a n c e . To consider  explain  i n T T and ir~ t o t a l c r o s s  the d i f f e r e n c e  sections,  +  u p s c a t t e r i n g i n terms of a p a r t i a l wave e x p a n s i o n .  The  ±  differential  cross  s e c t i o n f o r the s c a t t e r i n g of a p i o n  s t a t e , which may be i n t e g r a t e d  to y i e l d  the t o t a l  cross  one may  of either  charge  s e c t i o n , can be  written: da/an h(9)  | (e)  =  g I  |h!(e) |  +  2  .  2  and g ( 9 ) a r e the s p i n - f l i p and non s p i n - f l i p irp s c a t t e r i n g  amplitudes.  They a r e r e l a t e d to sums o f phase s h i f t s and Legendre  p o l y n o m i a l s , over the v a r i o u s index.  Tr+p can o n l y  contributing  da/dSl  do7dJ2  p a r t i a l waves.  I i s an i s o s p i n  e x i s t i n a pure i s o s p i n = 3/2 s t a t e , w h i l e ir~p can be  a m i x t u r e o f 3/2 and 1/2 s t a t e s .  and  |  (TT+P) =  1 (ir"p) = | j g  |g 2  3  Thus,  /  2  (6) |  2  3 / 2  + y g  +  | + 2  1  /  2  |h  (0) |  2  3 / 2  |j h  1 3  /  2  , 2 + - h  |  2  J  /  2  .  - 10 -  - 11 -  Near 180 MeV i n c i d e n t p i o n energy, amplitudes  a r e much l a r g e r  above that the r a t i o o f T T c r o s s s e c t i o n s independent At  lower  incident  cancellation.  of a n g l e , w i l l be 9.  that i f the 1/2 and 3/2 amplitudes  Fig.  the s i t u a t i o n a t a p p r o x i m a t e l y  1.4 i l l u s t r a t e s  f o r both 50 MeV and 160 MeV p i o n s .  energy.  energies.  F i g . 1.5 i l l u s t r a t e s  F o r 50 MeV  Thus, i t i s seen  i t rises  As noted  above, t h e i r + t o ir~  but i n c r e a s e w i t h a n g l e a t the  this ratio  t h a t low energy  - ,  50 MeV  the n^p d i f f e r e n t i a l c r o s s  for several incident  to 800 a t 1 8 0 ° . The average  p r o t o n s , and i r s w i t h neutrons, clear  are s i m i l a r i n  i n s i g n , then t h e r e i s a p o s s i b i l i t y o f  r a t i o i s c o n s t a n t a t t h e h i g h e r energy, lower  Thus, i t i s e v i d e n t from the  to ir~ t o t a l c r o s s s e c t i o n s , and d i f f e r e n t i a l  This i s i n fact  i n c i d e n t p i o n energy. sections  than the 1/2.  p i o n e n e r g i e s , t h i s w i l l no l o n g e r be the c a s e .  However, i t can be seen magnitude, but d i f f e r  +  because o f A p r o d u c t i o n , the 3/2  pion  v a l u e i s 20.  n+'s i n t e r a c t more s t r o n g l y w i t h  than v i c e - v e r s a .  Of c o u r s e i t i s not  t h a t t h i s s e n s i t i v i t y w i l l a l s o be p r e s e n t i n the case o f  pion-nucleus  scattering.  D i f f e r e n t combinations  give r i s e  to d i f f e r e n t  cancellation effects.  important  out t o f u r t h e r a n g l e s .  of p a r t i a l waves c o u l d  Coulomb e f f e c t s w i l l  But the p i o n - n u c l e o n  sensitivity  provides a strong motivation f o r i n v e s t i g a t i n g nuclear  scattering.  In f a c t , system. section  this  Evidence 4.2.4.  isovector sensitivity  be more  Is present i n the pion-nucleus  i n f a v o r of such a c o n c l u s i o n w i l l be p r e s e n t e d i n  -  Fig.  1.5  13  -  R a t i o of T t p / i r ~ p D i f f e r e n t i a l at S e v e r a l Energies +  Cross  Sections  - 14  1.4  Experimental  -  Considerations  In the p a s t , p i o n s c a t t e r i n g experiments have been performed a) N a l d e t e c t o r s , b) Ge Each of these p r e s e n t s  d e t e c t o r s , and i t s own  c) p l a s t i c  scintillator  i n t e r a c t i o n s w i t h the e l e c t r o n s present as a r e s u l t  converted  telescopes.  problems.  a) I n a Nal d e t e c t o r , a s t o p p i n g p i o n l o s e s energy  emitted  with:  through  i n the d e t e c t o r c r y s t a l .  Light  of the d e e x c i t a t i o n of the e l e c t r o n s i s c o l l e c t e d  I n t o an e l e c t r o n i c s i g n a l .  The  and  s i z e of the s i g n a l i s  p r o p o r t i o n a l to the amount of l i g h t , which i n t u r n i s p r o p o r t i o n a l to the amount of energy d e p o s i t e d by the p i o n . process  i s such  MeV.  all  but a few  the nature of the whole  t h a t the best energy r e s o l u t i o n o b t a i n a b l e w i t h  detectors, for incident 1.5  But  pions w i t h 50 MeV  kinetic  energy, i s  T h i s i s too l a r g e to s e p a r a t e the ground and n u c l e i , and  a s e r i e s of i n e l a s t i c  thus must be  scattering  and  i n t e r a c t d i r e c t l y w i t h the n u c l e i p r e s e n t  c a u s i n g the n u c l e i to d i s i n t e g r a t e .  nucleus.  to measure the k i n e t i c  T h i s i s due  i n atomic  i n the d e t e c t o r  the nucleus  had  These  pion  broken  would a l s o have to ensure t h a t a l l the fragments  incompatible.  orbits,  disintegrating  f o r the v a r i o u s b i n d i n g energy c o r r e c t i o n s .  are m u t u a l l y  to  crystal,  energy of the incoming  i n the d e t e c t o r , i n o r d e r to measure the t o t a l energy. requirements  S.  p i o n r e s t mass energy i s  would have to know e x a c t l y how  to account  other hand, one  The  - ,  the k i n e t i c energy of the fragments of the  In o r d e r  a c c u r a t e l y , one i n order  TT  +  into  excited states i n  experiments.  t h a t , u n l i k e i r ' s , s t o p p i n g TT~'S get c a p t u r e d  transformed  approximately  improved upon i n o r d e r to undertake  Furthermore, Nal d e t e c t o r s cannot be used f o r the f a c t  Nal  two  On  up,  the  stopped  -  15 -  b) In a Ge d e t e c t o r , the e l e c t r o n s w i t h which the s t o p p i n g interact  are gathered  detector crystal,  by means of an e l e c t r i c  and produce a c u r r e n t p u l s e  field  present  directly.  pions  w i t h i n the  Thus, o b t a i n a b l e  energy r e s o l u t i o n s a r e much lower than w i t h N a l ' s ; a p p r o x i m a t e l y f o r 50 MeV pions inelastic  being  typical.  T h i s i s c e r t a i n l y adequate f o r many  s c a t t e r i n g experiments.  used f o r T T  - ,  S either,  U n f o r t u n a t e l y , Ge d e t e c t o r s cannot be  f o r the same reasons  c) Of the d e t e c t o r s mentioned thus scintillator  as o u t l i n e d above.  f a r , o n l y the p l a s t i c  t e l e s c o p e can be used to d e t e c t i r s , as w e l l as - ,  a t e l e s c o p e c o n s i s t s o f a s e r i e s of t h i n p l a s t i c which o p e r a t e s  energy r e s o l u t i o n .  scintillators,  i n t o l i g h t , but w i t h much poorer  P i o n s a r e i d e n t i f i e d , and t h e i r e n e r g i e s  however, not by summing the h e i g h t s o f the s c i n t i l l a t o r  scintillator nature with  TT  + ,  Such  S.  each of  i n much t h e same way as the N a l d e t e c t o r , c o n v e r t i n g energy  d e p o s i t e d by p a s s i n g pions  measuring t h e i r  300 keV  range i n the t e l e s c o p e .  of the s e r i e s  they s t o p .  o f the s t o p p i n g process  intrinsic obtained,  s i g n a l s , but by  That i s , by n o t i n g i n which Unfortunately,  the s t a t i s t i c a l  means that the best r e s o l u t i o n o b t a i n a b l e  such d e t e c t o r s i s o f the order of 3 MeV.  This i s unacceptably  high  f o r most a p p l i c a t i o n s . The  d e t e c t o r chosen f o r the experiments d i s c u s s e d i n t h i s  the magnetic s p e c t r o m e t e r .  thesis i s  Such d e v i c e s have been used i n n u c l e a r  p h y s i c s , i n many a p p l i c a t i o n s , f o r many y e a r s , and a r e c a p a b l e o f providing  e x c e l l e n t energy r e s o l u t i o n s .  based on the f a c t  In p r i n c i p l e ,  their operation i s  t h a t charged p a r t i c l e s , moving through a magnetic  field,  16  -  -  e x p e r i e n c e a f o r c e p r o p o r t i o n a l to t h e i r momenta.  In the  simplest  a p a r t i c l e moving i n vacuum, w i t h i t s v e l o c i t y p e r p e n d i c u l a r d i r e c t i o n of a c o n s t a n t , a c i r c l e , whose r a d i u s  uniform d i p o l e f i e l d ,  i s r = p/qB.  o f the p a r t i c l e ' s p a t h , o b t a i n e d after  energy.  described  The  operation  i n s e c t i o n 2.2.  One  by d e t e c t i n g  both ir -  and  thus can  will  infinite  be  not  r e q u i r e the p a r t i c l e  be used w i t h e q u a l e f f i c i e n c y  for  to  detecting  fact  at the  which produce f i e l d s p a r t i c u l a r , one  can  one  must account  e n t r a n c e s and (Ste 65),  (Ban  66),  produced e x p e r i m e n t a l l y f o r the non-uniform  and  from good energy r e s o l u t i o n , t h e r e  r e q u i r e m e n t s f o r the d e t e c t i o n experiment.  fact  that  Fortunately,  c o n s t r u c t i n g magnets  quadrupole magnets which have the lenses  One  do  for  In  e f f e c t of  light.  are s e v e r a l  used f o r a low  other energy  pion  of these i s a l a r g e a c c e p t a n c e , to compensate  incoming pion  c a s e , t h i s r e q u i r e m e n t was  system to be  are  field  s u i t e d f o r s p e c i f i c a p p l i c a t i o n s are w e l l known. construct  and  complication  e x i t s of r e a l magnets.  beams of charged p a r t i c l e s , as  Apart  P a r t of the  t h a t the magnetic f i e l d s and  in  data i s q u i t e complicated,  f u r t h e r i n s e c t i o n 2.2.3.  techniques f o r designing  f o r the  not  thus i t s  will  notes, however, that s i n c e they do  of the spectrometer i s s t r a i g h t f o r w a r d  i n extent,  shapes o c c u r i n g  scattering  i t s momentum, and  of p o s i t i o n s e n s i t i v e d e t e c t o r s  i n p r a c t i c e the a n a l y s i s of the  be d i s c u s s e d  focussing  and  +  a r i s e s from the not  curvature  TT .  A l t h o u g h the use principle,  trajectory is  i t s p o s i t i o n before  gives  measure the p a r t i c l e ' s t o t a l energy, they do come to r e s t , and  the p a r t i c l e  the  Thus, a measurement of the  i t s passage through a d i p o l e f i e l d ,  kinetic  to  case of  met  f l u x e s are  relatively  by i n s t a l l i n g  low.  In the  present  a p a i r of quadrupole magnets  - 17 -  b e f o r e the bending  magnet.  These s e r v e t o f o c u s the beam, m a i n l y i n the  non-bend p l a n e , where the spectrometer d i p o l e gap l i m i t s the a c c e p t e n c e , and  thus i n c r e a s e the s o l i d a n g l e i n t o which p i o n s can s c a t t e r and s t i l l  be o b s e r v e d .  The name of the spectrometer, the QQD, i s taken from the  magnet c o n f i g u r a t i o n :  quadrupole-quadrupole-dipole.  - 18  -  CHAPTER I I EXPERIMENTAL DETAILS  The apparatus  purpose of the present chapter Is to p r o v i d e d e t a i l s of and  t e c h n i q u e s used  a r e the s u b j e c t of t h i s  t o perform  thesis.  ments i s s t r a i g h t f o r w a r d : one  the s c a t t e r i n g  experiments  In p r i n c i p l e , c a r r y i n g out  simply counts  the which  these e x p e r i -  the number of p i o n s which  s c a t t e r from a g i v e n n u c l e a r t a r g e t i n a g i v e n d i r e c t i o n .  There a r e ,  however, many p r a c t i c a l c o n s i d e r a t i o n s which must be taken  into  The  first  created  of these i s the source of i n c i d e n t p i o n s .  i n the i n t e r a c t i o n of 500 MeV  some p r o d u c t i o n t a r g e t . c h a n n e l or beamline, collected  and  experiments  d e t e c t e d , and  d i r e c t i o n can  target.  w i t h p i o n s from TRIUMF's M13  The  utilized  present  c h a n n e l , which i s  S e c t i o n 2.2  c i r c u i t s which p r o c e s s e d  deals  magnets, and  the s i g n a l s  i n conjunc-  the c o n f i g u r a t i o n of e l e c t r o n i c  from  these d e t e c t o r s .  There f o l l o w s  a d i s c u s s i o n of how  t h i s i n f o r m a t i o n i s used  p i o n s ' momenta, and  the v a r i o u s l i m i t a t i o n s of the whole system.  to c a l c u l a t e the s c a t t e r e d  c o n t a i n s a d e s c r i p t i o n of the p r e p a r a t i o n and  of the s c a t t e r i n g  be  f o r t h i s purpose i n the p r e s e n t e x p e r i -  I t s t a r t s w i t h a c o n s i d e r a t i o n of the d e t e c t o r s used  t i o n w i t h the s p e c t r o m e t e r  sition  be  t a r g e t n u c l e i , s c a t t e r e d p i o n s must  t h e i r e n e r g i e s or momenta determined.  S e c t i o n 2.3  a  2.1.  i n t e r a c t i o n with  w i t h the s p e c t r o m e t e r ments.  protons w i t h the n u c l e i c o n t a i n e d i n  toward a n u c l e a r s c a t t e r i n g  the s u b j e c t of s e c t i o n  are  Through the use of a s e r i e s of magnets, c a l l e d  were performed  After their  At TRIUMF, pions  the p i o n s emitted i n a p a r t i c u l a r  directed  account.  targets.  compo-  - 19 -  2.1  M13 Beamllne The M13 beamline,  Fig.  2.1.  Protons  a l o n g w i t h the QQD  spectrometer,  i s illustrated in  a c c e l e r a t e d to 500 MeV w i t h the TRIUMF c y c l o t r o n  pass  through  a p r o d u c t i o n t a r g e t , T l , where some of them i n t e r a c t and produce  pions.  Those IT'S e m i t t e d at 135° w i t h r e s p e c t to the incoming  are f o c u s s e d by the Q l and Q2 quadrupole  magnets, d e f l e c t e d  d i p o l e magnet, and come t o a d i s p e r s e d focus a t F l . Q3, Q4, and Q5, s e r v e s to produce another  p r o t o n beam  by the B l  A quadrupole  triplet,  d i s p e r s e d f o c u s at F2.  Finally,  the p i o n t r a j e c t o r i e s a r e bent by B2, and f o c u s s e d by Q6 and Q7, to form an a c h r o m a t i c  beam spot at the p i v o t p o i n t o f the s p e c t r o m e t e r ,  nuclear scattering  targets are placed.  P i o n s a r e not the o n l y p a r t i c l e s t o come down the M13 Other  charged  particles  of the same momentum coming from  t a r g e t w i l l a l s o be p r e s e n t . e's w i t h both i r + and ir~. Tl,  while  The u's come from  f o r by measuring the p a r t i c l e s '  beamline,  and y's and  the decay o f charged  time of f l i g h t  TT'S near  These can a l l be accounted from T l , which i s  w i t h the use of a c a p a c i t i v e probe i n the main p r o t o n  just before T l .  T h i s probe o u t p u t s a s i g n a l c o i n c i d e n t with the  passage o f a b u r s t o f p r o t o n s , which i n t u r n occurs corresponding  beamline.  the T l p r o d u c t i o n  These i n c l u d e protons w i t h i r + ,  the e's come from the decay o f ir^'s.  accomplished  where the  every 43.5 n s ,  t o the RF frequency of the TRIUMF c y c l o t r o n .  w i l l be addressed  This  topic  a g a i n i n s e c t i o n 3.1.1, which d e a l s with t h e  n o r m a l i z a t i o n o f the i n c i d e n t p i o n For many experiments,  flux.  the beamline may be thought  of s i m p l y as a  source o f p i o n s , w i t h no c o n s i d e r a t i o n g i v e n t o the q u a l i t y o f the beam. T h i s i s not so i n the p r e s e n t c a s e .  There a r e t h r e e beam p r o p e r t i e s which  F i g . 2.1  M13  Beamline  and QQD  Spectrometer  - 21 -  have a d i r e c t  i n f l u e n c e on QQD experiments:  a) The o v e r a l l  energy  r e s o l u t i o n which may be a c h i e v e d w i t h the s p e c t r o m e t e r depends on t h e energy  spread of the incoming beam, which thus must be s m a l l , b) Because  inelastic  scattering  c r o s s s e c t i o n s are s m a l l ( u s u a l l y < 1. mb/sr),  f l u x e s must be h i g h enough t o ensure that experiments  pion  can be completed i n  r e a s o n a b l e p e r i o d s o f time, c) Due to the l i m i t e d q u a n t i t i e s o f s e p a r a t e d i s o t o p e t a r g e t m a t e r i a l a v a i l a b l e , and the n o n - u n i f o r m i t y of the s p e c t r o m e t e r a c c e p t e n c e , the f i n a l beam spot s h o u l d be as s m a l l a s possible. cannot  be o p t i m i z e d The  For  These t h r e e c o n s i d e r a t i o n s a r e i n f a c t  c o u p l e d , and i n g e n e r a l  Independently.  t o t a l momentum acceptence o f t h e M13 channel i s Ap/p = 6.7%.  50 MeV p i o n s , 1% Ap/p i s e q u i v a l e n t t o 870 keV spread i n energy, and  so the momentum a c c e p t e n c e o f the channel must somehow be r e d u c e d . is  a c c o m p l i s h e d by means of m e c h a n i c a l  f o c i F l and F2. Unfortunately,  slits,  p o s i t i o n e d a t the d i s p e r s e d  The d i s p e r s i o n of the beam a t these p o i n t s i s 1.25 cm/%.  the t i l t  of the f o c a l planes i s such (81°) t h a t pions w i t h  a momentum spread g r e a t e r than 1% w i l l pass opening.  through a 1.25 cm  Two s e x t u p o l e magnets, SX1 and SX2, were i n s t a l l e d  c h a n n e l i n o r d e r t o s t r a i g h t e n the f o c a l p l a n e a t F2. as y e t proven u s e f u l because present  slit i n the  But these have not  of other not y e t f u l l y u n d e r s t o o d  effects  i n the channel.  One s h o u l d note a l s o t h a t t h e r e i s a l i m i t can be c l o s e d , and s t i l l  to how t i g h t l y  In e f f e c t ,  the s l i t s  reduce the momentum spread o f the p i o n beam.  There i s a m a g n i f i c a t i o n f a c t o r of = 1. from the T l t a r g e t focus.  This  to the F2  t h i s means that even a monochromatic beam o f p i o n s ,  e m i t t e d from T l , would n o t come to a p o i n t f o c u s , but would be spread out  -  over a space e q u i v a l e n t would  like  22  -  t o the s i z e of the p r o d u c t i o n  t o use as t h i n a p r o d u c t i o n  reduce the pion  target  target.  as p o s s i b l e .  f l u x , as does c l o s i n g down the s l i t s .  to use a t a r g e t w i t h as h i g h a Z as p o s s i b l e ,  to i n c r e a s e  pion  also  like  production.  have on t h e q u a l i t y  the main p r o t o n beam downstream o f T l . For  the experiments d e s c r i b e d  c o n f i g u r a t i o n was adopted: F2  But t h i s would  One would  But t h i s cannot be done because o f the e f f e c t i t would of  Thus, one  the T l t a r g e t was 10 mm g r a p h i t e ;  s l i t s were set- to 0.5 % Ap/p.  i r / s , and = 3 x l 0 +  ir~/s.  5  i n t h i s t h e s i s , the f o l l o w i n g  The r e s u l t i n g p i o n  f l u x e s were: =  2xl0  6  The energy spread o f the incoming beam, as  measured w i t h a Ge d e t e c t o r , was - 750 keV. scattering  the F l and  The beam spot a t the n u c l e a r  t a r g e t was » 20 mm FWHM i n both h o r i z o n t a l and v e r t i c a l  directions. H a l l probes were i n s t a l l e d nuclear  i n a l l M13 quadrupole magnets, and  magnetic resonance (NMR) probes i n the d i p o l e s  B l and B2, i n o r d e r  t o m o n i t o r magnet s t a b i l i t y , and ensure r e p r o d u c i b i l i t y of the magnet settings. magnetic  The energy of the p i o n beam was determined by s c a l i n g to the B l field.  T h i s had been c a l i b r a t e d w i t h an a - s o u r c e , a t the time  M13 was o r i g i n a l l y changes better  commissioned  i n the c h a n n e l mean t h a t than ± 2 . % .  Several  (0ra+ 8 1 ) .  Unfortunately,  t h i s c a l i b r a t i o n cannot be t r u s t e d t o  methods have been employed  improve on t h i s u n c e r t a i n t y ,  but as y e t none has been  I t may be o f i n t e r e s t to note t h a t improvements been made on an ongoing b a s i s consideration been i n s t a l l e d  since  have been performed. i n place  subsequent  i n an attempt to successful. to the system have  the experiments p r e s e n t l y A 3 mm  synthetic  o f the 10 mm g r a p h i t e  target  under  diamond t a r g e t has at T l .  Because of  - 23 -  the  h i g h e r d e n s i t y o f t h e diamond, p i o n f l u x e s w i l l n o t d e c r e a s e , but the  intrinsic  r e s o l u t i o n of the channel w i l l  d e t e c t o r s have been i n s t a l l e d  improve.  Position  i n p l a c e of the m e c h a n i c a l s l i t s  F2. Thus, the e n t i r e p i o n f l u x can be used  a t F l and  f o r experiments, s i n c e the  r e l a t i v e momenta o f the incoming p i o n s can be determined compensated f o r .  sensitive  and so  The s i z e of the beam spot a t the e x p e r i m e n t a l t a r g e t has  been reduced c o n s i d e r a b l y , by o p t i m i z i n g the s e t t i n g s o f t h e Q3-Q5 magnets.  F u r t h e r advances  will  c o n t i n u e to be made.  - 24 -  2.2  QQD  Spectrometer  The QQD beamline  Spectrometer  i n F i g . 2.1.  magnets, QT1  and  QT2,  i s shown i n p l a c e a t the end of the  The  spectrometer c o n s i s t s of two  and one d i p o l e magnet BT.  f o c u s s i n g , w h i l e QT2  f o c u s s e s i n the v e r t i c a l  to bend pions by 70°  to the l e f t ,  direction beam.  The  p r e s e n t l e n g t h i s 2.38  s p e c t r o m e t e r may direction,  be used  BT  w i t h the  spectrometer  The  M13  t a r g e t to the  f o c a l p l a n e , however,  a t an a n g l e o f 7 2 ° .  system was  the  as s h o r t as  from the s c a t t e r i n g  tilted  The  of p i o n s p a s s i n g through  The  p i o n beam  designed t o have an  and a momentum acceptence o f ±20%.  e x p e r i m e n t a l l y determined  i s designed  from 0° w i t h r e s p e c t t o the incoming  t o a p p r o x i m a t e l y 137°.  a c c e p t e n c e of 18 msr,  m,  The  beyond the l a s t wire chamber, and  The  v a l u e s w i l l be d i s c u s s e d i n s e c t i o n 3.1.5.  Detectors An  used  direction.  d e s i r a b l e to make the system  c e n t e r of the l a s t w i r e chamber.  2.2.1  QTl i s h o r i z o n t a l l y  i n the h o r i z o n t a l d i r e c t i o n .  In o r d e r t o keep l o s s e s from the decay  possible.  is  quadrupole  i s such as to a l l o w e v e n t u a l d i s p e r s i o n matching  s p e c t r o m e t e r s m a l l , i t was  M13  overview of the QQD  i n data a c q u i s i t i o n ,  spectrometer, i l l u s t r a t i n g  i s g i v e n i n F i g . 2.2.  u2, E l , E2, and E3 a r e p l a s t i c phototubes. which i s 12.7  A l l are 6.4 mm.  mm  scintillators  The d e t e c t o r s B l , B2, u l ,  (NE110), f i t t e d w i t h RC8575R  t h i c k , except B l , which i s 0.8  B l , B2, y l , and u2 were used  to monitor  p i o n f l u x , which i s d i s c u s s e d f u r t h e r i n s e c t i o n 3.1.1. signals  from E l , E2,  through  the s p e c t r o m e t e r .  a l l the d e t e c t o r s  mm,  and  E3,  the  incident  Coincident  and E3 were used to i d e n t i f y p a r t i c l e s p a s s i n g Three d e t e c t o r s were used  i n o r d e r to reduce  F i g . 2.2  QQD  Spectrometer with A s s o c i a t e d Det e c t o r s  -  the i n c i d e n c e of random and problem i n the o f f l i n e but  26  background e v e n t s .  The  d e s i r a b l e , i n order  on which a l l events were  d e t e c t o r s WC1,  counters.  Such events were not  a n a l y s i s , where they c o u l d be i d e n t i f i e d  t h e i r on l i n e e l i m i n a t i o n was  magnetic tapes  -  WC3,  WC4,  to save space on  are p o s i t i o n  the  sensitive  to a t TRIUMF as Wire Chambers, but  a r e more p r o p e r l y c a l l e d M u l t i Wire P r o p o r t i o n a l Counters responsibility  easily,  recorded.  and WC5  They are commonly r e f e r r e d  a  (MWPCs).  The  f o r the o p e r a t i o n and maintenence of these d e t e c t o r s  the major t e c h n i c a l c o n t r i b u t i o n of the a u t h o r  to the  was  present  experiments. The  chambers themselves were c o n s t r u c t e d a t the Workshop of  U n i v e r s i t y of C a r l e t o n ( B i r t 71), based on a d e s i g n adopted from U n i v e r s i t y of A l b e r t a ( G i l 84). Table  2.1.  wires.  The  of three p l a n e s of e q u a l l y spaced,  middle  (anode) i s kept  wire  t o the anode w i r e s  plane planes  ( x - d i r e c t i o n ) , a r e grounded.  69.7%  gas m i x t u r e ,  argon,  through  30%  nearest  The  A charged  liberated  i o n s , i n moving away from the anode w i r e ,  pulse s p l i t s ,  A l l the wires type d e l a y l i n e and  parallel  with a  c o n s i s t i n g of  particle  The  passing  resulting  Induce a p u l s e on  from each cathode plane (Bos+ 75).  specially  electrons accelerate  anode w i r e , c r e a t i n g an a v a l a n c h e .  cathode w i r e s .  i t s wires  to as magic gas,  freon.  the chamber I o n i z e s the gas.  onto a p r i n t e d c i r c u i t induced  of which has  chambers are f i l l e d  0.3%  thin  the o t h e r p e r p e n d i c u l a r  commonly r e f e r r e d  i s o b u t a n e , and  toward the n e a r e s t positive  The  and  parallel  at a p o s i t i v e h i g h v o l t a g e , w h i l e  ( c a t h o d e s ) , one  (y-direction),  the  exact s p e c i f i c a t i o n s a r e g i v e n i n  They c o n s i s t  the o t h e r two  prepared  The  the  After  the  are s o l d e r e d  r e a c h i n g i t , the  t r a v e l s to both ends of the d e l a y l i n e .  The  -  27  -  WC1 and WC3  WC4 and WC5  Anode Wire Thickness  20.3 um  20.3 um  Cathode Wire Thickness  63.5 um  63.5 um  Wire P l a n e Separation  4.76 mm  6.35 mm  Anode Wire Separation  1.0 mm  1.0 mm  Cathode Wire Separation  2.0 mm  2.0 mm  Number of Anode Wires ( y - d i r e c t i o n )  169  308  Number o f Cathode Wires ( x - d i r e c t i o n )  169  3 x 203  Operating Voltage ( w i t h Magic Gas)  4.3 kV  5.4 kV  T a b l e 2.1  S p e c i f i c a t i o n s f o r the QQD M u l t i Wire P r o p o r t i o n a l Counters  - 28  time d i f f e r e n c e locate  the  between the  p o s i t i o n of  I t i s to be particular,  the  noted  fairly  -  a r r i v a l of the initial  that  the e l e c t r o n a v a l a n c h e o c c u r s at  l o c a l i z e d p o s i t i o n around one  w i l l have a  c o r r e s p o n d s to  the  'picket-fence'  Some e f f o r t was the  put  Involved  l o c a t i o n s on  the  the  i n t o determining  the  into positions  soldering  of  introduction  altered  c h a r a c t e r i s t i c transmission  method f i n a l l y adopted i n v o l v e d  'picket-fence*  known d i s t a n c e s spectra  p o s i t i o n measurements.  positions  the  chamber edges.  each end  amplifier.  =100,  suit  the  sent.  of the  properties  the  This  fiducial  of the  a c o m b i n a t i o n of the  pion  of each d e l a y l i n e , The  o r i g i n a l l y developed a t Los better  method  beam, and  Alamos (Stu  of  present a p p l i c a t i o n .  p r o d u c i n g chamber p u l s e s >300 mV  The  but  fully  through a design  m o d i f i e d somewhat to  i n most c a s e s . rise  electron  into relative  a m p l i f i e r g a i n was  c a p a b l e of p r o d u c i n g output p u l s e s with 5 ns  lines.  y-direction  b u i l t based on a  74),  not  l o c a t i n g the known  s i g n a l s passed  a m p l i f i e r was  was  delay use  passage  wires  A b s o l u t e p o s i t i o n s were determined by  chambers w i t h the  differential  One  i n o r d e r to c o n v e r t time d i f f e r e n c e s  the  At  be  between peaks i n the  illuminating of  i n mm.  d e l a y l i n e s , a l o n g which s i g n a l s s i m u l a t i n g  however, because the  the  peak  ' f i d u c i a l ' w i r e s onto known  successful,  s o u r c e s and  or  best method f o r c a l i b r a t i n g  chambers c o u l d  The  y-direction  Each p i c k e t  of charged p a r t i c l e s through the  the  Thus, w h i l e  anode w i r e .  c o n v e r s i o n o f time d i f f e r e n c e s  investigated  to  a  anode w i r e .  c o n t i n u o u s , the  structure.  l o c a t i o n of an  i s used  charged p a r t i c l e ' s passage.  the x - d i r e c t i o n p o s i t i o n s p e c t r a w i l l be spectra  p u l s e at e i t h e r end  The  chosen to  amplifier  t i m e s , but  the  is  actual  be  -  29  s i g n a l s from the chambers had =15 original limits  -  ns r i s e times due  induced p u l s e s i n the d e l a y l i n e s .  It i s this  the l e n g t h of the d e l a y l i n e s which may  x-direction,  the WC4  and WC5  wide, were connected  t o d i s p e r s i o n of the factor  be used.  which  In the  cathode p l a n e s , which were a t o t a l of 600  to t h r e e s e p a r a t e d e l a y  lines.  The o p e r a t i n g v o l t a g e s f o r the chambers were determined by t h e i r e f f i c i e n c i e s a t v a r i o u s v o l t a g e s w i t h the use of a s o u r c e , and  two  plastic  scintillators.  chambers operated w i t h >96%  mm  1 0 6  Ru  measuring  electron  Under e x p e r i m e n t a l c o n d i t i o n s , a l l  efficiency.  A more d e t a i l e d d i s c u s s i o n of the  o v e r a l l s p e c t r o m e t e r e f f i c i e n c y w i l l be p r e s e n t e d i n s e c t i o n 3.1.3. It at  s h o u l d be n o t e d t h a t space charge e f f e c t s produce  s p e c t r o m e t e r a n g l e s <50°, where background  flux  through WC1 The  0.6  mm  t o >10  intrinsic  1+  in may  events r a i s e  WC1 the  second.  p o s i t i o n r e s o l u t i o n o f WC1  i n both the x and y d i r e c t i o n s .  were a p p r o x i m a t e l y 1.5 due  per  scattering  sparking i n  and 2.5 mm  For WC4  i n the two  and WC3  was  and WC5,  directions.  approximately  the r e s o l u t i o n s This  to s e v e r a l f a c t o r s , the major c o n t r i b u t i o n coming from the the l o n g e r l e n g t h s o f the d e l a y l i n e s used.  The  intrinsic  be e s t i m a t e d i n s e v e r a l ways: a) w i t h the use of h i g h l y  e l e c t r o n s o u r c e s ; b) from the widths of the peaks i n the  increase i s dispersion resolution  collimated  y-direction  ' p i c k e t - f e n c e ' s p e c t r a ; and c) from the w i d t h of the peak i n a spectrum the  sum  o f the times from each end o f a d e l a y l i n e .  peak must be c o r r e c t e d anode w i r e .  T h i s may  the  charged  original  f o r the i n i t i a l  drift  last  time of e l e c t r o n s t o the  be as long as 25 ns, depending p a r t i c l e ' s passage.  Note that t h i s  of  on the l o c a t i o n of  I t can be measured by o b s e r v i n g  i n d u c e d p u l s e s on the anode w i r e s d i r e c t l y ,  through a c a p a c i t i v e  filter.  - 30 -  2.2.2  E l e c t r o n i c s and Data A PDP  used  11/34  Acquisition  computer, r u n n i n g under the RSX  o p e r a t i n g system,  t o r e c o r d event by event spectrometer d a t a onto magnetic  later  detailed off line analysis.  d a t a a c q u i s i t i o n program DA from a predetermined  T h i s was  was  tape f o r  done u s i n g the s t a n d a r d TRIUMF  ( M i l 84), which responds  to an i n t e r r u p t  CAMAC c r a t e module, i n the p r e s e n t case a  (LAM)  C212  b i t - p a t t e r n u n i t , by r e a d i n g the i n f o r m a t i o n s t o r e d i n the o t h e r modules in  the c r a t e , TDCs, ADCs, and  a n a l y s i s was (Fer  a l s o performed  scalers.  Some immediate on l i n e  data  u s i n g a m o d i f i e d v e r s i o n of the program MULTI  79). A schematic diagram  experiments generated;  of the e l e c t r o n i c modules and  i s g i v e n i n F i g . 2.3.  Two  distinct  those i d e n t i f i e d as spectrometer  l o g i c used  i n the  types of LAMs c o u l d be  e v e n t s , o r those i d e n t i f i e d  as  beam sample e v e n t s . A s p e c t r o m e t e r event c o n s i s t e d o f the c o i n c i d e n c e E1*E2*E3*B1. that  the p u l s e s from these d e t e c t o r s were s e t In such a way  as to  always  take the a b s o l u t e t i m i n g from the l e a d i n g edge of the E l mean time. was  fluxes.  from  This  done so as to a l l o w f o r the e l i m i n a t i o n of B l from the c o i n c i d e n c e ,  s h o u l d i t prove  m),  Note  not to be f u n c t i o n i n g e f f i c i e n t l y a t h i g h i n c i d e n t p i o n  T h i s d i d not o c c u r .  Note a l s o t h a t because  E l , E2, and E3 were f i t t e d w i t h phototubes these phototubes  them independent  were passed  of t h e i r  a t both ends.  length (1. The  signals  through mean t i m e r s , i n o r d e r to make  of the l o c a t i o n o f a p a r t i c l e ' s passage  through  the  scintillator. The  s p e c t r o m e t e r event s i g n a l was  fanned  out t o p r o v i d e s t a r t s f o r  TDCs whose s t o p s came from the ends of the w i r e chamber d e l a y l i n e s .  This  - 31 -  E1R  2  Un  OSCRIMINATOR  D ED  MT  E1L  MEAN TIMER  E2R MT  BATE GENARATOR  00  E2L Un  E3R.  21  OR  BITO  E3R-  E3L-  >  E3L-  | \  2  B1  B2  AND  MT  M SPECTROMETER EVENT  MT lin  O—• rDlf  LAM  TDC starts ADC gdtes  BEAM SAMPLE EVENT  inhibit scalers  BIT1  '—I GG Is  T1 ION  T1  ^  CAMAC output rogistsr (comp busy)  C  T1 CAP PROBE  O  CAM AC ADC  O  CAMAC TDC STOP  •  VISUAL AND CAMAC SCALERS C A M A C BIT PATTERN UNIT  MWPC  SIGNALS  Fig.  2.3  Configuration  of E l e c t r o n i c s  - 32  TDC i n f o r m a t i o n was u s e d p a r t i c l e which  -  to determine  had p a s s e d  through  the position,  i n t h e MWPCs, o f t h e  the spectrometer.  N o t e t h a t t h e h e i g h t s o f t h e i n d i v i d u a l E l , E2,. a n d E3 s i g n a l s also recorded  i n ADCs.  of  loss  t h e energy  which  These s i g n a l s  o f the p a r t i c l e s  c o u l d be u s e d  to identify  were  c o u l d be summed t o p r o v i d e a m e a s u r e  passing through  pions which  the s c i n t i l l a t o r s ,  had decayed  t o muons i n t h e  spectrometer. Note a l s o that the i n d i v i d u a l provided a direct in  B l s i g n a l was r e c o r d e d  measure o f the f r a c t i o n o f the time  i n a TDC.  p i o n s were  c o n s e c u t i v e p r o t o n beam b u r s t s , a n d t h u s , t h e p r o b a b i l i t y  pions  i n t h e same beam b u r s t , f o r w h i c h  the incident  This  produced  o f h a v i n g two  f l u x m u s t be  corrected. The flight  p u r p o s e o f t h e beam s a m p l e c i r c u i t  of p a r t i c l e s  the i n c i d e n t  down t h e M13 c h a n n e l ,  beam c o u l d be d e t e r m i n e d .  c o i n c i d e n c e b e t w e e n B1«B2 a n d t h e o u t p u t was s e t t o =1. s , t h e r a t e a t w h i c h c o i n c i d e n c e would p r o v i d e a s t a r t  was t o m e a s u r e t h e t i m e o f  from which  the pion f r a c t i o n i n  A beam s a m p l e e v e n t  o f a g a t e g e n e r a t o r , whose w i d t h  t h e i n c i d e n t beam was s a m p l e d .  f o r a TDC, t h e s t o p c o m i n g f r o m  c a p a c i t i v e p r o b e i n t h e m a i n p r o t o n beam  t y p e was g e n e r a t e d ,  consisting  o f a L O G I C A L OR u n i t a n d g a t e g e n e r a t o r was u s e d  additional  events, u n t i l  t h e c u r r e n t one h a d b e e n p r o c e s s e d .  to inhibit  a l l scalers,  Such a a  line.  N o t e t h a t a s s o o n a s a LAM o f e i t h e r  s i g n a l was u s e d  consisted of a  thus  ensuring  a  to  circuit  inhibit  The same  t h a t no a d d i t i o n a l  d e a d - t i m e c o r r e c t i o n s n e e d t o be a p p l i e d t o t h e v a r i o u s beam  monitors.  - 33  2.2.3  -  Momentum D e t e r m i n a t i o n In the c o n s i d e r a t i o n o f the passage o f charged  system  of magnets, i t i s common p r a c t i c e  (see e.g.  s t a r t by d e f i n i n g a c e n t r a l t r a j e c t o r y through  particles  through  (Ban 6 6 ) , ( S t e 65)) to  the system.  The  c h a r a c t e r i s t i c s of the beam are then d e s c r i b e d by the d i s p l a c e m e n t s d i v e r g e n c e s of the outermost  rays away from the c e n t r a l one,  changes i n these c h a r a c t e r i s t i c s g i v e n by example, c o n s i d e r the two  related  and  'transfer matrices'.  and  the For  d i m e n s i o n a l case of a beam moving i n the  z - d i r e c t i o n from p o i n t 0 to p o i n t 1. p o i n t s by  a  The beam i s d e s c r i b e d at these  the column v e c t o r s ( x , d x / d z ) and 0  (x^dxj/dz).  Q  two  They are  by  where L i s the d i s t a n c e between z  Q  and  Z j .  Similarly,  f o r a beam p a s s i n g through a f o c u s s i n g quadrupole  where kq = ( B / a ) ( l / B p ) , L i s the e f f e c t i v e Q  magnet, B  Q  the c e n t r a l  at r a d i u s a, and  magnet i s  l e n g t h o f the  Q  i s the f i e l d  the t r a n s f e r m a t r i x  (Bp ) Q  quadrupole  i s the magnetic  rigidity  trajectory.  In a more g e n e r a l , t h r e e d i m e n s i o n a l c a s e , one must c o n s i d e r both horizontal  (x,dx/dz=9) and  divergenges.  vertical  (y,dy/dz=<J>) d i s p l a c e m e n t s  and  A l s o , the momentum spread of the beam, <5=Ap/p where c  p  c  of  - 34 -  is  the momentum o f the c e n t r a l r a y .  for  Thus, f o r the QQD spectrometer  example, the beam c h a r a c t e r i s t i c s a t t h e p o s i t i o n o f WC5 can be  related  t o those a t the s c a t t e r i n g t a r g e t :  / x \ / / 6 \ / y -I \ *5 / \ \ 6 / \  Rll R21 R31 R*l R61  5  5  5  5  R12 R22 R32 R^2 R62  R13 R23 R33 R43 R63  R14 R24 R34 R44 R64  R16 R26 R36 R46 R66  \ /x \ \ / eQ \ y. J\ t) J / \ 6„/ Q  (2.1)  Q  There a r e s e v e r a l computer programs a v a i l a b l e , most TRANSPORT in  system  (Bro+ 8 0 ) , which w i l l  notably  calculate transfer coefficients  (2.1)) f o r a system, g i v e n the magnet c o n f i g u r a t i o n s , f i e l d  strengths,  and  distances involved.  x ,  9Q , y , and <J> determined from p o s i t i o n s i n WC1 and WC3, (2.1) can be  0  Q  inverted  Once these  ( i . e . R's  t r a n s f e r c o e f f i c i e n t s a r e known, and  0  to f i n d  6Q, the momentum of the p i o n s c a t t e r e d from the n u c l e a r  target. Unfortunately, If  o n l y the f i r s t  i t i s r e w r i t t e n as  then  = f(rj(j)  the more g e n e r a l r  or,  (2.1) r e p r e s e n t s  ,  situation is  = f ( r j n ) + 8(rj0*k0)  i 5  o r d e r beam t r a n s f e r .  +  h  < j 0 r k 0 f c 0 ) + ••• > r  r  i n more c o n v e n t i o n a l n o t a t i o n , r  where  i  =  I [ ilrjl j j r  [rj_|rj],  r  I [r±|r-jr ]rjr jk  +  k  [ri|rjr ], k  k  +  £ [r jk£  ±  ' r ^ r ^ r jr r£, + ...  [ r i | r j r r £ ] , ... a r e f i r s t , k  order  transfer coefficients.  first  and second o r d e r c o e f f i c i e n t s , w h i l e  k  second, t h i r d , ...  The program TRANSPORT can o n l y c a l c u l a t e i n general, higher  order  c o e f f i c i e n t s may be e q u a l l y , or even more i m p o r t a n t .  In any case, one  cannot r e l y  s i n c e these a r e only  on c a l c u l a t e d v a l u e s of t h e c o e f f i c i e n t s ,  - 35 -  as good as the i n p u t d a t a , and cannot and  irregularities  system.  compensate f o r minor  misalignments  i n magnetic f i e l d s which a r e i n v a r i a b l y p r e s e n t i n any  Thus the t r a n s f e r c o e f f i c i e n t s must be determined  experimentally.  There have been two d i f f e r e n t approaches t o t h i s problem adopted group of people  i n v o l v e d w i t h the QQD e x p e r i m e n t s .  t h a t the two y i e l d One  similar  by the  A l l i n d i c a t i o n s are  results.  approach ((Sob 84a) and (Sob 8 4 b ) ) , which i s perhaps t h e more  s t r a i g h t f o r w a r d , i n v o l v e s measuring the v a r i a t i o n i n one c o o r d i n a t e w h i l e varying another,  directly.  But t h i s i n v o l v e s t a k i n g a l a r g e amount of  d a t a w i t h the s p e c t r o m e t e r . coefficients y , Q  <J>Q  [x|x] , [ x | x ] ,  , and 6  2  Q  F o r example, one c o u l d determine the [ x | x ] , e t c . by p l o t t i n g 3  a l l h e l d c o n s t a n t at z e r o , and f i t t i n g  with a high order polynomial. getting  sufficient  5  vs x , with 9 , Q  the r e s u l t i n g  curve  t h i s approach l i e s i n  Q  are i n f a c t  zero.  a l t e r n a t e approach ( ( G y l 84) and (Bar 8 4 ) ) , and t h a t adopted  the author  i s the f o l l o w i n g : Using the p i o n s c a t t e r i n g d a t a from a CH  t a r g e t , one i n i t i a l l y  c o n s t r u c t s an energy  o r momentum spectrum  TRANSPORT p r e d i c t i o n s f o r the t r a n s f e r c o e f f i c i e n t s .  v a l u e s of the c o e f f i c i e n t s , the widths  ground s t a t e ,  first  number of counts  to vary the  to any d e s i r e d o r d e r , t o s i m u l t a n e o u s l y  e x c i t e d s t a t e , and p r o t o n peaks.  1 2  C  A p r e s e l e c t i o n of  i n o r d e r to e l i m i n a t e background, and e q u a l i z e the  i n each peak, to ensure  Also, a p a r t i a l F test  2  u s i n g the  of the three peaks i n t h e spectrum, i . e . the  d a t a may be performed  by  One then runs a  computer program, which uses a m u l t i p l e r e g r e s s i o n t e c h n i q u e  minimize  Q  s t a t i s t i c a l accuracy while ensuring that a l l  c o o r d i n a t e s o t h e r than x The  The d i f f i c u l t y w i t h  x  proper  statistical  weighting.  i s a p p l i e d to each c o e f f i c i e n t , so t h a t those which  - 36  are s t a t i s t i c a l l y coefficients that  insignificant  found i n t h i s way  the c o e f f i c i e n t s  those of WCl I t was  and WC3,  are set to z e r o .  A t y p i c a l s e t of  are p r e s e n t e d i n T a b l e s 2.2  r e l a t e the c o o r d i n a t e s of WC4  indicating  sets of c o e f f i c i e n t s  c o e f f i c i e n t s does a f f e c t  2.2.4  2.3.  directly  target could  that they are c o r r e l a t e d .  c e r t a i n minimum peak w i d t h which cannot be improved  and  and WC5  r a t h e r than those of the i n i t i a l  found t h a t d i f f e r e n t  identical results,  -  Note to  position. produce  There i s a  upon.  The c h o i c e of  the shapes of the r e s u l t i n g peaks, however.  Resolution The narrowest  peak widths i n energy s p e c t r a accumulated  p r e s e n t experiments a r e =1.1  MeV  FWHM.  the  The c o n t r i b u t i o n o f the energy  spread of the i n c i d e n t p i o n beam from the M13 Assuming  during  channel i s =750 keV.  the c h a n n e l and spectrometer c o n t r i b u t i o n s combine i n  quadrature, t h i s =800 keV.  i m p l i e s t h a t the r e s o l u t i o n of the QQD  spectrometer  was  From the use of computer programs which s i m u l a t e the passage o f  p i o n s through the system, estimated  that  the major  i n c l u d i n g TRIUMF's REVMOC (KR 8 3 ) , i t can be c o n t r i b u t i o n to t h i s r e s o l u t i o n comes from the  e f f e c t of m u l t i p l e Coulomb s c a t t e r i n g o f the p i o n s i n the m a t e r i a l p r e s e n t i n the s p e c t r o m e t e r . At of 12.7  the time the experiments were performed, t h i s m a t e r i a l um  t h i c h mylar windows on WCl  windows on WC4 air  inside  and WC5,  and WC3,  50.8  um  thick  consisted  kapton  and the magic gas mixture i n the chambers.  the s p e c t r o m e t e r was  pumped out, and r e p l a c e d w i t h He  The  gas  before undertaking data c o l l e c t i o n . F u t u r e experiments w i l l be conducted w i t h the s p e c t r o m e t e r  totally  - 37 -  i  j  k  I  m  coefficient  0 0 0 1 0 0 0 0 0 1 1 0 0 0 1 0 0 0 0 0 1 1 2 0 0 0 0 1  0 0 0 0 0 0 1 1 2 0 1 0 0 0 0 0 0 0 1 2 0 1 0 0 0 0 1 0  0 0 1 0 0 2 0 1 0 0 0 0 0 1 0 0 1 2 0 0 1 0 0 0 0 1 0 0  0 1 0 0 2 0 1 0 0 1 0 0 1 0 0 2 1 0 1 0 0 0 0 0 1 0 0 0  0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2  4.906907 0.010630 1.203300 -0.348225 0.004083 -0.005827 -0.007299 -0.001407 0.002154 -0.002407 0.009717 -6.124691 -0.002581 0.043025 -0.011264 0.000119 -0.000003 -0.000019 -0.000189 0.000051 -0.000096 -0.000263 0.001364 0.025394 -0.000183 -0.000699 0.000169 0.000360  T a b l e 2.2  T y p i c a l Set of QQD T r a n s f e r C o e f f i c i e n t s f o r WC4 P o s i t i o n . Notation i s : = £ c o e f f i c i e n t ' X j i ' y j J *x ^» 3  J^^'Sg  111  -  38 -  i  j  k  I  m  coefficient  0 0 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 2 0 0 0 0 1  0 0 0 0 0 0 1 1 0 1 0 0 0 0 0 1 1 2 1 0 0 0 0 1 0  0 0 1 0 0 2 0 1 1 0 0 0 1 0 2 0 1 0 0 0 0 0 1 0 0  0 1 0 0 2 0 1 0 0 0 0 1 0 2 0 1 0 0 0 0 0 1 0 0 0  0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2  1.634761 -0.006957 0.564156 -0.290266 0.005915 -0.008948 -0.006424 -0.001811 0.001299 0.011112 -9.188643 0.000021 0.059224 0.000193 -0.000117 -0.000259 0.000087 0.000199 -0.000489 0.002178 0.020447 0.000119 -0.001601 0.000259 0.002622  T a b l e 2.3  T y p i c a l Set o f QQD T r a n s f e r C o e f f i c i e n t s f o r WC5 P o s i t i o n . Notation i s : Xg  = \ coefficient'Xji»yjJ'Xgk.  y3*"So  m  -  evacuated. drift  39 -  T h i s w i l l be accomplished  by r e p l a c i n g WCl and WC3 w i t h a  chamber, whose housing w i l l be a b l e t o w i t h s t a n d the p r e s s u r e  differential. Note t h a t  some e f f o r t was expended i n c o n d u c t i n g t e s t s o f the  p r e s e n t MWPCs w i t h reduced gas p r e s s u r e s . performed  consisting  0.2%  A l l i n o r d e r t o reduce  freon.  of 40% methane, 25% i s o b u t a n e , 34.8%  argon, and  the d e n s i t y and average atomic number  t h e gas, and t h e r e f o r e the m u l t i p l e s c a t t e r i n g .  found  were  ( G y l 84), i n which the s t a n d a r d magic gas m i x t u r e was r e p l a c e d  by a m i x t u r e  of  A l s o , experiments  Subsequently,  i t was  t h a t the MWPCs c o u l d be operated e f f i c i e n t l y w i t h a magic gas  m i x t u r e i n which a l l the argon was r e p l a c e d by h e l i u m . The 1984),  best combined s p e c t r o m e t e r - c h a n n e l r e s o l u t i o n seen t o date (May  under a c t u a l e x p e r i m e n t a l c o n d i t i o n s , i s 850 keV.  - 40 -  2.3  Target8 The  25.6 Max  mm,  2 6  M g t a r g e t was  and a u n i f o r m t h i c k n e s s of 0.300 g/cm .  i s o t o p i c a l l y pure. TRIUMF ( G y l 8 4 ) . unnecessary  The The  0,  2.8%  1 7  independent  0  and  1.9%  purchased  target holder.  i n the form of H 0  1 6  0.  T h i s has s u b s e q u e n t l y The  While  *0 8  heated,  still  t a r g e t was  25.4  quoted  been v e r i f i e d  injected into a  t h i c k n e s s than a l i q u i d .  at the top of the frame, which was  0.348 g/cm . 2  mm  t h i c k aluminum.  injected  through a s m a l l h o l e  l a t e r s e a l e d w i t h epoxy.  i n p l a c e , a d d i t i o n a l 12.7  um  0  i n the t a r g e t through  After  the  t h i c k aluminum windows  were g l u e d to the h o l d e r , i n order to prevent exchange of 1 8  The  windows were g l u e d , under t e n s i o n , over the area to be  o c c u p i e d by the t a r g e t m a t e r i a l , which was  atmosphere w i t h  by  Upon c o o l i n g , the s o l u t i o n forms a g e l , which i s  t a r g e t h o l d e r c o n s i s t e d of a frame of 6.35  t a r g e t m a t e r i a l was  95.3%  prepared as f o l l o w s : a  hot, the s o l u t i o n was  to m a i n t a i n a constant u n i f o r m  |im t h i c k mylar  to be  and mixed w i t h a s m a l l amount of  t h i c k n e s s of the g e l i n the t a r g e t h o l d e r was The  from Los Alamos  2  I t s i s o t o p i c c o m p o s i t i o n was  a n a l y s e s (Bar 84).  Agar (1.4% by w e i g h t ) .  more l i k e l y  Thus  or s u b t r a c t i o n s were needed.  t a r g e t was  0,  99.5%  s e l f - s u p p o r t i n g nature of the t a r g e t made i t  q u a n t i t y of t a r g e t m a t e r i a l was  prepared  It i s  the  same t a r g e t has been used on p r e v i o u s o c c a s i o n s at  ( B a t c h no. P4-H20-17). 1 8  H e i d e l b e r g , West Germany.  to mount i t i n a h o l d e r from which pions c o u l d s c a t t e r .  no background runs 1 8  f u r Kernphysik,  43 X  I t i s on l o a n from  2  Planck I n s t i t u t  The  a p i e c e of r o l l e d m e t a l , w i t h dimensions  1 6  0  the mylar windows.  i n the Background  runs were taken w i t h an empty t a r g e t h o l d e r , at a l l a n g l e s , i n order to s e p a r a t e the c o n t r i b u t i o n of the window m a t e r i a l from  t h a t of the  1 8  0  - 41 -  target  i n the accumulated The  1 2  C  spectra.  t a r g e t was i n the form of two s h e e t s of p o l y t h e n e  each o f which was 0.160 g/cm  thick.  2  s e v e r a l r e a s o n s : a) the spectrometer respect  2  2  Runs were taken w i t h t h i s t a r g e t f o r solid  a n g l e was n o r m a l i z e d w i t h  t o the Tf p c r o s s s e c t i o n s ; b) the CH +  2  t a r g e t s were l a r g e r than the  p i o n beam s p o t , w h i l e the other t a r g e t s were n o t . CH  (CH ),  Thus, by i n t e r s p e r s i n g  runs w i t h o t h e r t a r g e t r u n s , t h e s i z e o f t h e beam spot c o u l d be  measured, and accounted a t 4.4 MeV, determine  t h e ground  for.  c) Because the f i r s t  s t a t e was c l e a r l y  excited  2  1 2  C is  s e p a r a t e d , and c o u l d be used t o  the peak shape, which was then used i n f i t t i n g  w i t h o t h e r t a r g e t s , d) The CH  s t a t e of  data was used  t o determine  the d a t a QQD  taken  transfer  c o e f f i c i e n t s , as d e s c r i b e d i n s e c t i o n 2.2.3. All  t a r g e t s were mounted on a r e m o t e l y c o n t r o l l a b l e t a r g e t l a d d e r  ( G y l 84), which moved i n s i d e the s p e c t r o m e t e r s c a t t e r i n g chamber.  -  42  -  CHAPTER I I I EXPERIMENTAL ANALYSIS  In  section  2.2.3, the method used  p i o n p a s s i n g through sufficient  the QQD spectrometer  number o f pions have passed  c o n s t r u c t a p i o n energy  spectrum.  I t i s t h e number of counts  determine.  was o u t l i n e d .  through  the s p e c t r o m e t e r ,  one may  corresponds  the t a r g e t n u c l e u s i n a p a r t i c u l a r  i n each peak t h a t one wants t o  In o r d e r to compare the r e s u l t s o f the p r e s e n t  w i t h o t h e r s , however, or w i t h t h e o r e t i c a l c a l c u l a t i o n s , properly normalized.  of a  After a  Each peak i n t h e spectrum  to a p i o n h a v i n g s c a t t e r e d , and l e f t state.  f o r d e t e r m i n i n g the energy  The n o r m a l i z e d p r o b a b i l i t y  i n t o a g i v e n element o f s o l i d a n g l e i s c a l l e d  experiments  they must be  fora particle  the d i f f e r e n t i a l  to s c a t t e r cross  s e c t i o n , and i s g i v e n by: N =  dft where N  n c  scat »dJ2»ef f  d e t e c t e d , having l e f t  energy Nine  *  s  t  n  e  '  a t o m s  i s the number of pions s c a t t e r e d  s c a t  and  N  Ni »N  through  the spectrometer  the t a r g e t n u c l e u s i n a p a r t i c u l a r  state; t o t a l number o f pions i n c i d e n t  on t h e t a r g e t ;  i s the number o f t a r g e t atoms per cm , as seen by the 2  a t o m s  incoming dfi  pions;  i s t h e element o f s o l i d a n g l e subtended  by t h e s p e c t r o m e t e r ,  i n t o which the pions can s c a t t e r ; and  e f f i s the e f f i c i e n c y pions.  of the s p e c t r o m e t e r  system  f o r detecting  - 43 -  In s e c t i o n 3.1, with reference  each of the above f a c t o r s i s c o n s i d e r e d  to the e x p e r i m e n t a l l y  associated uncertainties. differential consideration  cross  determined v a l u e s ,  In s e c t i o n 3.2,  the t a b u l a t e d  s e c t i o n s measured i n the experiments  are presented.  and  i n turn; their  r e s u l t s f o r the under  -  3.1  C a l c u l a t i o n of Cross  3.1.1  Beam N o r m a l i z a t i o n  plastic  the incoming  scintillators  -  Sections  There were f o u r separate a) two  44  incident pion monitors.  ( B l and  B2)  The  main one  positioned directly  was  in line  with  beam on e i t h e r s i d e of the n u c l e a r s c a t t e r i n g t a r g e t , which  were used to measure the a b s o l u t e number of p a r t i c l e s p a s s i n g through target. 300  Then, b) two  mm,  small p l a s t i c  scintillators  ( p l and  u2)  separated  whose c e n t e r s were a l i g n e d at an angle of a p p r o x i m a t e l y  respect  to the incoming  beam.  Such a 'muon t e l e s c o p e ' may  9°  counting  the number of muons from p i o n decay which pass through  counters  (Wad  monitor  only,  But  c ) An  i n the p r e s e n t  i o n chamber and  case, i t was  p r o d u c t i o n t a r g e t were a l s o used as r e l a t i v e For most r u n s , r e l a t i v e monitors  the a b s o l u t e monitor  However, a t c e r t a i n spectrometer  the WC1 passed B2  housing through  counter.  the s c a t t e r i n g  had  B l or B2 would not problem was The effects  by the  Bl«B2 was  around the T l  used e x c l u s i v e l y . f o r these runs,  to remain c o n s t a n t  a n g l e s , <50°,  t a r g e t , but  to be used i n s t e a d . operate  not  no  two  relative  The the  to w i t h i n  the edges of QTl  i n t e r c e p t e d p a r t of the i n c i d e n t p i o n beam, a f t e r  In t h e s e c a s e s , B1»B2 was  other monitors  this  were found  with  monitors.  were c a l i b r a t e d a g a i n s t Bl«B2, and  r a t i o s of the v a r i o u s monitors ±2.%.  used as a  d) a Cerenkov counter  b e f o r e i t had  passed  longer r e l i a b l e ,  and  through one  and  i t had  of  There i s a l s o the p o s s i b i l i t y  e f f i c i e n t l y at high i n c i d e n t pion f l u x e s .  i n fact  by  be used to  measure the a b s o l u t e number of pions emerging from the beamline,  76).  the  the the that But  encountered.  t o t a l number of B1»B2 c o i n c i d e n c e s must be c o r r e c t e d f o r s e v e r a l  b e f o r e i t i s a measure of the a b s o l u t e number of p i o n s  passing  -  45  -  through the s c a t t e r i n g t a r g e t : i)  One  must account f o r the  b e a m l i n e , as w e l l as I T ' S .  and  u's  emerge from  As mentioned i n s e c t i o n s  2.1  and  accomplished by measuring the production and  target.  F i g . 3.1  demonstrates the  f r a c t i o n s are  =94%  and  =91%  raising  e's  illustrates  i i ) One  fractions.  and  l i t t l e energy i n t h a t  will  be  below.  I t r e l i e s on  The  have a l r e a d y  without  ns a p a r t ) , the  two  been  a  true  eliminated  the  s e v e r a l p i o n s may  fact  be  that  created  not be d i s t i n g u i s h e d ,  through and  c o n f i g u r a t i o n of e l e c t r o n i c s  first  one  passing  that of the f i r s t  pion.  successive  through the  the second s c a t t e r i n g i n t o the  B l time would be  +  method used to c o r r e c t f o r t h i s i s o u t l i n e d  the f a c t t h a t w i t h the  i n t e r a c t i n g , and  recorded of the  (43.5  T T and T T ~  r e l y i n g on  used i n data a c q u i s i t i o n , i f pions were produced i n two beam b u r s t s  e's.  or more p a r t i c l e s p a s s i n g  ps of each o t h e r w i l l  counted as one.  U ' S , and  spectra,  counter.  must account f o r the f a c t t h a t  B2 w i t h i n a few  time of f l i g h t  on B l , and  Two  2.2.2, t h i s i s  Note t h a t t h i s i s not  Some e's  the  of the p a r t i c l e s from the T l  typical  respectively.  i n a s i n g l e main p r o t o n beam b u r s t . Bl  e's  of T T ' S ,  the d i s c r i m i n a t o r t h r e s h o l d  lose r e l a t i v e l y  that  time of f l i g h t  clean separation  measure of the a b s o l u t e by  fact  The  proton  target  spectrometer, r a t i o of  the  the  areas  r e s u l t i n g peaks i n the B l time spectrum i s a d i r e c t measure of  the p r o b a b i l i t y of  observing  More s p e c i f i c a l l y ,  pions i n s u c c e s s i v e  for a  given  A = ( p i o n f l u x ) / ( c y c l o t r o n RF the p r o b a b i l i t y of o b s e r v i n g  frequency)  ,  n pions per beamburst i s :  P = (xn -*)/n! e  beam b u r s t s .  ,  so t h a t the p r o b a b i l i t y of o b s e r v i n g  1 or more p i o n s In two  successive  -  46 -  lOOOh  10  20  30  TIME (ns)  F i g . 3.1  Typical T T  +  and TT " Time of F l i g h t  Spectra  -  47  -  beam b u r s t s i s :  I  S = (  X  e"  n  / n! )  x  = ( 1 - e~*  2  )  .  2  n-1 This  i s the number measured e x p e r i m e n t a l l y , and  X = ln(l-/S)  .  The  correction factor  can be  solved f o r  t h a t needs to be a p p l i e d to the  measured B1«B2 i s :  I  n X  n  I  e~ x  n!  °=1  F =  I  X  n  e * -  / n!  n-1 A p l o t of S and F v e r s u s X i s g i v e n i n F i g . 3.2. v a l u e s f o r TT+ a r e :  S = 0.039  -»• F = 1.11  ,  and  S = 0.011  +  .  f o r ir":  T h i s i s an important  c o r r e c t i o n to make.  f o r p r e s e n t experiments fluxes,  F = 1.06  Because the T T  were a p p r o x i m a t e l y  +  -  I t i s p o s s i b l e f o r a p i o n emerging from  from 50 MeV respect  i n Appendix D.  the M13  to the i n c i d e n t  pion d i r e c t i o n ;  be mistaken  for pions.  the i r "  B2 d i p o l e  target.  P i o n decay  Roughly one h a l f of the muons  p i o n decay are emitted at a n g l e s between 15° and  Thus, some f r a c t i o n of the muons w i l l and  f l u x e s obtained  normalization.  magnet to decay b e f o r e r e a c h i n g the n u c l e a r s c a t t e r i n g k i n e m a t i c s are p r e s e n t e d  +  experimental  6 times h i g h e r than  t h i s c o r r e c t i o n a f f e c t s the r e l a t i v e T T / T T  iii)  Typical  18°  with  the o t h e r h a l f a t s m a l l e r a n g l e s .  pass  through  I t has been e s t i m a t e d  the B l and  B2  counters,  (Bar 84), from Monte  C a r l o s i m u l a t i o n s , t h a t the t o t a l e f f e c t on the f l u x , as measured by B1»B2 may  be as h i g h as 7%.  Because of the u n c e r t a i n t i e s i n v o l v e d i n making  such an e s t i m a t i o n , t h i s c o r r e c t i o n has data. will  In any be  c a s e , the e f f e c t  the same f o r T T  spectrometer  solid  +  and T T ~ .  angle.  not been a p p l i e d t o the  i s constant f o r a l l experimental  present runs,  I t i s e f f e c t i v e l y i n c l u d e d In the  and  - 48  F i g . 3.2  -  Pion Flux C o r r e c t i o n F a c t o r s  -  i v ) In the cases of both  1 8  49 -  0  and M g , t h e i n c i d e n t 26  at the t a r g e t p o s i t i o n s were l a r g e r  than the t a r g e t s  purpose  of extracting  through  the t a r g e t s , not that emerging  p i o n beam spots  themselves.  c r o s s s e c t i o n s , the r e l e v e n t p i o n f l u x i s t h a t from the M13 beamline.  The d i f -  f e r e n c e between these two numbers was measured by i n t e r s p e r s i n g CH  2  F o r the  t a r g e t s between the o t h e r s .  The CH  2  t a r g e t s were l a r g e r  runs w i t h  than the p i o n  beam s p o t s , and s c a t t e r i n g d a t a taken w i t h them c o u l d be used  to recon-  struct  over the  i n c i d e n t beam p r o f i l e s , which c o u l d then be i n t e g r a t e d  a r e a s o c c u p i e d by the s m a l l e r t a r g e t s . p r o f i l e s not changing  from one run to the n e x t .  that  t h e beam s p o t s remained  beam  time.  3.1.2  T h i s technique r e l i e s  c o n s t a n t throughout  on t h e beam  In f a c t , r e s u l t s the e n t i r e  show  experimental  Target Thickness The number r e q u i r e d i n the e x p r e s s i o n f o r the d i f f e r e n t i a l c r o s s  s e c t i o n i s the number of s c a t t e r i n g c e n t e r s per cm by t h e incoming  pions.  t r a n s m i s s i o n geometry. t a r g e t normals  i n the t a r g e t , as seen  2  A l l the p r e s e n t experiments  were performed  in a  For a g i v e n s p e c t r o m e t e r a n g l e 26, the s c a t t e r i n g  were i n c l i n e d  to the d i r e c t i o n o f t h e beam by an angle 6,  i n o r d e r to e q u a l i z e the amount of t a r g e t m a t e r i a l through which a s c a t tered pion passed.  The number of s c a t t e r i n g  c e n t e r s i s thus g i v e n by  N/cos0 , where N i s the number of s c a t t e r e r s per cm normal  i s parallel  i n the t a r g e t g i v e n by  t o the i n c i d e n t beam d i r e c t i o n .  2  i n a t a r g e t whose The f r a c t i o n a l  error  t h i c k n e s s due to the u n c e r t a i n t y i n the t a r g e t a n g l e i s  (tan9)A6,  w i t h A6 = ± 1 . ° .  The number o f s c a t t e r e r s per cm  2  f o r each o f the t a r g e t s used In the  - 50 -  present  experiments  i s g i v e n i n T a b l e 3.1.  F o r the H 0 t a r g e t , t h e con2  t r i b u t i o n of the window m a t e r i a l was taken i n t o account by s u b t r a c t i n g t h e background  measured i n the empty t a r g e t  frame r u n s .  F o r the purpose o f  a c c o u n t i n g f o r the i s o t o p i c i m p u r i t y of the t a r g e t ; t h e 0 , 1 8  1 7  0 , and  1 6  0  e l a s t i c c r o s s s e c t i o n s were a l l assumed to be o f s i m i l a r magnitude, so t h a t no c o r r e c t i o n was a p p l i e d ; f o r the i n e l a s t i c final  1 8  0 result  3.1.3  c r o s s s e c t i o n s , the  s c a l e d by l./(0.953 ± 0.02) .  Efficiency The  efficiency  different  effects:  background  events  spectrum  f a c t o r i n the e x p r e s s i o n on page 42 encompasses two  a) the i n h e r e n t e f f i c i e n c y o f the QQD MWPCs; and b) r e j e c t e d i n the o f f - l i n e a n a l y s i s , b e f o r e  peaks a r e i n t e g r a t e d .  a) Only those events a r e a n a l y z e d , i n which v a l i d obtained  energy  from a l l f o u r MWPCs.  signals are  Events i n which s i g n a l s from o n l y three o f  the MWPCs a r e o b t a i n e d , a r e used as a measure o f the i n e f f i c i e n c y of the f o u r t h chamber.  The product o f the i n d i v i d u a l e f f i c i e n c i e s o f the f o u r  w i r e chambers, as measured i n t h i s way, was t y p i c a l l y f o r both t r and I T . +  was  -  The i n t r i n s i c  96% f o r a l l r u n s ,  e f f i c i e n c y of a l l p l a s t i c  scintillators  assumed t o be 100%. b) Some p i o n s which s c a t t e r  i n t o t h e spectrometer w i l l decay  muons, b e f o r e p a s s i n g c o m p l e t e l y through t h e system.  As e x p l a i n e d i n  Appendix D, t h o s e p i o n s decaying b e f o r e , o r i n , the s p e c t r o m e t e r will  p r o b a b l y not pass through WC4 and WC5.  dipole,  Those d e c a y i n g a f t e r BT  probably w i l l ,  but r a t h e r than f a l l  spectrum,  form p a r t of a g e n e r a l background.  will  into  i n t o a p a r t i c u l a r peak i n t h e energy A l s o , p i o n s which  - 51 -  Target  CH  Nucleus  2  26  lH 2 6  M g  0  M g  N /  cm  error  2  1.38 x  10  2 2  ± 1.0 %  2.75 x  10  2 2  ± 1.0 %  6.95 x I O *  ± 1.0 %  1.05 x  2  10  2 2  ± 2.0 %  x 10  2 2  ± 2.0 %  H 0 2  2.10  Table  3.1  T h i c k n e s s e s , In S c a t t e r i n g Centers per cm , of E x p e r i m e n t a l S c a t t e r i n g T a r g e t s 2  -  52  -  a c t u a l l y pass through i n d i v i d u a l wires of the MWPCs w i l l to m u l t i p l e Coulomb s c a t t e r i n g .  be d e f l e c t e d due  These may a l s o end up c o n t r i b u t i n g  to t h e  background. It  i s d e s i r a b l e to e l i m i n a t e such background  e v e n t s , which  r e p r e s e n t some 12% o f the d a t a , i n o r d e r to o b t a i n b e t t e r f i t s peaks p r e s e n t i n t h e energy be accounted into  ground  they do a r i s e from p i o n s which  As was the case f o r f i n d i n g QQD t r a n s f e r approaches  o f people i n v o l v e d i n the QQD experiments  events.  t o the  A t the same time, these events must  (see s e c t i o n 2.2.3), s e v e r a l d i f f e r e n t  by t h e group  all  f o r p r o p e r l y because  the s p e c t r o m e t e r .  cients  spectra.  typically  scattered coeffi-  have been  adopted  t o e l i m i n a t e back-  The p r e s e n t data has been a n a l y z e d u s i n g a combination o f  these methods, which are o u t l i n e d below, and which a r e found t o g i v e  similar  results.  I ) The sum of the E l , E2, and E3 ADC s i g n a l s i s p r o p o r t i o n a l t o the energy  (AE) o f the p a r t i c l e s p a s s i n g through  loss  p o s s i b l e t o c o n s t r u c t an E-AE s c a t t e r p l o t using  the AE f o r each  determined will  from  It i s  of a l l events i n a g i v e n run, by  event, as d e f i n e d above, a l o n g w i t h the energy as  the p o s i t i o n s of the p a r t i c l e  form a d i s t i n c t  these c o u n t e r s .  i n t h e f o u r MWPCs.  band i n such a p l o t , w h i l e the background  Pions  events  will  be d i s t r i b u t e d more or l e s s randomly. i i ) WC4 and WC5 may be used i n d e p e n d e n t l y t o determine momentum 6 . Q  I f t h e two v a l u e s o f 6  Q  so o b t a i n e d a r e not i d e n t i c a l ( o r  w i t h i n 0.1% say) f o r a p a r t i c u l a r e v e n t , i t i s an i n d i c a t i o n event  the p a r t i c l e  t h a t the  i s a background one. i i i ) Having  determined  6  Q  from WC4 and WC5, i t i s p o s s i b l e to take  the average, and use t h i s number, a l o n g w i t h the WCl and WC3 p o s i t i o n s and  -  known t r a n s f e r c o e f f i c i e n t s , WC4  and WC5.  The  53  -  to c a l c u l a t e the p a r t i c l e  p o l a r angle between t h i s  one a c t u a l l y measured from the WC4 ted.  Events f o r which  calculated  and WC5  positions  t h i s p o l a r angle i s l a r g e  trajectory  through  t r a j e c t o r y and can a l s o be  (>3.5°  the  calcula-  say) a r e back-  ground . i v ) Rather  than s o l v e the two e q u a t i o n s  - f(*i.yi»* »y3»6o)  H x for 6  5  =  g(x ,y ,x ,y ,6 ) 1  2  = K  and determine the v a l u e of 6 the minimum v a l u e of x  3.1.4  3  3  0  Q  2  " f )  2  + (x  -  5  which r e s u l t s  g)2  ,  i n the minimum x ^  Events f o r  2  i s l a r g e can be i d e n t i f i e d  as  background.  Peak F i t t i n g The  energy r e s o l u t i o n of the QQD  p r e s e n t experiments  (=1.15 MeV)  was  guous i d e n t i f i c a t i o n o f the peaks ground  1  e x a c t l y , i t i s p o s s i b l e to d e f i n e  Q  X  which  »  3  and  first  excited  spectrometer system d u r i n g the  sufficient  to a l l o w f o r the unambi-  corresponding to s c a t t e r i n g  s t a t e s of  1 8  0  and  2 6  Mg.  I t was  t o the  insufficient,  however, to e n a b l e the peak areas to be e x t r a c t e d  from a s i m p l e summing of  the number of counts i n a c e r t a i n energy r e g i o n .  The computer  ZXMIN (IMSL 82) was determine t h e i r I t was  at  i n the energy s p e c t r a , and  thus  integrals.  found t h a t a simple Gaussian d i s t r i b u t i o n was  reproduce the shape form was  used to f i t the peaks  routine  of the i s o l a t e d  1 2  C  ground  s t a t e peak.  adequate Thus,  this  a l s o used f o r the other t a r g e t s , w i t h the peak widths h e l d  the v a l u e s needed  to f i t C , 1 2  to  and the r e l a t i v e energy s e p a r a t i o n s  fixed  -  54  -  between ground and e x c i t e d s t a t e s f i x e d a t t h e i r w e l l known The  statistical  e r r o r a s s o c i a t e d w i t h a peak a r e a  the square r o o t o f the t o t a l number o f c o u n t s . corresponding  t o the q u a l i t y of the f i t was a l s o  included.  1 2  3.1.5  +  1 8  0 , and  2 6  M g are  3.3, 3.4, and 3.5 r e s p e c t i v e l y .  Spectrometer Acceptence The  ir p  i n Figures  i s proportional to  An a d d i t i o n a l e r r o r  Examples of t y p i c a l energy s p e c t r a f o r C , presented  values.  QQD s p e c t r o m e t e r l a b o r a t o r y s o l i d  s c a t t e r i n g data,  from CH  2  a n g l e was o b t a i n e d  t a r g e t s , from s e v e r a l a n g l e s ,  by u s i n g  i n the  relation N  ~  Nine*N  a t o m s  scat .eff»(da/dfi)  s  where a l l the f a c t o r s have the meanings g i v e n on page 42, and d i s c u s s e d i n the p r e v i o u s cross  s e c t i o n s of t h i s c h a p t e r .  (da/dJ2)  i s the i r p +  s  differential  s e c t i o n i n the l a b o r a t o r y frame, as c a l c u l a t e d i n t h e c e n t e r  frame, a t the a p p r o p r i a t e  energy, u s i n g  o f mass  the phase s h i f t s of the ElOD  s o l u t i o n of Arndt and Roper (AR 8 2 ) , and k i n e m a t i c a l l y t r a n s f o r m e d . No e r r o r has been a s s o c i a t e d w i t h dJ2 due t o the i n t r i n s i c uncertainty  of t h e c a l c u l a t e d i r p c r o s s s e c t i o n s .  the a n g u l a r  r e g i o n of I n t e r e s t , the u p c r o s s  +  +  3% p e r MeV change i n I n c i d e n t p i o n energy. associated with (see due  t o the o t h e r  s e c t i o n s change by  Thus, an u n c e r t a i n t y  dQ because of the u n c e r t a i n t y  section 2.1).  Note however, t h a t over roughly of ±3% i s  i n the Incoming beam energy  T h i s has been added i n q u a d r a t u r e t o the u n c e r t a i n t i e s  factors.  Note a l s o t h a t t h e spectrometer acceptence depends on the l o c a t i o n  F i g . 3.3  Typical  1 2  C  Energy  Spectrum  F i g . 3.4  Typical  1 8  0  Energy  Spectrum  F i g . 3.5  Typical  2 6  M g Energy  Spectrum  - 58 -  on t h e s c a t t e r i n g lustrated  t a r g e t o f the o r i g i n a l s c a t t e r i n g  i n F i g . 3.6, which i s based  Because o f t h e v a r i a t i o n , the s o l i d averaged  over  incident  beam p r o f i l e .  thus the same s o l i d  dSl  2  angle as determined  0  and  2 6  Mg  Trp +  T h i s was done by  from s e l e c t e d p o r t i o n s of  t o the s i z e o f the o t h e r s .  The r e s u l t i s  = 12.7 ± 0.8 msr. It w i l l  be noted  t h a t t h i s number i s some 30% s m a l l e r than the  REVMOC p r e d i c t i o n , as g i v e n i n F i g . 3.6. fact it  t a r g e t s , weighed by the  f o r both.  events which o r i g i n a t e d  t a r g e t , corresponding  above Is i n f a c t  t a r g e t s were o f s i m i l a r s i z e , and  angle was determined  a c c e p t i n g o n l y those the CH  1 8  This i s i l -  on a REVMOC (KR 83) c a l c u l a t i o n .  the s i z e o f the n u c l e a r s c a t t e r i n g The  event.  t h a t pions d e c a y i n g  through  tially.  T h i s d i f f e r e n c e a r i s e s from the  i n the f r o n t end o f the s p e c t r o m e t e r  do not make  t o the back, and thus appear not t o have been a c c e p t e d  This effect  ini-  and o t h e r s ( s e e e.g. page 47) have been i n c o r p o r a t e d  i n t o d£2, which thus p l a y s the r o l e o f an e f f e c t i v e  spectrometer  solid  a n g l e , r a t h e r than a p u r e l y geometric one. The  number of p i o n s decaying  i n the f r o n t end o f the spectrometer  w i l l v a r y w i t h the s c a t t e r e d p i o n energy. in  the f a c t o r  p i o n energy  T h i s e f f e c t has been i n c l u d e d  ( d a / d f i ) , s i n c e T f p k i n e m a t i c s a r e such +  s  that the s c a t t e r e d  changes r a p i d l y w i t h the s c a t t e r i n g a n g l e .  T h i s i s not the  case f o r p i o n s c a t t e r i n g from h e a v i e r n u c l e i . Finally, and  note  t h a t comparisons made w i t h o t h e r e x p e r i m e n t a l  data,  t h e o r e t i c a l c a l c u l a t i o n s a r e made i n t h e p i o n - n u c l e u s c e n t e r of mass  frame. plied  Thus, the l a b o r a t o r y v a l u e o f dfi, as determined  above, i s m u l t i -  by the J a c o b i a n a p p r o p r i a t e t o the t r a n s f o r m a t i o n from l a b o r a t o r y t o  c e n t e r of mass frames f o r each i n d i v i d u a l  nucleus.  - 59 -  SOLID ANGLE SUBTENDED  45  —1111111111111111111 j i II 111111  -45  Fig.  -30  3.6  11111111111111111111111111111 -  "15 0 15 X D I R E C T I O N (BID)  30  45  REVMOC C a l c u l a t i o n o f V a r i a t i o n o f QQD S o l i d A n g l e over S c a t t e r i n g T a r g e t . Contours l a b e l l e d i n msr.  - 60 -  3.2  Results The v a l u e s  of the measured d i f f e r e n t i a l  i n T a b l e s 3.2 t o 3.7. Include  the v a r i o u s  (=3%), t a r g e t statistics. typically  sections  are presented  As mentioned i n s e c t i o n 3.1, the quoted  uncertainties associated  thickness  cross  (=2%), and s o l i d  F o r the e l a s t i c  normalization  a n g l e (=6%), as w e l l as  s c a t t e r i n g , the p u r e l y s t a t i s t i c a l e r r o r was  - 3.5 %, w h i l e f o r the i n e l a s t i c  except a t the backwardmost  w i t h beam  errors  angles.  s c a t t e r i n g i t was  = 10 %,  61  -  G  ir  -  c m  -  (deg)  da/dfi  (mb/sr)  49.0  4.18  +  0.43  59.1  2.74  +  0.28  69.2  2.35  +  0.30  79.2  3.77  +  0.32  92.7  6.07  +  0.55  102.7  6.58  +  0.57  112.6  6.86  +  0.61  122.6  7.28  +  0.63  132.3  6.11  +  0.56  49.0  9.23  +  1.03  64.1  2.32  +  0.35  79.2  3.85  +  0.44  89.2  5.82  +  0.56  92.7  6.41  +  0.64  102.6  8.19  +  0.78  112.6  8.03  +  0.76  122.6  8.56  +  0.79  132.3  7.55  +  0.71  T a b l e 3.2  Measured Cross S e c t i o n s f o r 1 2  C(TT,TT)  1 2  C  - 62 -  da/dfl  (mb/sr)  59.1  0.153  +  0.046  69.2  0.163  +  0.063  79.2  0.231  +  0.037  92.7  0.459  +  0.086  102.7  0.805  +  0.111  112.6  1.03  +  0.13  122.6  1.73  +  0.17  132.3  1.59  +  0.18  79.2  0.237  +  0.081  89.2  0.385  +  0.091  92.7  0.752  +  0.151  102.6  1.24  +  0.20  112.6  1.70  +  0.23  122.6  2.19  +  0.26  132.3  2.79  +  0.31  ©cm  (  d e  T a b l e 3.3  8)  Measured Cross 1 2  Sections f o r  C(TT,TT') C*(2 ,4.44) 1 2  +  - 63 -  e  c m  (deg)  do/dti  (mb/sr)  43.7  9.37  +  0.77  47.2  6.77  +  0.54  48.7  6.60  +  0.53  58.8  3.81  +  0.32  68.8  3.66  ±  0.30  78.9  5.08  +  0.41  88.9  6.63  +  0.59  92.4  6.98  +  0.58  102.4  7.97  +  0.65  112.4  7.65  +  0.64  122.3  7.16  +  0.59  132.0  5.72  +  0.49  48.7  14.07  +  1.17  63.8  4.39  +  0.39  78.9  7.80  +  0.65  88.9  9.67  +  0.84  92.4  10.64  +  0.89  102.4  10.79  +  0.89  112.4  9.14  +  0.77  122.3  6.61  +  0.58  132.0  3.89  +  0.36  Tf+  Table  3.4  Measured Cross 18  Sections for  0(TT,T0 0 18  64  -  do/dfi  (mb/sr)  48.7  0.150  +  0.026  58.8  0.117  +  0.019  68.8  0.182  +  0.023  78.9  0.194  +  0.022  88.9  0.282  +  0.048  92.4  0.439  +  0.058  102.4  0.667  +  0.073  112.4  0.685  +  0.081  122.3  1.06  +  0.10  132.0  1.18  +  0.11  63.8  0.258  +  0.051  78.9  0.653  +  0.084  88.9  1.26  +  0.16  92.4  1.32  +  0.16  102.4  1.56  +  0.16  112.4  2.16  ±  0.21  122.3  3.08  +  0.30  132.0  3.27  +  0.31  0cm  ir  -  -  ( e) d e  T a b l e 3.5  Measured Cross S e c t i o n s f o r 18  0(Tr,Tr') 0*(2+,1.98) 18  -  65  ©cm  (  d e  dff/dfi  8>  (m b / s r )  58.6  8.19  +  0.60  68.7  7.54  +  0.55  78.7  9.70  +  0.72  88.7  9.38  +  0.73  92.2  9.14  +  0.70  100.2  7.89  +  0.61  108.2  6.07  +  0.49  112.2  5.52  +  0.46  116.2  5.18  +  0.42  72.2  8.82  +  0.66  82.2  11.78  +  0.87  92.2  9.88  +  0.75  100.2  7.97  +  0.62  108.2  5.71  +  0.46  116.2  2.88  +  0.25  Cross  Sections  TT+  TT  -  —  Table  3.6  Measured 2 6  Mg(Tr,Tr) Mg 2 6  for  - 66 -  ©cm  1T+  IT  —  da/dti  < d e e)  (mb/sr)  58.6  0.34  +  0.04  68.7  0.50  +  0.05  78.7  0.82  +  0.07  88.7  1.29  +  0.13  92.2  1.43  +  0.14  100.2  2.52  +  0.22  108.2  3.48  +  0.30  112.2  2.77  +  0.25  116.2  3.08  ±  0.26  72.2  0.49  +  0.06  82.2  1.00  +  0.10  92.2  1.98  +  0.18  100.2  2.36  +  0.21  108.2  1.97  +  0.18  116.2  2.57  +  0.23  T a b l e 3.7  Measured Cross  Sections for  2 6Mg( TT.TT' )  2 6  Mg*(2  + >  1.81)  - 67 -  CHAPTER IV THEORETICAL DETAILS  In t h i s c h a p t e r , some r e l e v e n t calculations  d e t a i l s of the t h e o r e t i c a l  which have been performed f o r the a n a l y s i s  of t h e  experimental data are discussed. In s e c t i o n all  4.1.1 the MSU o p t i c a l p o t e n t i a l , which has been used f o r  calculations,  i s presented.  multiple  scattering  standard  texts,  There i s no l e n g t h y c o n s i d e r a t i o n  of  formalism, as t h i s can be found i n any number o f  e.g. (GW 64), (Tay 7 2 ) .  individual  terms i n c l u d e d  discussion  of t h e l i m i t a t i o n s of the MSU  R a t h e r , r e f e r e n c e i s made to the  i n the p o t e n t i a l .  In section  4.1.2 some  form, and o t h e r f o r m u l a t i o n s , i s  undertaken. In s e c t i o n sections vital  increased and  i s described.  input  sections  4.2.2 and 4.2.3.  and t h e i r p r o p e r t i e s ,  The r e s u l t s  s e n s i t i v i t y o f TT~ s c a t t e r i n g  4.2.4.  are discussed i n  of the c a l c u l a t i o n s  demonstrate the  to neutron t r a n s i t i o n  to p r o t o n t r a n s i t i o n d e n s i t i e s .  considered i n section  cross  The n u c l e a r t r a n s i t i o n d e n s i t i e s , which a r e a  to these c a l c u l a t i o n s ,  TT s c a t t e r i n g +  4.2.1 t h e c a l c u l a t i o n of i n e l a s t i c d i f f e r e n t i a l  densities,  These r e s u l t s a r e  - 68 -  4.1  Elastic  4.1.1  The MSU O p t i c a l The  the  Scattering  following  present  Potential  i s t h e g e n e r a l form of t h e o p t i c a l p o t e n t i a l used i n  analysis: 2EV  n  = A i p + A 6 p + A4P  + V (A3P+A3 6p+A p )  2  2  2 v  / +  A p  + A p + A  2  6  2  p = p  =  2  Aiv =  4TTCO/PI  a  A3  = -4irco(pi-l)/2pi  A4  = -4TTBOP2  A  =  n  6  4TTC / 2  2  =  5  V  =  2  5  P  y  (4.1)  7  4Treci(pi-l)/2pi *TTCO/P2  = -4TT(C +C )(p -l 0  Pn ~ Pp » where p  p r o t o n ground s t a t e m a t t e r d e n s i t i e s 2  A  4ifebipi  =  =  Ay  P 2  + Pp, and 6p  Pj and p  \ +  -4TT£CX/PI  2 V  A3 a  7  6p  2  A i = -4Trbopi A  2 v  \ 1 + (X/3)(A p +A p+A v'Sp)  V  6  with  2  v  n  2  2  )/2  P 2  .  and pp a r e t h e n e u t r o n and  respectively,  e = ±1 f o r i r * .  a r e k i n e m a t i c f a c t o r s g i v e n by  p  l  _ (1+E/m) " (1+E/mA)  a  n  d  _ (l+E/2m) 2 ~ (l+E/2mA) '  P  A l l c o e f f i c i e n t s a r e complex. E l a s t i c pion-nucleus scattering the  computer program  DWPI  t h i s form o f p o t e n t i a l . (EM  (EM 76), DWPI  cross sections  are calculated  with  which has been m o d i f i e d t o i n c o r p o r a t e  i s an e x t e n s i o n o f t h e e a r l i e r program  7 4 ) , and i s a l s o used f o r c a l c u l a t i n g i n e l a s t i c c r o s s s e c t i o n s .  P I R K  An  o u t l i n e o f the method of s o l u t i o n , a l o n g w i t h some m a t h e m a t i c a l d e t a i l s i s p r o v i d e d i n Appendix A.  -  The  b a s i c procedure  Klein-Gordon  69  -  i n v o l v e s a n u m e r i c a l i n t e g r a t i o n of t h e  e q u a t i o n , which f o r a f r e e p a r t i c l e (E -p )'P = m Y 2  It  i s common p r a c t i c e  p o t e n t i a l s with (E-V -V ), c  n  .  2  to i n c l u d e the e f f e c t s o f the Coulomb and n u c l e a r  the energy.  so t h a t  2  i s written:  That  i s , one r e p l a c e s E above by  the K l e i n - G o r d o n  e q u a t i o n becomes  ((E +V +V -2EV -2EV -V V -V V )-p )'P = m P 2  2  2  c  It  2  n  c  n  c  n  n  i s a l s o common p r a c t i c e t o n e g l e c t the V  this point.  Some c a l c u l a t i o n s  t a i n e d have been performed, Historically,  the f i r s t  2 n  , V V , and V V c  n  i n which the l a t t e r  however.  .  2,  c  These w i l l  n  c  terms at  two terms have been r e -  be d i s c u s s e d i n Chapter V.  p i o n - n u c l e u s o p t i c a l p o t e n t i a l which had  some s u c c e s s i n d e s c r i b i n g e l a s t i c s c a t t e r i n g d a t a was i n t r o d u c e d by Kisslinger  i n 1955 ( K i s 55).  I t i n c o r p o r a t e s the p i o n - n u c l e o n  scattering  amplitude: f = b in  Q  + bjt-x + (c +c t»x)k«k* 0  the standard m u l t i p l e s c a t t e r i n g  1  formalism.  To f i r s t  o r d e r , the  r e s u l t i n g o p t i c a l p o t e n t i a l may be w r i t t e n : 2EV The isovector  n  = b p + b 6p + V « ( c p + c 6 p ) V 0  coefficients b  x  Q  0  1  and b j a r e r e f e r r e d  s-wave parameters,  i s o v e c t o r p-wave parameters.  .  to as the i s o s c a l a r and  w h i l e C Q and C j a r e c a l l e d  the i s o s c a l a r and  The o r i g i n o f these names i s e v i d e n t  the form o f the p i o n - n u c l e o n s c a t t e r i n g a m p l i t u d e . coefficients  (4.2)  a r e o b t a i n e d from p i o n - n u c l e o n phase  from  The v a l u e s o f t h e shifts.  - 70 -  P i o n a b s o r p t i o n cannot  occur on a s i n g l e n u c l e o n ,  r e s t r i c t e d kinematic conditions. included i n (4.2). square  i s thus not e x p l i c i t e l y  Absorption i s u s u a l l y parametrized  i n terms of the  of the n u c l e a r d e n s i t y , assuming i t i s m a i n l y due t o two n u c l e o n  processes.  The o p t i c a l p o t e n t i a l i s then w r i t t e n : 2EV  The  Its effect  except under very  n  = b p + bjfip + B p Q  new parameters B  2  Q  Q  + V«(c p + c 6p + Cp )V 2  0  x  and C may be taken from  79), o r e x t r a p o l a t e d from  fits  t h e o r e t i c a l c a l c u l a t i o n s (CR  to p i o n i c atom widths  some t h e o r e t i c a l model f o r the energy  (SCM 80) assuming  dependence.  E r i c s o n and E r i c s o n (EE 66) p o i n t e d out t h a t another i n c l u d e d I n (4.2) i s t h a t of n u c l e a r p a i r c o r r e l a t i o n s . t h a t s h o r t range p a i r c o r r e l a t i o n s a r e important scattering;  giving rise  .  to a phenomenon analogous  e f f e c t not  They demonstrated  i n the m u l t i p l e t o one o c c u r i n g i n the  s c a t t e r i n g of e l e c t r o m a g n e t i c waves i n dense media which i s c a l l e d t h e Lorentz-Lorenz  effect.  Q u a n t i t a t i v e l y , the i n f l u e n c e on the p i o n - n u c l e u s  o p t i c a l p o t e n t i a l i s twofold. i s m o d i f i e d from  First,  the v a l u e of t h e s-wave parameter b  t h a t o b t a i n e d from p i o n - n u c l e o n  the p-wave terms a r e m u l t i p l i e d  2  0  may be e s t i m a t e d The  C l  ,  the L o r e n t z - L o r e n z - E r i c s o n - E r i c s o n (LLEE) term.  parameter X determines  Second,  by the f a c t o r :  (l+(X/3)(c p+ 6p+Cp ))-l which i s c a l l e d  phase s h i f t s .  The  the s t r e n g t h of the LLEE e f f e c t , and i t s v a l u e  theoretically  ( s e e (SCM 8 0 ) ) .  q u e s t i o n was r a i s e d by (SMC 79) as to whether the p-wave  a b s o r p t i o n term  should be i n c l u d e d w i t h i n the LLEE f a c t o r o r n o t . The  g e n e r a l form o f the o p t i c a l p o t e n t i a l used ( 4 . 1 ) , a l l o w s f o r both p o s s i b i l i t i e s .  i n the p r e s e n t a n a l y s i s , eqn.  One may s e t C =0.0, C *0.0 t o ?  n  Q  -  71  -  take the a b s o r p t i v e p-wave term o u t s i d e the LLEE f a c t o r , o r s e t Cg=0.0, Cj^O.O t o i n c l u d e i t . There i s some reason to b e l i e v e t h a t the l a t t e r procedure  i s the more c o r r e c t one ( C a r 84), and thus a l l the c a l c u l a t i o n s  discussed  i n Chapter  One f i n a l from  V were performed  effect  s e t e q u a l to z e r o .  i n c l u d e d i n the p o t e n t i a l  s c a t t e r i n g amplitude  (4.1) i s t h a t a r i s i n g  from the p i o n - n u c l e o n  p i o n - n u c l e u s c e n t e r of mass frames. 2  Q  the t r a n s f o r m a t i o n of the k»k' f a c t o r i n the p-wave  pion-nucleon  7 p  with C  and V p 2  2  .  I t was f i r s t  term  of the  t o the  T h i s i n t r o d u c e s terms p r o p o r t i o n a l t o  p o i n t e d out by T h i e s  (Thi 76).  I t introduces  no new parameters i n t o the p o t e n t i a l , and improves the agreement w i t h experimental  4.1.2  data.  Further Discussion T a b l e 4.1 p r e s e n t s the v a l u e s of the g l o b a l  p o t e n t i a l parameters of (CMS 8 2 ) . c a l c u l a t i o n performed V.  'Set E ' o p t i c a l  These formed the s t a r t i n g p o i n t f o r the  t o f i t the p r e s e n t d a t a , t o be d e s c r i b e d i n Chapter  The v a l u e s f o r these parameters were o b t a i n e d as f o l l o w s : ReB and ReC  were taken from phase s h i f t  theoretical calculations,  Imb and Ime were c a l c u l a t e d  v a l u e s , ImB and ImC were taken from  fits  from  to measured  a b s o r p t i o n c r o s s s e c t i o n s (Nak+ 8 0 ) , Reb and Rec were v a r i e d i n o r d e r to fit  the e x i s t i n g The  elastic  scattering cross s e c t i o n s .  f o l l o w i n g o b s e r v a t i o n s can be made about the Set E  a) t h e r e may be some q u e s t i o n ( J e n 84) as to the v a l i d i t y adopted  in fitting  themselves  of the approach  ImB and ImC to a b s o r p t i o n c r o s s s e c t i o n s .  warn t h a t q u e s t i o n s may a r i s e  r e s u l t i n g parameters.  parameters:  (CMS  82)  i n the i n t e r p r e t a t i o n of the  A l s o , the data of (Nak+ 80) seems to be i n  - 72 -  Coefficient  Units  Value  Re b  Q  fm  -0.061  Im b  Q  fm  0.006  Re b  :  fm  Im bj  fm  Re c  Q  Im c  Q  -0.13 -0.002  fm  3  0.70  fm  3  0.028 0.46 0.013  Re  c  1  fm  3  Im  Cj  fm  3  Re B  Q  fm  Im B  Q  fm *  0.11  2  fm  6  0.36  2  fm  6  0.54  Re C Im C  -0.02  k  1  \  T a b l e 4.1  1.4  (CMS 82) Set E Potential  Optical  Parameters  -  73  -  disagreement w i t h the more r e c e n t r e s u l t s o f (Nav+ 8 3 ) . the (CMS  82) f i t s were performed, the e x i s t i n g e l a s t i c  consisted V,  s o l e l y of i r  the i n c l u s i o n o f i r  +  scattering data.  -  s c a t t e r i n g r e s u l t s may  It has been p o i n t e d out (CMS form (4.1) has too many parameters (SMY  b) At the time cross  sections  As w i l l be d i s c u s s e d  i n Chapter  prove i m p o r t a n t .  82) that an o p t i c a l p o t e n t i a l o f the to be determined from e x p e r i m e n t a l d a t a  alone.  (SM 83) and  83) have e s t a b l i s h e d  the i n s e n s i t i v i t y of the  elastic  s c a t t e r i n g d a t a to the p o t e n t i a l s t r u c t u r e , which i s m a n i f e s t e d i n  the form o f c o r r e l a t i o n s between the c o e f f i c i e n t s of the p and p the  2  terms i n  potential. In f a c t ,  s e v e r a l a n a l y s e s of i r  performed w i t h o p t i c a l  +  elastic  s c a t t e r i n g data have been  p o t e n t i a l s of the form (4.2) (Ama+ 81), and o t h e r  few parameter  forms  varied  i n o r d e r to f i t the c r o s s s e c t i o n s .  freely  parameters d i f f e r  (Pre+ 8 1 ) .  In these a n a l y s e s , a l l c o e f f i c i e n t s were The r e s u l t a n t b e s t f i t  from n u c l e u s to n u c l e u s .  An a l t e r n a t e approach has been adopted by Friedman  ( F r i 83),  who  used an o p t i c a l p o t e n t i a l d e s c r i b e d by a F o u r i e r - B e s s e l s e r i e s , i . e . U(r) = I a and v a r i e d  the c o e f f i c i e n t s a  scattering data.  j (nirr/R )  n  The aim was  0  n  to f i t  c  ,  a l l the e x i s t i n g  elastic  to e x t r a c t as much i n f o r m a t i o n from the d a t a  as p o s s i b l e , w i t h o u t i n t r o d u c i n g any p o s s i b l y unnecessary The c o n c l u s i o n s i n c l u d e d : a) that the c h a r a c t e r i s t i c t y p i c a l of a that  -k p 2  the K i s s l i n g e r In l i g h t  - (l/2)V p 2  V»pV  shape which i s  term i s indeed r e q u i r e d by the d a t a ; and  type term i s a l s o n e c e s s a r y .  of the above c o n s i d e r a t i o n s ,  author i s the f o l l o w i n g :  assumptions.  the v i e w p o i n t adopted by the  A l l c a l c u l a t i o n s are performed w i t h the  b)  - 74 -  potential  (4.1).  As  i n d i c a t e d i n s e c t i o n 4.1.1  includes a l l physical effects pion-nucleus  problem.  the LLEE e f f e c t .  thought  this  form of  to be of importance  potential  i n the  T h i s i n c l u d e s the a n g l e t r a n s f o r m a t i o n terms and  A l s o , as i n d i c a t e d by the r e s u l t s of ( F r i 83),  g e n e r a l form o f the p o t e n t i a l i s t h a t r e q u i r e d by the e x i s t i n g p o t e n t i a l i s c a p a b l e of d e s c r i b i n g TT e l a s t i c +  nuclei  from  1 2  C  to  2 0 8  Pb  p o t e n t i a l s w i t h fewer The unshakable  parameters.  theoretical principles.  semi-phenomenological.  Certainly,  I t would be argued,  r e a c t i o n c r o s s s e c t i o n s , and  s e c t i o n s , the p o t e n t i a l  p o t e n t i a l i n both  made i n  however, t h a t f a c e d with  s i n g l e and  energy  double  the  pion-nucleus  charge  exchange c r o s s  can do a t the p r e s e n t  i n considering other  other p o s s i b i l i t i e s  on  e f f e c t s mean that i t i s  (4.1) i s the b e s t one  be gained  i s based  the a p p r o x i m a t i o n s  l a c k of e x t e n s i v e e x p e r i m e n t a l measurements of low  The  The  s c a t t e r i n g c r o s s s e c t i o n s on  c l a i m i s not made, t h a t the p o t e n t i a l (4.1)  and not much w i l l  data.  w i t h the same parameter s e t , which i s not t r u e of  i n c l u d i n g higher order multiple s c a t t e r i n g  total  the  time,  possibilities.  i n c l u d e f o r m u l a t i o n s of the  c o o r d i n a t e (JS 83) and momentum space  pion-nucleus  (LT 7 8 ) .  The  computer code LPOTT (Lan 82), u s i n g the l a t t e r  potential,  and  s c a t t e r i n g data f o r n u c l e i  i s found not  h e a v i e r than  1 2  p o t e n t i a l has to determine  C, no  to reproduce  t o w i t h i n f a c t o r s o f 2 or 3. free  i n p u t parameters,  off-shell  a b s o r p t i o n and  existing elastic  amplitudes,  and  has been run,  Note t h a t a l t h o u g h  the  i t does r e l y on p a r t i c u l a r models i n c l u d e the e f f e c t s of t r u e p i o n  Pauli exclusion.  A f o r m u l a t i o n of the p i o n - n u c l e u s s u c c e s s a t resonance  p o t e n t i a l which has met  p i o n e n e r g i e s i s the A-hole model (HLY  with great  77),  -  75  -  (Hir+ 7 9 ) . There, n u c l e a r medium e f f e c t s a r e p a r a m e t r i z e d phenomenological important  A-nucleus  spreading p o t e n t i a l .  i n terms of a  I t i s not c l e a r  the A i s a t 50 MeV i n c i d e n t p i o n energy.  how  A l r e a d y a t 100 MeV  t h e r e was some problem i n u s i n g the A-hole model to f i t the e l a s t i c and inelastic scattering some i n d i c a t i o n forthcoming,  cross sections f o r C 1 2  and  1 3  C  (Ant+ 8 3 ) .  There i s  (Mon 84) that c a l c u l a t i o n s f o r 50 MeV p i o n s w i l l not be  until  these problems a r e r e s o l v e d .  - 76 -  4.2  Inelastic  4.2.1  Calculation The  are  Scattering  cross  related  of Cross S e c t i o n s  sections  for scattering  to d i s c r e t e nuclear excited  t o sums of T-matrix elements.  states  The DWPI computer code, which  was used to p e r f o r m p i o n - n u c l e u s i n e l a s t i c s c a t t e r i n g c a l c u l a t i o n s , i s based on a s t a n d a r d d i s t o r t e d wave impulse a p p r o x i m a t i o n (DWIA) model f o r the is  T - m a t r i x element ( t r a n s i t i o n o p e r a t o r ) . included  inelastic  That i s , e l a s t i c s c a t t e r i n g  to a l l o r d e r s through the use of d i s t o r t e d waves, and the  t r a n s i t i o n i s treated  to f i r s t  Strongly c o l l e c t i v e excitations,  order  only.  such as the f i r s t  of many even-even n u c l e i , may be viewed as v i b r a t i o n s n u c l e u s as a whole, where the r a d i u s density  i s a function  2  +  and 3  -  or r o t a t i o n s  of angle.  states of the  The n u c l e a r  may then be expanded about a s p h e r i c a l d i s t r i b u t i o n , the  non-spherical part  giving  the c o u p l i n g  between the ground and e x c i t e d  states. More s p e c i f i c a l l y ,  p = p where  a y X  0  one can w r i t e :  + Ap = p  Q  +  I 8 F (r)Y "(n)a x  x  x  X l J  ,  (4.3)  i s a l i n e a r combination o f n u c l e a r e x c i t a t i o n c r e a t i o n and  annihilation operators.  The n u c l e a r o p t i c a l p o t e n t i a l may then be  expressed as: v  n It the  .  v v  (0)  +  n  v  (i) n  i s Ap , t h e deformed p a r t part  m  o f the d e n s i t y ,  of the o p t i c a l p o t e n t i a l g i v i n g  that  contributes  r i s e to nuclear  to  excitation.  - 77 -  For a t r a n s i t i o n  from a ground  s t a t e o f s p i n 0, t o an e x c i t e d  of a n g u l a r momentum J w i t h p r o j e c t i o n M, the T - m a t r i x element  state  i s related  to /  < LM | V  | 00 > f  ( 1 )  (  +  d r .  )  (4.4)  3  The "i"s a r e incoming and o u t g o i n g p i o n d i s t o r t e d waves, c a l c u l a t e d u s i n g the f u l l n u c l e a r o p t i c a l p o t e n t i a l , as d i s c u s s e d i n s e c t i o n 4.1.1. mathematical Appendix  details  relating  Some  t o the e v a l u a t i o n o f (4.4) a r e p r e s e n t e d i n  B.  F o r the e x p a n s i o n of the n u c l e a r d e n s i t y p r e s e n t e d i n ( 4 . 3 ) , < LM | Ap | 00 > = But  < LM | a  so t h a t  < LM | Ap  X  | 00 > =  y  | 00 > =  3 F (r)  I  f3 F Y x  6  L X  L  and  i s d i s c u s s e d f u r t h e r i n the next I t has been mentioned  « _ M  i s called  X  i s their  i n o r d e r to account  (4.5)  +  i n g e n e r a l , and p i o n s  the n e u t r o n s and the protons  Thus, the o r i g i n a l DWPI program has been a l t e r e d  Pn  =  That i s , s t a r t i n g  (P0p  +  P0n>  +  < Pp A  +  A  t r  v  (p_ ^ t r ') p  + ^(p_ t r n) v  y  =  Pn>  g F P P  to the  from t h e e x p a n s i o n >  can r e p e a t t h e arguments presented above, and end up w i t h P_ H v( r J) =  r  section.  f o r the n e u t r o n and p r o t o n c o n t r i b u t i o n s  P " Pp  Ptr^ ^ »  p r e v i o u s l y , t h a t one of the prime m o t i v a t i o n s  transition density separately.  one  .  the t r a n s i t i o n d e n s i t y  s e n s i t i v i t y t o both  p r e s e n t i n the n u c l e u s .  | 00 >  .  f o r p e r f o r m i n g s c a t t e r i n g experiments w i t h hadrons in particular,  y  ,  y  L  radial factor  L  < LM | a  u x  3 F (r)Y°(JJ)  The  L  x  + n 8 F n  - 78 -  4.2.2  Transition  Densities  Transition densities  are  perhaps most commonly encountered i n  c o n t e x t of d i s c u s s i o n s  of e l e c t r o n  the  s e c t i o n measured w i t h i n e l a s t i c  differential  is  (HB  The  general expression f o r electron  scattering  83):  do dfi a^'  cross  scattering.  .  , M  =  i s the Mott c r o s s  section, appropriate  to the  s c a t t e r i n g of  two  point  charged p a r t i c l e s , m u l t i p l i e d by a r e c o i l c o r r e c t i o n f a c t o r .  F 's  are  x  details  n u c l e a r charge and  of n u c l e a r  For  transition  the  i n n u c l e i , the  l o n g i t u d i n a l form f a c t o r  Fourier-Bessel  the  dominant  F C ( q ) , which x  t r a n s f o r m of the  nuclear  can  charge  density: F  This  form f a c t o r s , which account f o r  c o l l e c t i v e excitations  i s from the  e x p r e s s e d as  current  The  structure.  low-lying  contribution be  the  C x  /  (q) =  t r a n s i t i o n density  m a t r i x element o f the  p £(r)  j (qr) r dr  .  2  t  x  can be d e f i n e d  (Hei  83)  charge o p e r a t o r between i n i t i a l  as  the  and  reduced  final  nuclear  states: P *(r)  =  p  I  t  with Thus,  Ptr  =  e  ±  < f  f  5(r-  || r i  )  p  o p  microscopically of the  excited  Y  x  f  ±  overlap  state wavefunctions.  i f these w a v e f u n c t i o n s are s h e l l model f o r example.  ||  >  d r 3  ,  .  i s e s s e n t i a l l y a measure of the  ground s t a t e and  context  Q p  /  between n u c l e a r  I t can  be  calculated  known, or assumed, as i n  the  -  It  should  be p o i n t e d  out,  -  79  that a l t h o u g h t h e t r a n s i t i o n  introduced  i n the p r e c e d i n g  of nuclear  deformation, a m i c r o s c o p i c a l l y c a l c u l a t e d t r a n s i t i o n  parametrized  s e c t i o n with  i n t h e form of eqn.  reference  d e n s i t y was  t o a m a c r o s c o p i c model  ( 4 . 5 ) , c o u l d be used I n the DWPI  c a l c u l a t i o n , w i t h o u t a f f e c t i n g any of t h e d e t a i l s p r e s e n t e d The (BM  75),  factors  historical lies  i n the f a c t t h a t measurements of t h e l o n g i t u d i n a l form  d e n s i t y at the n u c l e a r  (Tas  s t a t e s suggest a s t r o n g  peaking of t h e t r a n s i t i o n  s u r f a c e , as i n t h e c a s e o f shape o s c i l l a t i o n s .  most f r e q u e n t l y used macroscopic model i s t h a t o f T a s s i e  56),  i n which t h e n u c l e u s i s d e s c r i b e d  irrotational fluid,  and g i v e s  6  t  po  as an  incompressible,  the r e s u l t :  p *(r) = where  i n Appendix B.  j u s t i f i c a t i o n f o r the use o f m a c r o s c o p i c models  for c o l l e c t i v e  The  density,  X  r  X  _  1  3p /3r 0  Is t h e ground s t a t e charge d i s t r i b u t i o n .  , Note that f o r a  G a u s s i a n ground s t a t e d e n s i t y d i s t r i b u t i o n , o r a two parameter Fermi Appendix C ) , t h i s form f o r the t r a n s i t i o n d e n s i t y , w i t h i d e n t i c a l t o the one u t i l i z e d r  was  X = 2, I s  i n t h e o r i g i n a l DWPI program:  Ptr( ) = As  (see  6  c  8p/9c  .  the q u a l i t y o f I n 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  improved, i t  found t h a t t h i s model d i d not a d e q u a t e l y reproduce the measured form Thus, f o r many n u c l e i , the parameters of t h e d e n s i t y p, whose  factors.  d e r i v a t i v e i s under c o n s i d e r a t i o n , a r e v a r i e d from t h e i r ground values,  and a d j u s t e d  expression  for p  t  r  to f i t the e x p e r i m e n t a l  presented  data.  state  In t h i s form, the  above i s no l o n g e r a n u c l e a r  model,  - 80 -  but merely  a convenient  p a r a m e t r i z a t i o n s f o r a s u r f a c e peaked shape  (HB 8 3 ) . It  should be mentioned a t t h i s p o i n t , t h a t i n c e r t a i n  models (BM 75), the parameter g can be r e l a t e d of the n u c l e a r shape.  I n l i g h t o f the f a c t  macroscopic  t o an i n t r i n s i c  that c e r t a i n other  deformation parameters  of the models must be v a r i e d i n o r d e r to o b t a i n agreement w i t h e x p e r i m e n t a l d a t a , one may be j u s t i f i e d s c e p t i c i s m as t o the v a l i d i t y of such a As  electron  correspondence.  s c a t t e r i n g measurements were extended  of momentum t r a n s f e r , i t was found from  i n m a i n t a i n i n g a c e r t a i n amount o f  the macroscopic  that even the a n a l y t i c  models was t o o l i m i t i n g  p r a c t i c e now i s t o express  to higher values  (Hei 81).  form  deduced  The common  the t r a n s i t i o n d e n s i t y as a F o u r i e r - B e s s e l  series: P where  X  n  t r  (r) =  i s the n  I  V ^ j ^ X ^ r / R )  e(R-r)  ,  zero o f the s p h e r i c a l B e s s e l f u n c t i o n  and R i s t h e r a d i u s beyond which p  t r  J _^(x) , x  ( r ) i s assumed t o be z e r o .  For the a n a l y s i s of the p i o n s c a t t e r i n g data on  1 8  0 and M g , which 26  were taken over a s m a l l range o f momentum t r a n s f e r , the macroscopic 8c9p/8c was used electron  exclusively.  s c a t t e r i n g data  form  T h i s shape i s c o n s i s t e n t w i t h the a v a i l a b l e  (Nor+ 82) and (Lee+ 74) f o r these  nuclei.  -  4.2.3  -  81  Neutron and P r o t o n M a t r i x Elements The  probability  f o r a nucleus  to undergo an e l e c t r o m a g n e t i c  t r a n s i t i o n of m u l t i p o l a r i t y X i s p r o p o r t i o n a l t o the square  of the m a t r i x  element of the X - m u l t i p o l e t r a n s i t i o n o p e r a t o r , taken between i n i t i a l and f i n a l nuclear states. m a t r i x element'.  T h i s m a t r i x element i s r e f e r e d  to as t h e 'proton  I t i s g i v e n by: Mp =  O f| | 0  \J±> ,  |  X p  Z 0  with  P  * =  I i  r j * Y U(%)  .  X  An e q u i v a l e n t (Hei 81) statement i s :  M P  -  / (Ptr)p  r  X  +  2  d  '  r  In t h e c o n t e x t o f e l e c t r o m a g n e t i c p r o p e r t i e s o f n u c l e i , the p r o t o n t r a n s i t i o n charge elements,  density.  (ptr^)p i s  One can a l s o d i s c u s s p r o t o n m a t r i x  however, i n r e l a t i o n to hadron s c a t t e r i n g .  In t h a t case,  n u c l e a r e x c i t a t i o n s a r e p r i m a r i l y due to the s t r o n g i n t e r a c t i o n , and (p £)p  s h o u l d be t a k e n as the proton t r a n s i t i o n matter  t  density.  This  distinction  i s f r e q u e n t l y not made i n v a r i o u s a n a l y s e s , and may make a  significant  difference.  Extending analogy  this  i d e a a b i t f u r t h e r , one can d e f i n e , i n complete  w i t h the above, the 'neutron m a t r i x ^  -  C  < iM J  element':  M i> J  •  N with  0 * =  or  M  n  n  =  I i  r * Y "(%) ±  / (P *) t  ,  X  n  r  X  +  2  dr  .  -  The  complete e x p r e s s i o n  8 2  -  f o r the e l e c t r o m a g n e t i c  transition  probability i s :  T h i s can be r e l a t e d  t o the t r a n s i t i o n  r a t e (BM 7 5 ) :  +1) F o r a s p e c i f i c n u c l e a r energy l e v e l ,  /io\ * 2  + 1  /l\  the t r a n s i t i o n r a t e i s r e l a t e d t o t h e  l e v e l width and l i f e t i m e by:  y(X)  = r/fi = l / x  m  In a c o l l e c t i v e model, based on n u c l e a r shape o s c i l l a t i o n s , the t r a n s i t i o n p r o b a b i l i t y may a l s o be r e l a t e d deformation  parameter B  x  (BM 75) to the t o t a l  nuclear  through:  R i s an e f f e c t i v e n u c l e a r r a d i u s , whose v a l u e depends on the d e t a i l s o f the model assumed.  Thus, one s h o u l d not expect  t o be a b l e to compare the  v a l u e o f the s c a l i n g parameter 6 (as d e f i n e d through the a n a l y t i c for  the t r a n s i t i o n d e n s i t i e s introduced  the above formula it  i n s e c t i o n 4.2.2) with  forms  the 6 i n  i n a model-independent f a s h i o n ; nor t o be a b l e to r e l a t e  to a p h y s i c a l deformation  of the n u c l e a r d e n s i t y .  - 83 -  4.2.4  S e n s i t i v i t y o f ir* I n e l a s t i c  Isovector  As mentioned i n s e c t i o n 4 . 2 . 1 , altered  t o accommodate the i n p u t  densities  Scattering  the DWPI computer code has been  of s e p a r a t e n e u t r o n and p r o t o n t r a n s i t i o n  i n the c a l c u l a t i o n of low energy i n e l a s t i c p i o n  scattering.  S i n c e such c a l c u l a t i o n s have not been performed elsewhere, i t i s o f some interest  to i n v e s t i g a t e  the r e s u l t s , even w i t h o u t r e f e r e n c e t o  experimental data. Fig.  4.1  illustrates  the d i f f e r e n t i a l  50 MeV ir— s c a t t e r i n g t o t h e 2 ^ in  this section w i l l  c9p /9c Q  p  state of Mg.  calculated for  A l l subsequent  2 6  figures The form  f o r both n e u t r o n and p r o t o n t r a n s i t i o n  i s t h e ground s t a t e d e n s i t y  Q  sections  r e f e r to c a l c u l a t i o n s f o r t h i s same s t a t e .  has been u t i l i z e d  densities.  cross  o f M g , of the Fermi form, 26  w i t h parameters as determined by ( G y l 8 4 ) . The is  g e n e r a l shape of the a n g u l a r d i s t r i b u t i o n s p r e s e n t e d  c h a r a c t e r i s t i c of the angular momentum t r a n s f e r i n the r e a c t i o n .  drop of the TT c r o s s -  negatively and  in Fig.  4.1  The  s e c t i o n a t back a n g l e s i s due to the f a c t t h a t the  charged p i o n  i s a t t r a c t e d by the Coulomb f i e l d  of the n u c l e u s ,  thus the TT i n t e r a c t i o n occurs a t a s l i g h t l y h i g h e r energy than the -  7T+.  The  s u r p r i s i n g feature  between the c r o s s  sections  o f F i g 4.1  i s t h e d i f f e r e n c e i n magnitudes  f o r the two p i o n  charge s t a t e s , even though the  c a l c u l a t i o n has been performed w i t h 8 n = 6 p = 0 . 5 0 , and  so t h a t  the n e u t r o n  p r o t o n t r a n s i t i o n d e n s i t i e s a r e o f s i m i l a r magnitude. F i g . 4.2  sections,  illustrates  t h e r a t i o o f TT t o i r + d i f f e r e n t i a l -  cross  c a l c u l a t e d w i t h s e v e r a l d i f f e r e n t v a l u e s o f 8 n and g , i n p  -  84 -  - 86 -  o r d e r t o determine what e f f e c t the h e i g h t o f the assumed d e n s i t i e s has. 8n=8p=0.65, similar  transition  One can see that f o r both 8 n =8 p =0.35, and  the a n g u l a r d i s t r i b u t i o n s o f the TT t o TT r a t i o s a r e very -  +  t o each o t h e r , and t o that noted i n F i g 4 . 1 .  The r a t i o reaches a  maximum o f a p p r o x i m a t e l y f i v e near 70 degrees i n the c e n t e r o f mass then drops t o u n i t y a t about  135 degrees.  s i m u l t a n e o u s a n a l y s i s o f TT and i r -  +  T h i s I s an i n d i c a t i o n that the  i n e l a s t i c scattering cross sections,  measured under s i m i l a r c o n d i t i o n s , w i l l prove r e l a t i v e l y any u n c e r t a i n t i e s  frame,  insensitive to  i n the a b s o l u t e n o r m a l i z a t i o n s of the d a t a p o i n t s .  One  a l s o notes t h a t i n the a n g u l a r r e g i o n > 80 degrees, a 15% change i n  Bp,  f o r f i x e d B n » causes  the r a t i o t o change by =200%.  i n d i c a t i o n t h a t w i t h the use o f TT* i n e l a s t i c the s e n s i t i v i t y  t o determine  F i g 4.3 i l l u s t r a t e s  s c a t t e r i n g , one may have  the r a t i o 3 n /Bp t o good a c c u r a c y .  the c a l c u l a t e d a n g u l a r d i s t r i b u t i o n s o f the  d i f f e r e n t i a l c r o s s s e c t i o n s themselves, Fig  T h i s i s an  f o r s e v e r a l v a l u e s of the B's. I n  4 . 3 ( a ) , B n i s h e l d c o n s t a n t a t 0.50, w h i l e B p i s assumed t o be  e i t h e r 0.35 o r 0.65. reversed.  I n F i g 4 . 3 ( b ) , the r o l e s o f B n and B p a r e  One notes t h a t i n c r e a s i n g Bn(B p ) w h i l e k e e p i n g  f i x e d produces d e c r e a s e i n the  a l a r g e i n c r e a s e i n the ir+(ir-)  curve.  Tf-(-rr+)  r e s u l t , but a much s m a l l e r  T h i s i s an i n d i c a t i o n t h a t n e g a t i v e  ( p o s i t i v e ) p i o n s are much more s e n s i t i v e t o the neutrons n u c l e i , through  their  Bp(B n )  (protons) i n  s e n s i t i v i t y t o the h e i g h t o f the n e u t r o n  (proton)  t r a n s i t i o n d e n s i t y , than they are to the p r o t o n s ( n e u t r o n s ) . As d i s c u s s e d i n Chapter 1.3, a s i m i l a r s e n s i t i v i t y case o f p i o n - n u c l e o n s c a t t e r i n g .  i s seen i n the  The case o f p i o n - n u c l e u s s c a t t e r i n g i s  s u f f i c i e n t l y more complex, however, that one c o u l d not a p r i o r i  assume  Fig.  4.3  50 MeV n* D i f f e r e n t i a l Cross M g ( 2 j ) f o r S e v e r a l Values  2 6  +  Sections for of 8  - 88 -  that such s e n s i t i v i t y would be p r e s e n t . 4 . 3 i n d i c a t e that  Fig.  In order First,  it is.  to q u a n t i f y  by f o l l o w i n g  The c a l c u l a t i o n s r e p r o d u c e d i n  this sensitivity,  one can proceed as f o l l o w s .  the d e t a i l s of the i n e l a s t i c c a l c u l a t i o n , as p r e s e n t e d  i n Appendix B, one notes that  i n the o r i g i n a l DWPI code, the parameter B  appeared as a m u l t i p l i c a t i v e f a c t o r i n every term c o n t r i b u t i n g t o the T - m a t r i x . Thus t h e d i f f e r e n t i a l c r o s s to 6 .  s e c t i o n s were d i r e c t l y  proportional  I n t h e a l t e r e d code, by i n t r o d u c i n g  2  Ptr = OpFp + M n ) » one  da/dn  would expect  where f and g r e p r e s e n t  = (r3 Zf(6) + 8 N g ( 8 ) ) p  n  2  ,  the a n g u l a r dependencies o f t h e p r o t o n and n e u t r o n  s c a t t e r i n g a m p l i t u d e s , which a r e not i d e n t i c a l . One  can now i n t e g r a t e the d i f f e r e n t i a l  t h i s a n g u l a r dependence, and w r i t e for  s c a t t e r i n g to a s p e c i f i c a+  and  similarly  tivity  f o r TT~.  b  cross  the t o t a l TT i n e l a s t i c +  p  +  to e l i m i n a t e  cross  section  s t a t e as  « (BpZb + + B N b + ) D  sections,  and b  n  n  provide  + n  2  (4.6)  ,  an i n d i c a t i o n of the s e n s i -  of t h e Tr s t o p r o t o n s and n e u t r o n s r e s p e c t i v e l y . +I  F i g . 4 . 4 ( a ) i s a p l o t o f \/a DWPI code, with 3 w i t h Bp h e l d  n  held  constant.  +  v s B , as c a l c u l a t e d w i t h p  constant.  F i g . 4.4(b) i l l u s t r a t e s  The s l o p e s  o f the c u r v e s i n F i g . 4 . 4 ( a ) and  4 . 4 ( b ) (note t h e d i f f e r e n t s c a l e s ) respectively.  I t i s evident  with increasing B  n  are p r o p o r t i o n a l t o b p  that b  p  i n d i c a t e s that b  n  +  +  >>b <0.  + n  .  interchanged.  Also,  / c r vs B , +  n  +  and b  + n  the d e c r e a s e of a  S i m i l a r c u r v e s and c o n c l u -  s i o n s may be drawn f o r the case o f ir~, w i t h neutrons  the a l t e r e d  the r o l e s o f p r o t o n s and  +  - 90 -  The  curves  i n F i g 4.4(a) f o r 8 <0.3, and i n 4.4(b) f o r a l l v a l u e s p  of 8 , a r e not s t r a i g h t  lines, indicating  n  i n w r i t i n g eqn (4.6) a r e not e x a c t .  t h a t the a p p r o x i m a t i o n s  made  In any case, because (4.6) i s o n l y a  p r o p o r t i o n a l i t y one c o u l d not use i t to o b t a i n n u m e r i c a l v a l u e s f o r the b ' s . One can however combine the r e l a t i o n s  Letting  b  + p  =b  and b = b ~ , one +  n  n  In F i g 4.5 c u r v e s  p  resulting  f o r TT and TT to o b t a i n : +  -  can r e w r i t e t h i s a s :  from the use o f t h i s r e l a t i o n f o r v a r i o u s  v a l u e s o f b / b p ~ , a l o n g with a shaded r e g i o n which r e p r e s e n t i n g t h e -  n  r e s u l t s o f the a l t e r e d that  DWPI c a l c u l a t i o n have been p l o t t e d .  b ~/b ~=bp- 7b +=-20. ,  n  p  r  T h i s i s a new r e s u l t . other nuclear probes.  One can compare i t t o v a l u e s o f b / b f o r n  electron scattering;  1.0 f o r a's;  above.  They a r e :  energy  pions i s c l e a r l y  ( i . e . =180 MeV)  0. f o r  0.83 and 0.95 f o r 800 MeV and 1 GeV  p r o t o n s c a t t e r i n g r e s p e c t i v e l y ; 3 (1/3) f o r low energy resonance  p  These v a l u e s have been computed by (BBM 8 1 ) , u s i n g  a f o r m u l a t i o n s i m i l a r t o the one p r e s e n t e d  and  I t i s evident  Tf~(Tr+).  protons  (neutrons)  The v a l u e f o r low energy  unique.  I t may be shown t h a t the p i o n s e n s i t i v i t y does n o t g r e a t l y depend on the d e t a i l s o f the o p t i c a l p o t e n t i a l used c o n s i d e r a t i o n s would n o t be m e a n i n g f u l , made to e x p e r i m e n t a l Chapter  V.  data.  i n the c a l c u l a t i o n s .  however, u n t i l  Such  some r e f e r e n c e i s  Thus f u r t h e r d i s c u s s i o n i s postponed  to  F i g . 4.5  a +/a~  vs B /B p  n  f o r 50 MeV it*  Scattering  to  2 6  Mg(2  + 1  )  - 92 -  CHAPTER V RESULTS AND CONCLUSIONS  In t h i s Chapter  chapter,  IV a r e  the o p t i c a l p o t e n t i a l c a l c u l a t i o n s d i s c u s s e d  a p p l i e d t o the measured d i f f e r e n t i a l c r o s s  S e c t i o n 5.1 d e a l s with e l a s t i c 1 8  0 and  2 6  Mg  scattering.  elastic  cross  data has been taken both at LAMPF, and at experiments. Master's  LAMPF r e s u l t s  thesis.  ongoing program to use  on d e t e r m i n i n g a b s o l u t e  used In the p r e s e n t  scattering  from a d j a c e n t  p a r t o f an  to determine  (e.g.  analysis  are  sections  are  has been p l a c e d  experiments w i t h the QQD  at  a r e g i v e n i n T a b l e 5.1.  fit  scattering.  is  The measured  by v a r y i n g the magnitudes o f  to the o p t i c a l model c a l c u l a t i o n s . It  distributions  those determined w i t h TT - s c a t t e r i n g  n e u t r o n and p r o t o n t r a n s i t i o n d e n s i t i e s  i n s e c t i o n 5.2.1.  ^»l^O,  are used i n  Note t h a t the n e u t r o n d e n s i t y  The d e n s i t y parameters  cross  isotopes  of c r o s s s e c t i o n s  normalizations, u n t i l  S e c t i o n 5.2 d e a l s w i t h i n e l a s t i c  presented  present  o n l y i n an u n p u b l i s h e d  however, r e l a t i v e l y l i t t l e emphasis  s p e c t r o m e t e r were u n d e r t a k e n .  input  to the  P a r t of the s t r e n g t h of t h i s program l i e s  S i n c e only the r a t i o s  t y p e of a n a l y s i s ,  differential  scattering  t h a t u n c e r t a i n t i e s due to the p i o n - n u c l e u s i n t e r a c t i o n can be  2 4 , 2 6 ^ ^ 32,34,36s) m  TRIUMF.  although  TRIUMF, p r i o r  low energy IT - e l a s t i c  e l i m i n a t e d by comparing s c a t t e r i n g  this  T h e r e are no  (Joh+ 79), ( G y l 84) were taken as  nuclear neutron d i s t r i b u t i o n s . i n the f a c t  sections,  (Daw 81) are a v a i l a b l e  TRIUMF data  sections.  50 MeV i r + s c a t t e r i n g on  t a r g e t s has not been measured p r e v i o u s l y .  p u b l i s h e d 50 MeV T T - a b s o l u t e  in  (i.e.  8  n  and 8p) , which  The r e s u l t s  found t h a t  of  the are  these f i t s  the r a t i o 8 / B n  p  are  - 93 -  Neutron  Proton  Charge  Nucleus  Density  c  12c  Gauss  1.687  1.067  1.595  1.067  1.595  1.067  18  Gauss  1.881  1.544  1.754  1.540  1.900  1.540  Fermi  3.050  .5234  2.896  .5238  2.967  .5491  Fermi  3.262  .5407  3.153  .5407  3.153  0  2 6  32  Table  M g  S  5.1  t or a  c  t or a  c  t or a  .5407  D e n s i t y D i s t r i b u t i o n Parameters Used i n the Present Analysis.  See Appendix C f o r F u r t h e r  Details.  - 94 -  e x t r a c t e d i n t h i s way used.  A more complete  i s independent o f the o p t i c a l p o t e n t i a l  d i s c u s s i o n of the model dependence of the p r e s e n t  r e s u l t s i s g i v e n i n s e c t i o n 5.2.2. M /Mp deduced n  parameters  A comparison  o f the v a l u e s o f  from the p r e s e n t r e s u l t s w i t h t h o s e from o t h e r  experiments i s made i n s e c t i o n 5.2.3.  - 95 -  5.1  Elastic The  from  1 2  Scattering  measured d i f f e r e n t i a l c r o s s  C and  scattering  1 8  sections  0 a r e p l o t t e d i n F i g . 5.1, and those f o r TT* e l a s t i c  from  Mg  2 6  and  3 2  S i n F i g . 5.2.  Note t h a t t h e S d a t a were 3 2  t a k e n w i t h t h e QQD s p e c t r o m e t e r , and a n a l y z e d are  included  f o r TT* e l a s t i c s c a t t e r i n g  here because of t h e i r r e l e v e n c e  by Sobie  (Sob 84a).  to the d i s c u s s i o n .  They The s o l i d  c u r v e s i n both f i g u r e s a r e based on t h e o p t i c a l model c a l c u l a t i o n s described evident  i n Chapter IV, with the Set E parameters o f (CMS 8 2 ) .  that  f o r C and 1 2  1 8  0 , the c a l c u l a t i o n s p r o v i d e  d e s c i p t i o n of both i r and TT c r o s s +  ir  +  E parameters i n c l u d e d  s c a t t e r i n g data,  and that  a reasonable  s e c t i o n s , w h i l e f o r M g and 26  d a t a appears to be reproduced much b e t t e r I t has been p o i n t e d  Set  -  out e a r l i e r ,  TT  -  cross The  S , the  -  that p a r t o f the d e r i v a t i o n o f t h e  a g l o b a l f i t t o a l l e x i s t i n g 50 MeV e l a s t i c that data c o n s i s t e d  s o l e l y o f it +  results. Before  -  possibility  3 2  than t h e T T .  some problem w i t h f i t t i n g i r r e s u l t s may be a n t i c i p a t e d . this question  It i s  Thus  pursuing  f u r t h e r , however, i t may be worthwhile t o c o n s i d e r the  that  sections  t h e d i f f e r e n c e i n the q u a l i t y o f the f i t s  to the T T  +  and  i s due to some e f f e c t of the Coulomb i n t e r a c t i o n .  dashed c u r v e s i n F i g u r e s  5.1 and 5.2 a r e t h e r e s u l t o f  c a l c u l a t i o n s i n which the Coulomb n u c l e a r  cross  terms have not been  dropped from the K l e i n - G o r d o n e q u a t i o n ( s e e p69).  I t i s evident  that  agreement w i t h the d a t a i s not improved. C a l c u l a t i o n s have been performed ( B a r 84) i n which  relativistic  Coulomb wave f u c t i o n s a r e used i n s t e a d o f the n o n r e l a t i v i s t i c  ones  included  from the  w i t h t h e DWPI code.  ones p r e s e n t e d i n F i g u r e s  These do not d i f f e r  5.1 and 5.2.  appreciably  Fig.  5.1  C and 0 E l a s t i c C r o s s S e c t i o n s C a l c u l a t i o n with Set E Parameters  1 2  1 8  F i g . 5.2  M g and S E l a s t i c Cross S e c t i o n s C a l c u l a t i o n with Set E Parameters  2 6  3 2  - 98  R e c e n t l y measured energy atoms have been found and  Tauscher  (ET 82)  -  s h i f t s and  l e v e l widths  i n high Z pionic  to be s e v e r a l times s m a l l e r than e x p e c t e d . suggest  that a l i k e l y  e x p l a n a t i o n a r i s e s from  c o n s i d e r a t i o n of the gauge i n v a r i a n c e of the e l e c t r o m a g n e t i c This generates  Ericson  an a d d i t i o n a l e l e c t r o m a g n e t i c p o t e n t i a l from  a  interaction. the  energy  dependence of the ir-nucleus s t r o n g i n t e r a c t i o n p o t e n t i a l , g i v e n by 4ir with  a = -0.044 T%~^«  a (1+u/M) p ( r ) V ( r ) c  C a l c u l a t i o n s performed  , w i t h the i n c l u s i o n of  such a term i n the p r e s e n t o p t i c a l p o t e n t i a l a r e v i r t u a l l y the curves p r e s e n t e d performed  i n f i g u r e s 5.1  and  i n which the v a l u e of a was  5.2.  and  to  C a l c u l a t i o n s have a l s o been  arbitrarily varied.  d i f f e r e n c e s i n the c a l c u l a t i o n s were not o b t a i n e d u n t i l i n c r e a s e d by a f a c t o r of « 20,  identical  then the change was  Significant a  had  been  not such as to  improve the f i t to the measured e l a s t i c c r o s s s e c t i o n s . I t i s p o s s i b l e t h a t the r e l a t i v e l y poor f i t to the d a t a i s due  to some as yet u n c o n s i d e r e d  present o p t i c a l p o t e n t i a l .  L a c k i n g any  2 6  M g and  than adopt  a complicated  c o e f f i c i e n t s to v a r y , i t was  d e c i d e d t o v a r y each of them, except  coefficient i s g i v e n by  i n Table 5.2,  r e q u i r e d to minimize  x /v, 2  -  themselves.  scheme f o r c h o o s i n g which combination  the u n v a r i e d ones f i x e d a t t h e i r Set E v a l u e s . are p r e s e n t e d  TT  t h e o r e t i c a l i n p u t , however, one i s  i s o v e c t o r parameters bj and c^, one at a time, f o r each n u c l e u s ,  procedure  S  e f f e c t , not i n c l u d e d i n the  f o r c e d to c o n s i d e r v a r i a t i o n s of the p o t e n t i a l parameters Rather  3 2  The  results  of  the leaving  of t h i s  which g i v e s the v a l u e of each and  the minimum x 2 / v  achieved,  x2  - 99 -  Best F i t Values f o r Parameters  Set E  12 c  180  26  -.060  s  3 2  M g  -.061  -.056  -.062  Q  .006  .005  .001  .019  .025  Re c  Q  .700  .672  .684  .601  .537  Im c  Q  .028  .054  .028  .158  .173  Re B  Q  -.020  .019  -.035  .020  .070  Im B  Q  .110  .098  .066  .174  .285  Re C  2  .360  .176  .299  -.330  Im C  2  .540  .647  .440  1.703  1.720  1.40  1.546  1.460  1.973  2.225  Re b  Q  Im b  . X  Best F i t Value  12  R e  Table 5 . 2  b  o  C  18  0  -.058  -.638  for x 2  26  32  M g  S  0.80  5.5  17.1  35.4  11.7  18.8  Im b  Q  1.13  4.2  Re c  Q  0.76  5.2  2.4  6.0  Im c  Q  1.02  5.5  2.4  2.0  Re B  Q  0.81  5.4  16.5  32.5  Im B  0  1.11  5.4  16.4  23.8  Re C  2  0.88  5.4  2.6  5.0  Im C  2  1.10  5.4  2.5  3.9  X  0.88  5.3  2.4  8.3  SET E  1.14  5.5  Best F i t Values  37.4  17.1  f o r O p t i c a l P o t e n t i a l Parameters,  One a t a Time, and R e s u l t i n g Minimum x 2 /  v  Values.  Varied Unvaried  C o e f f i c i e n t s Are L e f t F i x e d a t t h e i r Set E V a l u e s .  - 100 -  /dg/dncalc - da/d^meas\2 l>\  A da/dfi  /do/dacalc - dq/dft ^ \  + m  e  a  s  and v i s the number o f degrees  A da/dj^eag  m e a s  \  ( .!)  2  5  J TT  o f freedom.  I t i s e v i d e n t from T a b l e 5.2 t h a t the q u a l i t y o f t h e f i t s and  1 8  0 d a t a i s not s i g n i f i c a n t l y  p o t e n t i a l parameter.  to t h e  1 2  C  improved by v a r y i n g any one o p t i c a l  F o r the cases o f M g 2 6  and  3 2  S however,  significant  improvement can be o b t a i n e d by d e c r e a s i n g the s t r e n g t h s of the p-wave parameters Re c particular,  Q  o r Re C , o r i n c r e a s i n g t h e v a l u e s of Im c 2  i t appears  t h a t the c h o i c e of Im c  w i l l y i e l d a p o t e n t i a l which d e s c r i b e s scattering  simultaneously f a i r l y w e l l .  2 6  Mg  Q  and  3 2  o r Im C •  Q  2  = 0.165, o r Im C  2  In  = 1.711  S TT* e l a s t i c  The r e s u l t s o f such  calculations  a r e shown i n F i g . 5.3. T a b l e 5.3 p r e s e n t s the t o t a l r e a c t i o n c r o s s s e c t i o n s c a l c u l a t e d using is  some o f t h e parameter v a l u e s from T a b l e 5.2.  the increase of  v a l u e s of Im c (Fri  Q  f o r M g and 2 6  and Im C  2  3 2  S corresponding  mentioned above.  Of p a r t i c u l a r  interest  t o the i n c r e a s e d  I t has been p o i n t e d out by  83) t h a t i n c l u s i o n of measurements o f aj w i t h e l a s t i c  scattering  d a t a p r o v i d e s a n o t i c e a b l e improvement i n the a c c u r a c y of t h e model-independently  determined  considerable interest  potentials.  Such measurements would be of  i n the present case as w e l l .  Fig.  5.3  M g and S E l a s t i c Cross S e c t i o n s C a l c u l a t i o n w i t h 'Best F i t ' Parameters 26  3 2  - 102  Nucleus  Parameters  -  o*£ T T  C  -.056  120  162  b  0  Re  c  Q  .672  114  152  Re  B  Q  .019  120  162  Re  C  2  .176  114  151  1.546  108  144  169  246  Im  b  Q  .001  148  227  Im  B  Q  .066  155  234  242  383  Set E  2 6  M g  Re  c  Q  .601  220  341  Im  c  Q  .158  362  516  Re  C  2  -.330  220  340  Im  C  2  1.703  338  485  1.973  194  296  275  498  X  Set E  32  T a b l e 5.3  S  TT  156  Set E 0  aj  Re  X  18  (mb)  116  Set E  12  +  Re  c  Q  .537  239  412  Im  c  Q  .173  417  655  Re  C  2  -.638  242  417  Im  C  2  1.720  383  610  T o t a l R e a c t i o n Cross S e c t i o n s as C a l c u l a t e d Some of the Best F i t Values  f o r the  -  (mb)  Using  Coefficients.  -  5.2  I n e l a s t i c Scattering  5.2.1  Results The  outlined  c a l c u l a t i o n of i n e l a s t i c s c a t t e r i n g i n s e c t i o n 4.2.  8 c3p /3c = -8 r3p /3r n  n  n  n  proton t r a n s i t i o n and 8  TT  -  c r o s s s e c t i o n s has been  The present a n a l y s i s uses the m a c r o s c o p i c p  densities respectively.  p  n  and Pp a r e t h e n e u t r o n  p r o t o n ground s t a t e d e n s i t i e s (see T a b l e 5.1).  inelastic  The v a l u e s of 8 and n  i n order to o b t a i n a best f i t t o t h e TT and +  s c a t t e r i n g data s i m u l t a n e o u s l y .  t e r s were h e l d c o n s t a n t a t the v a l u e s used The b e s t f i t was taken  forms  and 8 c3pp/3c = - 8pr3pp/3r f o r the n e u t r o n and  were v a r i e d i n d e p e n d e n t l y  p  103 ~  to describe e l a s t i c  to be t h a t r e s u l t i n g  w i t h x /v d e f i n e d as f o r t h e f i t s 2  A l l o t h e r p o t e n t i a l paramescattering.  i n a minimum v a l u e o f x / » 2  to the e l a s t i c  scattering  v  (see eqn.  5.1). Fig.  5.4 i l l u s t r a t e s a t y p i c a l x  s t a t e of M g . 2 6  Each contour  2  2  c o n t o u r , whose b o u n d a r i e s  n  plot  f o r a f i t t o the 2 j  i n t h e middle  o f the centermost  p r o v i d e an i n d i c a t i o n o f the u n c e r t a i n t y a s s o c i -  a t e d w i t h the p r e s e n t r e s u l t s . stant values of 8 /3p«  contour  r e p r e s e n t s an i n c r e a s e o f 1 i n the v a l u e o f  The minimum v a l u e o f x A* l i e s  X /v.  2  The s t r a i g h t  I t w i l l be noted  ted i n the d i r e c t i o n o f c o n s t a n t 8 /3p« n  the r a t i o 8 / 6 p i s determined n  solid  l i n e s l i e along  t h a t the contours  con-  are elonga-  T h i s i s an i n d i c a t i o n t h a t  to a b e t t e r a c c u r a c y than t h e i n d i v i d u a l  v a l u e s of the 8's. T a b l e 5.4 g i v e s t h e b e s t f i t v a l u e s o f 8n» 3p» and B /3p f o r n  1 2  C,  1 8  0 , and M g 2 5  determined  as d e s c r i b e d above.  The measured  differen-  t i a l c r o s s s e c t i o n s , and the r e s u l t s of c a l c u l a t i o n s u s i n g t h e best f i t v a l u e s of g  n  and 8  P  are plotted  i n F i g u r e s 5.5, 5.6 and 5.7 f o r these  +  -  F i g . 5.4  104  -  Typical x P l o t f o r F i t of 8 and t o M g I n e l a s t i c Cross Sections 2  n  2 6  B  p  -  3  18  26  0  M g  105 -  M /M n  n  p  0.522  0.540  0.97 ± 0.08  0.97 ± 0.08  0.478  0.406  1.18 ± 0.10  1.81 ± 0.15  0.465  0.655  0.72 ± 0.06  0.90 ± 0.07  T a b l e 5.4  V a l u e s o f 8 , g , 8 / 8 , and n  M /M n  p  p  a s Determined Inelastic  n  p  by F i t t i n g  Scattering  Data.  -  106  -  - 107  Fig.  5.6  1 8  -  0(IT,TT ) ,  1 8  0*(2  + 1  ) I n e l a s t i c Cross  Sections  -  108  -  -  three  109 -  nuclei. T a b l e 5.4 a l s o l i s t s  the values o f t h e r a t i o o f n e u t r o n  m a t r i x elements (M /M ) f o r the 2 n  +  p  to proton  s t a t e s under c o n s i d e r a t i o n . As  e x p l a i n e d i n s e c t i o n 4.2.3, these a r e g i v e n by n  L  For  n  r > t  /  r l  (Ptr)p  the case o f C , w i t h N = Z, and p For  1 8  dr  *  1 2  expected. M /M  / (Ptr)n  n  d r  = p , M /M p  n  p  = B /B n  p  0 and M g , due t o d i f f e r e n c e s i n the o t h e r 2 6  factors,  i s o n l y p r o p o r t i o n a l to B /Bp.  p  n  5.2.2  Model Dependence Note t h a t t h e c a l c u l a t i o n s used  5.5, 5.6 and 5.7 a l l used  t o generate  t h e curves i n F i g u r e s  the Set E o p t i c a l p o t e n t i a l parameters.  been shown i n s e c t i o n 5.1, t h a t c a l c u l a t i o n s w i t h d i f f e r e n t describe  the TT* - M g e l a s t i c c r o s s s e c t i o n s b e t t e r . 2 6  in this section, dent  - 1. , as  t h a t the best f i t v a l u e f o r the r a t i o  o f the o p t i c a l p o t e n t i a l parameters used F i g . 5.8 shows the best f i t v a l u e s o f B  determined  parameters  I t w i l l be shown B /8p i s indepenn  t o perform  the c a l c u l a t i o n .  and Bp f o r M g , as 26  n  by making use of the method o u t l i n e d above, but u s i n g  v a l u e s o f t h e o p t i c a l p o t e n t i a l parameter Re c tions.  I t c a n be seen  I t has  Q  i n performing  different  the c a l c u l a -  that the i n d i v i d u a l best f i t v a l u e s o f B and n  Bp d i f f e r w i t h v a r i a t i o n s i n Re c , but i n such a way t h a t t h e r a t i o Q  B /8p remains a p p r o x i m a t e l y n  constant.  S i m i l a r graphs c a n be produced  f o r v a r i a t i o n s o f the other o p t i c a l p o t e n t i a l F i g . 5.9 shows a x  2  contour  parameters.  p l o t , s i m i l a r t o F i g . 5.4, b u t i n which  - 110 -  F i g . 5.8  V a r i a t i o n of  'Best F i t ' 6's  w i t h Changes i n Recj,  Fig.  5.9  X Contour P l o t U s i n g 'Best F i t ' O p t i c a l P o t e n t i a l Parameters 2  -  o n l y the centermost  112  -  contours a r e reproduced,  p o t e n t i a l parameters which gave good f i t s cross s e c t i o n s . determined  f o r the f o u r s e t s o f o p t i c a l  t o the M g e l a s t i c  scattering  26  I t i s e v i d e n t t h a t the b e s t  f i t v a l u e s o f B /8p, as n  w i t h c a l c u l a t i o n s u s i n g any o f these parameter s e t s , a r e  c o n s i s t e n t w i t h each other,and the Set E  with the r e s u l t s o f the c a l c u l a t i o n s u s i n g  parameters.  T h i s i s not meant t o imply t h a t the p r e s e n t f r e e of any model dependence. e f f e c t o f the d e t a i l s  But i t does seem t o i n d i c a t e t h a t the  of the pion n u c l e u s  i n t e r a c t i o n on the deduced  v a l u e s o f B /Bp> and thus M /Mp, a r e minimal n  results are entirely  n  when the s c a t t e r i n g  of i r + and i r - i s c o n s i d e r e d s i m u l t a n e o u s l y , as has been done a t p r e s e n t . It  s h o u l d be p o i n t e d out t h a t the assumption  present a n a l y s i s , be a d e q u a t e l y worthwhile for  t h a t the shapes of the neutron  d e s c r i b e d by the macroscopic  has been made i n the  t r a n s i t i o n d e n s i t i e s can  form c3p/8c.  a t some p o i n t to i n v e s t i g a t e what e f f e c t  t h e shape would have on the c a l c u l a t i o n s ,  I t may be  different  assumptions  or the deduced v a l u e s o f  M /M . n  p  A l s o , the a n a l y s i s o f p r o t o n s c a t t e r i n g data on M g 26  (Alo+ 81) has demonstrated the importance There i s reason  verified.  channel  effects.  t o b e l i e v e (DeT 84) t h a t such e f f e c t s would n o t a l t e r t h e  v a l u e of M /Mp deduced from n  o f coupled  (Bla+ 8 2 ) ,  the present d a t a , but t h i s  s h o u l d be  -  113  -  Comparison w i t h Other R e s u l t s  5.2.3  A summary o f the most r e c e n t l y determined the 2 ^ s t a t e s o f  1 8  0 and M g 2 6  v a l u e s f o r M /Mp, f o r n  i s g i v e n i n T a b l e 5.5, and the v a l u e s  p l o t t e d i n F i g . 5.10.  I t can be seen t h a t t h e p r e s e n t r e s u l t s a r e  c o n s i s t e n t w i t h almost  a l l the o t h e r s .  T h i s statement  may be s l i g h t l y  m i s l e a d i n g , however, s i n c e a l l o f t h e o t h e r v a l u e s a r e not i n agreement w i t h each o t h e r . f o r both  1 8  In p a r t i c u l a r ,  0 and ^Mg. 2  t h e m i r r o r n u c l e u s v a l u e seems t o be h i g h  The p r e s e n t r e s u l t has a l r e a d y been d i s c u s s e d i n  the p r e c e d i n g s e c t i o n s .  Perhaps i t would be a p p r o p r i a t e a t t h i s p o i n t t o  make a few comments on t h e o t h e r The  a n a l y s i s o f resonance  manner s i m i l a r is  determined  derived  It how  results. energy  p i o n s c a t t e r i n g i s not done i n a  t o t h e present a n a l y s i s .  Rather,  the r a t i o  n  from  i s not immediately  obvious how model dependent t h i s procedure  i t incorporates the d i f f e r e n t  scattering the 2  densities.  +  of 800 MeV p r o t o n  from **Mg and M g y i e l d s a v a l u e of 0.74 ± 0.12 f o r Mn/Mp 2  + X  i s , and  r a d i a l dependences o f t h e n e u t r o n and  (Bla+ 82) p o i n t out t h a t w h i l e t h e i r a n a l y s i s  the 2 ,  +  by i n t e g r a t i n g t h e l a b o r a t o r y c r o s s s e c t i o n s ; and M /Mp  proton t r a n s i t i o n  for  O"(TT~)/c(Tr )  2 6  s t a t e o f M g , i t a l s o r e s u l t s i n a v a l u e of 0.81 ± 0.11 f o r 26  s t a t e of t h e N = Z nucleus *Mg. 2l  -  114  -  18  Present  Result  Resonance Energy  Or,*')  26  0  1.81  ±  0.15  1.67  ±  0.15  a  1.58  ±  0.13  b  1.86  ±  0.16  b  1.58  ±  0.15  c  M g  0.90  ± 0.07  :  0.62  ±  0.14  d  800 MeV  (p,p')  0.74  ±  0.12  e  24 MeV  (p,p')  0.69  ±  0.17  f  24 MeV  (n,n')  1.02  ± 0.058  0.80  ±  0.17  1.03  ±  0.09  (<x,a')  1.5  h  M i r r o r Nucleus *-  0.50  2.07' ±  1  TheoryJ  T a b l e 5.5  ±  0.22  1.64  Comparison  0.83  of P r e s e n t l y Deduced V a l u e s o f  M /Mp w i t h those from o t h e r p r o b e s . n  is  Data  from a) (Ive+ 78), b) (Ive+ 79), c) (Lun+ 78),  d) (Wie+ 8 0 ) , e) (Bla+ 8 2 ) , f ) (Alo+ 8 1 ) , g) ( T a i + 83), h) (BBM j ) (Bro+ 8 2 ) .  79), i ) (Ale+ 8 2 ) ,  - 115 -  1  r—  1  I  i  i  —r  i .  present result  164-230 (TT.TT')  —  —  MeV  •  i  1  i  J  r.  1  —  i— 1,2  i  i  -#-  —w— • — j -  J  .4  5.10  *  i  ).±  2.0  i  1—  2.4  0 i  i  i  I  8 0 0 MeV  j  24 MeV  j  (a,a')  I  (  •  mirror  i  (7r,7r')  (p,p') (p,p')  (n,n') nucleus  theory  I  i l l i  MeV  24 MeV  i  i  i  1.2 M „ / M p for  Fig.  1 8  164  -U-  .8  J '.. 1.6  i  |  j  •  0  —  present result  — •  i  1  •  .8  i  9 ,—  nucleus  i  1  '  . i  L  M n / M p for -  i •  !  —  •- —  theory  .4  —  i  • — !  mirror  i  •  i  • i  (<*,<*')  0  i  1.6 2 6  i  i 2.0  Mg  Comparison o f Mn/Mp as Obtained U s i n g D i f f e r e n t Probes  2.4  -  It close  should be p o i n t e d  to the m i r r o r  inspection  116 -  out t h a t  the o n l y  r e s u l t which appears t o come  n u c l e u s v a l u e i s that o f ( T a i + 8 3 ) .  of the a n a l y s i s , however, r e v e a l s  m e r e l y a r e f l e c t i o n of the EM v a l u e i n p u t  that  A close  the r e s u l t quoted i s  i n t o the a n a l y s i s  i n the f i r s t  place. As that M  n  f o r the m i r r o r  nucleus r e s u l t i t s e l f ,  f o r a given nuclear state  the m i r r o r  nucleus.  M (  1 8  M (  1 8  p  s e c t i o n 4.2.3). that mirror  M (  1 8  n  i s e q u a l t o Mp f o r t h e same s t a t e i n  F o r example, n  Mp f o r a g i v e n s t a t e  i t i s based on the idea  0) _ M ( Ne) 1 8  p  0)  Mp(*°-0)  =  i s derived  *  from a measurement of i t s l i f e t i m e ( s e e  S h e l l model c a l c u l a t i o n s  0 ) = (0.89) M ( N e ) 1 8  p  (Bro+ 82),  however,  , due to Coulomb i n t e r a c t i o n s .  n u c l e u s r e s u l t quoted i n Table 5.5 ( A l e + 82) a l r e a d y  such a Coulomb c o r r e c t i o n . r e l i e d upon to g i v e v a l u e f o r M /M n  p  indicate The  incorporates  U n f o r t u n a t e l y , t h e same c a l c u l a t i o n which i s  the magnitude of the Coulomb c o r r e c t i o n p r e d i c t s a  which i s i n c o n s i s t e n t  w i t h the m i r r o r  nucleus r e s u l t .  - 117 -  REFERENCES  Ale+ 82  T.K. A l e x a n d e r , G.C. B a l l , W.G. D a v i e s , J.S. F o r s t e r , I.V. 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L e t t . 97B (80) 37.  - 123  -  APPENDIX A DETAILS OF ELASTIC CALCULATION  Differential  cross  s e c t i o n s are c a l c u l a t e d w i t h the computer  DWPI i n the f o l l o w i n g way: (E Letting  2  /^-V  2  one s t a r t s w i t h the K l e i n - G o r d o n e q u a t i o n  + V"! - 2 E V - 2 E V - p ) ? = m>? 2  C  p •»• i h V , and  m  =^k  + V  2  Assuming a p a r t i c u l a r form f o r V , n  Cju" + C u' 2  where the primes integrated point  - 2E(V +V )^u  2  C  numerically  one can r e w r i t e  The d i f f e r e n t i a l  cross  n  (A.2)  as (A.3)  3  to r .  (A.3) i s then radius  at which  the Coulomb wave  i s found, f o r each p a r t i a l wave.  section i s calculated  from  2  n  i s t h e Coulomb a m p l i t u d e , and f  n  = (1/ik) I (2i+l) exp(2ia ) f £  where o"£ i s the p o i n t  The  (A.2)  m  to a predetermined matching  c  wave, and  Y  = C u  da/da = | f ( 9 ) + f ( 9 ) |  c  j i  s h i f t 6JJ, between the f u n c t i o n u ( r ) and  f u n c t i o n c a l c u l a t e d without V  where f  n  denote d e r i v a t i v e s w i t h r e s p e c t  outward  the phase  ( A . l ) becomes  m  Y  (A.l)  2  n  Y •*• U£<r) Y ( Q ) ,  + £U+1)_^U£  code  f  £  total  =  charge Coulomb phase  £  P (cos9) £  ,  s h i f t of t h e l  t  n  (l/2)(exp(2iS£)-l). r e a c t i o n cross  ^ r e a c t i o n = <*/k ) I 2  section i s given ( *+D 2  C1"  by  |exp(2i6 ) | £  2  ) .  partial  - 124 -  In  the o r i g i n a l DWPI, the n u c l e a r o p t i c a l p o t e n t i a l 2EV  The  resulting  was taken t o be  = A p + A V » p V + A V p + A^V^p. 2  n  x  2  coefficients  3  i n (A.3) a r e :  - 1  C  :  = A p  C  2  = A p'  C  3  - A p' + k  2  2  - 2EV  2  2  + V, - A V p 2  C  2  3  - Aj_p - A ^ p .  r F o r the p o t e n t i a l b e g i n n i n g of Chapter  C, = A«-p  2  1  C  *  C  3  +  2  A  + V  2 c  - 2EV  5  P  P  L  lv  £(£+1) r-2  + (2A pp' + 5  ' (l+(XB/3)) y r B  B = AgP  and  B' = 2 A p p ' + AgP ' + 6  X  1  2  + A2P + A ^ S p  where  2  are g i v e n by:  '  - A p - A <5p - A ^ p  C  g i v e n a t the  - 1  t  +  2  analysis,  IV, the c o e f f i c i e n t s i n (A.3)  (l (XB/3))2 = k  i n the present  — —(l+(XB/3))  +  5  =  used  A (6p)* 2 V  2  - V (A p+A «p+A p ) 2  2  3  3  7  -  As d i s c u s s e d  125  i n the t e x t ,  -  some c a l c u l a t i o n s  which the Coulomb-nuclear c r o s s terms in  the K l e i n - G o r d o n e q u a t i o n .  following  2  C  3  _(l+(XB/3))  B  -  = k  A j 5 p 2  n  have been r e t a i n e d  c  in  2  C  +  2  A  [(l+(XB/3)) B'  [(l+(XB/3))  and  B' = 2 A p p ' + A j p ' +  I n c l u d i n g the  'Ericson  cases i s simple, i n v o l v i n g C  g  2  lv  + A p ^- 1  2  lt  2  3  3V  7  £(£+1)  + 2A.PP'  2  JD  V p  2E |_ (l+(XB/3))  5  A^op)'  gauge term', mentioned i n Chapter V, i n a l l  the a d d i t i o n  o f a term:  4ir3V p c  of ( A . 3 ) .  p(r)d r 3  / 0  _  5  i n a l l cases i s c a l c u l a t e d  Vc(r') =  2  ^  +  + Ajp + Ajy^P  6  coefficient  B  (A p+A 6p+A p +V (A p+A 6p+A p ))  5  B = Agp  c  ~ ~E [ ( l + ( X B / 3 ) )  2A pp'  2  where  2  J  P  2  VI 2E  P  v  1  V (l+(XB/3))  Er  5  E-V,  -  E-V, E  - 1  _  ,  + V . - 2EV  2  +  E-V,  c  n  The r e s u l t i n g e q u a t i o n (A.3) then has the  '  _ (l+(XB/3))2  +  V  c  r  -  E-V, C  (V V +V V )  coefficients: E-V  l  C  have been performed i n  =  ,  from:  , p(r)r dr  T  2  {  Z  p(r)r dr 2  to the  - 126 -  APPENDIX B DETAILS OF INELASTIC CALCULATION  The nucleus  differential  cross sections f o r pion i n e l a s t i c  with ground s t a t e s p i n 0  p r o j e c t i o n J and M are g i v e n  and  On  =  £ f(Ji) | T  |  to an e x c i t e d s t a t e w i t h s p i n and  +  by:  —  with  =  )" aoP»(cos6)  c o u p l i n g s of the i n t e r m e d i a t e angular momenta. 4.2.1, the T - m a t r i x element i s r e l a t e d  Apart  < LM  |  VC1)  the d i f f e r e n t p o s s i b l e As mentioned i n Chapter  to  | 00 > ?(+) d r . 3  from some a d d i t i o n a l angular momentum c o u p l i n g terms,  e x p r e s s i o n f o r T can be w r i t t e n as the sum  Ix  T «  with  ,  , where f ( J i ) i n c l u d e s a number of Racah  2  Clebsch-Gordon c o e f f i c i e n t s which r e p r e s e n t  / »<-)  s c a t t e r i n g from a  I  x  +  I  / uty*  =  2  + I  +  and  I  The  primed and  3  -  T  of t h r e e i n t e g r a l s , i . e .  ,  3  TD  ,(T)  this  u£  + )  dr  ,  „(TK*  /»„(+)  (X(X+l)-Jc.(JM-l)-rU'+l))  / u ^ , ) * DTD  u£  + )  dr  /(  .,(+)  —  ) TDD (  .  unprimed v a r i a b l e s r e f e r to outgoing  coordinates r e s p e c t i v e l y .  — J dr  and  incoming p i o n  - 127 -  Coulomb e x c i t a t i o n i s accounted f o r by i n c l u d i n g  x  where  -  "  (2x i) +  (r/R )  f(r) =  ' H'*  R e  r<R  x  c  (R /r)  i s the r a d i u s  c  In  So that  )  *  u  d  '  r  c  of the s p h e r i c a l charge d i s t r i b u t i o n .  = A p + A^'pV  n  :  + A V p  + AgV (Ap)  1  n  Aj_BF  ,  TDD =  A BF  ,  DTD =  A f „» A B f F"  BF  2 F  a  '  &U+I)F\ 2 ) »  3  f  i s the t r a n s i t i o n d e n s i t y ,  with respect For  2  , and thus  2  2  =  .  2  3  2EV ^= A ^ A p ) + A V»(Ap)V TD  where  r  the o r i g i n a l DWPI, the n u c l e a r o p t i c a l p o t e n t i a l was taken to be 2EV  and  (  term:  r>Rc  x + 1  c  and R  f  an e x t r a  and the primes i n d i c a t e d e r i v i t i v e s  to r .  the p o t e n t i a l used i n the present a n a l y s i s ,  g i v e n i n Chapter  4.1.1, one h a s :  2EVH ) =  Ai(Ap) + A i ( A ( 6 p ) ) + 2A p(Ap)  1  v  4  + V ( A ( A p ) + A ( A ( 6 p ) ) + 2A p(Ap) ) 2  3  3 v  / A (Ap)+A (A(6p))+2A p(Ap) * ^ (lf(X/3)(A A 6 A p )) 2  +  V  7  2v  6  2  2 P +  so t h a t  the f a c t o r s w i t h i n  2 v  P +  6  the i n t e g r a l s a r e :  2  +  2 A  \ 5P(AP)J V  ,  - 128 -  TD  = A i F + A i 6 F + 2A4pF  2 2v 6P (1+(X/3)(A2P+A «p A p^))2 A  T  D  D  =  6 F + 2 A  F + A  F  2v  and  DTD  =  3n n F  respectively.  +  2 A  5P  •  F  A ((6F)"+(2/r)(6F)'-X(X+1)6F/r ) 2  3 v  2A ((2Fp7r)+2F p +Fp") ,  n  d  ,  7  is: a  6  2  + The n o t a t i o n  +  (A3+2A7P)(F"+(2F'/r)-X(X+l)F/r ) +  where  ,  v  F = g F n  ^p p F  a  r  e  t  + g F  n  n  p  e  n  e  u  t  r  and  p  o  n  a  n  d  6F = B F n  n  - g F p  p  ,  proton t r a n s i t i o n d e n s i t i e s  - 129 -  APPENDIX C FORM OF NUCLEAR DENSITIES  The  following  a r e the two a n a l y t i c forms f o r n u c l e a r d e n s i t i e s  used i n the a n a l y s i s o f the data p r e s e n t e d i n t h i s t h e s i s :  GAUSSIAN:  -(r/c) P(r)  —, T [1 + t x ( r / c ) ] e (2+3a)(/Trc)  =  3  FERMI: (  r  )  =  P  °  1  +  P  °  3K 4Trc [l+(Trt/c) +(W/5)(3+10(1rt/c) +7(Trt/c),+ ]  The n o r m a l i z a t i o n s t o t a l number  neutrons) f o r p . n  2  exp((r-c)/t)  =  where  (the  L  1 + W(r/c) P  2  2  3  2  are such that  of p r o t o n s ) f o r pp,  2  4TT / p ( r ) d r = K; 3  and K = N  where K = Z  ( t h e t o t a l number of  - 130 -  APPENDIX D ir + w  Consider Fig. D.l.  the decay o f a p i o n i n t o a muon and n e u t r i n o , as shown i n  Four-momenta i n the l a b and c e n t e r o f mass frames a r e r e l a t e d  by a L o r e n t z b o o s t ,  K^ = &P±  i.e.  where  Y YS 0 0  A =|  and  y  KINEMATICS  =  »  6  =  (D.l) 0 0 1 0  Y3 Y 0 0  0 0 0 1  k^/cc,,  In the c e n t e r of mass frame, P2 = ( T  which i m p l i e s Using  the f a c t  that  xs^  1  py  2  = m = E  + P )2  P y  ,  v  + 2(E E  2 y  U  = E  2 v  V  + p  - m2  2 p  2 y  ) .  ,  u  (D.2) can be s o l v e d to y i e l d  PU and  =  ( TT ~ y )/ TT > m  2  m  2  2m  Ey = (ra^ + my_ )/2m 2  2  ir  ( D . l ) y i e l d s the f o l l o w i n g : OJ  and  u  = y ( E + p cos9) y  u  kyCOS<|>  = Y(8 y  kysin<()  =  E  +  pysin6  ,  PyCOS0)  ,  (D.2)  131 -  CENTER OF MASS FRAME  PIT  p  u  P  v  "  0,0,0 )  (  = ( Ey,pycos9,pysin8,0 )  = ( E ,-p cos9,-pySinG,0 v  u  )  LABORATORY FRAME  -  M —  ( <%,kTr»0,0  )  K  y  = { o)y . k y c o s ^ j .kysinijij ,0 J  K  v  = ( a« ,k cos<t> ,k sin<j> ,0 ) v  v  F i g . D . l TT •»• pv Decay i n Center o f Mass and L a b o r a t o r y Frames  2  v  2  - 132  F i g . D.2 for  -  i s a p l o t of the muon l a b energy  the decay of a 50 MeV  p i o n , and  v e r s u s the muon l a b a n g l e ,  demonstrates why  b e f o r e the QQD  spectrometer  I t can be seen  t h a t muons emitted at angles < 5° have e n e r g i e s < 17  or > 68 MeV,  and w i l l  d i p o l e get through  few p i o n s which decay  to the back w i r e  be bend s e v e r e l y by the d i p o l e magnet.  e n e r g i e s between 30 and  50 MeV  a r e emitted a t a n g l e s >  the above r e l a t i o n s , F i g . D.3  the r e l a t i v e of a 50 MeV  MeV,  Muons with  15°.  In the c e n t e r of mass frame, p i o n decay i s i s o t r o p i c . f a c t , and  chambers.  can be g e n e r a t e d .  Using It  this  illustrates  number of muons emitted at a g i v e n l a b a n g l e , f o r the decay pion.  T h i s s o r t of p i c t u r e may  of p i o n s which decay a f t e r the M13 by the B1»B2 f l u x  monitors.  be used  to e s t i m a t e the number  E2 d i p o l e magnet, but a r e s t i l l  counted  20  |—I—i—i—i—|—i—i—i—i—|—i—i—i—r—|—r—i—i—i—|—i—i—r—i—|—i—i—i—i—|—i—i—r  10  20  30  40  50  60  Muon Lab Energy (MeV)  F i g . D.2  E  u  vs <J> f o r 50 MeV IT'S y  70  80  F i g . D.3  Decay Muon Angular  Distribution  

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