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Calculated cross-sections of pion production by 450-mev protons on various nuclei. McMillin, Douglas John 1968

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CALCULATED  CROSS-SECTIONS OF PION PRODUCTION  BY 450-ME? PROTONS ON VARIOUS NUCLEI by DOUGLAS JOHN MCMILLIN  A T H E S I S SUBMITTED I N P A R T I A L FULFILMENT THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n t h e Department of PHYSICS  We a c c e p t t h i s t h e s i s a s c o n f o r m i n g t o t required  standard  THE UNIVERSITY OF B R I T I S H COLUMBIA September,  1963  OF  In  for  presenting  an  this  advanced  thesis  or  shall  I further  agree  for scholarly  Department  or by  publication  hits  make  i t freely  that  fulfilment  may  be  representatives.  thesis  Physics  The U n i v e r s i t y o f B r i t i s h V a n c o u v e r 8, . C a n a d a  2.6 S e p t e m b e r ,  Columbia  1968  the  granted  by  requirements,  for reference  t h e Head  shall  and  copying  i t i s understood  gain  1 agree  Columbia,  for extensive  for financial  permission.  of  British  available  permission  purposes  of this  without.my written  Department of  in partial  d e g r e e ' a t .the- U n i v e r s i t y o f  :that;the.Library,  Study.  thesis  of  of  this  my  that  n o t be  copying  allowed  - i i-  ABSTRACT  We c o n s t r u c t a model t o e x p l a i n t h e p r o d u c t i o n o f pions i n the bombardment by protons  o f v a r i o u s n u c l e i and we use  the model t o c a l c u l a t e r e l a t i v e c r o s s - s e c t i o n s f o r t h e p r o c e s s . The model assumes t h a t the i n c i d e n t proton i n t e r a c t s w i t h the t a r g e t nucleons  i n d i v i d u a l l y and t h a t the proton-nucleon  s e c t i o n can be used as a f r e e parameter. many important  cross-  The model accounts  for  n u c l e a r e f f e c t s , some f o r the f i r s t time i n  e x p l a i n i n g the A-dependence o f t h e p i o n y i e l d .  The e f f e c t s  i n c l u d e d a r e those due t o proton and pion a b s o r p t i o n , t o the background n u c l e a r p o t e n t i a l s and t o the s t r u c k - n u c l e o n momentum and density d i s t r i b u t i o n s .  We compute d i f f e r e n t i a l c r o s s - s e c t i o n s  i n s e v e r a l s p e c i a l cases and compare them w i t h experimental a t 450 MeV.  data  Agreement i s o n l y moderate, but i t i s as good as  any p r e v i o u s l y obtained and,  u n l i k e the e a r l i e r r e s u l t s , i t does  not depend on t h e assumption o f an absorbing neutron  blanket.  Our agreement depends i n s t e a d on the use o f a modern n u c l e a r r a d i u s and a reasonable  treatment  of pion absorption.  In t h i s  r e s p e c t our r e s u l t s confirm what e a r l i e r workers had assumed, t h a t a b s o r p t i o n i s the dominant f a c t o r c o n t r o l l i n g the protonnucleus p r o d u c t i o n o f p i o n s .  A l s o important  nucleon p r o d u c t i o n r a t e , a reasonable P o t e n t i a l e f f e c t s a r e important  i s the p r o t o n -  value o f which we assume.  because t h e b a s i c p r o d u c t i o n r a t e  and p i o n a b s o r p t i o n a r e both v e r y energy dependent.  The e f f e c t s  o f s t r u c k - n u c l e o n momentum and d e n s i t y d i s t r i b u t i o n , as we c a l c u l a t e them, a r e s m a l l a t t h e energy  considered.  - i i i  -  TABLE OF CONTENTS *>age ABSTRACT  i  i  L I S T OF FIGURES  i v  ACKNOWLEDGMENTS  v  CHAPTER  CHAPTER  CHAPTER  I -  INTRODUCTION  1.1  P i o n p r o d u c t i o n a n d t h e TRIUMF p r o j e c t  1  1.2  The b a s i c i n t e r a c t i o n  4  1.3  E x i s t i n g models  7  1.4  P r e s e n t program  10  I I - THE MODEL 2.1  Theory  13  2.2  An e s s e n t i a l s i m p l i f i c a t i o n  2.3  Parameters chosen  . . . .  18 22  I I I - THE CALCULATION 3.1  The i n t e g r a t i o n  27  3.2  R e s u l t s compared t o e x p e r i m e n t  28  3.3  R e s u l t s w i t h parameters v a r i e d  . . . . . . . . . . 3 5  3.4  Conclusion  37  REFERENCES  39  APPENDIX A  E x p e r i m e n t s on p r o t o n - n u c l e o n  pion production  . . 43  APPENDIX B  E x p e r i m e n t s on p r o t o n - n u c l e u s  pion production  . . 45  APPENDIX C  D e r i v a t i o n o f path  APPENDIX D  T r a n s f o r m a t i o n o f energy from l a b o r a t o r y frame t o r e s t frame o f a s t r u c k n u c l e o n . . . . . . . .  50  D e r i v a t i o n o f a n e x p r e s s i o n u s e d t o compute p i o n absorption coefficients .  51  O u t l i n e o f c o m p u t e r p r o g r a m PIPROD  53  APPENDIX E APPENDIX F  lengths i n a nucleus  47  - iv-  L I S T OF FIGURES page FIGURE 1 FIGURE 2  Model o f nucleus showing path o f i n c i d e n t and e m e r g i n g p i o n  proton 15  Cross-sections f o r the production  o f £3-MeV  positive  p i o n s a t 2 1 . 5 ° b y 450-MeV p r o t o n s  on v a r i o u s n u c l e i  29  FIGURE 3  More c r o s s - s e c t i o n s f o r p o s i t i v e p i o n p r o d u c t i o n  . . 32  FIGURE 4  Some c r o s s - s e c t i o n s f o r n e g a t i v e  . . 34  FIGURE 5  Cross-sections obtained  FIGURE 6  Geometry o f p a t h l e n g t h s i n a nucleus  pion production  by v a r y i n g parameters  . . .  36 48  ACKNOWLEDGMENTS  The author wishes t o thank, above a l l ,  Professor  E . W. Vogt who suggested t h i s i n v e s t i g a t i o n and p r o v i d e d i n v a l u a b l e guidance through a l l t h e phases o f i t s completion. Dr. Vogt's c o n t i n u e d i n t e r e s t and p a t i e n c e i n the matter a r e gratefully  acknowledged. He a l s o thanks L . Lam who s o r t e d out many a s p e c t s  o f the problem and Dr. P. C. Bhargava who a s s i s t e d i n t h e e a r l y stages o f t h e c a l c u l a t i o n .  Financial assistance  p r o v i d e d by t h e N a t i o n a l Research C o u n c i l o f Canada i s a l s o acknowledged.  - 1 -  CHAPTER I - INTRODUCTION  1.1  P i o n p r o d u c t i o n and the TRIUMF p r o j e c t I t i s now p o s s i b l e t o produce pions i n l a r g e q u a n t i t y .  The most important  p r o d u c t i o n process b r i n g s a beam o f protons  from an a c c e l e r a t o r i n t o c o l l i s i o n w i t h other nucleons which may be f r e e  (a hydrogen t a r g e t ) o r bound t o g e t h e r as a nucleus  carbon t a r g e t f o r example).  (in a  The beam energy a t which p i o n  p r o d u c t i o n can begin depends s t r o n g l y on the k i n d o f t a r g e t being used, but the energies a t which p i o n p r o d u c t i o n i s a t a l l e f f i c i e n t a r e g e n e r a l l y i n excess o f 3 50 MeV.  The r e c e n t  f e a s i b i l i t y o f h i g h beam c u r r e n t s a t these e n e r g i e s o f e f f i c i e n t p r o d u c t i o n has l e d t o the proposed c o n s t r u c t i o n o f meson " f a c t o r i e s " a b l e t o t u r n out pions and t h e i r decay products a t an unprecedented r a t e .  I t was one o f these p r o p o s a l s , namely the  TRIUMF p r o p o s a l ^ ^ ^ , which s t i m u l a t e d t h e present  investigation  i n t o the p i o n p r o d u c t i o n p r o c e s s . The  heart o f the TRIUMF p r o j e c t i s a s e c t o r - f o c u s s e d  c y c l o t r o n designed t o a c c e l e r a t e H A continuous  -  ions r a t h e r than  protons.  proton beam i s e x t r a c t e d by the simple and h i g h l y  e f f i c i e n t p r o c e s s o f m e c h a n i c a l l y s t r i p p i n g the two e l e c t r o n s from each H"~ i o n as i t g a i n s a predetermined  orbit.  In a d d i t i o n  to p r o v i d i n g an i n t e n s e beam o f r e l a t i v e l y high energy the d e s i g n f e a t u r e s permit easy energy v a r i a b i l i t y . variability  (though  o f prime i n t e r e s t i n proton  protons This  experiments).^  along w i t h the freedom o f choice o f a pion-producing  target  - 2 -  provides v a r i e t y i n the nature maximize t h e p i o n y i e l d , provided.  J u s t how t h i s To  of the pion y i e l d .  important  A way t o  t o many e x p e r i m e n t s ,  i s p o s s i b l e we s o o n make  i s also  clear.  s e l e c t a n u p p e r l i m i t t o p u t on t h e p r o t o n  e n e r g y t h e d e s i g n e r s o f TRIUMF b a l a n c e d o f machine c o s t , which f a v o u r s  t h e two i m p o r t a n t  l o w energy, and t h e p i o n  r a t e , which favours high energy.  beam factors  production  I n a carbon t a r g e t ,f o r  example, t h e p r o d u c t i o n r a t e a t s m a l l a n g l e s  (where t h e y i e l d i s  always g r e a t e s t ) i n c r e a s e s s h a r p l y w i t h proton  e n e r g y f r o m 350 t o  450 MeV, w h i l e b e y o n d 450 MeV i t i s r e l a t i v e l y  constant.  T h e r e f o r e , w h i l e t h e mean p i o n e n e r g y may become e v e r with i n c r e a s i n g proton  energy, there i s not a very l a r g e gain i n  t h e number o f p i o n s p r o d u c e d a t p r o t o n This o f course  greater  e n e r g i e s a b o v e 450  assumes t h a t t h e beam c u r r e n t i s c o n s t a n t  changing proton  energy.  Seeking  a maximum p i o n y i e l d  MeV. with  rather  t h a n a h i g h mean e n e r g y a n d a t t h e same t i m e a l l o w i n g a m a r g i n for  comfort  the planned  maximum p r o t o n beam e n e r g y . The  producing factors.  c h o s e t h e v a l u e o f 500 MeV f o r  t h e TRIUMF d e s i g n e r s  c h o i c e o f a n o p e r a t i n g beam e n e r g y a n d a p i o n -  t a r g e t i s made c o m p l i c a t e d Because t h e e l e c t r i c  l e s s probable  a t lower  by s e v e r a l a d d i t i o n a l  d i s s o c i a t i o n o f a n H~ i o n i s  energy t h e a c c e l e r a t o r can t o l e r a t e  l a r g e r beam c u r r e n t s a t l o w e r  energy.  Hence t h e p i o n  yield,  p r o p o r t i o n a l t o b o t h t h e beam c u r r e n t a n d t h e p r o d u c t i o n r a t e , c a n be g r e a t e r a t 450 MeV t h a n a t t h e maximum beam e n e r g y o f 500 MeV.  The y i e l d may become s t i l l  greater i f the proton  energy  -  i s f u r t h e r reduced.  3  -  Moreover, the p i o n y i e l d may  be enhanced by  the use o f a h e a v i e r t a r g e t m a t e r i a l s i n c e a h e a v i e r nucleus i s r i c h e r i n nucleons having h i g h i n t e r n a l momentum which can r a i s e the amount of energy a v a i l a b l e t o the p r o d u c t i o n r e a c t i o n .  This  enhancement i s e s p e c i a l l y n o t i c e d a t proton energies near the pion production threhhold.  A competing process however i s the  one i n which e i t h e r the i n c i d e n t proton or the emerging p i o n i s absorbed  or s c a t t e r e d by the nucleus h o s t i n g the p r o d u c t i o n event  and i n t h i s r e s p e c t a h e a v i e r host nucleus i s c l e a r l y not favoured.  Supporting t h i s t r e n d we  note t h a t a heavy nucleus  a l s o p r o v i d e s a l a r g e Coulomb p o t e n t i a l which serves t o lower i n c i d e n t proton energy. " n u c l e a r e f f e c t s " and beam energy  Determining  the  the balance o f these  i t s dependence on both n u c l e a r mass and  i s the concern o f t h i s t h e s i s , but the reason f o r  wanting t o be a b l e t o do so i s a t once e v i d e n t : i t w i l l the e f f i c i e n t adjustment  permit  o f machine parameters t o maximize the  TRIUMF p i o n y i e l d . Because o f the nature of the TRIUMF p r o j e c t we  are  i n t e r e s t e d i n the p r o d u c t i o n o f charged p i o n s a t s m a l l angles i n the bombardment of v a r i o u s n u c l e i by protons i n the energy around 450 MeV.  In the next chapter we  range  i n t r o d u c e a simple model  f o r the p r o d u c t i o n p r o c e s s , one which keeps separate i t s more important p h y s i c a l a s p e c t s , and  i n the chapter a f t e r t h a t we  d e s c r i b e c a l c u l a t i o n s made on the b a s i s o f t h i s model.  We  compute r e l a t i v e p r o d u c t i o n c r o s s - s e c t i o n s which agree w i t h a v a i l a b l e experimental c r o s s - s e c t i o n s t o such an extent as t o a f f o r d a f a i r degree o f confidence i n the assumptions we make i n  -  c o n s t r u c t i n g our model.  We  4  -  are a l s o a b l e t o d i s c o v e r which of  the p h y s i c a l aspects of the process dominate a t the e n e r g i e s c o n s i d e r e d . We  hope t h a t subsequently r e f i n e d v e r s i o n s of our  model can then be used t o p r e d i c t c r o s s - s e c t i o n s where none are y e t a v a i l a b l e e x p e r i m e n t a l l y and t h e r e b y become an a i d to more e f f i c i e n t p l a n n i n g i n the TRIUMF o r s i m i l a r p r o j e c t s .  1.2  The b a s i c  interaction  The p r o d u c t i o n of pions by proton-nucleus  collisions  and by proton-nucleon c o l l i s i o n s are r e l a t e d p r o c e s s e s .  At  e n e r g i e s above the p i o n p r o d u c t i o n t h r e s h h o l d both the i n c i d e n t proton wavelength  and the proton-nucleon c r o s s - s e c t i o n are s m a l l  enough t o permit the assumption  t h a t the proton i n t e r a c t s w i t h but  a s i n g l e t a r g e t nucleon t o produce nucleus as a whole.  a p i o n and not w i t h the t a r g e t  In t h i s approximation the proton-nucleus  c r o s s - s e c t i o n i s p r o p o r t i o n a l t o a corresponding proton-nucleon c r o s s - s e c t i o n and t o the number of those nucleons i n the t a r g e t n u c l e u s , although the nature of the p r o p o r t i o n a l i t y becomes complex as one admits the o p e r a t i o n o f n u c l e a r e f f e c t s such as those mentioned i n the p r e v i o u s s e c t i o n .  The  two-nucleon  c o l l i s i o n then i s a t the r o o t o f the p i o n p r o d u c t i o n p r o c e s s . The two-nucleon proton and produce  r e a c t i o n s t h a t begin w i t h a t l e a s t  a p o s i t i v e p i o n are as f o l l o w s :  P  +  P  p  +  p  n  +  p  one  - 5 -  There i s o n l y one such r e a c t i o n producing a negative  n  The  threshhold  +  rr-  p  +  p  +  pion, v i z :  p  energy i n t h e l a b o r a t o r y system i s 294  Me? f o r t h e  r e a c t i o n y i e l d i n g deuterium and 253 MeV f o r the o t h e r s .  The  d i f f e r e n t i a l c r o s s - s e c t i o n s f o r the r e a c t i o n s a r e p o o r l y known experimentally and  (experiments(2-11)  a  r  e  t h e e x i s t i n g t h e o r i e s ( 1 2 - 1 6 ) ^-^ch  summarized i n Appendix A) m  i g h t be used t o compute  c r o s s - s e c t i o n s a r e e i t h e r l a r g e l y phenomenological or incomplete. Consequently we a r e able t o c a l c u l a t e only r e l a t i v e nucleus c r o s s - s e c t i o n s , u s i n g t h e proton-nucleon as f r e e parameters t o a d j u s t our r e s u l t s .  proton-  cross-sections  I t should be noted  however t h a t the combined c r o s s - s e c t i o n f o r the f i r s t two r e a c t i o n s l i s t e d above i s approximatly t e n times t h e c r o s s section f o r the t h i r d . production state.  T h i s i s t h e c o r r e c t r e s u l t ( ^ ** f o r p i o n  proceeding through an intermediate  3-3  Hence we assume t h a t f o r t h e p r o d u c t i o n  resonance of positively-  charged pions i t i s the protons o f t h e t a r g e t nucleus which a r e the most important. The  nuclear  e f f e c t s which i n f l u e n c e the nature o f the  p r o p o r t i o n a l i t y between t h e proton-nucleus c r o s s - s e c t i o n and the number o f t a r g e t nucleons a r e : 1) the momentum e f f e c t wherein the momentum o f a s t r u c k nucleon w i t h i n i t s nucleus a l t e r s t h e amount o f energy a v a i l a b l e t o the production angle o f the p i o n produced.  r e a c t i o n and both the energy and e x i t J u s t how the p i o n y i e l d i s a f f e c t e d  I  -  6  -  depends on t h e nature o f t h e v a r i a t i o n o f the proton-nucleon c r o s s - s e c t i o n w i t h proton and p i o n energy and p i o n angle i n the regions of i n t e r e s t .  I n t e r n a l momentum i s e s p e c i a l l y  important  near t h e p i o n p r o d u c t i o n t h r e s h h o l d which, because o f i n t e r n a l momentum, i n a nucleus may be as low as 170 Mev"; 2) t h e a b s o r p t i o n - s c a t t e r i n g e f f e c t wherein t h e nonp a r t i c i p a t i n g nucleons  o f t h e s t r u c k n u c l e u s , mere s p e c t a t o r s i n  the p r o d u c t i o n event, p r o v i d e a background o p t i c a l p o t e n t i a l which a l t e r s t h e p i o n y i e l d by a l l o w i n g f o r a b s o r p t i o n w i t h i n the nucleus  o f e i t h e r t h e incoming proton o r t h e outgoing p i o n o r , i n  the case o f d i f f e r e n t i a l c r o s s - s e c t i o n s , f o r s c a t t e r i n g o f pions away from t h e energy and angle o f i n t e r e s t . r e s u l t s on t h e proton-nucleus  Experimental  p r o d u c t i o n o f pions as a f u n c t i o n  o f i n c r e a s i n g n u c l e a r mass (the experiments(1^~30) i n Appendix B) show a decrease  a  r  e  summarized  i n production e f f i c i e n c y , i . e . , i n  the number o f pions produced p e r t a r g e t nucleon, the o p e r a t i o n o f an a b s o r p t i o n e f f e c t .  consistent with  The n u c l e a r cross-:  s e c t i o n s a r e found t o i n c r e a s e more and more s l o w l y w i t h A u n t i l o f t e n t h e r e i s no i n c r e a s e a t a l l ; 3) p o t e n t i a l e f f e c t s wherein t h e amount o f energy a v a i l a b l e f o r p i o n p r o d u c t i o n and t h e energy o f the p i o n as i t i s seen o u t s i d e the nucleus a r e a l t e r e d by t h e presence o f background Coulomb and n u c l e a r p o t e n t i a l s , i . e . , by the r e a l p a r t o f the o p t i c a l p o t e n t i a l .  These p o t e n t i a l s a l s o a f f e c t the process  o f a b s o r p t i o n , which i s u s u a l l y energy dependent, and change the paths taken by incoming and outgoing p a r t i c l e s  (refraction);  -  4) d e n s i t y e f f e c t s  7  -  w h e r e i n t h e momentum,  s c a t t e r i n g and p o t e n t i a l e f f e c t s ,  as w e l l as t h e  absorption-  simple  p r o b a b i l i t y o f f i n d i n g a n u c l e o n t o s t r i k e , e a c h become o f p o s i t i o n w i t h i n t h e t a r g e t n u c l e u s ; and 5) m i s c e l l a n e o u s e f f e c t s  finally  w h i c h as f a r as our work i s  concerned a r e r e l a t i v e l y u n i m p o r t a n t and w i l l further.  Included i n these are  among t h e t a r g e t n u c l e o n s  functions  the  effects  (important  n o t be  discussed  of correlation  i n very l i g h t nuclei)  and  the P a u l i i n h i b i t i o n o f r e a c t i o n s which would leave a nucleon i n an a l r e a d y o c c u p i e d s t a t e scattering 1.3  (important at  low energies  large  angles).  Existing  models  V a r i o u s models t a k i n g n u c l e a r e f f e c t s have been used t o e x p l a i n t h e proton-nucleus  into  observed A-dependence o f  production of pions.  The e a r l i e s t w o r k o f i n t e r e s t  Gasiorowicz^^  who e x p l a i n s t h e  t o us i s t h a t  fashion,  and assumes, f u r t h e r m o r e ,  nuclear surface  excess neutrons which,  pion  of  production of low-energy pions  He c o n s i d e r s ,  there e x i s t s at the  the  cross-  b y 2 4 0 - a n d 340-MeV p r o t o n s . the absorption effect  account  To compute r e l a t i v e  y i e l d s t h e y a l l assume c o r r e s p o n d i n g p r o t o n - n u c l e o n sections.  and  a shell  i n an a p p r o x i m a t e  or blanket o f  that the  s i n c e the m a j o r i t y o f p o s i t i v e pions  produced i n proton-proton c o l l i s i o n s , p r o d u c t i o n volume and r e d u c e s t h e  encloses the  pion yield  are  effective  o b t a i n e d from a  heavy n u c l e u s r e l a t i v e t o t h a t o b t a i n e d from a l i g h t e r one.  His  - 8 -  c a l c u l a t e d c r o s s - s e c t i o n s agree w e l l w i t h t h e t h e n - a v a i l a b l e experimental  ones, but because o f u n c e r t a i n t y i n h i s chosen  p a r a m e t e r s h e c a n d r a w no c o n c l u s i o n a b o u t t h e e x i s t e n c e o r t h e non-existence  o f a neutron  blanket.  Gasiorowicz  discusses the  e f f e c t o f t h e Coulomb p o t e n t i a l , b u t o n l y o n t h e r e l a t i v e of p o s i t i v e and negative  shapes  pion energy s p e c t r a .  M e r r i t t and H a m l i n ^ ' ) formulate 2  an A-dependent  a t t e n u a t i o n f a c t o r u s i n g t h e n u c l e a r r a d i u s and energy-dependent p r o t o n a n d p i o n mean f r e e p a t h s a s p a r a m e t e r s w h i c h t h e y t o o b t a i n approximate agreement w i t h t h e i r e x p e r i m a n t a l  adjust cross-  s e c t i o n s f o r p o s i t i v e p i o n p r o d u c t i o n By 335-MeV p r o t o n s . f o r Gasiorowicz  As  t h i s agreement i s i n c o n c l u s i v e because t h e v a l u e s  o f t h e parameters used a r e u n c e r t a i n and t h e data p o i n t s a r e few. They c o n s i d e r o n l y t h e a b s o r p t i o n  effect.  H e n l e y ^ 3 ) d i s c u s s e s and then n e g l e c t s t h e e f f e c t o f p i o n a b s o r p t i o n on p i o n y i e l d .  H i s main concern  i susing the  e f f e c t o f s t r u c k - n u c l e o n momentum t o e x p l a i n e x p e r i m e n t a l energy s p e c t r a from carbon. has  pion  S i m i l a r u s e o f t h e momentum e f f e c t  b e e n made b y o t h e r s , e . g . R o s e n f e l d ^ . > Imhof e t a l . ( ^ )  explain t h e i r experimental  u s e t h e model o f G a s i o r o w i c z t o  r e s u l t s a t 3 4 0 MeV a n d c o n s i d e r t h e  momentum e f f e c t t o e x p l a i n t h e p i o n y i e l d t o carbon.  i n c r e a s i n g from  lithium  A l l c a l c u l a t i o n s a r e crude. Ansel'm and S h e k h t e r ^ ' ' ) d e r i v e an e x p l i c i t  expression  f o r an A-dependent a t t e n u a t i o n f a c t o r assuming t h a t p r o t o n and  -  9  -  p i o n a b s o r p t i o n i s energy independent.  They are a b l e t o f i t  curves to most o f the t h e n - a v a i l a b l e experimental  data, but  they  do not c o n s i d e r what v a l u e s f o r the a b s o r p t i o n parameters might be r e a l i s t i c .  They c o n s i d e r no e f f e c t other than a b s o r p t i o n .  The most e x t e n s i v e and  s u c c e s s f u l o f the  c a l c u l a t i o n s i s t h a t o f L i l l e t h u n ^ - ^ who energy-dependent treatment concept  combines an  previous improved  o f the a b s o r p t i o n e f f e c t w i t h  the  o f a neutron b l a n k e t as proposed by G a s i o r o w i c z .  Pion  a b s o r p t i o n c o e f f i c i e n t s are c a l c u l a t e d f o r v a r i o u s n u c l e i from o p t i c a l p o t e n t i a l s g i v e n by Frank e t a l . ^ ^ J . parameters are i n g e n e r a l q u i t e reasonable  Lillethun's  and h i s c a l c u l a t e d  p i o n y i e l d s agree w e l l w i t h a l a r g e amount o f experimental at  450 MeV  data  and t h i s agreement, as L i l l e t h u n shows, depends v e r y  s t r o n g l y on the neutron b l a n k e t assumption.  Lillethun also  t r e a t s the Coulomb and n u c l e a r p o t e n t i a l s , i g n o r i n g r e f r a c t i o n . In summary then the a b s o r p t i o n e f f e c t has been most o f t e n used t o o b t a i n agreement between t h e o r y and experiment, but t h i s agreement i s never c o n c l u s i v e without concerning  the assumption  the e x i s t e n c e o f an absorbing neutron b l a n k e t .  The  e f f e c t o f i n t e r n a l nucleon momentum has not been d i s c u s s e d i n c o n e c t i o n w i t h the g e n e r a l problem o f A-dependence; n e i t h e r has the e f f e c t of u s i n g a nucleon d e n s i t y d i s t r i b u t i o n t h a t does not have square edges.  P o t e n t i a l e f f e c t s have been i n c l u d e d i n o n l y  one model, t h a t of L i l l e t h u n . played by the important  We  must conclude  t h a t the r o l e s  p h y s i c a l f a c t o r s c o n t r o l l i n g the p i o n  p r o d u c t i o n process are not w e l l  understood.  -10-  In p a s s i n g we note t h a t models not o f the type outl i n e d above have been used by S e r b e r ^ ^ ^ , M e t r o p o l i s et  a l . ^ 9 ) ^  and M a r g o l i s ^ ^ ^ and t h a t these have met w i t h o n l y l i m i t e d amounts o f s u c c e s s . the p r e s e n t  1.4  T h e r e f o r e they are o f l i t t l e  interest to  investigation.  P r e s e n t program I t i s the o b j e c t i v e o f the present i n v e s t i g a t i o n t o  c a l c u l a t e some r e l a t i v e d i f f e r e n t i a l c r o s s - s e c t i o n s f o r charged p i o n p r o d u c t i o n i n the proton-nucleus r e a c t i o n a t e n e r g i e s around 450 MeV.  We  must assume t h a t the f r e e two-nucleon  though not w e l l known can be used as a parameter, may  cross-section  w i t h which we  normalize our r e s u l t s t o experimental v a l u e s . The model we  s h a l l use i s o f the g e n e r a l k i n d we  have been d e s c r i b i n g , the k i n d having the most success so f a r . I t makes the b a s i c assumption  o f i n d i v i d u a l proton-nucleon  inter-  a c t i o n and, u n l i k e those d e s c r i b e d , i t accounts f o r a l l the n u c l e a r e f f e c t s which might be important a t the energy of interest  (the e f f e c t s l i s t e d as l ) - 4 ) i n s e c t i o n 1.2).  c a l c u l a t i o n we make attempts a c t u a l l y are  The  t o s o r t out those e f f e c t s which  important. I t seems p a r t i c u l a r l y important t o a v o i d the neutron-  blanket assumption  f o r two reasons.  F i r s t , f a i r agreement w i t h  experimental r e s u l t s has a l r e a d y been obtained by L i l l e t h u n used t h i s assumption  and,: second, we f e e l t h a t while c e r t a i n  who  - 11 -  evidence  (the i s o t o p i c s p i n term o f the n u c l e a r o p t i c a l model  p o t e n t i a l ) may i n d i c a t e an excess o f neutrons a t the n u c l e a r sur f a c e t h e a c t u a l d i s t r i b u t i o n i s n e a r e r t o the one w i t h no excess than t o the opposite extreme w i t h complete proton and neutron separation.  We t h e r e f o r e regard the h y p o t h e s i s o f Gasiorowicz  and L i l l e t h u n as a r t i f i c i a l . In  t r e a t i n g p r o t o n and p i o n a b s o r p t i o n i t i s most  convenient t o use the same a b s o r p t i o n c o e f f i c i e n t s as L i l l e t h u n , but important  c o r r e c t i o n s must be made.  We d e s c r i b e these  c o r r e c t i o n s and a p p l y them i n our c a l c u l a t i o n .  The a b s o r p t i o n  c o e f f i c i e n t s were computed f o r protons from experimental proton and proton-neutron  proton-  c r o s s - s e c t i o n s (about the same a t our  energy) and f o r p i o n s from mean f r e e paths f o r e i t h e r a b s o r p t i o n or  inelastic scattering.  By u s i n g these c o e f f i c i e n t s we there4-  f o r e ignore the e f f e c t s o f r e f r a c t i o n  ( e l a s t i c s c a t t e r i n g ) and  the p o s s i b i l i t y t h a t pions may be s c a t t e r e d from o t h e r e n e r g i e s and angles i n t o the r e g i o n o f i n t e r e s t . e f f e c t s are s i m i l a r l y ignored. s m a l l c r o s s - s e c t i o n s however.  Multiple  scattering  The i g n o r e d i n t e r a c t i o n s have The p i o n mean f r e e paths were i n  t u r n computed from o p t i c a l model p o t e n t i a l s obtained from twobody s c a t t e r i n g phenomena assuming t h a t two-body f o r c e s are not a p p r e c i a b l y m o d i f i e d w i t h i n the n u c l e u s . c o n s i s t e n c y i n method.  Hence there i s some  Pion a b s o r p t i o n gets s p e c i a l a t t e n t i o n  i n our c a l c u l a t i o n s i n c e i t i s both g r e a t e r and more  energy  dependent than proton a b s o r p t i o n and s i n c e , as i t t u r n s out, the e f f e c t o f a b s o r p t i o n i t s e l f i s g r e a t e r than the other e f f e c t s .  - 12 -  In d e a l i n g w i t h t h e momentum e f f e c t we make t h e assumption t h a t t h e momentum d i s t r i b u t i o n has a n e g l i g i b l e  effect  on both t h e energy o f t h e p i o n s produced and on t h e angle o f t h e i r emission.  The j u s t i f i c a t i o n f o r t h i s seemingly b o l d  assumption i s d i s c u s s e d i n d e t a i l a t the time we make i t .  It i s  a n e c e s s a r y one t o make and i t l e a d s us t o suspect t h a t t h e Adependence o f the p i o n y i e l d i s not as s e n s i t i v e  t o t h e type o f  momentum d i s t r i b u t i o n as i t i s t o the mean struck-nueleon energy, a t our v a l u e o f proton energy anyway.  kinetic  In our work we  t h e r e f o r e c o n s i d e r o n l y one d i s t r i b u t i o n , t h a t o f a Fermi g a s . The  d e n s i t y e f f e c t s which we c o n s i d e r a r e those o f  u s i n g a Saxon-Woods shape on t h e d e n s i t y d i s t r i b u t i o n , i . e . , o f g i v i n g the nucleus a d i f f u s e  edge, and t h a t o f reducing t h e b a s i c  n u c l e a r r a d i u s from the l a r g e r v a l u e s used by e a r l i e r workers t o the v a l u e now g e n e r a l l y accepted. t u r n s out t o be an e s p e c i a l l y  The l a t t e r c o n s i d e r a t i o n . ;  important  one: i t l i b e r a t e s , as we  show, the model from the need o f a neutron l i b e r a t i o n , t o g e t h e r w i t h the r e l a t e d  blanket.  This  confirmation that  t i o n i s t h e dominant n u c l e a r e f f e c t i n proton-nucleus  absorp-  pion  p r o d u c t i o n a t 450 MeV, c o n s t i t u t e s the main r e s u l t o f the i n v e s t i g a t i o n t o which we now t u r n .  CHAPTER I I - THE MODEL  2.1  Theory The t h e o r y and assumptions u n d e r l y i n g our model are  s t a t e d most d i r e c t l y by a mathematical  e x p r e s s i o n f o r the p r o t o n -  nucleus c r o s s - s e c t i o n f o r p i o n p r o d u c t i o n .  This cross-section  i s a d i f f e r e n t i a l c r o s s - s e c t i o n which i n the l a b o r a t o r y system depends on the energy E  o f t h e i n c i d e n t p r o t o n , on the angle ¥ JL  o f p i o n emission r e l a t i v e t o the i n c i d e n t proton beam, and on the energy E^ of t h e p i o n e m i t t e d . [1J  (7(E  Ijf,^)  -  The  //G (T f  expression f o r i t i s : ¥,T„,k>exp(-/n ds - / n ^ d s ) .  .#(r).f(k,r).d3r.d3k, where Of i s the p i o n p r o d u c t i o n c r o s s - s e c t i o n , i n the l a b o r a t o r y system, f o r protons on " f r e e " nucleons, i . e . , i t i s what we have, been c a l l i n g the corresponding proton-nucleon  cross-section.  It  i s assumed f o r p o s i t v e pions t o be the proton-proton c r o s s s e c t i o n and f o r negative pions the proton-neutron c r o s s - s e c t i o n . I f we  i g n o r e r e f r a c t i o n e f f e c t s Of depends on the same angle Ijf o f  p i o n emission as 0" depends on.  Of a l s o depends on p r o t o n  k i n e t i c energy Tp and p i o n k i n e t i c energy T^. i n s i d e the t a r g e t n u c l e u s , which d i f f e r from t h e i r e n e r g i e s o u t s i d e by  potential  terms depending on p o s i t i o n r i n the nucleus and the mass number A o f the n u c l e u s .  F i n a l l y , Of depends on the momentum v e c t o r k  o f the p a r t i c u l a r nucleon s t r u c k s i n c e we are i n the l a b o r a t o r y frame.  We  have more t o say about t h i s l a t e r .  The other terms  - 14 -  i n the i n t e g r a n d o f [1] account  f o r the v a r i o u s n u c l e a r  effects  mentioned i n the I n t r o d u c t i o n . The  e x p o n e n t i a l term i n the i n t e g r a n d o f [1]  f o r the a b s o r p t i o n and  scattering effects.  np and n,,. are  r e s p e c t i v e l y proton and p i o n a b s o r p t i o n c o e f f i c i e n t s . depend on the corresponding k i n e t i c energy, T  or T^,  p  d e n s i t y jo a t p o i n t s s along Sp and  l o c a l nucleon  Since we  and on  the  proton and  the  have ignored r e f r a c t i o n these paths are  the s t r a i g h t l i n e s i l l u s t r a t e d i n F i g u r e 1.  I f we  adopt a  s p h e r i c a l l y p o l a r c o o r d i n a t e r e f e r e n c e ( F i g u r e 1) and p o i n t o f proton-nucleon  They  s,,. which denote  i n t h a t order the paths f o l l o w e d by the incoming outgoing p i o n .  accounts  c o l l i s i o n by r  (r,Q,<{>) we  =  label  the  can d e r i v e  (Appendix C) e x p l i c i t e x p r e s s i o n s f o r the d i s t a n c e s t r a v e l l e d the proton and the p i o n i n s i d e the t a r g e t nucleus.. i s the r a d i u s beyond which we  Where R  by m a x  assume n e g l i g i b l e a b s o r p t i o n these  e x p r e s s i o n s are r e s p e c t i v e l y : [2] and, where [3]  Sp  =  r.cosO  X  =  cos9*cos!jF  =  - r-X  s„  +  +  In our model a b s o r p t i o n ( n  (R^ax + (R  p  ~ r sin 9)£ 2  sin9«sinty'Cos<f>, 2 a x  and n  n  - r  2  ( l - X ))£. 2  a l s o account  of s c a t t e r i n g ) i s averaged over these path The  2  nucleon d e n s i t y d i s t r i b u t i o n /o[r)  n u c l e u s , which we  kinds  lengths.  t o the t o t a l number o f p o s s i b l e p i o n producers  and neutrons  for certain  i s normalized  i n the t a r g e t  assume are protons f o r p o s i t i v e p i o n p r o d u c t i o n  f o r negative pion production.  Thus y / b ( r ) . d 3 r i s  -  15  -  FIGURE 1  Model o f n u c l e u s showing p a t h o f i n c i d e n t p r o t o n and emerging p i o n . P r o t o n e n t e r s n u c l e u s a t A, t r a v e l s d i s t a n c e Sp, a n d s t r i k e s a n u c l e o n a t r w h e r e i t c r e a t e s a p i o n . The p i o n t h e n t r a v e l s d i s t a n c e s a n d l e a v e s n u c l e u s a t a n g l e Ijl. Paths remain i n plane p a r a l l e l t o t h a t of paper. Coordinate system i s i n d i c a t e d . ff  - 16 put equal t o e i t h e r Z o r N  =  A - Z accordingly.  The n u c l e o n  momentum d i s t r i b u t i o n f ( k , r ) i s assumed t o d e p e n d on p o s i t i o n r through the l o c a l nucleon d e n s i t y /o(r). normalized t o unity, i . e . ,  /f(k,r)»d3k=  f ( k , r ) i s always;!  1.  B e f o r e d e s c r i b i n g a s i m p l i f i c a t i o n we make o f f l ] we review the approximations which p r o p e r t h e o r y a n d comment  d i s t i n g u i s h our model from a  on t h e e x t e n t t o which  improvement o v e r i t s e a r l i e r v e r s i o n s .  our model i s an  The b a s i c  approximations  i n h e r e n t t o [1] a r e t h o s e d u e t o : 1) t h e a s s u m p t i o n proton-nucleon  collisions  t h a t p i o n s a r e produced  ( t h e h i g h energy  small cross-sections f o r proton-nucleon energy  justify this  o f t h e p r o t o n and t h e  interaction at this  approximation);  2) t h e a s s u m p t i o n nucleus  i n s i n g l e r.  that only the protons o f a t a r g e t  can c o n t r i b u t e t o t h e p r o d u c t i o n o f p o s i t i v e p i o n s  (this  we assume b e c a u s e t h e n e u t r o n c o n t r i b u t i o n i s known t o be down b y one  order o f  magnitude);  3) t h e n e g l e c t o f n u c l e a r d e t a i l w h i c h we assume when we u s e t a r g e t - n u c l e o n momentum a n d d e n s i t y d i s t r i b u t i o n s  (many  s i m i l a r c a l c u l a t i o n s , e . g . o n e l e c t r o n s c a t t e r i n g , meet w i t h a remarkable  amount o f s u c c e s s u s i n g t h i s t r e a t m e n t ) ; 4)  the neglect o f incoming  and o u t g o i n g  particle  r e f r a c t i o n w h i c h we assume when we t a k e Sp a n d s ^ t o b e s t r a i g h t l i n e s a n d when we i g n o r e e l a s t i c n,r ( t h i s a p p r o x i m a t i o n  s c a t t e r i n g i n c o m p u t i n g np a n d  i s p a r t i a l l y j u s t i f i e d by the high  energy  o f b o t h t h e p r o t o n and t h e p i o n and b y t h e s m a l l c r o s s - s e c t i o n  -  for  elastic  17  -  scattering); 5) t h e a s s u m p t i o n t h a t 0~f i s a known f u n c t i o n o f T _ ,  f,  TJJ,  next  a n d k (we do n o t i n f a c t  know t h i s f u n c t i o n , b u t i n  the  s e c t i o n we d e s c r i b e h o w , b y m a k i n g s e v e r a l a p p r o x i m a t i o n s we c a n move ( T o u t s i d e t h e  not l i s t e d here, and t r e a t the  f  integral sign i n  [1]  c a l c u l a t i o n p h e n o m e n o l o g i c a l l y ) ; and  6) the neglect and t h o s e e f f e c t s  o f such t h i n g s as m u l t i p l e  scattering  m e n t i o n e d a s m i s c e l l a n e o u s i n s e c t i o n 1.2  (we  s i m p l y c o n s i d e r these unimportant t o our c a l c u l a t i o n ) . Our r e s u l t s w i l l ment may n o t be w a r r a n t e d . results  confirm,  i n d i c a t e t h a t a more a c c u r a t e The p i o n p r o d u c t i o n p r o c e s s ,  i s dominated b y p i o n a b s o r p t i o n and t h e  production rate,  same a p p r o x i m a t i o n s a s we l i s t  developed of the  n  t  above.  The most  o  [1])'.  (we c a n do t h i s b y p u t t i n g  He l i m i t e d t h e i r d e n s i t y d i s t r i b u t i o n  t o h a v i n g a s q u a r e s h a p e (he p u t />(r) n u c l e a r r a d i u s and t o z e r o beyond)  equal to a constant  and chose t o use a  cases h i s choice of parameters d i f f e r e d  s i g n i f i c a n t l y f r o m o u r s , w h i c h we d i s c u s s i n a l a t e r  up t o  square-  shaped n e u t r o n b l a n k e t w h i c h i s not a p a r t o f our m o d e l . number o f o t h e r  make  e a r l i e r m o d e l s i s t h a t o f L i l l e t h u n who i g n o r e d  t h e momentum o f t h e s t r u c k n u c l e o n s 0*3(0) i  our  both o f w h i c h are accounted f o r by our model.  e s s e n t i a l l y the  =  as  basic  The e a r l i e r m o d e l s ( d e s c r i b e d i n s e c t i o n 1 . 3 )  f(k)  treat-  In a quite  section.  a  - 13 2.2  An e s s e n t i a l s i m p l i f i c a t i o n Our  model as expressed by [1] cannot be used t o  c a l c u l a t e a c r o s s - s e c t i o n u n t i l t h e dependence o f Of on Tp, If, T^,  and k i s known.  I t i s because t h i s dependence i s not known  t h a t we must make the f o l l o w i n g  simplification.  Consider t h e momentum i n t e g r a l o f [ 1 ] , v i z : C4]  CT (r) k  =  /0 (T ,f,T ,k).f(k).d3k. f  p  f r  In a d d i t i o n t o angle t h e f r e e c r o s s - s e c t i o n , Of, depends on the k i n e t i c energies  T  p  and T,,. and on t h e momentum k, a l l o f which we  assumed were measured r e l a t i v e t o a l a b o r a t o r y frame.  Alter-  n a t i v e l y , t h e f r e e c r o s s - s e c t i o n depends on an angle ljP and on k i n e t i c energies  T  s t r u c k nucleon. [5]  p  and T^, a l l measured i n the r e s t frame o f t h e  We can w r i t e  CTffTp^T^k)  :  0f(T ,ip,T;,0), p  showing t h a t the f r e e c r o s s - s e c t i o n does not, i n the moving frame, depend e x p l i c i t l y on momentum s i n c e by d e f i n i t i o n k  T =  0.  This  does not help us i n t e g r a t e however s i n c e there i s an i m p l i c i t Independence i n the k i n e t i c energies formation  T  p  and T£, i . e . , i n the t r a n s -  back t o t h e l a b o r a t o r y frame.  The t r a n s f o r m a t i o n i s  g i v e n by the f o l l o w i n g r e l a t i v i s t i c r e l a t i o n s : [6]  T  p  -  ( T T - kp.kj/m + T. + T  z  ( V  p  p  and [7]  T»  - k^.kj/m  +  (u/m)T  + T„,  - 19 -  w h i c h a r e d e r i v e d i n A p p e n d i x D. k i n e t i c energy corresponding corresponds  t o T^, i  t o k a n d kj^ i s t h e momentum w h i c h  p or IT,  =  T i s the struck-nucleon  All unprimed q u a n t i t i e s a r e i n  the l a b o r a t o r y system; m i s t h e nucleon  r e s t mass (933 MeV) a n d  u i s t h e p i o n r e s t mass (140 M e V ) . L e t u s i n t r o d u c e t w o more  approximations:  1) t h e n e g l e c t o f t h e e f f e c t o f t h e s t r u c k - n u c l e o n momentum, k, o n t h e a n g l e  o f p i o n emission, which l e t s us w r i t e  U  i s a t l e a s t p a r t i a l l y j u s t i f i e d by t h e  f =  U ( t h i s approximation  high incident proton  energy, w h i c h keeps s m a l l .the.rianguiarJspread  c a u s e d b y c h a n g i n g k, a n d b y t h e f r e e c r o s s - s e c t i o n , w h i c h v a r i e s o n l y moderately w i t h p r o d u c t i o n a n g l e ) ; and 2) t h e n e g l e c t o f t h e e f f e c t o f t h e s t r u c k - n u c l e o n momentum, k, o n t h e e n e r g y o f t h e p i o n e m i t t e d , w h i c h l e t s u s p u t T£  =  T ( t h i s approximation f T  the f i r s t ) .  has a j u s t i f i c a t i o n  The s e c o n d a p p r o x i m a t i o n The  spread  in' pion k i n e t i c  similar t o that of  s h o u l d be d i s c u s s e d . energy caused by changing  k i s n o t t o o l a r g e i f compared t o t h e c o r r e s p o n d i n g i n e f f e c t i v e proton  energy.  that f o r the high proton reasonably  caused  I t c a n be s e e n ( b y u s i n g numbers)  energy being  c o n s i d e r e d and f o r a  h i g h p i o n e n e r g y (30 MeV s a y ) T^ i s a l w a y s b e t t e r  a p p r o x i m a t e d b y T„ ( e q u a t i o n  [7]) t h a n Tp* i s b y T  even though i n t h e extreme cases  d i f f e r e n c e i s not always g r e a t .  p  ( e q u a t i o n [6])  o f head-on and t a i l - o n  a t h i g h T (where n e i t h e r a p p r o x i m a t i o n  important  spread  collision  i s v e r y good) t h e  The d i f f e r e n c e becomes more  h o w e v e r when we n o t e t h a t t h e f r e e c r o s s - s e c t i o n f o r  -  -  20  p i o n p r o d u c t i o n i s , but not without e x c e p t i o n , more s e n s i t i v e t o a s m a l l change i n T  than i t i s t o a s i m i l a r one i n T^.  p  Thus  our model w i l l , by u s i n g the above approximations, account f o r the momentum e f f e c t o n l y as i t a l t e r s the amount o f energy a v a i l a b l e t o the p r o d u c t i o n r e a c t i o n . listed  approximations  i n t h i s s e c t i o n do not a f f e c t the model i n any other Putting f  Taylor series i n C*]  The two  = f,  =  T , and expanding ff  about the p o i n t T  ^(Tp.f.Vk)  =  way.  [5] by a  g i v e s us  p  0 X ^ , 1 ^ , 0 )  = OfCTp.^.O) «. +  dfffCTp^T^OMTp  + A l t e r n a t i v e l y , we  can expand [5]  - T ) p  *  ... i n T^, ¥', and T^ and, i n the  approximation d e s c r i b e d above, drop terms c o n t a i n i n g dOf/df, (¥' - ¥ ) , dOf/dT^, or (T,» - T ^ ) .  The r e s u l t i s the same.  If  we drop h i g h - o r d e r terms and put [8j  i n t o [ 4 ] , u s i n g [6] t o note  C9)  rs,  /  (T  p  - T )-f(k)-d>k p  where y i s the f a c t o r (T  + m)/m  r  and T i s the l o c a l mean v a l u e o f  the s t r u c k - n u c l e o n k i n e t i c energy, we UOJ  Q (r) k  z  (J (T . y . T J f  +  get the r e s u l t  that  dffftTp.f.Tj^. dT P v  We  have stopped i n d i c a t i n g e x p l i c i t l y the f a c t t h a t k'  our l a s t approximation we  =  0.  remove the r-dependence from a l l  As terms  -  in  [ 1 0 ] except  Tp  and 1  n  2 1 -  T b y assuming t h a t , f o r t h e purpose o f computing  from given values o f E  p  a n d E„. o n l y , t h e n u c l e a r  p o t e n t i a l s d e p e n d j u s t on mass number A . [I]  We f i n a l l y f i n d  that  c a n b e w r i t t e n i n t h e f o l l o w i n g manner: (nEp.V.Ej  [II]  =  C (T p . Y . T j . l ! f  •  d0- (T  ^T^.Tlz,  f  d T  p  where  I±  z  f e x p ( - ^ h d s - ^ n ^ d s ) «o(r).d3r  and  12  = f T ( r ) . e x p ( - / n _ d s - / n ^ d s ) «>o(r) «d3r.  p  °p"  °1T  T h i s i s t h e e x p r e s s i o n on w h i c h o u r c a l c u l a t i o n s are; b a s e d . The  proton-nucleus  i s now e x p r e s s e d rather than  cross-section f o r pion  production  i n t e r m s o f mean s t r u c k - n u c l e o n k i n e t i c  i n t e r m s o f a p a r t i c u l a r momentum d i s t r i b u t i o n .  T h i s s i m p l i f i e d v e r s i o n o f o u r m o d e l m u s t be u s e d u n t i l a s we h a v e more i n f o r m a t i o n a b o u t t h e b a s i c production process. for  energy  time  proton-nucleon  I n [ l l ] the proton-nucleon  p i o n p r o d u c t i o n and i t s f i r s t  such  cross-section  derivative with respect t o  p r o t o n e n e r g y a p p e a r a s c o e f f i c i e n t s t o i n t e g r a l s w h i c h c a n be e a s i l y evaluated. and  I n o u r c a l c u l a t i o n we e v a l u a t e b o t h  use the c o e f f i c i e n t s t o adjust r e s u l t s t o experimental  on t h e p r o t o n - n u c l e u s our  I ^ a n d I2  calculated  production pf_pion&  The A - d e p e n d e n c e o f  c r o s s - s e c t i o n s enters only through  1-^ a n d I2:  cannot take i n t o account t h e A-dependence e n t e r i n g through c o e f f i c i e n t s a p p e a r i n g w i t h them. any  d e p e n d e n c e on i n c i d e n t p r o t o n  through  the coefficients.  data  We c a n n o t t a k e  into  energy s i n c e i t enters  we the:  account only  Our l a c k o f knowledge c o n c e r n i n g t h e  - 22 -  free production  process a l s o keeps us from knowing j u s t when the make i n going from [1]  assumptions we  I f we wish t o ignore  t o [11]  need o n l y s e t T  =  as 0.  r e s u l t w i l l be the same model we would get by p u t t i n g f ( k ^  o^'(0). i n t o Cl].  With the a p p r o p r i a t e  o f yo(r) we  choice  have the model o f L i l l e t h u n , but without the neutron  2.3  invalid.  the momentum e f f e c t , as w e l l  the approximations made i n t h i s s e c t i o n , we The  might be  The  =  would  blanket.  parameters chosen To c a l c u l a t e a proton-nucleus c r o s s - s e c t i o n f o r p i o n  production  [11]  using  we  must s p e c i f y a b s o r p t i o n  coefficients, a  d e n s i t y d i s t r i b u t i o n , a momentum d i s t r i b u t i o n (a mean k i n e t i c energy at l e a s t ) , and  a f r e e proton-nucleon c r o s s - s e c t i o n .  f r e e c r o s s - s e c t i o n we  cannot s p e c i f y e x a c t l y hence we  calculate  o n l y r e l a t i v e proton-nucleus c r o s s - s e c t i o n s u s i n g the f r e e s e c t i o n as a n o r m a l i z i n g we  comment on how  parameter.  the value we  cross-  When d i s c u s s i n g our r e s u l t s  must assume f o r the f r e e  s e c t i o n compares with:the p o o r l y known experimental The  The  d e n s i t y d i s t r i b u t i o n t h a t we  cross-  one.  choose has  the  f a m i l i a r Saxon-Woods form, v i z : [12]  Mr)  -  >0 .(l o  +  i n which the n u c l e a r r a d i u s R  exp((r - R ) / a ) ) - l , ( f o r nucleon d e n s i t y and  the  o p t i c a l p o t e n t i a l s o f protons and  pions) i s the u s u a l o p t i c a l  model r a d i u s R  1.25  r^A / 1  =  3  with r  Q  =  fermis.  T h i s i s the  -  value used i n our work. l a r g e r value  (r  0 =  1.35  23  -  L i l l e t h u n and e a r l i e r workers used a fm.)  and because the p i o n p r o d u c t i o n r a t e  i n a heavy nucleus i s l a r g e l y c o n t r o l l e d by p i o n a b s o r p t i o n the difference i s significant.  Our r e s u l t s show t h i s .  R i s the  r a d i u s where the d e n s i t y i s 50% o f i t s c e n t r a l value and i s not t o be confused w i t h R x >  t  n  r a d i u s beyond which a b s o r p t i o n i s  e  ma  assumed n e g l i g i b l e . nucleus.  The two  c o i n c i d e o n l y i n a square-edge  F o r our d i f f u s e - e d g e nucleus we  a r b i t r a r i l y set R ax m  t o be the r a d i u s where the d e n s i t y has f a l l e n t o 10% o f i t s c e n t r a l v a l u e so t h a t R  and R t o g e t h e r d e f i n e the value S  m a x  2»  =  •(Rmax - R) which i n t u r n d e f i n e s the s u r f a c e t h i c k n e s s constant = S/(4»ln3).  a  For a we  have chosen a standard value o f 0 . 5 5  Our r e s u l t s are not v e r y s e n s i t v e t o a. constant o  use i n the i n t e g r a n d s o f [111 >o  0  =  £> »N 0  The n o r m a l i z a t i o n  i s v a r i e d a c c o r d i n g t o the context o f i t s use.  0  f>  t a 0  fm.  k e s the form o  Q  For  /o »Z or  =  0  depending on whether p o s i t i v e or negative p i o n produc-  t i o n i s b e i n g c o n s i d e r e d , where >o  0  [13]  >6  0  =  has the standard  value  (3/(4frR )).(l + tr a /R ). 3  2  2  2  When used alone /o£ normalizes the d e n s i t y d i s t r i b u t i o n t o u n i t y . F o r use w i t h o p t i c a l p o t e n t i a l s and a b s o r p t i o n c o e f f i c i e n t s , i n equations l i k e V ( r ) - / o ( r ) . V form fi - > ° Q  ,v 0  0  and n ( r )  =  / o ( r ) . n , /o Q  ° l > where V o l i s the n u c l e a r volume  Q  takes the  (4/3JtR . 3  F o r the l o c a l mean k i n e t i c energy o f the t a r g e t nucleons we  choose a value corresponding t o the momentum d i s t r i -  b u t i o n o f a Fermi gas a p p r o p r i a t e t o the d e n s i t y > o ( r ) , v i z :  - 24  T(r)  [14] where  0.6T (r),  =  Tf(r)  f  i O A f r ) ) /3 2  =  i s t h e F e r m i e n e r g y a t r a n d p{r) The energies  proton  of interest  i n a l l our  I t was  proton-neutron  i s normalized  t o u s : we  have chosen n  cross-sections.  fm.~^ by  from t o t a l proton-proton  P i o n a b s o r p t i o n on t h e  other  the t a r g e t nucleus.  At any  kinetic  point r inside  i s related  Eff  where V ( r ) c  =  1„  +  V (r) c  In our  calculation V (r) c  t i o n t h a t Z protons radius R  c  =  r  i s t h e Coulomb p o t e n t i a l  part of the pion o p t i c a l p o t e n t i a l , energy.  V (r,T  +  [16]  V (r)'  and  -v" (r.)  c  c  d e p e n d i n g on p i o n  =  =  f .  we  m  therefore  Ze£._J^.(3 -  0  Z§£.1 4rre r  the numerical  the  take  r2/R2)  f o r  r  for r  <  R  >R„, c  value  of  1.44.  real  kinetic  i s computed u n d e r t h e  0  w h e r e e /(4ffC ) h a s  i s the  r  are d i s t r i b u t e d uniformly inside  1.07AV3  by  ),  f f  v" (r,Tn.)  and  the  to i t s total  e n e r g y , i . e . , i t s k i n e t i c e n e r g y o u t s i d e t h e n u c l e u s , B,,., [15]  and  h i g h l y energy dependent.  the p i o n k i n e t i c energy, T ^ ,  nucleus  N,  the  P i o n a b s o r p t i o n c o e f f i c i e n t s d e p e n d on p i o n energy i n s i d e  Z or  0.182  t o be  T h i s i s a l s o the value used  computed by him  hand i s s t r o n g and  to either  a b s o r p t i o n i s r e l a t i v e l y weak a t  calculations.  Lillethun.  -  assumpcharge  -  25 -  Values o f V ^ r , ^ ) v s . T et a l .  f  f  have been computed by Frank  We use t h e i r v a l u e s c o r r e c t e d by a f a c t o r o f (  r 0  /  r 0  )^>  where r£ i s t h e b a s i c n u c l e a r r a d i u s assumed by Frank e t a l . and r  Q  i s our v a l u e o f 1 . 2 5 fm.  Frank e t a l . i m p l i c i t l y assume a  v a l u e o f 1.4-1 fm. f o r r£ when they s e t A t h e i r Table I I ) . (3)) R s A* A A " ^ (1.41 fm.).  =  1 ( c f . the c a p t i o n t o  A i s d e f i n e d by the r e l a t i o n  (their  equation  , where X' i s the Compton wavelength f o r pions  3  The c o r r e c t i o n which we make t o V  s i n c e A e n t e r s i n t o the e x p r e s s i o n g i v i n g V ( c f . Frank e t a l . , t h e i r equation  (13)).  r  r  i s appropriate  as an i n v e r s e cube  We a l s o a p p l y a  c o r r e c t i o n f o r l o c a l nucleon d e n s i t y , i o ( r ) , normalized volume ( c f . t h e i r equation  (1)) and by manipulating  to nuclear  [15] compute  the p i o n k i n e t i c energy, T^, a t r from the given p i o n t o t a l energy, E ^ . Having thus counted T^ a t a p o i n t r we can i n t e r polate f o r n  ff  u s i n g t h e v a l u e s o f n„. v s . T , , . a l s o g i v e n by Frank  et a l . (they g i v e v a l u e s o f p i o n mean f r e e path v s . H ,  but these  are j u s t r e c i p r o c a l s o f the a b s o r p t i o n c o e f f i c i e n t s ) .  Three  n  c o r r e c t i o n s must be a p p l i e d t o the i n t e r p o l a t e d value: 1) a c o r r e c t i o n f o r u n i t s , i n v e r s e Compton wavelengths t o i n v e r s e f e r m i s ( c f . Frank e t a l . , t h e i r Table I ) ; 2) a c o r r e c t i o n f o r l o c a l nucleon  d e n s i t y , /o(r), normalized  t o volume; and 3) the  c o r r e c t i o n mentioned i n connection w i t h V r concerning t h e b a s i c nuclear radius, r  Q  .  The l a s t c o r r e c t i o n , a p p l i e d t o n^, i s  again one o f (r^/r^)-? obtained from an i n s p e c t i o n o f Frank e t a l . , equations  ( 6 ) , (7) and ( 9 ) .  T h e i r equation  ( 9 ) , g i v i n g n^., i s  -  26 -  i s d e r i v e d u s i n g a model o f Brueckner e t a l . ^ ' which assumes v  x  t h a t p i o n a b s o r p t i o n by n u c l e a r matter i s p r o p o r t i o n a l t o the p i o n capture equation  c r o s s - s e c t i o n o f a deuteron.  We r e - d e r i v e t h i s  i n Appendix B t o show i t s i n v e r s e cube dependence on A,  not shown e x p l i c i t l y by Frank e t a l . A l t e r n a t i v e l y , we have taken f o r our pion c o e f f i c i e n t s those v a l u e s l i s t e d by L i l l e t h u n  ( i n h i s Table V ) ,  who a l s o uses the data o f Frank e t a l . , but without c o r r e c t i o n s d e s c r i b e d above. technique  absorption  a p p l y i n g the  L i l l e t h u n uses an i n t e r p o l a t i o n  which seems t o be d i f f e r e n t from ours s i n c e , i f f o r no  o t h e r reason, h i s computed v a l u e s o f n  ff  are not a t a l l smooth  f u n c t i o n s o f A and we t h i n k they should be. Nuclear  s i z e i s important  t o the a b s o r p t i o n e f f e c t i n  two ways: 1) the d i s t a n c e a p a r t i c l e t r a v e l s i n s i d e a  nucleus  depends d i r e c t l y on the n u c l e a r s i z e ; and 2) the a b s o r p t i o n c o e f f i c i e n t depends on l o c a l nucleon depends on the n u c l e a r s i z e .  d e n s i t y , which i n "turn  The f i r s t dependence i s r o u g h l y  l i n e a r i n the n u c l e a r r a d i u s and the second i s one i n v o l v i n g the i n v e r s e cube o f the n u c l e a r r a d i u s . t i o n depends on the product  Hence, s i n c e t o t a l absorp-  o f d i s t a n c e t r a v e l l e d and a b s o r p t i o n  c o e f f i c i e n t , we have the seemingly c u r i o u s r e s u l t t h a t the s m a l l e r o f two e q u a l l y massive n u c l e i i s the s t r o n g e r  absorber.  I t must not be f o r g o t t e n however t h a t the s m a l l e r nucleus  i s also  the s m a l l e r t a r g e t , i . e . , t h a t the i n t e g r a t i o n s o f [ l l ] are over a s m a l l e r volume.  The net balance  p i o n y i e l d we c a l c u l a t e i n a s p e c i a l  o f these and other e f f e c t s on case.  -  27  -  CHAPTER I I I - THE CALCULATION  3.1  The i n t e g r a t i o n Our  the  calculations required the numerical evaluation o f  i n t e g r a l s 1 ^ and I  o f equation  2  [11].  T h i s we d i d u s i n g t h e  U n i v e r s i t y o f B r i t i s h C o l u m b i a ' s I B M 7 0 4 4 c o m p u t e r a n d a FORTRAN I V p r o g r a m c a l l e d PIPROD, t h e l o g i c o f w h i c h i s o u t l i n e d i n A p p e n d i x F.  I n t h i s program t h e i n t e g r a n d s o f 1 ^ and I  each evaluated throughout  the nucleus  (10~  1 3  are  over a network o f p o i n t s ,  none o f w h i c h i s s e p a r a t e d f r o m i t s n e a r e s t n e i g h b o u r than a f r a c i o n o f a fermi  2  cm.).  b y more  The i n t e g r a t i o n i s  c o m p l e t e d b y a n a p p r o p r i a t e number o f S i m p s o n ' s r u l e a p p r o x i mations.  The e r r o r i n t r o d u c e d b y e a c h a p p r o x i m a t i o n  i s esti-  mated b y r e - c a l c u l a t i n g t h e p a r t i c u l a r i n t e g r a l over a c o a r s e r network o f p o i n t s obtained by d o u b l i n g the s e p a r a t i o n d i s t a n c e . In  o u r c a l c u l a t i o n t h e coarse and f i n e i n t e g r a l s d i f f e r e d by an  amount t h a t was u s u a l l y l e s s t h a n t w o p e r c e n t a n d v e r y more t h a n f i v e p e r c e n t The  of the f i r s t ,  i . e . , the fine  seldom  integral.  t r u e e r r o r , i . e . , t h e d i f f e r e n c e between t h e f i n e and a n  i n f i n i t e l y f i n e i n t e g r a l , a t e a c h s t a g e s h o u l d a l w a y s be much s m a l l e r t h a n t h e one e s t i m a t e d  i n t h e above manner.  At each p o i n t o f t h e network t h e paths have t o be d e t e r m i n e d .  s  p  and s ^  A t each o f s e v e r a l p o i n t s ( s e p a r a t e d by  l e s s than fermi) along s  n  "the p i o n o p t i c a l p o t e n t i a l  }  V , and t h e  p i o n a b s o r p t i o n c o e f f i c i e n t , n , r , h a v e t o be d e t e r m i n e d . did  r  T h i s we  b y f i t t i n g a t h i r d o r d e r p o l y n o m i a l t o t h e known d a t a o f  -  28  -  Frank e t a l . ( d i f f e r e n t a t each p o i n t s i n c e we  i n c l u d e a density-  c o r r e c t i o n ) and i n t e r p o l a t i n g a t the p i o n k i n e t i c energy interest  (which a l s o v a r i e s from p o i n t t o p o i n t )  The  of nn- along Sfr, l i k e the i n t e g r a t i o n o f 1^ and J-2, was  of  averaging done i n  the Simpson's r u l e approximation w i t h an e r r o r e s t i m a t i o n . V a r i a t i o n s run on the PIPROD program are not cussed i n t h i s t h e s i s .  V a r i a t i o n s were run however and  outcomes c o n s t i t u t e the s u b j e c t matter  3.2  R e s u l t s compared t o  of s e c t i o n  distheir  3.3.  experiment  A t y p i c a l r e s u l t o f our main c a l c u l a t i o n i s i l l u s t r a t e d i n F i g u r e 2, f o r the case i n which we have i n c i d e n t of 450 MeV  producing p i o n s of 83 MeV  l a b o r a t o r y system. Lillethun.  The  The  protons  a t an angle o f 21.5° i n the  experimental c r o s s - s e c t i o n s are those o f  c a l c u l a t e d curves have been normalized t o the  experimental value a t  2  ?A1.  The  curve marked L i s L i l l e t h u n ' s  p u b l i s h e d r e s u l t , c a l c u l a t e d from h i s model w i t h an absorbing neutron b l a n k e t .  We were a b l e t o d u p l i c a t e t h i s r e s u l t i n our  work by u s i n g L i l l e t h u n ' s neutron b l a n k e t and parameters,  these  including his calculated absorption c o e f f i c i e n t s . The  curve marked 1^  i n F i g u r e 2 i s our r e s u l t ,  cal-  c u l a t e d u s i n g our model as expressed by [11] and the parameters and c o r r e c t i o n s d e s c r i b e d i n s e c t i o n 2 . 3 .  We  have p l o t t e d the  i n t e g r a l I]_ o n l y , thereby i g n o r i n g the e f f e c t of a momentum distribution.  The  i n t e g r a l I 2 , which a l l o w s us t o take a  - 29 -  300  C T |JB/MEV/ST. £ , + = 8 3 MEV Ep =450MEV  200  100  -  M A S S NO.. A 0  0  FIGURE 2  100  200  C r o s s - s e c t i o n s f o r the p r o d u c t i o n o f 33-MeV p o s i t i v e pions a t 21.5* by 450-MeV protons on v a r i o u s n u c l e i .  - 30 -  momentum e f f e c t i n t o account, plotted  curve,  c o i n c i d e s almost e x a c t l y w i t h the at 7 A 1 .  when normalized  The r a t i o I 2 / I 1  2  F  O  R  the momentum d i s t r i b u t i o n o f a Fermi gas i s shown as an i n s e t t o Figure 2 . .  The i n d i c a t i o n  i s t h a t the e f f e c t o f a momentum  d i s t r i b u t i o n on the r e l a t i v e p i o n y i e l d i s s m a l l , although the e f f e c t may be important  when choosing n o r m a l i z i n g f a c t o r s  dOf/dTp t o compare with experimental production o f pions.  0~f and  data on the proton-nucleon  Our model i s not y e t a b l e t o t r e a t  pion  p r o d u c t i o n a t proton e n e r g i e s near the p r o d u c t i o n t h r e s h h o l d where t h e momentum e f f e c t i s c e r t a i n t o be more important,  but a t  the energy o f 4 5 0 MeV i t suggests  effect.  t h a t we may n e g l e c t t h i s  We have thus i g n o r e d t h e e f f e c t o f a momentum d i s t r i b u t i o n i n p l o t t i n g our r e s u l t s .  There a r e two other A-dependent  e f f e c t s which we have i g n o r e d , one i n d e r i v i n g p l o t t i n g our c u r v e s .  They a r e : 1)  neutrons on p o s i t i v e  pion production  e f f e c t by i n c l u d i n g  [ 1 1 ] and one i n  the e f f e c t o f t a r g e t - n u c l e u s (we could estimate  i n our r e s u l t the f a c t o r  this  (10«Z + N)/11.A«Z  which removes t h e n o r m a l i z a t i o n imposed on >o(r) and weights the proton-proton  and proton-neutron  free  c r o s s - s e c t i o n s i n :,.  accordance w i t h p i o n p r o d u c t i o n proceeding mediate 3 - 3 resonnance s t a t e :  through an i n t e r -  i t would r a i s e a l l p i o n y i e l d s , but  would e s p e c i a l l y favour high v a l u e s o f A, by 5% i n -^Pb as shown 2<  i n F i g u r e 2 f o r example); and t h e r e i s 2 ) the e f f e c t due t o the c o e f f i c i e n t s Of, dOf/dTp and 7 , which have an A-dependence t h a t e n t e r s through t h e p o t e n t i a l  terms used t o r e l a t e k i n e t i c  energy  i n the nucleus w i t h t o t a l energy o u t s i d e ( t h i s e f f e c t i s hard t o  -  31  -  even e s t i m a t e , but i t p r o b a b l y lowers s l i g h t l y the r e l a t i v e  yield  from a heavy nucleus a t our e n e r g i e s ) . The v a l u e f o r Of which we  i m p l i c i t l y assume i n  n o r m a l i z i n g our r e s u l t o f F i g u r e 2 t o the experimental c r o s s 27 section of  A l i s 31.0  microbarns/MeV/steradian.  i n t e g r a l 12 i n t o our r e s u l t we  P u t t i n g the  can lower t h i s value t o n e a r l y  25 ub/MeV/st. by t a k i n g f o r d O f / d T / 0 f the value 0.015, which i s p  the v a l u e o f the l o g a r i t h m i c d e r i v a t i v e a t 450 MeV  taken from .  data on the t o t a l c r o s s - s e c t i o n s 0(pp-*rr+d) and 0"(pp-»- tr+pn) v s . p r o t o n energy as summarized by M c l l w a i n e t a l . ^ ^ . 2  Pondrom^  uses 450-MeV protons t o o b t a i n d i f f e r e n t i a l c r o s s - s e c t i o n s a t  20°27» f o r the r e a c t i o n pp-* tr+pn.  His v a l u e i s almost  / s t . a t the p i o n energy o f 122 MeV,  t h i s b e i n g the k i n e t i c  needed by a p o s i t i v e p i o n i n s i d e an ^A1 2  seen o u t s i d e w i t h an energy o f 83 MeV.  I f we  energy  nucleus i f i t i s t o be Gell-Mann and W a t s o n ^ ^ x  i n d i c a t e t h a t 70% of the p o s i t i v e pions coming from nucleon r e a c t i o n s a t 300-400 MeV  8 ub/MeV/  proton-  are due t o the r e a c t i o n pp-** d. +  assume t h a t t h i s i s a l s o t r u e a t 450 MeV  and t h a t t e n times  the number t h a t come from pn-*tr+nn come from the two pp r e a c t i o n s , then we  can estimate a p o s s i b l e experimental value f o r Of o f j u s t  over 31 ub/MeV/st.  T h i s l a s t v a l u e i s v e r y u n c e r t a i n , but i t i s  nonetheless c l o s e t o our assumed v a l u e and f o r t h a t reason i t i s encouraging. Illustrated pion p r o d u c t i o n at  450  s c a t t e r i n g a n g l e s as  i n F i g u r e 3 are MeV, b u t  indicated.  at  some r e s u l t s  on  positive  o t h e r p i o n e n e r g i e s and The  pion  e x p e r i m e n t a l p o i n t s and  the  - 32 -  1  1  1  i  E = 450MEV P  Y=21.5° 150  E^F 166 MEV  o-o-L-^-  50  co >  LU  I  0  I  1  I  CO  b  Y = 2i.5° £ ^ = 2 3 6 MEV  50  ')  ——•  ——°  r>.  ^^xy-^ I  0  I  IjJ  50  Q  u  6  \  I  I  l  I  = 60°  i  100  I . °  *- —  E^r 14 9 MEV  ^  FIGURE 3  .  o  200  More c r o s s - s e c t i o n s f o r p o s i t i v e p i o n p r o d u c t i o n .  - 33 -  curves marked L a r e once a g a i n those o f L i l l e t h u n .  What we have  s a i d about t h e r e s u l t i n F i g u r e 2 can a l s o i n g e n e r a l be s a i d about the r e s u l t s i n F i g u r e 3. Some r e s u l t s on n e g a t i v e p i o n p r o d u c t i o n a t 450 MeV are g i v e n i n F i g u r e 4.  Pion e n e r g i e s and angles a r e i n d i c a t e d  and a g a i n t h e experimental p o i n t s and t h e curves marked L a r e due to L i l l e t h u n .  I n the case o f n e g a t i v e pions the remark we made  i n c o n n e c t i o n w i t h the r e s u l t i n F i g u r e 2 concerning a c o r r e c t i o n f a c t o r f o r t h e t a r g e t - n u c l e u s neutron c o n t r i b u t i o n t o the pion y i e l d no l o n g e r a p p l i e s s i n c e the neutrons o n l y t h e neutrons, a r e c o n s i d e r e d . b u t i o n whatever.  o f the t a r g e t , and  There i s no proton  contri-  The remark made about the A-dependence o f t h e  n o r m a l i z i n g parameters s t i l l  applies.  Agreement between our c a l c u l a t e d curves and the data o f L i l l e t h u n i s seldom e x c e l l e n t and i t i s never r e a l l y I t i s o f t e n as good as any p r e v i o u s l y o b t a i n e d .  poor.  We show i n the  next s e c t i o n how the a b s o r p t i o n e f f e c t , e s p e c i a l l y p i o n absorpt i o n , dominates the proton-nucleus p r o d u c t i o n p r o c e s s .  It  t h e r e f o r e seems l i k e l y t h a t the use o f d i f f e r e n t , more modern data on p i o n a b s o r p t i o n might l e a d t o d i f f e r e n t and perhaps b e t t e r agreement than we o b t a i n .  Hence agreement between our.  c a l c u l a t e d r e s u l t s and those o f experiment as being c r i t i c a l t o our model.  should not be viewed  The importance  o f our r e s u l t s  i s o f another k i n d : good agreement p r e v i o u s l y depended on the neutron b l a n k e t assumption the next  section.  and now i t does not.  We show why i n  - 34 -  1  30 U  20  :  E =450MEV P  IjJ = 21.5 132 MEV  10  0  .10  =21.5 E^.1.66 MEV  LU CO  D  10  0 10  l|l=2f.5°,  E^200MEV  0 Y = 60  99MEV 10  FIGURE 4  Some c r o s s - s e c t i o n s f o r n e g a t i v e  pion  production.  -  3.3  -  35  R e s u l t s w i t h parameters v a r i e d  In  o r d e r t o d e t e r m i n e t h e manner i n w h i c h t h e v a r i o u s  p h y s i c a l f a c t o r s c o n t r o l l i n g the proton-nucleus process  pion  production  a f f e c t t h e A - d e p e n d e n c e o f t h e p i o n y i e l d , we c a l c u l a t e d  c r o s s - s e c t i o n s u s i n g parameters other than those  described i n  section 2 . 3 .  The r e s u l t s o f t h e s e  in  As f o r F i g u r e 2 we h a v e c o n s i d e r e d 450-MeV  F i g u r e 5.  calculations are i l l u s t r a t e d protons  83-MeV p o s i t i v e p i o n s a t 2 1 . 5 ° a n d a g a i n c u r v e , L i s  producing  Lillethun s 1  c a l c u l a t e d r e s u l t , w h i c h we w e r e a b l e t o d u p l i c a t e  using a neutron  b l a n k e t and h i s parameters, these  calculated absorption  including h i s  coefficients.  Curve I i s t h e r e s u l t o f u s i n g a n e u t r o n  b l a n k e t and  L i l l e t h u n * s parameters, but using our i n t e r p o l a t i o n technique t o calculate pion absorption coefficients. e f f e c t o f removing t h e neutron s i d e r i n g the protons  C u r v e I I shows t h e  b l a n k e t , i . e . , t h e e f f e c t o f con-  and neutrons  uniformly distributed within  t h e n u c l e a r r a d i u s o f L i l l e t h u n , u s i n g t h e same p a r a m e t e r s a n d i n t e r p o l a t i o n technique  a s we u s e d f o r c u r v e  I.  C u r v e I I I , on  t h e o t h e r h a n d , was o b t a i n e d b y u s i n g a v a l u e o f 1 . 2 5 f m . f o r r a Saxon-Woods nique  d e n s i t y d i s t r i b u t i o n , and o u r i n t e r p o l a t i o n  t o compute p i o n a b s o r p t i o n .  s e c t i o n 2 . 3 were a l l a p p l i e d .  o b t a i n e d when, s t i l l  The c o r r e c t i o n s d e s c r i b e d i n  Hence c u r v e  F i g u r e 2 , t h e momentum e f f e c t s t i l l  tech-  being  I I I i s our r e s u l t i n  ignored.  Curve IV i s  i g n o r i n g t h e momentum e f f e c t , we remove t h e  d i f f u s e edge f r o m t h e m o d e l f o r c u r v e  I I I by s e t t i n g a  =  0.  Q  ,  - 36  -  7 z  FIGURE 5  Cross-sections  o b t a i n e d by v a r y i n g  parameters.  -  -  37  Otherwise t h e models used f o r curves The s u g g e s t i o n  I I I and IV a r e i d e n t i c a l .  h e r e i s t h a t , l i k e t h e e f f e c t o f a momentum  d i s t r i b u t i o n , the e f f e c t o f a d e n s i t y d i s t r i b u t i o n i s secondary when c o m p a r e d t o t h a t o f a b s o r p t i o n .  The i m p o r t a n c e o f t h e  a b s o r p t i o n e f f e c t i s i n d i c a t e d by t h e f a c t t h a t without  absorp-  t i o n t h e p i o n y i e l d becomes d i r e c t l y p r o p o r t i o n a l t o Z ( c u r v e 5).  in Figure  The i m p o r t a n c e o f n u c l e a r  Z  size t o the absorption  e f f e c t i s i n d i c a t e d by t h e f a c t t h a t t h e r e d u c t i o n o f t h e b a s i c nuclear radius, r IV)  lowers  Q  , f r o m 1 . 3 5 fm. ( c u r v e  the r e l a t i v e pion y i e l d  I I ) t o 1 . 2 5 fm.  i n a heavy nucleus  by almost  e x a c t l y t h e same amount a s d o e s t h e i n t r o d u c t i o n o f a n neutron  blanket  (curve  I).  That our agreement w i t h  (curve  absorbing  experiment  i s n o t a s g o o d a s L i l l e t h u n ' s t h e r e f o r e seems t o be due o n l y t o the  f a c t t h a t t h e method we u s e t o i n t e r p o l a t e f o r p i o n  absorp-  t i o n c o e f f i c i e n t s i s d i f f e r e n t t h a n t h e one u s e d b y h i m . we t o u s e L i l l e t h u n ' s m e t h o d i n c a l c u l a t i n g o u r c u r v e would obtain h i s r e s u l t .  3.4  And w i t h o u t  a neutron  Were  I I I we  blanket.  Conclusion We  of pions  have c o n s t r u c t e d a model t o e x p l a i n t h e  i n t h e bombardment b y p r o t o n s  o f v a r i o u s n u c l e i a n d we  have used t h e model t o c a l c u l a t e r e l a t i v e process.  production  cross-sections f o r the  The m o d e l a s s u m e s , among o t h e r t h i n g s , t h a t t h e  incident proton  i n t e r a c t s w i t h but a s i n g l e target nucleon  produce a p i o n and n o t w i t h t h e t a r g e t n u c l e u s assumes t h a t t h e p r o t o n - n u c l e u s  as a w h o l e .  cross-section f o r pion  to It  production  - 33 c a n be c a l c u l a t e d k n o w i n g t h e b a s i c p r o t o n - n u c l e o n  cross-section  f o r p i o n p r o d u c t i o n a n d t h e number o f t a r g e t n u c l e o n s , c e r t a i n important  nuclear e f f e c t s are taken  provided  i n t o account.  n u c l e a r e f f e c t s accounted f o r by o u r model a r e those  The  due t o  p r o t o n and p i o n a b s o r p t i o n , t o n u c l e a r p o t e n t i a l s , and t o t h e s t r u c k - n u c l e o n d e n s i t y a n d momentum d i s t r i b u t i o n s . Because o u r knowledge o f t h e b a s i c  proton-nucleon  p r o d u c t i o n r a t e i s l i m i t e d , we c a n o n l y c a l c u l a t e r e l a t i v e s e c t i o n s and use t h e b a s i c r a t e as a f r e e parameter.  cross-  We do t h i s  i n s e v e r a l s p e c i a l cases  a n d compare o u r r e s u l t s w i t h d a t a  e x p e r i m e n t a t 4 5 0 MeV.  Agreement i s o n l y m o d e r a t e , b u t i t i s a s  g o o d a s a n y p r e v i o u s l y o b t a i n e d .and, u n l i k e t h e e a r l i e r it  results,  d o e s n o t d e p e n d on t h e r a t h e r a r t i f i c i a l a s s u m p t i o n o f a n  absorbing use  from  neutron  blanket.  O u r a g r e e m e n t d e p e n d s i n s t e a d on t h e  o f a modern n u c l e a r r a d i u s a n d o n a r e a s o n a b l e  pion absorption.  treatment  of  I n t h i s r e s p e c t o u r r e s u l t s c o n f i r m what t h e  e a r l i e r w o r k e r s had assumed, t h a t a b s o r p t i o n , e s p e c i a l l y  pion  a b s o r p t i o n , i s t h e dominant p h y s i c a l f a c t o r c o n t r o l l i n g t h e proton-nucleus  production of pions.  At high proton  energies  we  f i n d t h a t t h e e f f e c t s o f a s t r u c k - n u c l e o n momentum d i s t r i b u t i o n are a l l but n e g l i g i b l e pion y i e l d , although  i n determining  t h e A-dependence o f t h e  t h e y may become more i m p o r t a n t  a t lower  e n e r g i e s n e a r t h e p i o n p r o d u c t i o n t h r e s h h o l d w h e r e we hope t o extend  the present  investigation.  Density d i s t r i b u t i o n  effects  we f i n d a r e l i k e w i s e a l m o s t n e g l i g i b l e a t h i g h e n e r g i e s .  These  are our c o n c l u s i o n s .  - 39 REFERENCES  Reference  ( 1 ) d e s c r i b e s the TRIUMF  (Tri-University  Meson F a c i l i t y ) p r o p o s a l , which we mention i n Chapter (1)  I.  TRIUMF P r o p o s a l and Cost E s t i m a t e , E . W. Vogt and J . J . B u r g e r j o n , eds., U n i v e r s i t y of B r i t i s h Columbia (1966), w i t h annual  supplements.  References  (2-11) d e s c r i b e experiments on the proton-  nucleon p r o d u c t i o n o f p i o n s , which we (2)  summarize i n Appendix A.  M. M. B l o c k , S. Passman, and ¥ . W.  Havens, J r . , Phys. Rev.  88, 1239 (1952). (3)  R. H. March, Phys. Rev. 120,  1874  (4)  A. H. R o s e n f e l d , Phys. Rev. £6, 130 (1954).  (5)  L. G. Pondrom, Phys. Rev.  (6)  Y. M. Kazarinov and Y. N. Simonov, Sov. J . N u c l . 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R e v . £ 6 , 140  (1954).  a n d T. H. F i e l d s , P h y s . R e v . 12_Z, 239 The are discussed  (1962).  calculations described i n thetext o f this thesis  i n t w o o t h e r w o r k s , w h i c h we h a v e n o t m e n t i o n e d .  They a r e t h e f o l l o w i n g : (44)  D. J . M c M i l l i n , P. G. B h a r g a v a , L . Lam, a n d E . W. V o g t , C a n . J . P h y s . 4.6, 1141 ( 1 9 6 8 ) ; a n d  (45)  L . Lam, M. S c . T h e s i s , U n i v e r s i t y o f B r i t i s h C o l u m b i a  (1968).  APPENDIX A EXPERIMENTS ON PROTON-NUCLEON: PION PRODUCTION  We l i s t  below t h e values  i n t h e t e x t ) a t which a d i f f e r e n t i a l like  (p + n u c l e o n  values  o f E , ¥, a n d E ^ ( a s d e f i n e d p  cross-section f o ra reaction  t r - +• n u c l e o n s ) h a s b e e n m e a s u r e d . A l l  a r e i n t h e l a b o r a t o r y frame u n l e s s  otherwise indicated.  En  Err  Year  Ref.  Reaction  Block  1952  (2)  PP  381  90  0-95  March  I960  (3)  PP  420  65  23-76  Rosenfeld  1954  (4)  PP  440  55  50-75*  tt  tt  90  15-58  tt  n  124  12-23  450  30°14  tt  tt  20°27«  40-147  n  n  13°14'  45-152  600  0  100-max  tt  20  100-max  tt  0  100-max  tt  20  100-max  654  5.6  all  et a l .  Pondrom  1959  (5)  Kazarinov and  PP  MeV  ?  Author(s)  fT-  -»•  Simonov  1967  (6)  pn  -*  Heer e t e a l .  1966  (7)  PP  •  tr+pn  rr+»-  tt  pn  Heifer et a l .  1961  (3)  a-  4  PP  MeV  deg.  f  42-135  600  PP  •* ft+pn^  tt  56  all  PP  •* TT+d^  tt  56  all  n  56  all  pn  -  44 -  Author(s)  Year  Ref.  Reaction  Bp MeV  Vovchenko  1966  (9)  pp -* rr+  655  Yale r e p o r t  1964  (10)  pp + «+  660  (11)  pp •* "+  725  Haddock e t a l . 1964 @  ¥ deg.  Err  MeV  19.5 0  62% p o l a r i z e d protons  $. centre-of-mass system $  from a C H 2 - C  subtraction  &  from a CD -CH  2  #  inH  %  in D  2  2  and D  2  subtraction  both  2  Some e a r l y r e f e r e n c e s Watson^ ^ and by 1/f  Mandelstam^ -^. 1  a r e l i s t e d by Gell-Mann and  - 45 -  APPENDIX B EXPERIMENTS ON PROTON-NUCLEUS PION PRODUCTION  We l i s t b e l o w t h e v a l u e s o f E , ¥, a n d E ^ ( a s d e f i n e d p  i n the t e x t ) a t which a d i f f e r e n t i a l like  (p + nucleus  cross-section f o r a reaction  rr± +. n u c l e u s ' ) h a s b e e n m e a s u r e d . A l l  v a l u e s a r e i n t h e l a b o r a t o r y frame u n l e s s o t h e r w i s e  indicated.  ?  tr +  Err  Author(s)  Year  Ref.  Target(s)  MeV  Clark  1952  (13)  Be t o Pb  240  135®  +  40  n  tt  35*  +  40  L i t o Pb  340  135  +  36  Li toC  n  135  +  63  Imhof e t a l .  1957  (19)  deg.  MeV  Sagane a n d Dudziak  1953  (20)  Be t o Pb  340  90  +  12.5-33  Hamlin e t a l .  1951  (21)  Be t o Pb  340  0  +  53  (22)  Be t o B b  335  0  +  34-129  0  +  52-147  M e r r i t t e t a l . 1955  Tt  Block et a l .  1952  (23)  D t o Pb  331  90  +  20-120  Rosenfeld  1954  (24)  C only  440  90  +  30-125  Lillethun  1961  (25)  Be t o U  450  21.5  +  83-236  21.5  -  132-200  C only  21.5  +  44-max  Be t o U  60  +  149  C toU  60  -  99  60  +  44-max  C only  »  - 46 -  Author(s)  Year  Ref.  Reaction  Heer e t a l .  1966  (26)  Ep  rr  Err  MeV  deg.  Be t o Pb  600  0  tt  tt  21.5  100-max all  MeV  +  100-max  Heifer et a l .  1961  (27)  C only  654  56  Meshkovskii e t al.  1959, 195S, 1957  (28)  C only  660  19.5  +  100-max  L i t o Cu  tt  45  +  70-max  n  tt  45  -  90-max  45  +•  153  Meshcheriakov et a l .  Ag, Pb  1957, 1956  (29)  C  660  24  +  50-max  Haddock e t a l . 1964  (30)  G  725  0  +  50-max  @  e v e r y t h i n g between 130 and 150  <fr  e v e r y t h i n g between 3 0 and 50  Summaries a r e a l s o given by L i l l e t h u n  (25) v  J  ' and by Heer  (7) et a l , ' . u  report< ). 1 0  A r a t h e r d e t a i l e d r e f e r e n c e l i s t appears i n t h e Yale  -  47 -  APPENDIX C DERIVATION OF PATH LENGTHS IN A NUCLEUS  In F i g u r e 6 A i s the p o i n t o f proton i s the p o i n t o f p i o n p r o d u c t i o n . center.  Hence OA Sp  [C.l]  =  =  R  m a x  , OB  ~ r.cosO  +  R  where oc i s the angle AOD.  The o r i g i n 0 i s a t the  r , and AB  =  m a x  e n t r y and B  s  =  p  r  =  nuclear  and we have  .cosoc ,  From the t r i a n g l e AOD, s i n c e AD  =  r . s i n O , we get (C.2]  R  m a x  (r.sinO)  =  Solving [ C 2 ] f o r R m a x  , c o s o 6 ,  +  2  (R  .cosoo ) . 2  m a x  ( p o s i t i v e r o o t ) and p u t t i n g the  r e s u l t i n t o [ C . l ] then g i v e s us [C.3]  s  p  s  r-cosO  +  (Rmax " r s i n 0 ) ^ . 2  2  I f we denote the angle COE by jQ, then CE i s g i v e n by [C.4] S i n c e OF [C.5] Squaring  R  =  max'  s i  V  r» sinO* eos<j>  +  s^sinf.  r»cosO + Sn-'cosf t r i a n g l e FOE e s t a b l i s h e s 2 (Rmax* / ) 003  =  9  (r»sinO* sin^))  2 2 sin/9 + cos /S r 1, a q u a d r a t i c equation s„  where  X  +  (r«cosO  the r e s u l t t o [ C . 5 ] g i v e s us,  [ C . 4 ] and adding  [C6]  2  =  - r-X  =  cosO*cosf  ±  (RLx +  --r (l 2  i n Srr having the - X )) , 2  1  sinO *sinI)F •cos4>.  2 Sn-• c o s y ) . since roots  - 48  -  Geometry o f p a t h l e n g t h s i n a  nucleus.  - 49 -  I t i s e a s i l y shown t h a t only the r o o t i n [C.6J u s i n g the + s i g n always g i v e s s„ ^ 0 . desired results  With t h i s i n mind we have the  ([2] and [3] o f the t e x t ) i n [C.3] and  [C.6].  - 50 -  APPENDIX B TRANSFORMATION OF ENERGY FROM LABORATORY FRAME TO REST FRAME OF A STRUCK NUCLEON  I f we l e t P  (p_, i E p ) and K  =  (k, i E ) represent t h e  =  k  four-momenta o f p a r t i c l e s having r e s t masses o f m  and m  p  k  respec-  t i v e l y , then we can w r i t e [D.l]  (P + K )  (£ + k )  2 =  = where T  p  and T  (j2 + k )  p  -  2  (T + T p  )  k  2  + (m  k  + m )) , 2  p  k  Assume t h a t the q u a n t i t i e s  s i d e o f [ D . l ] a r e measured i n - t h e  laboratory  Then i f primed q u a n t i t i e s a r e measured i n t h e r e s t frame  of the p a r t i c l e described [D.2]  (B + E  are k i n e t i c energies.  k  on t h e r i g h t - h a n d frame.  -  2  (P + K )  by K we can a l s o w r i t e (p_' k')  2 =  -  2  +  = s i n c e by d e f i n i t i o n k'  =  £** T  ~  <p T  (E + E p  )  k  2  (»p • k>> >  +  m  2  0.  k =  I f we equate [ D . l ] and [D.2] and expand terms u s i n g 2 standard [D.3]  r e l a t i o n s l i k e p_ T  p  =  T  p  +  2 T  p  + 2m T  (T T  k  - £.k)/m  =  p  p  p  we get d i r e c t l y t h a t  k  +  which has [ 6 ] and [ 7 ] o f t h e t e x t as s p e c i a l  (m /m )T , p  k  cases.  p  -  51 -  APPENDIX B DERIVATION OF AN EXPRESSION USED TO COMPUTE PION ABSORPTION COEFFICIENTS  To e x p l a i n the a b s o r p t i o n o f pions i n n u c l e a r matter Brueckner e t a l . ^ D assume t h a t the a b s o r p t i o n per nucleon i n any nucleus  i s p r o p o r t i o n a l t o the known capture  pions by deuterons.  cross-section of  L e t t i n g P be the p r o p o r t i o n a l i t y constant  we can then w r i t e the n u c l e a r c r o s s - s e c t i o n f o r a b s o r p t i o n as  [E.l]  (J  n-P-a ,  =  d  where n i s the number o f a b s o r b i n g nucleons i n the nucleus  and 0^  i s the deuteron pion-capture  (with  cross-section.  I f we assume  Brueckner e t a l . ) t h a t the b a s i c p i o n a b s o r p t i o n r e a c t i o n s a r e j u s t the i n v e r s e s o f the b a s i c p i o n p r o d u c t i o n r e a c t i o n s (some o f which we l i s t  i n the t e x t , s e c t i o n 1.2) then f o r , p o s i t i v e  pions  we have 0"^ i CT(TT+d * pp) and n f'N, i . e . , we have a b s o r p t i o n by neutrons m o s t l y .  F o r n e g a t i v e pions we have 0"^ ^ 0"(rr-d » nn)  and n ± Z, a b s o r p t i o n by protons'mostly. i  Using the model d e s c r i b e d above we can w r i t e the p i o n a b s o r p t i o n c o e f f i c i e n t , d e f i n e d as a b s o r p t i o n per u n i t volume, as (E.2)  n  a  =  n-?.(7  .  d  (4/3)"R  3  To use [E.2] i n computing values o f n , Frank e t a l . ^ ? ) take f o r a  (J^ ( c f . t h e i r footnote  (13)) the values  - 52 -  CE.3]  0"  d  (4.45/q)-(0.14 + q ) , 2  =  w h e r e q i s t h e maximum c e n t e r - o f - m a s s p i o n , i n u n i t s o f u«c. formula  momentum a v a i l a b l e t o t h e  This they o b t a i n from the semi-empirical  o f Gell-Mann and W a t s o n ^ .  I t c a n a l s o be o b t a i n e d  f r o m R o s e n f e l d ' s ^ ^ e x p r e s s i o n f o r 0~  10  p r i n c i p l e o f d e t a i l e d balance  = 0 ( p p + n-+d) a n d t h e  i n a low energy  approximation.  F r a n k e t a l . assume, i n a c c o r d a n c e w i t h B r u e c k n e r f  =  4 a n d t h e y have u s e d ( t h e i r e q u a t i o n  radius R  (fi/(u«c))'A'A /-^. 1  =  0"ci a n d R i n t o  et a l . ,  that  (13)) f o r t h e n u c l e a r  P u t t i n g t h e above e x p r e s s i o n s f o r  (E.2) g i v e s u s a n e x p r e s s i o n f o r t h e p i o n  absorp-  tion coefficient,v i z .  [E.4]  n  a  =  1 A  A  =  u - c . l 0.107.(0.14 + q ) . 2  -h  A  q  which i s t h e Frank et a l . expression ( t h e i r equation t h e 1/x^  f a c t o r made e x p l i c i t .  (9)) w i t h  Frank e t a l . i m p l i c i t l y s e t  A z 1 when t h e y compute v a l u e s o f n„.  =  n  a  + n , where n s  s  i s the  c o e f f i c i e n t f o r i n e l a s t i c p i o n s c a t t e r i n g . ( a l s o depending on a f a c t o r o f 1/X ). 5  I n d e r i v i n g [ E . 4 l we c o n s i d e r e d o n l y p o s i t i v e  pion  absorption.  Brueckner  e t a l . q u o t e r e f e r e n c e s w h i c h e x p l a i n why  we may e x p e c t  t h e a b s o r p t i o n o f n e g a t i v e p i o n s t o be s i m i l a r .  - 53 -  APPENDIX F OUTLINE OF COMPUTER PROGRAM PIPROD  The f o l l o w i n g i s a sketch o f a FORTRAN IV program w r i t t e n t o n u m e r i c a l l y i n t e g r a t e equation [11] o f the t e x t . The symbols used below a r e d e f i n e d i n the t e x t . PIPROD Read constants E , 1|J, np, r , a, e t c . ff  0  Read Frank data on mfp v s . T„. and V  r  v s . T„. f o r a range o f T„.. 3  C o r r e c t Frank data f o r u n i t s (mfp) and f o r ( r / r r 0  Calculate n  ff  = 1/mfp v s . T  n  f o r range o f T„..  DO f o r each A a v a i l a b l e : Read A and Z. C a l c u l a t e R, R  m a x  , R , and / D . Q  c  C a l c u l a t e number o f r - s t e p s and exact step A r . Set r - 0. DO f o r each r up t o R  m a x  :  C a l c u l a t e number o f 9-steps and exact A9. Set 9 = 0 . DO f o r each 9 up t o it; C a l c u l a t e number o f $>-steps and exact Set  <j)  A<t>.  0.  =  DO f o r each < > f up t o rr : C a l c u l a t e s^. C a l c u l a t e number o f s-steps and exact As. Set s  =  0.  (mfp, V ) . r  1 2  3  54  -  4  DG f o r each s up t o s^.: C a l c u l a t e r , <o, and V . c  C a l c u l a t e E„.  =  T,, + V (T!„)*/o  Interpolate f o r T  r  •• •• V  c  v s . T,,. f o r range o f T^.  a t E„. o f i n t e r e s t .  n  I n t e r p o l a t e f o r n^. a t t h i s T . n  C o r r e c t t h i s n,,. by /O. S t o r e c o r r e c t e d n^. a s , say, A v s . s. L Change s by As. Integrate  /A(s)ds.  Check e r r o r and w r i t e i f c r i t i c a l . C a l c u l a t e exp(/A(s)ds) and s t o r e a s , say, B v s . <J>. L- Change cj> by A<1>. I n t e g r a t e /B(<J>) d<b. Check e r r o r and w r i t e i f c r i t i c a l . Calculate Sp. Calculate  2sin0  «exp(-s »np) • p  /B{ty)dty.and  Store a s , say, C v s . 0 . L Change G by AG. I n t e g r a t e /C(Q)d6. Check e r r o r and w r i t e i f c r i t i c a l . C a l c u l a t e o, T. C a l c u l a t e r />./C(Q)dG and r >o./C(G)dG.f and 2  2  S t o r e a s , say, D and E v s . r . L Change r by A r . Integrate  I i = / D ( r ) d r and I  2  = /E(r)dr.  Galaulate I 2 A 1 . Store each o f 1 ^ , I , and I / I i v s . A. 2  2  -  -  55  1 Calculate normalization constants i f A appropriate. - Change  A.  Normalize  1^,  I2.  W r i t e 1-^,  I2,  and  normalized  1^,  I2  vs.  A.  W r i t e E,,., U, n o r m a l i z a t i o n c o n s t a n t s , and  other data of  interest.  END Subroutines: SIMPLE - Simpson's r u l e  integration.  TINT - t h i r d o r d e r p o l y n o m i a l  interpolation.  Comments: 1)  For the f i r s t  interpolation  T„. v s . E^. i s s i n g l e - v a l u e d . 2)  The  (for T  ff  i n the s-loop) check t h a t  I t i s f o r the Frank et a l . data.  <J>-integration i s o n l y b e t w e e n 0 and tt b e c a u s e o f s y m m e t r y  i n Q> ( h e n c e t h e f a c t o r 2 i n C ( 0 ) ) . 3)  E r r o r i n i n t e g r a l s i s e s t i m a t e d by r e - i n t e g r a t i o n u s i n g a doubled step-length.  

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