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On the interaction of low energy pions with nuclei Scherk, Leonard Raymond 1969

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5u^  ON THE INTERACTION 03? LOW ENERGY PIONS WITH NUCLEI  LEONARD RAYMOND SCHERX B. Sc.,' U n i v e r s i t y o f B r i t i s h Columbia, 1965 M. So. U n i v e r s i t y o f B r i t i s h Columbia. 1967  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n the Department o f PHYSICS  We accept t h i s required  t h e s i s as conforming  to the  standard  THE UNIVERSITY OF BRITISH COLUMBIA J u l y , 1969  In  presenting  this  an a d v a n c e d d e g r e e the I  Library  further  for  agree  in p a r t i a l  fulfilment  of  at  University  of  Columbia,  the  make  that  it  freely  permission for  representatives.  this  thes.is  It  for. f i n a n c i a l  is  of  gain  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, Canada  Date  September B,  Columbia  1969  for  extensive by  the  shall  not  the  requirements  reference copying of  Head o f  understood that  written permission.  Department  British  available  s c h o l a r l y p u r p o s e s may be g r a n t e d  by h i s of  shall  thesis  I agree and  that  Study.  this  thesis  my D e p a r t m e n t  copying or  for  or  publication  be a l l o w e d w i t h o u t  my  i i  ABSTRACT The  o p t i c a l i n t e r a c t i o n o f low energy  with n u c l e i i s discussed. s i n c e the n u c l e a r  f ^ 3 0 MeV)  pions  In p a r t i c u l a r , i t i s shown t h a t ,  d e n s i t y enters the low energy  pion-nucleus  i n t e r a c t i o n i n a very d i r e c t manner, t h i s i n t e r a c t i o n prov i d e s a s e n s i t i v e means o f i n v e s t i g a t i n g such p r o p e r t i e s o f the n u c l e a r d e n s i t y . a s the d i f f u s e n e s s o f the n u c l e a r s u r f a c e . A.geometric d i s c u s s i o n o f the s t r u c t u r e o f the low energy pion-nucleus  i n t e r a c t i o n i s given which emphasizes the analogy  hetween adding  the s c a t t e r e d p i o n waves i n the n u c l e a r medium  and  adding  electric  potentials i n a classical  The  parameters o f the o p t i c a l p o t e n t i a l which r e p r e s e n t s the  i n t e r a c t i o n are taken authors  dielectric.  to he those c a l c u l a t e d by e a r l i e r  who have used a m u l t i p l e - s c a t t e r i n g formalism to  deduce the d e t a i l s  o f t h e - o p t i c a l i n t e r a c t i o n from a micro-  s c o p i c p o i n t o f view.  The i n t e r a c t i o n i s s t r o n g l y momentum-  dependent and the l o c a l p a r t o f the i n t e r a c t i o n i s r e p u l s i v e . I t i s shown t h a t i n o p t i c a l s c a t t e r i n g and a b s o r p t i o n , the resonance aspects  o f the problem depend mainly  upon the h e i g h t o f the l o c a l p o t e n t i a l b a r r i e r  only  (^15  MeV)  because o f the long wavelength o f the p i o n i n s i d e the n u c l e u s . The  o p t i c a l a b s o r p t i o n o f low energy pions i s shown to be  sensitive  to the d i f f u s e n e s s o f the n u c l e a r s u r f a c e through  the s t r o n g s u p p r e s s i o n  o f momentum-dependent a b s o r p t i o n near  the top o f the p o t e n t i a l b a r r i e r .  I t i s argued t h a t low  energy pions can t h e r e f o r e be used to r e s o l v e the c o n f l i c t s which p r e s e n t l y e x i s t i n the i n f o r m a t i o n a v a i l a b l e from  iii several  experiments concerning the d i s t r i b u t i o n o f neutrons  i n the n u c l e a r The  surface.  i d e a s developed i n d i s c u s s i n g  the o p t i c a l a b s o r p t i o n  of pions are extended to the e x c i t a t i o n by pions o f r o t a t i o n a l states the  i n strongly  deformed n u c l e i .  e x c i t a t i o n cross s e c t i o n s  interactions  obtained  depend s t r o n g l y  l e a r density The  obtained with more c o n v e n t i o n a l  from the pion-nucleus o p t i c a l i n t e r upon such c h a r a c t e r i s t i c s o f the nuc-  as i t s s u r f a c e  thickness.  r o t a t i o n a l model o f s t r o n g l y  assumed and an a n a l y s i s made i n the D i s t o r t e d that  unlike  (such as the Coulomb i n t e r a c t ! o n ) , the e x c i t a t i o n  cross sections action  I t i s shown t h a t ,  the s e n s i t i v i t y  deformed n u c l e i i s  o f the e x c i t a t i o n cross s e c t i o n s i s  Wave Born Approximation.  I t i s shown  to the n u c l e a r s u r f a c e t h i c k n e s s i n  the  e x c i t a t i o n cross sections  arises  from the s u p p r e s s i o n o f  the  e x c i t a t i o n processes due to the momentum-dependent i n t e r -  a c t i o n near the top o f the p o t e n t i a l b a r r i e r . that  the e x c i t a t i o n o f r o t a t i o n a l l e v e l s i n s t r o n g l y  nuclei the  I t is' suggested deformed  by pions can t h e r e f o r e be used to c a r e f u l l y examine  d i s t r i b u t i o n o f neutrons i n the s u r f a c e s o f s t r o n g l y  deformed  nuclei.  iv • TABLE OF CONTENTS Page CHAPTER 1  INTRODUCTION  1  CHAPTER 2  THE LOW ENERGY PION-NUCLEUS INTERACTION  12  2-1  Structure  17  2-2  A Microscopic Interaction  4  o f the I n t e r a c t i o n Derivation  o f the  25  t  2- 3 CHAPTER 3 •  Experimental V e r i f i c a t i o n  39  THE OPTICAL PROPERTIES OP LOW ENERGY - PTONS IN SPHERICAL NUCLEI  43  3- 1  A Formalism For A n a l y z i n g O p t i c a l S c a t t e r i n g and A b s o r p t i o n  50  3-2  Q u a l i t a t i v e Features o f Low Energy P i o n Optics i n N u c l e i  57  3-3  Pion-Hucleus O p t i c a l P o t e n t i a l  60  3-4  Numerical D i s c u s s i o n  64  3- 5  Charge Exchange and other E x o t i c i n the E r i c s o n s ' P o t e n t i a l  CHAPTER 4  Terms  . 90  EXCITATION OF ROTATIONAL LEVELS IN DEFORMED NUCLEI BY PIONS  94  4- 1  A Review o f the R o t a t i o n a l Model and a D i s c u s s i o n o f Pion E x c i t a t i o n  98  4-2  DWBA Formulae f o r P i o n E x c i t a t i o n Cross S e c t i o n s  105  4-3  Numerical D i s c u s s i o n  118  CONCLUSIONS  136  CHAPTER 5 BIBLIOGRAPHY APPENDIX A  . 144 EQUATIONS FOR THE INTERIOR LOGARITHMIC  146  DERIVATIVES A„i  Integral•Equation  for Absorption  A-2  Uniform D i s t r i b u t i o n  146 .148  V  Page NOTES Oil EVALUATION OP PION EXCITATION CROSS SECTIONS  150  B-l  E v a l u a t i o n o f DWBA M a t r i x  150  B-2  E v a l u a t i o n o f Electromagnetic P o t e n t i a l For a Deformed Charge D i s t r i b u t i o n  155  THE NUCLEAR OPTICAL .MODEL AND WAVE. i« PROPERTIES: BARRIER PENETRATION, REFLECTION, ABSORPTION, AND RESONANCE by G, Michaud, L. Scherlc, and E. Vogt  155  APPENDIX B  APPENDIX C  Elements  LIST OF FIGURES  Nuclear D e n s i t y and Square o f Nuclear Densi ty  Ca  4 0  Calculated E l a s t i c Scattering Jt 4  Sections  r  Ca  Cross  4 0  C a l c u l a t e d T o t a l A b s o r p t i o n Cross S e c t i o n s W  *  ca  4 0  Dependence o f Real Phase S h i f t s upon Surface Parameter  TV* +  Ca  4 0  Dependence o f Real Phase S h i f t s upon Potential  Barrier  Tt +• Ca  Dependence o f Real Phase S h i f t s upon 40 Radius  71  +  Ca  40  Real Logarithmic ^ D e r i v a t i v e s Imaginary Phase S h i f t s  K"" +• C a 1  R a t i o s o f Imaginary Phase ^  +  Ca  3T  Shifts  4 0  s-wave A b s o r p t i o n  7T •* C a ^ f  4  s-wave Imaginary'Phase S h i f t s f o r >~  f  Ca  4 0  and  K  r  +  Ca  4 0  s-wave Imaginary Phase S h i f t s f o r Tr-  +- P b  2 0 8  and  ^  -  Ca  4 0  s-wave Imaginary Phase S h i f t s f o r f + 208 _ 7T-+-Pb 208  K  p b  a n a  +• Ca 4 0  vii Figure  .  11  Imaginary Phase S h i f t s  13a  Reduced T o t a l E x c i t a t i o n Gross S e c t i o n s 7X + A l +  12b  4  2 0 8  and  Y  13a  7t >  Sections  f  E x c i t a t i o n Gross  Al  Reduced D i f f e r e n t i a l  n  r  ±  E x c i t a t i o n Cross  T J  2  5  Values o f R a d i a l I n t e g r a l s  15a  R a d i a l I n t e g r a l s I n s e n s i t i v e to Surface K  r  Radial Integrals  +  Al  T  +• T J  128  129  2 5  Momentum-Dependent R a d i a l I n t e g r a l s K  127  2 3 8  Momentum-Dependent R a d i a l I n t e g r a l s • • 7C  126  8 5  I n s e n s i t i v e to Surface  X* *• U  Parameter  16b  +- A l  123  8  Relative  Parameter.  122  2 5  14  16a  121  n  Sections  15b  120  + 238  Reduced D i f f e r e n t i a l .  87  8 5  Reduced T o t a l E x c i t a t i o n Cross S e c t i o n s 7i  13b  71 + P h  Page  2 3 8  120 '  viii ACMOWLEDGEkENTS The  author would l i k e  to express h i s deep a p p r e c i a t i o n  t o h i s Teacher and Research S u p e r v i s o r , Only those who have had the good fortune  P r o f e s s o r E. W. Vogt. to have p e r s o n a l l y  known or have worked with P r o f e s s o r Vogt w i l l the extent  to which h i s immense patience  of  prolific  and p e r c e p t i v e  suggestions  appreciate  and p e r s o n a l  ness, h i s thorough knowledge o f a l l aspects his  fully  kind-  o f p h y s i c s , and  a s s i s t i n the s o l u t i o n  one's'problems. The  author would a l s o l i k e  t o thank Dr. P. C. Bhargava  f o r many h e l p f u l d i s c u s s i o n s .and  he would l i k e  to thank Hr.  C. T. T i n d l e f o r making a v a i l a b l e h i s computer program f o r s p h e r i c a l Coulomb The  functions.  author a l s o expresses h i s a p p r e c i a t i o n to the N a t i o n a l  Research C o u n c i l o f Canada f o r p r o v i d i n g f i n a n c i a l through N. R. C. S c h o l a r s h i p s .  assistance  CHAPTER 1  1  INTRODUCTION  I n this: t h e s i s i t i s our purpose t o d e s c r i b e the prope r t i e s o f the low energy  (^50 MeV) pion-nucleus  optical  i n t e r a c t i o n and to suggest and examine some o f the ways i n which t h i s i n t e r a c t i o n can be used ture o f n u c l e i .  to i n v e s t i g a t e  the s t r u c -  Because a t r a n s p a r e n t c o n n e c t i o n e x i s t s  between the o p t i c a l i n t e r a c t i o n o f a p i o n w i t h a nucleus and  the elementary  i n t e r a c t i o n s o f the p i o n with the con-  s t i t u e n t nucleons, the p r o p e r t i e s o f t h i s  optical  interaction  are r e l a t e d i n a very d i r e c t way to such macroscopic p r o p e r t i e s as the n u c l e a r d e n s i t y .  nuclear  In t h i s t h e s i s we examine  the r o l e o f the n u c l e a r d e n s i t y both i n determining the o p t i c a l s c a t t e r i n g and a b s o r p t i o n o f pions i n n u c l e i and a l s o i n determining the e x c i t a t i o n by pions o f r o t a t i o n a l s t a t e s i n s t r o n g l y deformed n u c l e i .  We show t h a t these  p r o c e s s e s depend s t r o n g l y upon the d e t a i l s o f the d e n s i t y o f nucleons i n the s u r f a c e o f the n u c l e u s . suggest  We t h e r e f o r e  t h a t pions provide an i d e a l probe f o r measuring such  aspects o f n u c l e a r s t r u c t u r e as the d i s t r i b u t i o n o f nucleons, i n p a r t i c u l a r , neutrons  i n the n u c l e a r s u r f a c e .  D e t a i l s o f the neutron d i s t r i b u t i o n i n the n u c l e a r s u r face are n o t e a s i l y a v a i l a b l e from more c o n v e n t i o n a l i n t e r actions  (such as the nucleon-nucleus  i n t e r a c t i o n ) where the  c o n n e c t i o n between the m i c r o s c o p i c and macroscopic  aspects  o f the problem i s l e s s d i r e c t ; i n f a c t , i n f o r m a t i o n about the s u r f a c e d i s t r i b u t i o n o f neutrons a v a i l a b l e from s e v e r a l experiments  i n nuclei  currently  i s n e i t h e r very w e l l  8  determined consistent,  nor are the r e s u l t s o f these experiments We suggest that the c o n f l i c t s i n t h i s  mutually important  a s p e c t o f n u c l e a r s t r u c t u r e can be l a r g e l y r e s o l v e d by means o f experiments  which invoke the o p t i c a l p r o p e r t i e s  such as those to be d i s c u s s e d i n t h i s The  of pions,  thesis.  d i r e c t manner i n which the macroscopic  aspects o f  the n u c l e u s , such as the n u c l e a r d e n s i t y , e n t e r the p i o n nucleus o p t i c a l i n t e r a c t i o n a r i s e s from the t r a n s p a r e n t conn e c t i o n between the o p t i c a l i n t e r a c t i o n and the s c a t t e r i n g and a b s o r p t i o n processes a s s o c i a t e d w i t h the elementary nucleon  (and two-nucleon) s c a t t e r e r s whose average  s c a t t e r i n g nucleus i t p u r p o r t s to r e p r e s e n t .  over the  This s i n g u l a r  d i r e c t n e s s i n the r e l a t i o n s h i p o f the macroscopic  to the  m i c r o s c o p i c i n t e r a c t i o n s o f pions i n n u c l e i makes the p i o n nucleus o p t i c a l i n t e r a c t i o n o f i n t e r e s t as one o f the few examples o f t r a c t a b l e many-body s c a t t e r i n g problems as w e l l as an e x c e l l e n t t o o l f o r s t u d y i n g the macroscopic o f the nucleus.  The s i m p l i c i t y i n the case o f low energy  pion scattering arises lengths  properties  from the s h o r t p i o n - n u c l e o n s c a t t e r i n g  ( ~ 0 , 1 fm. ) combined w i t h the s m a l l mass o f the p i o n  r e l a t i v e t o the s c a t t e r e r s  (~l/7).  I n these c i r c u m s t a n c e s ,  the p i o n i n t e r a c t s i n d i v i d u a l l y w i t h each nucleon and, because it  o f the s m a l l energy  i s relatively  t r a n s f e r s i n elementary  o b l i v i o u s to the dynamics which govern the  motions o f the s c a t t e r i n g n u c l e o n s the impulse  collisions,  approximation  0  T h i s allows one to make  (making t h e , s c a t t e r i n g m a t r i x  a b l e ) and g r e a t l y s i m p l i f i e s  tract-  the Green's f u n c t i o n s which d e s c r i b e  the p r o p a g a t i o n o f the p i o n i n the n u c l e a r medium; the dynamics of the problem are thus taken care o f  through  known one-body o p e r a t o r s and  the kinematics are d e s c r i b e d  by u s i n g f a m i l i a r techniques  f o r adding m u l t i p l y s c a t t e r e d  waves.  This i s to be c o n t r a s t e d with the other f a m i l i a r  example o f a s t r o n g l y i n t e r a c t i n g p r o j e c t i l e , the where c o n t r a r y c o n d i t i o n s h o l d and macroscopic  nucleon,  the c o n n e c t i o n between  and m i c r o s c o p i c processes i s clouded by the  i n t r i n s i c manner i n which the many-body aspects of the s c a t t e r e r m a n i f e s t themselves  i n the o p t i c a l p r o p e r t i e s o f the  n u c l e a r medium. Apart  from the simple connections which e x i s t between  the low energy  pion-nucleus i n t e r a c t i o n and i t s elementary  o r i g i n s , the i n t e r a c t i o n i t s e l f possesses  a s t r u c t u r e and  form which make i t o f i n t e r e s t i n i t s own  right.  The b a s i c  s t r u c t u r e o f the i n t e r a c t i o n i s r e a d i l y seen by n o t i n g t h a t , at  the e n e r g i e s of i n t e r e s t , only s- and p-waves enter the  elementary  s c a t t e r i n g processes.  adding these waves i s analogous a r i s i n g from the charges tric. in  and  The geometric  to adding the p o t e n t i a l s  dipoles i n a c l a s s i c a l  The s-waves add l i k e the p o t e n t i a l s due  the d i e l e c t r i c  pion-nucleus  problem o f  dielec-  to the  charges  and l e a d to a l o c a l c o n t r i b u t i o n to the  optical potential.  n u c l e a r momentum f i e l d t i o n o f the e l e c t r i c  The p-waves modify  o f the p i o n i n analogy  field  due  the  intra-  to the m o d i f i c a -  to the d i p o l e s i n a d i e l e c t r i c ;  the r e s u l t of the p~waves i s t h e r e f o r e to c o n t r i b u t e a momentumdependent term to the pion-nucleus o p t i c a l p o t e n t i a l .  In  fact,  i f we i n c l u d e the e f f e c t s o f the short-range  correlations  o f the nucleons w i t h i n the n u c l e u s , we f i n d t h a t the modification  o f the i n t r a - n u c l e a r momentum f i e l d  due to the  p-waves i s n o n - l i n e a r i n the n u c l e a r d e n s i t y , i n analogy  to •  the well-known Lorenz-Lorentz e f f e c t i n dense o p t i c a l media. S i n c e the Lorenz-Lorentz e f f e c t a f f e c t s the s t r u c t u r e of  the o p t i c a l i n t e r a c t i o n , i t r a i s e s hopes that p i o n o p t i c s  might p r o v i d e a t o o l f o r i n v e s t i g a t i n g the e l u s i v e of  nature  the. short-range c o r r e l a t i o n s o f the nucleons w i t h i n the  nucleus.  I n f a c t , these hopes are c o n s i d e r a b l y minimi zed  s i n c e the wavelength o f a low energy p i o n i s much l o n g e r than the range o f the c o r r e l a t i o n s so t h a t i t i s an i n s e n sitive  t o o l f o r measuring t h e i r s t r u c t u r e .  Thus, while the  p i o n - n u c l e u s i n t e r a c t i o n may be s t r o n g l y a f f e c t e d even i n i t s s t r u c t u r e by the a n t i c o r r e l a t i o n s  o f nucleons  at short  distances  (which removes the s e l f - e x c i t a t i o n o f the s c a t -  terers),  i t i s probably i n s e n s i t i v e  to their d e t a i l s .  IFever  t h e l e s s , c a r e f u l measurements o f o p t i c a l l y s c a t t e r e d pions might provide i n f o r m a t i o n about these c o r r e l a t i o n s which can p r e s e n t l y be obtained  only i n very i n d i r e c t ways ( f o r i n -  s t a n c e , from the s t u d i e s o f n u c l e a r m a t t e r ) . While  the analogy with e l e c t r o m a g n e t i c theory p r o v i d e s  the b a s i c s t r u c t u r e o f the pion-nucleus i n t e r a c t i o n , a quant i t a t i v e estimate  o f the macroscopic  parameters from the  m i c r o s c o p i c dynamics r e q u i r e s a c o n s i d e r a b l y more d e t a i l e d calculation.  A thorough  study o f t h i s problem has been g i v e n  by E r i c s o n and E r i c s o n (1966).  They have used  the techniques  of m u l t i p l e  scattering  theory to c a l c u l a t e  form o f the i n t e r a c t i o n t a k i n g rections and  associated  including The  the macroscopic  i n t o account kinematic  cor-  w i t h the motion o f the s c a t t e r i n g  terms which simulate o p t i c a l  nucleons  absorption.  o r i g i n o f the a b s o r p t i v e terms i n the E r i c s o n s '  potential  l i e s i n r e a l p i o n a b s o r p t i o n on "elementary" two-  nucleon s c a t t e r e r s  (pions,  b e i n g bosons, need not be con-  s e r v e d ) ; kinematics i n h i b i t p i o n a b s o r p t i o n by a s i n g l e l e o n and the s m a l l mass o f the p i o n i n h i b i t s n u c l e a r t i o n through k i n e t i c exchanges. the  nuc-  excita-  The a b s o r p t i v e aspects o f  pion-nucleus o p t i c a l i n t e r a c t i o n are thus connected to  elementary processes as d i r e c t l y as the r e f r a c t i v e  aspects;  t h i s i s an enormous s i m p l i f i c a t i o n over the a b s o r p t i v e  aspects  of the nucleon-nucleus o p t i c a l i n t e r a c t i o n which a r i s e  solely  from k i n e t i c exchanges between the  p r o j e c t i l e and the  s c a t t e r e r s . (and, hence, i n t r i n s i c l y i n v o l v e aspects o f the s c a t t e r e r ) .  In a d d i t i o n  to a b s o r p t i o n , the E r i c s o n s f i n d that terers  to t h e i r  contribution  these two-nucleon  a l s o make important c o n t r i b u t i o n s  properties  the many-body  scat-  to the r e f r a c t i v e  o f the o p t i c a l i n t e r a c t i o n , p a r t i c u l a r l y to the  l o c a l i n t e r a c t i o n where a s t r o n g c a n c e l l a t i o n s-wave p i o n - n u c l e o n s c a t t e r i n g  occurs i n the  lengths.  With these e f f e c t s taken i n t o account, the E r i c s o n s f i n d that  the o p t i c a l s c a t t e r i n g  o f pions from n u c l e i  can be sum-  marized by a Schroedinger e q u a t i o n which has the s t r u c t u r e predicted  by the e l e c t r o m a g n e t i c analogy  ( i n c l u d i n g the Lorens-  Lorentz' e f f e c t ) and whose parameters are t r a n s p a r e n t l y  connected  6 to p i o n s c a t t e r i n g l e n g t h s from nucleons addition, their  and  deuterons.  optical interaction includes optical  terms which account  f o r q u a s i - e l a s t i c charge  isospin  exchange pro-  cesses and i n c l u d e s a h y p e r f i n e term which takes i n t o the s p i n of ..the t a r g e t nucleus.  They f i n d  i n t e r a c t i o n i s c o n s i s t e n t w i t h the  In  account  t h a t the r e s u l t i n g  ( e s s e n t i a l l y zero energy)  7t-mesic x-ray data hut, u n f o r t u n a t e l y , the i n t e r a c t i o n cannot he t e s t e d  at h i g h e r e n e r g i e s u n t i l  the advent  of more  intense-^pion beams (such as the Triumf F a c i l i t y proposed  for  U.B.C. ).  I t i s r e a s o n a b l e , however, to expect p o t e n t i a l should adequately  t h a t the E r i c s o n s '  d e s c r i b e the pion-nucleus  i n t e r a c t i o n up to 30 or 40 MeV  where the zero-energy  optical descrip-  t i o n o f the elementary  s c a t t e r i n g processes begins to  and where the e f f e c t s  of the 3 - 3 pion-nucleon resonance  the n a i v e treatment  fail make  o f the m u l t i p l e s c a t t e r i n g problem i n v a l i d  T h i s l i m i t i s to some e x t e n t s e t by the Fermi motion of the nucleons  and  the parameters of the p o t e n t i a l may  energy-dependent at the h i g h e r e n e r g i e s . would expect  t h a t the b a s i c s t r u c t u r e  should be c o r r e c t and zero-energy  Nevertheless,  T h i s seems a reasonable procedure  one  of the i n t e r a c t i o n  i n our c a l c u l a t i o n s we  parameters suggested  be somewhat  have used  the  by the E r i c s o n s ' a n a l y s i s . s i n c e , i n any event,  the  p o t e n t i a l has not been t e s t e d a t h i g h e r e n e r g i e s and i t would not seem worthwhile to attempt determine  without some reasonable e m p i r i c a l b a s i s  to untangle i t s exact  the v a r i o u s second  details.  order e f f e c t s which  In the present t h e s i s we have taken the E r i c s o n s f o r the o p t i c a l pion-nucleus i n t e r a c t i o n both to d e s c r i b e p i o n o p t i c s  form  1  and we have attempted  i n the low energy r e g i o n and to  •'I elucidate ceptible the  those f e a t u r e s o f n u c l e a r s t r u c t t i r e which are susto i n v e s t i g a t i o n  w i t h the p i o n probe, i n p a r t i c u l a r ,  d e n s i t y o f neutrons i n the n u c l e a r s u r f a c e .  character of pion optics  resides  i n the m i l d l y  The b a s i c repulsive  l o c a l p o t e n t i a l , i n the s t r o n g momentum-dependence of the interaction,  and i n the f a c t that pions are very  absorbed by n u c l e i .  These c h a r a c t e r i s t i c s  lead  strongly to o p t i c a l  phenomena which are q u a l i t a t i v e l y q u i t e d i f f e r e n t  from those  encountered w i t h more c o n v e n t i o n a l p r o j e c t i l e s . In p a r t i c u l a r , the resonance aspects of the problem at the  energies o f i n t e r e s t  presence o f a p o t e n t i a l  (£z30  MeV) are governed by the  barrier  f^-15 MeV) which, combined  w i t h the s m a l l p i o n mass, y i e l d s a p i o n wavelength w i t h i n the nucleus which i s c o n s i d e r a b l y g r e a t e r than the n u c l e a r dimensions;  the resonance aspects o f the problem a r e , t h e r e f o r e ,  rather insensitive  to the d e t a i l s o f the i n t e r a c t i o n  governed mainly by the s i z e  and are  o f the nucleus and by the proper-  t i e s o f the e l e c t r o s t a t i c b a r r i e r which surrounds i t . c o n t r a s t s w i t h the s t r o n g resonances c h a r a c t e r i s t i c attractive  potentials  encountered i n n u c l e a r o p t i c s .  a b s o r p t i o n o f pions a l s o are  o f the The  presents some unusual f e a t u r e s :  absorbed not only through a l o c a l i n t e r a c t i o n  primarily  This  (which depends  upon the q u a n t i t y o f matter present) but a l s o  a. momentum-dependent i n t e r a c t i o n  pions  through  (which depends s e n s i t i v e l y  8  upon the d i s t r i b u t i o n  of t h i s m a t t e r ) .  Because pions are  absorbed p r i m a r i l y i n the n u c l e a r s u r f a c e , the energy dependence o f p i o n a b s o r p t i o n c r o s s s e c t i o n s y i e l d s d e t a i l e d i n formation The  about the p e r i p h e r a l ' nuclear  density.  d e n s i t y o f nucleons i n the n u c l e a r  s u r f a c e has  been an e l u s i v e q u a n t i t y so t h a t the r e l a t i v e l y d i r e c t simple  technique  o f measurement provided  by pion  represents  one  of the most u s e f u l aspects  Of course,  the i s s u e i s somewhat complicated  pion absorption terers.  long and  absorption  of pion o p t i c s . by  the f a c t  that  takes p l a c e p r i m a r i l y on two-nucleon s c a t -  However, p r o v i d e d  adequate account i s taken  of the  long-range c o r r e l a t i o n s of nucleons i n the nucleus  (which are  q u i t e w e l l understood),  density  and  the connection  the a b s o r p t i o n r a t e s can be  precision.  between the  obtained w i t h  I n f a c t , the E r i c s o n s have shown i n t h e i r c a l c u -  l a t i o n s that the a b s o r p t i o n r a t e s are simply square of the d e n s i t y and  r e l a t e d to the  they have found these  r a t e s to be i n good agreement with quired  considerable  to e x p l a i n the widths o f  absorption  the a b s o r p t i o n r a t e s r e -  7T-niesic x - r a y s .  o p t i c a l p i o n measurements of the n u c l e a r  A t any  d e n s i t y should  rate, be  a c o n s i d e r a b l e improvement over measurements w i t h nucleons (which s t r o n g l y p o l a r i z e the t a r g e t ) and measurements based on the b r a n c h i n g  ratios  a s s o c i a t e d with kaonic  the a n a l y s i s i s confused Of c o u r s e ,  by complicated  atoms (where  atomic p h y s i c s ) .  the i n t e r e s t i n f i n d i n g the n u c l e a r  surface  d e n s i t y i s to o b t a i n s e p a r a t e l y the s u r f a c e d i s t r i b u t i o n protons  and neutrons.  the charge d e n s i t y  of  T h i s i s immediately accomplished s i n c e  (or d i s t r i b u t i o n  of protons) i s well-known  9  both  from e l e c t r o n s c a t t e r i n g experiments  from  >/-mesic x-ray a n a l y s i s .  and neutron  and, more r e c e n t l y ,  The i n f o r m a t i o n about p r o t o n  d i s t r i b u t i o n s then p r o v i d e s a new  dimension  in  the t e s t s of n u c l e a r models (such as the s h e l l model) and i s also necessary i n f o r m a t i o n i n o b t a i n i n g precise r a t e s f o r various processes associated with the nucleus -  (such as p i o n pro-  d u c t i o n r a t e s from h i g h energy p r o t o n s ) . In  a d d i t i o n to o p t i c a l p i o n a b s o r p t i o n , a second  nique can be used  to u t i l i z e  tech-  the momentum-dependence o f the  p i o n - n u c l e u s i n t e r a c t i o n i n o b t a i n i n g the s u r f a c e f e a t u r e s of  deformed n u c l e i .  In the s t r o n g - c o u p l i n g model, a deformed  nucleus i s d e s c r i b e d as the product t r i n s i c wave f u n c t i o n ;  of a r o t a t i o n a l and i n -  the p i o n f i e l d  can be used  to e x c i t e  the r o t a t i o n a l s t a t e s o f such a nucleus i n the same way both the e l e c t r o s t a t i c  field  field  In f a c t , i n the D i s t o r t e d Wave Born  have been used.  Approximation,  (Coulomb e x c i t a t i o n ) and  as  the p i o n e x c i t a t i o n amplitudes  sums of the Coulomb e x c i t a t i o n and  are  neutron  essentially  local excitation  a l o n g w i t h a moment urn-dependent e x c i t a t i o n amplitude  amplitudes arising  from the momentum-dependent i n t e r a c t i o n . The simple D W B A i n t e g r a l s which r e l a t e the e x c i t a t i o n c r o s s s e c t i o n s t o the I n t e r a c t i o n  (and, hence, the d e n s i t y )  then a l l o w a d i r e c t c o n n e c t i o n between the c r o s s s e c t i o n s and the p r o p e r t i e s o f the i n t r i n s i c wave f u n c t i o n of the n u c l e u s . The  Coulomb and  l o c a l e x c i t a t i o n amplitudes, b e i n g i n t e g r a l s  over the n u c l e u s , are o n l y moderately of  s e n s i t i v e to the  details  the n u c l e a r s u r f a c e ; the momentum-dependent e x c i t a t i o n  amplitude, however, i s very s e n s i t i v e to these d e t a i l s  since,  10  !  near the top o f the p o t e n t i a l h a r r i e r , the momentum v a r i e s r a p i d l y both with energy and across the n u c l e a r Thus the a n a l y s i s  o f the energy dependence o f the e x c i t a t i o n  by pions o f r o t a t i o n a l s t a t e s  i n deformed n u c l e i y i e l d s de-  t a i l e d i n f o r m a t i o n about the s u r f a c e s i c wave  surface.  features  o f the i n t r i n -  function.  A more complete d i s c u s s i o n  o f the t o p i c s i n t h i s  Intro-  d u c t i o n i s g i v e n i n the remaining c h a p t e r s o f the t h e s i s . A review and i n t e r p r e t i v e d i s c u s s i o n the  o f the c o n s t r u c t i o n  of  pion-nucleus o p t i c a l p o t e n t i a l i s g i v e n i n Chapter 2 .  I n Chapter 3 t h i s i n t e r a c t i o n i s used t o analyze the e l a s t i c s c a t t e r i n g and t o t a l absorption, cross  sections  o f pions from  v a r i o u s s p h e r i c a l n u c l e i w i t h p a r t i c u l a r emphasis on the r o l e o f the n u c l e a r s u r f a c e .  I n Chapter 4 the e x c i t a t i o n o f r o t a -  t i o n a l l e v e l s i n deformed n u c l e i by pions i s d i s c u s s e d i n DWBA as a technique f o r i n v e s t i g a t i n g the s u r f a c e o f the i n t r i n s i c wave f u n c t i o n s  features  o f deformed n u c l e i .  Chapter 5  p r e s e n t s a review o f the t h e s i s and a summary o f the c o n c l u sions  reached i n the e a r l i e r This  chapters.  thesis also includes,  o f a paper now submitted  as Appendix C, a p r e p r i n t  for publication  o f which the author  o f t h i s t h e s i s i s a co-author and which r e p r e s e n t s a separate problem i n which the author became i n t e r e s t e d iod  d u r i n g the per-  o f h i s i n t e r e s t i n the pion-nucleus problem presented i n  the body o f t h i s t h e s i s .  This  problem was prepared i n c o l -  l a b o r a t i o n with P r o f e s s o r E , w . Vogt, the author's Research t  S u p e r v i s o r , and w i t h Mr. Georges Michaud, a v i s i t i n g  student  .1  1 ' t  i' I  11  In astrophysics  from the C a l i f o r n i a I n s t i t u t e o f Technology.  This paper i n v e s t i g a t e s the o p t i c a l i n t e r a c t i o n s between n u c l e i and  heavy i o n s and -  diffuse nuclear  cross sections.  p a r t i c u l a r l y examines the e f f e c t s of the  surface  .  '  upon the s c a t t e r i n g and  These cross sections are o f  i n t e r e s t i n determining  i  absorption  considerable  the r a t e s of v a r i o u s processes  en-  countered i n a s t r o p h y s i c s . The  author's i n t e r e s t i n t h i s problem arose from h i s  e a r l i e r i n t e r e s t i n alpha-decay r a t e s and h i s main c o n t r i butions  are  to the  d i s c u s s i o n of the resonance p r o p e r t i e s  r e a l p o t e n t i a l s i n the presence o f l a r g e Coulomb b a r r i e r s , these b e i n g contained  i n Sec.  4 o f the  paper.  of  CHAPTER S  THE LOW ENERGY PION - NUCLEUS INTERACTION  I n the p r e s e n t chapter we d i s c u s s the c o n s t r u c t i o n o f the low energy  (^50 MeV) pion-nucleus o p t i c a l  interaction  »  4  from the elementary processes o f pion-nucleon and p i o n deutron s c a t t e r i n g .  A comprehensive study o f t h i s  problem  and a review o f the e a r l i e r work by other authors has been g i v e n i n an a r t i c l e by M. E r i c s o n and T. E. 0. E r i c s o n and  (1966)  our d i s c u s s i o n i s to a l a r g e e x t e n t a summary o f t h e i r  d i s c u s s i o n and r e s u l t s .  A more g e n e r a l d i s c u s s i o n o f the  i n t e r a c t i o n o f pions w i t h n u c l e i can be found i n a r e c e n t review a r t i c l e by K o l t u n  (1969),  The pion-nucleus i n t e r a c t i o n i s d i s t i n g u i s h e d o t h e r complex i n t e r a c t i o n s which admit tion it  such as the nucleon-nucleus  an o p t i c a l  interaction  from most descripi n that  can e a s i l y be c o n s t r u c t e d from the m i c r o s c o p i c pion-nucleon  and p i o n - d e u t e r o n s c a t t e r i n g s  (whose average i t r e p r e s e n t s ) .  This s i m p l i f i c a t i o n a r i s e s i n p a r t i n t h a t , because the 4  ,  p i o n - n u c l e o n s c a t t e r i n g s l e n g t h s are s m a l l (^0.1 pared t o the i n t e r - n u c l e o n s e p a r a t i o n (^1.4  fm. ) com-  fm.) and because  the p i o n mass i s s m a l l ( ~ l / 7 the nucleon mass) , one can assume t h a t a p i o n s c a t t e r s from a nucleon i n s i d e the nucl e u s i n the same way i n which i t s c a t t e r s from a nucleon i n f r e e space;  t h a t i s , one can make the impulse  approximation.  In a d d i t i o n , the s m a l l mass o f the p i o n allows a c o n s i d e r able s i m p l i f i c a t i o n o f the Green's f u n c t i o n which d e s c r i b e s p i o n propagation w i t h i n the nucleus s i n c e one may then  largely  neglect The  the e f f e c t s o f i n t e r m e d i a t e n u c l e a r e x c i t a t i o n s .  problem i s thus e s s e n t i a l l y reduced to the  problem of c a l c u l a t i n g the o f massive s c a t t e r e r s .  total scattered  In f a c t , s i n c e  t e r i n g s are mainly s-wave s c a t t e r i n g s charges) and one  p-wave s c a t t e r i n g s  can c a l c u l a t e the  same way  i n which one  c a l c u l a t e s the  from t h i s p o i n t  at low  energies  (resembling e l e c t r i c  scat-  dipoles),  wave i n e x a c t l y total electric  the  potential  I t i s the purpose of t h i s chapter  to p r e s e n t a d e r i v a t i o n o f the low potential  wave from a system  (resembling e l e c t r i c  t o t a l scattered  i n a d i e l e c t r i c medium.  classical  energy pion-nucleus o p t i c a l  o f view and  d i r e c t manner i n which i t i n v o l v e s  to demonstrate  the  density  o f the  the scat-  t e r i n g nucleus. I t was  r e a l i z e d by Peaslee  (1952) that the  o f p i o n s from complex n u c l e i could obtained simply by making the summing the scatterers.  singly scattered In f a c t , by  t e r i n g o f 62 MeV  p i o n waves from the  applying  MeV.  below t h i s energy both s- and  waves can be  at l e a s t q u a l i t a t i v e l y  impulse approximation and  his analysis  pions by carbon, he  nucleon resonance a t 200  •account i n the  be  scattering  This  elementary  to the  i n f e r r e d the l e d him  pionthat  taken i n t o  while h i g h e r p a r t i a l  neglected.  His r e s u l t s were extended to i n c l u d e m u l t i p l e i n g s by K i s s l i n g e r  (1955) who  formalism o f Watson (1953) and write  3-3  scat-  to conclude  p-waves must be  elementary s c a t t e r i n g s  by  used the m u l t i p l e Francis  and  scatter-  scattering  Watson (1953) to  the p i o n - n u c l e u s o p t i c a l i n t e r a c t i o n i n terms of a  14 pseudopotential„ While K i s s l l n g e r n e g l e c t e d a b s o r p t i o n and the e f f e c t s  o f nucleon-nucleon c o r r e l a t i o n s , h i s treatment  showed the d i r e c t manner i n which the n u c l e a r d e n s i t y  enters  the  o f the  i n t e r a c t i o n and c o r r e c t l y p r e d i c t e d  i n t e r a c t i o n showing i t to c o n s i s t  the s t r u c t u r e  of a l o c a l interaction  a r i s i n g from the elementary s-wave s c a t t e r i n g s momentum dependent i n t e r a c t i o n  --- and a  a r i s i n g from the elemen-  *  t a r y p-wave s c a t t e r i n g s . the  The s t r o n g momentum dependence o f  i n t e r a c t i o n suggested by K i s s l i n g e r was a l s o shown by  Frank e t a l , (1956) who evaluated both the r e a l and absorpt i v e terms o f the p o t e n t i a l i n an i n f i n i t e medium o f uncorrelated The  nucleons. basic  elucidated  structure  by c o n s i d e r i n g  o f the pion-nucleus i n t e r a c t i o n i s the analogy between adding the  elementary s- and p-wave s c a t t e r i n g s and  adding e l e c t r i c p o t e n t i a l s  i n the n u c l e a r medium  and f i e l d s i n a d i e l e c t r i c  w i t h free charges • — the. geometry o f the two problems i s e s s e n t i a l l y the same.  Baker e t al„ (1958) used c l a s s i c a l  arguments e s s e n t i a l l y o f t h i s kind derivation  to p r e s e n t a s i m p l i f i e d  o f the K i s s l i n g e r p o t e n t i a l and K r o l l  (1961),  r e a l i z i n g the e l e c t r o m a g n e t i c analogy, suggested that should be a m o d i f i c a t i o n f i e l d since ties  o f the i n t r a - n u c l e a r  there  p i o n momentum  such an e f f e c t e x i s t s i n the r e f r a c t i v e proper-  o f dense o p t i c a l media (the Lorenz-Lorentz e f f e c t ) .  However, the e l e c t r o m a g n e t i c analogy does not seem to be written  i n d e t a i l i n the l i t e r a t u r e and -~- because o f i t s  h e u r i s t i c value i n r e v e a l i n g  the s t r u c t u r e  of the pion-nucleus  15 interaction  we discuss  i t i n some d e t a i l .  A d e r i v a t i o n o f the low energy pion-nucleus i n t e r a c t i o n has  r e c e n t l y been g i v e n by M. E r i c s o n and T. E. 0. E r i c s o n  (1966) which takes i n t o account both a b s o r p t i o n correlations.  They derive  the m u l t i p l e  and n u c l e a r  s c a t t e r i n g equations  f o r a p i o n i n the nucleus by a method s i m i l a r to Lax's t r e a t metn (1951) o f m u l t i p l e basic  Their  i d e a i s t o use a systematic expansion i n t o h i g h e r  c o r r e l a t i o n functions truncated the  s c a t t e r i n g i n c l a s s i c a l media.  order  between the s c a t t e r e r s which they have  at p a i r c o r r e l a t i o n s .  They f i n d i n t h i s way that  short-range a n t i c o r r e l a t i o n s o f nucleons lead  to a mod-  i f i c a t i o n o f the i n t r a - n u c l e a r momentum f i e l d i n analogy to the  Lorenz-Lorentz e f f e c t i n a dense o p t i c a l medium.  make the impulse approximation t a k i n g t e r i n g lengths  the s- (p-)wave s c a t -  (volumes) to be those obtained  n u c l e o n s c a t t e r i n g ; they then i n c l u d e  They  from  kinematic  pion-.  corrections  f o r the binding' and Fermi motion o f the nucleons and a l s o extend t h e i r c a l c u l a t i o n s t o take account of the e f f e c t s o f s p i n and i s o s p i n . Their  c a l c u l a t i o n i s further  distinguished  by the f a c t  t h a t , i n a d d i t i o n to a c c o u n t i n g f o r the p i o n s c a t t e r i n g from n u c l e o n s c a t t e r e r s , they a l s o take i n t o account p i o n  scattering  from two-nucleon (or quasi-deuteron) s c a t t e r e r s .  I n t h i s way  they a r e able  which i s  to d e s c r i b e  pion o p t i c a l absorption  dominated by the a b s o r p t i o n  o f a p i o n on two nucleons.  In  a d d i t i o n , they f i n d t h a t the two-nucleon s c a t t e r e r s make an important c o n t r i b u t i o n  to the l o c a l i n t e r a c t i o n where a s t r o n g  16  c a n c e l l a t i o n occurs i n the s-wave s c a t t e r i n g l e n g t h s . resulting potential is  found  which c o n t a i n s no  free parameters  to he i n good e m p i r i c a l agreement with the  and widths  Their  o f x-rays a s s o c i a t e d w i t h 7C -mesic  .—  shifts  atoms.  T h i s t r a n s p a r e n t c o n n e c t i o n which e x i s t s between the elem e n t a r y s c a t t e r i n g processes and  the pion-nucleus o p t i c a l  a c t i o n i s to be c o n t r a s t e d with the nucleon-nucleus interaction.  inter  optical  In the l a t t e r case, the i n t e r a c t i o n s between  the s c a t t e r e d nucleon and t a r g e t nucleons w i t h i n the nucleus depend s t r o n g l y upon the p r o p e r t i e s of the n u c l e a r medium, both because of the long nucleon-nucleon s c a t t e r i n g lengths and because o f the l a r g e r n u c l e o n mass; a c t i o n s can no simple way  hence, t h e i r  inter-  longer be expected to be d e s c r i b e d i n such a  as was  p r o v i d e d to us i n the pion-nucleus case  by the impulse approximation.  I n a d d i t i o n , i n nucleon- ,  nucleus s c a t t e r i n g , we can no l o n g e r i n t r o d u c e the  technical  s i m p l i f i c a t i o n s W h i c h were allowed us i n pion-nucleus s c a t t e r i n g by the s m a l l p i o n mass and, i n g e n e r a l , both  inter-  mediate n u c l e a r e x c i t a t i o n s and kinematic c o r r e c t i o n s f o r r e c o i l o f the nucleon t a r g e t must be taken i n t o  account.  The pion-nucleus o p t i c a l i n t e r a c t i o n t h e r e f o r e i n v o l v e s the m&croscopic p r o p e r t i e s o f the s c a t t e r i n g nucleus i n a much more e x p l i c i t manner than the nucleon-nucleus  optical  inter-  a c t i o n so t h a t pions are much b e t t e r t o o l s f o r i n v e s t i g a t i n g such p r o p e r t i e s o f the nucleiis as the n u c l e a r d e n s i t y . I n the present chapter our purposes F i r s t l y , we w i l l show t h a t the s t r u c t u r e  w i l l be two-fold. o f the pion-nucleus  17  i n t e r a c t i o n c a n be s i m p l y u n d e r s t o o d problem  from  the f a m i l i a r  o f a d d i n g p o t e n t i a l s and f i e l d s i n a c l a s s i c a l  dielectric  S e c o n d l y , we w i l l  c u l a t i o n i n a manner w h i c h  discuss the E r i c s o n s ' c a l -  emphasizes the m i c r o s c o p i c  The i n t e r a c t i o n w i l l  aspects  of  the problem.  in  l a t e r c h a p t e r s w i t h e m p h a s i s on how i t c a n be u s e d a s a  t o o l t o i n v e s t i g a t e macroscopic  t h e n be i n v e s t i g a t e d  p r o p e r t i e s o f the nucleus,  p a r t i c u l a r l y the nuclear density. 2.1  S t r u c t u r e o f the I n t e r a c t i o n In  sical  t h e p r e s e n t s e c t i o n we p r e s e n t a s i m p l i f i e d  d e r i v a t i o n of the pion-nucleus  emphasizes the s t r u c t u r e on t h e a n a l o g y  optical interaction  o f t h e i n t e r a c t i o n and w h i c h  e l e c t r i c p o t e n t i a l s and f i e l d s i n a dense  free charges  dipoles.  The g e o m e t r i c  t e r e d p i o n wave f r o m  and  iso-  i n a d i e l e c t r i c medium; t h e  p-wave s c a t t e r i n g s , b e i n g d i p o l e i n c h a r a c t e r , l o o k  adding  i s based  The i d e a i s t h a t s-wave s c a t t e r i n g s , b e i n g  tropic, look l i k e  electric  which  b e t w e e n a d d i n g s - and p-waves i n t h e n u c l e a r  medium and a d d i n g dielectric.  clas-  problem  like  o f adding the s c a t -  t h e n u c l e a r medium i s t h e n t h e same a s  t h e t o t a l e l e c t r o s t a t i c p o t e n t i a l f r o m a homogeneous  isotropic  d i e l e c t r i c medium i m m e r s e d i n a n e l e c t r i c  Of c o u r s e , a d e r i v a t i o n o f t h i s nafvet?  of geometrical solutions  type possesses  and c a n n o t  field.  a l l the  be e x p e c t e d  to  q u a n t i t a t i v e l y i n c l u d e t h e d y n a m i c s o r quantum a s p e c t s o f the problem; realized and,  i n a d d i t i o n , the technique  of solution  (though n o t s p e c i f i c a l l y s t a t e d ) by other  h e n c e , we p r e s e n t n o t h i n g r e a l l y new.  was authors  Nevertheless, a  18 s o l u t i o n o f t h i s type ledge o f elementary  which s p e c i f i c a l l y invokes a know-  e l e c t r o m a g n e t i c theory - —  e r a b l e value i n understanding-the We  s h a l l f i r s t review  i s of consid-  s t r u c t u r e o f the i n t e r a c t i o n .  the c a l c u l a t i o n o f the e l e c t r o -  s t a t i c p o t e n t i a l from a homogeneous and i s o t r o p i c immersed i n a c o n s t a n t e l e c t r i c i d e n t i f y t h i s p o t e n t i a l and  field.  We  dielectric  s h a l l then simply  the c o r r e s p o n d i n g e l e c t r i c  field  w i t h the p i o n o p t i c a l wave f u n c t i o n and the p i o n o p t i c a l momentum, f i e l d ,  respectively,  l e t us d e f i n e the macroscopic  p r o p e r t i e s o f the  t r i c by d e f i n i n g i t to have a free charge a polarization J _ E ^ i ® U(r),  i s then the sum  dielec-  d e n s i t y , ef°(r_) , and  t o t a l p o t e n t i a l at a point r ,  of the p o t e n t i a l . U ( r ) . a s s o c i a t e d  w i t h the e x t e r n a l e l e c t r i c , f i e l d p l u s the p o t e n t i a l , due  to the  (2-la)  U^fr),  dielectric: Ufr) -  U (r) 0  +  U (r)  .  a  The l a t t e r term i s well-known from elementary  electromagnetic  theory:^ (2-lb)  U  d ( r )  -  U P -  ep(r')  JL-ZL  4-  p  ..  (rM'V  / i  r - r  j ^ T l J l f  [epff t  )  g  o  (__p r  ») +  P ( . E  where we have d e f i n e d the free p a r t i c l e Green's (2-lc)  g]Jr_-r ) = T  1  £7T  e  ik  l^'f  Ir - r  £  j  V  g  0  (r-r  function,  r  A d i s c u s s i o n o f the macroscopic p r o p e r t i e s of d i e l e c t r i c s can be found i n most textbooks on c l a s s i c a l e l e c t r o m a g n e t i c theory ( f o r i n s t a n c e , c . f . Jackson (1962)).  «)J  19  Now,  i f the  d i e l e c t r i c were homogeneous and i s o t r o p i c  on the m i c r o s c o p i c we  s c a l e as w e l l as on the macroscopic s c a l e ,  would have t h a t the e f f e c t i v e f i e l d  was  simply  the macroscopic e l e c t r i c  at a d i p o l e , E g f f ( r ) ,  field  i n the  dielectric,  .  (no c o r r e l a t i o n s ) .  E(r): (2-2)  e  f  However, i f the —-  U(r)  d i p o l e s are a n t i c o r r e l a t e d i n the  that i s , i f two  dielectric  d i p o l e s cannot e x i s t at the same p o i n t  the e f f e c t i v e m i c r o s c o p i c E(r).  y  l f ( £ ) = E(r) =  field  w i l l i n general  differ  from  This e f f e c t i s e s s e n t i a l l y the well-known Lorenz-  L o r e n t z e f f e c t (although, refer  strictly,  t h i s term i s used to  to the r e s u l t a n t e f f e c t on the r e f r a c t i v e index: o f  the medium)„ To c a l c u l a t e the L o r e n z - L o r e n t z e f f e c t , l e t us note t h a t the p o l a r i z a t i o n i s simply  first  r e l a t e d to the e f f e c t i v e  field: (2-3) where  P (£) ?  -  *> (r) 2 d  e f f  #  Tf. i s the molecular p o l a r i z a h i l i ty and  dipole density.  Now  macroscopic f i e l d  the e f f e c t i v e f i e l d  plus the  field  zation: (-> 2  (r_)  4  i s the sum  a r i s i n g from the  £  e f f  (r) =  E(r)  +  latter,  E^fr)  polari-  .  l e t us choose a sphere o f r a d i u s ,  a, s u r r o u n d i n g the d i p o l e so t h a t no other  and  o f the  .  To c a l c u l a t e the  sphere  -^P^(r) i s the  (which we  can  do s i n c e  assume t h a t o u t s i d e  d i p o l e s are i n the  the d i p o l e s are  anticorrelated)  the sphere the medium i s homogeneous  20 and  isotropic.  The e l e c t r i c  field  a r i s i n g from the p o l a r i .  z a t i o n i s then e a s i l y c a l c u l a t e d . n  S p h e r i c a l C a v i t y Surrounding a Dipole i n a D i e l e c t r i c Referring  to the above s k e t c h and n o t i n g  the charge on the surface normal), we have (2-5)  o f the c a v i t y  (n i s the outward  that (-P.n) (-ndS)  E  -P  t h a t -P/n i s  p cose n an.  = ~? p  Thus, from Eqs. 2-3 and 2-4, (2-6a) —eff  —  —  —  rn[  3 (2-6b)  Q  1 . L r ; ( r ) Substituting  d  E(r)  ~  these r e l a t i o n s i n - E g . .  e q u a t i o n f o r the p o t e n t i a l : o  (zero-range anticorrelations)  ~  P(r) 3  E(r)  2 - l a , we have an i n t e g r a l  21 (2-7)  U(r) = U ( r ) Q  1  Eqs.  —  47T ] d r  5  e/(rj)g (rlr') 0  _ -Mr/ (r5  2u(^)V' (r-r') g 0  d  2-6 are o f t e n used i n elementary  o p t i c s to explain  t h e . n o n - l i n e a r dependence o f the r e f r a c t i v e index o f an opt i c a l medium upon i t s d e n s i t y effect.  T h i s i s e a s i l y seen by remembering from  Lorenz-Lorentz elementary  theory t h a t the molecular p o l a r i z a b i l i t y , J£ ,  electromagnetic is  the c l a s s i c a l  d e f i n e d by the e q u a t i o n , (2-8a)  and  P-'JCE,  t h a t the r e f r a c t i v e index o f the medium, n, i s g i v e n i n  terras o f X (2-8b)  by the r e l a t i o n , n  8  = 1+ 4 7 T X ,  so t h a t (from Eq. 2-6b), (2-8c)  o  n^ =  1+  r P " -cl  47r2 a  1  For our purposes,  -  B*J>  d  we w i l l be i n t e r e s t e d i n the geometric  m o d i f i c a t i o n o f the i n t r a - n u c l e a r p i o n momentum f i e l d  sug-  gested by Eqs, 2-6; the " r e f r a c t i v e index" o f the p i o n i n the nucleus  does not bear  the same analogy  to the e l e c t r o m a g n e t i c  case because the p i o n o p t i c a l wave and the e l e c t r i c  potential  are not d e s c r i b e d by the same e q u a t i o n s . We w i l l now show t h a t the o p t i c a l l y s c a t t e r e d p i o n wave from a nucleus i s d e s c r i b e d by an e q u a t i o n l i k e Eq, 2-7 provided'  22 we i d e n t i f y the o p t i c a l wave with the e l e c t r o s t a t i c p o t e n t i a l and  provided we i d e n t i f y the s-wave s c a t t e r i n g with the  e l e c t r i c charge and the p-wave s c a t t e r i n g with the molecular, polarizability  ( f o r s i m p l i c i t y , we w i l l n e g l e c t the p i o n  charge). To see the analogy, l e t us f i r s t note t h a t the s c a t t e r i n g amplitude o f a p i o n from a nucleon has the g e n e r a l  form  ( n e g l e c t i n g e x p l i c i t r e f e r e n c e to the i n t e r n a l pion and nucleon co-ordinates (2-9)  f =  where the f i r s t  and talcing Ti s c = 1 ) ,  b + cK»._k  ,  term i s the s-wave s c a t t e r i n g amplitude and  k i s the  the second term i s the p-wave s c a t t e r i n g amplitude; i n i t i a l p i o n momentum and It the f i n a l p i o n momentum. 1  Now the wave, d $ s c ( r ) ,  s c a t t e r e d from an element o f  n u c l e a r densi t y , j> (r') d r , a t the p o i n t , r', i s the product . 1  o f the amplitude f o r f i n d i n g a p i o n a t r ,  described by the  T  p i o n o p t i c a l wave, $ f r ' ) , with  the s t r e n g t h o f the s c a t t e r i n g ,  f p ( r ' ) d r ' , and the s p h e r i c a l l y outgoing wave from the s c a t terer, g j j E - r ' ) , t i o n o f Eq. (2-10)  where g ^ i s the f r e e - p a r t i c l e Green's func-  2-lc: .ikjr-r'l  = - ^ T l g ^ r - r M b $(r» ) f ( r ')dr  1  • - i J t g ^ f r - r M o k ' - k J ( r ) ? ( £ ' ) dr f  1  „  Nov? (2-11)  Z'gij£-£') •=  -ii'Sic(£-£')  and we may c o n v e n i e n t l y d e f i n e the e f f e c t i v e i n t r a - n u c l e a r  23 momentum d e n s i t y a r r  E^fr')  (2-12)  T  -  -ik_$(r')  .  E _-_-(r') i s r e l a t e d t o the macroscopic momentum d e n s i t y , e  (2-13)  E(r)  through E q s  w  i 2 $(r)  5  2-2 and 2-6.  ,  Eq. 2-10 can then be r e w r i t t e n  i n the form o f Eq. 2 - l b :  (2-lb ) 1  $  s c  ^)  -4^*v[b§(rM/(rMgiJr-rM  s  with " p o l a r i z a t i o n " P ( r ) = -c/(r) E _ e f f ( r )  (2-3')  .  Assuming an i n c i d e n t plane wave, f (r)  (2-14)  0  -  e  1  ^ ,  the t o t a l wave becomes £(r) = e  (2-15)  Thus  1  to the extent  quantum aspects  - 47rjdr [bf ( r ' )f(r M g ^ r - r ' )  ^  t h a t we can n e g l e c t the s t r u c t u r e and  o f the s c a t t e r e r  the p i o n s c a t t e r i n g  w i t h i n the n u c l e a r medium i s e q u i v a l e n t to the c l a s s i c a l optical  s c a t t e r i n g w i t h i n a d i e l e c t r i c medium, p r o v i d e d  identify (2-16a)  the s-wave s c a t t e r i n g w i t h the e l e c t r i c e  )  and p r o v i d e d we i d e n t i f y molecular  b|(r')  we  charge,  ,  the p-wave s c a t t e r i n g w i t h the  polarizability,  (2-16b)  ^  -c  We can w r i t e the i n t e g r a l Schroedinger i n conventional  Equation 2-15  d i f f e r e n t i a l form by o p e r a t i n g on both s i d e s  24 wi th the operator 0  1  V  + k  2  .  2  Then (2-17)  (V  2  + k ) J(r)  = -47lb^(r • )$(r )  2  +  2 »4*c/(r < ) E  (r ) :  e f f  I n an u n c o r r e l a t e d n u c l e a r medium, we have from Eq„ 2-2 that 4  —  (2-2')  i  i  ' ^Q±f^S)  ~  ^  n 0  correlations)  so t h a t  ( V + k )J ( r )  (2-18a)  = -4nb/>(r )<jj (r) f ^ .4Xc/(rJ V $ ( r ) ,  2  (no c o r r e l a t i o n s ) which i s the form o f the pion-nucleus i n t e r a c t i o n d e r i v e d by K i s s l i n g e r  (1955); i n a n u c l e a r medium w i t h  a n t i c o r r e l a t i o n s we have the Lorenz-Lorentz (2-6«)  E (r) = -exx —  \  f f  l  n  effect,  / T E r r ) , <zero-range c/lr) • . » a n t i c o r r e l a t i ons) P  + , - - — •  «->  so t h a t (2-18b)  ( V + k ) $ ( r ) z -47ibp(r)|(r) a  3  ,  . 4Jic.p(r] I f f  (zero-range  _ ^. .  cj>(r) anticorrelations)  which i s the m o d i f i c a t i o n t o the K i s s l i n g e r gested  short-range  by K r o l l  equation:";sug-  (1961) and shown i n d e t a i l by the E r i c s o n s  (1966). I n the present s e c t i o n we have demonstrated the s t r u c t u r e o f the pion-nucleus i n t e r a c t i o n by a c l a s s i c a l and geometric  technique  f o r adding the elementary  s c a t t e r e d waves  from a ( m a c r o s c o p i c a l l y ) homogeneous and i s o t r o p i c n u c l e a r  25 medium. T. E. 0. E r i c s on  (1967) has a l s o c o n s i d e r e d the c l a s s i c a l  problem hut from a p o i n t  o f view which emphasizes the r o l e  o f the m i c r o s c o p i c c o r r e l a t i o n s  o f the s c a t t e r e r s .  His tech-  nique i s to expand the o p t i c a l wave i n terms o f the h i g h e r order c o r r e l a t i o n f u n c t i o n s  between the s c a t t e r e r s ,  c a t i n g the r e u l t i n g h i e r a r c h y tions.  trun-  o f equations a t p a i r c o r r e l a -  He then f i n d s the U s s l i n g e r  e q u a t i o n , 2-18a, i n the  absence-of c o r r e l a t i o n s and the l o r e n z - L o r e r i t z e f f e c t , Eq. 2-18,; i n the case o f zero-range a n t i c o r r e l a t i o n s . advantage o f E r i c s o n J s  technique i s t h a t i t g e n e r a l i z e s i n  an obvious way to i n c l u d e the  the s t r u c t u r e  quantum aspects o f the problem.  been made by the E r i c s o n s c u l a t i o n i n the next 2.2  A Microscopic  o f the s c a t t e r e r ana  This g e n e r a l i z a t i o n has '  (1966) and we d i s c u s s  Derivation  o f the I n t e r a c t i o n ;  o f the p i o n - n u c l e u s i n t e r a c t i o n u s i n g  involve  only  their cal-  section.  I n the p r e c e d i n g s e c t i o n , we have d e r i v e d ture  The  the s t r u c -  arguments which  the geometry o f adding c l a s s i c a l waves.  p r e s e n t s e c t i o n we w i l l d i s c u s s  I n the  a m i c r o s c o p i c and quantum  mechanical treatment o f the problem which has been g i v e n by T. E. 0. E r i c s o n and ( h i s wife) M. E r i c s o n tant feature introduce  'be  The impor-  o f t h e i r c a l c u l a t i o n i s t h a t i t enables one to  the s t r u c t u r e  for pion absorption. a point  (1966).  o f the s c a t t e r e r and the mechanism  We s h a l l d i s c u s s  t h e i r c a l c u l a t i o n from  o f view which emphasizes the assumptions which must  made i n order to o b t a i n the s i m p l i f i e d p o t e n t i a l o f the  p r e v i o u s s e c t i o n and which p o i n t s out the c o r r e c t i o n s which i t can be  by  improved.  For the purposes low energy p i o n may  o f i t s i n t e r a c t i o n s with nucleons, a  be regarded as a l i g h t boson w i t h zero  s p i n and three charge s t a t e s  ( JI , 71°, X ). +  Moreover, u n l i k e  the nucleon-nucleon i n t e r a c t i o n , the low energy  pion-nucleon  i n t e r a c t i o n i s d e s c r i b e d by very s h o r t s c a t t e r i n g l e n g t h s ; f o r i n s t a n c e , the NN s i n g l e t s c a t t e r i n g length i s about fm. and the  the NU  -24  t r i p l e t s c a t t e r i n g l e n g t h i s about 5 fm. while  IT i s o s p i n doublet and quadruplet s c a t t e r i n g lengths  are both o f the order o f 0,2  fm.  I t i s mainly t h i s p r o p e r t y  o f the p i o n , combined w i t h the s m a l l p i o n mass ("^1/7 the nucleon mass), which allows us to simply c o n s t r u c t the average pion-nucleus i n t e r a c t i o n from the elrnentary p i o n - n u c l e o n s c a t t e r i n g data; on the other hand, the c o r r e s p o n d i n g  problem  f o r the nucleon o p t i c a l p o t e n t i a l i s almost u n t r a c t a b l e .  Nucleons's view o f a nucleus  (zero energy)  Pion's view o f a nucleus  To see how the s m a l l pion-nucleus  s c a t t e r i n g lengths  s i m p l i f y the problem, i t i s h e l p f u l to c o n s i d e r the zeroenergy s c a t t e r i n g ; we can then r e p l a c e the a c t u a l i n t e r a c t i o n s by e q u i v a l e n t hard  sphere i n t e r a c t i o n s having  the s c a t t e r i n g l e n g t h  (above s k e t c h ) .  the r a d i u s o f  For a nucleon  pro-  j e c t i l e , the spheres are l a r g e r than the i n t e r - n u c l e o n sepa r a t i o n w i t h i n the nucleus  (-^1.4 fm.) , while  f o r a pion  they are much s m a l l e r . We'can i n t e r p r e t t h i s as meaning t h a t a nucleon  sim-  u l t a n e o u s l y i n t e r a c t s w i t h many nucleons w i t h i n a n u c l e a r medium so t h a t the nucleon-nucleon s c a t t e r i n g operator i s a many-body operator  depending upon the p o s i t i o n s and motions  o f many other nucleons.  On the other hand, a low energy p i o n  i n t e r a c t s with only one nucleon  a t a time, and i t i s reason-  able to assume t h a t the pion-nucleon  s c a t t e r i n g operator  w i t h i n a n u c l e a r medium i s the same as i n f r e e space, part i c u l a r l y so s i n c e the p i o n i s l i g h t and does not f e e l the dynamics o f the nucleus.  T h i s assumption i s commonly known  as the impulse approximation (and i s a l s o . v a l i d a t very e n e r g i e s when the p a r t i c l e wavelength becomes s m a l l e r the i n t e r - p a r t i c l e The  high  than  spacing).  s m a l l mass o f the p i o n a l s o s i m p l i f i e s the c a l c u l a -  t i o n i n the f o l l o w i n g two ways. c o l l i s i o n s with the nucleon  F i r s t l y , i n i t s elementary  s c a t t e r e r s , the k i n e t i c  exchanges  of the p i o n with the s c a t t e r e r s are small so that we can e s s e n t i a l l y n e g l e c t the e f f e c t s o f i n t e r m e d i a t e excitations.  nuclear  In f a c t , i n t h i s approximation we c o n s i d e r the  28 elementary  c o l l i s i o n s to be e s s e n t i a l l y e l a s t i c , "which r e s u l t s  i n an enormous s i m p l i f i c a t i o n o f the Green's f u n c t i o n d e s c r i b i n g the p r o p a g a t i o n of the p i o n w i t h i n the n u c l e u s .  The  E r i c s o n s have estimated the c o r r e c t i o n s from i n t e r m e d i a t e e x c i t a t i o n s and A second  f i n d them to be  negligible.  simplification results  s c a t t e r e r s to be massive.  from c o n s i d e r i n g the  T h i s g r e a t l y s i m p l i f i e s the mul-  t i p l e s c a t t e r i n g equations s i n c e i n the s c a t t e r i n g we  do not have t o account  T h i s assumption  f o r the motion o f the  nucleons.  can be improved by making the f o l l o w i n g two  kinematical corrections:  f i r s t l y , we must c o r r e c t f o r the  f a c t t h a t the t a r g e t nucleons energy  processes  of about 25 MeV  are i n motion having a k i n e t i c  (Fermi -motion); secondly, we must  c o r r e c t f o r the r e c o i l o f the bound n u c l e o n s .  The E r i c s o n s  have estimated both c o r r e c t i o n s , f i n d i n g them to be o f the order o f 10$.  .  To make these i d e a s more e x p l i c i t , we w i l l E r i c s o n s ' technique problem.  We  '  s  d e s c r i b e the  f o r s o l v i n g the m u l t i p l e s c a t t e r i n g  w i l l not give a d e t a i l e d e x p o s i t i o n o f the  d e r i v a t i o n s , but w i l l content ourselves w i t h s t a t i n g the highlights We  of the  calculation.  begin by n o t i n g t h a t the p i o n wave,(|/^. , (which,  course, i s an operator i n the space  of the t a r g e t nucleons)  s a t i s f i e s the e q u a t i o n (Goldberger & Watson (2'-19a)  U/T  = X ,  4.  where the " e f f e c t i v e wave'*,  1  of  lira  i s the. sum  (1964)), 1  T , \jf.  o f the i n c i d e n t  "wave.Xj^ / a n d the  i  t  n  •  .  i  :  J(=i)=l  HJJ i s the n u c l e a r H a m l l t o n i a n  ~  and  . number E ^ the. t o t a l operator  from the i will  /0>  ' :  the p i o n k i n e t i c  A i s the atomic number o f the s c a t t e r e r ;  state,  • .  the s c a t t e r e d waves from a l l s c a t t e r e r s except  •  We  J •  29  energy;  k i s t h e . p i on wave  energy o f the system; Tj[ i s the s c a t t e r i n g t  h  nucleon  0  assume the nucleus  , and w i l l  to i n i t i a l l y be i n the  apply the  o p e r a t o r s , Eq„  system i n the c o n f i g u r a t i o n space o f the p i o n . are then e q u i v a l e n t to the h i e r a r c h y o f  (2-20a) (/£(r)|0?=  X  (*)  Q  \0?  2-19,  ground  to the  Eqs.  2-19  equations,  t f j j g " ( r ,r') f  (r' , r ' W  (r " )  1  ±  ±  x /07  (2-20b) (jA(r)J07=  X (r)IO> 0  T  X  Q  ( r )  dr'dr";  +  £1 ( S ( r , r ' ) f J r , r ' ') W.  (2-20c) l ^ ; i ( r ) =  '"  . ( r ' M /0>dr'dr  ti  |0> -f.(r'MjO>dr'dr"  where we  have i n t r o d u c e d  an obvious n o t a t i o n to keep t r a c k o f  4  the n u c l e o n  co-ordinates.  (2-21a)  Here  ; g(r,r ) = f  . lim  ^ r | -  —s  1 lim <r|  L—,  n  J  k  M  a  2 /n><n| I r > 1  Jr 7-|n><nl T  so is  the t o t a l pion-nuoleus Green's f u n c t i o n , where we  have  d e f i n e d the o n e - p a r t i c l e Green's f u n c t i o n s , 4  (2-21b)  •  <  g fr,r')  and where the sum  l i m <r j  s  n  /  r  *  >  i s oyer a l l n u c l e a r s t a t e s , / n>.  s i m p l i c i t y , we have ignored particle  JL_i_——  For  the p i o n charge i n the  Green's f u n c t i o n , Eq. 2-21b  (and,  one-  for n o t a t i o n a l  convenience, have suppressed the i n t e r n a l c o - o r d i n a t e s the p i o n ) .  In Eq. 3-20  we  have a l s o d e f i n e d the  of  matrix  elements o f the s c a t t e r i n g operator, (2-22) We  fi(r,r')  =  <r|Ti/£>  have mentioned i n e a r l i e r paragraphs that we  s i m p l i f y the Green's f u n c t i o n , Eq. 2-21, mass "of the p i o n and o p e r a t o r , Eq„ 2-22,  t h a t we  can  because o f the  light  can s i m p l i f y the s c a t t e r i n g  mainly because o f the small s c a t t e r i n g  l e n g t h s which d e s c r i b e the pion-nucleon  interaction.  Because of the s m a l l value of the p i o n mass, elementary k i n e t i c exchanges are s m a l l so that the propagation p i o n i s not s e n s i t i v e to the i n t e r m e d i a t e excitation. to be  Thus we may  of the  states o f nuclear  choose the elementary  collisions  elastic,  (2-23)  ' g (r,r') ^  g (r,r')  n  Q  whence the Green's f u n c t i o n , Eq. 2-21, (2-24)  gfr.r ) = 1  g  Q  l.r ,r') X| n>&|  simply =  becomes  g (r,r*) I 0  N  ,  where  i s the i d e n t i t y o p e r a t o r i n the space  o f the nucleus.  T h i s approximation, o f course, r e p r e s e n t s an enormous s i m p l i f i c a t i o n since i t l a r g e l y decouples the s t r u c t u r e the s c a t t e r e r  from the o p t i c a l s c a t t e r i n g .  To see how  the impulse approximation s i m p l i f i e s the  s c a t t e r i n g o p e r a t o r , %(£.,£.')» © w  elements i n the c o n f i g u r a t i o n  w i l l w r i t e i t s matrix  space  of the nucleons  s i n g the i n t r i n s i c nucleon c o - o r d i n a t e s ) , approximation we (2-25)  of  </£;  have  (suppres-  From the impulse  that,  R i . R E ^ o . R A l T i l ^ ;  ^ W ^ i - ^ V  c  * % - ^ V ° °  B  ^ ^ i - i - S i - i  1  where  h e i n g the free space s c a t t e r i n g operator o f the p i o n from the i ^ k nucleon. reduced  Hence, the many-body s c a t t e r i n g operator i s  to a known two-body s c a t t e r i n g operator.  The s c a t t e r i n g m a t r i x i s f u r t h e r s i m p l i f i e d by the as sump t i o n that (2-26)  the nucleons are massive: f ^,R .;r i  i  , t  R^)  5  f. (r ; -R^;  r \£  ±  ) /(^-Ef ).  • Since i n the impulse approximation the s c a t t e r i n g  oper-  a t o r i s simply the free space s c a t t e r i n g operator o f a p i o n from a nucleon, i t i s e a s i l y evaluated because i t s m a t r i x elements i n momentum space are (at low e n e r g i e s ) simply the s-wave and p-wave s c a t t e r i n g  amplitudes:  32 (2-27)  f . ^ ) S <t'|T |k> i  -  b^b^JT-j^  (s-wave)  (p-wave) Here b , "b  are r e l a t e d to the s-wave pion-nucleon s c a t t e r i n g  Q  and c . o „ . d , a, are r e l a t e d t o the p-wave s c a t ' * :. O* 1 0' 1  lengths, to  t e r i n g volumes ( c . f . E r i c s o n and E r i c s o n . ( 1 9 6 6 ) ) ;  the s p i n -  f l i p term has no s-wave c o n t r i b u t i o n s i n c e the p i o n has s p i n zero.  I n Eq„ 2-27, T and  o f the p i o n and the i  t  J  l  are the isospi.n operators  nucleon, r e s p e c t i v e l y .  As we have  o f t e n noted, i t i s important to r e t a i n the p-wave c o n t r i butions,  even a t low e n e r g i e s ,  because o f the l i n g e r i n g  e f f e c t s o f the 3-3 r e s o n a n c e ; t h e h i g h e r p a r t i a l waves make negligible  contributions.  Employing the assumptions i n Eqs. 2-23 to 2-27, the o p t i c a l pion wave, (2-28)  f ( r ) = <0|||£(r) ) 0 ^ ,  can be evaluated i n a s t r a i g h t f o r w a r d The  Ericsons  (1966) have performed the nuclear  find multiple for multiple (2-29a)  averages and  s c a t t e r i n g equations which are e s s e n t i a l l y those s c a t t e r i n g i n a c l a s s i c a l media (Lax (1951)):  J(r)  = Xjr) x  (2-29b)  manner from Eqs. 2-20.  (r) =  ^J'^Li^Lijj •J  r  B in-L iL\-Li>  (r")dr'dr  L*-L^  ,)f  Q  •  n  XJx)/^^ ;r )dr^g (r-r ') 8  x  f(r.?-r„; r " - r ' J  1  0  1  (r"  )dr*dr";  33  Jrg.^W  (2-290)  x  Here ^> is  (r^)  s  ^o^+^^s^l^^sjTgo^^ ; r"-r_„) J  ,f(r -r T  #  (r")dr'dr" *  i s the average d e n s i t y o f t h e nucleus ;/>  the p r o b a b i l i t y d e n s i t y f o r f i n d i n g a p a r t i c l e  there i s a p a r t i c l e a t r ;  5  ?  (r^jr^)  at  when  (r"0 i s the sum a t r " o f  the i n c i d e n t wave and the s c a t t e r e d waves from a l l s c a t t e r e r s except, the  one a t r_g, s u b j e c t to the knowledge t h a t there i s  a s c a t t e r e r at r ^ . These equations  The other  terms have a s i m i l a r meaning.  d e s c r i b e e s s e n t i a l l y the same problem that  we gave a geometric s o l u t i o n f o r i n Sec. 2-1 and they have been d i s c u s s e d  i n v a r i o u s approximations by E r i c s o n (1967).  We s h a l l now wish to t r u n c a t e at  some p o i n t and  the h i e r a r c h y o f Eqs. £-29  s o l v e the coupled  equations.  i f we took (2-30) in  For instance -  $  Ll  ( r ) = X (r) ~ o -  Eq„ 2-29a, i t i s seen t h a t we would simply have made the  f i r s t Born approximation.  That t h i s i s not a good approx-  i m a t i o n can be seen from the dominance o f the s- and p-waves in  the pion-nucleon As  scattering  amplitudes.  a second attempt - — one which i n c l u d e s m u l t i p l e  s c a t t e r i n g — . we might take 4  (2-31)  \  § (r) = J ( r ) -1  T h i s i s i d e n t i c a l to our geometric d e r i v a t i o n i n the f i r s t s e c t i o n i n which we n e g l e c t e d  nuclear  correlations; i n fact.  34 t h i s i s seen e x p l i c i t l y to be the case s i n c e t e r m i n a t i n g the h i e r a r c h y at Eq„ relations;  2-29a i s e q u i v a l e n t to n e g l e c t i n g the c o r -  Of c o u r s e , t h i s approximation leads to the  K i s s l i n g e r Eq„ 2-18a,  From our present p o i n t of view, we  note t h a t the primary handicap' i n t h i s approximation i s that ?  r  from J ( r ) a t r by a wave which i s s i n g u l a r  (r) d i f f e r s  at r . To a v o i d t h i s s e l f - e x c i t a t i o n of the s c a t t e r e r , we  go  to the -next approximation and s e t (2-32)  I  (r) = -2'-l .  i n Eq. 2-29b„  \  (r) ~2  Since r ^ and r ^ are g e n e r a l l y f a r a p a r t i n  the n u c l e u s , we  do not expect the wave  <F (r) to depend £-2 ~l ~ ;  very s e n s i t i v e l y  on the exact p o s i t i o n o f r _  In f a c t , i f  l0  2) *  we w r i t e the j o i n t p r o b a b i l i t y d e n s i t y ,  i  n  t  e  r  m  s  o f the p a i r c o r r e l a t i o n f u n c t i o n , G ( r . , r ) , P  (2-33)  \P(rQ;^i)  = / (£a >  )  ( 1  +  Gf  £i»£a  )  }  *  we can remove the u n p h y s i c a l divergence a t r = v by "—1 ~<d 0  ducing the zero-range a n t i c o r r e l a t i o n of two (2-34)  G(r  ,r ) 2  -1  =  intro-  nucleons,  „  I n f a c t , i t i s t h i s c h o i c e o f the p a i r c o r r e l a t i o n  function  4  which leads to the Lorenz-Lorentz e f f e c t , as we our geometric The  foiznd i n  d e r i v a t i o n (Eq. 2-18'b) .  E r i c s o n s have shown that r e p l a c i n g the  c o r r e l a t i o n f u n c t i o n , Eq. 2-34, range, leads to e f f e c t s  with a f u n c t i o n o f f i n i t e 2  of order (§ K)  o f the c o r r e l a t i o n f u n c t i o n and  zero-range  where ^ i s the range  K i s the wave number o f the  35 p i o n i n the n u c l e a r medium. r e g i o n i s thus n e g l i g i b l e than one  The  s t r u c t u r e o f the c o r r e l a t i o n  f o r wavelengths a p p r e c i a b l y l a r g e r  f e r m i ; on the other hand, t y p i c a l wavelengths o f  low energy pions are s e v e r a l times the n u c l e a r Hence, we expect to l e a m l i t t l e  dimensions,,  about the s t r u c t u r e o f the  c o r r e l a t i o n f u n c t i o n from low energy p i o n scattering,,  Tn  f a c t , the E r i c s o n s have estimated these range c o r r e c t i o n s find  them t o be  of the order o f 10%,  and  T h i s i s i n c o n t r a s t to  the measurements o f m o l e c u l a r c o r r e l a t i o n s through the s c a t t e r i n g of slow neutrons by a c r y s t a l l a t t i c e ;  here the neutron-  n u c l e a r s c a t t e r i n g lengths are a l s o much s m a l l e r than the i n t e r - m o l e c u l a r s p a c i n g , but the neutron wavelength a comparable value We have now  (Van Hove  i s of  (1954)),  d i s c u s s e d the connections between the micro-  s c o p i c quantum d e s c r i p t i o n of the pion-nucleus i n t e r a c t i o n and  the geometric  d e r i v a t i o n g i v e n i n the p r e v i o u s s e c t i o n  (Sec,  2-1),  In a d d i t i o n , we  range  c o r r e c t i o n s which must be made to the pion-nucleus  i n t e r a c t i o n , Eqs, 2-18,  have d i s c u s s e d the kinematic and  I t now  remains  to d e s c r i b e the  E r i c s o n s ' technique f o r d e s c r i b i n g o p t i c a l p i o n a b s o r p t i o n . There are e s s e n t i a l l y three mechanisms by which a low energy  o p t i c a l p i o n can be absorbed:  f i r s t l y , i t can be  i n e l a s t i c a l l y s c a t t e r e d by the nucleus; s e c o n d l y , b e i n g a boson, i t can be absorbed  by a s i n g l e n u c l e o n , i t s r e s t mass  b e i n g transformed i n t o the k i n e t i c energy o f the t h i r d l y , i t can be absorbed the nucleus i s suppressed so that  by two nucleons.  due  scatterer;  Excitation of  to the s m a l l mass o f the p i o n ,  e f f e c t s o f the f i r s t mechanism may  be n e g l e c t e d .  36 In a d d i t i o n , the a b s o r p t i o n by  one nucleon  r e q u i r e s a nuc-  l e a r momentum o f about 500 MeV/c from kinematic while 850  considerations  the Fermi momentum o f the nucleons i s of the order  of  MeV/c; hence, one-nucleon a b s o r p t i o n i s a l s o s t r o n g l y i  4  suppressed.  Thus only the process  o f two-nucleon absorp-  t i o n c o n t r i b u t e s s i g n i f i c a n t l y to the o p t i c a l  absorption.  To s t r i c t l y account f o r two-nucleon p i o n a b s o r p t i o n , should  use  t i o n and  field  t h e o r e t i c techniques  d e s t r u c t i o n of pions.  a d e t a i l e d model o f the nucleus to  describe  the simple  use  (such as the s h e l l model)  As  orders  f o r the  d i s c u s s e d by the E r i c s o n s , t h i s with much success;  a more phenomenological approach, and and  i n fact,  absorption  of~~magnitude s m a l l e r than  T h i s then would seem s u f f i c i e n t  done t h i s by extending  obviates  obtained  i n t h i s manner, Spec t o r (1964) .found  r a t e s n e a r l y two observed.  In a d d i t i o n ^ we would need  macroscopic d e s c r i p t i o n which we  method of approach has not met proceeding  to d e s c r i b e the c r e a -  the kineriiatics of the a b s o r p t i o n , which  r e f r a c t i v e processes.  we  those  j u s t i f i c a t i o n to the E r i c s o n s have  r e f o r m u l a t i n g the  quasi-deuteron  model o f Brueckner e t a l . (1951), In the Brueckner model, a b s o r p t i o n i s considered take p l a c e at zero energy with the two  absorbing  to  nucleons  in  a r e l a t i v e s - s t a t e * the two-nucleon a b s o r p t i o n  amplitude  is  then i n t r o d u c e d  accounting  for  as a phenomenological q u a n t i t y  a l l the short-range  a b s o r p t i o n processes.  Hence, the  problem o f o b t a i n i n g a d e s c r i p t i o n which i s s a t i s f a c t o r y a field  t h e o r e t i c p o i n t o f view i s l a r g e l y t r a n s f e r r e d to  a j u s t i f i c a t i o n of these  phenomenological q u a n t i t i e s .  from  37  The  E r i c s o n s have g e n e r a l i z e d the Brueckner model (and  i t s r e f o r m u l a t i o n by E c k s t e i n  (1963))  i n two ways.  Firstly,  they i n t r o d u c e an e l a s t i c s c a t t e r i n g amplitude to d e s c r i b e p i o n s c a t t e r i n g by a two-nucleon s c a t t e r e r ; f o l l o w i n g Eckstein litude  they measure the parameters o f t h i s amp-  (1963),  from c o n s i d e r a t i o n s o f p i o n p r o d u c t i o n i n M c o l -  lisions. included  Secondly, they extend the Brueckner model, which only s-wave p i o n a b s o r p t i o n , to i n c l u d e p~wave  pion absorption.  The r e s u l t i s t h a t the two-nucleon  amplitude c a n be w r i t t e n i n the form, (2-35)  f.  .=  <T(r,-r.) J ( r - i ( r , + r , ) ) (B + C k'-k  -+ ( B 5  where B.. an C m  m  c  +- C k'. k)j~(T» t. ) ( T ' t .)-f (T-t^) (T»t. A ) , 5— — L a -J J 1.J  are l i n e a r combinations o f the v a r i o u s  amplitudes o f the angular momentum and i s o s p i n s t a t e s o f the  (7I2W)  (KW)  system.  amplitude,  Since Eq.  2-35  i s o f the form o f the  Eq. 2 - 2 7 , we c a n simply account f o r p i o n  o p t i c a l a b s o r p t i o n by adding the two-nucleon amplitude o f Eq. 2 - 3 5 t o the one-nucleon amplitude o f Eq. 2 - 2 7 . Of course,  the two-nucleon amplitudes w i l l a l s o make  c o n t r i b u t i o n s t o the r e f r a c t i v e aspects  o f the problem  through the r e a l p a r t s o f the parameters B^ and C ; o .r m m these  r e a l p a r t s are to some extent induced  t i o n , and are n o t very w e l l determined.  ,  by the absorp-  This i s o f par-  t i c u l a r i n t e r e s t t o the s_wave s c a t t e r i n g , where a s t r o n g c a n c e l l a t i o n occurs Of course,  i n the one-nucleon amplitude,  f o r the two-nucleon processes,  b  Q  the two-  38 nucleon density replaces  the  one-nucleon d e n s i t y .  To  a  good approximation the two-nucleon d e n s i t y can simply  he  r e p l a c e d by the square o f the d e n s i t y ; however, i n the case of s-waves, where the s c a t t e r i n g lengths  strongly  c a n c e l , the c o r r e c t i o n for long-range c o r r e l a t i o n s should be i n c l u d e d .  The  E r i c s o n s show that the c o r r e c t i o n term  i s a l s o p r o p o r t i o n a l to the square o f the d e n s i t y ; f o r l a t e r purposes we t i v e processes and we  w i l l only r e q u i r e the  our  f a c t t h a t absorp-  are p r o p o r t i o n a l to the square of the  w i l l not i n c l u d e t h i s c o r r e c t i o n i n our  density  subsequent  discussion. t  We  s h a l l now  simply  s t a t e the r e s u l t which the  sons o b t a i n f o r the pion-nucleus r e s u l t i s e s s e n t i a l l y t h a t the o p t i c a l equation, (2-36a) ( V -f k 2^ ) S  t  iT if f ) '  a  2"  n  l(£J 1  Their  pion wave s a t i s f i e s  the  -uil(r) -m (r) 2  S  + n  interaction.  Eric-  2^  -i47fd tf(r) X 0  I<£) ,  2  + i d i ^ r ) +n (r)) 2  where  m, (r) = 4 j f ( l )  (2-36b)  Mr) +  M„ +.  (2-36c)  2 r  m  () £  ( 1  + i(n  (r) = 4 n ( l +- J^Z 8-.. 2M  2M-  (r)  +  I'VP^"  n (r)) 8  8B (T.t)fT-  2  -°  H  ~  X-ri.(n (r) sr Q 1 — n  pfr) )  A  B P (r)  )  1  IT  '=1=  0  5  +  t ) / ( r )  *  ~~  n (r)) 2 — _ Q  -1  2 <  P  2 ^  39 (2-36d)  n, (r) =  47i(l + ^  ^  1  (2-36e)  )  %  •  C /(r)  n ( r ) = ""4*(1+3L) • ~ • 2M  0  0  A  8C (T t ) ( T t ) / ( r )  +  5  T  The e l e c t r o s t a t i c p o t e n t i a l , V i n c l u d e d i n Eqs. 2-36, charge  o f the next  -Comparing Eqs. interaction retains  c  (?) *  w  h  i  c  h  Experimental  ©  n a v  e  n  o  d i s c u s s the e l e c t r o s t a t i c  t  potential  chapter.  2-36  and  Eqs. 2-18,  we  see t h a t the  the simple form p r e d i c t e d by the geo-  m e t r i c arguments o f the previous s e c t i o n 2-3  w  i s to he evaluated from the n u c l e a r  d i s t r i b u t i o n ; we  i n Sec. 3-3  A *  (Sec.  2-1).  Verification  I n the present s e c t i o n we  w i l l b r i e f l y review  the  e x p e r i m e n t a l c o r r o b o r a t i o n o f the E r i c s o n s ' p o t e n t i a l , Eq. 2-36.  V i r t u a l l y the o n l y data a v a i l a b l e i n the  energy r e g i o n i s t h a t from the 7T -mesic x-ray (which are a t e s s e n t i a l l y is  to analyze the s h i f t s  zero energy) „ and  w i t h the p o t e n t i a l , Eq. 2-36,  the widths  The  experiments technique  o f these x-rays  and to compare the r e s u l t i n g  parameters w i t h the t h e o r e t i c a l l y p r e d i c t e d values. g e n e r a l remark might be (1967) f i n d periment,  low  A  that the E r i c s o n s (1966) and E r i c s o n  theory to be i n s a t i s f a c t o r y agreement with  ex-  the t h e o r e t i c a l and e m p i r i c a l parameters g e n e r a l l y  agreeing to b e t t e r than 50%,  We  shall  only c u r s o r i l y re-  view t h e i r d i s c u s s i o n here. The  e s s e n t i a l i de as" i n v o l v e d i n analj/zing the TT -mesic  x-ray data f o r the parameters of the pion-nucleus  potential  40  are as f o l l o w s .  Since a p i o n i s much h e a v i e r than  an  e l e c t r o n , a p i o n i n a 7T-mesic atom spends much more time i n the nucleus, than does an e l e c t r o n o f c o r r e s p o n d i n g quantum numbers i n a c o n v e n t i o n a l atom.  The energy  o f a 7C -mesic atom are thus c o n s i d e r a b l y m o d i f i e d those which we would expect netic  forces.  from the p u r e l y  from  electromag-  The e f f e c t o f the r e a l terms i n the  optical  p o t e n t i a l i s to s h i f t the energy l e v e l s from those calculated  from  levels  values  the e l e c t r o m a g n e t i c i n t e r a c t i o n and  the  e f f e c t o f the a b s o r p t i v e terms i s to broaden the l e v e l s . Thus we  can measure the r e a l  terms i n the o p t i c a l  potential  by a n a l y z i n g the l e v e l s h i f t s o f the ir-mesic x-rays and  we  can measure the imaginary terms i n the o p t i c a l p o t e n t i a l by a n a l y z i n g t h e i r A simple way widths,  widths. to analyze the r e a l l e v e l s h i f t s and  or imaginary  plex l e v e l s h i f t s , A E  the  l e v e l s h i f t s , i s to c a l c u l a t e the comn l  ,  o f the l e v e l  (n,l), i n first-order  p e r t u r b a t i o n theory:  here  <j> fr) i s the wave f u n c t i o n of the p i o n i n the nl  orbit  w i t h p r i n c i p a l quantum number, n, and angular momentum, 1, calculated  from the e l e c t r o m a g n e t i c i n t e r a c t i o n ; m(r)  and  n ( r ) s c h e m a t i c a l l y r e p r e s e n t the l o c a l and momentum-dependent p o t e n t i a l s i n Eq. 2-36 How  ( n e g l e c t i n g the hyper f i n e  a t the x-ray e n e r g i e s  (^-1  MeV)  term).  the i n t r a - n u c l e a r  p i o n wavelength i s l o n g compared to the n u c l e a r  dimensions  41 ana the s wave f u n c t i o n (1=0)' v a r i e s s l o w l y with r a d i u s . r  The complex s-wave l e v e l s h i f t s are thus dominated by the l o c a l i n t e r a c t i o n , m(r) , i n Eq. 2-37, s i n c e the g r a d i e n t term which enters the momentum-dependent i n t e r a c t i o n i s small.  By a n a l y z i n g the s-wave s h i f t s we t h e r e f o r e o b t a i n  the l o c a l i n t e r a c t i o n , m(r) , and, hence, the m i c r o s c o p i c s-wave p i o n - n u c l e o n and pion-deuteron parameters, B  , i n Eq. 2-36.  "D and  For h i g h e r p a r t i a l waves, $ ]_(£), i s a n  much s t r o n g e r f u n c t i o n o f r a d i u s (since i t l o o k s , l i k e a s p h e r i c a l B e s s e l f u n c t i o n o f order,1) so t h a t , from the complex l e v e l s h i f t s o f h i g h e r angular momentum s t a t e s , we can o b t a i n the momentum-dependent i n t e r a c t i o n , n ( r ) , and, hence, the microscopic p-wave p i o n - n u c l e o n and p i on-deuteron parameters,  c  m  and C , o f Eq. 2-36. Xfl  In a c t u a l p r a c t i c e , the E r i c s o n s have n o t used the p e r t u r b a t i o n formula, Eq. 2-37, to analyze the 7T-mesic x-ray data but r a t h e r have obtained a n a l y t i c the l e v e l s h i f t s  formulae f o r  ( f o r a uniform p o t e n t i a l ) by matching  wave f u n c t i o n s c a l c u l a t e d i n s i d e well-known Whittaker  the p o t e n t i a l with the  f u n c t i o n s (bound-state  spherical  Coulomb f u n c t i o n s ) which d e s c r i b e the p i o n wave outside the potential. The v a l i d i t y  o f the E r i c s o n s ' p o t e n t i a l i s t e s t e d by  comparing the p i o n - n u c l e o n and pion-deuteron lengths  scattering  (or volumes) obtained from a n a l y z i n g the TT-mesic  x-ray data w i t h Eq. 2-36 w i t h those obtained from the elemen t a r y experiments.  The most important u n c e r t a i n t y l i e s i n  the r e a l p a r t o f the l o c a l i n t e r a c t i o n , where a l a r g e can-  • cellation For  f l : 1 0 ) occurs i n the s-wave s c a t t e r i n g  lengths.  t h i s reason terras o r d i n a r i l y of order 10'fo  motion c o r r e c t i o n s , %he  ,  4a  r e a l part  the f i n i t e  the Fermi  c o r r e l a t i o n length  corrections,  o f the two-nucleon s c a t t e r i n g amplitude  become o f f i r s t  order importance.  these c o r r e c t i o n s ,  the E r i c s o n s  w i t h the e m p i r i c a l  values  i^ZOfo)  N e v e r t h e l e s s , making  f i n d remarkable agreement as c a l c u l a t e d  from the  Is l e v e l s h i f t s i n L i and F, although t h i s may be somewhat fortuitous.  The l o c a l i s o t o p i c s p i n term i n the l o c a l i n t e r -  a c t i o n i s found to have, a s i m i l a r e m p i r i c a l l o c a l absorption and  validity.'  term i s measured from the I s l e v e l widths  i s found to be i n agreement to about 50^.  local  The  The non-  terms are measured from the Sp and 3d l e v e l  shifts  and  widths  (where momentum-dependent e f f e c t s predominate)  and  are found i n s i m i l a r agreement to the l o c a l terms. A good deal  o f t h i s - i m p r e c i s i o n may l i e i n the r a t h e r  crude data obtained from the x-ray measurements and i n the only p o o r l y  determined e m p i r i c a l  input scattering  lengths  rather  than i n the p r e c i s i o n o f the technique o f c a l c u l a -  tion.  A t any r a t e , the p o t e n t i a l would seem to be s u f -  f i c i e n t l y well v e r i f i e d its  properties  investigate  further.  ( a t l e a s t a t zero energy) to study In the subsequent chapters we w i l l  the p r o p e r t i e s  o f the o p t i c a l i n t e r a c t i o n and  suggest how i t may be used to measure the d i f f u s e n e s s the n u c l e a r  surface.  of  43  CHAPTER 3  THE OPTICAL PROPERTIES 01 LOW ENERGY, PIONS IN SPHERICAL NUCLEI  I n the p r e v i o u s chapter we have shown t h a t the s t r u c t u r e o f the pion-nucleus understood  o p t i c a l i n t e r a c t i o n i s quite well  and we have suggested  t h a t i n the low energy  r e g i o n (^30 MeV) a t l e a s t the q u a l i t a t i v e  f e a t u r e s o f the  i n t e r a c t i o n are known w i t h reasonable c e r t a i n t y . poses i n the p r e s e n t chapter are t w o - f o l d : wish  Our pur-  f i r s t l y , we  to- use t h i s i n t e r a c t i o n to study the o p t i c a l p r o p e r t i e s  o f low energy  pions i n s p h e r i c a l n u c l e i ; s e c o n d l y , we wish  to study the e f f e c t s o f the d i f f u s e n e s s o f the n u c l e a r s u r face on s c a t t e r i n g and a b s o r p t i o n c r o s s s e c t i o n s .  In a  l a t e r chapter we w i l l use these r e s u l t s to i n t e r p r e t the behaviour pion  o f deformed n u c l e i i n the presence  o f the o p t i c a l  field. I n t h i s chapter our f i r s t o b j e c t i v e i s to d i s p l a y the  v a r i o u s o p t i c a l processes a s s o c i a t e d w i t h the low energy pion-nucleus i n t e r a c t i o n .  We s h a l l be p a r t i c u l a r l y  concerned  w i t h those processes which concern the wave p r o p e r t i e s o f the p i o n i n s i d e  the n u c l e u s , namely, resonance  Since we are a t low e n e r g i e s , i t i s convenient  and a b s o r p t i o n . to demonstrate  these p r o p e r t i e s by making a p a r t i a l wave a n a l y s i s . f o l l o w the f a m i l i a r technique  We then  o f d e f i n i n g an i n t e r f a c e i n  the r e g i o n o f the n u c l e a r s u r f a c e and match wave f u n c t i o n s c a l c u l a t e d i n s i d e the i n t e r f a c e  from the o p t i c a l p o t e n t i a l  w i t h e x t e r i o r wave f u n c t i o n s which depend only upon the e l e c t r o s t a t i c and angular momentum b a r r i e r s surrounding the  44  nucleus  (spherical  Coulomb  T h i s allows us i n terms o f the the low  functions).  to s t a t e  p a r t i a l wave phase s h i f t s  i n t e r i o r logarithmic derivatives  which, f o r  energy pion-nucleus i n t e r a c t i o n , have a simple  instructive  form:  the  r e a l p a r t s o f the  ithmic  derivatives,  o f the  optical interaction,  insensitive o f the  the  to the  which d e s c r i b e the  details  are  over l o c a l and  can  e x p l i c i t l y state  of the nucleus;  derivatives,  be w r i t t e n i n the  and  examine the top  of the  3-2  we  form of a  treating  uniform d e n s i t y .  low  In the  t e r i n g and  a b s o r p t i o n are  ("•^15 MeV)  a r i s i n g from the  local potential  the  we  of  barrier  repulsive  the  (^r30  MeV)  potential  b e i n g very l o n g so  resonance p r o p e r t i e s are  mined mainly by the  the  details  the  s i z e .of the of the  wavelength i n the  scattering  barrier  nucleus deter-  to  have an a t t r a c t i v e  a s h o r t nucleon wavelength i n s i d e  f a c t , the  scat-  nucleus r a t h e r  interaction, (in contrast  nucleon-nucleus s c a t t e r i n g where we  the  local interaction.  pion  that  of  nucleus as having a  energy r e g i o n  governed by  discussion  This b a r r i e r r e s u l t s i n . the  In  that  relative effects  present a q u a l i t a t i v e  pion-nucleus i n t e r a c t i o n  e n t i a l and  the  p i o n momentum i s s m a l l .  In Sec.  than by  because  momentum-dependent a b s o r p t i o n so  these processes near the where the  energy r e g i o n )  i n t e r i o r logarithmic  which d e s c r i b e a b s o r p t i o n , can sum  low  optical potential  l o n g p i o n wavelength i n s i d e  imaginary p a r t s o f the  logar-  resonance p r o p e r t i e s  ( i n the  of the  interior  and  the  resonance p r o p e r t i e s o f s-waves are  pot-  nucleus). quite  45 i n s e n s i t i v e to the i n t e r a c t i o n while the resonance p r o p e r t i e s o f h i g h e r p a r t i a l waves depend mainly o f the p i o n i n s i d e the nucleus dependent i n t e r a c t i o n .  a r i s i n g from the momentum-  This l a t t e r e f f e c t i s expected  angular momentum must "be conserved i n t e r f a c e while  upon the e f f e c t i v e mass  i n c r o s s i n g the  since  nuclear  the mass o f the p i o n a p p a r e n t l y changes.  N e v e r t h e l e s s , because o f the l o n g p i o n wavelength, we little  learn  about the s t r u c t u r e of the n u c l e a r i n t e r f a c e  for  instance', the d i f f u s e n e s s o f the n u c l e a r s u r f a c e  from  the. e f f e c t s of resonance upon o p t i c a l s c a t t e r i n g experiments. Of course, we  can go to h i g h e r energies where the  wavelength o f the p i o n i s s h o r t e r and resonance e f f e c t s  are  more dependent upon the d e t a i l s o f the p o t e n t i a l , but we encounter two  difficulties:  f i r s t l y , except i n very  then  light  n u c l e i , s e v e r a l p a r t i a l waves e n t e r the problem so t h a t i t is  difficult  to untangle  these v a r i o u s p a r t i a l waves i n a  c o n v i n c i n g manner; s e c o n d l y , m i c r o s c o p i c processes  and  d i s c u s s e d i n Sec.  i s no  2-2  the simple  connection  o p t i c a l phenomena which we longer v a l i d .  we  might expect  have  However, from our  d i s c u s s i o n o f the s t r u c t u r e o f the pion-nucleus i n Sec, 2-1,  between  interaction  t h a t the only e f f e c t  of the  latter  d i f f i c u l t y i s to make the parameters o f the p o t e n t i a l energy dependent.  In f a c t , K i s s l i n g e r  (1955) was  able to show t h a t 12  the s t r o n g back s c a t t e r i n g of 62 MeV  pions i n C  could  be e x p l a i n e d when proper  taken o f the  nuclear  surface.  account was  N e v e r t h e l e s s , as we w i l l now  show  5  low energy  c a l p i o n a b s o r p t i o n cross s e c t i o n s should provide  only  opti-  a much more  46 d i r e c t method f o r measuring the n u c l e a r s u r f a c e and one which can he used even i n heavy n u c l e i . The  basic i d e a o f using absorption cross s e c t i o n s to  measure the n u c l e a r  dxffuseness  i s the f o l l o w i n g .  a b s o r p t i o n o f pions a r i s e s both from the l o c a l and to  the momentum-dependent i n t e r a c t i o n .  The o p t i c a l  interaction  The a b s o r p t i o n due  the l o c a l i n t e r a c t i o n i s s e n s i t i v e only t o the t o t a l amount  o f matter p r e s e n t and i s r a t h e r i n s e n s i t i v e t o i t s d i s t r i b u t i o n . -.On the other hand, as the p i o n e n t e r s and  the nucleus  encounters the r e p u l s i o n o f the l o c a l p o t e n t i a l , the i n -  t e r n a l p i o n momentum becomes s m a l l so t h a t momentum-dependent processes  are b o t h suppressed, and,  a t the same time, very  energy dependent; i n f a c t , the. p i o n momentum v a r i e s r a p i d l y over the n u c l e a r s u r f a c e so t h a t the i n t e r a c t i o n i s - v e r y s e n s i t i v e t o the n u c l e a r dependent processes processes,  diffuseness.  Since the momentum-  are comparable i n magnitude t o the l o c a l  a b s o r p t i o n c r o s s s e c t i o n s have c o n s i d e r a b l e s t r u c -  t u r e near the t o p o f the p o t e n t i a l b a r r i e r ; i n f a c t ,  this  s t r u c t u r e c l e a r l y m i r r o r s the d i f f u s e n e s s o f the n u c l e a r face.  Of course,  our arguments s t r i c t l y apply  which have no angular  only to s-waves  momentum; however, the e f f e c t s  l a r momentum f o r h i g h e r p a r t i a l waves i s simply  sur-  o f angu-  to " r a i s e "  the b a r r i e r y i e l d i n g a s i m i l a r s o r t o f s t r u c t u r e only a t h i g h e r energy.  Since we are a t low e n e r g i e s , only a few par-  t i a l waves e n t e r the problem and t h e i r untanglement i s r e l a t i v e l y straightforward.  O p t i c a l p i o n a b s o r p t i o n should  there-  fore be a good t o o l f o r measuring the n u c l e a r d i f f u s e n e s s i n  • 4 7 a large variety of nuclei. One o f the major reasons  f o r w i s h i n g to know the d i f f u s e -  ness o f the n u c l e a r s u r f a c e i s t h a t i t allows us t o i n f e r the neutron d e n s i t y i n the n u c l e a r s u r f a c e .  I n f a c t , the n u c l e a r  d e n s i t y i s simply the sum o f the proton and neutron and, s i n c e the proton d e n s i t y i s well-known from s c a t t e r i n g and JJ -mesic x-ray experiments (1969), Wu (1967)), we can e a s i l y i n f e r  densities  electron  (Devons and Duerdoth  the l a t t e r .  This e l u -  s i v e q u a n t i t y i s o f c o n s i d e r a b l e i n t e r e s t t o the theory o f n u c l e a r s t r u c t u r e ( f o r i n s t a n c e , the shell-model) and r e s u l t s currently available  from v a r i o u s experiments  are not mut-  u a l l y c o n s i s t e n t , as d i s c u s s e d below. One technique neutrons  f o r measuring the s u r f a c e d i s t r i b u t i o n o f  has been to observe  i n v o l v i n g v a r i o u s elementary  the branching r a t i o s particles.  f o r events  Auerbach e t a l .  (1968) have r e a n a l y z e d the 700 MeV p i o n charge-exchange data o f Abashian  e t a l . (1956) and f i n d  t h a t the rms neutron  r a d i u s agrees w i t h the rms proton r a d i u s t o w i t h i n a few percent  ( g e n e r a l l y being s m a l l e r ) unless one admits  peaked neutron  sharply  d i s t r i b u t i o n s i n the n u c l e a r s u r f a c e ,  Burhop  (1967) has analysed the decay products r e s u l t i n g from the capture o f kaons i n atoms and f i n d s t h a t the r e s u l t s can be accommodated by choosing the rms r a d i u s o f the proton and neutron  d i s t r i b u t i o n s t o be the same but with the neutron  s u r f a c e t h i c k n e s s 50$ g r e a t e r than t h a t o f the proton  dis-  t r i b u t i o n ; however, as Bugg e t a l , (1969) have pointed out,' these r e s u l t s are somewhat clouded by the complicated p h y s i c s which'-'enters' the, problem.  atomic  48 Temmer (1966) suggested  t h a t the observed  Coulomb  displacement e n e r g i e s between i s o b a r i c analog s t a t e s i n lead i s o t o p e s was  evidence f o r a neutron excess i n heavy n u c l e i ;  ITolen e t al„  (1968) extended  these r e s u l t s and  found the  rms  n e u t r o n r a d i u s to be about a p e r c e n t l a r g e r than that o f the. charge r a d i u s .  .This i s i n agreement w i t h r e s u l t s u s i n g  o p t i c a l s c a t t e r i n g of nucleons:  Greenlees e t a l . (1966)  measured the rms r a d i u s o f the neutron d i s t r i b u t i o n by ing  the'rms r a d i u s o f the mass d i s t r i b u t i o n to be  given.by  the s p i n - o r b i t term i n the o p t i c a l p o t e n t i a l ; E l t o n deduced the d i s t r i b u t i o n s obtained  from a n a l y s i s  choos-  (1968)  from the s h e l l - m o d e l p o t e n t i a l s  o f the s i n g l e - p a r t i e l e  and  single-  S0 8 h o l e s t a t e s i n Pb  .  Both authors found t h a t the neutron  d i s t r i b u t i o n i s up to t e n p e r c e n t g r e a t e r than the proton distribution.  We  l a r g e l y resolved of  f e e l t h a t these disagreements simply by measuring  can be  the o p t i c a l a b s o r p t i o n  low energy pions i n heavy n u c l e i where the n u c l e a r d e n s i t y  e n t e r s the i n t e r a c t i o n i n a much more d i r e c t manner. U n f o r t u n a t e l y , a t the p r e s e n t time there i s e s s e n t i a l l y no  o p t i c a l s c a t t e r i n g data a v a i l a b l e  energy r e g i o n ( < 50 MeV). to  d i s c u s s one example  We  f o r pions i n the  low  t h e r e f o r e though i t u s e f u l  (the' s c a t t e r i n g o f p o s i t i v e pions  from  40v Ca  ) from the p o i n t o f view  be analyzed. for  We  first  from which an experiment  might  g i v e a n u m e r i c a l d i s c u s s i o n of r e s u l t s  the E r i c s o n s ' p o t e n t i a l d e s c r i b e d i n Sec. 2-2''taking i n t o  account the e l e c t r o s t a t i c charge  p o t e n t i a l a r i s i n g from the n u c l e a r  d i s t r i b u t i o n and n o t i n g r e l a t i v i s t i c  corrections  which  49 may become important contained  a t higher e n e r g i e s ;  i n Sec. 3-3.  While t h i s  this discussion i s  zero-energy p o t e n t i a l may  n e g l e c t some energy dependence o f the parameters o f the p o t e n t i a l a t h i g h e r e n e r g i e s , i t should able d e s c r i p t i o n o f e m p i r i c a l  provide  a reason-  processes.  In Sec. 3-4 we use t h i s p o t e n t i a l t o c o n s t r u c t the e l a s t i c s c a t t e r i n g and a b s o r p t i o n c r o s s s e c t i o n s o f pions from a l i g h t nucleus  such as C a ^ ,  We d i s c u s s these  cross  sections, i n terms o f a p a r t i a l wave a n a l y s i s and we use the formalism  developed i n Sec.  3-1 to provide  an e x p l a n a t i o n o f  the p a r t i a l wave phase s h i f t s i n terms o f the i n t e r n a l arithmic  derivatives.  log-  We examine the o p t i c a l p r o p e r t i e s o f  the p o t e n t i a l f o r t h i s q u a n t i t a t i v e example i n terms o f the qualitative  f e a t u r e s which we d e s c r i b e f o r a uniform  b u t i o n i n Sec. 3-2.  distri-  In a d d i t i o n , we suggest a procedure  f o r o b t a i n i n g the empiricar-parameters the experimental, c r o s s s e c t i o n s .  o f the p o t e n t i a l from  We a l s o extend our d i s -  c u s s i o n to examine the e f f e c t s o f the d i f f u s e n e s s o f the nuclear  s u r f a c e on the o p t i c a l c r o s s s e c t i o n s , p a r t i c u l a r l y  the a b s o r p t i o n cross s e c t i o n s .  We then extend our d i s c u s s i o n  to the p r o p e r t i e s o f p i o n s c a t t e r i n g and a b s o r p t i o n i n heavy n u c l e i , choosing  as an example the nucleus  Pb ^ . 2  8  In Sec. 3-5 we b r i e f l y mention some o f the terms o f order A""*" and higher which e n t e r the E r i c s o n s described i n Eq, 2-36.  1  potential  These i n c l u d e i s o s p i n terms, which  m a c r o s c o p i c a l l y account f o r s i n g l e and double charge exchange, and  terms which depend on the nuclear  s p i n and lead to  50 hyper fine these  terms  mining been  s p l i t t i n g i n  the  are  optical  omitted  3-1  present  optical  p a r t i a l  interface each  inside  the  i n  outside  the  wave  the  i n t e r i o r  The  them  to  to  pion-nucleus  and  instructive: of  the  This  the  nuclear  are  of  to  the  are  for  the  our  i n t e r i o r  interaction as  we  logarithmic  show  the  derivatives.  i n  the  Then, functions potential; matching  and  outgoing  interaction to  are  express specific  shift  function  optical  potential  formulae  and,, where  we necessary,  potentials.  i s  that  logarithmic these  choose  by  The  potentials  formulae  f i r s t  us  and  for  sections  wave  the  which  to  and  optical  allows  factor  momentum-dependent of  to  radial  quantities  specific  thesis.  cross  incoming  have  formalism  defined  procedure  local  a  deter-  they  surface.  describe  penetration  which  the  then  i n  the  i s  pion-nucleus  which  logarithmic  advantage  access  the  shifts  terms  well-known  the  parts  from  the  calculate  functions  i n  the  extend  we  of  functions  quantities  are  technique  phase  --~  describe  Our  region  Because  Scattering  analysis.  interface.  barrier  w i l l  of  absorption  wave  shifts  we  role  nuclei,  and  wave,  and  direct  i n  scattering  the  Coulomb  phase  section  interface  i n t e r i o r  spherical  we  wave  p a r t i a l  p a r t i a l  derive  pions  Optical •  an  to  of  Analyzing  a  the  subordinate  A Formalism for Absorption  by  these  properties  a  spectrum.  discussion  analyzing  the  play  x-ray  remaining  the  from  and  7i-mesic  the  In  for  small  the  they  derivatives,  us  derivatives.  quantities next  allow  are  section,  which  For  both the  describe  simple real  the  51 resonance aspects o f the problem, are roughly constant o f the long p i o n wavelength i n s i d e  because  the nucleus; the imaginary  p a r t s are r e l a t e d i n a very d i r e c t way t o a b s o r p t i o n . i n the d i s c u s s i o n of the l a t e r s e c t i o n s , we w i l l the r o l e  Hence,  emphasize  o f the l o g a r i t h m i c d e r i v a t i v e s i n understanding the  p a r t i a l wave phase s h i f t s ;  the c o n n e c t i o n between these  con-  cepts and the e m p i r i c a l data i s the s u b j e c t o f the present section. A s p h e r i c a l l y symmetric  and momentum-dependent  such as t h a t which d e s c r i b e s the pion-nucleus  potential,  optical  inter-  a c t i o n i n s p h e r i c a l n u c l e i , has the g e n e r a l form,  (3-la)  VCr)  -^Y^(r)  B  V  + V (r) -f V(r) + IW(r) .  Here,  (3-lb) • * ( r ) =  •= *°(r) -t-irt^r)  e  1 _  0  1  '  r  i  3 is  the e f f e c t i v e s t r e n g t h o f the. momentum-dependent  i n c l u d i n g the term  )~^~ a r i s i n g  ( 1 -  interaction,  from the Lorenz-  •o L o r e n t z e f f e c t , where  <*(r) r e p r e s e n t s terms i n Eq„ 2-36, n ^ ( r )  and n ( r ) , which are p r o p o r t i o n a l to the d e n s i t y (or square p  of.the d e n s i t y ) ; V" (r) i s the e l e c t r o s t a t i c p o t e n t i a l , V ( r ) c  is  the r e a l  l o c a l p o t e n t i a l , and W(r) i s the a b s o r p t i v e  potential.  The n o n - r e l a t i v i s t i c Schroedinger  d e s c r i b e s the p i o n - n u c l e u s (1 2) 2 -  K  2  ~ V  and  the t a r g e t nucleus.  equation which  i n t e r a c t i o n then has the form '  ^)(r) -f o r ( ) | ( r )  E i s the r e l a t i v e energy  local  r  s  Ej>(r_)  (andjJ' the reduced  ; mass) o f the p i o n  Eq.  3 - 3 can be reduced  making the p a r t i a l wave (3-3)  Mr)  =  to a s e t of r a d i a l equations by  expansion, + 1) i  gkrtZJl  J  * t ±  4=0  Yjfe)  where Jt i s the o r b i t a l angular momentum. 3-3  ,  r  Substituting Eq.  i n E q . 3 - 3 , we deduce the r a d i a l Schroedinger  f o r each angular momentum v a l u e , i ? ,  equations,  f o r the r a d i a l p i o n .  waves, U J J (r)  - '' - \ (l-fcf(r)lai'fr) '. "L. ^frlu,' (r)  (3  4  2.V  +  r^  2  [y ( )+V(r)+iWfr) u.fr) * c  Eu,(r)  r  ^  r  f J5 ^ 0, 1, 2, ... )  To begin our c a l c u l a t i o n , we w i l l  first  d e f i n e the bound  dary c o n d i t i o n s on E q . 3-4Tby assuming the u s u a l experimental s i t u a t i o n o f a beam o f p a r t i c l e s d e s c r i b e d by an i n c i d e n t d i s t o r t e d plane wave,  <J> : Q  ( 3 - 5 )  fcf^)  =  \ P  ^hn(ZSl  iV^(V)Yj(e)  + 1)  JL = 0  ;  here Pi>(/,?l) i s the r e g u l a r s p h e r i c a l Coulomb wave f u n c t i o n , i • j? <3A =  arg V  C  + 1-f 171) 2 . 6 "  c  -f  0  is  X tan £ sal s - 1  the usual Coulomb phase s h i f t , and jp and 7^ are the con-  v e n t i o n a l Coulomb  parameters,  p = kr;  n\ =  where k i s the free space k  - s T 3 ^  z  2 Z  n  e  vj  2S  wave-number,  53 % i s the charge o f the t a r g e t nucleus and z the charge o f the p i o n (1,0, or - 1 ) . The technique i s to now s o l v e Eq. 3-4 i n s i d e some matchi n g r a d i u s , R , which i s chosen s u f f i c i e n t l y c l o s e to the Q  nucleus so as to exclude b a r r i e r p r o p e r t i e s but s u f f i c i e n t l y f a r removed  so as t o e s s e n t i a l l y i n c l u d e a l l aspects o f the  potential.  S u b j e c t t o Eq. 3-5, we then o b t a i n the f a m i l i a r  c o n t i n u i t y c o n d i t i o n s a t the matching r a d i u s f o r each p a r t i a l wave and i t s d e r i v a t i v e i n terms o f the (complex) phase  e  MV  (3-6a)  2 i < < e  0,(/ ) 0  r  u ^ )  ;  (/  g kR  0  I  (3-6b)  i;(^ )  e  0  8 1  ^0;(/ ) n  =  U'(7) ;  shifts  o'  _  = d d/  where (3-7a)  M^)  e^MG^)  1 I F , f/ )) .  (3-7b)  0^(/ )  e- MGj(/ )  4 i^f^)) J  Q  1,r  0  o  are the e x t e r i o r s p h e r i c a l incoming and outgoing Coulomb waves, r e s p e c t i v e l y : G^{P^)  i s the i r r e g u l a r  spherical  Coulomb wave f u n c t i o n . Of c o u r s e , Eqs. 3-6 are t r i v i a l l y s o l u b l e f o r the t o t a l phase s h i f t s , oty,  The problem i s t o re-express the s o l u t i o n  i n a form which e x p l i c i t l y s t a t e s the o p t i c a l p r o p e r t i e s o f the pion-nucleus i n t e r a c t i o n .  To do t h i s we need only note  that the p a r t i a l wave phase s h i f t , ^ (3_8) . o ( j s < r / +  (//H^)  , i s the sum o f two terms  ,  where <f^ i s the Coulomb phase s h i f t a s s o c i a t e d with the l o n g c  54  range  Coulomb p o t e n t i a l and where  (^i+±J^)  i s the phase  s h i f t which a r i s e s from the short-range pion-nucleus tial.  poten-  I t i s the l a t t e r phase s h i f t which c o n t a i n s the op-  t i c a l p r o p e r t i e s o f the p o t e n t i a l and i t i s t h i s phase s h i f t which we wish to e v a l u a t e . To perform t h i s e v a l u a t i o n , i t - i s account  These b a r r i e r s have essen-  two e f f e c t s which are s p e c i f i c  interaction. for  first  f o r the Coulomb and angular momentum b a r r i e r s which  surround the t a r g e t n u c l e u s . tially  convenient to  to the short-range  F i r s t l y , we must, f o r each p a r t i a l wave, account  the quantum mechanical  p e n e t r a t i o n o f the p i o n through  these b a r r i e r s which we do by i n t r o d u c i n g the c o n v e n t i o n a l penetration  factors,  -p  (3-9a) P„f/)  =  ±o  Secondly, we must account f o r the phase o f the waves i n going through the b a r r i e r s , which i s taken care o f by the shift ( 3  *  #  functions , 9 l 3 ) s ( /  >)  -  /ofe^W^o)  +  GJCPQIG, (/» )] 0  It  i s convenient to note t h a t these q u a n t i t i e s are summarized  in  the e x t e r i o r d e r i v a t i v e q u a n t i t y ,  The term " s h i f t f u n c t i o n " i s used f o r a s l i g h t l y d i f f e r e n t q u a n t i t y i n the -matrix theory of n u c l e a r r e a c t i o n s (Vogt (1962)), I t d i f f e r s from our s h i f t f u n c t i o n mainly by the a d d i t i o n o f a boundary c o n d i t i o n number.  55  V-^)' ~  (3-10)  fo  *  g ty> \  -  -J~-2-~  The p r o p e r t i e s o f the short-range t i r e l y contained  .  )  iv.lp  0  p o t e n t i a l are en-  i n the i n t e r i o r l o g a r i t h m i c d e r i v a t i v e s ,  the r e a l p a r t , cfg , c o n t a i n s  the resonance p r o p e r t i e s o f  the i n t e r a c t i o n and the imaginary  p a r t , 7Cg , the a b s o r p t i v e  loroperties. The match o f the i n t e r i o r and e x t e r i o r wave f u n c t i o n s i s provided  by Eqs„ 3-6 which can now be c o n v e n i e n t l y r e w r i t -  ten i n the form, (3 12) 0  gi«feo 0' ) <  _ :  0  "  '  frlrp)  - MA))'  Now, as i s e a s i l y shown from Eqs. 3-7,  MTol  (3ll3)  !  .  '  -  A  I  C  ' _  where <5^ is the Coulomb phase s h i f t c  < " 3  1 4 )  J l / =  t a n - ^ ' i  )  2  I  N  J  E  (Eq. 3-5 f f . ) and '  '  i s the well-known hard-sphere phase s h i f t which accounts f o r the p o s i t i o n o f the matching r a d i u s with r e s p e c t to the origin. (3-15)  D e f i n i n g the p a r t i a l wave t r a n s m i s s i o n T  =  s  1  -  e" ^ 4  functions,  ,  and using Eqs. 3-10, 3-12, and 3-13, we see t h a t (3-l6a)  .  h  <  2  .  „ ! aP»(«5 (<5} _  - Sg) )2 2 „ p 2 +  x  O  56 (3-16b)  The  real part,  ^  , o f the p a r t i a l wave phase s h i f t s  accounts  f o r the r e f l e c t i v i t y and resonance p r o p e r t i e s o f the o p t i c a l potential;  the imaginary p a r t , P $ , o r , a l t e r n a t i v e l y , the  transmission  f u n c t i o n , T^, accounts f o r a b s o r p t i o n ,  stating  the net f l u x of p a r t i c l e s through the n u c l e a r i n t e r f a c e . We w i l l show i n subsequent s e c t i o n s that the r e a l o f the i n t e r i o r  l o g a r i t h m i c d e r i v a t i v e , <Xe , which  the resonance aspects o f the problem, has r a t h e r properties:  part  contains  simple  f o r s-waves i t i s n e a r l y u n i t y and f o r higher  p a r t i a l waves i t depends s e n s i t i v e l y only upon the e f f e c t i v e mass of the pion i n s i d e the nucleus. deserves f u r t h e r  The a b s o r p t i v e  term, 7Q ,  comment.  We show i n Appendix A, Sec. A-1, that  7fy can,  f o r each  p a r t i a l wave, be w r i t t e n i n terms o f an i n t e g r a l over the wave f u n c t i o n s :  C l e a r l y , the f i r s t  A\  term i s the l o c a l  0  absorption  term i s the momentum-dependent a b s o r p t i o n .  Hence, we can  e a s i l y segregate and d i s p l a y those absorptive from the momentum-dependent;'interaction.  and the second  effects arising  57  3-2  Qualitative  Features o f Low Energy Pion O p t i c s i n Nuclei  I n the previous s e c t i o n we have derived a formalism f o r describing  the o p t i c a l p r o p e r t i e s  o f pions i n n u c l e i .  In  the  p r e s e n t s e c t i o n we w i l l use t h i s formalism to i n v e s t i g a t e  the  qualitative  f e a t u r e s o f the pion-nucleus  interaction.  These f e a t u r e s are mainly governed by the long p i o n wavelength within potential  the nucleus a r i s i n g from the r e p u l s i v e  local  (and the s m a l l mass o f the pion) and the s t r o n g  energy dependence o f processes a r i s i n g from the momentumdependent i n t e r a c t i o n when we are near the top o f the potential  barrier. Because o f the l o n g p i o n wavelength, o p t i c a l resonance  properties terer  tend to depend mainly upon the s i z e o f the s c a t -  and the e f f e c t i v e mass o f the p i o n i n s i d e  rather  than upon the d e t a i l e d  i n the n u c l e a r s u r f a c e . tion exhibits  structure  the nucleus  o f the i n t e r a c t i o n  On the other hand, o p t i c a l absorp-  considerable structure  dependent i n t e r a c t i o n s i n c e  through the momentum-  the p i o n momentum i n the nucleus  v a r i e s r a p i d l y near the top o f the p o t e n t i a l b a r r i e r . make these p o i n t s c l e a r e r we w i l l i n t h i s s e c t i o n the  nucleus to have_ a uniform d e n s i t y ,  o b t a i n simple a n a l y t i c We w i l l extend discussion  consider  i n which case we can  formulae f o r the s c a t t e r i n g  parameters.  our r e s u l t s to d i f f u s e n u c l e i i n our numerical  i n Sec. 3-4.  3Por a uniform n u c l e a r density, o f r a d i u s , tial  To  R , the potenQ  parameters o f Eqs. 3-1 have the r a d i a l dependence,  58 (3-18a)  <  c* (r) e  = 0  V(r)  (3-18b)  r ^ R, , -"o r^R  ,  0  ;  r 0  W(r)  (3-18c)  e lo  ~«Ro  .  -V/ , o '  =  r>R„ ; r^R  • o *  0  Ro'  lo »  o-  anfl W  p  have been chosen to be p o s i t i v e .  We then have from E q . 3-4 that the wave f u n c t i o n f o r the p i o n inside  the nucleus i s  (3-19)  u^(r) =  Ajgrj^(br),  r<R , 0  where the i n t e r n a l wave number, b, i s g i v e n by the r e l a t i o n , (3-20)  (V  c  being  Vc  (E  the average value  - V 0^ 4 iW0')  v  x  o f the e l e c t r o s t a t i c p o t e n t i a l  w i t h i n the n u c l e u s ) . As we show i n Appendix A, Sec, A-2, the l o g a r i t h m i c d e r i v a t i v e at R (3-21)  Q  MH ) 0  i s given by the r e l a t i o n , -  (14  «*)  rfp  h<**)  04(br)  r=R,  Now, near the top o f the p o t e n t i a l b a r r i e r , E ^ ( V _ + V ) , ^ 0 so t h a t bR  0  i s small  (W  0  i s small).  Thus, we may make an  asymptotic expansion 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 s : (3-22) 4+2 (bR ) 3 (bRj 0  (an-i)::  IT o  0  Tn  t h i s case, Eq. 3-21 has the approximate form (to order  (bR )  2  0  ),  (3-23)  -[  _ *  or,  •  :  from Eqs.  '  •  sZ  /,  R  2-*+3  0  '  3-11, 3-18, and 3-20, '•  (3-24a)  ir v ) R  n  (a-t (3-24b)  7Tj(R )'= 0  ^  f  0  +  n  0  3-24a, i t i s seen that  2  the resonance aspects  o f the problem, which are contained i n the l o g a r i t h m i c atives,  <5~JI  , are i n s e n s i t i v e  '  2^W  (24-f-3)h  From Eq.  + 3)  3  deriv-  to the d e t a i l s o f the. p o t e n t i a l '  f o r s-waves and, f o r h i g h e r p a r t i a l waves, depend only upon the  r e a l p a r t o f the e f f e c t i v e s t r e n g t h o f the momentum-  dependent i n t e r a c t i o n ,  °^^ » i n s i d e 0  the nucleus.  We t h e r e f o r e  do n o t expect to f i n d i n the case o f a d i f f u s e - e d g e nucleus that  these q u a n t i t i e s  depend very s e n s i t i v e l y upon the  d i f f u s e n e s s o f the n u c l e a r s u r f a c e ; i n ' f a c t , we w i l l show t h i s t o be the case i n Sec. I t i s also  3-4.  seen from Eq. 3-24b that s-wave a b s o r p t i o n ,  which i s d e s c r i b e d by the l o g a r i t h m i c d e r i v a t i v e , 7t , i s , Q  near the top o f the p o t e n t i a l interaction.  This arises  b a r r i e r , dominated by the l o c a l  since,  near the top of the b a r r i e r ,  the  p i o n momentum i s s m a l l .  The momentum-dependent processes  are  therefore strongly  the  b a r r i e r and, as we w i l l show i n Sec.  energy dependent i n the v i c i n i t y o f 3-4, the d e t a i l s o f  60 t h i s energy dependence provide a s e n s i t i v e t o o l  f o r measuring  the d i f f u s e n e s s o f the nuclear s u r f a c e . Since the e f f e c t s i m p l y to r a i s e for  of i n c r e a s i n g the angular momentum i s  the p o t e n t i a l h a r r i e r , we  expect  a b s o r p t i o n i n v o l v i n g higher p a r t i a l waves there i s a  s i m i l a r s t r u c t u r e to t h a t demonstrated here now  a t a h i g h e r energy.  U n f o r t u n a t e l y , we  demonstrate t h i s a n a l y t i c a l l y s i n c e we simple will  to f i n d t h a t  expansions  f o r s-waves, only cannot so  can no longer  use  f o r the s p h e r i c a l B e s s e l f u n c t i o n s .  t h e r e f o r e s a t i s f y o u r s e l v e s w i t h demonstrating  n u m e r i c a l l y i n a subsequent s e c t i o n , Sec. 3-4. we w i l l  easily  f i r s t r e q u i r e n u m e r i c a l values f o r the  We  this  To do  this  optical  potential.  3-3  Pion-Nucleus In the  Optical Potential  f i n a l s e c t i o n s o f t h i s c h a p t e r , we w i l l demon-  s t r a t e the o p t i c a l p r o p e r t i e s of the pion-nucleus i n a more q u a n t i t a t i v e way. a detailed  from the XT-  2-2.  take to be  the  T h i s p o t e n t i a l would seem  4  I  (Sec. 2-3)  To t h i s purpose we w i l l r e q u i r e  o p t i c a l p o t e n t i a l which we w i l l  E r i c s o n s ' p o t e n t i a l o f Sec.  interaction  mesic x-ray  and  data to be q u i t e good at low  at higher energies  (^30  MeV)  energies  i t should  still  reasonably r e p r e s e n t the o p t i c a l i n t e r a c t i o n , a t l e a s t i f one  allows  f o r some energy dependence i n i t s parameters.  a d d i t i o n , we must i n c l u d e the e l e c t r o s t a t i c p o t e n t i a l from the n u c l e a r charge d i s t r i b u t i o n and we ativistic energies.  In  arising  a l s o mention r e l -  e f f e c t s o f the p i o n which become important  at'higher  61 To c o n s i d e r the r e l a t i v i s t i c  e f f e c t s , we f i r s t  note  t h a t , s i n c e the p i o n i s s p i n l e s s , i t s o p t i c a l wave f u n c t i o n should s a t i s f y a Klein-Gordon  equation o f the form,  (3-25) •hS 2y2  -/2 2  c  ![(v. (E+^c2)  +  C  Here E i s the k i n e t i c energy  - V (r) -Iff: 0  4  o f the p i o n , jJ i s i t s (reduced)  r e s t mass, V (r) i s the e l e c t r o s t a t i c p o t e n t i a l a r i s i n g the n u c l e a r charge  <j>(r) = 0  from  d i s t r i b u t i o n , a n d l T f r ) i s the momentum-  dependent short-range i n t e r a c t i o n o f the p i o n with the nucl e u s d i s c u s s e d i n Chapter  2 and Eq. 3-1.  r e w r i t t e n i n the convenient  iii v  (3-26)  s  +  2^  Eq. 3-25 can be  form.  (E - V ( r ) - V ( r ) ) i - H  E  - Vc(r)  -ir(r)  o  x <j> (r)  =  0  From Eq. 3-26 we see t h a t t o order 4  ••  4  E - Y ( r ) -'Vffr)  ^  c  2JJG  the Klein-Gordon Sehroedinger  20:MeV  ^ 2 8 0  C  e q u a t i o n reduces  0 1  MeV  to the n o n - r e l a t i v i s t i c  equation,  (3 ?7) ' -~2 2 " ~ V . <j)(r)  4-  (V (r)4V(r))|(r) « Ej.fr)  Hence, n e g l e c t i n g r e l a t i v i s t i c  c  c o r r e c t i o n s w i l l make parameters  of the p o t e n t i a l energy-dependent t o the order o f 10$. the parameters are probably somewhat energy event, we have simply n e g l e c t e d these The  .  electrostatic potential,  dependent i n any  corrections.  v" fr), c  Since  f o r a uniform  dis-  t r i b u t i o n i s well-known and, as we show i n Appendix B, Sec. B-2  62 i s given t o good approximation  (even f o r d i f f u s e - e d g e n u c l e i )  by the formula, (3-28) V (r) c  zZe R c  8  zZe  8  Lr where R  c  ,  r?R,  i s the rms r a d i u s o f the charge d i s t r i b u t i o n , Z i s  the n u c l e a r charge, and z i s the charge o f the p i o n . As d i s c u s s e d i n Chapter  2, we w i l l  take the short-range  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 ,*\T(r) , to be the E r i c s o n s ' p o t e n t i a l g i v e n i n Eq. 2-36 and Eq. 3-1.  We w i l l  only  c o n s i d e r s p h e r i c a l n u c l e i h a v i n g zero s p i n and i s o s p i n ; i n any event, c o r r e c t i o n s  f o r these e f f e c t s are s m a l l ( o f order  A"*"*") except i n very l i g h t n u c l e i .  Eq. 2-36 can then be  r e w r i t t e n i n the form o.f Eq. 3-2:  3  -y  |V(r) c  + V ( r ) + iW(r)| $>(r)  =  where (3-29b)  V(r) =  V j>(r) Q  2  o (3-29c)  o eff  U  ;  B  W(r) r - W ^ ( r ) , 8  W  0  (3-29d)  M  <*(r) =  r | | 47T(l __^ +  -K&g(r)  )Im(B )(^)0  -io< j!>lr) Io  4  E^(r)  63  ^Io 0  z  47T(l. ^ L . 81% +  )l (G )(JL.)  6  0  m  m  *°  We have here i n c l u d e d the c o n v e r s i o n f a c t o r s from the u n i t s IT = C = 1 used i n the E r i c s o n s  1  paper to e.g. s. u n i t s .  In c a l c u l a t i o n s with Eq. 3-29 we have used  the e x p e r i -  mental values f o r the parameters given by the E r i c s o n s (1966) (3-30a)  Cb )eff  ~  -0.0Z8;  (3-30b)  Im(B )  =  0.028;  =  0.208;  =  0.134.  (3-30c)  0  0  (c )  .  0  (3-30d)  e  f  f  Im(G ) 0  We have chosen the n u c l e a r d e n s i t y i n Eq. 3-89 to have a Saxon-Woods d i s t r i b u t i o n : (3-31)  Pn • 1  4- e-  a l/3  where ' a i s the d i f f u s e n e s s parameter, R  = 1.05 A '* fm. ,  1  J>  and  i s normalized  t o the atomic  0  number.  With these values f o r the parameters,  we f i n d t h a t  t y p i c a l values o f the p o t e n t i a l i n s i d e a nucleus a r e , (3-32a)  V  (3-32b)  W  (3-32c)  <*  0  Q  Rq  ^  15 MeV  ;  ^  7 MeV  ;  -1.5  ;  ^  4  (3-32d) . *  , T  ^  '_o 5  Of course, the exact values depend somewhat on the p a r t i c u l a r nucleus through the choice o'"f the d e n s i t y f u n c t i o n , y ( r ) .  64 3-4  lamerical Discussion  .  In the p r e v i o u s s e c t i o n s we have q u a l i t a t i v e l y d e s c r i b e d the o p t i c a l s c a t t e r i n g and nuclei.  a b s o r p t i o n of low energy  In p a r t i c u l a r , we have noted  pions i n  t h a t resonance  proper-  t i e s tend to be r a t h e r i n s e n s i t i v e to the d e t a i l s o f the pote n t i a l near the top o f the p o t e n t i a l b a r r i e r because of the l o n g p i o n wavelength. that o p t i c a l  On the  other hand, we  have  suggested  a b s o r p t i o n e x h i b i t s c o n s i d e r a b l e s t r u c t u r e near  the top- of the p o t e n t i a l b a r r i e r ent i n t e r a c t i o n ; i n f a c t , we  through the momentum-depend-  have suggested  that t h i s  t e r e f f e c t i s s t r o n g l y dependent on the d e t a i l s  lat-  of the n u c l e a r  s u r f a c e and, hence , p r o v i d e s . a s e n s i t i v e t o o l f o r measuring the d i f f u s e n e s s of the n u c l e a r s u r f a c e . In the present s e c t i o n we w i l l make these concepts more q u a n t i t a t i v e by employing d i s c u s s e d i n Sec. 3-3.  the pion-nucleus  We  will  a n a l y s i s of one example 4  worthwhile  and  might be analyzed.  f o r d e s c r i p t i v e purposes  (^30  MeV).)  We w i l l  pions with  from the p o i n t of view (This seems  s i n c e there i s p r e s e n t l y  e s s e n t i a l l y no experimental data i n the energy interest  potential  f i r s t provide a r a t h e r d e t a i l e d  the i n t e r a c t i o n s of  a l i g h t n u c l e u s , such as C a ^ from which an experiment  optical  then extend  region of  these i d e a s to the  i n t e r a c t i o n s of pions with a heavy nucleus, such as P b ^ . 2  Prom the experimental  p o i n t of view, the raw  analyzed i s provided by the d i f f e r e n t i a l e l a s t i c c r o s s s e c t i o n s and have c a l c u l a t e d  8  data to be scattering  the t o t a l a b s o r p t i o n cross s e c t i o n s .  these c r o s s s e c t i o n s f o r the example of  We  65 40 p o s i t i v e pions on Ca employing the p o t e n t i a l o f Eqs. 3-S8 ana  3-29 i n Eq. 3-4 to c a l c u l a t e the r a d i a l p a r t i a l wave  functions. calculated and  The p a r t i a l wave phase s h i f t s , <^ + i/fy , are then from Eqs. 3-6 f a t an asymptotic matching  the d i f f e r e n t i a l  absorption the  cross  radius)  e l a s t i c s c a t t e r i n g c r o s s s e c t i o n and  s e c t i o n are c a l c u l a t e d , r e s p e c t i v e l y ,  from  formulae:  f3-33a)  2  71  d_6elf0)  d-n.  ,  ^  6  - i T l l o g i s i n 2)  2ksin*f  6  1 V / . ,\ i f f a ^ 1 (2*+l)e k i=0  Ki^)  ' _ sin(«5+i^e)  2 x P^fcose) €*£>  (s-33b)  <5  abs  =  fam)^  .  Of course, i n p r a c t i c e the experimental phase s h i f t s are t o be  deduced by a n a l y z i n g  the e m p i r i c a l  cross s e c t i o n  data  w i t h Eqs. 3-33. Since one o f our aims i s to d i s p l a y the dependence o f o p t i c a l cross sections  on the d i f f u s e n e s s  s u r f a c e , we have performed  of the n u c l e a r  our c a l c u l a t i o n s f o r both a sharp-  edge  fdiffuseness  parameter, a=0.1 fm.) nucleus and f o r a  "more  conventional  fa=0.5 fm. ) d i f f u s e - e d g e  of these two d e n s i t i e s  nucleus,  A plot  f o r a Saxon-V/oods d e n s i t y d i s t r i b u t i o n  fEq, 3-31') w i t h the above s u r f a c e appropriate  to C a  i n P i g . 1.  V/e have i n t h i s * ' f i g u r e a l s o p l o t t e d the squares  4 0  fR = 1.05 A ^  parameters, and a r a d i u s  1  0  3  fm. = 3.59 fm.) i s g i v e n  o f the d e n s i t i e s upon which the o p t i c a l a b s o r p t i o n  depends.  67  S u b s t i t u t i n g these d e n s i t i e s i n the p o t e n t i a l o f Eq. 3-29,  we have c a l c u l a t e d the d i f f e r e n t i a l e l a s t i c s c a t t e r i n g  c r o s s s e c t i o n and . t o t a l a b s o r p t i o n iety  of energies.  Pig.  1)  c r o s s s e c t i o n f o r a var-  The r e s u l t s ( f o r the two d e n s i t i e s o f  are presented i n F i g . 2a ( d i f f e r e n t i a l e l a s t i c  t e r i n g cross s e c t i o n )  and F i g . 2b ( t o t a l a b s o r p t i o n  scat-  cross  section) „ T t i s seen from F i g . 2 a that the d i f f e r e n t i a l s c a t t e r i n g cross s e c t i o n s possess c o n s i d e r a b l e and  are quite s e n s i t i v e t o the d i f f u s e n e s s  surface. energies due  The s t r o n g  elastic  structure  o f the n u c l e a r  d e s t r u c t i v e i n t e r f e r e n c e a t higher  between the Rutherford  s c a t t e r i n g and the s c a t t e r i n g  t o the s t r o n g p a r t o f the pion-nucleus i n t e r a c t i o n  (middle and lower f i g u r e s ) a r i s e s from the i n t e r f e r e n c e between the p-wave  (and higher  p a r t i a l wave) s c a t t e r i n g  which i s dominated by the a t t r a c t i v e momentum-dependent potential  ( c , f . Eq, 3-24)  with the s c a t t e r i n g due to the  Coulomb p o t e n t i a l - — which i s r e p u l s i v e . the i n t e r f e r e n c e i s c o n s t r u c t i v e  A t lower  (top f i g u r e ) s i n c e ,  the Coulomb i n t e r a c t i o n , the s - w a v e ' i n t e r a c t i o n ( c . f . Sq. 3-24).  These  energies like  i s repulsive  p r o p e r t i e s o f the s t r o n g pion-nucleus  i n t e r a c t i o n are demonstrated more e x p l i c i t l y i n F i g s . 3 where we have p l o t t e d the r e a l p a r t i a l wave phase s h i f t s , the s-wave phase s h i f t s  are seen to be negative  to a repulsive Interaction) d-wave phase s h i f t s )  •  (corresponding  and the p-wave phase s h i f t s (and  are seen to be p o s i t i v e  tp an a t t r a c t i v e i n t e r a c t i o n ) .  (corresponding  I t i s seen t h a t a t l a r g e  68  ELASTIC CROSS  Fl G. 2 a  SCATTERING SECTIONS  20 7T* +  Ca  E " 15-MeV a = 0.5 Fm  10 TOTAL RUTHERFORD 0  180  E = 23M*V  ,^40  0.1 F  -oho  JL=*  0  E = 31 M e V 400 0.1  fm  200  °0°  30°  60°  JL  90°  e  120°  150°  I8CP  69  FIG.  T O T A L ABSORPTION CROSS S E C T I O N S  2b  r  40  600  s s —  500 s  y  /  •N.400 -a  b  _  y  300 -  s  y  s  y  y  /  s / /  /  /  /  / —  /  /  200 / / /  '  /  100  /  a = 0.5 fm a. = 0 . | F m  1 1 1 /  0  1/  0  •1  i  i  10  E  l  20  MeV  i 25  —~  30  72  4  FIG, *  +  REAL LOGARITHMIC  DERIVATIVES  40  Ca  EQ  0  E  3-24a  5  20 M  eV  25  30  74  75  FIG.  . RATIOS OF I MAG I N A R Y PHASE SHIFTS  6  o  Ol  0  0  10  15 E  20  MeV  25  30  76 angles  there i s a l s o (as expected) i n t e r f e r e n c e between  the s-wave s c a t t e r i n g and  the s c a t t e r i n g s of the higher  par-  t i a l waves. The  dependence o f the e l a s t i c s c a t t e r i n g on the  fuseness o f the n u c l e a r s u r f a c e i n F i g , 2a does not  difdepend  on the resonance p r o p e r t i e s o f the p o t e n t i a l but r a t h e r i t depends on the d i f f e r e n c e i n the d i f f r a c t i o n a s s o c i a t e d the  d i f f e r e n c e i n the a b s o r p t i o n between the d i f f u s e - e d g e  sharp-edge c h o i c e s  f o r the n u c l e a r d e n s i t y .  i n F i g . 3a, where we with  see  t h a t the r e a l phase s h i f t s  two  p o t e n t i a l s are n e a r l y the same, and  where we  see  that the corresponding  are q u i t e d i f f e r e n t . on the  i n F i g . 5,  imaginary phase  s t r o n g dependence of  associated  shifts  absorption  d i f f u s e n e s s parameter i s a l s o demonstrated i n F i g . Eb,  where we We  The  and  This i s shown  the  have p l o t t e d the w i l l now  t o t a l a b s o r p t i o n cross s e c t i o n s .  address ourselves  to the problem of o b t a i n i n g  the e m p i r i c a l parameters of the pion-nucleus Eq.  with  3-29,  potential,  from the phase s h i f t s which describe  s c a t t e r i n g and  absorption  cross sections..  the  L e t us  elastic first  .  enumerate the e s s e n t i a l s i x parameters o f the p o t e n t i a l assuming the Saxon-Woods d e n s i t y d i s t r i b u t i o n , Eq. (1) The  real local interaction, V ;  i n t e r a c t i o n , W; Q  ^  °^Ro*  (5) The  (2) The  0  '^  ie  (3) The  3-31:  absorptive  local  r e a l momentum-dependent s t r e n g t h ,  a b s o r p t i v e momentum-dependent s t r e n g t h ,  n u c l e a r r a d i u s parameter, R ; Q  (6) The n u c l e a r  » . surface  parameter, a. Before  proceeding,  i t w i l l be convenient to r e c a l l Eq,  3-  i n which we  s t a t e d the p r o p e r t i e s  of the i n t e r i o r  logarithmic  d e r i v a t i v e s near the top o f the l o c a l p o t e n t i a l "barrier f o r a uniform d e n s i t y (3-24a)  distribution:  c^(R )  '=  0  1  +  L * §  - V-  )  0  .  (3-241)')  In ent  L jf*?  XjfHj  Eqs.. 3-24,  the  2  density)  and  0  <^-j-  0  oZg  p r o p e r t i e s are  differ  (which are p r o p o r t i o n a l  3-24,  but,  determined by  to f i r s t  t h e i r values  the  i  optical  3-24  t o t a l change i n the e f f e c t i v e mass an i n t e g r a l  surface. *  For s-waves (4=0)  (V  Lorenz-  That i s , the o p t i c a l prop-  t  almost u n i t y and,  the  i n s i d e the nucleus  o f the p i o n i n e n t e r i n g the nucleus which i n v o l v e s over the. n u c l e a r  the  (as can be seen from Eq.  s h a l l p r e s e n t l y show).  e r t i e s are s e n s i t i v e to the  through the order,  to  I t i s the  Of course,  vary i n the s u r f a c e  where the d e n s i t y i s constant as we  <*L®  which are important i n determining  L o r e n t z e f f e c t (Eq. 3-lb)  and  and  Q  s c a t t e r i n g , as can he seen from Eq. strengths  2  of the momentum-depend-  through the Lorenz-Lorentz e f f e c t .  e f f e c t i v e strengths  effective  (2-0 + 3)ft  ^  R  i n t e r a c t i o n i n s i d e the nucleus, c<^  2  Q  >o Q (2^+3)^  u  the e f f e c t i v e strengths  from the s t r e n g t h s  V )R  Q  i t i s seen from Eq.  to f i r s t  3-24a that <?  order, depends only on V  Q  Q  is  and'R  i s f i x e d bv the n u c l e a r charge d i s t r i b u t i o n ) . ^ Vv'e c . if 'IMs r e s u l t i s to be c o n t r a s t e d w i t h the behaviour o f l o g a r ithmic derivatives f o r . i n t e r a c t i o n s i n v o l v i n g strongly a t t r a c t i v e p o t e n t i a l s (such as the nucleon-nucleus i n t e r • a c t i o n ) where the i n t e r i o r l o g a r i t h m i c d e r i v a t i v e s are r a p i d l y v a r y i n g f u n c t i o n s of the energy. Tn f a c t , i t i s t h i s p r o p e r t y of the d e r i v a t i v e s which g i v e s r i s e to the s t r o n g resonance p r o p e r t i e s c h a r a c t e r i s t i c o f such p o t e n t i a l s  78 have p l o t t e d  <T  0  f o r "both the d i f f u s e - e d g e and  sharp-edge  d e n s i t i e s of F i g . 1 i n F i g . 4 (upper f i g u r e ) . from F i g . 4 t h a t  CT  0  i s rather insensitive  I t i s seen  to the  diffuse-  ness of the n u c l e a r s u r f a c e ; i n f a c t , i n both cases the l o g a r i t h m i c d e r i v a t i v e i s seen to he i n c l o s e agreement with the approximate How  value c a l c u l a t e d  from Eq. 3-24a (dashed).  the r e a l p a r t o f the s-wave phase s h i f t , <P . depends 0  m a i n l y on C5~ (and the h a r r i e r p r o p e r t i e s ) s i n c e , i n g e n e r a l , 0  i s s m a l l ( c . f . Eq. 3 - l 6 a ) . like  <5", 0  We  i s rather insensitive  t h e r e f o r e expect that  f  Qt  to the d i f f u s e n e s s o f the  n u c l e a r edge; t h i s i s shown f o r our p r e s e n t example i n F i g . 3a On the other hand we expect J*  (upper f i g u r e ) . s e n s i t i v e l y on V  0  t h i s i s demonstrated  and  (which determines the p e n e t r a t i o n ) ; i n F i g s . 3b and. 3c  (upper  Q  V  Q  R. Q  see from Eq. 3-24a that  depends s e n s i t i v e l y only <x'j| l o g a r i t h m i c d e r i v a t i v e , cTji) tion).  The i n s e n s i t i v i t y o f  and R  (through the i n t e r i o r Q  JJI to  (through-barrier penetrad i f f u s e n e s s parameter,  and the r e a l p o t e n t i a l , V , i s demonstrated (lower f i g u r e s ) ;  the s e n s i t i v i t y to R  F i g . 3c (lower f i g u r e s ) . of  figures),  Hence, <F depends s e n s i t i v e l y only on  For h i g h e r p a r t i a l waves, we  3b  to depend  (which determines the h e i g h t of the poten-  t i a l h a r r i e r ) and R  respectively.  0  0  a,  i n F i g s . 3a and  i s demonstrated i n  We have demonstrated  the s e n s i t i v i t y  <5~2_ (and, hence, /]_) to c * ^ i n F i g . 4 (lower f i g u r e ) ; i t  i s seen that value  <5^ i s i n c l o s e agreement with the  (dashed) c a l c u l a t e d  approximate  from Eq. 3-24a f o r both the sharp-  7 9  edge and d i f f u s e - e d g e p o t e n t i a l s . I t i s seen from the above d i s c u s s i o n that an a n a l y s i s o f the r e a l p a r t s o f the phase s h i f t s , meters  V-.,  o»  R , and  cxcf  o*  h  , y i e l d s the para-  o f the p i o n nucleus p o t e n t i a l  Ro  r e l a t i v e l y independent  o f the other parameters, i n p a r t i c u l a r ,  independent o f the d i f f u s e n e s s parameter, a, Hear the top of the p o t e n t i a l b a r r i e r momentum-dependent a b s o r p t i o n processes  (V + V ^  2 6 MeV);  are suppressed.  Hence, near the top o f the b a r r i e r , the imaginary l o g a r i t h m i c derivative, J* , 0  7T and, t h e r e f o r e , the i m a g i n a r y phase 0  (Eq. 3-16), are s e n s i t i v e only to the l o c a l  parameter, W . 0  This i s demonstrated  where wehave p l o t t e d the r a t i o s appropriate  t o the sharp-edge  shifts,  absorption  i n F i g . 6 (upper f i g u r e )  o f the imaginary phase  and d i f f u s e - e d g e  shifts  densities of  F i g , 1; i t i s seen t h a t , near the top o f the p o t e n t i a l b a r r i e r (^26 MeV) , ' i f . ^ l .  Hence, 1SL can be obtained  from the  a b s o r p t i o n c r o s s s e c t i o n s near the top o f the b a r r i e r  (which  i s known from a n a l y s i s o f the r e a l p a r t s o f the phase  shifts,  I t i s t h e r e f o r e p o s s i b l e to o b t a i n values meters V , R , p^fj » Q  Q  0  a n < 3  ^  0  f o r the para-  r e l a t i v e l y independent o f the  nature o f the n u c l e a r s u r f a c e .  Now,  as we s h a l l s h o r t l y  d i s c u s s i n more d e t a i l and as can be seen from F i g , 5, the energy dependence o f the imaginary p a r t s o f the phase Pj(  shifts,  , i s a s t r o n g ' f u n c t i o n of the d i f f u s e n e s s parameter, a,  through the momentum-dependent a b s o r p t i o n . values which we have p r e v i o u s l y obtained  Hence, using the  f o r the other  para-  80 m e t e r s o f t h e i n t e r a c t i o n , we c a n a n a l y z e J^,  t h e phase  shifts,  f o r t h e d i f f u s e n e s s p a r a m e t e r , a , and t h e a b s o r p t i v e  t e r m i n t h e momentum-dependent i n t e r a c t i o n , ^ f  0  .  Of c o u r s e , b y f o l l o w i n g t h i s p r o c e d u r e w h i c h we h a v e outlined  only very approximate values  parameters o f the p o t e n t i a l .  are obtained  Of c o u r s e ,  better  f o r the values  c a n be f o u n d by r e p e a t i n g t h e a n a l y s i s w i t h t h e v a l u e s found from the f i r s t  attempt.  p r e l i m i n a r y use o f t h e s e  to s i m u l t a n e o u s l y  meters which enter to experimental  at least a  p r o c e d u r e s w o u l d seem t o be  more p h y s i c a l l y i n s t r u c t i v e attempting  I n any e v e n t ,  and more e f f i c i e n t vary  then  both simply  the s e v e r a l ( s i x ) para-  the i n t e r a c t i o n to o b t a i n a best f i t  cross sections  (when s u c h c r o s s s e c t i o n s  become a v a i l a b l e , ) , , - A s we h a v e n o t e d  i n P i g s , 5 and 6 , t h e o p t i c a l  t i o n d e s c r i b e d by t h e i m a g i n a r y  phase s h i f t s , ^ ,  depend s t r o n g l y on t h e t h i c k n e s s o f t h e n u c l e a r This r e s u l t s  Now, b e c a u s e a d i f f u s e - e d g e of matter barrier  a t lower  nucleus  f o r the former than  f a c t , the s t r u c t u r e o f the i m a g i n a r y  diffuseness this  barrier,  suppressed. a g r e a t e r amount  d e n s i t i e s than a square-edge nucleus, the  appears lower  inary interior  contains  should  surface.  s i n c e , near the t o p o f the p o t e n t i a l  the momentum-dependent a b s o r p t i o n i s s t r o n g l y  absorp-  the latter.and,; i n  phase s h i f t s  ( o r imag-  l o g a r i t h m i c d e r i v a t i v e s ) c l e a r l y mirrors the  o f the n u c l e a r s u r f a c e .  We h a v e a l r e a d y  seen  e f f e c t i n P i g , 5 w h e r e we have p l o t t e d t h e i m a g i n a r y  phase s h i f t s , surface  f o r two p o s s i b l e c h o i c e s  t h i c k n e s s g i v e n i n P i g , 1„  f o r the nuclear  We now e x h i b i t t h e  81  82  •  \  i  s p e c i f i c e f f e c t o f the momentum-dependence more c l e a r l y "by making use o f Eq„ 3-17 which allows  us to segregate  the e f f e c t s  o f the l o c a l and momentum-dependent a b s o r p t i o n : ( 3  "  1 7  '  7t,(R ) '. -  ^ 2 .  h(")\ dr  WW  o f  8  • lmfH H  h &  0  E  , (Local)  0  (Momentumdependent)  I n F i g . 7 w e have used Eq_. 3-17 t o p l o t the l o c a l and momentum-dependent (dashed) a b s o r p t i o n  (solid)  f o r the two den-  s i t i e s o f F i g . 1 ( a = 0.5 f m . , l e f t - h a n d s i d e ; a = 0.1 f m . , right-hand the  side).  For r e f e r e n c e purposes, we have i n c l u d e d  t o t a l l o c a l p o t e n t i a l b a r r i e r , V (r) •+ V (r) , and the c  i n c i d e n t energy o f the p i o n  (marked  T  0  E ). T  I n P i g . 7a and 7b, we have compared s-wave a b s o r p t i o n at  an energy below the b a r r i e r f o r the d i f f u s e - e d g e  edge n u c l e a r d e n s i t i e s o f F i g . 1, diffuse-edge  and sharp-  I t i s seen t h a t , f o r the  d e n s i t y , most pions are absorbed a t n e a r l y  zero  momentum so t h a t momentum-dependent a b s o r p t i o n i s s t r o n g l y '  suppressed  (the momentum i s roughly  •  •  .2  p r o p o r t i o n a l to JE-Vc ( r ) - V ( r |)J  o n the other hand, f o r the sharp-edge d e n s i t y , pions  are b e i n g  absorbed i n s i d e the b a r r i e r a t higher momentum so that momentumdependent a b s o r p t i o n i s c o n s i d e r a b l y l e s s suppressed.  We see  from F i g , 7c and 7d t h a t , b y , s i m i l a r arguments, momentumdependent a b s o r p t i o n i s suppressed f o r both choices o f the  83 d e n s i t y near the top o f the h a r r i e r . see  that, for energies  In P i g . 7e and 7 f we  above the b a r r i e r , momentum-dependent  a b s o r p t i o n i s more suppressed  f o r the sharp-edge d e n s i t y  (where a b s o r p t i o n takes place i n s i d e the b a r r i e r ) than f o r the d i f f u s e - e d g e  density  (where, because there i s more s u r -  f a c e , a b s o r p t i o n takes place a t lower d e n s i t y and higher momentum). Por h i g h e r p a r t i a l waves, t h i s simple somewhat more complicated momentum b a r r i e r s . f o r ^ ^ 0, the angular  a n a l y s i s becomes  through the presence o f angular  F i r s t l y , i t i s seen from Eq. 3-17 t h a t ,  there i s a c o n t r i b u t i o n to the a b s o r p t i o n from terms which enter the momentum-dependent  a c t i o n as w e l l as from the r a d i a l terms; l i k e absorption,  these  the l o c a l  terms are i n s e n s i t i v e to the d i f f u s e n e s s  o f the n u c l e a r s u r f a c e . enter  inter-  In a d d i t i o n , the r a d i a l terms which  the momentum-dependent i n t e r a c t i o n f o r the h i g h e r  p a r t i a l waves see a h i g h e r p o t e n t i a l b a r r i e r through the appearance o f the c e n t r i f u g a l p o t e n t i a l ; we t h e r e f o r e expect the c o n t r i b u t i o n to the momentum-dependent a b s o r p t i o n  from  the r a d i a l terms to e x h i b i t a s i m i l a r dependence on the p o t e n t i a l b a r r i e r t o that shown i n P i g . 7 f o r s-waves, only now a t h i g h e r e n e r g i e s . case from P i g . 5 .  I n f a c t , t h i s i s seen t o be the  We w i l l  d i s c u s s these  e f f e c t s more e x p l i c -  i t l y i n Sec. 4-3 where s i m i l a r e f f e c t s are found i n the c r o s s s e c t i o n s which d e s c r i b e the. e x c l t a t i o n o f r o t a t i o n a l l e v e l s i n deformed n u c l e i . Now, as we have p r e v i o u s l y d i s c u s s e d , we know the h e i g h t o f the p o t e n t i a l b a r r i e r i n s i d e o f the nucleus  from  84  FIG. 8  S-WAVE IMAGINARY PHASE SHIFTS  FIG.  0.15  A 0.1 0  0.05  0.15H  O.IOh  0.0 5 h  S - W A V E IMAGINARY ' P H A S E SHIFTS  9  jC +  Plo  87  80 Jj;,  our a n a l y s i s of the r e a l p a r t of the phase s h i f t s , which d e s c r i b e the resonance the other hand- we phase s h i f t s , ^  aspects of the problem.  see here .that the imaginary p a r t of the  , are determined  potential barrier  by the behaviour  o f the  (and, hence, the n u c l e a r d e n s i t y ) i n the  r e g i o n o f the n u c l e a r s u r f a c e . sible  On  I t . s h o u l d t h e r e f o r e - b e pos-  from o p t i c a l s c a t t e r i n g and a b s o r p t i o n  w i t h pions to very a c c u r a t e l y determine  experiments  the d i f f u s e n e s s  of the. n u c l e a r s u r f a c e . In the preceding d i s c u s s i o n , we have noted effect was  that the main  of the e l e c t r o m a g n e t i c i n t e r a c t i o n upon our  i n a d j u s t i n g the e f f e c t i v e h e i g h t of the p o t e n t i a l  rier.  In the case  an average  of C a \ 4 (  value i n s i d e the nucleus  of about 10 MeV.  r a t h e r than p o s i t i v e pions t h a t the only important on our a n a l y s i s would be lower.  We  pions effect  t h a t the b a r r i e r would appear  20  T h i s e f f e c t i s shown i n F i g . 8, where we have  compared the s-wave imaginary phase s h i f t s , J^ , 0  r e a c t i o n s 7T + C a . a n d +  sharp-edge  bar-  the e l e c t r o s t a t i c p o t e n t i a l has .  t h e r e f o r e would expect i f we were to use negative  MeV  analysis  40  (dashed) and  TT*" +• C a  4 0  employing  f o r the  both the'  diffuse-edge (solid) nuclear densities  of our previous c a l c u l a t i o n s In a s i m i l a r manner, we  (Fig. 1). expect  that i f we go to h e a v i e r  n u c l e i , the only major changes i n our a n a l y s i s w i l l be to the l a r g e e l e c t r o s t a t i c b a r r i e r s the short-range atomic number).  (as seen  from Eq.  pion-nucleus i n t e r a c t i o n i s i n s e n s i t i v e We  two  have demonstrated  due 3-29, to  this effect i n Fig. 9  89  where  we  f o r the  compare the  reactions  calculations two  are  from the  t  TT " 1  Ca  4 0  and  TT* •+• P b ° . 2  for Saxon-Woods d e n s i t i e s  c h o i c e s f o r the  (dashed) and  s-wave imaginary phase s h i f t s ,  fm.  difference  (solid).  barrier  i n p e n e t r a t i o n , the  through the  with  fm.  I t i s seen t h a t ,  going to a h e a v i e r nucleus i s to r a i s e sive  5-31)  d i f f u s e n e s s parameter, a = 0.1  a - 0.5  difference  apart  only e f f e c t the  of  effective  i n the  ,  Both  8  (Eq.  ^  repul-  electrostatic  potentials. In are  so  f a c t , in. heavy n u c l e i large  that,  potential  strated  i n F i g . 10 where we  and  Tf  might be  attractive.  +  Pb  .  The  depth of the  f o r there to be  Of c o u r s e , as we of the the  r e a c t i o n s 17"*"+ P b ^ 2  average e l e c t r o s t a t i c p o t e n t i a l  b a r r i e r i s s i m i l a r to 16 MeV. potential  suggested by  For  these estimates  p i o n mass i s  s t r o n g resonance e f f e c t s  at low  too energies.  have d i s c u s s e d i n Chapter 3,.the parameters  pion-nucleus p o t e n t i a l ,  potential  the  the  b a r r i e r , are  i n p a r t i c u l a r , the h e i g h t  at present not  very w e l l  of  deter-  mined. Another e f f e c t of the  electrostatic potential  in  case of n e g a t i v e pions i s t h a t i t s u f f i c i e n t l y reduces angular momentum b a r r i e r s the  8  inside  whereas.our choice for  (roughly, - 5MeV), i t would seem that light  that  p o ft  short-range p o t e n t i a l net  potentials  This e f f e c t i s demon-  compare the  the nucleus i s s i m i l a r to 31 Mev,  the  electrostatic  for negative pions, i t i s possible  the net  —  the  problem,  the the  so .that many p a r t i a l waves enter  V/e have demonstrated t h i s e f f e c t i n P i g ,  11.  90  for  the r e a c t i o n  TJ  Pb  2 0 8  „ flower  figure).  l i n g o f the p a r t i a l waves i s then a formidable  The untangproblem.  Of course, i n the case o f p o s i t i v e p i o n s , the converse e f f e c t a p p l i e s s i n c e the e f f e c t o f the e l e c t r o s t a t i c  pot-  e n t i a l i s to i n c r e a s e the p e n e t r a t i o n f a c t o r s ( P i g . 11, upper f i g u r e ) Prom the a n a l y s i s presented t h a t we have c l e a r l y thickness  i n t h i s chapter, we b e l i e v e  demonstrated t h a t the n u c l e a r s u r f a c e  p l a y s an important  r o l e i n determining  p t i o n r a t e s of pions i n n u c l e i , t h a t , on the b a s i s o f these  the absor-  Tn f a c t , we would suggest  r e s u l t s , i t i s o f great  interest  to measure low energy p i o n o p t i c a l data, p a r t i c u l a r l y  pion  a b s o r p t i o n , as a means to measuring the d e n s i t y o f nucleons i n the n u c l e a r s u r f a c e . should  Accurate  measurements o f t h i s  f i n a l l y r e s o l v e the c o n s i d e r a b l e  the s u r f a c e d i s t r i b u t i o n  controversy  over  o f neutrons i n heavy n u c l e i which  we have d i s c u s s e d i n the i n t r o d u c t i o n to t h i s 3-5  nature  chapter.  Charge Exchange and other E x o t i c Terms i n the E r i c s o n s ' Potential' " I n the p r e s e n t s e c t i o n we w i l l b r i e f l y mention some i  terms o f order A"  and higher which enter  the'Ericsons'  p o t e n t i a l , Eq;. 2-36, and which we have u n t i l now from o u r - d i s c u s s i o n .  omitted  These terms consist o f i s o s p i n  terms,  which m a c r o s c o p i c a l l y account f o r s i n g l e and double charge exchange between n u c l e a r analog  s t a t e s , and terms a r i s i n g  from the s p i n o f the nucleus, which lead to hyper f i n e t i n g i n the TT-mesic  x-ray  spectrum.  Hence, while  terms make s m a l l c o n t r i b u t i o n s t o the pion-nucleus  split-  these optical  91 i n t e r a c t i o n , they lead, to q u a l i t a t i v e l y  different sorts of  e f f e c t s which are both i n t e r e s t i n g and e a s i l y measured. The problem  o f charge exchange has long been o f i n t e r -  e s t s i n c e the p i o n , u n l i k e most other f a m i l i a r p a r t i c l e s , has three charge s t a t e s ~_-  elementary  jfi jt* 71". Hence,  i n a d d i t i o n to s i n g l e charge exchange, there are the i n t e r e s t i n g processes o f double charge exchange by which, f o r example, one can r e a c h n u c l e a r i s o b a r s o f g r e a t e r (or l e s s e r ) charge than have h e r e t o f o r e been a v a i l a b l e . The  first  approaches  to charge exchange were m i c r o s c o p i c  i n nature i n which, so to speak,,  one t r a c e s the path o f the  p i o n through the nucleus and attempts  to account f o r the  p a r t i c u l a r nucleon where charge exchange occurred. these purposes  For  one r e q u i r e s a m i c r o s c o p i c d e s c r i p t i o n o f the  nucleus which i s u s u a l l y taken to be that g i v e n by the s h e l l model.  The p i o n - n u c l e o n s c a t t e r i n g operators are o f t e n taken  to be the free space s c a t t e r i n g operators (the impulse  approx-  imation) and the p r o p a g a t i o n o f the p i o n w i t h i n the nucleus is  o f t e n t r e a t e d by making some average o f the i n t e r m e d i a t e  e x c i t a t i o n e n e r g i e s so that one can reduce the m a n y - p a r t i c l e Green's f u n c t i o n to a one-body Green's f u n c t i o n f o r the pion  (the " c l o s u r e approximation"; c . f . Eq, 2 - S l a ) .  lations  of t h i s type have been attempted;by  Calcu-  s e v e r a l authors;  a summary of the c u r r e n t s t a t u s o f both s i n g l e and double charge exchange r e a c t i o n s , a l o n g w i t h a review o f the e a r l i e r c a l c u l a t i o n s and experiments, can be found i n a r e c e n t review a r t i c l e by K o l t u n (1969)  A second manner i n which one can account f o r charge, exchange i s through an o p t i c a l p o t e n t i a l , such as the E r i c sons' o p t i c a l p o t e n t i a l o f Eq„. 3-36.  Such an o p t i c a l  descrip-  t i o n s t a t e s that charge exchange can only occur through those nucleons which c o n t r i b u t e to the macroscopic  isospin  of the n u c l e u s , that i s , between i s o t o p i c analog s t a t e s ; charge exchange which takes place i n t h i s manner i s o f t e n referred  to as q u a s i - e l a s t i c  s c a t t e r i n g s i n c e only the  charge 'of the nucleus i s changed i n the s c a t t e r i n g process. As an example o f q u a s i - e l a s t i c s c a t t e r i n g , l e t us con(C  s i d e r the A = 14 i s o s p i n t r i p l e t , these n u c l e i have c l o s e d  s-,/r>  1  4  , I  1  ,0  4  o r b i t s and p  ,  1 4  ).All  o r b i t s ; the  3/2  remaining two nucleons are i n p ^ g o r b i t s and are coupled to s p i n , J = 0 , and i s o s p i n , T ~ 1, w i t h M M  T  = 0 for N  1 4  , and %  = 1 for 0 . 1 4  T  - -1 f o r C  Quasi-elastic  1 4  ,  scat-  t e r i n g can only account f o r changes i n 1A<% and, hence, only i n v o l v e s the VxfZ le° ; "the other hand, we might w e l l expect charge exchange to occur on the p _ nucleons or n u c  n s  o  n  /  o  3/ «  even the s-^g nucleons  ( o f course, charge exchange i s more  l i k e l y i n open then c l o s e d s h e l l s tions).  from e n e r g e t i c c o n s i d e r a -  I n f a c t , from c o n s i d e r a t i o n s only o f q u a s i - e l a s t i c  s c a t t e r i n g , we would conclude t h a t charge exchange i s f o r bidden i n n u c l e a r s t a t e s  o f i s o s p i n , T = 0, such as the  13  ground  state of C  „  An o p t i c a l d e s c r i p t i o n o f charge ex-  change, such as that g i v e n by the E r i c s o n s , i s t h e r e f o r e only o f value i n c o n s i d e r i n g q u a s i - e l a s t i c s c a t t e r i n g between n u c l e a r analog s t a t e s .  93 As  can be seen from Eq. 2-36, the E r i c s o n s  1  potential  accounts f o r o p t i c a l charge exchange processes a r i s i n g i n a v a r i e t y o f ways.  F i r s t l y , there are s i n g l e charge exchange 4  processes a r i s i n g from "both the s-wave (terms i n b-j_) and p-wave  (terras, i n c^) i n t e r a c t i o n s  ( i n c l u d i n g a charge ex-  4  change terra through the Lorenz '-Lorentz e f f e c t ) .  Double  charge exchange can occur e i t h e r through repeated a p p l i c a t i o n o f these vector c o u p l i n g s or d i r e c t l y through the t e n s o r coupling  (terms i n Bg and Cg) which a r i s e  nucleon p r o c e s s e s .  from the two-  C a l c u l a t i o n s w i t h charge exchange  inter-  a c t i o n s b e i n g d e s c r i b e d by an o p t i c a l p o t e n t i a l , such as the E r i c s o n s ' , are found to be i n moderate agreement with experiment  (Koltun (1969)).  I n a d d i t i o n to terms accounting f o r i s o s p i n , the E r i c sons' p o t e n t i a l a l s o takes i n t o account the c o u p l i n g between the angular momentum o f the i n c i d e n t p i o n and the n u c l e a r spin  (terms i n d ) Q  the h y p e r f i n e i n t e r a c t i o n .  p i o n i n t e n s i t i e s are too low to observe hyper f i n e  At present splitting  i n the TT-mesic x-ray s p e c t r a , but i n t e r e s t i n g i n f o r m a t i o n about these terms should be obtained once i n t e n s e p i o n beams (such as those provided by the proposed bee ome av ai1able„  Triumf f a c i l i t y )  94  CHAPTER 4-  '  In the  THE  EXCITATION 0? ROTATIONAL LEVELS IN DEFORMED NUCLEI BY PIONS  l a s t chapter'we have examined the  t i e s of pions i n n u c l e i that  the  chapter we f i e l d may the  i n p a r t i c u l a r , we  o p t i c a l a b s o r p t i o n of pions i n n u c l e i  l y upon the  by  and,  o p t i c a l proper-  d e t a i l s o f the will  be  nuclear surface.  d i s c u s s a second way  have shown depends s t r o n g -  In the  i n which the  present  pion o p t i c a l  used to examine n u c l e a r s u r f a c e f e a t u r e s ,  excitation  of r o t a t i o n a l states  i n strongly  namely,  deformed  nuclei. C o n v e n t i o n a l techniques f o r s t u d y i n g deformed the  e x c i t a t i o n of r o t a t i o n a l l e v e l s by  or nucleon f i e l d s ; levels  the  and  integrals  only to the  From such experiments on  d e t a i l e d i n f o r m a t i o n about the  over the  diffuseness  Our  of the  quadropole "and  d e t a i l s of i t s shape, such as  thickness.  nucleus  nuclear  deformed n u c l e i , we  pole deformations o f the n u c l e a r d e n s i t y , about the  excited  average n u c l e a r deformation  which are i n s e n s i t i v e to the  edge.  electromagnetic  e l e c t r o m a g n e t i c decay of these  e s s e n t i a l l y involve  which are s e n s i t i v e  nuclei  but  the  learn  higher we  multi-  learn  little  nuclear surface  purpose i n t h i s chapter i s to show that  the  e x c i t a t i o n o f r o t a t i o n a l l e v e l s i n deformed n u c l e i by  pions  ---  call  which we  w i l l henceforward  "pion e x c i t a t i o n "  somewhat l o o s e l y )  ( i n analogy with Coulomb e x c i t a t i o n )  does i n t r i n s i c l y i n v o l v e density  (and  because of the  nucleus i n t e r a c t i o n .  the  surface f e a t u r e s o f the  momentum-dependent p a r t Our  of the  nuclear pion-  technique for demonstrating t h i s  '95  Is to segregate  the Coulomb and l o c a l e x c i t a t i o n  .] '  processes  i n " p i o n e x c i t a t i o n ' * from the momentum-dependent e x c i t a t i o n processes  by. employing the D i s t o r t e d 17ave Born Approximati on.  The i n f o r m a t i o n obtained  from p i o n e x c i t a t i o n , when  !  combined with the a n a l y s i s o f o p t i c a l a b s o r p t i o n experiments fas d i s c u s s e d i n Chapter study  3 ) , should  allow a comprehensive  o f the n u c l e a r s u r f a c e i n deformed n u c l e i .  Since the  charge d i s t r i b u t i o n i n deformed n u c l e i i s becoming i n c r e a s i n g l y well-known from jJ-mesic  x-ray experiments,  this  should  y i e l d q u a n t i t a t i v e i n f o r m a t i o n about the s u r f a c e d i s t r i b u t i o n s of protons and neutrons p o s s i b l y , the angular  i n such n u c l e i  dependence o f these  including,  distributions.  This i n f o r m a t i o n should be o f c o n s i d e r a b l e value i n r e f i n i n g the m i c r o s c o p i c  d e s c r i p t i o n s o f deformed n u c l e i and i n com-  p a r i n g the e x t e n t to which these d e s c r i p t i o n s correspond to the m i c r o s c o p i c  descriptions of spherical  nuclei.  Our p i c t u r e o f a s t r o n g l y deformed nucleus i s t h a t o f the strpng-.couplingor r o t a t i o n a l model i n which the nucleus i s viewed as an falmost) s p h e r o i d a l body r o t a t i n g i n space f c . f . Sec. 4 - 1 ) .  The e s s e n t i a l i d e a i s t h a t when the nucleus  i s immersed i n - a p o t e n t i a l creases  field  the speed o f r o t a t i o n i n -  i . e . we e x c i t e a r o t a t i o n a l degree o f freedom  o f the nucleus; i n g e n e r a l , as we w i l l d i s c u s s i n Sec. 4 - 1 , we may n e g l e c t e x c i t a t i o n s o f other  ( v i b r a t i o n a l and i n t r i n s i c )  degrees o f freedom, so that we can d i r e c t l y connect s e c t i o n s f o r e x c i t i n g r o t a t i o n a l s t a t e s to such aspects o f the nucleus as the nuclear d e n s i t y .  the c r o s s  macroscopic  96 For i n s t a n c e , the e x c i t a t i o n of r o t a t i o n a l l e v e l s due to the e l e c t r i c  f i e l d "between the p r o j e c t i l e and  the deformed  t a r g e t nucleus i s the well-known phenomena of Coulomb e x c i t ation  (Alder and  Winther  o f r o t a t i o n a l l e v e l s we  (1966) ),  From Coulomb e x c i t a t i o n  can i n f e r the s t a t i c m u l t i p o l e moments  o f the deformed nucleus and,  hence, the average deformation  o f the n u c l e a r charge d i s t r i b u t i o n  (which we  expect to be  s i m i l a r to the average deformation o f the mass d i s t r i b u t i o n ) ; i n fact", r e c e n t experiments on i n e l a s t i c t e r i n g have allowed pole moments with The  the e v a l u a t i o n o f even, the h i g h e r  considerable  to a l o c a l  scat-  multi-  p r e c i s i o n (Bernstein  e x c i t a t i o n o f r o t a t i o n a l l e v e l s due  i n t e r a c t i o n has  alpha p a r t i c l e  (1969)). strong-  a l s o been i n v e s t i g a t e d by bombarding deformed  n u c l e i w i t h neutrons  (Chase, W i l e t s , and  Edmonds  (1958)),  However, the e x c i t a t i o n s of r o t a t i o n a l l e v e l s with e i t h e r the Coulomb or neturon f i e l d s e s s e n t i a l l y i n v o l v e i n t e g r a l s over the nucleus and,  hence, are r a t h e r i n s e n s i t i v e to the  d e t a i l s o f the n u c l e a r  surface.  I n the case o f the n u c l e a r  charge d e n s i t y t h i s i s not  too d i s a p p o i n t i n g s i n c e the s u r f a c e  d e t a i l s can be  moderately w e l l from other  such as the  x-ray experiments The  sources,  obtained  p-mesic  (Devons and Duerdoth (1969); Wu  (1967)),  d e t a i l s o f the n u c l e a r mass d e n s i t y , however, are  e a s i l y obtained  from c u r r e n t experiments; i n f a c t , as  discussed i n Chapter 3,  the s u r f a c e  we  d i s t r i b u t i o n o f neutrons  i s a c o n t r o v e r s i a l i s s u e even i n s p h e r i c a l n u c l e i c purpose'of t h i s chapter  not  i s to demonstrate t h a t the  The surface  97  f e a t u r e s o f the mass d e n s i t y distribution) ation  are s u s c e p t i b l e  (as w e l l  basic  t o i n v e s t i g a t i o n by p i o n e x c i t -  as by o p t i c a l a b s o r p t i o n ) because o f the mo-  mentum dependent pion-nucleus Our  (and, hence o f the neutron  interaction.  i d e a i s the same as i n our study o f p i o n  o p t i c a l a b s o r p t i o n i n Chapter 3:  near the top o f the poten-  t i a l b a r r i e r a r i s i n g from the l o c a l pion-nucleus the momentum-dependent processes are s t r o n g l y in  interaction  suppressed;  f a c t , t h e i r v a r i a t i o n w i t h energy i n t h i s r e g i o n should  strongly  r e f l e c t the d e t a i l s o f the n u c l e a r s u r f a c e because  o f the v a r i a t i o n o f momentum across the n u c l e a r  surface.  Of c o u r s e , the " p i o n e x c i t a t i o n " c r o s s s e c t i o n s include  also  Coulomb e x c i t a t i o n , due t o the p i o n charge, and l o c a l  e x c i t a t i o n , due to the l o c a l p i o n - n u c l e u s i n t e r a c t i o n .  A  g r a p h i c technique f o r s e g r e g a t i n g these processes from the i n t e r e s t i n g momentum-dependent processes i s p r o v i d e d by employing  the D i s t o r t e d  Wave Born Approximation.  As we d i s c u s s  i n Sec. 4-1, we expect the DWBA to be reasonably good the e x c i t a t i o n ric  cross s e c t i o n s  cross section  since  are much l e s s than the geomet-  o f the n u c l e u s .  In DWBA the amplitude f o r e x c i t a t i o n i s p r o p o r t i o n a l to integrals  over the p i o n o p t i c a l p o t e n t i a l and the p i o n o p t i c a l  wave f u n c t i o n s .  The t o t a l amplitude i s then e s s e n t i a l l y a  sum o f three terms a r i s i n g from the three terms i n the o p t i c a l potential: static  a Coulomb e x c i t a t i o n term, a r i s i n g from the e l e c t o  potential;  a l o c a l e x c i t a t i o n term, a r i s i n g from the  l o c a l pion-nucleus i n t e r a c t i o n ;  a momentum-dependent term,  98  a r i s i n g from the momentum-dependent -.pion-nucleus i n t e r a c t i o n . We can  t h e r e f o r e c l e a r l y demonstrate the  e f f e c t s o f the momen  turn-dependent i n t e r a c t i o n by examination o f the  corresponding  DWBA amplitude.. Tn Sec.. 4-1 we review the deformed n u c l e i states  r o t a t i o n a l model o f s t r o n g l y  and we d i s c u s s t h e ' e x c i t a t i o n  i n such n u c l e i by pions.  cross sections  of rotational  I n Sec.. 4-2 we o b t a i n  the  for these e x c i t a t i o n processes i n DWBA, I n  Sec.  4-.3 we employ the  Ericsons'  Sec.  3-3 t o evaluate these formulae for a r e a l i s t i c  nucleus i n t e r a c t i o n and, dependence o f the  potential  o f Chapter 2 and  hence, t o demonstrate the  pion e x c i t a t i o n cross sections  pionstrong  on the  n u c l e a r s u r f a c e parameters. '4-1  A Review o f the R o t a t i o n a l Pion E x c i t a t i o n  Model and a D i s c u s s i o n o f  In the p r e s e n t s e c t i o n we f i r s t b r i e f l y review the t i o n a l model of s t r o n g l y excitation we the  deformed n u c l e i .  We then d i s c u s s th  o f r o t a t i o n a l s t a t e s i n such n u c l e i  suggest that  these processes can  be w e l l  rota-  by pions and  described by  techniques o f DWBA. To understand  the  s t r o n g - c o u p l i n g or r o t a t i o n a l model,  i t i s worthwhile to review our b a s i c actions  of a constituent  Prom s t u d i e s  nucleon i n i t s . n u c l e a r environment.  p r i n c i p l e , a nucleon i n n u c l e a r matter moves  average p o t e n t i a l  i s rather  inter-  of n u c l e a r matter we know that, because of the  Pauli exclusion i n the  knowledge o f the  insensitive  o f the  to the  surrounding nucleons and  details of their  positions  and motions.  Hence, i n a f i n i t e nucleus we expect  n u c l e o n to move e s s e n t i a l l y  i n a single-particle  the other hand, the average  potential  move i s determined  orbit.  by the n u c l e a r d e n s i t y which i s simply  from the s i n g l e - p a r t i c l e the average  single-particle  On  i n which the nucleons  the sum over the nucleus o f the nucleon d e n s i t i e s  finding  each  orbits.  potential  orbits  obtained  Hence, the problem o f  and the problem  are coupled.  o f f i n d i n g the  The way out o f t h i s  apparent " c h i c k e n or egg" conundrum i s provided by the vari a t i o n a l p r i n c i p l e - which says that the r e s u l t a n t wave f u n c t i o n must r e p r e s e n t a s t a t e  nuclear  o f minimum energy.  In a nucleus w i t h many nucleons i n an u n f i l l e d s h e l l , i t turns out that because o f the i n t e r a c t i o n core nucleons, which tend to p o l a r i z e  between the e x t r a -  the n u c l e a r c o r e , and  because the n u c l e a r core which, l i k e a c l o s e d s h e l l n u c l e u s , p r e f e r s to be s p h e r i c a l , i s s t r o n g l y deformed.  the e q u i l i b r i u m  Such a deformed nucleus e x h i b i t s  l e c t i v e modes o f o s c i l l a t i o n , r o t a t i o n s addition  to the s i n g l e - p a r t i c l e  basic s h e l l structure The  which determines  excitations  a r i s i n g from the  aspects and the r o t a t i o n a l  to a rotation  o f the average  the s i n g l e - p a r t i c l e  considerably simplified s u f f i c i e n t l y quickly they respond  and v i b r a t i o n s , i n  a r e s t r o n g l y coupled s i n c e a r o t a t i o n  nucleus corresponds  col-  o f the n u c l e u s .  single-particle  o f the problem  shape o f the nucleus  orbits.  o f the  potential  The problem i s  by n o t i n g t h a t the nucleons  compared- to the r o t a t i o n a l  adlabatically  aspects  orbit  motion  to the changing average  that  potential.  j  100  Thus the t o t a l wave f u n c t i o n  of the nucleus may be  treated  a d i a b a t i c a l l y and taken to i n c l u d e the product o f an i n t r i n s i c wave f u n c t i o n ,  2C, d e s c r i b i n g  the shape and s t r u c t u r e  nucleus, and a r o t a t i o n a l wave f u n c t i o n , orientation also  exhibits  shape. age  i n space.  In addition  potential  D, d e s c r i b i n g  to r o t a t i o n ,  small c o l l e c t i v e vibrations  Since these v i b r a t i o n s  o f the its  the nucleus  about i t s  equilibrium  only s l i g h t l y modify the aver-  seen by the nucleons, they are only weakly  couple t o the s i n g l e - p a r t i c l e and r o t a t i o n a l  aspects o f the  problem and we can account f o r them s e p a r a t e l y w i t h , say, a v i b r a t i o n a l wave f u n c t i o n ,  ^» iij. v  Thus f o r the moment  we can take the t o t a l n u c l e a r Wave f u n c t i o n schematic  in  form,  '  fact, vibrational  -  excitations  g e n e r a l l y occur a t c o n s i d -  e r a b l y h i g h e r e n e r g i e s then the r o t a t i o n a l which we s h a l l be i n t e r e s t e d from our d i s c u s s i o n  to have the  ftaking  excitations i n  and we w i l l simply omit them the nucleus to always be i n i t s  v i b r a t i o n a l ground, s t a t e ) . The  m o t i v a t i o n o f our c a l c u l a t i o n i s now c l e a r .  purpose i s to i n v e s t i g a t e  the e x c i t a t i o n  Our  of r o t a t i o n a l  states,  D, through the o p t i c a l pion-nucleus i n t e r a c t i o n which, from our  discussion  i n Chapter 2, i n v o l v e s the n u c l e a r mass  d e n s i t y , _?(r) , i n a very d i r e c t manner. w i l l show i s that  the e x c i t a t i o n  In f a c t , what we  o f these r o t a t i o n a l  states  depends s e n s i t i v e l y on the s u r f a c e d e t a i l s of„P(r) through the momentum-dependent pion-nucleus i n t e r a c t i o n .  Since J>{v)  101 . i s simply  the sum  of the s i n g l e - p a r t i d e d e n s i t i e s obtained wave f u n c t i o n , OC  from the i n t r i n s i c  , and  i n 3C  s u r f a c e d i s t r i b u t i o n of protons  s i n c e we  know the  from other experiments  (such as the JJ -mesic x - r a y experiments) , t h i s allows us to s t a t e s e p a r a t e l y the s u r f a c e d i s t r i b u t i o n o f neutrons protons.  When combined with  the r e s u l t s  of o p t i c a l  experiments, as d e s c r i b e d i n Chapter 3, t h i s should  and  absorption allow  a comprehensive i n v e s t i g a t i o n o f the n u c l e a r s u r f a c e i n deformed n u c l e i and  an accompanying i n c r e a s e i n our under-  s t a n d i n g o f the m i c r o s c o p i c for i n the i n t r i n s i c Before tion  a s p e c t s . o f the problem accounted  wave f u n c t i o n , X  proceeding, we  .  r e q u i r e a more complete d e s c r i p -  of the n u c l e a r wave f u n c t i o n which we  w r i t t e n i n Eq, 4-1. model can be physics  found  have s c h e m a t i c a l l y  A d e t a i l e d d e r i v a t i o n o f the i n most elementary textbooks  ( P r e s t o n (1962)) ""and we  w i l l satisy  rotational  on n u c l e a r  ourselves  with  a brief description. The b a s i c p o s t u l a t e o f the a d i a b a t i c assumption I s t h a t the t o t a l n u c l e a r Hamilton!an, h, can be w r i t t e n i n the (4-2)  h = H  intl  , ( x ' ) -f T  form,  ,  r o t  ' 4  where % t r ^ n  x  T  ^  i s  »  s a  y»  a  s h e l l - m o d e l Hamiltonian  to a deformed average p o t e n t i a l and sic of  co-ordinates, x , T  the r o t a t i o n a l  and  where T  r o t  appropriate  depending upon the i s the k i n e t i c  intrin-  energy  motion.  For our purposes, we  w i l l be i n t e r e s t e d i n the  (large)  c l a s s o f deformed n u c l e i which are a x i a l l y symmetric, e x h i b i t i n g p r i m a r i l y s p h e r o i d a l deformations.  The  these  rotation'  ICS o f such a body i s analogous to the c l a s s i c a l j>recession o f a symmetric top ( G o l d s t e i n (1950)).  We t h e r e f o r e might  expect the k i n e t i c energy operator to have the form, m r  'where  o  t  _  h  _ ^2  "  ?J(J)  v  i s the "moment o f i n e r t i a " o f the n u c l e u s , depend'  ****  i n g upon the average deformation,  , and where I i s the  t o t a l angular momentum operator o f the nucleus. a l i z e d e i g e n s t a t e s , D, o f T the  The norm-  are then e a s i l y shown to have  r o t  form,  (4-4)  .  where  <=<  =  Tor,.-,!*  n  I  = ( <* , J , ^ ) are the E u l e r angles 5  K  .  T  which r o t a t e the s p a c e - f i x e d axes,  .  .  .  ( G o l d s t e i n (1950))  ( x , y , z ) , z b e i n g i n the  d i r e c t i o n o f the a x i s o f q u a n t i z a t i o n , i n t o the body-fixed axes,  (1,8,3), 3 b e i n g the symmetry a x i s o f the n u c l e u s .  oDcKjj.) (Preston  i s the m a t r i x element of the r o t a t i o n m a t r i x , (1962)) c o r r e s p o n d i n g to the eigenvalues I , M, and  K o f the operators I , I , and I z  respectively.  Clearly,  these are the q u a n t i t i e s which we-expect to be constants o f the motion from the analogy with the c l a s s i c a l  symmetric  top. In a d d i t i o n , s i n c e the nucleus i s a x i a l l y  symmetric,  we expect i t s s p i n i n the 3 - d i r e c t i o n , say, _Q. , to a l s o be a constant o f the motion so t h a t the e i g e n s t a t e s of the i n t r i n s i c Hamilton!an,  H  j _ ^ , ^ ^ » have the form n  : r  x I  "X^x ), 1  where T are the remaining quantum numbers r e q u i r e d t o s p e c i f y  103 the i n t r i n s i c n u c l e a r  state.  I t c a n - e a s i l y be shown that  a x i a l symmetry r e q u i r e s t h a t X l = K. . The n u c l e a r wave  func-  t i o n t h e r e f o r e has the form,  -  ( 4  "  %  5 a >  =  f i g *  A c t u a l l y , Eq. 4-5a c o r r e c t l y r e p r e s e n t s s t a t e only i f K = 0; f o r higher  the n u c l e a r  o f K we r e q u i r e  values  combinations o f s t a t e s l i k e Eq. 4-5a i n order o f d e f i n i t e p a r i t y (Preston  T * ' ° . o ) .  :  (1962)).  linear  to have  states  The r e s u l t i s then  o f the form, 4  (4-5b) L16J1 . 2  m , — ii.  j  . (  K  k  0 )  ( -f or — depending on whether the s t a t e has p o s i t i v e or nega t i v e p a r i t y , r e s p e c t i v e l y ) where we have expanded 'JC'K. terms o f i t s angular momentum sub s t a t e s ,  "3C..  ^  c  Prom Eqs. 4-5 i t i s seen that a g i v e n i n t r i n s i c "^CJJ , may e x i s t i n various  r  Such a s e t o f r o t a -  t i o n a l s t a t e s i s c a l l e d a " r o t a t i o n a l band". shown t h a t , i f K - 0, I * i n c r e a s e s u  - 0* 2"^ 4*", ...  o f one  deformed  i n steps  I t i s easily  o f two  eg.  while i f K § 0, locincreases i n steps  Ioj = K, K>1,  the s i g n a t u r e  state,  s t a t e s o f r o t a t i o n as p r e s c r i b e d  by the r o t a t i o n a l wave f u n c t i o n s , S)T^ „ Mil  l  E-+2, ... .  This band s t r u c t u r e i s  o f the energy l e v e l spectrum o f a s t r o n g l y  nucleus.  in  104  Our  technique  f o r e v a l u a t i n g the r o t a t i o n a l e x c i t a t i o n .  c r o s s s e c t i o n s w i l l he to use the D i s t o r t e d Wave Born Approximation  ( c . f . K i k u c h i and Kawai  In DWBA we see the nucleus  (1968), Messiah  (1963)).  i n i t s ground s t a t e as being im-  mersed i n the e l a s t i c a l l y s c a t t e r e d irion f i e l d which we c a l culate  from the pion-nucleus  i n t e r a c t i o n , say t h a t  described  by the E r i c s o n s ' p o t e n t i a l discussed i n Chapter 2 and Sec. 3 - 3 . . The  pion f i e l d  then causes the nucleus  to i n c r e a s e i t s r o t a -  t i o n , e x c i t i n g i t to some higher s t a t e i n the r o t a t i o n a l band.  Before  proceeding  t o evaluate  the e x c i t a t i o n c r o s s  s e c t i o n s i n DWBA we must f i r s t ask o u r s e l v e s about the v a l i d i t y of this  approximation.  We argued i n Chapter 2 t h a t the a b s o r p t i v e e f f e c t s due • to e x c i t a t i o n w i t h i n the i n t r i n s i c wave f u n c t i o n were small compared to r e a l p i o n a b s o r p t i o n i n the E r i c s o n s ' p o t e n t i a l ) .  (which i s taken i n t o account  In a d d i t i o n , , we do not expect  a b s o r p t i v e e f f e c t s o f v i b r a t i onal s t a t e s to be important they  occur a t c o n s i d e r a b l y higher  energies  since  then r o t a t i o n a l  states. The  question  o f the v a l i d i t y  l a r g e extent be rephrased  o f DWBA can t h e r e f o r e to .a  by e n q u i r i n g what the a b s o r p t i v e  e f f e c t s o f r o t a t i o n a l e x c i t a t i o n are on the ground s t a t e wave function  t h a t i s , we wish to determine the c o u p l i n g o f  the r o t a t i o n a l channels.  I f i t i s s m a l l , we may simply use  DWBA;-if i t i s l a r g e  (as i t i s i n the e x c i t a t i o n o f r o t a t i o n a l  s t a t e s by n e u t r o n s ) ,  the d i s t o r t e d wave c a l c u l a t e d from the  o p t i c a l pion-nucleus  interaction  e f f e c t s o f -channel-coupling)  (which does not i n c l u d e the  does not a c c u r a t e l y d e s c r i b e the  105 pion  f i e l d seen by the nucleus i n i t s ground s t a t e and we  must r e s o r t to some coupled channel approximation. . The e s s e n t i a l c o n d i t i o n f o r the therefore  validity  o f DWBA i s  that the e x c i t a t i o n c r o s s s e c t i o n s be much s m a l l e r  than the geometric c r o s s s e c t i o n 4JIR , where R i s the o f the t a r g e t nucleus. • As we w i l l show i n the d i s c u s s i o n '(Sec. 4-2),  radius  numerical  the e x c i t a t i o n c r o s s s e c t i o n s are a  few m i l l i b a r n s w h i l e the geometric c r o s s s e c t i o n i s a few barns.  This c o n d i t i o n would t h e r e f o r e seem t o be w e l l  f i e d and DWBA should is  t o be c o n t r a s t e d  be good a t l e a s t t o f i r s t  order.  one  o f the same order  (Chase,  (1958)).  Even i f i t should  t u r n out t h a t one might l a t e r wish  to take more c a r e f u l account o f the c o u p l i n g  o f the r o t a t i o n a l  channels, the DWBA c a l c u l a t i o n s are o f c o n s i d e r a b l e s i n c e they allow  section  as the e x c i t a t i o n c r o s s s e c t i o n s so t h a t ,  must r e s o r t to some coupled-channel approximation  W i l e t s , and Edmonds  This  w i t h the e x c i t a t i o n o f r o t a t i o n a l s t a t e s  by neutrons where i t i s found.that the geometric c r o s s is  satis-  interest  us to s e p a r a t e l y s t a t e the e x c i t a t i o n pro-  cesses due t o the Coulomb, l o c a l , and momentum-dependent pion-nucleus i n t e r a c t i o n s ; these e f f e c t s are not e a s i l y separated 4-2  i n , say a coupled-channel approximation. DWBA Formulae f o r Pion E x c i t a t i o n Cross  In the present  Sections  s e c t i o n we d e r i v e i n the D i s t o r t e d Wave  Born Approximation -the  formulae f o r the e x c i t a t i o n o f r o t a -  t i o n a l s t a t e s i n s t r o n g l y deformed n u c l e i by pions.  A gen-  e r a l d i s c u s s i o n o f DWBA can be found i n many t e x t books on  106  nuclear physics and we  (KLkuchi  and  Kawai (1968)-, Preston  quantum mechanics (Messiah  (1963)),  only c u r s o r i l y review t h e - s u b j e c t and  view o f the problem at hand, to present  We  nucleus Sec.  purposes,  from the p o i n t  then use  the DWBA  these  expressions  In Sec. 4-3 we  formulae n u m e r i c a l l y  will  for a r e a l i s t i c  pion-  p o t e n t i a l (the E r i c s o n s ' p o t e n t i a l o f Chapter 3  3-.3)  of  formulae s u i t a b l e for e v a l u a t i n g and i n t e r p r e t i n g  the p i o n e x c i t a t i o n c r o s s s e c t i o n s . evaluate  For our  (1963))  with p a r t i c u l a r emphasis upon the r o l e  and  o f the nuc-  l e a r surface thickness. To begin w i t h , kinetic  energy operator, T^,  t a r g e t nucleus We w i l l and  l e t us consider  a p i o n , d e s c r i b e d by a  to be i n c i d e n t on a deformed  d e s c r i b e d by the Hamiltonian,  assume that the pion i n t e r a c t s with  the nucleus  4-3, •• through  average p o t e n t i a l , " \ f ( r ) , such as the momentum-dependent  E r i c s o n s ' p o t e n t i a l d i s c u s s e d i n Chapter 3. that r e a l p i o n a b s o r p t i o n and  imaginary  We  w i l l assume  a b s o r p t i v e e f f e c t s due  many-body s t r u c t u r e o f the nucleus  able  h, of Eq,  part of this p o t e n t i a l ;  to the  are accounted f o r i n the the  only channels  to the problem are then through those  avail-  c o l l e c t i v e modes  o f o s c i l l a t i o n which can be e x c i t e d by the average i n t e r a c t i o n , i n p a r t i c u l a r , the r o t a t i o n a l  channels.  In our c a l c u l a t i o n s we w i l l assume the deformed nucl e u s to be a x i a l l y symmetric.  To evaluate  c r o s s s e c t i o n s i t i s then convenient body-fixed (4-6)  the e x c i t a t i o n  to expand ~[f(r)  frame i n a s e r i e s o f s p h e r i c a l h a r m o n i c s  i n the 8  107  where the body-fixed co-ordinate system (r-^, 0^, ^) is chosen to have its origin at the center of the nucleus and its polar axis along the nuclear axis of symmetry. We can then divide ~V(r) into the spherically symmetric potential, 1/" (r), and the perturbing potential, 0  (4-7)  -V'(r) = £ l # r ) Y £ ( e ) , b  b  which is responsible for the nuclear excitations. The total Hamiltonian of the system can now be written in the 'form, (4-8a)  H  E +--V* (r) x  ,  where the unperturbed Hamiltonian is (4-8b)  H  X  =  H  0  +  V^(r)  and where the Hamiltonian in the absence of interaction is (4-8c)  H  =  T  r  4-h.  Before proceeding, i t will be convenient to define 'the ei gens tat es of the Hamiltonians of Eqs. 4 - 8 . Let us first define the eigenstates, J j , of E , the Hamiltonian in the a  absence of interaction: (4-9)  J  a  =  ^e ^"1  where the nuclear states, ik • r e  , are given by Eq. 4 - 5 . , and  _a — is the pion wave function, k being the incident a  pion momentum (for notational convenience, we will treat the incident pion as a plane wave and neglect terms in log(3k r ) a  which arise from the long-range effects of the pion-nucleus interaction; the generalized discussion is essentially the same) „  108 The  t o t a l s t a t e o f the system i s represented "by the  wave f u n c t i o n , ,\u  , which i s t h a t e i g e n s t a t e o f the t o t a l  H a m i l t o n i a n , H, o f t o t a l energy, c o n t a i n s an incoming  E, which a s y m p t o t i c a l l y  plane wave i n the i n c i d e n t channel,  w i t h p i o n momentum, k , and outgoing waves i n : a l l  ,  other  channels, say ^ , w i t h c o r r e s p o n d i n g p i o n momentum, say k : H ty *' =  U-lOa)  1  E  ;  M / W s-^, Ta r —>•<> %  (4-10D)  ik°r„,  „ (+•;,  -ik^r^T . e '  *  r*  a  r Here  fil'f-Ap)  channel,  a  r  i s the s c a t t e r i n g amplitude  from the i n c i d e n t  , i n t o the channel,^3 .  We can d e f i n e s t a t e s , OC ,  f o r the unperturbed  a  t o n i a n , Hj_, s i m i l a r  to the t o t a l s t a t e ,  Hamil-  •, o f Eqs. 4-10,  except t h a t JQ^' c o n t a i n s only an outgoing wave i n the i n c i d e n t .  EL  channel, «*• ^ s i n c e H-j_, h a v i n g s p h e r i c a l symmetry, c o n t a i n s no mechanism f o r e x c i t i n g a r o t a t i o n a l s t a t e .  We can then  write  (4-lla) where  h^'ir^) i a  a " =  i s the o p t i c a l wave f u n c t i o n o f the p i o n s a t i s -  f y i n g the e q u a t i o n , (4-llb) and  .  (T^ +  ^(rJJ^Vr*) =  h a v i n g the asymptotic  (4-llc)  iK^r, r., -><»  *  ,  form,  i,, Ta — < <  (E - 6*  ^C4; .  A  J-a^  .  . e"  i k  a r  .  .r* » •  109 here the p i o n wave number i s g i v e n by the r e l a t i o n ,  (4-iia)  <  , '  a  h  where ^ i s the reduced mass o f the p i o n . The DWBA .now says the following:. {f $*)  f^yf-M  I f the amplitudes  i n Eq. 4-10 b are s m a l l _ — i . e . , i f the  e x c i t a t i o n c r o s s s e c t i o n s are s m a l l  we may omit  from  [|/ those terms which lead to a s y m p t o t i c a l l y outgoing waves a  i n an e x c i t e d channel: i t i s then reasonable a l s o to n e g l e c t the a b s o r p t i v e e f f e c t s  on f^f-^*<) o f the c o u p l i n g between a  the i n c i d e n t channel,  , and the e x c i t e d channels, V  t  in  it)  .  Since the p e r t u r b a t i o n term, i f M r ) ,  i s s m a l l com-  pared to I T ( r ) we may a l s o n e g l e c t i t s e f f e c t s on the f  e l a s t i c s c a t t e r i n g and thus take  (4-13)  t£(-"*)  =  -  Hence, DWBA says t h a t , t o good approximation,  (4-13)  (J/^  =  *  •a  We c a l l X l ^ t h e " d i s t o r t e d .wave", a. I t i s to be noted that " X ^ c o n t a i n s almost a l l the e f f e c t s / a o f the pion-nucleus i n t e r a c t i o n .  We t h e r e f o r e expect DWBA  to be a much b e t t e r approximation than plane wave Born Approximation, where we take  -  ^  , the i n c i d e n t plane  wave; i n f a c t , i n pion-nucleus s c a t t e r i n g , the e l a s t i c a l l y s c a t t e r e d wave i s s t r o n g l y d i s t o r t e d and we do not expect the l a t t e r approximation to be good a t a l l . I t remains now only t o e v a l u a t e the e x c i t a t i o n c r o s s sections.  From•elementary quantum mechanics (Messiah (1963)),  110 the e x c i t a t i o n c r o s s s e c t i o n i s g i v e n by (  4  -  1  4  )  aer-*b  '  a  ^  , ...  /  12  Gl  k  W  n  i  ^  ..|-a-*b|  a where the m a t r i x elements,  \ T a  _yt,» o f the s c a t t e r i n g m a t r i x ,  T, are g i v e n by the e x p r e s s i o n s , T  = <5 l li >  a^b  T  b  =  a  <$„ |(TT fr)+TP(r))j||/ ^ 0  I t i s then a well-known r e s u l t '4-16)  T  a  (4-17a> (4-17b) f 4  ~  1 7 a )  (1968))  i  n  E  that, |V' (r)|l|/ W>  0  The d i s t o r t e d wave, " X ^ , t i o n s analogous  (Messiah  '. < - J „ |TT -fr) | ' ^ > + <X'  ^  .  B  a  3 . 4-16 s a t i s f i e s  equa-  to Eqs. 4-11:  ah'  =  (T, +ir v  I (r^))^!^)  -  b  ay(E'-  =  (E  - ^ ^ ( r ^  ;  e,) '  -h  except that i t i s asymptotic to incoming r a t h e r than outgoing s c a t t e r e d waves:  •>V  '  e  ~* ~  p  +  -77- i  here k, i s the momentum i n the d i r e c t i o n of the s c a t t e r e d p i o n wave.  We w i l l d e f e r g i v i n g e x p l i c i t e x p r e s s i o n s f o r  ,<j>^' and ffl u n t i l we are ready to evaluate the m a t r i x elem1  ents o f Sq, 4-16 ( c . f . Eq, 4-30 f f. ).  I l l  Now, of  if  i f  (r)  and  , we have from-the s p h e r i c a l  and from Eqs. 4-5 and 4-11  <J KW|y '  (4-18)  I f we  ~i> £  0  a  W  symmetry  that i n Eq. 4-16,  V '..  >  0  f u r t h e r make the DWBA, Eq. 4-13, we have from Eqs. 4-14  4-16  that  (4-19)  d£a->c  k  x . 2  -,  (  *a Eq.  4-19 can be evaluated i n a s t r a i g h t f o r w a r d manner and  the  details  of  this  o f the e v a l u a t i o n have been r e l e g a t e d to Appendix B  thesis.  As i s shown i n Appendix B, Sec. B - l , the r e s u l t u n p o l a r i z e d beam o f pions  (where we average over  f o r an  initial  s t a t e s and. sum over f i n a l s t a t e s ) i s that (4-20)  d£-a*c i  d  where A ^  X  :  y  -2.1  are s t a t i s t i c a l  „  \2 * o\ £  . ^  f 7  £ 1  I  fl^2 ex  a c  c o e f f i c i e n t s d e s c r i b i n g the r o t a -  t i onal band, (4-21a)  (4-21b) and  A**  =  —  =  0 .  r r — : Q K + 1  ( J  even)  ;  (7-  odd)  ;  where' G^ac are m a t r i x elements over the p i o n c o - o r d i n a t e s ,  Before proceeding to evaluate E q . 4-22  f o r the matrix  I I S  elements  ^ ac  o r f r ) and We  will  we  s h a l l need to d e s c r i b e the p o t e n t i a l s  I T ' ( r ) o f Has.  4-6  and 4-7  i n more  detail.  take the p i o n - n u c l e u s i n t e r a c t i o n to be d e s c r i b e d by  the E r i c s o n s ' p o t e n t i a l of Eq. • Now  we  2-36.  know from e l e c t r o m a g n e t i c experiments,  such as  Coulomb e x c i t a t i o n , that a s t r o n g l y deformed nucleus i s n e a r l y s p h e r o i d a l so t h a t we  can w r i t e the n u c l e a r d e n s i t y  as (4-23.)  /(r)  J> (  =  r ( ) /  x  ^  where J> (r) I s , say, the Saxon-Woods d e n s i t y of Eq. The  average  3-31.  deformation parameter, J? , i s t y p i c a l l y s m a l l  (^.0.3) so t h a t we  can take  (4-24)  As can be seen from P i g . 1 , will  Sec. 3-4, J> (r)-'-\- p  (r) , and  /  simply make t h i s replacement  i n Eq. 3-29.  We  can  we then  take the pion-nucleus p o t e n t i a l to be given by (4-25)  -V (r)  =  - -  V'  ~^[7T  2  +  V  c  (r).+V(r)+iW(r)  where (4-26a)  c<(r)  ::  ac.(r)  (4-26b)  V(r)  :  V (r)  (4_26c)  W(r)  =  here  o  W (r) o  < ^ ( r ) , V ( r ) , and W (r) 0  vMr)  Yg(6 )  ;  Jfr W«(r) Y°(e )  ;  b  0  Q  -  are the p o t e n t i a l s  b  (r),'V(r) ,  113 and Wfr) i n Eqs. 3-29, evaluated  The Coulomb p o t e n t i a l . V (r) , i s  i n Appendix B, Sec. B-2, where we  Ze V (r) = —  (4-27b)  p3  2  "Yg.fe )  +  (here we have  treated  that  1 /r\ 2  "  c  find  b  the charge d e n s i t y  ,  . r-R ;  as uniform and o f  radius,-. R). We t h e r e f o r e  have  iTfr) o  -  ,  ' 2 ^ - 1- V • 1  •  (4-28)  that  r  >  ^ o ' ~TT f  -  ^  r  0  )  1  ( r ) 1  .'. 3 + V (-r) 4- V ( r ) +- i W ( r ) c  is  Q  ,  Q  the o p t i c a l p o t e n t i a l o f Eq, 3-29 and t h a t the p e r t u r b i n g  potential i s (4-29a)  f (r)Yg(e ) + V  -V'(r) =  b  L  - % (r) D  Y°(^)  2  4-f (r)YX) c  where (4-29b) is  f (r) C  =  O  ~rV*(r)  irW'(r) o  the l o c a l i n t e r a c t i o n , where ,  (4-29c)  is  -  „  , . _  Ti  the momentum-dependent  r  2  a  .  o  <  0 ^ )  =  T-fTtT-F  strength,  and where  r > R ; f  c  ( r )  =  -  j5 W  4  K  3  ,  r < R ;  114 is  the electromagnetic Of c o u r s e , we  interaction.  should s t r i c t l y have c a l c u l a t e d  t i a l harmonics, l / ^ f r ) , c a l c u l a t i o n has  from Eq. 4-6.  the  poten  For i n s t a n c e , such a  been performed by Chase, W i l e t s , and  Edmonds  (1958) for the e x c i t a t i o n of r o t a t i o n a l s t a t e s i n deformed nuclei  by neutrons;  they f i n d t h a t ' e s t i m a t e s such as  made i n d e r i v i n g Eqs. 4-28 accuracy,  and  4-29  are o f only moderate  n e v e r t h e l e s s , f o r our purposes we  simply wish to  demonstrate the e f f e c t of the n u c l e a r s u r f a c e on the t i o n cross s e c t i o n s and  f o r t h i s purpose these  should be q u i t e s a t i s f a c t o r y .  those  excita-  estimates  A more accurate  calculation  would, o f course, take a more c a r e f u l account  o f the  poten-  tials. Since we gies  (^30  are i n t e r e s t e d  i n p i o n e x c i t a t i o n at low  MeV) , only a few p a r t i a l waves e n t e r the problem  (as can be seen from F i g s . 3, 5, and i t w i l l be convenient the o p t i c a l waves, 4-17  ener-  and  to make a p a r t i a l wave expansion  ^a^sJ »  an<3  i t can be shown (Messiah  =g  (4-30)  11 o f Sec. 3-4)  2  tfc^?)  •  of  Prom Eqs. 4-11  and  (1962)) that  ^ - A , ^  tff}  )Y?(£)I>  (  k  :  r  )  ,  where ^ + iJlf are the p a r t i a l wave phase s h i f t s ( and F^(k;r) i s the r e g u l a r s p h e r i c a l Coulomb f u n c t i o n ) . Hence, (4-31)  tf"'(r) 'k -  =  A *('-r) It rt  A t the energies which we we  . .  :  w i l l consider  (several  can n e g l e c t the e f f e c t s o f n u c l e a r e x c i t a t i o n  .and take  k.  G  = k  In a d d i t i o n ,  we w i l l  take k , a  MeV)  (Cv^^l the beam  Me  115 d i r e c t i o n , to be along the (space-fixed) n-axis and we w i l l henceforward suppress the i n d i c e s a, c, f o r example, t a k i n g A  /  k = k = k  c c  to be the d i r e c t i o n of the s c a t t e r e d wave. to  (4-32a)  ( 4  "  =  j>(r)  f(r) =  f(lr)  ^(r)  3 2 b )  =  1 F  = - 277 ^  V+l  Then,  ~  *0(r)  ;  ,  ,  u*(r)  where the p a r t i a l waves, u ^ ( r ) , are s o l u t i o n s o f the r a d i a l Schroedinger  Eqs. 3-4.  C o n s i d e r i n g Eqs. 4-29 and 4-32, we can r e w r i t e Eq. 4-20 i n the convenient (4-33a)  form,  acr,->y _  j3  ;  0  2 A  d XL  ^^?  r  e  <  3  a  where (4-33b)  d6"red  here, (4-34)  I*  *  f JJ \  = Jar  Z  W  $(-rJ^n" dCT  As can be seen from Eq. 4-34', the r o t a t i o n a l band  A  Y^fe.^^r)  .  red  —-  (to the extent  Z  i s independent o f to which we can n e g l e c t  the energies a s s o c i a t e d with n u c l e a r e x c i t a t i o n s ) and o f the n u c l e a r deformation  (to the extent Eq. 4-29 c o r r e c t l y do" red  r e p r e s e n t s the p e r t u r b i n g p o t e n t i a l ) and i t i s  '  which  we s h a l l c o n s i d e r i n our c a l c u l a t i o n s . The  evaluation of  simply i n v o l v e s the s u b s t i t u t i o n  ,of the p a r t i a l wave expansion,  Eq. 4-32, f o r ij> (r) and the  116 substitution  The  E q . 4-34. then  of  the  perturbing  integration  straightforward  result  of  the  over  a n d we  calculation  potential, the  can  by  Eq. 4--S9, i n  angular  variables  conveniently  defining  the  state  is  the  following  radial  integrals. Firstly,  (4-35a) arising radial  there  ,««  .  Fj*  from the  are  radial  f  =  jdr  local  arising  essentially  two  /  L  interaction;  there  are  similar  .,  F ^  f r o m the  4  iy(r) f ( r ) u ^ f r )  integrals,  (4-35b)  integrals  •= J d r  l y f r ) , f f r ) u ^ fr)  electromagnetic kinds  momentum-dependent  of  ,  c  interaction.  radial  integrals  interaction:  firstly,  Tliere  arising there  are  from  are  the  radial  integrals  r arising  from  secondly,  there  (4-35d)  arising  the  the  these  cross sections, equation # Note  are  that,  term i n  radial  jo IP.**  from  With  radial  ' =  u  terms  definitions Eq.  from Eq.  4-33,  • ^  momentum  r  operator;  integrals  f „ jdr «  angular  the  i'u;  i t are  4-34-7 t h e  f  f  r  f  ( r )  in  the  is  found  given  x  ^(r)  )  momentum  that  i n terms  integrals  are  operator  the of  differential I"" by  over  the  uvfrK  not  117  (-)  (4-36)  P"'  MD,9  f  J  k  JI(i+i)  1  '. *  /2^n isTTi M)\  n  L  J  ( Vi'W'+l)-^-!)  (/+!)'->/C"+l) ^ 24  IjiM+l  ' Eq. 4-33 i s e a s i l y i n t e g r a t e d a t i o n cross s e c t i o n ,  Y  ;(k)  2  the r e s u l t  <2J?-/^/-l|i-l^  [ Si 1 ^  t o give  *  )J  the  total.excit-  being  (4-37a) * JaMi</i00120/<tf i.00120?  '0' (4-37b) where /M»  (4-38a)  *  =  ^//(i'+i)  -^l)<2i-H*0>  ^JY(-*'-»-l) - -v(^ + l ) < - 8 i - ^ l H . 1~7 + JiV'-H)  - W ^ - l ) < ?J^X-l|-f - 1 > ] . (  I t i s seen i n Eqs„ 4-36 and 4-37 that we have succeeded  118 I n s e p a r a t i n g those e x c i t a t i o n processes which a r i s e  from the  momentum-dependent i n t e r a c t i o n from those which a r i s e  from *  the l o c a l and e l e c t r o m a g n e t i c i n t e r a c t i o n ; i n f a c t , the momentum-dependence is contained i n the i n t e g r a l s , F^-p  .  p  I n the next s e c t i o n we w i l l provide numerical  estimates  o f the i n t e g r a l s , . E q . 4-35, and the c r o s s s e c t i o n s , Eqs. 4-33, 4-36 and 4-37, "based on the E r i c s o n s ' p o t e n t i a l o f Chapter 3 and 4-3  Sec. 3-3. Numerical D i s c u s s i o n In the p r e s e n t s e c t i o n we present a numerical d i s c u s s i o n  o f p i o n e x c i t a t i o n "based and  on the DWBA c r o s s s e c t i o n  formulae  the r a d i a l i n t e g r a l s which we have d e r i v e d i n the l a s t .  section.  In these c a l c u l a t i o n s , our main o b j e c t i v e i s to  demonstrate the s e n s i t i v i t y  o f the e x c i t a t i o n c r o s s s e c t i o n s  to the d i f f u s e n e s s of the n u c l e a r s u r f a c e .  As d i s c u s s e d  i n the p r e v i o u s s e c t i o n , t h i s s e n s i t i v i t y i s contained i n the r a d i a l i n t e g r a l s , F ^ in  r  which a r i s e  from the r a d i a l terms  the momentum-dependent p a r t of the pion-nucleus  To show t h a t the experimental q u a n t i t i e s , we  first  data can he analyzed  interaction f o r these  c o n s t r u c t the t o t a l and d i f f e r e n t i a l  e x c i t a . t i o n c r o s s s e c t i o n s and demonstrate t h a t these cross s e c t i o n s are s e n s i t i v e to the d e t a i l s o f the n u c l e a r s u r f a c e . We then show t h a t t h i s s e n s i t i v i t y i s contained i n the r a d i a l i n t e g r a l s , F^p  r  and t h a t i t can be understood  i n terms o f  the s u p p r e s s i o n o f momentum-dependent processes near of the p o t e n t i a l b a r r i e r Chapter 3 ) .  the top  f i n analogy with our d i s c u s s i o n i n -  119 For  the purposes of the p r e s e n t d i s c u s s i o n , we have  the  potential  to be that, given by Eqs. 4-28  4-2  where we have expanded  the  average n u c l e a r deformation p a r a m e t e r , ^  terms of order f we have tion  (Eq.  9  retaining  only  For the parameters of the i n t e r a c t i o n , from the E r i c s o n s '  (1966) which we have d i s c u s s e d i n Sec. 3-3  calcula-  o f the  prev-  We have then i n t r o d u c e d the d i f f u s e n e s s o f  n u c l e a r s u r f a c e by employing a Saxon-Woods d i s t r i b u t i o n 3-31)  f o r the n u c l e a r d e n s i t y . '  s i t y we have our  o f Sec.  the a c t u a l p o t e n t i a l i n terms o f  used the values obtained  ious chapter. the  .  and 4-29  chosen  chosen R  calculations  a = 0.5  fm.  and a = 0.1 lines).  1/3  = 1.05  0  In the Saxon-Woods den-  A '  fm. and we have performed  f o r two choices o f the s u r f a c e  ('which i n our f i g u r e s we fm.  parameter,  denote by s o l i d  (which i n our f i g u r e s we  lines)  denote by dashed  We have performed our c a l c u l a t i o n s f o r both a l i g h t OK  n u c l e u s , where we have chosen parameters; a p p r o p r i a t e to A l and f o r a heavy n u c l e u s , where we have chosen parameters  ,  2 38 a p p r o p r i a t e t o IT In F i g s . 12a and 12b we have p l o t t e d e x c i t a t i o n cross sections, C 5 ^ j e (  tively, sections, (4-37b) where J$  ^red ^ C5£MT,  s  r  elated  for A l  ?  the reduced  and 11"° , r e s p e c -  to the a c t u a l e x c i t a t i o n  by Eq. 4~37b o f Sec. = ^ A^^" 2  e d  total  cross  4-2:  5  i s the average n u c l e a r deformation parameter and  A^< i s a s t a t i s t i c a l c o e f f i c i e n t (Eq. 4-21) d e s c r i b i n g the i n i t i a l and f i n a l r o t a t i o n a l s t a t e s i n v o l v e d i n the e x c i t a t i o n . 2 T y p i c a l l y J 3 ^ o . 3 and A i s a number always l e s s than u n i t y Ve<  120  REDUCED T O T A L EXCITATION CROSS S E C T I O N S  F I G . 12a  ;f+AI  0.20  2 5  / /  c  0.15 4.  0.10  a  = 0.5 Fm  a = Oo I Fm 0.05  0  0  20 MeV  35  PIG  12b  REDUCED T O T A L E X C I T A T I O N CROSS SECTIONS  71^  U  2  3  @  E,  MeV  '  i2a  FIG.  13a  REDUCED EXCITATION  >  + Al  DIFFERENTIAL CROSS S E C T I O N S  25  30  20  i  IOI  O ot 75 a = 0.5 Fm a = 0.1 F m  50  25  01 0  E = 35 MeV  (  30  (  60°  90°  e  120°  15 0°  I80  c  123  FIG.  i  j  0°  13b  I  30°  REDUCED EXCITATION  I  60°  J  9 0°  .  9  DIFFERENTIAL CROSS SECTIONS  I  120°  !_  150°  I  18 0°  ( f o r example, A +^ + 6  ='  2  i . ).  Tt i s t h e r e f o r e seen from  F i g s . 12 t h a t , t y p i c a l l y , <5^->~s  <:  0.03  barns,  considerably  l e s s than the geometric c r o s s s e c t i o n of a few barns. our previous  d i s c u s s i o n , t h i s would seem s u f f i c i e n t  From  justi-  f i c a t i o n to.employ the DWBA. I t i s seen from F i g s . 12.that, i n both l i g h t and  heavy  n u c l e i , even the t o t a l e x c i t a t i o n cross s e c t i o n s are s e n s i t i v e to the  diffuseness  of the n u c l e a r  surface.  p r a c t i c e , the t o t a l c r o s s s e c t i o n s can be measured a c c u r a t e l y by measuring the d e - e x c i t a t i o n  quite In  very  "tf-rays when the  e x c i t e d r o t a t i o n a l s t a t e decays so that these e f f e c t s can measured w i t h c o n s i d e r a b l e o f the  diffuseness  energies,  precision.  Although the e f f e c t s  of the surface become greater, at  the simple connection  higher'  between the d e n s i t y and  pion-nucleus i n t e r a c t i o n many then break down, as was cussed i n Chapter 2.  Nevertheless,  be  i f our present  the  dis-  understand-  i n g o f the i n t e r a c t i o n i s even q u a l i t a t i v e l y c o r r e c t , the e x c i t a t i o n c r o s s s e c t i o n s should f o r i n v e s t i g a t i n g the In F i g s . 13a  and  e n t i a l cross s e c t i o n s for A l  and u  i n Figs.. 12a  provide  a powerful t o o l  d e t a i l s of the n u c l e a r 13b  surface.  we have p l o t t e d the reduced  ( f o r the i n e l a s t i c a l l y s c a t t e r e d  c o r r e s p o n d i n g to the t o t a l cross  and  12b,  differ-  respectively.  pions)  sections  Measurement o f the  di 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 v o l v e s measuring the i n e l a s t i c a l l y s c a t t e r e d pions and measuring the the  i s probably somewhat more d i f f i c u l t  t o t a l cross sections.  than  I t i s seen, however, that  d i f f e r e n t i a l c r o s s s e c t i o n s have a s t r u c t u r e which i s  125 strongly  dependent upon the d i f f u s e n e s s  surface;  i n P i g . 13a (upper f i g u r e )  be  i t i s seen that  the case even when the t o t a l c r o s s s e c t i o n s  much s m a l l e r s e n s i t i v i t y . since in  o f the n u c l e a r  lision  The l a r g e back s c a t t e r i n g  tially  the nucleus  ( 6 = 180° ) than i n a g r a z i n g c o l -  to analyze the c r o s s s e c t i o n s  d e t a i l , i t w i l l be u s e f u l cussion  (Eqs. 4-35) that  i n more  to remember from our e a r l i e r d i s pion e x c i t a t i o n arises  four separate processes; f i r s t l y ,  from essen-  there i s e x c i t a t i o n  to the l o e a l p i o n - n u c l e u s i n t e r a c t i o n , accounted  the i n t e g r a l s , F^; s e c o n d l y , there i s "Coulomb due  arises  ( 0 = 0 ° ) .  Before we proceed  due  exhibit a  i t i s more probable f o r the p i o n to e x c i t e  a head-on c o l l i s i o n  t h i s may-  for i n  excitation"  t o the e l e c t r o s t a t i c p o t e n t i a l , accounted f o r i n the  integrals, F  ; t h i r d l y , there i s e x c i t a t i o n due to the  angular terms i n the momentum-dependent i n t e r a c t i o n , accounted for  i n the i n t e g r a l s , P ^  ^; f i n a l l y , there i s e x c i t a t i o n due  to the r a d i a l terms i n the momentum-dependent accounted f o r i n the i n t e g r a l s , 3?. _ „ MD r  interaction,  I n the subsequent A  9  discussion  we w i l l show t h a t  the f i r s t  three i n t e g r a l s are  i n s e n s i t i v e to the d e t a i l s o f the n u c l e a r s u r f a c e and that the  s e n s i t i v i t y demonstrated i n P i g s .  m a i n l y i n the i n t e g r a l s , tive in  to show that  r  „  12 and 13 i s contained  Tt w i l l  then be our objec-  t h i s s e n s i t i v i t y can e a s i l y be understood  terms o f the s u p p r e s s i o n o f pion momentum  o f the p o t e n t i a l In P i g .  the t o p  barrier.  14 we have p l o t t e d  radial integrals  near  the absolute values o f the  f o r the';local i n t e r a c t i o n , j F-^ j, at 35 MeV  126  Fl r  14  R E L A T I V E VALUES OF RADIAL INTEGRALS  E = 35 MeV  TC+25Al  E  = 3 5 MeV  U  0-2  2-2  1-3  3-3  238  129  FIG.  16.  RADIAL  INTEGRALS  130  FIG.  16b  RADIAL  INTEGRALS  f o r the allowed angular momentum t r a n s i t i o n s between the i n i t i a l s t a t e p a r t i a l wave o f angular momentum, J[ , and the f i n a l s t a t e p a r t i a l wave o f angular momentum, |'„ restricted precise  our c a l c u l a t i o n to X, |'  (although a more  j  c a l c u l a t i o n would take i n t o account the h i g h e r  p a r t i a l waves). the  ^ 3  We have  I t i s seen that-the  main c o n t r i b u t i o n to  e x c i t a t i o n a r i s e s from the. i n t e g r a l F-^ and the secondary L  contribution  from the i n t e g r a l F ^ ( o r , e q u i v a l e n t l y ,  F^),  JJ  Hence, even a t moderately high e n e r g i e s ,  1/  only a few p a r t i a l  waves e n t e r the problem; o f course, as we go t o lower energies the r e l a t i v e dominance o f the i n t e g r a l s , F-*-^, i n c r e a s e s . will  therefore  here r e s t r i c t  form F""^ and 3?^; 1  We  o u r s e l v e s to i n t e g r a l s of the  i n any event, the g e n e r a l i z a t i o n  to i n t e g -  r a l s i n v o l v i n g p a r t i a l waves o f h i g h e r angular momentum w i l l be  apparent from our d i s c u s s i o n . In F i g . 15a and 15b we have p l o t t e d  the r e l a t i v e mag-11' n i t u d e s o f the absolute values o f the i n t e g r a l s , 3?^,.-.3? , 11 p-R ? 38 and 3? ' , f o r A l and U" , r e s p e c t i v e l y . I t i s seen 1  1  9  4 0  '  X l i s p \J  t  from F i g s . 15 t h a t these r a d i a l i n t e g r a l s are quite i n s e n s i t i v e to the d i f f u s e n e s s  parameter, a, even i n the case o f  25 Al  , which i s mostly s u r f a c e .  This a r i s e s s i n c e  these  q u a n t i t i e s i n v o l v e i n t e g r a l s over the nucleus and are sensitive  only  to the amount o f n u c l e a r matter present  rather  than to i t s d i s t r i b u t i o n . surface ana  A s i m i l a r i n s e n s i t i v i t y to the e'Jt ' ' 0" i s found i n g e n e r a l f o r the i n t e g r a l s F , P , T  V.e • In F i g . 16a and 16b "(upper f i g u r e s ) we have p l o t t e d  •'  (on the same s c a l e ) the corresponding  values o f the  integrals  a r i s i n g from the r a d i a l term i n the momentum-dependent i n t e r action,, I ' l l ^  I  t i s seen t h a t , u n l i k e the other  integrals,  these i n t e g r a l s depend very s t r o n g l y on the d i f f u s e n e s s p a r a meter, a.  I n f a c t , as we w i l l now show, the behaviour o f  these i n t e g r a l s can be understood the behaviour  i n an analogous manner to  o f momentum-dependent a b s o r p t i o n discussed i n  Chapter 3. •In F i g . 16a we have p l o t t e d the r a d i a l i n t e g r a l s | F ! 1 (upper f i g u r e ) and J F ^ ' 2  Al  ..  for  Al  in  Since  I  (lower  the parameters o f the pion-nucleus  are s i m i l a r to those  2 5  f i g u r e ) a p p r o p r i a t e to potential  o f Ca, , which we discussed 40  connection w i t h o p t i c a l a b s o r p t i o n i n Sec. 3-4, we w i l l .  i n t e r p r e t the e f f e c t s found here i n terms o f the explanations given  there. For i n s t a n c e , the behaviour ' ,  of  P? MD,r 2  p r o p e r t i e s o f the r a d i a l s-wave f u n c t i o n . top o f the  depends on the Now, near the  p o t e n t i a l b a r r i e r , the r a d i a l momentum a s s o c i a t e d  w i t h the s-wave becomes s m a l l .  For the d i f f u s e - e d g e poten-  t i a l , most o f the i n t e r a c t i o n takes p l a c e i n the s u r f a c e o f the nucleus where the n u c l e a r d e n s i t y i s lower; hence, f o r the d i f f u s e - e d g e p o t e n t i a l , the e f f e c t i v e b a r r i e r lower.  Thus, we expect  value o f tial  appears  a t lower .-energies to f i n d t h a t the  |F 9- I i s l e s s i f we assume a d i f f u s e - e d g e I MD,rl , 2  T  poten-  than i f we assume a sharp-edge p o t e n t i a l and, a t h i g h e r  e n e r g i e s , we expect  to f i n d  that i t i s g r e a t e r .  This i s i n  analogy with our d i s c u s s i o n o f momentum-dependent a b s o r p t i o n  133 in Ca  which, we  4 0  have d i s c u s s e d i n F i g . 7.  shown here i n F i g . 16a w i t h the imaginary in Ca  (lower  This e f f e c t i s  f i g u r e ) and i s to he compared  s-wave phase s h i f t  for o p t i c a l  d i s c u s s e d i n Sec.  3-4,  the e f f e c t ' o f i n c r e a s i n g  t h e r e f o r e expect at h i g h e r energies  behaviour f o r the i n t e g r a l . F H * MD,r 02 e n e r g i e s f o r the i n t e g r a l • F,!: ° * MD,r i n F i g . 16a As we  to see  discussed  potential.  for P b ^  0 8  i n Sec. 3-4  due  ( F i g . 8), the  i s an i n c r e a s e i n  to an i n c r e a s e i n the e l e c t r o s t a t i c  This e f f e c t i s seen i n F i g . 16b  shown that even a t h i g h e r e n e r g i e s both the  where i t i s integrals  3?0S ^ are more s t r o n g l y suppressed f o r a than for a sharp-edge p o t e n t i a l .  at lower energies course,  might now  responding  13.  have presented o f these easily  The  attenuation  We  to the cross s e c t i o n s c o r -  r a d i a l i n t e g r a l s which we  have given i n  b e l i e v e t h a t i n the d i s c u s s i o n which we  here we  have c l e a r l y shown that the  dependence  c r o s s s e c t i o n s on the n u c l e a r s u r f a c e a r i s e s  understandable way  interaction. qualitative  of  of p e n e t r a t i o n .  return b r i e f l y  to these  12 and  diffuse-edge  which i s seen In the diagram i s due,  to the e f f e c t s  We  a similar  (upper f i g u r e ) .  the e f f e c t i v e b a r r i e r  potential  barrier.  , as was found at lower * . This e f f e c t i s shown  major e f f e c t i n going to a heavy nucleus  Figs.  absorption !  the angular momentum i s to i n c r e a s e the p o t e n t i a l  and  i  ( F i g . 5).  4 0  As we  We  !  I n f a c t , our  i n an  through the momentum-dependent d i s c u s s i o n depended only upon the  f e a t u r e s o f the pion-nucleus  interaction'---  the  134 presence  o f a p o t e n t i a l b a r r i e r and a s t r o n g momentum-depend-  ent i n t e r a c t i o n . the  We h a v e n o t i n v e s t i g a t e d  cross sections i n d e t a i l since,  of  the p o t e n t i a l  are not very well  a b l y somewhat e n e r g y - d e p e n d e n t ) treatment o f the p o t e n t i a l approximately correct.  the  firstly,  the parameters  determined  (and are p r o b -  and, s e c o n d l y , our rough  ( E q s . 4-28  and 4-29)  Future c a l c u l a t i o n s  p r o b a b l y employ p o t e n t i a l s iments  the s t r u c t u r e o f  i s only  of this  type  o b t a i n e d from s c a t t e r i n g  ( s u c h as those s u g g e s t e d i n C h a p t e r  will  exper-  3) and w i l l  treat  p e r t u r b i n g p o t e n t i a l i n a more p r e c i s e manner ( s u c h a s  t h a t s u g g e s t e d by C h a s e , W i l e t s , and Edmonds is  w o r t h w h i l e , however, to mention  is  seen i n these c r o s s s e c t i o n s . Tt  ing  (1958)).  one f u r t h e r e f f e c t  i s s e e n i n F i g . 12 t h a t , i n t h e c a s e o f A l  the d i f f u s e n e s s parameter,-a,  2  5  ,  It which  increas-  decreases the t o t a l cross  ? ^8 s e c t i o n w h i l e , i n the case o f U is  enhanced.  the  , the t o t a l  cross section  On t h e o t h e r h a n d , we know f r o m F i g . 16  radial integrals, F ^  r  that  the o n l y a s p e c t s o f the prob-  lem which a r e s e n s i t i v e t o the d i f f u s e n e s s parameter are g e n e r a l l y s u p p r e s s e d a t l o w e n e r g i e s f o r a" d i f f u s e - e d g e potential Al  8  5  )  Eq.  r  i n the case o f  o  The is  . no n02 ( w i t h the minor e x c e p t i o n o f F ^  e x p l a n a t i o n o f t h i s behaviour o f the cross sections  understood  from the f o r m u l a f o r the t o t a l  cross section.  4-37, i n w h i c h i t i s s e e n t h a t t h e c r o s s s e c t i o n p»o •  i n t e r f e r e n c e b e t w e e n t h e a m p l i t u d e s . F, j ; '  involves  and t h e o t h e r  IVLD , r  a m p l i t u d e s which e n t e r the e x c i t a t i o n process  ( F ^ , 3?f^ , and  135 ^MD  0*°' ^  amplitudes  n  ^  a c  i  ^ » by comparing the absolute values o f these  on F i g s , 15a and 15b and on F i g s , 16a and 16b,  respectively,  (which have been drawn to the same s c a l e ) i t i'j  is  c l e a r t h a t the e f f e c t s o f F,,^ are seen p r i m a r i l y MD,r . * i n t e r f e r e n c e s with the more dominant terms, J  / 25*. $ Hence, i n F i g , 12a ( A l ) the decrease i n F j ^  r  i through' to  with  i n c r e a s i n g d i f f u s e n e s s parameter repre-sents a decrease i n c o n s t r u c t i v e i n t e r f e r e n c e ; i n F i g , 12b (U  ) i t represents  a decrease i n d e s t r u c t i v e i n t e r f e r e n c e and, hence, an enhancement o f the c r o s s section,,  It i s difficult  to see through  the complex a l g e b r a i n Eq„ 4-37 and t o present these ments i n a more s a t i s f a c t o r y form.  argu-  However, a more c a r e f u l  a n a l y s i s o f e f f e c t s such as-these w i l l  be r e q u i r e d i n any  event when more s a t i s f a c t o r y values f o r the pion-nucleus pote n t i a l become a v a i l a b l e .  136 CHAT^TSR 5  CONCLUSIONS  I n t h i s t h e s i s we tant  f e a t u r e s o f the  i n p a r t i c u l a r , we a c t i o n as a t o o l  have attempted to d e l i n e a t e the  low energy pion-nucleus  have emphasized  impor-  interaction  the usefulness  and,  of this  f o r i n v e s t i g a t i n g the s t r u c t u r e of the  internuclear  surface. In  Chapter 2 we  the pion-nucleus nucleon  and  have i n v e s t i g a t e d the c o n s t r u c t i o n of  o p t i c a l p o t e n t i a l from the elementary p i o n -  pion-deuteron  processes;  we  have shown t h a t a  transparent  connection  microscopic  d e s c r i p t i o n s of p i o n i n t e r a c t i o n s i n n u c l e i  in  e x i s t s between the macroscopic  and and,  f a c t , t h a t the p o t e n t i a l which d e s c r i b e s the macroscopic  i n t e r a c t i o n i n v o l v e s the n u c l e a r  d e n s i t y i n a very  direct  manner. In Chapter 3 we  have used t h i s p o t e n t i a l to . i n v e s t i g a t e  the o p t i c a l p r o p e r t i e s o f pions i n n u c l e i and we t h a t the e l a s t i c  s c a t t e r i n g and  depend s t r o n g l y upon the  have shown  a b s o r p t i o n cross s e c t i o n s  details  of the n u c l e a r • s u r f a c e  the momentum-dependent a b s o r p t i o n .  In Chapter 4 we  through  have shown  t h a t the c r o s s s e c t i o n s f o r the e x c i t a t i o n o f r o t a t i o n a l s t a t e s in  deformed n u c l e i through the  o p t i c a l pio.n-nucleus  interaction,  depends s t r o n g l y on the n u c l e a r s u r f a c e through momentumdependent e x c i t a t i o n f o r analogous reasons to those which were d i s c u s s e d i n Chapter 3 f o r o p t i c a l a b s o r p t i o n . cluded pions  from our c a l c u l a t i o n s a n d : d i s c u s s i o n provide  an i n t e r e s t i n g new  i n p a r t i c u l a r , they  provide  We  have con-  that low  energy  probe of n u c l e a r s t r u c t u r e ;  a d i r e c t means o f measuring the  137  1  d i s t r i b u t i o n o f nucleons i n the n u c l e a r s u r f a c e . In Chapter 2 we have d i s c u s s e d o p t i c a l p o t e n t i a l which describes  the c o n s t r u c t i o n o f an  the low energy pion-nucleus  o p t i c a l i n t e r a c t i o n i n terms o f elementary pion-nucleon and pion-deuteron s c a t t e r i n g and a b s o r p t i o n  processes.  I n par-  t i c u l a r , we have shown that t h i s i n t e r a c t i o n i n v o l v e s the density  o f the nucleus i n a very d i r e c t manner.  In Sec. 2-1 we have d i s c u s s e d e x h i b i t i n g the s t r u c t u r e  a geometric technique f o r  o f the low energy'pion-nucleus op-  t i c a l p o t e n t i a l ; we have argued t h a t , s i n c e the elementary p i o n - n u c l e o n and pion-deuteron s c a t t e r i n g s are s-wave and p-wave s c a t t e r i n g s , the elementary s c a t t e r e d waves should add to form the t o t a l s c a t t e r e d wave i n an analogous way to that i n which  one adds the e l e c t r i c p o t e n t i a l s o f d i p o l e s and  charges i n a c l a s s i c a l d i e l e c t r i c .  We have shown that the  s-wave s c a t t e r i n g s add l i k e the p o t e n t i a l s from f r e e charges in  a dielectric  potential; field  and lead to a l o c a l term i n the pion-nucleus  the p-wave s c a t t e r i n g s modify the p i o n momentum-  i n the n u c l e u s , i n analogy t o the manner i n which the  d i p o l e s modify  the e l e c t r i c  field  i n a d i e l e c t r i c , and lead  to a momentum-dependent term i n the pion-nucleus i n t e r a c t i o n which i s analogous  to the Lorenz-Lorentz e f f e c t i n a dense  c l a s s i c a l medium. We have d i s c u s s e d  the d e r i v a t i o n o f the o p t i c a l p o t e n t i a l  from a m i c r o s c o p i c and quantum mechanical p o i n t o f view i n Sec. 2-2 where'we have reviewed an e a r l i e r d i s c u s s i o n due to E r i c s o n and E r i c s o n  (1966).  I've have, - shown that the many-  138 body aspects of the problem t i o n s mainly through  are reduced  to t r a c t a b l e  propor-  the s h o r t pion-nucleon s c a t t e r i n g l e n g t h s  and the s m a l l mass of the p i o n .  We have shown t h a t  these  p r o p e r t i e s allow us to make the impulse approximation, so we may  that  use the known f r e e space pion-nucleon s c a t t e r i n g amp-  l i t u d e s , and  allow us to s i m p l i f y  the Green's f u n c t i o n s which  d e s c r i b e the p r o p a g a t i o n of the pion i n the n u c l e u s , s i n c e the elementary elastic.  p i o n c o l l i s i o n s may  be t r e a t e d  Prom these approximations, and  as being e s s e n t i a l l y the assumption  that  the nucleoli s c a t t e r e r s are massive, we were able to perform the n u c l e a r averages  and  for the p i o n analogous i n a c l a s s i c a l medium.  to o b t a i n m u l t i p l e s c a t t e r i n g  to those  equations  describing multiple scattering  I n f a c t , we were able to show that  these equations could be reduced  to c o n v e n t i o n a l Schroedinger  equations f o r the p i o n i n ' w h i c h the o p t i c a l p o t e n t i a l i n v o l v e d the d e n s i t y o f the n u c l e u s ' i n a very d i r e O t manner and which had  the s t r u c t u r e p r e d i c t e d by the e l e c t r o m a g n e t i c  I n Sec, 2 - 3 we  presented the r a t h e r sparse experimental  f i c a t i o n of the parameters JC - mesic x-ray In Chapter  analogy. veri-  o f t h i s p o t e n t i a l provided by the  experiments. 3 we  used  t h i s p o t e n t i a l to examine.the  t i c a l p r o p e r t i e s o f pions i n n u c l e i and  to i n v e s t i g a t e  r o l e o f the n u c l e a r s u r f a c e i n determining e l a s t i c and a b s o r p t i o n c r o s s s e c t i o n s ,  Tn Sec,  3-1  opthe  scattering  we have made a  p a r t i a l wave a n a l y s i s of the c r o s s s e c t i o n s and we have r e viewed  the d e r i v a t i o n of well-known formulae which express  the  139 p a r t i a l wave phase s h i f t s i n terms o f the i n t e r i o r mic  d e r i v a t i v e s ; where necessary,  mulae  logarith-  we have extended these  for-  (which are c o n v e n t i o n a l l y w r i t t e n f o r l o c a l i n t e r a c t i o n s )  to momentum-dependent i n t e r a c t i o n s . In Sec.- 3-2 we have used the formalism Sec.  3-1 and the assumption o f a uniform  developed i n  nuclear  d e n s i t y to  o b t a i n a n a l y t i c formulae which d e s c r i b e the o p t i c a l p r o p e r t i e s .of the pion-nucleus onance -aspects  interaction.  We have shown t h a t the r e s -  o f the problem, which are contained  i n the  r e a l parts o f the i n t e r i o r l o g a r i t h m i c d e r i v a t i v e s , are i n sensitive  to the d e t a i l s o f the i n t e r a c t i o n because o f the  l o n g p i o n wavelength near the top o f the p o t e n t i a l b a r r i e r . In f a c t , we have shown t h a t the resonance aspects lem  o f the prob-  depend only upon the h e i g h t o f the p o t e n t i a l b a r r i e r and,  f o r h i g h e r p a r t i a l waves, the r e a l term i n the momentum-dependent i n t e r a c t i o n .  We have a l s o shown t h a t , near the top o f  the p o t e n t i a l b a r r i e r , the momentum-dependent a b s o r p t i o n c e s s e s , which are described  by the imaginary  pro-  p a r t s o f the l o g -  a r i t h m i c d e r i v a t i v e s , are s t r o n g l y suppressed due to the small momentum o f the p i o n . In Sec. 3-3 we have s t a t e d numerical ameters o f the pion-nucleus  values  f o r the par-  o p t i c a l p o t e n t i a l which we have  taken to be the zero-energy values g i v e n by E r i c s o n and E r i c s o n (1966).  In Sec. 3-4 we have used t h i s p o t e n t i a l  assuming a d i f f u s e - e d g e n u c l e a r d e n s i t y to g i v e a more quantitative  d i s c u s s i o n o f the i d e a s developed i n Sec. 3-2.  We  have shown t h a t the r e a l p a r t s o f the p a r t i a l wave phase s h i f t s  140 depend r i e r the  sensitively  (through  gest  a  of  the  aspects  of  part  the  of  the  from  and  part  of  shown,  the by  nuclear  comparing  effective nuclear  density) of  lower  potential  energy On  from  contain  have  can  be  elastic  i n  by  a  the  barrier  nucleus  for  seen  imaginary  shifts,  the  concluded  accurately scattering  very  We. h a v e  segregating  the  the  absorption  with  absorption  real  part  resonance that  the  determined  we  the of  know  the  from  absorption  a  to  The  than the  the of  lower to  strong  a  wave  sharpof  the  inside phase We  nuclear  sections.  sup-  occurs  height  of  the  the  problem.  the  have  nuclear  density)  measurements cross  We  sharp-edge  is  partial  of  thickness  imaginary;  therefore  nuclear  aspects  the  diffuse-edge  pions.  an  absorp-  proportional  absorbed  hand,  provided  local  i n  diffuse-edge is  upon  derivative.  found  (which  hence,  determines  sensitively  surface.  conventional  other  and  which  diffuse-edge'nucleus  (and, the  shown  logarithmic  the  sug-  the  by  barrier  to  that  have  depends  of  results  absorptive  absorption  effect  penetration  the  phase  more  these  upon  of  momentum-dependent  nucleus.  which  the  a  potential  pression  edge  and  the  and  investigation  nuclear  the  problem)  bar-  data.  we  i n t e r i o r  potential  empirical  momentum-dependent  s-wave  that  used  effect  the  the  We h a v e  and  the  of  (through  analyzing  wave  height  nucleus  thorough  this  density  density,  a  of  the  aspects.of  diffraction,  of  the  the  problem  diffuseness  tion  a  p a r t i a l  explanation  "the  for  made  the  absorption  for  problem).  procedure  We h a v e  upon  resonance  radius- chosen  aspects  at  the  only  shifts there-  surface  pion-nucleus We  have  '  ' shown that  -  141  the main e f f e c t on the a n a l y s i s  cross sections  o f using p o s i t i v e  o f pion-nucleus  or negative p i o n s , or o f  u s i n g d i f f e r e n t n u c l e i , i s simply t o a d j u s t the e f f e c t i v e h e i g h t o f the p o t e n t i a l h a r r i e r seen by the p i o n through the  electrostatic interaction.. In Sec. 3-5 we have b r i e f l y d i s c u s s e d the i s o s p i n and  h y p e r f i n e terms which enter the pion-nucleus i n t e r a c t i o n . In Chapter 4 we have examined the e x c i t a t i o n o f r o t a t i o n a l l e v e l s .in deformed n u c l e i by pions and we have shown unlike  e x c i t a t i o n processes a r i s i n g from more c o n v e n t i o n a l  interactions, sitively Sec.  that,  the e x c i t a t i o n process found here depends sen-  upon the s t r u c t u r e  of the n u c l e a r s u r f a c e .  4-1 we have b r i e f l y reviewed the d e s c r i p t i o n  In  of strongly  deformed n u c l e i p r o v i d e d by the r o t a t i o n a l model and we have d i s c u s s e d the v a l i d i t y o f the D i s t o r t e d tion  description  Wave^'Born Approxima-  o f pion e x c i t a t i o n processes.  I n Sec. 4-3  we have employed the DWBA to w r i t e the p i o n e x c i t a t i o n ampl i t u d e s i n a form which s e p a r a t e l y segregates the e f f e c t s of Coulomb e x c i t a t i o n ,  l o c a l e x c i t a t i o n , and momentum-dependent  excitation.  • - •  In Sec. 4-3 we have used the E r i c s o n s ' Sec.  3-3 to c a l c u l a t e  realistic ential. is and  the e x c i t a t i o n cross s e c t i o n s  diffuse-edge p o t e n t i a l We have shown that  strongly  affected  p o t e n t i a l of  and f o r a sharp-edge pot-  the s t r u c t u r e  o f the c r o s s  by the t h i c k n e s s o f the n u c l e a r  that t h i s s t r u c t u r e  for a  sections  surface  i s mainly contained i n the r a d i a l  i n t e g r a l s which d e s c r i b e . t h e momentum-dependent  excitation.  142 We have shown that is  the e f f e c t o f the d i f f u s e n u c l e a r s u r f a c e •  to lower the e f f e c t i v e p o t e n t i a l  b a r r i e r so that  the s t r o n g  s u p p r e s s i o n o f momentum-dependent e x c i t a t i o n occurs at a lower energy f o r a d i f f u s e - e d g e nucleus than f o r a sharp-edge nucleu In  f a c t , t h i s i s ' i n analogy to our d i s c u s s i o n  o f momentum-  dependent a b s o r p t i o n i n Sec. 3-4 and we have found here, as we found t h e r e , t h a t the main e f f e c t o f c o n s i d e r i n g  higher  p a r t i a l waves i s to r a i s e the e f f e c t i v e b a r r i e r seen by the p i o n through the c e n t r i f u g a l p o t e n t i a l  and t h a t  e f f e c t of considering  or pions o f d i f f e r e n t  different nuclei  the main  charges i s to change the e f f e c t i v e b a r r i e r through the e l e c t r o static potential.  We have concluded from these r e s u l t s  pion e x c i t a t i o n cross sections for i n v e s t i g a t i n g  the s t r u c t u r e  provide a s e n s i t i v e  technique  o f the n u c l e a r s u r f a c e i n  deformed n u c l e i , The  •  •  general c o n c l u s i o n s of t h i s t h e s i s  energy pions have O p t i c a l p r o p e r t i e s  ,  '  .  i n n u c l e i which are  from those encountered with more c o n v e n t i o n a l  different  interactions  because o f the appearance o f a short-range p o t e n t i a l a strongly  action. that  momentum-dependent c o n t r i b u t i o n  These p r o p e r t i e s  the n u c l e a r d e n s i t y  .  are that low  e a s i l y understood and which are q u a l i t a t i v e l y quite  and  that  barrier  to the i n t e r -  o f the i n t e r a c t i o n , and the f a c t enters the pion-nucleus o p t i c a l  pot-  e n t ! el i n a very d i r e c t manner, allow one to i n v e s t i g a t e aspects o f n u c l e a r s t r u c t u r e e a s i l y accessible,, elastic  scattering  which have p r e v i o u s l y  Tn p a r t i c u l a r , an a n a l y s i s  n o t been  o f pion-nucleus  and a b s o r p t i o n c r o s s s e c t i o n s  and o f p i o n  143 e x c i t a t i o n c r o s s s e c t i o n s i n deformed n u c l e i should r e v e a l d e t a i l e d i n f o r m a t i o n ah out the d i s t r i b u t i o n o f nucleons i n the n u c l e a r s u r f a c e , i n f o r m a t i o n c r u c i a l to a refinement o f the d e t a i l e d models o f n u c l e a r s t r u c t u r e .  144 BIBLIOGRAPHY" Abashian, A., C o o l , R„ , ana C r o n i n , J.W., 855 (1956).  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New  York,  K i s s l i n g e r , L., Phys. Rev.  1968)  98, 761  (1955).  K o l t u n , L.S., "Advances i n Nuclear P h y s i c s , Volume 3", eds. H Baranger and E, Vogt. (Plenum P r e s s , New York, 1969) K r o l l , N.M. , c i t e d i n E d e l s t e i n , R . M . , Baker, P.W. , and Rainwater, J . , Phys. Rev. 122, 252 (1961). s  Lax, M. , Revs. Mod..  Phys. 23, 287  (1951).  Messiah,'A., "Quantum Mechanics, Volume 2". (Addison-Wesley,. Reading, Mass., 1962) Nolen, -J.A. J r . , S c h i f f e r , J.P. , and W i l l i a m s , N. ,Phys, L e t t e r s 27B, 1 (1968). P e a s l e e , D. C. , Phys. Rev.  E57, 862  (1952).  P r e s t o n , - M o A. , " P h y s i c s o f the Nucleus". (Addison-Wesley, Reading, Mass., 1962) S p e c t o r , R.M.,  Phys. Rev.  129, 413  (1964).  Temmer, G.M., " I n t e r n a t i o n a l N u c l e a r P h y s i c s Conference, G a t l i n h u r g , Tennessee, Sept. 12-17,.1966, 223." (Academic P r e s s , New York, 1967) Van Hove, L. , Phys. Rev. Vogt, E.W.,  Revs. Mod.  95_, 249  Phys. 34,  Watson, K.M. , Phys. Rev.  89, 575  (1954). 723  (1962).  (1953).  Wu, C.S., " I n t e r n a t i o n a l N u c l e a r P h y s i c s Conference, G a t l i n h u r g , Tennessee, Sept. 18-17, 1966, 409" (Academic P r e s s , New York, 1967)  146 APPENDIX A  EQUATIONS FOR THE INTERIOR LOGARITHMIC DERIVATIVES  In the present  appendix, we w i l l be concerned with e v a l -  u a t i n g the i n t e r i o r l o g a r i t h m i c Eq.  3-13.  F i r s t l y , we w i l l  t h e i r imaginary p a r t s , functions, u^(r). uni form n u c l ear  derivatives,  given i n  f i n d an i n t e g r a l s o l u t i o n f o r  Tty ,  i n terms o f the p a r l a l wave  Secondly, we w i l l e v a l u a t e  7^ f o r a  d e n s i t y i n terms o f the i n t e r i o r wave  func-  tions.. A-1  I n t e g r a l Equation f o r A b s o r p t i o n In Eq. 3-4 we have w r i t t e n r a d i a l Sehroedinger equa-  tions  f o r the p a r t i a l waves, u ( r ) :  2^  ••*  Zp  +  4-  M±H r  tt. Zp  (1 + ^ ( r ) ) u , ( r ) e  2  ( )  u  r  r =  We w i l l now use t h i s equation f o r the a b s o r p t i v e  r  Y  U  Euj,(r)  f ) + V ( r ) + - i W ( r l u«(r)  [  +  *  c  J  ( J? = 0, 1, S, ...)  to o b t a i n an i n t e g r a l equation  term, 71^ , o f the l o g a r i t h m i c d e r i v a t i v e ,  Jj{ ° Le t us f i r s t de f i n e the quanti t y , (A-i)  f (,).: t  .  i  i  ^  M  )  ^  _  I t i s then e a s i l y shown from Eq. 3-4 t h a t  147 (A-8)  f'fr)  .  = (l+ * (r)) e  [y (  ^  r  H  ,i2  c  2  r ) +  v(r)+lW(r)|  L  -  k<  J  (r)  L e t us a i v i d e and  fjj(r) i n t o i t s r e a l and  imaginary  f^fr):  (A-3)  f f r ) = fff.r) +  if^(r)  4  Then, from E q .  A - 2 and  #  remembering Eq.  3-1*0,  (A'-4)  . af  = <^(r)M i l l  J( ) r  dr  Now,  * f W(r)  +  •  r  r  1 1  # 2  1 21  .  from Eq.  A-1  uj{(r)  ^  (and s u p p r e s s i n g r ) , !k +  u<> ,  e  P  '  A  -  6  Eq.  A-5  dfJ  )  .  dr  e  i n Eq.  A-4,  M + l ), r  As i s e a s i l y checked, a s o l u t i o n integral  equation  o*^  -  2  81  Substituting  ( i )  v*  JL ( 1 4 ^ ) ( 5 L )  f  parts, f^(r) ,  we  ^^-  t  #  2  e  u. UJ0  have t h a t  (suppressing r ) u,' 2  4  u^  h  to E q ,  A-6  i s g i v e n by  dr Ju  2 {l  the  148 (A-7)  f  ^W(r') Tf  0  dr  L 1(1+-1)  dr  u (r)  -r  0  Now  outside the i n t e r a c t i o n r e g i o n , say at the matching e r a d i u s , R . oC (r) vanishes. In a d d i t i o n , we have from * o , Eq. A-1 7fy  to J± (Eq, 3-11). Ro  t h a t f (r) i s then equal  i s g i v e n hy (A-8) r  ,  ( R  the  ) =  relation,  -  /°W(r) u,(r)| dr 3  0  {  Ju«(Ro)| )  0  °^?(r)  a  ar  2  u^fr) —^—  u^(r) r  MM A-2  Thus  J  p  dr  2  Uniform D i s t r i b u t i o n We w i l l here derive the  with. Eq.  3-4  us  d e f i n e the  first (A-9)  f o r a uniform  g^(r)  logarithmic derivative associated d i s t r i b u t i o n , fc. f, Eq.  3-18).  quantity  = (1 4-< (r) )  ujr)  d  r  e  u jjfr)/r I t i s then e a s i l y shown t h a t Eq. • 3-4 terms o f the R i c a t t i  equat'on,.  cam  be r e w r i t t e n i n  Let  149 (A-10)  gj(r)  k  (l + * e f r ) )  =  _  2  HitH.. r  r f  2 g$(r)  +  l£ -h  jV f ) ! V ( r ) f i W ( r L  2  r  c  (r) 1-f- oce(r)  Now, (A-ll)  from. Eq,. 3-4, i t i s seen t h a t i n s i d e  0  ^ ( r ) = Ajprj^br),  where A^ i s a n o r m a l i z a t i o n b r  constant  H- •< g  J  0  and. where o o  7  From Eq. A-10 i t i s seen t h a t the c o e f f i c i e n t the  R,  differential  equation  functions of  c o n t a i n no s i n g u l a r i t i e s so t h a t  lira r -» R  Thus, (A-13)  d dr /r  Ujj f r )  r=R.  0  0  ,d  fo(br) fy.(br)  or, r e w r i t i n g Eq. A-13 in/teimis o f fA-14)  >^(R ) « 0  (1 4- *|)  rd_^(br) dr_  TbrT'  r=R,  r=R  o »  150 APPENDIX B  NOTES ON EVALUATION OF PION EXCITATION CROSS SECTIONS  In the present Appendix we w i l l evaluate  some quant-  i t i e s which are r e q u i r e d i n the e v a l u a t i o n o f the p i o n e x c i t a t i o n , cross s e c t i o n s given i n Sec. 4-2. we provide  In Sec. B - l ,  d e t a i l s o f the e v a l u a t i o n o f the matrix  elements  which enter the p i o n e x c i t a t i o n c r o s s s e c t i o n formulae. Sec. B-2, we evaluate  In  the e l e c t r o s t a t i c p o t e n t i a l f o r a u n i -  form deformed charge d i s t r i b u t i o n . B-l  E v a l u a t i o n o f DWBA M a t r i x  Elements  In the t e x t , Sec. 4-2, we have shown t h a t i n DWBA the c r o s s s e c t i o n s f o r the e x c i t a t i o n o f a r o t a t i o n a l s t a t e i n a s t r o n g l y deformed nucleus are g i v e n by the e x p r e s s i o n , (4-19)  ajXV  .  k  where, t a k i n g the example (4-5a)  N  2  ,  '  o f a K - 0 band  -  M  v  i . 2  ( c . f . Eq. 4-5),  23^+ 1  en _ 2  = ( , P, V ) are the E u l e r angles  here  ( i n the conventions  o f G o l d s t e i n . (1950)) which r o t a t e the s p a c e - f i x e d (x,y,z) i n t o the body-fixed  axes  axes  (1,2,3) (see t e x t ) ;  and $^(r) are s o l u t i o n s o f Eqs. 4 - l i b and 4-17b, 'respectively; (4-7)  and 1T'(r) i s the p e r t u r b i n g p o t e n t i a l : ' .  ^  We begin o\ir e v a l u a t i o n by r e w r i t i n g Eq. 4-7 i n the  151 s p a c e - f i x e d c o - o r d i n a t e system,,  To do t h i s l e t us  first  note t h a t (B-1)  ^e ,* ) b  b  =  l ^ ^ f ^ )  <(e,*)  where f r , 6, (J>) are the s p h e r i c a l c o - o r d i n a t e s i n the spacef i x e d system and in  f r ^ , 0^,  ^)  the body_fixed system.  are the s p h e r i c a l c o - o r d i n a t e s  Both systems are chosen to have  t h e i r o r i g i n s at the c e n t e r o f the nucleus so that  r  =  r^.  Using Eg., B-1, we can w r i t e Eq, 4-7 i n the form,  We s h a l l now  assume t h a t we have an u n p o l a r i z e d beam  i n which case we average over i n i t i a l s t a t e s and sum final states.  To do t h i s we see from Eq, 4-21  that we  r e q u i r e m a t r i x elements o f the form ( t a k i n g K .= 0 ) : (B-3a)  -tt<  drd^dx'  $> (r) 2Iy +1 0  SI**1" .  81'  4? ^ 2.  r  ^  a*'  ^-^TfctjM'MftK  ac  where (B_3b)  8I*+1 "Vrf  ; M  f  M  8/i x  ^k  s  81*+1  J L en* i  Jwo' V (  ^ *o  (o<  k>  over  $  first  152  Remembering the r e l a t i o n (c.f. P r e s t o n  o$. V l l  $ l  J 3  k  E  r r 22  Z to  we f i n d t h a t  K  <Pi 3 3 M  K  S v  "—  ^  J  2J +1  k  (1962)),  i JpMiMpJ J J , >  1  2  1  2  2  3  3  ( a f t e r some elementary m a n i p u l a t i o n s ) ,  <:^ioo|i o>  x  ,  Now, averaging over i n i t i a l s t a t e s f i n a l s t a t e s , we have  and summing over  that  i  ^  Uh J  2im  2  k  U*h / S  a  -3I.C-+  1  M.M* '  ^acWac  M  M  .°W. .M W;M»M C  M  where  (B1 )  AI<  6  '=  ^100/ L O *  " ( z - o )  -  • '  + 1 For  hands where Z £ 0 we need only r e p l a c e  Eq_ 4-5a  w i t h Eq_ 4-5b o f the t e s t i n the above d e r i v a t i o n . 0  B  It is  then e a s i l y shown t h a t the g e n e r a l i z a t i o n o f Eq. B - 6 i s  g i v e n by the r e l a t i o n ,  ~  At*  (B-7b)  B-2  ~  —  ^  (  0 ,  7  (7.  Q  V  e  n  )  .  odd) „  E v a l u a t i o n o f E l e c t r o m a g n e t i c P o t e n t i a l f o r a Deformed Charge D i s t r i b u t i on .In the present s e c t i o n , we w i l l e v a l u a t e the e l e c t r o -  magnetic p o t e n t i a l , V fr) , f o r a deformed charge d i s t r i b u t i o n o f uniform charge density,^pfr)„ electromagnetic  theory, 1  (B-8) V fr) C  As i s known from elementary  r  1  TY*  t  = f7 £ > £ J r' - r f  ~  •  a  so t h a t , because of the i n t e g r a l , we make n e g l i g i b l e  errors  by n e g l e c t i n g the d i f f u s e n e s s o f the n u c l e a r edge. We  take the charge d i s t r i b u t i o n to have a s p h e r o i d a l  deformation, as i n the t e x t (B-9a)  (Eq. 4-23):  • 1 + J»"*§(e )_ 1)  ^r_2 f o r a uniform (B-9b)  distribution,  J" fr)  -  /> H(R-r) 0  ,  where . (B'-9c)  n  '--  '  If  X )' s  1  and where H(R-r) i s the well-known. H e a v i s i d e f u n c t i o n .  Prom  154 B - 9 a and  Eqs„  B-9b  t  (B-10) y(r)  PH(R-r)  -  *  4- ^ B r / ^ f R - r )  § ^o p  (  V  k> 2 Y  (  .'  e  whence, (B-ll) 0  To evaluate  0^  y  |r - P'J  the angular I n t e g r a l s i n Eq, B - l l , i t i s  c o n v e n i e n t to note the i d e n t i t y , $  (B-12)  I^TTl where  =  jlo ^ - f ^ m r j *  (r^. } i s the s m a l l e r  U s i n g t h i s i d e n t i t y , we (B-13a) d J2. r  Ji  ( l a r g e r ) o f r and R„ find  that  4c K  r <r  :  1  _ r  4Jt r and  (e»,At)  r  yr  1  ;  that  (B-13b) 2 4n r ' 5" ^  VpJtA)  ,  r>r»  S u b s t i t u t i n g Eqs, B _ 1 3 i n Eq, B - l l , and n o t i n g Eq„ B-9, we  find  that  APPENDIX C  The N u c l e a r O p t i c a l Model and Wave P r o p e r t i e s : B a r r i e r P e n e t r a t i o n , R e f l e c t i o n , A b s o r p t i o n and Resonance  by  Georges Michaud, Department o f Astronomy C a l i f o r n i a I n s t i t u t e of Technology  and  Leonard Scherk and E r i c h Vogt, P h y s i c s Department U n i v e r s i t y o f B r i t i s h Columbia  -  1  -  ABSTRACT  A d e t a i l e d study optical.model  o f t h e wave p r o p e r t i e s o f t h e n u c l e a r  i s presented  to e l u c i d a t e the problem of b a r r i e r  p e n e t r a t i o n by c h a r g e d p a r t i c l e s  a n d t o remove some o f t h e  mystique o f o p t i c a l model c a l c u l a t i o n s . and  the concomitant penetration  The wave  a r e most s t r a i g h t f o r w a r d , f o r  square w e l l s f o rwhich the resonance, r e f l e c t i o n are  easily  ascribed to separate  p r o p e r t i e s o f more g e n e r a l  properties  factors.  diffuse-edge  and p e n e t r a t i o n  We show t h a t t h e wave optical potentials  a s i m i l a r s i m p l i c i t y b y t h e c o n s t r u c t i o n o f an e q u i v a l e n t well  (ESW'J w h i c h h a s t h e same r e s o n a n c e , p e n e t r a t i o n  f a c t o r s as t h e o p t i c a l p o t e n t i a l b u t w h i c h d i f f e r s factor.  A general  c o n s t r u c t i o n o f t h e ESW  to the f o l l o w i n g problems:  (1) t h e v e r y  square absorption  in i t s reflection  i s given  narrow  and  achieve  a n d we a p p l y i t  single-particle  resonances of r e a l o p t i c a l p o t e n t i a l s which occur a t energies f a r b e l o w t h e Coulomb b a r r i e r ( 2 ) in  absorption  t h e p r e s e n c e o f b a r r i e r s ; . (3) t h e c a l c u l a t i o n o f  cross  sections a t a s t r o p h y s i c a l energies  employing o p t i c a l models f i t t e d value  o f the nuclear  .analysis of nuclear penetration properties. absorption in  the nuclear  radius  penetration)  energies;  appropriate  square w e l l f a i l s  may a t t a i n i n t h e d i s t a n t " t a i l "  to y i e l d  to the  a l l the  the corresponding  o f the imaginary  radius.  o f the term  r e a c t i o n rates can y i e l d  about the b e h a v i o u r o f the n u c l e u s a t d i s t a n c e s  beyond.the normal n u c l e a r  ('-I) t h e  I n some c a s e s o f e x t r e m e b a r r i e r  F o r example, cases a r e d e s c r i b e d where the b u l k  the o p t i c a l p o t e n t i a l :  information  at higher  sections  absorption  (extreme b a r r i e r  a n d sum r u l e , l i m i t s  reactions.  the e q u i v a l e n t  to data  cross  much  -  1.  2  -  Introduction The  particularly  '  behaviour those  o f most n u c l e a r r e a c t i o n s a t low energy -  o f i n t e r e s t f o r a s t r o p b y s i c a l systems - i s  d o m i n a t e d b y Coulomb a n d a n g u l a r treatments''"'"^  momentum b a r r i e r s .  o f such r e a c t i o n s employed a simple p i c t u r e , the  "black nucleus"  o r " b l a c k box" p i c t u r e .  In this picture a  b o m b a r d i n g p a r t i c l e was v i e w e d a s p a s s i n g and  Early  through  known Coulomb  a n g u l a r momentum b a r r i e r s up t o t h e n u c l e a r r a d i u s .  n u c l e a r r a d i u s i t was a b s o r b e d b y t h e " b l a c k b o x " .  A t the  This early  p i c t u r e was b a s e d o n t h e s h o r t r a n g e a n d g r e a t s t r e n g t h o f t h e nuclear forces. During  t h e l a s t two d e c a d e s a m i c r o s c o p i c m o d e l o f  n u c l e a r s t r u c t u r e h a s emerged w h i c h h a s changed o u r v i e w o f n u c l e a r r e a c t i o n s and o f b a r r i e r p e n e t r a t i o n . and  I n spite of the short  g r e a t s t r e n g t h o f n u c l e a r f o r c e s i t has been found t h a t i n z e r o -  o r d e r n u c l e i may b e r e g a r d e d moving i n o r b i t s o r s h e l l s of f i n i t e  range.  a s composed o f n e u t r o n s  The s i n g l e - p a r t i c l e o r b i t s o r s h e l l s  a t low e n e r g y e x h i b i t " g i a n t " resonances''^  o f the " s i n g l e - p a r t i c l e  The  neutrons  a t the p o s i t i o n  l e v e l s o f the average p o t e n t i a l w e l l .  r e s o n a n c e s a r e b r o a d e n e d by n u c l e o n - n u c l e o n energy, the observed  a r e those  The v e s t i g e s o f s u c h a  s i n g l e - p a r t i c l e p i c t u r e remain i n nuclear r e a c t i o n s . protons  and p r o t o n s  and i n t e r a c t i n g w i t h moderate p o t e n t i a l s  of an a p p r o p r i a t e average p o t e n t i a l w e l l .  and  range  resonances resemble  The  i n t e r a c t i o n and, a t h i g h those  o f t h e Ramsauer-  Townsend. e f f e c t i n t h e e l e c t r o n bombardment o f a t o m s .  The  o p t i c a l model o f n u c l e a r r e a c t i o n s accounts f o r  7 the observed s i n g l e - p a r t i c l e e f f e c t s . the  I n the o p t i c a l model,  i n t e r a c t i o n o f a bombarding p a r t i c l e and a t a r g e t n u c l e u s i s  depleted  i n terms o f a complex p o t e n t i a l w e l l .  the p o t e n t i a l  refracts  a b s o r b s them.  The r e a l p a r t o f  t h e i n c o m i n g .waves a n d t h e i m a g i n a r y  The m a g n i t u d e o f c r o s s s e c t i o n s i s d e t e r m i n e d b y  the magnitude and shape o f t h e o p t i c a l p o t e n t i a l . and  angular  part  the whole r e a l  potential  o f t h e s y s t e m i s t h e sum o f t h e r e a l p a r t o f t h e o p t i c a l  potential  with  momentum b a r r i e r s  t h e .Coulomb p o t e n t i a l  the present nucleus  and c e n t r i p e t a l p o t e n t i a l .  paper i s t o understand the formation  i n the presence o f b a r r i e r s  absorption yields  are present  When Coulomb  dominate t h e behaviour  o f t h e compound  and o p t i c a l p o t e n t i a l s :  o f waves by t h e i m a g i n a r y  t h e compound n u c l e u s  Our a i m i n  formation  part of the o p t i c a l cross section;  o f t h e wave a m p l i t u d e s  the  potential  the barriers  i n the absorbing  region. The  formation  o f t h e compound n u c l e u s  has the f o l l o w i n g '  aspects  which a r e t r e a t e d i n subsequent s e c t i o n s o f t h i s paper.  1)  On t h e one h a n d . t h e o p t i c a l m o d e l i s a wave m o d e l a n d  deals with  the simple  p r o p e r t i e s o f wave p e n e t r a t i o n ,  reflection  and a b s o r p t i o n .  refraction,  On t h e o t h e r h a n d , t h e o p t i c a l  potential  7 in  common u s a g e  h a v e many p h e n o m e n o l o g i c a l p a r a m e t e r s .  t o u n d e r s t a n d how v a r i a t i o n s it  .is u s e f u l  to describe  i n the parameters a f f e c t  the r e l a t i o n s h i p  t h e b a s i c wave p r o p e r t i e s .  I n order  cross  sections  b e t w e e n t h e p a r a m e t e r s and.  I 1 1  -  II  -j  -  II 2)  Many o f t h e r e a c t i o n s  o c c u r a t such low energy t h a t laboratory.  the d e s i r e d  the nuclear  affecting 3)  they are r a r e l y measured i n the  t h e more a b u n d a n t d a t a a t h i g h e r  an e x t r a p o l a t i o n for  astrophysics  I n s u c h c a s e s i t i s t e m p t i n g t o u s e an o p t i c a l  model which f i t s calculate  of .interest i n  rates.  formula.  energies to  Thus t h e o p t i c a l m o d e l s e r v e s  Our a n a l y s i s  o p t i c a l model a l l o w s  o f t h e wave  us t o u n d e r s t a n d t h e f a c t o r s  the e x t r a p o l a t i o n . The e a r l y b l a c k  n u c l e u s models a r e s t i l l  v e r y l a r g e number o f r e a c t i o n r a t e s calculation.  frequently  2 a n d s e c . 3)  We  square w e l l s . 9  theory of resonance r e a c t i o n s  p o w e r f u l framework f o r d e s c r i b i n g n u c l e a r  been s u c c e s s f u l l y a p p l i e d  to analyzing  and  o f the s t a t i s t i c a l  to the cross  sections  cross  c o n n e c t i o n between  8  and  cross  d e v e l o p such, r e l a t i o n s h i p s  as part- o f a g e n e r a l  d i f f u s e edge p o t e n t i a l s ' a n d  The  cumbersome.  n u c l e u s model and t h e c o r r e s p o n d i n g  o f the o p t i c a l model.  'I)  too  a  i n a single  u s e f u l t o know t h e r e l a t i o n b e t w e e n t h e  s e c t i o n o f the b l a c k  (see.  are required  The o p t i c a l m o d e l i s t h e n o f t e n  i s therefore  sections  as  properties  employed, p a r t i c u l a r l y i n a s t r o p h y s . i c a l c a l c u l a t i o n s where  It  '  i s a very  general  reactions:  i t has  compound n u c l e u s theory of  resonances nuclear  r e a c t i o n s " " " i n w h i c h a v e r a g e s a r e made o v e r t h e compound n u c l e u s 1  resonances.  I t provides a jjustification''*  f o r the d e s c r i p t i o n o f  the  average cross  the  r e s o n a n c e t h e o r y h a s some o f H i e u n d e s i r a b l e  early black  sections  1  box p i c t u r e :  d e f i n i t e nuclear  radius.  b y an o p t i c a l m o d e l p o t e n t i a l .  i trelies  explicitly  features  Nonetheless, o f the  on. t h e u s e o f a  Many o f t h e common r e s u l t s o f t h e r e s o n a n c e  theory  imply  a s q u a r e edge t o t h e n u c l e a r  surface.  I n an  earlier  32 article  one  o i us  tried  o p t i c a l m o d e l c o u l d be  t o show how  t h e d i f f u s e edge o f  accommodated i n the g e n e r a l  the  resonance  theory j  f o r neutron  reactions.  That accommodation i s extended i n  present  article  angular  momentum b a r r i e r s .  5)  In seeking  p h y s i c a l ideas value  to cover  to d e s c r i b e  of the  reactions fermis.  r e a c t i o n s i n v o l v i n g Coulomb  a l l nuclear reactions with  o p t i c a l p o t e n t i a l we  o f the n u c l e a r r a d i u s .  the s i m p l e  A2  ( s e c . 1)  need to re-examine  the  which  black nucleus  p i c t u r e was'employed to d e s c r i b e n u c l e a r 3 3 / 3 l / " * c h o i c e o f r a d i u s was R = l . ' l (A^ + A^ )  the u s u a l  H e r e A.j i s t h e a t o m i c w e i g h t o f t h e The  target nucleus  nuclear  o p t i c a l m o d e l i s d e t e r m i n e d , q u i t e s e n s i t i v e l y by d i f f r a c t i o n patterns  of e l a s t i c  much s m a l l e r , e.g.  R = 1.25  and  the  I n the s e v e r a l decades d u r i n g  t h a t of the bombarding p a r t i c l e .  t o be  the  ] /3 A£ fermis.  measured i n e l a s t i c  s c a t t e r i n g data  f o r nucleons i t s value ( I n t u r n , the n u c l e a r  and  radius of the f i t s  the to  the  and  i t turns  out  is  approximately  charge r a d i u s  scattering i s smaller s t i l l ;  the  as  charge  radius  1/3 i s R = 1.09  A^  fermis.  The  s m a l l difteren.ee of t h i s . r a d i u s  the normal r a d i u s o f the n u c l e o n - n u c l e u s number o f e f f e c t s ;  s u c h as  scope of the p r e s e n t recent is  evidence suggesting  t h e ' r i g h t one.  Why  decades, p a r t i c u l a r l y the  work).  was  interaction  lies  from  in a  core p o l a r i z a t i o n , w h i c h are beyond  the  T h e r e i s an o v e r w h e l m i n g amount o f t h a t the s m a l l e r o p t i c a l model the  radius  e a r l i e r model wrong f o r s e v e r a l  f o r r e a c t i o n r a t e s i n v o l v i n g b a r r i e r s where  c r o s s s e c t i o n , d e p e n d s q u i t e s t r o n g l y on  the  choice, of  nuclear-  -  6  -  radius?  Our e l u c i d a t i o n o f t h e o p t i c a l p o t e n t i a l s u g g e s t s  a fairly  universal explanation.  i m p l i e d a square n u c l e a r  The e a r l y b l a c k - n u c l e u s  model  edge a n d t h e r e f o r e h a d an u n r e a l i s t i c  amount o f wave r e f l e c t i o n .  I t compensated f o r t h i s  o f a b s o r p t i o n by a n a p p r o p r i a t e  attenuation  increase i n the nuclear  radius.  Some p r e l i m i n a r y r e s u l t s o f o u r p r e s e n t i n v e s t i g a t i o n ] 3  were d e s c r i b e d b r i e f l y  i n an e a r l i e r p a p e r .  I n our present  p a p e r we t r y t o " g i v e a c o m p l e t e a c c o u n t o f a l l a s p e c t s o f b a r r i e r p e n e t r a t i o n a n d t h e o t h e r wave p r o p e r t i e s w i t h i n t h e s c o p e o f t h e optical  model. We b e g i n o u r a n a l y s i s ( S e c . 2) w i t h a c o m p a r i s o n  wave p r o p e r t i e s f o r d i f f u s e - e d g e square w e l l s .  o p t i c a l p o t e n t i a l s and f o r s i m i l a r  The m o t i v e f o r b r i n g i n g s q u a r e w e l l s , i n t o t h e a n a l y s i s  .is t w o - f o l d :  first  of a l l ,  the v a r i o u s  w e l l c a n be w r i t t e n i n t h e s i m p l e ,  cross s e c t i o n s f o r a square  familiar  f o r m s w h i c h make p o s s i b l e  e a s y e v a l u a t i o n o f t h e wave p r o p e r t i e s ; s e c o n d l y , t h e b a s i c wave p r o p e r t i e s a r e e a s i l y  f o r a square w e l l  separated - barrier  occ\irs o n l y beyond the square w e l l r a d i u s , r e f l e c t i o n square w e l l r a d i u s and a b s o r p t i o n well radius.  The c o m p a r i s o n y i e l d s  ( S e e . 2)  i s the absorption  o p t i c a l p o t e n t i a l an  e q u i v a l e n t s q u a r e w e l l c a n be d e f i n e d u n i q u e l y .  its  occurs a t the  the dominant r e s u l t o f our  i t i s found t h a t f o r each d i f f u s e - e d g e  of the diffuse-edge  penetration  and resonance w i t h i n t h e square  The v e h i c l e f o r t h e c o m p a r i s o n  cross section. work:  o f the  The  p o t e n t i a l h a v e t h e same s i m p l e  cross  sections  f o r m as t h o s e o f  e q u i v a l e n t s q u a r e w e l l w i t h a l l t h e b a s i c wave p r o p e r t i e s  separated.  clearly  The o n l y q u a n t i t a t i v e d i f f e r e n c e b e t w e e n t h e wave  p r o p e r t i e s o r c r o s s s e c t i o n s o f t h e two w e l l s i s shown t o r e s i d e in  t h e p e n e t r a t i o n and s h i f t f u n c t i o n s .  The c o n v e n t i o n a l  square  w e l l p e n e t r a t i o n and s h i f t  f u n c t i o n s apply  j '  to the diffuse-edge  w e l l when t h e y  a r e m u l t i p l i e d by a r e f l e e t i o . n f a c t o r w h i c h depends  on  t h i c k n e s s ' a n d r e d u c e d mass b u t n o t on t h e c h a r g e ,  the surface  energy or angular  momentum o f . t h e b o m b a r d i n g p a r t i c l e .  Thus t h e  b r e a k d o w n o f t h e o p t i c a l m o d e l i n t o b a s i c wave p r o p e r t i e s i s achieved  by means o f e q u i v a l e n t s q u a r e w e l l s . . I n S e c . 3 we c a r r y o u t , f o r d i f f u s e - e d g e  potentials, and  the e x p l i c i t  determine  optical  c o n s t r u c t i o n o f e q u i v a l e n t square w e l l s  the corresponding  reflection factors.  I n S e c . 1 we  show t h a t o u r a n a l y s i s o f wave p r o p e r t i e s b a s e d on t h e of absorption wells: given  j  cross  sections  ( S e c . 2) a l s o a p p l i e s t o p u r e l y  the sharp resonances o f a diffuse-edge approximately  behaviour  r e a l p o t e n t i a l are  i n terms o f the sharp resonances o f the e q u i -  v a l e n t s q u a r e w e l l when t h e p e n e t r a t i o n f a c t o r a n d s h i f t of the l a t t e r we d e s c r i b e  real  a r e m u l t i p l i e d b y a known r e f l e c t i o n  the behaviour  of general  s e c t i o n s and t h e c i r c u m s t a n c e s  function  factor.  In Sec. 5  o p t i c a l model a b s o r p t i o n  u n d e r w h i c h t h e r e a r e some  cross  departures  o f t h e wave p r o p e r t i e s f r o m t h o s e w h i c h may be d e a l t w i t h i n . t e r m s of e q u i v a l e n t square w e l l s .  I n S e c . 6 we u s e o u r a n a l y s i s o f t h e  b a s i c wave p r o p e r t i e s o f t h e o p t i c a l p o t e n t i a l t o d i s c u s s t h e uncertainties  i n the conventional  approach o f a s t r o p h y s i c s  problem o f extreme b a r r i e r p e n e t r a t i o n . number o f c o n c l u s i o n s the n u c l e a r  I n S e c . 7 we d i s c u s s a  r e s u l t i n g from o u r work i n c l u d i n g t h e p l a c e o f  radius i n the analysis of nuclear  The  to the  o p t i c a l model has a f i r m  reaction  foundation  data.  only f o r nucleon  r e a c t i o n s b u t i t h a s a l s o b e e n f o u n d t o be a v e r y u s e f u l t o o l f o r d e s c r i b i n g heavy i o n r e a c t i o n s .  I t i s more a m b i g u o u s f o r h e a v y  t h e r e s o n a n c e e f f e c t s w h i c h a r e the s t r o n g e s t  ions  s i g n a t u r e o f the o p t i c a ' '  model are the  largely missing  strong  absorption  i n this  and  the  importance f o r heavy i o n s . the b e s t simple t h e mass and properties s e r v e as we  but  i t can  diffuse nuclear  T h e r e f o r e the  accommodate  edge w h i c h a r e  of  o p t i c a l model i s perhaps  model, f o r d e a l i n g w i t h h e a v y i o n s .  Because of  '  c h a r g e o f t h e h e a v y i o n s many o f t h e b a s i c wave  a r e more c o m p l e x t h a n f o r n u c l e o n s .  a u s e f u l measure o f the  s h a l l seele t o be  with  case  general  a p a r t i c u l a r 'case:  success'of  most of our  the  our  Such  reactions  approach.  results w i l l  i n t e r a c t i o n of alpha  be  Although  illustrated  p a r t i c l e s with  32 S.  This  o f the  example o f f e r s extreme b a r r i e r p e n e t r a t i o n  astrophysical reaction rates  to w h i c h our  and  is  typical  r e s u l t s might  be  applied. Recently an  a modified  version  extreme case of b a r r i e r p e n e t r a t i o n  of our  m e t h o d s was  - that of alpha  applied  to.  decay i n heavy  ]' I nuclei. 2.  Other a p p l i c a t i o n s are  s u g g e s t e d i n the  Comparison of the A b s o r p t i o n Cross S e c t i o n s of O p t i c a l P o t e n t i a l s w i t h those of Square W e l l s In order  t h e wave p r o p e r t i e s  c h o o s i n g and  s h a l l compare them w i t h a r e much more p e r s p i c u o u s .  parameterizing  p o t e n t i a l which are  of i n t e r e s t to  terms of the  VO)  •= - V  of d i f f u s e -  square w e l l s We  begin  t h o s e terms o f the modern n u c l e a r  F o r m o s t s c a t t e r i n g and principal  Diffuse-Edge  t o d i s p l a y t h e b a s i c wave p r o p e r t i e s  edge o p t i c a l p o t e n t i a l s we which  following sections.  absorption, cross  (1 + e ^ ~ o ^ ) " R  a  by optical.  us.  o p t i c a l p o t e n t i a l may  r  for  3  1 W(r)  be  sections  the  written  (1)  Here V  q  i s the " d e p t h " o f t h e r e a l p a r t o f t h e p o t e n t i a l and has  a value i n the neighbourhood more f o r h e a v y 1.25 A  ] /3  ions; R  o f 50 MeV  i s t h e n u c l e a r r a d i u s whose v a l u e i s a b o u t o  fm f o r n u c l e o n s  i  (where A i s t h e a t o m i c w e i g h t o f t h e  t a r g e t n u c l e u s ) and a s l i g h t l y the  f o r n u c l e o n s and c o n s i d e r a b l y  l a r g e r v a l u e f o r heavy  ions;  a is  " s u r f a c e t h i c k n e s s " w h i c h h a s a v a l u e o f a b o u t 0.5 fm f o r  n u c l e o n s and heavy  i o n s ; I W(r) i s t h e i m a g i n a r y term o f t h e o p t i c a l  p o t e n t i a l which leads to absorption. c h o s e n _ t o have a shape  either like  .W(r)' = - W  (1 +  e  The i m a g i n a r y t e r m i s u s u a l l y  that of the r e a l  ^  V  ^  )  "  term,  (2)  1  w h i c h i s c a l l e d "volume a b s o r p t i o n " o r , a l t e r n a t i v e l y , W(r) i s chosen  t o be l a r g e r  i n the region of the n u c l e a r surface  choice i s c a l l e d "surface absorption"). d i s c u s s i o n we w i l l in  S e c . 5 we w i l l  W(r) has  I n t h e main p a r t o f o u r  c h o o s e v o l u m e a b s o r p t i o n i n o r d e r t o be d i s c u s s i n some d e t a i l  on t h e wave p r o p e r t i e s .  specific:  the. e f f e c t o f t h e s h a p e o f  The d e p t h p a r a m e t e r , W  , typically  a v a l u e o f 2-5 MeV f o r n u c l e o n s a n d somewhat l a r g e r v a l u e s f o r  heavy  ions. We h a v e c h o s e n  t o c o n c e n t r a t e on t h e p r i n c i p a l  (1) , o f t h e o p t i c a l p o t e n t i a l b e c a u s e  For  example,  JL . _s (where  we  properties.  the s p i n - o r b i t c o u p l i n g term which i s p r o p o r t i o n a l t o X i s the o r b i t a l  makes t h e o p t i c a l . m o d e l p h a s e =  terms,  the other terms, which  n e g l e c t , do n o t m o d i f y o u r m a i n r e s u l t s a b o u t wave  Q  (this  + s) .  a n g u l a r momentum a n d £ t h e s p i n ) s h i f t s ' d e p e n d on b o t h J and j  Our t r e a t m e n t below  m u s t t h e n a l s o be c a r r i e d o u t  s e p a r a t e l y f o r e a c h v a l u e o f X. a n d j . - S i m i l a r l y  the i s o t o p i c  spin  t e r m when i m p o r t a n t makes i t n e c e s s a r y t o t r e a t n e u t r o n s a n d p r o t o n s  o n a common f o o t i n g a n d l e a d s contributions We c o u l d  s u c h as t h e "cjuas.iclas.tic  section  ( p , n)  t h e wave p r o p e r t i e s  only with  we w i l l  ignore  such  However  refinements  t h e dominant terms o f ( 1 ) .  The o p t i c a l p o t e n t i a l o f  (1) ( o r Saxon-Woods p o t e n t i a l  w h i c h t h e p a r t i c u l a r w e l l shape o f  (1) i s f r e q u e n t l y  be shown t o e x h i b i t a l l o f t h e b a s t e wave p r o p e r t i e s : penetration,  r e s o n a n c e , r e f l e c t i o n and a b s o l u t i o n .  straightforward absorption  manifestation  cross  ;  reactions.  a l s o accommodate s u c h t e r m s i n o u r a n a l y s i s .  to c l a r i f y and d e a l  to unusual cross  o f these p r o p e r t i e s  s e c t i o n and t h e r e f o r e  called)  will  barrier  The m o s t occurs i n the  we c h o o s e t o b e g i n o u r  analysis with i t . For  any r e a c t i o n c h a n n e l  cl (  p r o d u c t s and. t h e i r s t a t e o f e x c i t a t i o n ) p o t e n t i a l w i t h phase s h i f t s  '  S  labels the p a i r of reaction described  the absorption  by an o p t i c a l  cross  section i s :  CT.(abs) •= Or A ) £ , ( 2 * + 1) T (oQ  (3)  2  where T><)  S i -  The T f ' i ) a r e c a l l e d , n u c l e a r properties mission  of the absorption  functions  2i6 e  2 (<!)  W  transmission cross  or, equivalently,  functions.  s e c t i o n are those o f the transthose o f t h e phase  Therefore the numerical c a l c u l a t i o n of the absorption is  straightforward  equation containing the  A l lof the  - one n e e d s m e r e l y t o i n t e g r a t e  shifts. cross  section  the Schroedinger  the o p t i c a l p o t e n t i a l to a large  distance  where  d e c o m p o s i t i o n , o f t h e wave f u n c t i o n , i n t o i n c o m i n g a n d o u t g o i n g  waves y i e l d s t h e phase s h i f t .  B u t t h e n u m e r i c a l s o l u t i o n does n o t  display  t h e wave p r o p e r t i e s .  transmission  functions  The depth - ( V  q  To s e e t h e s e we f i r s t  o f a complex  transmission  * For a derivation  o f a complex  square w e l l o f  R ^ , may be w r i t t e n *  o f t h i s r e s u l t and a d e t a i l e d d i s c u s s i o n  nuclear transmission  functions  'I  1"  P, 1  (I - S f J ;  e  + P,f,J ) m  i s the usual penetration  of  s e e p a g e s 281-29 5 o f r e f e r e n c e 10  • ^ ( O "=  Here  square w e l l p o t e n t i a l .  function  H- i W ) , a n d r a d i u s  examine t h e  >  _ 2  :  1  () 5  + O^T  +  ^^'V  f a c t o r used i n n u c l e a r  reaction  studies,  where  F, a n d G  are, respectively,  C o u l o m b wave f u n c t i o n s s h i f t functions  the regular  o f t h e c h a n n e l <X .  of nuclear reaction /  and i r r e g u l a r  The  are the usual  theory  r F^ dE, / d r + r G., dG^ / d r j  S. = - b  i n which the b ^  (7)  R., a r e boundary  resonance p r o p e r t i e s f  (  f  + i f  t h e wave f u n c t i o n way  • •  condition  of the w e l l related  numbers d e t e r m i n e d , b y t h e  (see b e l o w ) .  F i n a l l y , we h a v e  to the l o g a r i t h m i c  i n s i d e the square w e l l , j  derivative of  (\- ) ' ^ r  n  ^  e  following  The f  are complex numbers because the K . are -  2 m . .  K =  The q u a n t i t i e s f.,  }  '-\\  (E + V •  . 2  refer  "  i  + i WJ/ * '  0  -  "  Q  (9)  .  j  to the p r o p e r t i e s o f the square w e l l  w h i l e the q u a n t i t i e s P  and S, r e f e r  The t r a n s m i s s i o n f u n c t i o n s  (5)  to the e x t e r n a l  properties.  c l e a r l y e x h i b i t a b s o r p t i o n and  b a r r i e r penetration i n separate f a c t o r s . resonance  comple)  .The m a n i f e s t a t i o n  and r e f l e c t i o n r e q u i r e s f u r t h e r To e x h i b i t the " r e s o n a n c e s "  of  analysis.  o f the t r a n s m i s s i o n  functions  o f the square w e l l we take the r e a l p a r t o f the p o t e n t i a l and f i n d . the n o r m a l modes or r e s o n a n c e s .  The n o r m a l modes are s o l u t i o n s  the Sehroed.inger e q u a t i o n , w h i c h vanish, a t the o r i g i n and  satisfy  a s u i t a b l e boundary c o n d i t i o n a t the square w e l l r a d i u s . w r i t e the s o l u t i o n s as A c o n s t a n t and K , ( px v  ~ f 2 m . : h ~ L o-  i , (K , r) where A (E „ + V ) P'o.i  ~) 2  J  of  I f we  . i s a normalizing i s the d i s c r e t e  value  o f the wave number f o r w h i c h the f o l l o w i n g boundary c o n d i t i o n i s satisfied: r d j / d r = b ^ -} x  A  J  (10)  r = R ,  The c o r r e c t v a l u e o f the boundary c o n d i t i o n number b ^ i s w h i c h makes the s h i f t f u n c t i o n ( 7 ) ]  v a n i s h a t the energy  that  of  2  i n t e r e s t . ""  The normal modes have s i n g l e p a r t i c l e F J  The f  '= pA  K 2 m ^  2  pi  , were g i v e n q u i t e g e n e r a l l y  B e s s e l f u n c i : i o n s o f complex arguments:  i n terms o f  spherical  a l t e r n a t i v e l y we can g i v e  them i n terms o f the n o r m a l modes o r . r e s o n a n c e s the p o t e n t i a l .  energies  By u s i n g the completeness  o f the r e a l p a r t of  of the resonances we  find:  - 13 2 f  f  Re  l  m  ,  -  '  ,  P stands  for  2  W  -I-  Xp  -I W  f  p  1  o  s.fEndE-  J  2  (11)  r  2  v  - E )  P  :~  °  (E, -E o) Ap  P  2  v  w  —'-l'  ( E ^ E ) Y,  =  (E where  Y  =  v*.  y  E' - E  ~ oo  0  the p r i n c i p a l v a l u e  of  the  improper  integral  2  a n d wher.e Y, i s the s i n g l e p a r t i c l e reduced w i d t h . I f we c h o o s e AP t h e b o u n d a r y c o n d i t i o n number t o be b - - X - w h i c h w o u l d be t  correct  for  2  neutral particles  2  Yjjp =  "11 /m ,  .  The q u a n t i t y  the  strength  The n u c l e a r a resonance  data.  structure.  -  then  U s i n g the  (S) - o r i n t h e  There  i s much e v i d e n c e  for  closer  usually  ions  so  that  absorption cross  are  expected  of-W  the  are  exhibits  of f^'  in  n  section  to appear  them i n l o w e n e r g y  the v a l u e s  together  (11) c l e a r l y  corresponding value  also  values  is  (11)  function of  these resonances  For heavy  of E  s^ d e f i n e d by  that  absorption.  exhibit  limit  function.  strength  transmission function  (3) - shows the  low energy  2  r e f e r r e d , t o as  the  i n the  larger  d a t a does n o t  in  nucleon and  the  usually  resonances. The r e f l e c t i v i t y  penetration  factor.  of  the  s q u a r e w e l l c a n be t r a c e d  For X ~ 0 absorption of n e u t r a l  to  the  particles  we h a v e no Coulomb o r a n g u l a r momentum b a r r i e r s a n d t h e r e f o r e t h e p e n e t r a t i o n f a c t o r , (6) , becomes P ~ k R . Near t h r e s h o l d , the o ^ ci 1 s t r e n g t h f u n c t i o n remains f i n i t e b u t ' P v a n i s h e s as E . This r  v  J  2  vanishing of P approaching  o  c a n be  traced  to  total  the square w e l l r a d i u s  reflection  from e i t h e r  of  side.  t h e wave  -  -  1.1-1  Having displayed absorption, resonance the  and. r e f l e c t i o n  f o r square  corresponding quantities  results  above.  I t might appear  decomposition of There  the  i s no f e a t u r e  not apply  diffuse-edge reasonable  the  of  evaluate  t h e m o d i f i c a t i o n o f P,  f  , for  the  and e v a l u a t e  decompose  exhibit  resonances  the  matching radius? functions,  wave  the as  above.  energies,  a  w h i c h does  radius  by the  logarithmic  choose  of  the  derivative,  that matching r a d i u s .  factors,  reside  to  the  strength  d e p e n d on w e l l  will  Indeed,  o f any w e l l  B u t , how do we c h o o s e  etc.  a  (taking  tail  logarithmic derivative  to these questions  properties?  our a n a l y s i s  of  properties. Before  we i l l u s t r a t e  a d a p t i n g our e x p o s i t i o n to diffuse-edge  the  decomposition  (5) by t h e " b l a c k n u c l e u s " at  (S)  that  How do t h e p e n e t r a t i o n  resonance  I n our answers  the  complex p o t e n t i a l a t  we c a n e v e n  for  the  in ( 5 ) .  We c a n a l w a y s  a n d S, X  K  complex p o t e n t i a l ) ,  to begin w i t h  decomposition of  P . a n d S,  compare  wells with  t r a n s m i s s i o n f u n c t i o n as of  penetration,  w e l l s we now seele t o  t o an a r b i t r a r y w e l l s h a p e .  matching radius, account  for  barrier  all.  waves f o r  In this r <  case i t .  is  of  the  potentials  t r a n s m i s s i o n f u n c t i o n as  model - which i s not a p o t e n t i a l assumed t h a t t h e r e - i Kr  R e p l a c i n g j . by e  in  we t h e n h a v e OU  KR^  (8)  are  only  in  model  incoming  and c h o o s i n g b  = 0  - 15 -  Using  this value  of f  cU  (5) we o b t a i n ' t h e b l a c k  xn  nucleus  t r a n s m i s s 1 o n f u n c t i on s  (1 -I- P f / K R J  Uk/K ( 1 + k/K) At  low e n e r g i e s  b l a c k nucleus  + (s^/Kiy  2  (HI)  2  f o r s-wave  2  (k<< K) a l l t h e t r a n s m i s s i o n f u n c t i o n s o f t h e  model  (even those  f o r s-wave n e u t r o n s )  are very  s m a l l b e c a u s e o f wave r e f l e c t i o n o r wave p e n e t r a t i o n . energies  (P ~->kR  ,. k — > K) t h e b l a c k n u c l e u s  f u n c t i o n s a p p r o a c h t h e i r maximum v a l u e Like reflection  than a r e a l nucleus.  is  of unity.  which i s treated l i k e  The b o m b a r d i n g wave  a resonating cavity.  t u n e d so t h a t waves p r o p a g a t e i n w a r d t h e g i v e n wave p r o p a g a t i o n  this  i s not easily  difficult  number K.  accomplish J u s t as i t  c a v i t i e s which a r e tuned i n a c e r t a i n  potentials  f o rwhich the black nucleus  energies.  The b l a c k n u c l e u s  t o make  optical-model  conditions apply  ata l l  transmission functions completely  resonances of the square-well:  functions oscillate  The c a v i t y  The t u n i n g t o  d u p l i c a t e d by a p o t e n t i a l w e l l m o d e l .  to build resonating  approaches  a t t h e c a v i t y e n t r a n c e , R.  way f o r a l l w a v e l e n g t h s , i t i s i m p o s s i b l e  the  h a s more '  To u n d e r s t a n d t h e wave a n a l y s i s  theory.  with  is  transmission  we c a n c o m p a r e i t t o a wave g u i d e problem"'""'  c l a s s i c a l electromagnetic  the nucleus  At higher  the square w e l l model, the b l a c k nucleus  of the black nucleus in  neutrons.  the square w e l l  about t h e b l a c k ' n u c l e u s  lack  transmission  transmission  function  - l.G -  ! " ii  i i f we c h o o s e K t o h a v e t h e same v a l u e f o r b o t h .  !  The two  t r a n s m i s s i o n f u n c t i o n s h a v e t h e same mean v a l u e .  We 'show  below t h a t the t r a n s m i s s i o n f u n c t i o n s o f a r e a l i s t i c  nuclear  o p t i c a l p o t e n t i a l h a v e a l a r g e r mean v a l u e  o f the  than both  '  ' i a b o v e m o d e l s b e c a u s e t h e d i f f u s e n u c l e a r edge g i v e s reflection  than  a square edge.  same u n r e a l i s t i c  The b l a c k n u c l e u s  less  s u f f e r s the  r e f l e c t i o n b e c a u s e o f the' s u d d e n c h a n g e i n t h e  wave number a t R,.  Thus t h e " b l a c k n u c l e u s " model f a i l s  c o n s i d e r a b l e degree i n i t s main o b j e c t i v e  to a  of optimizing nuclear  absorption. Much o f t h e a b o v e wave a n a l y s i s c a n be a d a p t e d a t o n c e to  a realistic  decomposition  nuclear optical potential.  than  P,, S , a n d f X  < are modified,  i s no l o n g e r o b v i o u s .  still  and t h e c h o i c e  aA  x  of a matching r a d i u s a t which these  an a r b i t r a r y  (5)  o f t h e t r a n s m i s s i o n ' f u n c t i o n as i n  a p p l i e s except  evaluated  F o r example, the  -.  .  t h r e e q u a n t i t i e s a r e t o be  The d e c o m p o s i t i o n  i s valid for  choice o f matching r a d i u s but f o r our purposes a  well-defined choice w i l l r a d i u s we c a n f i n d  t u r n o u t t o be u s e f u l .  F o r an a r b i t r a r y  P,, a n d S. f r o m t h e l o g a r i t h m i c d e r i v a t i v e o f X  X  an o u t g o i n g wave Cl, , d[  •  ! kZ? = rdF, /dr I r dG, /dr +  =  S  - l b ,  -I- I P„  ^  (for  f a t the matching  s i m p l y by m a k i n g t h e a n a l y t i c c o n t i n u a t i o n o f t h e l o g a r i t h m i c derivative is  still  t o t h e m a t c h i n g r a d i u s as' i n d i c a t e d .  g i v e n by  (8)  i f , i.Ln  Similarly,  (8) , we r e p l a c e j  f  w  by t h e  .  r)  (15)  radius)  - 17 -  appropriate  r e g u l a r wave f u n c t i o n o f t h e o p t i c a l  T h e r e a r e two i m p o r t a n t choice  o f matching radius  potential.  f a c t o r s w h i c h n a r r o w down t h e  i n s u c h a wave a n a l y s i s .  F i r s t of a l l ,  i the n u c l e a r typically  surface  a/R  in  of r e a l i s t i c potentials i s s t i l l (1) h a s a v a l u e  o f 0.1.  quite  sharp:  Secondly, the  resonance c o n d i t i o n s  t h e m s e l v e s do n o t a l l o w much f r e e d o m o f  Choice..  t h e n u c l e u s a s a wave g u i d e we t h i n k o f  Considering  s e t t i n g up n o r m a l modes o r r e s o n a n c e s . definition which  I f we i n c l u d e i n o u r  o f t h e wave g u i d e p a r t o f t h e t r a n s m i s s i o n  l e a d i n t o i t t h e n t h e r e s u l t i n g l a r g e •- a r t i f i c a l l y  dimension o f the guide give  i t unusual properties.  one-mode a p p r o x i m a t i o n  Breit-Wigner radius  formula  i s poor.  b r e a k s down.  to l i e reasonably  close  large -  E v e n when o u r  w a v e l e n g t h i s c l o s e t o t h a t o f one o f t h e a r t i f i c i a l the  channels  n o r m a l modes  I n the nucleus the o n e - l e v e l We m u s t c h o o s e t h e m a t c h i n g  to R .  12 I n an e a r l i e r work neutrons there  e x i s t e d a very  one o f u s showed t h a t f o r s-wave simple  correspondence between the  o p t i c a l p o t e n t i a l (1) and. a s q u a r e w e l l , o f v e r y radius.  F i g . 1 shows t h e i m p o r t a n t  With the r a d i i so  features  nearly  t h e same  o f the correspondence.  t h e same, t h e d e p t h o f t h e s q u a r e w e l l i s a d j u s t e d  that the resonances near zero  With this choice  i t turns  e n e r g y c o i n c i d e f o r t h e two w e l l s .  out that a l l of the resonance  energies  and  a l l o f t h e r e d u c e d w i d t h s o f t h e two w e l l s a r e t h e same.  two  w e l l s h a v e e x a c t l y t h e same r e s o n a n c e p r o p e r t i e s .  But the  penetration  f u n c t i o n s a n d s h i f t f u n c t i o n s o f t h e two w e l l s a r e  different.  F o r the square w e l l the bracketed  equal b  The  term o f  (7) i s  t o X a t low energy and t h e r e f o r e i t i s n a t u r a l '  t o choose  = 0  s o t h a t S. = 0 .  For the diffuse-edge  well,  one  obtains  - 18 -  b, . by c a l c u l a t i n g  the a n a l y t i c c o n t i n u a t i o n o f the b r a c k e t e d  t e r m a n d by c h o o s i n g b , s o t h a t S .. v a n i s h e s a t low e n e r g y . correspondence of F i g . 1 i s obtained w i t h penetration factors  this  choice.  The  The  o f t h e too w e l l s d i f f e r by a c o n s t a n t f a c t o r , 1(3  f, .f  which  i s n e a r l y energy  As shown f i r s t by P e a s l e e  ,  c a n be w e l l a p p r o x i m a t e d by f — TT K a c o t h (IT K a)  w h i c h has  a v a l u e o f a b o u t 2.5  common u s e f o r n u c l e o n s . in  independent.  The  f o r the o p t i c a l p o t e n t i a l s i n f a c t o r f accounts f o r the  r e f l e c t i o n o f the d i f f u s e - e d g e and For  (16)  difference  square-edge.  s~wave n e u t r o n s t h e c o r r e s p o n d e n c e b e t w e e n  the  o p t i c a l p o t e n t i a l and i t s e q u i v a l e n t s q u a r e w e l l e x t e n d s t o a l l c a s e s , from r e a l w e l l s w i t h narrow w i t h b r o a d peaks  r e s o n a n c e s t o complex  i n the t r a n s m i s s i o n f u n c t i o n s .  s q u a r e w e l l r e s u l t s a p p l y .to d i f f u s e w e l l s the  factor f.  real  the d i f f u s e - e d g e w e l l s .  b e c a u s e we  merely  The  i s established  c o r r e s p o n d e n c e o f F i g . 1. was I t e x t e n d s t o complex  potentials  c a n i n c l u d e an i m a g i n a r y t e r m o f s q u a r e s h a p e  (12) a p p l y t o t h e d i f f u s e - e d g e w e l l a l s o .  i m a g i n a r y term i s o f l i t t l e reflection  consequence  from the i m a g i n a r y term  because  The  shape  there i s  i n the .  Thus  of the  little  (see, however, Sec. 5 b e l o w ) .  t r a n s m i s s i o n f u n c t i o n s a r e much s m a l l e r t h a n u n i t y  w e l l approximated  multiply  the  l o g a r i t h m i c d e r i v a t i v e s i m p l y by r e p l a c i n g E by E -- i W  (11) a n d  the  a l s o i f we  Thus t h e w h o l e wave a n a l y s i s  established f o r purely real wells. too  I n a l l cases the  s q u a r e w e l l p e n e t r a t i o n f u n c t i o n a n d s h i f t f u n c t i o n by  reflection for  potentials  they can  Mien be  by T U  'ITT  P, s., f  (17)  Here they  clearly  show t h e common r e s o n a n c e a n d p e n e t r a t i o n  factors of diffuse-edge  and square w e l l s and the r e f l e c t i o n  f a c t o r w h i c h d i s t i n g u i s h e s them. The e a r l i e r  a n a l y s i s a p p l i e d t o s-wave n e u t r o n s and  .  it  i s n o t a t a l l c l e a r t h a t i n t h e p r e s e n c e o f Coulomb b a r r i e r s  it  i s p o s s i b l e t o decompose t h e t r a n s m i s s i o n f u n c t i o n s i n t o  p r o p e r t i e s i n q u i t e t h e same s i m p l e  I t t u r n s o u t t o be s o  for  protons  are  shown w e l l b e l o w t h e Coulomb b a r r i e r f o r d-wave p r o t o n s .  radii  as we show o n F i g . 2.  way.  wave  Here t r a n s m i s s i o n  functions The  o f a l l t h e w e l l s a r e t h e same b u t t h e w e l l d e p t h s a r e  a d j u s t e d * , t o make r e s o n a n c e e n e r g i e s  coincide roughly.  * The a d j u s t m e n t p r o c e d u r e i s t h e f o l l o w i n g : p a r t o f the o p t i c a l p o t e n t i a l the J  Although  I f V(r) i s the r e a l  V(r) r dr i s taken  t o be  o  constant there  for a l l potentials.  i s s t r o n g b a r r i e r p e n e t r a t i o n h e r e the r e l a t i o n between the  various  transmission functions i s s t i l l It  i s n o t d i f f i c u l t t o u n d e r s t a n d why  works f o r protons. potential  that of (17). t h e same p r e s c r i p t i o n  F i g . 3 shows t h e sum o f t h e n u c l e a r  f o r the o p t i c a l model and i t s e q u i v a l e n t square  The p o i n t i s t h a t t h e Coulomb p o t e n t i a l v a r i e s l i t t l e region  (encircled)".  Within  exactly like  case.  approximate  'Then o u r r e f l e c t i o n  t h a t f o r s-wave n e u t r o n s o f n e g a t i v e  w h e r e B 'is t h e b a r r i e r h e i g h t . this  constant.  well.  i n the surface  t h e s u r f a c e r e g i o n we m i g h t  t h e Coulomb p o t e n t i a l a s b e i n g is  a n d Coulomb  The e a r l i e r  problem  energy E - B  a n a l y s i s would apply to  - 20 -  '  I  I  • I t was s u r p r i s i n g t o u s t h a t t h e s i m p l e between d i f f u s e - e d g e down c o m p l e t e l y  w e l l s and square w e l l s  f o r heavy i o n s .  functions  factor i n this  varies strongly with  '-IS +  functions  He.  The e x p e c t e d to  reflection  (17) .  i s much l a r g e r .  But the  Moreover i t  t h e e n e r g y a n d d e p e n d s s t r o n g l y on t h e  a n g u l a r momentum. There i s . a simple  heavy ions We  appeared t o break  ?  c a s e i s f = '1.6 a c c o r d i n g  •ratio of transmission  orbital  lor  relationship  3 i g . '1 shows a s i m i l a r r a t i o o f  32 transmission  I  i s restored  explained  way i n w h i c h t h e ' w a v e a n a l y s i s f o r  to the s i m p l i c i t y  that the matching radius  of that f o r nucleons.  f o r the o p t i c a l p o t e n t i a l  had  t o be c h o s e n t o be r e a s o n a b l y c l o s e t o R , t h e m i d p o i n t o f  the  surface .  B u t i t d o e s n o t h e e d t o be e x a c t l y R  • n e x t s e c t i o n we show t h a t i f we c h o o s e R , = R cA o /S.R c l e a r l y as and  defined  those o f  and e v a l u a t e d  - we a g a i n  . I n the o + /JS R - w i t h  g e t r e s u l t s as  simple  ( 1 7 ) . The r o l e o f /\R i s i m p o r t a n t f o r h e a v y  Ions  n o t f o r n u c l e o n s b e c a u s e t h e more m a s s i v e h e a v y i o n s h a v e wave  . f u n c t i o n s which, o s c i l l a t e more r a p i d l y ~ t h e "wave g u i d e "  commences  earlier. ' 3-  The C o n s t r u c t i o n For  o f Eqiafval.ent Square W e l l s .  a r b i t r a r y o p t i c a l p o t e n t i a l s of the kind  wish to construct  equivalent  (1) we  s q u a r e w e l l s . (ESW) i n o r d e r  may d i s p l a y t h e b a s i c wave p r o p e r t i e s to the approximate view o f nuclear Coulomb b a r r i e r s h a v e no g r e a t diffuse-edge  (ESW)  o f the p o t e n t i a l .  reactions  depicted  that  According  on F i g .  e f f e c t on t h e r e f l e c t i o n o f a  optical potential.  we  N e i t h e r , do a n g u l a r momentum  3,  barriers  as we  shall  wave p r o p e r t i e s  show b e l o w .  The  problem  of displaying  i s t h e n r e d u c e d t o t h e c a s e w h e r e we  neutral particles.  12  A c c o r d i n g t o e a r l i e r work  the  h a v e s-wave  t h e ESW  f o r the  i c a s e o f s-wave n e u t r a l p a r t i c l e s the p a r t i c l e s potential.  i s independent  or o f the a b s o r p t i v e (imaginary) term i n the  We  c a n t h e r e f o r e c o n s t r u c t t h e ESW  o f z e r o - e n e r g y s-wave n e u t r a l p a r t i c l e s o f the o p t i c a l p o t e n t i a l . o f t h e ESW  The  i n o n l y the r e a l  to d i s p l a y  i s u n i v e r s a l and  radius (1)  has  fix  .  On  only-two parameters:  has  S i n c e t h e ESW  i t s depth  and i t s  the o t h e r hand the r e a l p a r t of the o p t i c a l V , Q  R  and a.  We  n e e d two  is  potential  conditions  to  .  e x a m p l e , i f we  at zero-energy:  do, i n p r o c e e d i n g t h r o u g h t h e p e r i o d i c  o c c u r a t z e r o - e n e r g y w h e n e v e r t h e r a d i a l wave  z e r o d e r i v a t i v e beyond the w e l l . '  a f f e c t t h e s c a t t e r i n g c r o s s s e c t i o n we the resonances  for  k e e p t h e d e p t h o f e i t h e r w e l l f i x e d and v a r y  ( a s we w o u l d  resonances  i n the  proceed, f o r a l l masses, i n  Both w e l l s can have resonances  radius  to the  o f . m a s s m,  shown f o r n e u t r o n s on F i g . 1.  three parameters: and. R  we  t o what e x t e n t i t  t h e c o n s t r u c t i o n o f t h e ESW  r e a l p a r t o f t h e o p t i c a l p o t e n t i a l we  p u r e l y r e a l i t has  factor  In later sections  c a s e o f z e r o - e n e r g y s-wave n e u t r a l p a r t i c l e s ,  t h a t was  part  t h e b a s i c wave p r o p e r t i e s f o r a l l c a s e s .  Having reduced  t h e way  scattering  are expected to apply to a l l  t h e o t h e r , more c o m p l i c a t e d c a s e s as w e l l . show t o w h a t e x t e n t t h e ESW  u s i n g the  of  optical  d e p t h , r a d i u s and r e f l e c t i o n  c o n s t r u c t e d i n t h i s way  e n a b l e s us  o f the energy  i n t h e two w e l l s  to have e q u i v a l e n t p r o p e r t i e s .  t o be Making  the table)  function  S i n c e the resonances  greatly  must choose the p o s i t i o n  of  t h e same i f we w a n t t h e w e l l s the p o s i t i o n s o f the  resonances  -  22 -  c o i n c i d e f i x e s one o f t h e two p a r a m e t e r s A second  o f t h e ESW.  c o n d i t i o n i s o b t a i n e d from the w i d t h s o f the  s c a t t e r i n g resonances.  A conventional, diffuse-edge p o t e n t i a l  h a s much l a r g e r w i d t h s t h a n a s q u a r e w e l l , a f a c t w h i c h associate with reflection. with  A=  F o r e a c h w e l l we c a n w r i t e  we the width  0 i n o u r c a s e , w h e r e n i s t h e number o f n o d e s o f  t h e s-wave r e s o n a n t wave f u n c t i o n )  as a p r o d u c t  of a penetration  f a c t o r and a r e d u c e d w i d t h P w h e r e t h e two f a c t o r s r a d i u s , R.  o  2 no  (18)  a r e each e v a l u a t e d a t a s u i t a b l e  F o r the square w e l l  c h o o s e R = R, a n d f o r t h i s  I  Y  matching  i t i s n a t u r a l and n e c e s s a r y t o  c h o i c e we h a v e P  .  = kR.. a n d Y  o  1  = ii  '  no  F o r t h e d i f f u s e - e d g e p o t e n t i a l we c a n e a s i l y  e x h i b i t a resonant-  wave f u n c t i o n - as on F i g . 5 - by v a r y i n g R  until  has in  t h e wave  /mR-, .  1  function  z e r o d e r i v a t i v e f a r beyond the w e l l , b u t t h e r e i s a r b i t r a r i n e s s t h e c h o i c e o f the. m a t c h i n g r a d i u s R .  The w i d t h i t s e l f i s  i n d e p e n d e n t o f the c h o i c e o f R b u t the p e n e t r a t i o n f a c t o r and reduced w i d t h a r e n o t . A t t h i s s t a g e we make a c h o i c e * . We w i s h t o a s s o c i a t e * The c h o i c e i s n o t e n t i r e l y an a r b i t r a r y one - no more t h a n t h e choice o f the matching r a d i u s i n the R~matrix -theory, f o r which the r a d i u s m u s t be t h a t o f t h e a c t u a l n u c l e u s i f t h e r e s o n a n c e are t o have r e a s o n a b l e convergence  formulae  properties.  the d i f f e r e n c e i n widths - the r e f l e c t i o n - w i t h  the d i f f e r e n c e i n  p e n e t r a t i o n o f t h e two w e l l s a n d t h e r e f o r e we c h o o s e a m a t c h i n g R, s u c h t h a t t h e o p t i c a l p o t e n t i a l a n d i t s ESW h a v e t h e same w i d t h s , a t t h e common m a t c h i n g r a d i u s .  Fixing  radius,  reduced  t h e p o s i t i o n a n d reduced.  w i d t h o f -the ESW d e t e r m i n e s V , a n d R.,  . •  The a p p l i c a t i o n o f t h e a b o v e two c o n d i t i o n s c o n s t r u c t i o n o f t h e ESW  i s i l l u s t r a t e d on F i g . 5 .  o p t i c a l p o t e n t i a l we f i x V J  o  and a and t h e n v a r y R o J  to the  For a given to find a l l  of  the resonances.  R  a n d t h e c o r r e s p o n d i n g r e s o n a n t wave f u n c t i o n s a s shown on t h e  q  figure.  We t h e n h a v e a d i s c r e t e s e t o f v a l u e s o f  N e x t , we b e g i n w i t h a s q u a r e w e l l w i t h an i n i t i a l  choice  of  R ^ ( ^''RQ) J a n d f i - V . ^ s o t h a t we a l s o g e t a z e r o - e n e r g y  in  t h e s q u a r e w e l l h a v i n g t h e same number o f n o d e s a s t h e d i f f u s e  resonance  x  edge r e s o n a n c e .  For this i n i t i a l  c h o i c e we compare r e d u c e d  of  both w e l l s  a t t h e same m a t c h i n g  r a d i u s , R(= R j ) .  is  t h e r a d i a l wave f u n c t i o n o f t h e r e s o n a n c e  If  ^  /  /  widths N  O  (  R  0  the reduced w i d t h i s  ,1.2  g i v e n by  It  i s the square  at  the matching  o f t h e a m p l i t u d e o f t h e n o r m a l i z e d wave radius.  I f t h e two w e l l s  reduced width f o r the i n i t i a l  coincide  exactly) u n t i l  do n o t h a v e t h e same  c h o i c e o f R , we v a r y R^  a d j u s t i n g V.^ so t h a t t h e resonance they do.  function  positions  (always  o f t h e two w e l l s  I n t h i s way we a c h i e v e a u n i q u e  •value o f t h e d e p t h and r a d i u s o f t h e o p t i c a l p o t e n t i a l . c o n s t r u c t i o n holds only f o r those values o f R r e s o n a n c e , b u t we c a n i n t e r p o l a t e  which  This  correspond to  the c o n s t r u c t i o n i n between  resonances. The r e f l e c t i o n c o n s t r u c t i o n o f t h e ESW. |~  o  factor  i s a d i r e c t r e s u l t o f t h e above  I t c a n be d e f i n e d , b y  (diffuse well)  =  f  f  (square w e l l )  (20)  -  Since  2'-l  -  t h e r e d u c e d w i d t h s o f t h e two w e l t s  are- t h e same  this  yields P  The  (diffuse well)  O  (20)  comparison of  -  a n d (21)  f  P  q  employed  (square well)  - f K R.  (21)  'natural* widths - not \ \  the  observed widths o f narrow resonances.  We  discussion i n Sec. 1 o f the r e l a t i v e values widths f o r various • We directly latter  a complete  o f observed and n a t u r a l  wells.  can evaluate  from  give  the d i f f u s e - w e l l penetration  (6) i f . we r e p l a c e  F^ a n d G , by F^ a n d  a r e the- r a d i a l wave f u n c t i o n s  behave a t l a r g e d i s t a n c e  factor where the  o f the o p t i c a l p o t e n t i a l which  l i k e usual  r a d i a l wave f u n c t i o n s  - F^  and Gj,, r e s p e c t i v e l y - i n t h e a b s e n c e o f a p o t e n t i a l . I n o u r c a s e ' F = s i n k r and G = c o s k r so t h a t a t z e r o e n e r g y F = 0 and o o ^ o G  o  =1.  T h e r e f o r e we  f =  [G  find  O  (R) / G  o  (R)j  =  2  - C r W ^ ^ W  1  (G  ^  2  / G (R) )  O  2  O  =  ^ d l f f ^ ^ V 4 u u a r e ^ ^ 7  (22) since G  i s p r o p o r t i o n a l t o t h e d i f f u s e - w e l l r e s o n a n t wave  o f F i g . 5 a n d t h e r e s o n a n t wave f u n c t i o n s normalized  function  o f t h e two w e l l s a r e  t o t h e same a m p l i t u d e a t t h e m a t c h i n g r a d i u s  on F i g .  5.  Then f i s j u s t t h e s q u a r e o f t h e a m p l i t u d e r a t i o o f t h e r e s o n a n t wave f u n c t i o n s  at large distance,  as i n d i c a t e d on t h e f i g u r e .  I f we remember t h e d e f i n i t i o n f a c t o r and employ t h e c o n v e n t i o n a l 12  (see  (6) o f  the.penetration  d e f i n i t i o n o f an i n c o m i n g w a v e ,  ('-I-1) b e l o w ) , we c a n g i v e  a s i m i l a r p h y s i c a l meaning t o  25 -  the r e f l e c t i o n conventional  f a c t o r which i s valid, at f i n i t e  The  p e n e t r a t i o n f a c t o r may b e w r i t t e n lc R  For  energy.  a diffuse-edge  (1J.J)  •1  w e l l we r e p l a c e X y b y i t s a n a l y t i c a l  continuation I ^ , yielding  )<R ( I , I/O "  wh'ere  f  intensity  1  I , I * / 1,1* X  The  = f kR ( I , I*)  1  A  A  A. 1  o f t h e I n c o m i n g wave a t l a r g e d i s t a n c e n e e d e d t o  y i e l d a g i v e n . i n t e n s i t y a t the matching radius i s f times f o r a square w e l l than f o r a d i f f u s e w e l l . t h a t we d e t e r m i n e i s r e l a t e d t o wave  The r e f l e c t i o n f a c t o r  reflection.  I n t h e a b o v e c o n s t r u c t i o n o f an ESW mass m a n d a n a r b i t r a r y parameters  o p t i c a l p o t e n t i a l only  (V > R , a , m) a r e i n d e p e n d e n t .  f o r a p a r t i c l e of two o f t h e f o u r  This  c a n be seen  t h e r a d i a l e q u a t i o n w i t h t h e o p t i c a l p o t e n t i a l V, o f  11! 2m  At zero in  energy  dr  -  dx  (1) •  (23)  , , (r-Rl7a 1 + e *• o  2  t e r m s o f too d i m e n s i o n l e s s  ?  from  V ^ 3 /  the right-hand, s i d e vanishes  d \\J  bigger  a n d we c a n w r i t e (23)  parameters  (K v  o  R. ) o'  1 -.- e* - - M V x  ]  (211) a)  -  where  x  = r/R  Ko  = 2m  2  Thus R  Q  determines  The  two  and  t h e ESW  covered  26  "  V o/ l i  (25)  (26) \ J  2  o n l y t h e s c a l e o f t h e a b s c i s s a on F i g . 5.  important parameters are K L  o  R  f o r a l l c h o i c e s of these  and R / a .  o  o  two  I f we  find  p a r a m e t e r s t h e n we  c a r r i e d out  t h e c o n s t r u c t i o n o f t h e ESW  and  a s s o c i a t e d r e f l e c t i o n f a c t o r f o r a l a r g e range of v a l u e s q  R  and  a/R  applications positions  .  The  depth  o f t h e ESW  •- i t m u s t a l w a y s be  at the r i g h t  i s not  e n e r g i e s , (see Sec.  g i v e n v e r y n e a r l y by and  Rj  differs  M-) and  ^  V  Q  R.  important  results  i n terms of the r a d i u s d i f f e r e n c e  which i s a l s o dimensionless. AR/R  w h i c h we  o b t a i n e d from  p o l a t i o n s between resonances.  .  q  The  a p p r e c i a b l y from  R  o  the of  i n many resonant  i s always  found  2  usually  R  important  a d j u s t e d to produce the  2 t o be  have  a l l cases. We  K  f  R  ESW .  3?adius i s  We  present  the  /SR/R v  o  •  F i g . 6 shows t h e v a l u e s o f f the zero energy resonances F i g . 6 i s an  and  and  inter-  enlargement of a s m a l l 13  r e g i o n o f a. c o r r e s p o n d i n g the enlargement g i v e n here most  applications.  map  g i v e n by us  i n a preliminary report  c o n t a i n s t h e r e g i o n o f t h e map  used i n  >  A l t h o u g h o u r wave a n a l y s i s o f p o t e n t i a l w e l l s taken the r e f l e c t i o n f a c t o r  has  t o be e n e r g y i n d e p e n d e n t , t h e  o f wave p r o p e r t i e s o f p o t e n t i a l s  analysis  i n terms o f the e q u i v a l e n t  square  w e l l c a n be g e n e r a l i z e d a n d i m p r o v e d by a l l o w i n g f t o be e n e r g y  |  dependent. In  F i g . (7) we  see t h e e x t e n t t o w h i c h  f a c t o r i s independent of energy.  The  reflection  the.reflection factor  c a l c u l a t e d by s u p p o s i n g an i n c o m i n g wave a t i n f i n i t y , calculating  the  and.  V1  and  ratio  f  w h e r e I„ DW  was  ESW  / DW 1  V  ,  a r e e v a l u a t e d a t R = 0.1 -  R^,,,,. ESW  To e l i m i n a t e any p o s s i b l e e f f e c t o f a b s o r p t i o n i n t h e b a r r i e r , made o u r c a l c u l a t i o n s by s u p p o s i n g t h a t , f o r b o t h t h e d i f f u s e square w e l l s ,  a b s o r p t i o n o c c u r r e d o n l y ' a t R. - 0 and. t h a t  i n c o m i n g wave was is  completely absorbed there.  The  The  and  the  reflection  t h e n s e e n t o be m o d e r a t e l y d e p e n d e n t o f e n e r g y .  we  factor  energy  d e p e n d e n c e i s a l s o a f u n c t i o n o f . t h e r e d u c e d mass; i t i s l a r g e r f o r the of  heavier particles.  We  have a l s o checked t h a t the energy  t h e r e f l e c t i o n f a c t o r was  within  10%  or so, whatever  not a f u n c t i o n of charge.  dependene  I t i s t h e same  t h e Coulomb b a r r i e r , f o r a g i v e n r e d u c e d  mass. B e f o r e a p p l y i n g t h e ESW  i n some d e t a i l  b a s i c wave p r o p e r t i e s o f v a r i o u s p r o b l e m s we a n d d i s c u s s some s p e c i a l , c a s e s . we  can r e s o l v e  potential with w i t h Z\R  no  to o b t a i n the.  g i v e a s i m p l e example 32  F o r our s t a n d a r d example o f  t h e d i f f i c u l t i e s .of F i g . '-I. F o r a known a d i f f u s e - e d g e we  now  c o n s t r u c t t h e ESW  l o n g e r z e r o b u t f i x e d , by F i g . 6.  The  S  +<&  optical as  results  above, a r e shown  -  on  Fig-  8.  Nov;  the  28  r a t i o of  transmission"  energy, i s r o u g h l y e q u a l to the from  (17).  There are  the  functions,  reflection factor  corrections  o f w h i c h i s a b s o r p t i o n i n the energy v a r i a t i o n of  -  however, the  barrier.  at  as  low  expected  most i m p o r t a n t  I t largely cancels  reflection factor  so  that  the  the  notion  of  a c o n s t a n t r e f l e c t i o n f a c t o r becomes a g o o d a p p r o x i m a t i o n . corrections  to the  i n Sec.  The  5.  successful  r e f l e c t i o n f a c t o r w i l l be  p r e s c r i p t i o n f o r the  in  ESW  further  The  discussed  i s obviously  quite  this"case.  12 For Fig.  p r o t o n s , a l p h a p a r t i c l e s and  6 to c o n v e n t i o n a l o p t i c a l p o t e n t i a l s .  w h i c h shows t h e of  the  the  C n u c l e i we  value of  f and  bombarding n u c l e u s .  figure  to the  /^R  The  can  T h i s i s done on  f o r t h e s e p a r t i c l e s as  r e s u l t s f o r . f are  also  approximate r e f l e c t i o n f a c t o r ,  apply Fig.  a  9  function  compared  (3.6), d e r i v e d  on by  P e a s l e e f o r s-wave n e u t r o n s . I.  Single  P a r t i c l e Resonances For The  in  Sec.  3 can  potentials wells.  The  most s t r a i g h t f o r w a r d be  applied  i s f o r the ESW  was  to d i s p l a y  Real  case i n which the  also  one  Coulomb b a r r i e r s , much as  we  go  now  still  and  add  absorption  i n t e r e s t h e r e i s t o show how prescription applies r e s u l t s w i l l be  of nuclear reactions  (Sec.  the  7)  In  of  optical  the  potential  resonances energy  later  picture  (Sec.  of  and the  sections S).  The  neutral-particle  potential with  i n generalizing  constructed  i n analyzing  nuclei.  to  the  to f i n i t e  we'll a z e r o - e n e r g y  to a r e a l i s t i c  useful  go  m i g h t do  of alpha p a r t i c l e s from L i g h t  further  ESW  resonances of r e a l  c o n s t r u c t e d from a matching of We  scattering  the  wave p r o p e r t i e s  elastic scattering  neutral, p a r t i c l e s at zero energy. add  Potentials  barriers.  f o r m a l many-channel  t o take i n t o a c c o u n t the  finite  The theory surface  29  thickness  of  nuclei.  We b e g i n w i t h a. r e a l n u c l e a r p o t e n t i a l w h i c h m i g h t  be  32 suitable expect V L  for o  S -l- C\  C h o o s i n g t h e Saxon-Woods f o r m 1 /3 75 MeV, R t o b e * 1.25 A +1.6 o  t o be a b o u t  * This value analyses  of  the r a d i u s i s  + A?"^ )  common].y u s e d f o r  (A^ " H- A^  data  - is  )  neither  .Its  o p t i c a l model  difference  from  o r f r o m t h e more f u n d a m e n t a l r a d i u s  J  1.09  and a  _ /  of a l p h a - p a r t i c l e r e a c t i o n s .  1.25  (1) we  .  - which i s  s u g g e s t e d by e l e c t r o n  j u s t i f i e d by t h e  of  scattering  data nor Important f o r  our  discussion. .  t o be 0.5 'fm.  We f i x  a at  t h i s , value  and R  at  5.5685 f ni.  The  o Coulomb b a r r i e r h a s alpha-particle For E  purposes  res  =3.0  a height  resonances  of  of  8.3 MeV a t R. .  Therefore  a t a few MeV e n e r g y w i l l  be v e r y  i l l u s t r a t i o n we c h o s e a f i x e d r e s o n a n c e  MeV, a n d v a r i e d V  a single-particle,  o  i n the v i c i n i t y of  s-wave r e s o n a n c e .  F i g . 10a  any  energy,  75 MeV t o shows  narrow.  obtain  such a  r e s o n a n t wave f u n c t i o n o b t a i n e d f o r V angles where•Coulomb section  is  that  of  da'  . _  cross Fig. is  no  is  ilk  the  r- 2 no  1 2  level  s e c t i o n near 10a.  scattering  the B r e i t - W i g n e r f o r m u l a  d.Owhere  (E  v  no  for  the energy  this  -E- A  shift.  The s i n g l e - p a r t i c l e  1.5 keV w i d e  = 61.9 MeV. A t b a c k w a r d o is-minimum the s c a t t e r i n g c r o s s  case.  ) no'  2  + ~ 4  P no  F i g . 10b shows of  the  2  the  scattering  r e s o n a n t wave f u n c t i o n  resonance  of  of  the o p t i c a l model w e l l  - 30 -  In f i n d i n g the s i n g l e - p a r t i c l e m o d e l p o t e n t i a l we h a v e c o m p l e t e l y the  potential.  t h e ESW. and  T h e r e f o r e we  In. t h e p r e s e n t  a depth  resonance with  c a n a t once u s e F i g . 6 t o c o n s t r u c t  i n order  has a r a d i u s  that i ty i e l d  f i v e n o d e s a t 3.0 MeV.  The ESW  wave f u n c t i o n a r e shown on F i g . 10a a n d t h e scattering  cross  o f 6.25 fm  an a l p h a - p a r t i c l e  and i t s r e s o n a n t  corresponding  s e c t i o n on F i g . 1 0 b .  Clearly,'both resonances.  optical  s p e c i f i e d the parameters o f  c a s e t h e ESW  = S6.3 MeV  r e s o n a n c e o f an  t h e o p t i c a l p o t e n t i a l a n d i t s ESW  The r e s o n a n c e o f t h e ESW  i s considerably  exhibit  narrower  t h a n t h a t o f t h e o p t i c a l m o d e l , a f a c t w h i c h m i g h t be a s c r i b e d t o reflection.  We w i s h t o e n q u i r e  model resonance p r o p e r t i e s resonance p r o p e r t i e s  i n some d e t a i l how  c a n be d e s c r i b e d  the o p t i c a l  i n t e r m s o f t h e ESW  a f t e r t a k i n g account of r e f l e c t i o n .  t o do s o we n e e d t o f a c t o r y . t h e r e s o n a n c e w i d t h s i n t o factors of  a n d ' r e d u c e d w i d t h s a n d we n e e d t o a n a l y z e  the l e v e l  potential shift and (see (6)  I n such a f a c t o r i z a t i o n  - the " t a i l "  and p e n e t r a t i o n  (6) r e s p e c t i v e l y .  f a c t o r s from t h e i r  penetration  of the o p t i c a l  - modifies  conventional  a n d (7) i f we r e p l a c e  F  and G  , t o t h e mach.ing r a d i u s .  1n o • =  2 P o  by t h e i r  f  still  the  f o r m s , (7)  A r e - d e r i v a t i o n of the Brelt-Wigner and S  order  the behaviour  a part  beyond the matching r a d i u s  3?ef. 1 2 , f o r e x a m p l e ) shows t h a t P  F, a n d G, have  shift.  In  formula  are given  by  analytic continuations,  D e n o t i n g t h e s e b y P, a n d S^ we  y  2  no  v  (29) '  J u s t as i n S e c . 3, t h e m a t c h i n g r a d i u s o f b o t h w e l l s f o r t h e factorization  I s t h e ESW  r a d i u s R.j = G.25  fm.  The  reduced  w i d t h s a r e o b t a i n e d f r o m t h e r e s o n a n t wave f u n c t i o n s by ( 1 9 ) . In turn,  t h e r e s o n a n t wave f u n c t i o n i s f o u n d t o be  f u n c t i o n w h i c h behaves the p o t e n t i a l .  The  like  boundary  (7) i s s t i l l  s h i f t function vanish at E = E  res  .  t h e i r r e g u l a r s o l u t i o n G„ c o n d i t i o n number, b„ t o be c h o s e n X  faJ  need  f a r beyond  , o f the  level  t o make t h e l e v e l  shift  H o w e v e r b o t h P,. and S,  e n e r g y a n d t h e r e f o r e we -  t h a t wave  are f u n c t i o n s o f the  A  f  to r e t a i n both  no  and ^  i n the  no  Breit-Wigner formula. • There  i s no d o u b t a.bout t h e v a l i d i t y  of the  one-level  32 a p p r o x i m a t i o n i n our case. 10  MeV  The  0\ r e s o n a n c e s a r e  S +  about  a p a r t c o m p a r e d t o t h e w i d t h o f 1.5'keV. The  energy dependence o f the l e v e l s h i f t  c a n be  taken  i n t o a c c o u n t a p p r o x i m a t e l y b y u s i n g a T a y l o r " s e r i e s f o r t h e s ! i i f." 1 a b o u t t h e e n e r g y and r e t a i n i n g o n l y t h e l o w e s t n o n - z e r o  term  dA no  Combining  dE  (31) w i t h  (28) we  f o r m u l a i n the f o l l o w i n g  der  d-o. The  =  x  (E - E )  (31)  E = E no can w r i t e the s i n g l e - l e v e l  resonance  way  1  (i _ ^  (*r•~ n 0  )  no (  ,)  E :  dE  2  E  )'  no  2  -i-  fr 1  2  (32)  no  e f f e c t o f the energy dependence o f the l e v e l s h i f t  on  ] 7  c r o s s s e c t i o n s was  first  n o t e d by Thomas  and  i s described i n d e t a i l  32  by B r e i t "  who  0  has  a l s o computed t h o s e c a s e s where a p p r o x i m a t i o n 37  (31) f a i l s .  Thomas  approximation  l i a s s u g g e s t e d t h a t i f (31) were a g o o d  (as i t s h o u l d be f o r t h e e x t r e m e l y n a r r o w  resonances ii  w i t h i n a Coulomb b a r r i e r )  t h a t the energy dependence of the  level  s h i f t s be a s s o c i a t e d w i t h t h e r e d u c e d w i d t h s (Y< )  ~  2  Y  / [Y-  2  d  A  1  n o  "1  :  no In for  f a c t , he  (33) ;  then found t h a t the reduced widths of h i s c a l c u l a t i o n  a s q u a r e w e l l w i t h Coulomb b a r r i e r w e r e i n c l o s e a g r e e m e n t w i t h 2  t h o s e a p p r o p r i a t e t o a s q u a r e w e l l w i t h o u t Coulomb b a r r i e r W i t h t i l l s a s s o c i a t i o n we may the  calculation i ' 'no  —  2 P  n a t u r a l l y d e f i n e the o b s e r v e d w i d t h s o f  (Y ) no^ 2 1  o  v  2  =_  /rno '//I;rf i 1 -  so t h a t the s i n g l e - l e v e l f o r m u l a a t t a i n s in  the absence of l e v e l  ^  =  '  »  comparing  p o t e n t i a l a n d i t s ESW  .  2  , 2 ;  E  1(  no  4  J  v  no  J  1  ( o r more p r e c i s e l y  f l  ii)  f  y  /  •«  E k  2 -  (r - ) cl;i;1  :r  !  2  3w  -  / (Y  Y E g w  )2  V  F  A  =  1.93.  I  -  2  2  be  M.'l .at The  r a t i o s are:  *)  optical  t h e r e a r e a number o f r a t i o s w h i c h m i g h t  f r o m F i g . ( 7 ) ) f o u n d f o r o u r p r o b l e m f r o m F i g . 6.  •  (3.1)  p ~ 2  the resonance p r o p e r t i e s o f the r e a l  r e l a t e d t o t h e r e f l e c t i o n f a c t o r o f 5.2 3 MeV  no  the s i m p l e form a p p r o p r i a t e  'no>  . -v  In  " &-A -' dE  shifts  -1  d.n  ( i . e . - - ! ! /mR  ,  2 )  -  • -.p  p  „  iv)  f  ,,  ^  p  1  33  / p  _  " ^^no '^ 1  A -  vl)  f  (  y  t  )  2  =r  (Y<  Q  -  f f  l f f  /  1  =  0.727  v  /dE) I ~ ' M t = E no r  )  2  / ( Y ^ /  Tp,,, -  -  1-38  M..M6 32  where  t h e numbers  The f i r s t  ratio,  refer  to our s p e c i a l case o f  'I  "S +  f ' , of the observed widths d i f f e r s  a p p r e c i a b l e amount f r o m 5.2 w h i l e t h e r a t i o ,  (35)  no  d i f 3 ?  f  .  ' ^ ) ; •• E  _____ no  v)  / d l  p 31.1.  He a t 3 by  MeV.  an  the zero-energy, z e r o - b a r r i e r r e s u l t o f  f , o f t h e one-body w i d t h s  /  i s in. m o d e r a t e l y  good agreement w i t h t h e s e r e s u l t s and i n e x c e l l e n t agreement w i t h t h e one o b t a i n e d a t E = 3.0 MeV remembered,  I t must  be  however, t h a t i n t r o d u c t i o n o f the s u r f a c e e f f e c t s . i n t o  the channel widths w i l l , energy  (from F i g . (7)) .  i n g e n e r a l , i n v o l v e an a d j u s t m e n t  dependence o f the l e v e l  shift  (f'- f  f) .  f o r the  It  is  a s i m p l e way  interesting  into  d A  d  :  the  level  i  / di; I IL = .L no  f  f  '"d ,U  (  K  )  :  -  R  n l V )  u  =  to  potential,  •  •  a p p l i e s we h a v e ,  near  (suppressing  indices  the  (R ) )  calculation,  f ,  dA  ,  an a r b i t r a r y m a t c h i n g r a d i u s ,  the-one-body  (R) / a  enters  f  / dE  K S W  in  E = E. no  no  - o°  a n d G^ i s  since  -_&  resonance,  a ^  =  F  (R )  2."  ^Sw"  Y  n  ^;( )  (36)  s  ,„  analytic  is  tbe  R  ~2  the  approximation Moreover,  that of  the  we h a v e ( 3 6 ) .  diff_(v  .  e n e r g y , E (j ,  (R) .  the r a d i u s  (R^,)  g  y  shown  is  q  a ^ ^  =  at  continuation of  (R) = 7^ ^  ]  normalized  the o n e - l e v e l  easily  dG,/dr)j  (KCT^R)  l T  i-  r  a ^  where  d i f f " ^  Y  the  that  (R^) -  F  u  Since  ( n , £ )) i t  (R) / a °  n;;,(R)^  t h e nth. i -wave r e s o n a n c e  i r r e g u l a r A -wave Coulomb f u n c t i o n .  (a  f  d u ^ / d ^ ^  wave f u n c t i o n a p p r o p r i a t e of  the f a c t o r  let  IdE  where R i s  that  shift*  r  no  * To show t h i s  to note  a  diff  „, 2~  ESW  \  S  W  (  V  *ESW (  V  32  In  fact  be  3.26 w h i l e  of t h e of  i n the  real  example f  has  T  the v a l u e  diffus.e-edge  the parameters  of  p'  Miff  S -I- c< t h e  of  its  "  3.22 o f  of  (36) t u r n e d o u t  (35) .  The o b s e r v e d  p o t e n t i a l c a n t h e n a g a i n be s t a t e d equivalent f  :  ratio  r  square  widths terms  well  1 - f ^ f r mn or ; 1 J  " d lE  in  to  IE = E no  (37)  35  by the p o s i t i v e  s o l u t i o n o f the s i m p l e q u a d r a t i c e q u a t i o n  c f'  2  - f  + (f - c f)  = 0  where  (38)  .  dA  c ~  H e n c e we may in  state  E S W  no dE  E = E. no  the widths  terms o f the p a r a m e t e r s  (37) o f t h e d i f f u s e - w e l l  o f the square i n t e r a c t i o n  entirely  and o f the  standard' r e f l e c t i o n f a c t o r , f , o b t a i n e d from the z e r o - e n e r g y z e r o barrier  c a s e .*. ' 1 '-I  * I n an e a r l i e r i t was manner.  c a l c u l a t i o n on a l p h a - d e c a y r a t e s I n h e a v y  o f h e u r i s t i c v a l u e t o d e f i n e an ESW To h a v e c h o s e n  t h e ESW  . that found f o r n e u t r a l p a r t i c l e s shown i n t h e p r e s e n t a n a l y s i s ) d i v i d e d between the i n t e r i o r aspects was  (penetrabilities)  in a slightly  t o have the parameters  i n Sec. 3 would have y i e l d e d a r e f l e c t i o n  nuclei  factor  different  suggested  consistent  a t z e r o e n e r g y b u t ( a s we  •  with have  t h i s r e f l e c t i o n would have been  a s p e c t s ( r e d u c e d w i d t h s ) and  of'the alpha-decay problem.  Our  t h a t p r e v i o u s a u t h o r s had. t r e a t e d t h e p e n e t r a b i l i t i e s  b u t bad. n o t a c c o u n t e d f o r t h e e n h a n c e m e n t  exterior point correctly  o f t h e wave f u n c t i o n i n  t h e n u c l e a r s u r f a c e e f f e c t e d by t h e d i f f u s e n u c l e a r e d g e ; i n f a c t , t h i s enhancement earlier  e x p l a i n e d t h e anomalous r a d i i w h i c h had p l a g u e d  alpha-decay rate c a l c u l a t i o n s  the n u c l e a r i n t e r i o r  t o be s q u a r e ) .  b y c o n s t r u c t i n g an "ESW" manner o f S e c . 3:  (which had g e n e r a l l y T h i s was m o s t e a s i l y  from f i r s t p r i n c i p l e s ,  displayed  that i s , i n the  t h e r a d i u s a n d d e p t h o f t h e ESW  /  taken  were chosen  so  that  t h e r e s o n a n t wave f u n c t i o n s  o f the d i f f u s e and s q u a r e  w e l l h a d t h e same number o f n o d e s a n d t h e same a m p l i t u d e a t the  ESW  radius.  considerably  The r a d i u s  larger  o f t h i s ESW was f o u n d t o be  than that  of the d i f f u s e - w e l l , explaining  much o f t h e a n o m a l y i n .the p r e v i o u s  •  Complex O p t i c a l We w i s h  calculations.  Potentials  to extend our treatment o f b a r r i e r  to t h e case o f complex o p t i c a l p o t e n t i a l s . barriers  and r e a l o p t i c a l p o t e n t i a l s  barriers with the  penetration  As i n t h e case o f  ( S e c . '1) t h e c o m b i n a t i o n o f  c o m p l e x p o t e n t i a l s w i l l b e shown t o e x h i b i t many o f  s i m p l e wave p r o p e r t i e s  all.  !  p r e s e n t w h e r e t h e r e a r e no b a r r i e r s a t  A g a i n t h e r e a r e some s t r a i g h t f o r w a r d  s i m p l e wave p r o p e r t i e s  modifications  o f the  b r o u g h t about by t h e i n t r o d u c t i o n  of barriers.  32 As  i n our e a r l i e r d i s c u s s i o n ,  illustrate  the main In  we s h a l l u s e t h e e x a m p l e  S + C\ t o  points.  the absence o f b a r r i e r s  the a d d i t i o n  o f an i m a g i n a r y  term t o a r e a l o p t i c a l model p o t e n t i a l i n t r o d u c e s a b s o r p t i o n , i t b r o a d e n s t h e wave r e s o n a n c e s , b u t h a s a n a l m o s t n e g l i g i b l e 12 on wave r e f l e c t i o n heavy ions the  - i n d e e d , t h e a b s o r p t i o n and s c a t t e r i n g  of  imaginary  magnitude compared t o  Therefore the r e f l e c t i o n from the imaginary  complex p o t e n t i a l w e l l the  The i m a g i n a r y t e r m f o r n u c l e o n s a n d e v e n f o r  i s n o r m a l l y chosen t o have a s m a l l  r e a l term.  small  .  effect  term.  cross section  term i s of a  a r e r e l a t i v e l y i n s e n s i t i v e t o t h e shape o f  F o r t h e same r e a s o n s we e x p e c t t h e same  kind  c h a n g e s when, i n t h e p r e s e n c e o f b a r r i e r s , we a d d an i m a g i n a r y  •term t o t h e o p t i c a l p o t e n t i a l :  a b s o r p t i o n enters the p i c t u r e , the  r e s o n a n c e s b r o a d e n b u t t h e wave r e f l e c t i o n s h o u l d r e m a i n r e l a t i v e l y  unchanged. At  lev? e n e r g y ,  themselves most d i r e c t l y Fig.  8 we  showed  t h e wave r e f l e c t i o n i n nuclear  a straightforward  properties  transmission  manifest  functions.  On  a p p l i c a t i o n of the zero-energy, 32  zero-barrier The  results to the transmission  success  of the model  transmission In  order  functions  to elucidate  function's  functions  of Sec. 3 appears  than  f o r the widths  t o what  extent  t o be g r e a t e r discussed  the ratio  should, be r e l a t e d t o t h e r e f l e c t i o n  calculated  that  ratio  over  a wide  range  d i f f e r e n t Coulomb b a r r i e r s .  results,  examine  a  d i f f u s e absorbing  reflection an  properties  imaginary  diffuse  edge,  the normal  means  of  the  found  radius  i n this I f we  way  2.  adopt  with  the appropriate  functions  the complex phase  the phase  shifts  functions.  of  p o t e n t i a l has a may  take  are calculated  shifts  place  (5)  exactly  a t complex  to a.point  well  s o l u t i o n i s matched  The e x a c t  transmission  t h e i r wave p r o p e r t i e s  the form  then  that the  are calculated directly  equation  the numerical  can have  Sec.  and  radius.  transmission  where  Coulomb wave  shall,.find  i f the absorbing  i n t e g r a t i o n o f t h e wave  nuclear  standard  In turn,  those  a square  f r a c t i o n of the absorption  nuclear  ('-I) f r o m  potentials. numerical  nuclear  and f o r r e a c t i o n s  To a n a l y z e  We  have  a r e n e g l i g i b l y a f f e c t e d by t h e p r e s e n c e  an i m p o r t a n t  The  transmission  f a c t o r , we  of introducing  to the p o t e n t i a l .  potential but that,  much b e y o n d  by  part  the'effect  f o r the  i n S e c . 4.  of the  of energies  involving widely we w i l l  "S + C< .  for  mo d i f i c a t i o n s m a d e f o r d i f f u s e - e d g e  beyond on t o  functions  exhibited  f o r the transmission  from  as i n  function  -  potentials,  2 - then a simple  as d i s c u s s e d i n Sec. condit.ion numbe3: vanish.  i s one which makes the s h i f t f u n c t i o n , S  With t h i s c h o i c e we  -  I S defined  where TT  •.  x  r  I  CI+T;  = ti tr f p .  A  ;  have  AD  (39)  2  by  •  r  c h o i c e of the boundary  s  , A  Here P^ 'is the p e n e t r a t i o n f u n c t i o n , s ^ the s t r e n g t h f u n c t i o n , and  2.  f the r e f l e c t i o n f a c t o r d i s c u s s e d i n Sec. The  result  (39) f o r the form of the t r a n s m i s s i o n f u n c t i o n  e x p l a i n s the r e s u l t s of P i g . 8 ,  both the simple  mission  and  f u n c t i o n s a t low  at h i g h e r  energy.  At  low  energy  r a t i o of t r a n s -  a l s o the c o r r e c t i o n s p e r t a i n i n g  energy, f a r below the b a r r i e r , 7.' /'I 'is A,  much s m a l l e r than u n i t y so t h a t T , " V , . . diffuse-edge equal  t r a n s m i s s i o n f u n c t i o n to t h a t of the ESW  to the r e f l e c t i o n f a c t o r .  At energies  Then the r a t i o of  the  should  be  T h i s i s found, to be so in. F i g . 8 .  approaching the top of the Coulomb b a r r i e r i t i s not  the  t r a n s m i s s i o n f u n c t i o n s themselves but r a t h e r the r a t i o o f 1'^ which should equal f .  A g a i n t h i s i s found to be v e r i f i e d by F i g . 8 .  r e s u l t s and many other s i m i l a r r e s u l t s which we.obtained but show here j u s t i f y our c l a i m t h a t the ESW b o t h the Coulomb and F i g . 5 and ESW  should be  These  cannot  independent of  the angulai? momentum b a r r i e r s . A comparison of  F i g . 8 shows the importance o f /^R  i i i the c h o i c e of  the  f o r heavy i o n r e a c t i o n s . A l t h o u g h the dominant, wave p r o p e r t i e s of complex w e l l s  a3."'e g i v e n i n the d i s c u s s i o n above, there are some minor and some  - 39  -  major c o r r e c t i o n s i n c e r t a i n cases.  Two  minor.corrections  32 (each l e s s from  than  resonance  10% i n our s t a n d a r d e f f e c t s - and  p a r t o f the p o t e n t i a l . in  a s l i g h t energy  (39)  is fitted  from  The  S +  reaction)  r e f l e c t i o n by  former  the  arise  imaginary  of these manifests  itself  d e p e n d e n c e o f t h e ' s t r e n g t h f u n c t i o n when  to c a l c u l a t e d  transmission functions.  Although  m i n o r f o r most heavy i o n r e a c t i o n s the resonance c o r r e c t i o n s c a n become v e r y imaginary  term  l a r g e i f the v a l u e of W  is  l e v e l - s p a c i n g i n the r e a l p a r t of the  r e f l e c t i o n i n d u c e d by W  interest.  I t c a n be  to replace V  the r e f l e c t i o n  of  i n t h e p o t e n t i a l , becomes s m a l l e r t h a n  single-particle The  , the depth  r  by  i s s m a l l i n a l l cases  of  the the well. practical.  shown t h a t a r o u g h e s t i m a t e o f t h i s 2 2(V^ + W  ')  2  i n Peasl.ee's f o r m u l a ,  effect  (16), f o r  factor.  A major c o r r e c t i o n to o p t i c a l model a n a l y s e s a b s o r p t i o n deep w i t h i n t h e b a r r i e r w h i c h f a r b e l o w t h e Coulomb b a r r i e r .  occurs  I t f o l l o w s from 1 9  o f G r e e n ' s t h e o r e m t o t h e wave e q u a t i o n  concerns  at energies an  application  t h a t the t r a n s m i s s i o n  f u n c t i o n , Tj. , i s e x a c t l y p r o p o r t i o n a l t o t h e i n t e g r a l o f i m a g i n a r y p a r t To f t h e  (4-1)  vj/ i s t h e r a d i a l wave f u n c t i o n ,  by n u m e r i c a l i n t e g r a t i o n o f t h e wave e q u a t i o n w i t h t h e  of  (Ml)  the  potential  where c i s a c o n s t a n t and  potential.  very  'In c e r t a i n c a s e s  obtained  optical  i m p o r t a n t c o n t r i b u t i o n s t o the  occur w e l l beyond the normal n u c l e a r r a d i u s .  I  The  integral  poi.nl: i s  ~ MO  straightforward. increases  VJl.thin t h e b a r r i e r  exponentially:  jL^f{  c>.c  K  •  =  ( 2m ™  l<A<)  (B - E)  ll.  where B i s the h e i g h t  1  Therefore  the  (43)  a, f a l l s  c o n t r i b u t i o n to the a b s o r p t i o n  the b a r r i e r becomes i m p o r t a n t  approaches the value, of a fm  f o r a we  find  2Ka  I f we  from the  at energies take  mass) and  .  region  such t h a t  a standard  (B~E)  value  is in  2K of  reduced  MeV.  a b s o r p t i o n w i t h i n the b a r r i e r i s • o f g r e a t  f o r many h e a v y i o n r e a c t i o n s .  The  o f f as e  = 0.23\/m (B--E) w h e r e m i s t h e  mass ( i n u n i t s o f t h e p r o t o n The  ('12)  --  o f the b a r r i e r and E i s the e n e r g y .  p o t e n t i a l (2) , f o r r - R  0.5  2  "  J  absorbing  within  t h e wave f u n c t i o n  I t s m a g n i t u d e c a n be  importance  estimated  by  32 means o f  (Ml) .  the a b s o r p t i o n  I n our occurs  standard  b e y o n d t h e ESW  d o e s n ' t obey the n o r m a l r u l e s : due of W  t o r e f l e c t i o n and .  mission  case of  S + c<  radius.  i t i s not  a t 3 MeV  This  s u b j e c t to a c o r r e c t i o n  t o remember t h a t t h e e f f e c t on  f u n c t i o n s of a b s o r p t i o n  i n the b a r r i e r  top  o f the b a r r i e r  t e n d s t o i n c r e a s e T^ w h i l e  the  the  A s we  absorption  value  transto go  i n the  that down  barrier  the energy dependence of f d e c r e a s e s  F o r many commonly u s e d o p t i c a ] , p o t e n t i a l s t h e cancel.  the  i s opposite  o f the energy dependence of the r e f l e c t i o n f a c t o r . i n energy from the  of  absorption  i t tends to i n c r e a s e l i n e a r l y w i t h  I t i s important  16%  two  effects nearly  M o r e i m p o r t a n t l y , i t s u g g e s t s t h a t some a s t r o p h y s i c a l  T^.  -  reaction rates w i l l  be d o m i n a t e d by t h e t a i l  p a r t of the o p t i c a l potential.. loses i t su t i l i t y energies it  I n s u c h c a s e s t h e o p t i c a l mode].  that absorption  can p r o v i d e  d e n s i t i e s f a r beyond  11 f o r t h e  at high  -1- 1 5 a .  cross  section  information  Conversely measurements  about  at twice  of b a r r i e r  absorption  d e c a y o f " "P •  the nuclear  i s shown on  absorption  i n t e g r a t e d wave f u n c t i o n s  Here  the  radius, at values  Many o p t i c a l m o d e l p r o g r a m s  t h r o w away t h e b a r r i e r  nuclear  radius..  208' Pb -l-CX r e a c t i o n a t an e n e r g y o f 8 MeV 212  to the i n v e r s e of the alpha important  important  the nuclear  • An extreme example  to R  imaginary  to a s t r o p h y s l c a l r a t e s f a r below the b a r r i e r .  a t low e n e r g i e s  is  of the  i n e x t r a p o l a t i n g f r o m measurements  r a i s e s the prospect  Fig.  -  Ml  appropriate  absorption  o f r comparable  i n c u r r e n t use would  -  b e c a u s e t h e y m a t c h 'the n u m e r i c a l l y  t o Coulomb f u n c t i o n s a t t o o s m a l l a. r a d i u s .  -  •6.  i|2 -  The Use o f t h e O p t i c a l M o d e l i n A s t r o p h y s i c s C u r r e n t e f f o r t s by a s t r o p h y s i c i s t s t o d e t e r m i n e t h e  l a t e s t a g e s o f e v o l u t i o n o f t h e s t a r s and t o d e t e r m i n e t h e e a r l y history  of the universe  according  t o the " B i g Bang" t h e o r y  require  t h e k n o w l e d g e o f a l a r g e number o f c h a r g e d p a r t i c l e r e a c t i o n  rates.  20 According  to the "Big.Bang" theory  emerged from thermodynamic e q u i l i b r i u m a t v e r y (T^,  10^  °K)  and w o u l d have c o o l e d  amount o f n u c l e a r the  transmutations  energy r e l e a s e  the nuclear nuclear  during  matter w o u l d have high  down a s i t e x p a n d e d .  occurring  that time,  The  as t h e m a t t e r e x p a n d e d ,  and the temperature a t which'  r e a c t i o n r a t e s become t o o s l o w t o a l l o w  reactions  temperatures  any more  t o o c c u r a l l d e p e n d s e n s i t i v e l y on c h a r g e d  particle reaction rates.  On t h e o t h e r  hand, once the s t a r s , i n '  t h e i r deep i n t e r i o r s , h a v e b u r n e d a l l t h e i r h y d r o g e n a n d a l l t h e i r ' helium,  ' C and  " 0 a r e e x p e c t e d t o burn, u n t i l ,  after  their  28  exhaustion, S i decomposes i t s e l f i n t o a l p h a p a r t i c l e s w h i c h , 28S i s e e d n u c l e i , f o r m t h e i r o n p e a k 21 adding themselves to The ^ C 2  + ^"C 2  and t h e ^0  + "^0  r e a c t i o n s have been  22 23 studied as  i n the l a b o r a t o r y  they are expected to occur a t i n the s t a r s .  standing .  with  of the p h y s i c s  of the r e a c t i o n s  accuracy, or at l e a s t , to evaluate  extrapolation. not  '" , b u t n o t a t q u i t e as l o w an e n e r g y  Many o t h e r  One n e e d s an u n d e r -  to extrapolate  downwards  the u n c e r t a i n t i e s o f the  needed charged p a r t i c l e  b e e n m e a s u r e d , e s p e c i a l l y ( 0(, p) , (C* > n)  reactions  ('X, y )  and  have (p,y)  28 reactions  f o r t a r g e t n u c l e i w i t h masses a r o u n d and above  Some o f t h o s e r e a c t i o n s p r o b a b l y w i l l ' involve unstable  target nuclei.  Si.  n e v e r be m e a s u r e d s i n c e  Theoretical determinations  t h o s e r a t e s , b a s e d on t h e o p t i c a l m o d e l h a v e b e e n vised by physicists.  they .  of astro-  - '13 -  In p r i n c i p l e equation for  they  could, have s o l v e d the  o f t h e Woods-Saxon  each n u c l e u s .  Due  o p t i c a l model f o r each channel  and  t o t h e l a r g e number o f r a t e s n e e d e d ,  have r e l i e d upon s q u a r e w e l l , b l a c k n u c l e u s accuracy  Schroedinger  calculations.  w i t h which the e q u i v a l e n t square w e l l replaces  w e l l v i n d i c a t e s t h e i r e f f o r t s , b u t a l s o shows between the r a d i u s o b t a i n e d  by f i t t i n g  By f i t t i n g  (n,  p) r e a c t i o n s  and  (c^jV) r e a c t i o n s  Truran  experimental  (for 3 7 < ^  the r e l a t i o n s h i p  s c a t t e r i n g data  t o an  (for A  60) a n d f o r (p ,-X. ) >  24 e t a l . d e t e r m i n e d a r a d i u s R = 1.2  (A  target.  ] /3 o  i s t h e mass number o f t h e p r o j e c t i l e • .  o  (P>y)  35) t o b l a c k n u c l e i c r o s s  v  where A  cross  r e a c t i o n cross sections f o r  ^  t a r  The  a diffuse  o p t i c a ] , m o d e l a n d t h e r a d i u s one s h o u l d u s e i n a b s o r p t i o n sections.  they  sections,  ] /3 + A' ) I  fm  J  and A , t h a t o f t h e 1 •  F r o m e l e c t r o n a n d e l a s t i c s c a t t e r i n g d a t a , one  rather  1/3 expects R  R  = 1.2 5 A  = 1.09 A " * ' ^ + 1.6  should  fermis f o r protons  and n e u t r o n s and. ,  fm f o r a l p h a p a r t i c l e s .  a d d t o o b t a i n t h e ESW  radius  From F i g .  / \ R = 0.1fm f o r p r o t o n s  n e u t r o n s and. £ R = 0.7fm f o r a l p h a p a r t i c l e s . by Truran nucleon  c h a n n e l s and v e r y  closely  probably  t h e same f o r a l p h a  obtained  channels.  comes f r o m t h e a r t i f l e i a l  r e f l e c t i o n o c c u r r i n g i n the b l a c k nucleus.  I f the r e f l e c t i o n  f a c t o r i s used to m u l t i p l y the p e n e t r a b i l i t y , f i t t h e ( n , p) r e a c t i o n m e a s u r e m e n t s  channel  The r a d i u s  and  e t a l . i s s e e n t o be l a r g e r t h a n t h e s c a t t e r i n g one f o r  P a r t o f the discrepancy  to  9 , one  r a d i u s i s a l s o reduced however.  the r a d i u s needed  i s smaller.  The  alpha  We h a v e made c a l c u l a t i o n s  •I'l  11  -  u s i n g d i f f u s e well, t r a n s m i s s i o n f u n c t i o n s , o f the r e a c t i o n s 27 A  2' I (]?/•'•) 25  mental R for  31  M£  a n (  l  P  28  (lV '0  S i . Comparing w i t h  %  cross sections points  = 1.09 A ^ ^  J  + 1.6 + .4 f  t o an a l p h a  the.expert-  channel  i n good agreement w i t h  radius the radius  particle scattering. It  i s then p o s s i b l e t o r e l a t e  t h e p a r a m e t e r s t o be  used i n r e a c t i o n c r o s s section, c a l c u l a t i o n s t o those  obtained  L|  f r o m e l e c t r o n , meson, p, n and Formulas l i k e  C~60 o f F o w l e r a n d H o y l e  P by f P ,  replaces  He s c a t t e r i n g b y n u c l e i . 2 (5 may be u s e d i f one  t h a t i s , i f one m u l t i p l i e s  the p e n e t r a b i l i t y ,  o r e c j u i v a l o n t l y t h e s t r e n g t h f u n c t i o n , by t h e r e f l e c t i o n The  radius  t o be u s e d i s t h e n t h e one o b t a i n e d  experiments plus use  Eq.  C-GO  t h e J^R,  from F i g .  o f R e f . 26 t o e s t i m a t e  6  factor.  from s c a t t e r i n g  or F i g .  6 .  One c a n  the u n c e r t a i n t y i n the  r e a c t i o n r a t e s c a u s e d by t h e u n c e r t a i n t y i n t h e p a r a m e t e r s o f the  optical  model.  Changing the depth, V , o f the p o t e n t i a l has only a second order due  effect:  to r e f l e c t i o n  f on V .  the f i r s t  and they  order  a r e c a n c e l l e d by t h e d e p e n d e n c e o f  The e f f e c t o f t h e s u r f a c e  I n c r e a s i n g i t by 2 5 % w i l l b u t , more i m p o r t a n t ,  e f f e c t s i n E q . (C-GO) a r e  t h i c k n e s s , a , i s more p r o f o u n d .  g e n e r a l l y i n c r e a s e t h e f f a c t o r by 2 5 % 32 4  w i l l , s o m e t i m e s , a s f o r " "S +  He,  increase  /\R  by a f a c t o r o f 2 (see F i g . 6 ) . A l l i m p o r t a n t  can  then.be r e l a t e d t o t h e r a d i u s , R , o f the d i f f u s e w e l l , and o 32 4  to  t h e /\R n e e d e d t o o b t a i n t h e r a d i u s o f t h e ESW.  uncertainties  F o r " "S 1  q a t T - 3.0 x 10* °K, i . t can'" e a s i l y be c a l c u l a t e d , u s i n g e q .  He,  C-GO,  t h a t c h a n g i n g t h e r a d i u s f r o m R = 5.6 t h e r e a c t i o n r a t e by r a d i u s and  fm  a f a c t o r o f 5.0.  the s u r f a c e  t o R = 6.6  Uncertainties in  t h i c k n e s s t h e n seem t o  u n c e r t a i n t i e s o f a f a c t o r o f 5.0  fin i n c r e a s e s the  introduce  i n the r e a c t i o n r a t e s i n v o l v i n g \  L|.  He  channels . I n the p r e c e d i n g  d i s c u s s i o n , i t was  assumed t h a t  o p t i c a l model i s a p r o p e r r e p r e s e n t a t i o n of the channels. is  on  We  can  a l s o s t u d y how  the u n d e r l y i n g p h y s i c s .  optical  this  n  M  = - V  ,  -I- e  (1  Q  end,  ( r - R ) / a .~1 o v)  v  we  ...  - l W  J  h a v e v a r i e d a... w h i l e k e e p i n g  of  <C ' cr  h a v e u s e d as  our  .  °  a„  , +  (1  e  constant.  ( r - R ) / a . -1 o ' w)  v  J  This  represents  V  o f t h e ways i n w h i c h t h e commonly u s e d o p t i c a l mode], c a n  modified and  s e n s i t i v e the v a l u e  To  W  one  reaction  potential,  V(r)  a n d we  He  the  the  to represent  d i f f e r e n t h y p o t h e s e s as  l o c a t i o n o f the  compound n u c l e u s  t o matte]?  be  distribution  formation. 32  N u m e r i c a l c a l c u l a t i o n s have been c a r r i e d out l|,|  T  +e\  and  1 2  C  +  1 2  C  w i t h a„  / a„. V  are presented  on F i g .  12  good a p p r o x i m a t i o n  to the  completely  a^.  i f a^ >" ;  .  The  r a t i o s T„  W  The  low  can be  t h e n e a s i l y be  an u n d e r e s t i m a t e  expected  important  particle  seen f o r  energy cross s e c t i o n obtained w i t h  nuclei with  large Z  (Z >  / a ) /T , w  V'  W  (ESW)  A  i t b r e a k s down  c r o s s s e c t i o n i s then dominated  i n t h e b a r r i e r as  t o be  S + CX ,  Whereas the e q u i v a l e n t s q u a r e w e l l i s a  ] 2  absorption  (a„  A  d i f f u s e w e l l f o r a^ = a  The  for  by  specially 20)'.  by  ] 2  C +  C on  Fig.  the b l a c k n u c l e u s  a n o t h e r f a c t o r o f 5. f o r alpha p a r t i c l e s  However, the data  .  could  This  is  i n c i d e n t on  available for  c h a n n e l s i n r e a c t i o n c r o s s s e c t i o n s a t low  13  alpha  energy i s f o r  '16  -  31  P  (p,o'.)  Since  -  27  St,  the Q v a l u e o f those  reactions i s positive,  transmission f u n c t i o n f o r protons  i s approximately  a t a g i v e n e n e r g y i n t h e compound n u c l e u s , particles.  The  reasonably  alpha p a r t i c l e  large.and i s not  the equal,  to that f o r alpha  !  transmission function i s  expected  t o show a b s o r p t i o n i n  the b a r r i e r ,  w i t h b l a c k n u c l e i c r o s s s e c t i o n s and mass numbers c o m p l e t e l y the b a r r i e r . estimates  of alpha p a r t i c l e The  C +  s e c t i o n o f the  a f a c t o r o f 5.  A  J  the cross,  the d e t a i l e d ' s h a p e o f  the  20% i n c r e a s e i n the d i f f u s e n e s s parameter  C +  ' C s y s t e m c o u l d be h o p e d t o p e r m i t  recent results a  of Ref.  3  MeV  the  22 may  indicate  any  that  extrathe  tool.  Concerning  Nuclear  Reactions  a n a l y s i s o f the r o l e of the o p t i c a l p o t e n t i a l i n  b a r r i e r p e n e t r a t i o n suggests nuclear -reactions. reactions,  sensitive  2  Conclusions  Our  i n d i c a t e how  C r e a c t i o n i s on  o p t i c a l model i s too crude General  ft-W.  Only a model t h a t would r e p r e s e n t c l o s e l y  ]2  ^•  at A  p o t e n t i a l i n c r e a s e s the c r o s s s e c t i o n a t G ^  o f the imaginary  The  channels  reliable  12  o p t i c a l model chosen.  polations.  data i s needed to p e r m i t  C results  C 1  p h y s i c s of the  of absorption i n  12  ]2  by  n e g l e c t s the p o s s i b i l i t y  .More e x p e r i m e n t a l  ]2  then e x t r a p o l a t i n g to h i g h e r  the v a l u e  a number o f g e n e r a ] , c o n c l u s i o n s  They c o n c e r n  about  the r o l e o f the r a d i u s i n resonance  of. t h e n u c l e a r r a d i u s i n • r e a c t i o n s and  o f a b s o r p t i o n c r o s s s e c t i o n s , f a r below the b a r r i e r  to probe  the the  use  tall  of the nuclear d e n s i t y The  to  distribution.-  many-channel theory  h a v e a much more c o m p l i c a t e d  of resonance r e a c t i o n s  geometry  than the  one-channel  p o t e n t i a l - s c a t t e r i n g p r o b l e m whose wave p r o p e r t i e s we in  Sec. 4.  We  s h a l l show t h a t i t i s r e a s o n a b l e  •scattering matrix  appears  described  t o decompose t h e  i n a manner s u c h t h a t e a c h r e a c t i o n  channel  possesses the one-channel p o t e n t i a l - s c a t t e r i n g p r o p e r t i e s . we  c a n vise o u r e a r l i e r  results.  Then  r e s u l t s f o r t h e o n e - c h a n n e l c a s e t o remove  many o f t h e a r t i f i c i a l theory  1  ""square-well" aspects  A similar  treatment  of the  resonance  f o r nuclear reactions  without  ]2 b a r r i e r s 'was  g i v e n i n an e a r l i e r p a p e r by  The framework  general  theory  Vogt.  o f n u c l e a r reactions'""""' p r o v i d e s  a  i n w h i c h a l l c r o s s s e c t i o n s c a n be d e s c r i b e d , i n t e r m s  of l e v e l parameters. parameters  The  level  r e l a t i o n b e t w e e n c r o s s s e c t i o n s and  i s made i n two s t e p s .  First  t h e c r o s s s e c t i o n , XT'  ,,  cc' for  an i n i t i a l  channel  of s t a t i s t i c a l  . "  channel  c' , i s w r i t t e n i n t e r m s  s p i n f a c t o r s and c o l l i s i o n m a t r i x  the c o l l i s i o n m a t r i x level  c and. a f i n a l  components, U  components.  ,, a r e w r i t t e n i n t e r m s  Next of  parameters c  c  t .  =  e  "where the  1  (  ^ c  +  (g  - c') n  are phase  shifts  + i£  c c t  X  X  l  r|  and t h e A ^ ,  c  r| , A c  X X I  )  -  (15)  the components o f a  A  m a t r i x whose i n v e r s e i s (  A  _  L  > X X T  =  (E  X  -  E)S  A  A  ,  +  A  A A  ,  -  (|)|  X  X  l  (-16)  - MS  -  T h i s form o f the framework i s c o m p l e t e l y g e n e r a l :  i t applies  to a l l approximate forms from the Breit-W.igner Formula Hauser-Feshbaeh theory. parameters  The q u e s t i o n i s ,  to the  how do t h e l e v e l  c h a n g e when a r e a c t i o n c h a n n e l i s a s s u m e d t o i n c l u d e  an a v e r a g e Saxon-Woods  potential?  The s q u a r e w e l l ordinary reaction  interaction  i s easily  adapted t o the  theories.  F o r a s q u a r e w e l l , b o t h t h e p a r t i a l w i d t h s and t h e level shift  c a n be f a c t o r e d i n a manner w h i c h s e p a r a t e s o u t t h e  many-body f e a t u r e s , o f t h e p r o b l e m a n d w h i c h c l e a r l y wave p r o p e r t i e s - a s s o c i a t e d w i t h t h e a v e r a g e  r;= c  A = A  d i s p l a y s the  interaction:  m  » r' - v »; <yV p  8Ac  - I >c  S  S ,  P X  ^  ____  The s p e c t r o s c o p i c f a c t o r s ,  (Y*V  .  '  (.18)  I  A  ^, a r e e s s e n t i a l l y  c o e f f i c i e n t s w h i c h measure t h e p r o b a b i l i t y  statistical  f o r f i n d i n g the  compound n u c l e u s i n t h e p a r t i c u l a r mode s p e c i f i e d by t h e c h a n n e l number, c.  S i n c e t h e y d e p e n d on a v e r a g e s o v e r t h e n u c l e a r v o l u m e  they are i n s e n s i t i v e the e f f e c t s the problem,  to the d e t a i l s of the n u c l e a r surface.  o f t h e s u r f a c e i n f l u e n c e o n l y t h e one-body a s p e c t s o f the s i n g l e p a r t i c l e widths  s i n g l e - p a r t i c l e reduced widths  (7  from a square average i n t e r a c t i o n  )  .  or the corresponding Therefore the t r a n s i t i o n  t o a Saxon-Woods  a f f e c t s only the s i n g l e - p a r t i c l e width. is  Thus  interaction  The s i n g l e - p a r t i c l e w i d t h ,  a f f e c t e d , by t h e d i f f u s e - e d g e i n t h e manner d i s c u s s e d , i n S e c . 2  a n d S e c . 3.  Thus t h e i n s e n s i t i v i t y  of the spectroscopic factors to  t h e p r e c i s e v a l u e o f t h e m a t c h i n g r a d i u s makes o u r wave  analysis  _ 4 9 _  apply  i  t o each r e a c t i o n channel s e p a r a t e l y . The  conventional  :  treatment o f nuclear  r e a c t i o n s by  the black-box o r resonance t h e o r i e s i s e s s e n t i a l l y treatment. this  Because o f t h i s f a c t  the n u c l e a r  a  radius  t r e a t m e n t t o f i t o b s e r v e d r e a c t i o n r a t e s was  square-well 'j  required i n artificially  3 large. for  F o r s e v e r a l decades  nuclear  the usual value  r e a c t i o n s was R = l.'l-  ' ( A ^  radius  •  + A^  3  o f the nuclear  )  F  M  where A ^ i s a t o m i c w e i g h t o f t h e t a r g e t n u c l e u s and A ^ t h a t o f t h e bombarding-particle.  On t h e o t h e r h a n d t h e u s u a l  optical  model  radius f o rnucleons i s R = 1.25 A ] / and  f o r heavy  fm  3  (50).  ions R - 1.25 ( A * / + A ? / )  fm  (51)  R = 1.09 ( A ^  fm  (52)  3  or  3  perhaps  For b o t h nucleons and heavy ions  3  + A ^  3  )  the d i f f e r e n c e i n the r a d i i i s  u s u a l l y b e t w e e n one a n d two f e r m i s .  I t i s our conclusion  larger' r a d i i were a r e s u l t o f t h e square w e l l Before  e m b a r k i n g on an e x p l a n a t i o n  that the  treatment. o f the d i f f e r e n c e s  b e t w e e n o l d a n d new r a d i i we make some r e m a r k s on t h e c u r r e n t The  charge r a d i i  of nuclei,  fashions.  a s m e a s u r e d by e l e c t r o n s c a t t e r i n g a n d  1/3 m e s i c atoms a r e 1.09 x A nuclear  density  fermis.  I f t h i s i s taken to r e f l e c t the  as w e l l as t h e charge then a n u c l e o n s h o u l d ' f e e l the  same r a d i u s - t h e same v a l u e s h o u l d a p p l y  1.09  (A-j  ion reactions.  Now  + A.-,  ) s h o u l d t h e n be t h e r a d i u s f o r h e a v y  t h e r e a r e some a d d i t i o n a l e f f e c t s w h i c h  t o i n c r e a s e the o p t i c a l model r a d i u s s l i g h t l y slightly  d e p e n d e n t on t h e s h e l l s t r u c t u r e .  • p o l a r i z a t i o n - an i n c o m i n g toward  it.  nucleus  nucleon  pulls  The s e c o n d i s t h e n e u t r o n  •- an e x c e s s  The f i r s t  the t a r g e t  excess  i s core nucleons  i n the surface o f a  which l e a d s t o t h e i s o t o p l c s p i n term S u c h a t e r m means t h a t t h e d e n s i t y  beyond the charge.  Both  .justifies  uncertainty  i n the r a d i i  e f f e c t s determine  involved.  t h e s m a l l amount by w h i c h  exceed the charge r a d i i .  T h e r e i s some  as w e l l .  the product V  1% - we know o f no e x p e r i m e n t  q  i n the  extends  e f f e c t s c a n be e s t i m a t e d  a n d d e p e n d on t h e p a r t i c u l a r n u c l e u s  magnitude roughly radii  o f these  tend  a n d t o make i t  optical potential..  roughly  potential.  ] /~  l/3  Similarly  to the o p t i c a l  Although 2 R  very  only  Their current  experimental  t h e wave r e s o n a n c e  a c c u r a t e l y - perhaps to  which unambiguously determines  r a d i u s o f 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  the  to anything  like ,  ] /3 t h i s accuracy.  Therefore  not only a reasonable By  t h e same t o k e n ,  the choice t o r nucleons  o f 1.25 A  c h o i c e b u t a l s o has about a 10% u n c e r t a i n t y .  a.n a l p h a p a r t i c l e  r a d i u s o f 1.6 i s a l s o  reasonable  From o u r a n a l y s i s o f wave p r o p e r t i e s we s e e t h a t a r e s e v e r a l ways i n w h i c h square w e l l modifies difference  the t a i l  (Fig. 5  there  t h e change from an o p t i c a l p o t e n t i a l  the r a d i u s .  First  of.all,  to a  there i s the  i n r e f l e c t i o n b e t w e e n t h e w e l l s w h i c h c a n be c o m p e n s a t e d  f o r by a d i f f e r e n c e i n r a d i i ; in  is  secondly,  t h e r e i s t h e wave  oscillation  o f the r e a l pari; o f the o p t i c a l p o t e n t i a l which  leads  ) t o a d i f f e r e n c e i n r a d i u s between, t h e o p t i c a l p o t e n t i a l and.  its  e q u i v a l e n t square w e l l ;  tail  of the  imaginary  thirdly,  p a r t o f the  there i s a b s o r p t i o n  d e p e n d e n c e on  the n u c l e a r  i t has  out  turned  cross s e c t i o n s f o r which  are  Vary through, t h e p e r i o d i c t a b l e  size-resonance ( s e e F i g . 2) .  and  o t h e r heavy i o n s a l l the s i z e - r e s o n a n c e  out  - particles  reaching  needs t o examine  r a d i u s i s unambiguous.  that there  the n u c l e u s  are  For  the  1  as  the  e f f e c t s which For  alpha  e f f e c t s are  absorbed.  particles washed  I t has  diffuse-edge reactions.  no  model f o r alpha p a r t i c l e  shown ( i n S e c .  t o be  that of a  o p t i c a l model p o t e n t i a l s u i t a b l e f o r alpha We  m i g h t t h e r e f o r e be be  appropriate  o f F i g . 9 has  c h o s e n t° be  a value  a l p h a p a r t i c l e s has  tempted to say  square  o f a b o u t 0.5  reflection factor.  a. v a l u e b e t w e e n 3 and  To  t h a t the  But  black-  5.  directly  can  t h u s i n c r e a s i n g the p e n e t r a t i o n f a c t o r .  square  such a  the choice  reflection factor for The  increase The  transmission  p r o p o r t i o n a l to  enhance the b l a c k - n u c l e u s  a f a c t o r o f 3 t o 5 we  particle  In such a choice  fermis.  From F i g . 9 the  f u n c t i o n s f a r below the b a r r i e r are  the  t h a t o f the e q u i v a l e n t  optical potential.  i g n o r e s wave r e f l e c t i o n .  f u n c t i o n s by  2)  absorption  g i a n t r e s o n a n c e s t r u c t u r e b u t n e i t h e r does  radius should  w e l l o f the AR  as w e l l  reactions.  s t r e n g t h f u n c t i o n was  nucleus  evidence"  l a r g e r a d i u s came f r o m c h a r g e d p a r t i c l e r e a c t i o n s  neutron  well.  2  Early 27  I n the b l a c k - n u c l e u s the  j  nucleons  i  for  the  potential.  A reassessment o f the o l d a n a l y s e s only the h e a v y i o n a b s o r p t i o n  in  the  transmission the n u c l e a r  radius,  required increase  in  -  the  radius  i s a b o u t 0.5  ' The p e n e t r a t i o n  l  52 -  fermis*.  f a c t o r a t an e n e r g y , E , f a r b e l o w t h e Coulomb  b a r r i e r , B, d e p e n d s on t h e n u c l e a r P  k 11 G ~  2  ( k R)  oc  radius roughly  k R  "  2  k  as  R  where  For  a l p l i a ' p a r t i c l e s k t y p i c a l l y has values  b e t w e e n 1.0 a n d 2.0.  Thus t h e wave o s c i l l a t i o n . («/\R i ; 0.5) a n d t h e wave together  a c c o u n t f o r t h e 1.0 fm d i f f e r e n c e b e t w e e n t h e  r a d i u s and t h a t o f modern o p t i c a l p o t e n t i a l s . of alpha  reflection black-nucleus  The e a r l y a n a l y s i s  decay r a t e s i n heavy n u c l e i r e q u i r e d s i m i l a r  anomalous  l a r g e r a d i i w h i c h have been b r o u g h t i n t o agreement w i t h modem " 1  values  by our-wave a n a l y s i s . "  a t a l l t h a t any n u c l e a r radii  of nuclear The  reflection  the i n d i v i d u a l  large  particle  structure.  sum-rule l i m i t s  square w e l l values  i s no e v i d e n c e  r e a c t i o n r a t e s r e q u i r e anomalously  o r t h a t they throw i n t o q u e s t i o n  picture  the  I n our view there  a s s o c i a t e d w i t h reduced widths are  w h i c h n e e d t o be m o d i f i e d  factor of a diffuse-edge  well.  t o take  i n t o account  According  to  (20)  a b o v e , f o r a n y r e a c t i o n c h a n n e l we c a n w r i t e a s i n g l e p a r t i c l e w i d t h  r V = 2P f (.v4V  (53)  as  -  S3  -  Where F, i s the-: c o n v e n t i o n a l p e n e t r a t i o n f a c t o r ,  ( 6 ) , and  (y  )"  I  the s i n g l e - p a r t i c l e  reduced width  of the w e l l .  E q u a l l y , we  can  write  (yV  f(V  =  X  where f i s the r e f l e c t i o n the s i n g l e - p a r t i c l e  h  p  )  2  ffi M 2  =  (SM)  2  f a c t o r d i s c u s s e d a b o v e and  reduced width  ( y ' 1\) 2 i s I SW  o f the e q u i v a l e n t square w e l l  (ESW)  2  It  i s f y ) which, has t h e c o n v e n t i o n a l s u m - r u l e • v a l u e o i SW di /mR ^ w h e r e R i s t h e r a d i u s o f t h e ESW. Therefore a true ' o o 2  particle  l e v e l s h o u l d ' h a v e a r e d u c e d width, e x c e e d i n g  s u m - r u l e l i m i t by  the r e f l e c t i o n  a v a l u e b e t w e e n two applies  and  five.  Our  to a l l charged p a r t i c l e  and. g i v e s q u a n t i t a t i v e e s t i m a t e s F a r below the b a r r i e r optical potential  can  f a c t o r f which  the  single  conventional  typically  has  w o r k shows t h a t t h i s f a c t o r  r e a c t i o n s as w e l l as  to-neutrons  of f. the a b s o r p t i o n i n the  t a i l of  the  dominate the whole a b s o r p t i o n p r o c e s s .  have warned o f the dangers o f u s i n g c o n v e n t i o n a l o p t i c a l to  c a l c u l a t e r e a c t i o n r a t e s w h i c h a r e d o m i n a t e d by  We  c a n , h o w e v e r , t u r n t h e a r g u m e n t a r o u n d and  We  models  their  tails.  suggest that  the  measurement o f a b s o r p t i o n c r o s s s e c t i o n s f a r below the b a r r i e r used to determine n u c l e a r p r o p e r t i e s at l a r g e r a d i i . p r o b l e m o f c o n s i d e r a b l e i m p o r t a n c e i n modern n u c l e a r 28 29 because of f i s s i o n isomerism , m e s o n i e atoms " and question of nuclear study  i t i s clear  clusters  t h a t any  i n r e g i o n s o f low  nuclear density  This  be  is a  physics the  density.  general From  (whether nucleons  our or  -  other clusters)  extending  absorption process . to perform  -  t o l a r g e r a d i i , can dominate the  1 1 w o u l d be v e r y i n t e r e s t i n g a n d v a l u a b l e  n u c l e a r r e a c t i o n s w i t h i n t e n s e beams a t l o w e n e r g i e s  to examine, s a y , the p r o b a b i l i t y in  5'-l  o f alpha p a r t i c l e  absorption  t h e b o m b a r d m e n t o f many l i g h t n u c l e i . ' S i m i l a r l y ,  w i t h • p r o t o n . o r h e a v y i o n beams m i g h t a l s o y i e l d  experiments  anomalously l a r g e  cross sections.  36 20 2M I s " 0 much more t i g h t l y b o u n d t h a t Ne o r Mg?  Does t h e d i s t a n t  tail  nucleons  o f the n u c l e a r d e n s i t y i n the l a t t e r  or alpha p a r t i c l e  clusters? 12  I t i s likely  12  that the k i n d  22  of s u r p r i s e s found  i n the  many o t h e r c a s e s .  The r e s u l t s may a l s o c h a n g e many o f o u r n o t i o n s  concerning  reaction rates.  stellar  C +  present  C reactions  will  abound i n  References  G. B r e . i l : a n d E . P . W i g n e r ,  i I  P h y s . R e v . 49", 5 1 9 , 642  H. A. B e t h e , R e v . Mod. P h y s . 9 , 69 V.W.  Weisskopf  a n d D.H.  Ewing,  a n d J . B l a t t a n d V.W.  (1936).  (1937).  P h y s . R e v . 57 , 472 Weisskopf,  (1940)  Theoretical  N u c l e a r P h y s i c s (John W i l e y & Sons,  I n c . , New  Y o r k , 1952) . H.H.  Barschall,  P h y s . R e v . 8 6 , 431 ( 1 9 5 2 ) .  see, f o r example,  E . A l m q v i s t , D.A.  a n d E. V o g t , Conference b y D.A.  Bromley, J . Kuchner  i n Proceedings of the I n t e r n a t i o n a l  on N u c l e a r S t r u e t u r e , K i n g s t o n , e d i t e d  Bromley  a n d E. V o g t  ( U n i v e r s i t y o f Toronto  P r e s s , T o r o n t o , Canada, 1 9 6 0 ) , p. 736. K.W.  McVoy, P h y s . L e t t e r s 91  1 7 , 42 ( 1 9 6 5 ) , A n n a l s o f P h y s i c s 4 3 ,  (196 7) .  H. F e s h b a c h ,  C.E. P o r t e r a n d V.W.  W e i s s k o p f , Phys. R e v / 96,  448 (19 54) . G.R..  Satchler, i n International Nuclear Physics  C o n f e r e n c e , G a t l i n b u r g , e d i t e d b y R.I,. B e c k e r e t a l . (Academic  P r e s s , New York,. 196 7) , p . 1.  L . E i s e n b u d a n d E.P. W i g n e r , 281  • A.M.  P r o c . N a t l . Acad.  ( 1 9 4 1 ) ; E.P. W i g n e r ,  S c i . U.S. _27,  P h y s . Rev.' 70 > 15 ( 1 9 4 6 ) , Phy  Rev.  7 0 , 606 ( 1 9 4 6 ) ; E.P. W i g n e r a n d L . E i s e n b u d , P h y s  Rev.  7 2 , 29 ( 1 9 4 7 ) ; T. T e i c h m a n n a n d E.P. W i g n e r , ' P h y s  R e v  - 1Z»  1 2 3  0° -) • r,;  L a n e and'D. R o b s o n , P h y s . R e v . 1 5 1 , 774  (1966).  10.  E.W.  Vogt, i n Advances  i n N u c l e a r P h y s i c s , V o l . 1, e d i t e d  by M. B a r a n g e r a n d E.W.  Vogt,  (Plenum P r e s s ,  New  Y o r k , 1968) , p . 2 6 1 . 11.  A.M.  L a n e , R.G.  Thomas  a n d E.P. W i g n e r , P h y s . R e v . 9 8 , 693  ,  (1955) . 12.  E.W.  V o g t , R e v . Mod. P h y s . 34, .723 ( 1 9 6 2 ) .  13.  E.W.  V o g t , G . J . M i c h a u d a n d H. R e e v e s , P h y s . L e t t e r s 1 9 , 570 (1965) .  14.  L . S.cherk a n d E.W.  15.  N. A u s t e r n , A. ' P r a k a s h a n d R.M. 253  V o g t , Can. J . P h y s . 4 6 , 1119 (1958) . D r i s k o , Ann. Phys.  (1966) .  16.  D.C.  17.  R.G. Thomas, P h y s . R e v . 8JL, 148 ( 1 9 5 1 ) .  18.  G. B r e i t , H a n d b u c h d e r P h y s . , V o l . X L I , ( S p r i n g e r  P e a s l e e , N u c l . P h y s . _3, 255 (1957) .  Berlin, 1 9 . . M.A.  (N.Y.) _39,  Verlag,  1.9 59) , p . 1.  P r e s t o n , P h y s i c s -of t h e N u c l e u s  (Addison Wesley  Publ.  Co. , New Y o r k , 1962) . 20.  R.V. Wagoner, W.A.  21.  J.W.  F o w l e r a n d F. H o y l e , A p . J . _148, 3 (1967) . •  T r u r a n , A.G.W. Cameron a n d A. G i l b e r t , 563  ( 1 9 6 6 ) ; D. B o d a n s k y ,  Phys. Rev. L e t t e r s  D.D.  C a n . J . P h y s . _44,  C l a y t o n a n d W.A.  2 h , 161 (1968)  a n d W.D.  Fowler,  Arnett, '  E x p l o d i n g S t a r Models and Supernovae, K e l l o g  Preprint,  1968. 22.  J.R. P a t t e r s o n , H. W i n k l e r and C.S. Z a i d i n s ,  t o be p u b l i s h e d  (1969) . 23.  J.R. P a t t e r s o n , H. W i n k l e r a n d H. S p i n k a , P r i v a t e (1968).  communication  21 1  J.W.  T r u r a n , C . J . H a n s e n , A.G.W. Cameron and A. Can. J . P h y s .  151  Gilbert,  (1966) .  25.  P.M.  E n d t and C. V a n  26.  W.A.  F o w l e r a n d F. H o y l e , A s t r o p h y s . J . S u p p l . 9 1 , 201  .27.  der Lenn, N u c l . Phys. A105,  1,  s e e , f o r e x a m p l e , J . P . B l a s e r , F. Boehm, P. M a r m i e r D.C.  P e a s l e e , P h y s . A c t a 24, 3 ( 1 9 5 1 ) .  discussion  A  1967. (196'l)  and  complete  of e a r l y n u c l e a r r e a c t i o n s i s f o u n d i n  B l a t t a n d W e i s s k o p f s b o o k , r e f . 3. 28.  V.M.  S t r u t i n s k y , N u c l . P h y s . A_95, !20 l  ( 1 9 6 7 ) ; E.  a n d J . P . T h e o b a l d , N u c l . P h y s . A112, 29.  S. D e v o n s and I . ' D u e r d o t h , A d v a n c e s Vol.  2, e d i t e d by M.  P u b l . Co., New  603,  Mlgneco 1968.  i n Nuclear Physics,  B a r a n g e r a n d E.W.  Y o r k , 1 9 6 9 ) , p . 29 5.  Vogt,  (Plenum  F i g i i r e C ap 11 ons  Figure  1.  Comparison of  the low l y i n g r e s o n a n t s t a t e s  square w e l l and o f well for radius  the c o r r e s p o n d i n g d i f f u s e  s-wave n e u t r o n s .  occur at very  and w i t h v e r y n e a r l y  It  then p o s s i b l e  the resonance  between  of resonance the  two w e l l s  o n l y i n t h e p e n e t r a b i l i t y and s h i f t  Figure  2.,.  Comparison of the s u r f a c e functions  energy  (except  at E  -- 53 MeV)  reproduce  of a diffuse is  a.  Because  c l o s e .to 1.,  it  well.  then contained  functions.  of protons for d i f f e r e n t values  thickness  are  t h e same  f o r a square w e l l to  the i n t e r i o r p r o p e r t i e s The d i f f e r e n c e  closely  resonances  t h e same r e d u c e d w i d t h  the r e d u c e d w i d t h o f is  both  a r e s o n a n c e ' w i t h t h e same r e d u c e d w i d t h  t h e two w e l l s  for  edge  a d j u s t e d so t h a t  a t E = - O.M-3 MeV, a l l t h e o t h e r l o w l y i n g in  the  Once t h e d e p t h a n d  of the square w e l l are  w e l l s have  of  of  the t r a n s m i s s i o n  i s necessary  to o b t a i n the r e f l e c t i o n f a c t o r s .  to  The ' T  ?  compare are  o b t a i n e d f r o m t h e t r a n s m i s s i o n f u n c t i o n s by u s i n g T« = ^2  (  a  / ( I l-t;/ !)  .  1  " 0)  It  by a f a c t o r  one g e t s an e x c e l l e n t The I n t e r i o r o f similar  the  resonance  is  seen  t h a t by m u l t i p l y i n g  f (a.) , i n d e p e n d e n t o f approximation. t o T ^  (a ¥ 0) •  two w e l l s m u s t t h e n h a v e  properties.  Only the  energy,  very  penetrabilities  j i I  vary.  I t then, seems p o s s i b l e , a s i t was f o r  neutrons,  to replace  a d i f f u s e w e l l , by a s q u a r e  well. !  Figure  3.  The s h a p e o f t h e r e a l p o t e n t i a l f o r s-wave  protons  32 ( • S + p) .  The  Coulomb p o t e n t i a l (V ) d o e s a d d a  r a d i u s dependent term.  However  i t sradius  dependence  i s much w e a k e r t h a n t h e r a d i u s d e p e n d e n c e o f t h e nuclear p o t e n t i a l (V^) • Coulomb not  field  In f i r s t  adds o n l y a c o n s t a n t  the  term w h i c h does  c h a n g e t h e r e f l e c t i o n p r o p e r t i e s o f t h e two w e l l s .  Protons  should  behave l i k e n e g a t i v e  As i s t h e case f o r n e u t r o n s , to f i n d  should  energy  i t should  an e q u i v a l e n t s q u a r e w e l l  particles.  This  approximation  neutrons.  t h e n be p o s s i b l e  0^^.,)  f o r charged  The d i f f e r e n c e b e t w e e n t h e two w e l l s  again  reside only  in. the r e f l e c t i o n  i s v i n d i c a t e d b y F i g . 2.  In this  factor.  example, the  n u c l e a r p o t e n t i a l i s c h o s e n t o h a v e a Saxon-Woods s h a p e w i t h d e p t h SO MeV,  r a d i u s 3.96fm,  t h i c k n e s s O.Sfm. Figure  4.  and. s u r f a c e •  '  R a t i o o f t r a n s m i s s i o n f u n c t i o n s o f an o p t i c a l p o t e n t i a l  4 suitable for  thickness.  32 S f o r two v a l u e s  He -I-  (  a  =  w e l l s h a v e t h e same V either  o f the surface  -^"V'^J^Q ( a = 0 . 0 ) ) .  The two  a n d t h e same R . I t seems o o t h a t t h e i n t e r i o r o f t h e two w e l l s show d i f f e r e n t  1!  resonance, b e h a v i o u r  or that d i f f e r e n t  phenomena'  o c c u r u n d e r t h e Coulomb b a r r i e r .  It will  in  the  the n e x t  variations  sections of  t h a t most o f  be  seen  energy  t h e r a t i o comes f r o m c h o o s i n g  w e l l s w h i c h d o n ' t have  t h e same I n t e r i o r  Absorption. I n the b a r r i e r w i l l dominant only at very  be s e e n  low energy  (E <  two  properties.  to  become  1 MeV i n  this  case) .  Figure  5.  A diffuse well  (SOLID)  and the  functions.  and a s q u a r e  The d e p t h and. r a d i u s o f  c h o s e n so t h a t  that  the  that  the  diffuse  function, of of  the  the  well  a t E = 0 , and :  that resonance  be t h e  same  w e l l at the square w e l l  The i n t e r i o r p r o p e r t i e s be t h e same as  wave  the square  i t had a resonance  reduced width of of  potential  c o r r e s p o n d i n g s-wave z e r o - e n e r g y  were  as  (DASHED)  of  the  two w e l l s w i l l  can be s e e n f r o m F i g . 1 . diffuse w e l l continues  square w e l l r a d i u s .  radius. then  The wave to r i s e  As d i s c u s s e d  outside  i n the  text  this  c a n be r e l a t e d t o t h e p e n e t r a b i l i t y o f . t h e  diffuse  well  and the d i f f e r e n c e s  diffuse  potential  between  c a n a l l be r e l a t e d  the square  to the  and  ratio  w h e t h e r we be s t u d y i n g n a r r o w resonances the  t h e Coulomb b a r r i e r  transmission functions  (Sec. have  f a r below  5) .  The d i f f u s e  of  complex  (Sec.  l  l)  or  potentials  p o t e n t i a l has been chosen  to  a Saxon-Woorls • s h a p e w i t h i n t e r i o r wave number K ,  r a d i u s R , and s u r f a c e o  thickness  a.  I  I i  Figure  6.  The v a l u e o f A R / R f,  as a f u n c t i o n  and o f the r e f l e c t i o n f a c t o r  o f a/R  o.  o  T h i s i s t h e most f r e q u e n t l y o f R e f . 1.3. is referred  2m V R  a n d K R /IT ( -  -  used s e c t i o n  R and f a l l o w  determine the cross sections  7.  calculations varying the  7  1  j  one t o  and the r e s o n a n c e w i d t h  from the r a t i o o f the t r a n s m i t t e d  f o r a g i v e n i n c o m i n g wave.  2  the reader-  a r e known.  The e n e r g y v a r i a t i o n o f t h e r e f l e c t i o n f a c t o r calculated  2  of F i g . 1  o f d i f f u s e w e l l s , once t h o s e o f square w e l l s Figure  /1I. TT J ) .  o o  F o r a w i d e r range o f v a l u e s , t o R e f . 13. /\  2  was waves  We h a v e made o t h e r  t h e Coulomb p o t e n t i a l , b u t k e e p i n g  r e d u c e d mass a n d t h e n u c l e a r p o t e n t i a l t h e same.  The r e f l e c t i o n f a c t o r was t h e n f o u n d t o d e p e n d on (B-E) the  only,  where B i s t h e h e i g h t o f t h e b a r r i e r i n  diffuse well-  The r e f l e c t i o n f a c t o r  takes i t s  maximum v a l u e a t t h e e n e r g y E = B ^ ^ , w h e r e maximum v a l u e o f t h e d i f f u s e b a r r i e r . reflection  i s the  L e t t i n g the  f a c t o r v a r y with, energy w i l l ,  i n some c a s e s ,  improve the a c c u r a c y o f the e q u i v a l e n t square w e l l model (not  f o r the c a l c u l a t i o n o f transmission  however) .  functions  The e n e r g y v a r i a t i o n o f t h e r e f l e c t i o n f a c t o r  d e p e n d s m a i n l y on t h e r e d u c e d mass a n d t h e f i g u r e  may  be  u s e d t o e s t i m a t e i t , by i n t e r p o l a t i o n , f o r c a s e s  not  c o v e r e d by o u r c a l c u l a t i o n s .  the  e n e r g y s c a l e , i s g i v e n b y I'  12 Note- t h a t  for  = E + 10.0 i d  C  -I-  J  ^S  32 C + " S MeV.  Figure  8.  The r a t i o  of diffuse-edge  o p t i c a l mode], t r a n s m i s s i o n  functions  to'equivalent square-well  transmission  ' • • 32 f u n c t i o n s a s a f u n c t i o n o f t h e e n e r g y f o r 'S + ^ ;  The  .  d i f f u s e p o t e n t i a l i s t h a t o f F i g . 'I a n d t h e  I  - e q u i v a l e n t square w e l l has been c a l c u l a t e d from Fig.  6 (/^ R = 0.S8 fm) . ' The r e f l e c t i o n - f a c t o r  ( f = '1.62) i s t h e r a t i o  obtained  a t zero  t h e a b s e n c e o f Coulomb a n d a n g u l a r It /\E  energy i n  momentum  barriers.  i s s e e n t h a t i f one c h o o s e s a c o r r e c t v a l u e o f f o r the e q u i v a l e n t square w e l l the observed  reflection barriers  i s insensitive  t o charge and c e n t r i p e t a l  over a wide range o f e n e r g i e s  c l o s e agreement w i t h contrast,  the r e f l e c t i o n  factor; i n  i t was s e e n i n F i g . 'I f o r a d i f f e r e n t  square, w e l l  ( A R = 0) t h a t t h e o b s e r v e d  depended s t r o n g l y on a l l t h e s e lines  and i s i n  give the corresponding  reflection  factors.  ratios  (The d o t t e d  for.  - *> 7f  *\ J_j /  c o n n e c t e d t o t h e t r a n s m i s s i o n f u n c t i o n s by t h e 1 a p p r o x i m a t e r e l a t i o n T^ ~ 1 ^ / ( 1 +  Figure  9 . ,aR a n d i: f o r p , n ,  ]2  •I He a n d  C i n c i d e n t on t a r g e t  nuclei* o f d i f f e r e n t masses. used had R 1.25  (12 / 1  o 3  2 '/^) .  = 1.2 5 h]'/ , T 3  + A ^ / ) and V  The d i f f u s e p o t e n t i a l s  1.6 + 1.25 A , ) / , and T 3  3  q  = 50 MeV, 75 MeV a n d 100 MeV, ] 2  •for n u c l e o n s , and  alpha  particles  and.  C respectively  a = 0. 5fm i n a l l e a s e s . A R a n d f w e r e  obtained.  f r o m F i g . 6. line  (- - ~) t h e a p p r o x i m a t i o n  Peaslee is  Figure  Wo h a v e a l s o p l o t t e d w i t h  f o r s-wave n e u t r o n s  a dashed  to f obtained  (16) .  The a g r e e m e n t  s e e n t o be q u i t e c l o s e f o r n e u t r o n s a n d  1 0 . The r e s o n a n t and  s-wave f u n c t i o n s  the' e l a s t i c  (Fig.  10b)  by  protons.  ( u p p e r p a r t o f F i g . 10a)  s c a t t e r i n g cross  sections  f o r a d i f f u s e p o t e n t i a l (SOLID)  appropriate  32 to the r e a c t i o n well the  (DASHED)  "S +  and i t s e q u i v a l e n t  ( l o w e r F i g . 10a) . The r a t i o  cross sections i s simply  square  (3.2'l) o f  r e l a t e d t o t h e wave  r e f l e c t i o n f a c t o r (5.2) c a l c u l a t e d i n S e c . 3 t h r o u g h the energy dependence o f t h e l e v e l s h i f t . from the resonant  wave f u n c t i o n s t h a t  I t i s seen  the•reflection  o f t h e d i f f u s e w e l l i s now d i s t r i b u t e d b e t w e e n t h e interior  (resonant)  and e x t e r i o r  p r o p e r t i e s o f the w e l l ;  (penetration)  nevertheless,  t h e sum o f t h e  effects  i s w h a t we w o u l d e x p e c t f r o m t h e z e r o - e n e r g y  barrier  c a l c u l a t i o n s . The d i f f u s e w e l l h a s a S a x o n -  Woods s h a p e w i t h , d e p t h - 6M.9 MeV, r a d i u s and  surface  5.5685 fm.,  t h i c k n e s s 0.5 fm.; t h e e q u i v a l e n t  square  w e l l i s f o u n d t o h a v e d e p t h - 56.3 MeV a n d r a d i u s 6.25 fm.. Figure  11. F r a c t i o n o f the absorption  as a f u n c t i o n o f r a d i u s  f o r d i f f u s e p o t e n t i a l s . w i t h parameters to the r e a c t i o n s (DASHED) .  2 0 8  ]?b  -f-  c <  (SOLID) and.  appropriate 3 2  S +  The r a d i i u s e d f o r t h e Saxon-Woods w e l l s  are  indicated  by v e r t i c a l l i n e s .  i n the b a r r i e r  greatly  the extreme b a r r i e r s  ( ~  increased-by  7 MeV)  i f we  use  for  3 2  S  a l l .absorption  potential  +  and  the n a t u r e  is,  I f the  absorbed, b e f o r e  12.  ( F i g . a)  fm  for  2 0 8  Pb  barrier  badly  ) . i n the  d e s c r i b e d by  barrier cross  the  figure well  s i n c e most p a r t i c l e s w o u l d the n u c l e a r  functions for a variety  absorption  of barriers  i s s e e n t o depend  s t r o n g l y on  the s u r f a c e  potential.  In f a c t , even a s l i g h t i n c r e a s e i n  ( F i g . b)  barrier  absorption;  insensitive  t h i c k n e s s o f the  imaginary  i s seen to c o n s i d e r a b l y on  the  enhance  the  the o t h e r hand, i t i s r a t h e r  t o t h e r e d u c e d mass ( F i g . c) .  .these cases the  be  surface.  Even i n the p r e s e n c e o f moderate  barrier  fm  Hence,  the e q u i v a l e n t square  reaching  the b a r r i e r  the  o f 5.568  +  potential  absorption  Ratios of transmission reactions.  i n the  i n describing absorption  i n fact, physical,  model would f a i l  associated  Saxon-Woods s h a p e f o r  imaginary  region i s c r i t i c a l sections..  occurs  be  energies  (with a r a d i u s ^ R ,  o f 8.8  o f the  i s seen to  I n f a c t , a t low  a conventional  . imaginary  Figure  relative  amount o f a b s o r p t i o n  w i t h heavy n u c l e i .  '. '  The  In a l l  e q u i v a l e n t s q u a r e w e l l mode], i s s e e n  to have c o n s i d e r a b l e  validity  conventional  t h i c k n e s s f o r the  surface  p r o v i d e d we  choose a  imaginary  I . •  •  I I  i .  i  potential the  (a^ =  = 0.5  fm).  In the presence  extreme b a r r i e r s a s s o c i a t e d w i t h heavy  r e a c t i o n s and nuclei  alpha  ( F i g . d)  particle  i t i s seen t h a t the  thicknesses.  interest  equivalent  even f o r  Hence, i t i s of  h i g h energy data  to the  i f one  crucial imaginary  wishes to  energies  !  conventional  t o u n d e r s t a n d ' the d e t a i l s o f the  p o t e n t i a l i n the b a r r i e r  ion  s c a t t e r i n g from heavy  "square w e l l model f a i l s b a d l y surface  of  extrapolate  of i n t e r e s t  to  astrophysics.  Figure  13.  F r a c t i o n o f the a b s o r p t i o n  as a f u n c t i o n o f t h e  f o r a d i f f u s e p o t e n t i a l whose r e a l and  radius  imaginary  parts  h a v e Saxon-Woods s h a p e s w i t h p a r a m e t e r s a p p r o p r i a t e 12  the r e a c t i o n absorption  occurs  12  C +  "C .  f o r s-waves  shows t h a t t h e  The  comparison of  (SOLID) and  relative  i n the b a r r i e r  heightened. VI  which  i s ' i n c r e a s e d , when t h e b a r r i e r  of the  imaginary  absorption.  H e n c e , low  amount o f  energy a b s o r p t i o n  d e t a i l e d i n f o r m a t i o n about the  a b s o r p t i v e p o t e n t i a l i n the n u c l e a r e n e r g y i s '!  MeV.)  is •  surface  p o t e n t i a l (DOTTED)  enhances the r e l a t i v e  yield  g-waves (DASHED)  amount o f a b s o r p t i o n  considerably  may  the  I t i s a l s o seen t h a t i n c r e a s i n g the  thickness a ,  to  barrier  cross  sections  shape o f  surface.  (The  the  A.E.C.L. Rcf. it A-2792-K  NORMALIZED  RESONANCE  STATES  ENERGY  L E V E L S (MeV)  REDUCED  WIDTH  10  8 6  2h  0.01 I 20  J - . 24  I  I  28.  32  I 36  I 40  |. .  co  CO  st  CVJ  O  POTENTIAL •  •i  I  , MeV I  WAVE I  FUNCTION  r , fm.  CD  

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