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On alpha-decay in heavy nuclei Scherk, Leonard Raymond 1967

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ON ALPHA-DECAY IN HEAVY NUCLEI  by  LEONARD RAYMOND SCHERK B.Sc., University of B r i t i s h Columbia, 1965  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n the department of Physics  We accept this thesis as conforming to the required standard  THE UNIVERSITY OF BRITISH COLUMBIA March, 1967  In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that tfu: Library shall make it freely available for reference and study  I further agree that permission for extensive copying of this  thesis for scholarly purposes may be granted by the Head of my Department or by his representatives..  It is understood that copying  or publication of this thesis for financial gain shall not be allowed without my written permission.  n • • PHYSICS Department of The University of British Columbia Vancouver 8, Canada f  Date  OflSul 6y  If / 7  ABSTRACT The a l p h a - p a r t i c l e reduced widths f o r the ground  state  i n P o ^ ^ a r e c a l c u l a t e d on the b a s i s of the n u c l e a r s h e l l 1  model, employing  the technique of Harada, but t r e a t i n g the  n u c l e a r s u r f a c e i n a more d i r e c t manner.  I t i s contended  t h a t the c a l c u l a t i o n s of p r e v i o u s a u t h o r s , who have g e n e r a l l y used a square-edge  nucleus and a Coulomb b a r r i e r rounded-off  by the n u c l e a r p o t e n t i a l of Igo, have, e s s e n t i a l l y , used the e q u i v a l e n t square-edge  nucleus model of Vogt.  T h e i r J.W.K.B.  estimate of the b a r r i e r p e n e t r a b i l i t i e s i s checked by an anal y t i c c a l c u l a t i o n i n Chapter 3 and i s found to be r e a s o n a b l e . It  i s shown i n Chapter 4 t h a t , i n the s c a t t e r i n g of an a l p h a -  p a r t i c l e from the ground s t a t e of P b ^ , the d i f f u s e n u c l e a r o a  edge c o n s i d e r a b l y enhances  the one-body reduced widths and,  i n a d i r e c t manner, that i t s i m i l a r l y enhances d i f f e r e n t i a l elastic scattering cross-section. ner, i t i s demonstrated e q u i v a l e n t square-edge  the one-body In t h i s man-  that the r a d i u s i n v o l v e d i n the nucleus model must be c o n s i d e r a b l y  l a r g e r than that of the d i f f u s e - e d g e nucleus to which i t corresponds. validity nuclei  T h i s i s shown d i r e c t l y i n Chapter 5, where the  of the e q u i v a l e n t square-edge  i s examined.  I t i s contended  nucleus model i n heavy  that t h i s e x p l a i n s the  l a r g e r a d i i found i n p r e v i o u s c a l c u l a t i o n s .  T h i s i s demon-  s t r a t e d d i r e c t l y by r e p e a t i n g the c a l c u l a t i o n of Harada w i t h the d i f f u s e n u c l e a r edge being i n t r o d u c e d i n a d i r e c t manner. A l t h o u g h the e f f e c t s of c o n f i g u r a t i o n mixing have not been d i r e c t l y examined, i t has been concluded that s h e l l model  iii c a l c u l a t i o n s can explain the major part of the empirical decayrates provided manner.  that the nuclear surface i s treated i n a d i r e c t •  iv TABLE OF CONTENTS Page CHAPTER 1  INTRODUCTION  1  CHAPTER 2  A DERIVATION OP THE DECAY CONSTANT  7  CHAPTER 3  EXPERIMENTAL REDUCED WIDTHS  3-1  THE REACTION P o ^  3- 2  COMPARISONS WITH J.W.K.B. AND SQUAREWELL ESTIMATES  24  INDEPENDENT-PARTICLE MODEL DECAY RATES OF P o  29  4- 1  ONE-BODY REDUCED WIDTHS AND EXPERIMENTAL SPECTROSCOPIC FACTORS  32  4-2  ONE-BODY DIFFERENTIAL ELASTIC SCATTERING CROSS-SECTION OF F b 0 8 FOR AN ALPHAPARTICLE AT 90°  41  4-3  HARADA'S FORMULA FOR THE INDEPENDENTPARTICLE MODEL REDUCED WIDTHS  43  4- 4  NUMERICAL  47  CHAPTER 4  » ^208  21  1 2  +  2  2  2 1 2  2  CM  INDEPENDENT-PARTICLE MODEL  REDUCED WIDTHS CHAPTER 5  THE EQUIVALENT SQUARE-EDGE NUCLEUS MODEL  52  5- 1  THE EQUIVALENT SQUARE-EDGE NUCLEUS MODEL  53  5-2  APPLICATIONS TO HEAVY NUCLEI .  54  CHAPTER 6  : CONCLUSIONS  62  BIBLIOGRAPHY  65  APPENDIX A  HARADA'S FORMULA FOR THE REDUCED WIDTH FOR ALPHA-PARTICLE DECAY IN EVEN-EVEN NUCLEI  66  APPENDIX B  TALMI TRANSFORMATION COEFFICIENTS  74  APPENDIX C  CALCULATION OF THE ONE-BODY DIFFERENTIAL ELASTIC SCATTERING CROSS-SECTION  80  LIST OF TABLES  Page TABLE 1  EXPERIMENTAL REDUCED WIDTHS  25  TABLE 2  ONE-BODY POTENTIALS  37  TABLE 3  EXPERIMENTAL SPECTROSCOPIC FACTORS  38  TABLE 4  OVERLAP INTEGRALS  49  TABLE 5  COMPARISON OF DECAY RATES  50  TABLE 6  CALCULATED SPECTROSCOPIC FACTORS  50  VI  LIST OF FIGURES Page FIGURE 1  SURFACE POTENTIAL  FIGURE 2  COMPARISON OF DECAY-RATE PARAMETERS WITH SQUARE-WELL AND J . W. K. B. ESTIMATES (Fb208)  FIGURE 3  SQUARE AND DIFFUSE POTENTIALS  FIGURE 4  RESONANT WAVE FUNCTIONS FOR SQUARE AND DIFFUSE POTENTIALS ( P b )  35  FIGURE 5  RESONANT WAVE FUNCTIONS: WELL DEPTH  EFFECT OF  39  FIGURE 6  RESONANT WAVE FUNCTIONS: SURFACE THICKNESS  EFFECT OF  40  FIGURE 7  ONE-BODY SCATTERING CROSS-SECTION  FIGURE 8  BARRIER ABSORPTION  FIGURE 9  REFLECTION FACTOR  lPb  2 0 8  J  26  (Pb  2 0 8  27  )  34  2 0 8  (Pb (Pb  2 0 8  2 0 8  )  ) "  (Pb  2 0 8  ) 42 59  •  60  vii ACKNOWLEDGEMENTS I wish to thank P r o f e s s o r E . V/. Vogt f o r h i s c o n t i n u a l encouragement and generous a s s i s t a n c e i n the s o l u t i o n of t h i s problem.  T h i s t h e s i s was done while the author was supported  by a b u r s a r y from the N a t i o n a l Research C o u n c i l of Canada.  CHAPTER 1  INTRODUCTION  The aim the  of t h i s t h e s i s i s to estimate the  independentrparticle  r a t e s i n heavy n u c l e i . has had  t h i s aim  A considerable  amount of e a r l i e r work  (1964)), hut past estimates of  a l p h a - p a r t i c l e decay r a t e s have tended to be  two  factors:  The  smaller  the  f a c t o r , which accounts f o r  the many-body a s p e c t s of the n u c l e a r  problem; b) a one-body  which accounts f o r the average i n t e r a c t i o n of  the a l p h a - p a r t i c l e with the daughter n u c l e u s . much of the remaining d i s c r e p a n c y the  than  the  c a l c u l a t e d r a t e s are a product of  a) a s p e c t r o s c o p i c  decay constant,  to which  model can e x p l a i n a l p h a - p a r t i c l e decay  ( c . f . Liang  observed decay r a t e s .  extent  We  between the  find  that  calculated  and  observed decay r a t e s can be removed by a more d i r e c t and  more accurate  treatment of the nuclear  l a t i o n of the  one-body decay constant.  that, apart  surface We  i n the  calcu-  w i l l demonstrate  from a d i f f e r e n c e i n t h e i r r e f l e c t i o n p r o p e r t i e s ,  a heavy nucleus having a d i f f u s e - e d g e  behaves l i k e a  e r a b l y l a r g e r nucleus having a square-edge i n the  consid-  analysis  of a l p h a - p a r t i c l e decay r a t e s or a l p h a - p a r t i c l e s c a t t e r i n g cross-sections„  (The decay r a t e s are r e l a t e d to the  tering cross-sections  i n a simple manner; the decay  i s p r o p o r t i o n a l to the width of the e l a s t i c  scatconstant  scattering cross-  section.) Vogt (1962) has  suggested that one  r e p l a c e a conventional" d i f f u s e - e d g e square-edge nucleus" He  considers  which he  may,  in  general,  nucleus with an  defines  "equivalent  i n the f o l l o w i n g manner.  a simple one-body model i n which the  incident  p a r t i c l e i s s c a t t e r e d by a p o t e n t i a l w e l l . problem without the many-body aspects;  T h i s i s the n u c l e a r  i n f a c t , the s p e c t r o -  s c o p i c f a c t o r (which accounts f o r the many-body aspects of the problem) i s r a t h e r  i n s e n s i t i v e to the nature of the nuclear  s u r f a c e and to the s i z e of the nucleus. diffuse-edge  nuclear  He r e p l a c e s the  p o t e n t i a l w e l l with an " e q u i v a l e n t  square-  edge w e l l " whose r a d i u s and depth a r e chosen so that i t exh i b i t s a resonance a t the resonance energy of the d i f f u s e edge w e l l and so. that the resonant wave f u n c t i o n of the squareedge w e l l s a t i s f i e s  the f o l l o w i n g c o n d i t i o n s :  a) i t has the  same number of nodes as the resonant wave f u n c t i o n of the diffuse-edge  w e l l ; b) i t has the same amplitude as the r e s o -  nant wave f u n c t i o n of the d i f f u s e - e d g e the  square-edge w e l l .  w e l l a t the r a d i u s of  Now the one-body s c a t t e r i n g width (or  decay constant) i s a product of two f a c t o r s :  a) a one-body  reduced width, which depends only upon the amplitude of the resonant wave f u n c t i o n a t the nuclear  radius  the  b) a p e n e t r a b i l i t y , which  i n t e r n a l a s p e c t s of the n u c l e u s ) ;  (and, hence, on  depends only upon the r e f l e c t i v e p r o p e r t i e s of the p o t e n t i a l b a r r i e r outside  of the n u c l e a r  radius.  Apparently the only  d i f f e r e n c e i n the one-body problem between the d i f f u s e - e d g e w e l l and the corresponding e q u i v a l e n t  square-edge w e l l i s the  anomalous r e f l e c t i o n of the l a t t e r ; t h i s d i f f e r e n c e i n r e f l e c t i o n i s assigned  to a " r e f l e c t i o n f a c t o r " .  Vogt (1962) has found f o r neutron s c a t t e r i n g , and Vogt, Michaud, and Reeves (1965) f o r a l p h a - p a r t i c l e s c a t t e r i n g from l i g h t n u c l e i , that the e q u i v a l e n t  square-edge w e l l has a con-  s i d e r a b l y l a r g e r r a d i u s than the corresponding well; i n general,  diffuse-edge  i t has about the same depth as the d i f f u s e -  edge w e l l and the r e f l e c t i o n f a c t o r i s found to be s m a l l . In f a c t , the parameters of the e q u i v a l e n t are  square-edge w e l l  found to be r a t h e r i n s e n s i t i v e to the nature of the r e a c -  t i o n and to the many-body a s p e c t s of the problem so that the diffuse-edge nucleus. apply  nucleus does behave l i k e a l a r g e r square-edge  In t h i s t h e s i s , we w i l l  show that these r e s u l t s  to the a l p h a - p a r t i c l e decay r a t e s of heavy n u c l e i ; i n  f a c t , we w i l l  show that the p r e v i o u s  a l l y used t h i s e q u i v a l e n t  c a l c u l a t i o n s have gener-  square-edge w e l l , thus h i d i n g the  true r a d i u s of the decay problem. I t i s of c o n s i d e r a b l e  value  to show that the d i s c r e p a n c i e s  between the e m p i r i c a l and the c a l c u l a t e d a l p h a - p a r t i c l e decay r a t e s found i n the e a r l i e r c a l c u l a t i o n s have l a i n  i n the method  of c a l c u l a t i o n r a t h e r than i n the i n d e p e n d e n t - p a r t i c l e assumptions.  The f a i l u r e  of the i n d e p e n d e n t - p a r t i c l e  would imply t h a t the model does not i n t r o d u c e  model model  sufficient  cor-  r e l a t i o n s i n t o the n u c l e a r wave f u n c t i o n s  to account f o r the  observed c l u s t e r i n g i n t o a l p h a - p a r t i c l e s .  The f a c t that the  P a u l i E x c l u s i o n P r i n c i p l e becomes l e s s i n h i b i t i v e i n the surface r e g i o n makes i t seem n a t u r a l to a t t r i b u t e much of the c l u s t e r i n g to the nuclear preted  the discrepancy  surface.  Wilkinson  between the e m p i r i c a l and c a l c u l a t e d  a l p h a - p a r t i c l e decay r a t e s as evidence that describe  (1961) has i n t e r -  one cannot  easily  the n u c l e a r wave f u n c t i o n i n the surface r e g i o n by  shell-model  wave f u n c t i o n s ;  i n f a c t , he has suggested  that  -  i t might he necessary to r e s o r t to a phenomenological model.  A n a t u r a l way  i n which  4  -  cluster  to introduce c o r r e l a t i o n s  into  the i n d e p e n d e n t - p a r t i c l e model wave f u n c t i o n s i s through conf i g u r a t i o n mixing; Harada found t h i s enhances magnitude.  We  (1961) and Zeh and Hang (1962) have  the c a l c u l a t e d decay r a t e s by an order of  contend t h a t the remaining d i s c r e p a n c y between  the e m p i r i c a l and the c a l c u l a t e d decay r a t e s can be removed by a c o r r e c t i n t e r p r e t a t i o n of the. r a d i u s i n v o l v e d i n the c a l c u l a tions. The procedure used i n c a l c u l a t i n g the a l p h a - p a r t i c l e r a t e s i s f a m i l i a r from the theory of n u c l e a r r e a c t i o n s .  decay In  Chapters 2 and 3 of t h i s t h e s i s , we have d e r i v e d a formula f o r the decay constant by t h i s procedure. oped  However, we have d e v e l -  the decay constant from a p o i n t of view which i s appro-  p r i a t e to the decay problem r a t h e r than to the s c a t t e r i n g problem. oped Zeh  The formula f o r the decay constant has been d e v e l -  i n t h i s manner by s e v e r a l other authors, p a r t i c u l a r l y (1963) and Mang (1964). Igo (1959) has used the e l a s t i c  s c a t t e r i n g data to empir-  i c a l l y determine the m o d i f i c a t i o n of the Coulomb f i e l d s i n the n u c l e a r surface by the r e a l , average i n t e r a c t i o n of an a l p h a p a r t i c l e w i t h a heavy n u c l e u s .  We  to round o f f the Coulomb b a r r i e r s . ~  use Igo's n u c l e a r p o t e n t i a l The p e n e t r a b i l i t y through  a rounded-off Coulomb b a r r i e r can be determined quite  easily  and, i n Chapter 3, the p e n e t r a b i l i t y i s e v a l u a t e d f o r e i t h e r 208 the s c a t t e r i n g from P o  2 1 2  .  of an a l p h a - p a r t i c l e from Pb  or of i t s decay  The r e s u l t s are compared w i t h those obtained from  the J.W.K.B. and square-well  estimates;  have been used i n most of the previous We have p r e v i o u s l y noted  the l a t t e r  estimates  calculations.  that most of the e f f e c t s of the  n u c l e a r s u r f a c e upon the a l p h a - p a r t i c l e decay r a t e s are a s signed to the one-body decay constants r a t h e r than spectroscopic f a c t o r s .  to the  In Chapter 4, we i n v e s t i g a t e the e f f e c t  of the n u c l e a r s u r f a c e upon the one-body decay c o n s t a n t s . comparing the resonant wave f u n c t i o n of a d i f f u s e - e d g e w i t h t h a t of a square-edge nucleus  In f a c t ,  nucleus  of the same r a d i u s , we show  that the d i f f u s e - e d g e enhances the reduced body problem.  By  widths of the one-  i t i s seen from these resonant wave  f u n c t i o n s t h a t the d i f f u s e - e d g e nucleus  corresponds  s i d e r a b l y l a r g e r e q u i v a l e n t square-edge n u c l e u s .  to a con-  This sug-  g e s t s t h a t the one-body s c a t t e r i n g c r o s s - s e c t i o n s should be c o n s i d e r a b l y enhanced by the d i f f u s e - e d g e ; i n f a c t , we show t h i s i n a d i r e c t manner by c a l c u l a t i n g the d i f f e r e n t i a l tic  elas-  s c a t t e r i n g c r o s s - s e c t i o n f o r the s c a t t e r i n g of an a l p h a -  p a r t i c l e from a p o t e n t i a l w e l l a p p r o p r i a t e to a P b  2 0 8  nucleus.  We i n c l u d e the many-body aspects of the decay problem by r e p e a t i n g the i n d e p e n d e n t - p a r t i c l e model c a l c u l a t i o n of pip  Harada (1961) f o r the a l p h a - p a r t i c l e decay r a t e of Po  .  Harada has used a square-edge w e l l of r a d i u s ten f e r m i s to to e v a l u a t e the amplitude  of the one-body resonant wave func-  t i o n s ; he has rounded-off  the Coulomb b a r r i e r w i t h  p o t e n t i a l of Igo (1959).  He has chosen h i s model wave func-  the n u c l e a r  t i o n s to be i n f i n i t e harmonic o s c i l l a t o r wave f u n c t i o n s and has determined the harmonic o s c i l l a t o r  s i z e parameter by  taking  the amplitude of the mode of the r e l a t i v e motion to he  equal to the amplitude of the one-body square-well tions.  While i t i s not obvious that t h i s procedure  represents that  wave  a square-well  func-  really  c a l c u l a t i o n , i t might be suspected  i t does s i n c e the e f f e c t s of the i n t e r n a l modes a r e r a t h e r  i n s e n s i t i v e to t h e . s i z e parameter. c a l c u l a t i o n i s considerably  The'radius of Harada's  l a r g e r than those b e l i e v e d  typi-  c a l of heavy n u c l e i ; i n f a c t , we contend that he has essent i a l l y used the e q u i v a l e n t diffuse-edge but  replace  well.  square-edge w e l l of a  In Chapter 4, we repeat h i s c a l c u l a t i o n  the one-body wave f u n c t i o n s of the square-edge  w e l l w i t h the one-body wave f u n c t i o n s edge w e l l .  smaller  of a smaller d i f f u s e -  We show that the decay constant  manner f o r Po  c,-L  obtained  * i s comparable to that which Harada has ob-  t a i n e d w i t h the l a r g e r square-edge w e l l . . In f a c t , we c u l a t e the e q u i v a l e n t  of the two w e l l s  ceding  cal-  square-edge w e l l of a t y p i c a l d i f f u s e -  edge w e l l i n Chapter 5 and f i n d radii  i n this  that the d i f f e r e n c e i n the  i s s i m i l a r to that found i n the p r e -  decay r a t e c a l c u l a t i o n s .  In a d d i t i o n to d i s c u s s i n g a l p h a - p a r t i c l e decay, we use the b a r r i e r p e n e t r a t i o n cesses.  c a l c u l a t i o n s to study a b s o r p t i o n  In Chapter 5, we give  a l e n t square-well absorption  of a Pb  the extent  model i s found to apply nucleus.  pro-  to which the equivto the a l p h a - p a r t i c l e  The e f f e c t of the s t r e n g t h of  the o p t i c a l model potential i n the e l e c t r o s t a t i c b a r r i e r i s also  studied. A summary of the preceding  Chapter  6.  c a l c u l a t i o n s i s given i n  CHAPTER 2  A DERIVATION OF THE DECAY CONSTANT  The systems which w i l l he d i s c u s s e d i n t h i s t h e s i s are nuclear  systems which decay by means of simple a l p h a - p a r t i c l e  disintegration.  The formalism f o r d e s c r i b i n g such systems i s  f a m i l i a r from the theory of n u c l e a r r e a c t i o n s ; i n f a c t , Thomas (1954) has d e r i v e d the decay constant of such a system by the standard techniques of n u c l e a r r e a c t i o n theory.  The purpose  of the d e r i v a t i o n p r o v i d e d i n t h i s t h e s i s i s to employ these techniques i n a s i m i l a r manner, but from a p o i n t of view which i s more n a t u r a l to the study of decay problems. The most obvious method i n which to d e s c r i b e a decaying system would be to f i n d  the wave f u n c t i o n of the system a t  some convenient time, say t . Q  I f a t t h i s time i t i s found  that the system i s d e s c r i b e d by the quantum numbers,71 ( t ) , 0  then the temporal e v o l u t i o n of the system i s governed by the Schroedinger E q u a t i o n ; (2.1)  £11^. I71.(t}> = Til7l(t)>.  (Here H i s the H a m i l t o n i a n of the system and "h i s the crossed P l a n c k ' s constant.) A n a t u r a l choice f o r t f o r m a t i o n of the system. tive,  would be a t a time p r i o r to the  If t  0  were chosen s u f f i c i e n t l y  nega-  one could t h i n k of the system as c o n s i s t i n g of the wave  packets of the c o n s t i t u e n t p a r t i c l e s , w i t h n e g l i g i b l e  overlap  and i n t e r a c t i n g through slowly v a r y i n g p o t e n t i a l s , and as conv e r g i n g on the p o i n t a t which the system formed; the dynamics of the wave packets (at the speeds of i n t e r e s t ) are then Newtonian so thatlTl ( t )7 may be determined p r e c i s e l y . 0  - a The d i f f i c u l t y h a v i n g converged  i n t h i s modus operandi i s t h a t , a f t e r  and formed,  the system does not decay f o r a  time many orders of magnitude longer than the t y p i c a l of a c o n s t i t u e n t nucleon's motion w i t h i n the formed The  period  system.  temporal f l u c t u a t i o n s of the wave f u n c t i o n of the formed  system, which a t f i r s t were s t r o n g l y c o r r e l a t e d  to the forma-  t i o n process, a f t e r a time "become s t a t i s t i c a l . It i s , t h e r e f o r e , more expedient to choose  t  a f t e r the  0  f o r m a t i o n process and to c o n s t r u c t a s t a t i s t i c a l l y  determined  wave f u n c t i o n of the c o n s t i t u e n t nucleons, the nucleons being assumed to be w i t h i n the n u c l e a r volume. be hoped to r e a l i z e t h i s s t a t i s t i c a l  In p r a c t i c e ,  i t may  d i s t r i b u t i o n by forming  the n u c l e a r wave f u n c t i o n s from models such as the independentp a r t i c l e model. formed  F o r t h i s reason, the wave f u n c t i o n of the  s t a t e i s commonly r e f e r r e d  to as the "parent nucleus"  wave f u n c t i o n . Care must be e x e r c i s e d i n i n t e r p r e t i n g t h i s s t a t e approximation.  stationary  Such s t a t i o n a r y wave f u n c t i o n s are i r r e g -  u l a r a t i n f i n i t y , while the parent nucleus wave f u n c t i o n i s not only r e g u l a r a t i n f i n i t y the n u c l e a r r a d i u s .  but e s s e n t i a l l y vanishes outside  T h e r e f o r e , the approximation i s u s e f u l  only w i t h i n the volume of the formed  system.  Assuming a parent nucleus a t t = 0, and assuming two fragment break-up,  i t would seem reasonable to choose the  wave f u n c t i o n of the decaying system to be of the form,  2£ai Sa » ) t  x ,s ,x ,s ,...x ,s ) 1  1  deb  2  clC2  2  ( .t) 6  A  A  + ...  V (qi)^c (q^) i c l  2  <  ,  C l  c (R. ). t  2  where  (xi»a.i,x ,Sg, ... , 2 L A » ^ A )  =  2  ••• and where ( x ^ s j the i  t  h  a  ^ (xi..§.]_,Xg  •  x  A  , s  A  , 0 ) ,  are the s p a t i a l and s p i n c o - o r d i n a t e s of  nucleon, r e s p e c t i v e l y .  the wave f u n c t i o n s of the f i r s t c-j_ and c  * * •  Here, (j/ (q ) n d ( ^ ( q ^ ) are a  Cl  1  and second decay  a r e the a p p r o p r i a t e quantum numbers.  2  C 2  a s t a t i o n a r y s t a t e of the r e l a t i v e motion;  fragments; ^  (R, 6) i s  c  £ i s the r e l a t i v e  energy and R the s e p a r a t i o n of the fragments. A theory of alpha-decay has been developed of  view by Mang ( i 9 6 0 ) .  heavy n u c l e i , he f i n d s (2.3) In  fact,  a(t) #  EXP(  from t h i s p o i n t  In a p p l i c a t i o n s to the alpha-decay of that, (6 - i*)t).  the formula which he o b t a i n s f o r 2f i s e s s e n t i a l l y the  same as that which w i l l be obtained i n the c a l c u l a t i o n to be discussed. At decayed  large p o s i t i v e  times, the parent nucleus w i l l have  and should be d e s c r i b a b l e as an a l p h a - p a r t i c l e and a  daughter nucleus a t some l a r g e r e l a t i v e  separation.  has noted that, i n a s u i t a b l y s m a l l space-time  Zeh (1963)  r e g i o n , the  time dependence of the parent nucleus wave f u n c t i o n may be taken to be that of E q . (2.3) and the wave f u n c t i o n s i n each channel of break-up, (2.4)  G  C l C 2  (R.t) «  <c  lC2  R|71 (t)> ,  may be taken to be those of p u r e l y outgoing f l u x . t h e s i s we s h a l l f o l l o w t h i s approach. wave f u n c t i o n w i l l  In t h i s  The parent nucleus  be expanded i n the n u c l e a r volume i n terms  - 10 of a complete set of s t a t i o n a r y s t a t e s which e x p l i c i t l y  include  the alpha-decay channels; the boundary c o n d i t i o n s a t the nuc l e a r r a d i u s w i l l be chosen so that the channel wave f u n c t i o n s are a s y m p t o t i c a l l y required  waves of outgoing f l u x .  To determine the  expansion c o e f f i c i e n t s , i t w i l l be assumed that the  parent nucleus wave f u n c t i o n can be d e s c r i b e d e n t - p a r t i c l e model. and  by the independ-  Of course, t h i s procedure i s well-known  has o f t e n been used i n the a n a l y s i s of n u c l e a r  reactions.  In the c o n s t r u c t i o n of the parent nucleus wave f u n c t i o n , it will  be assumed that n o n - l o c a l and many-body p o t e n t i a l s  can be n e g l e c t e d  and that the H a m i l t o n i a n of the system, which  w i l l be assumed to c o n s i s t of A nucleons (N neutrons and Z protons),  can be w r i t t e n i n the form,  A (2.5)  H  H »  i #  where, (2.6)  ' H i - ^ V ^  +  * 4 .V-'l.^>. j  1  Here,  "Vi are i* * 1  s  (ii»s- ), ±  the s p a t i a l and s p i n c o - o r d i n a t e s , 1  r e s p e c t i v e l y , of the  nucleon; the proton and neutron masses have been assumed  to be equal. The functions  c o n s t r u c t i o n of the i n d e p e n d e n t - p a r t i c l e i s well-known (Preston  (1962)).  model wave  One c a l c u l a t e s the  s p a t i a l average of the p o t e n t i a l term i n Eq, (2.6) a t the i nucleon; from t h i s average p o t e n t i a l one d e r i v e s nucleon o r b i t a l s .  the one-  The parent nucleus wave f u n c t i o n i s then  formed of l i n e a r combinations o f the S l a t e r Determinants of  ^  - l i the assumed c o n f i g u r a t i o n ; the l i n e a r combinations  are chosen  a p p r o p r i a t e to the quantum numbers of the parent n u c l e u s . The  r e s i d u a l i n t e r a c t i o n does not, i n g e n e r a l , v a n i s h ; i f i t  can be t r e a t e d as a s m a l l p e r t u r b a t i o n , one can i n t r o d u c e the r e s u l t a n t mixing of the c o n f i g u r a t i o n s by the techniques of perturbation theory. To f i n d  the nature of the s t a t i o n a r y s t a t e s of the Kam-  i l t o n i a n commensurate with the system a f t e r decay, i t i s desirable  to i n t r o d u c e the approximations  i n a manner convenient  f o r c o n s t r u c t i n g a l p h a - p a r t i c l e and daughter tions.  nucleus wave f u n c -  In the d i s c u s s i o n of t h i s chapter, i t w i l l be assumed  that the a l p h a - p a r t i c l e and daughter  nucleus wave f u n c t i o n s  have been antisymrnetrized w i t h r e s p e c t to the interchange of proton or of neutron c o - o r d i n a t e s ; the parent nucleus wave f u n c t i o n to be c o n s t r u c t e d from these wave f u n c t i o n s w i l l , t h e r e f o r e , only be p a r t i a l l y antisymrnetrized. However, i t w i l l be shown that the parent nucleus wave f u n c t i o n can be regarded as a s o l u t i o n of a Schroedinger E q u a t i o n having a complex eigenvalue  (Eq. (2.25)).  Since the Hamiltonian i s  symmetric: i n the interchange of the c o - o r d i n a t e s of any two identical particles,  the completely antisymrnetrized wave  f u n c t i o n w i l l be an e i g e n s t a t e of the above Schroedinger E q u a t i o n i f the p a r t i a l l y antisymrnetrized wave f u n c t i o n s are solutions.  Only the l a t t e r f a c t w i l l be used i n the argu-  ments to follow.' It w i l l be assumed that the system of nucleons by  labelled  (1234) c o n s t i t u t e s the a l p h a - p a r t i c l e , where 1 and 2 are  protons and 3 and 4 are neutrons; the remaining  nucleons,  -  (56...A), w i l l be taken as c o n s t i t u t i n g ' t h e daughter I t i s convenient  to make the  and  a  s  8 2 , 8 ^ , 8 4 ) ;  =  -  nucleus.  definitions,  2L« = l2Ll.2£2-t2£3.2L4); s.^  12  -  s  2LA);  12.5.2i6 (85,85  8A);  to introduce the r e l a t i v e c o - o r d i n a t e s , -  (2.7)  R.CM  L2 L  4 (x]_+  +  (ii +  A"  =  2L2  ( L i . L  2  . L  3  (2L5  =  J Li  2  + 2L  6  L i . 1-2»  I-3)  * ' * + 2LAJ I  2Li - 2L I  =  2  +  ) ;  R  =  R ,  2  -  R<r  •  (The J a c o b i a n of the t r a n s f o r m a t i o n (x1.X2.2L3.2L4) (R-.  +  C3 = i ( 2 L i 2£ ) " -H2L3 + 2L4);  3  =  R ,  ^ 2L4 ) I  2L + *' * + 21A)  2£ ~ 2L4 ;  =  2S.3  —^  unity.)  i s  A f t e r decay, the n u c l e a r system can be d e s c r i b e d by a "one-body" model; by t h i s we mean that the i n t e r a c t i o n b e t ween the daughter  nucleus and. the a l p h a - p a r t i c l e i s a func-  t i o n only of t h e i r s e p a r a t i o n . formed i n the center-of-mass ^(M) and  frame,  RcM  V  If the c a l c u l a t i o n s are per-  • .  i f A i s s u f f i c i e n t l y l a r g e that the r e c o i l i n the  daughter  nucleus wave f u n c t i o n can be n e g l e c t e d , 2lA^4)m then one (2.8)  R  —f  can w r i t e , H  + H  =  +" H««- ,  f  where,  u. )a) ^ 9  •  1  i  1  - | i v  L  ; +  *Ai4 v 1  l j  (v .^j 1  l  - 13 c) IW -  V  +  &  U ( R ) .  Here, (2.10) M . 4 l A = 4 ±  ; P  m  ZJ  1  . I  2  . , j  a  J .  are the reduced masses w i t h r e s p e c t to the r e l a t i v e nates R, average  respectively; U ( R )  £ , and ^ , 2  i n t e r a c t i o n between the decay  The  co-ordi--  r e p r e s e n t s the  fragments.  e f f e c t of the r e s i d u a l i n t e r a c t i o n between the decay  fragments  can be accounted  f o r by t a k i n g U(R)  to be an  optical  model p o t e n t i a l : (2.11)  = V ( R ) + iW(R).  U(R)  (V and V/ are r e a l . )  It  i s well-known that the complex term, W(R),  of  the e f f e c t s of the r e s i d u a l i n t e r a c t i o n ; i t e s s e n t i a l l y  vanishes o u t s i d e of the n u c l e a r volume.  simulates many  In p r a c t i c e ,  both  y(R) and W(R).are regarded as being phenomenological;  i n genr  eral,  of the  they w i l l be dependent upon the r e l a t i v e energy  decay fragments  and upon the channels being c o n s i d e r e d .  We  w i l l n e g l e c t t h i s dependence. In  t h i s t h e s i s , we w i l l be concerned, only with the  spher-  208 i c a l n u c l e u s , Pb  .  U(R)  between the decay fragments  then depends only upon the d i s t a n c e so that the s t a t e s of r e l a t i v e  motion are s o l u t i o n s of the Schroedinger E q u a t i o n , (2.12) ( - ^ V The  "T* U(R))(|/(R) » £ l | / ( R ) . R s o l u t i o n s of Eq. (2.12) have the form, 2  2p  iy ( R ) • «  (2.13)  Y ^ L  (  R  where Y ^ f (~L_) i s a s p h e r i c a l harmonic and U R of the r a d i a l Schroedinger E q u a t i o n , L  L  (2.14)  d ' uT(R) -h 2Jl dR " • •• = £ u (R).  - h. 2  2  2  u  L  L  (U(R)-f- h  2  2JJ  T  u  (R) i s a s o l u t i o n  L(Lfl) ) R 2  u (R) L  =  '"  - 14 The  e i g e n s t a t e s of the t o t a l Hamiltonian, H, a r e then of the  form,  T  where \ij .  and (il .'  m  momenta,  a r e standard e i g e n s t a t e s of the angular  and j. , resps c t i v e l y , and are s o l u t i o n s of the f  Schroedinger  Equations,  (2.16) a)  H.l}^,  = ^  1  7* and TJr are the remaining  ^  ;  quantum numbers r e q u i r e d to s p e c i f y  the s t a t e s completely. The many-body aspects of the decay problem are w i t h i n the n u c l e a r volume. alpha-decay  F o r the purposes  r a t e s , i t i s convenient  t i o n s i n the f o l l o w i n g manner.  contained  of a n a l y z i n g  to d e f i n e a set of f u n c -  In each channel, one can d e f i n e  a boundary c o n d i t i o n number, b ; c  (2.17) Here, R  (R d u ( R ) ) * u ( R ) *c dR R=R i s taken as the r a d i u s of the parent nucleus; the u  c  8  Q  0  f u n c t i o n s , u ( R ) , a r e taken to be normalized c  will  later  in R^R . Q  show that the boundary c o n d i t i o n s can be chosen  i n a manner which i s n a t u r a l to the decay problem.) boundary c o n d i t i o n s , together w i t h the d i f f e r e n t i a l Eq.  (2.14), y i e l d a set of s.olutions of Eq.  complete w i t h i n R < R . Q  u£(R)  (We  are complete,  equation,  (2.14) which are  These s o l u t i o n s may be taken to be,  , where n i s the number of nodes of u£ .  s o l u t i o n s of E q .  These  Since the  (2.16) (which are d e f i n e d over a l l space)  one can d e f i n e a complete set of f u n c t i o n s i n  the f o l l o w i n g manner.  One d e f i n e s the channel wave f u n c t i o n  - 15 to be, (2.18) l | /  = |  c  0  ip£  EM  m < l  m,M (  <i,L  '  J  L  M  ?  V  ^  Y  L  .  L  I  where c denotes the channel quantum numbers, ( j ^ ,m ,%  ,  x  L,J,M). U.w)  ,  The s e t of f u n c t i o n s , l ( J « « • 1|/o. n o  i s then complete w i t h i n R  R  so that one can make the expan-  0  sion,  where, (2.21)  C  n c  = $V  ^fo  Q  Nuclear systems e x h i b i t i n g alpha-decay t y p i c a l l y have l o n g l i f e t i m e s whence i t might be expected that they should, i n some sense, behave  like a stationary state.  To see i n what  sense t h i s i s t r u e , i t w i l l be assumed that the c o e f f i c i e n t s , a ( t ) and ' c c ^ » ) b  t  1  (2.22)  2  o  f  E<  1» ( » )» 2  2  a  r  e  o  f  t  h  e  form,  a ( t ) = EXP(-~ E t ) E X P ( - | - t ) ; Q  ^ W ' ^  =EXP(-lE t)  (1-EXP(  0  (This temporal dependence  -|  t)b  C i C 2  has, i n f a c t , been j u s t i f i e d  t h e o r e t i c a l c o n s i d e r a t i o n s by Mang (i960).)  (*)... from  Accepting Eq.(2.22),  the time-dependent wave f u n c t i o n i s of the form, (2.23) t£ (t) = EXP(- £ E t) ( 1 P E X P ( - I t ) Q  "ft  '••  (1 - EXP(-  J  t)V  d  ).  where tp ^ r e p r e s e n t s the system a f t e r decay.  The Schroedinger  E q u a t i o n of the' system then becomes',  Since  E t)f  i T i ^ ( t ) - ( E - i * ) ? ( t ) + i^EXP(4 =Hl?(t). 0 "bt ~& i s , i n g e n e r a l , s m a l l , a good approximation to the  (2.24)  Q  d  - 16 parent nucleus wave f u n c t i o n should be obtained by s o l v i n g the complex eigenvalue (2.25)  H?  0  problem,  = (E - i*)fP . 0  0  W r i t t e n i n the form,  it  i s seen that ~$ can be t r e a t e d as a p e r t u r b a t i o n with the  z e r o t h order approximation  of E q . (2.25) being a s t a t i o n a r y  state. From E q . (2.25), (2.26)  -2i#P = 0  if Hf - M t 0  Q  D e f i n i n g the decay constant by, (2.27) employing  "X-f£. E q . (2.9) and Green's Theorem, and i n t e g r a t i n g Eq.  (2.26) over a l l space, Ro (2.28) J = i  _ jf ^ o " ^o Vo  H  ^ o  H  H  r  JR^dRdldx^dadJl 7?  3 h V7 U < • •*< 3 l  (2) n^T  — ' j , ^ ' ,TJ. n j : m;V j,7, L J ^ j ; ^ •)  ) X (  ./a (  R  e  C  a  " J ^ 5 . °n^X  l  l  j, ^ LJM  . ..x r O ^ L m ^ M j JM>|1/^ fx( £ ? T m ^ . ) m^ <J(f ' ° _ i m  1  A  i»r.tT..T»MI  • • «x IJ^'m; )dJ ds., x  5  x  - 17 R = K  Eq.  0  ,  (2.28) i s of the form,  (2)  (3)  ? " L/RJ=K. I c ' d ^ : . =  R  0  where j_ denotes the p r o b a b i l i t y c u r r e n t i n the s p e c i f i e d channels.  F o r heavy a l p h a - e m i t t e r s , decay through channels  other than simple alpha-decay i s g e n e r a l l y n e g l i g i b l e terms  (1) and (2) i n E q . (2.28) can be n e g l e c t e d .  so that  E q . (2.28)  can then be w r i t t e n i n the form, (2 ?Q) "AX = {2.29} - --i lji ^ ^ n . . SC* i c SCi ' c  (n u * ^ n« c  d u  l U  A n a t u r a l choice o f the boundary  u . d u ° >) c  -  R = R  O  .  c o n d i t i o n s i s to match  the r a d i a l wave f u n c t i o n s i n each channel onto the wave of p u r e l y outgoing f l u x i n that channel.  This i s accomplished  by choosing, (2.30) S  c  b  and P  c  c  .= S ( 6 , R ) + i P ( 6 , R ) ; e  c  0  C  c  0  (6 = E C  - E^-E j )  Q  are. the n u c l e a r s h i f t f u n c t i o n and the n u c l e a r  penetrability: (2.31)  s  C  ( £  .  , R  C  R  0  .  + iP (e ,R )  )  c  G  G  Here F  c  and G  c  C  C  C  =  0  . . .  ( *  C  , R )  -f- i F » ( 6 c , R ) I  U  C  , R )  -h i F ( 6 C , R ) c  IR=R  0  .  a r e those s o l u t i o n s of E q . (2.14) which a r e  asymptotic to the r e g u l a r and irregular.Coulomb f u n c t i o n s , J5'  c  and G , i n the channel c. c  (The one-body p o t e n t i a l i s essen-  - 18 t i a l l y the Coulomb potential outside of the nuclear surface.) The o p t i c a l model potential, U(R), is essentially phenomenological.  I t s behaviour  in the nuclear surface has been  determined from the e l a s t i c scattering data by Igo (1959), but the depth of the r e a l and imaginary volume i s largely a r b i t r a r y . and 5-10 MEV  terms within the nuclear  (Values of 100-150 MEV  f o r Y/(R) are currently fashionable.)  f o r V(R)  To the  extent that the r e s i d u a l interaction can be neglected, we need only r e t a i n the r e a l term, V(R), in Eq.  (2.14). It would  then seem natural to choose the depth of the well i n such a . manner that, for some n , 0  (2.32) £ £  = £  q  .  c  This one-body wave function should then represent many of the r a d i a l properties of the actual wave function of the system.  decaying  (The a r b i t r a r i n e s s in the number of nodes r e f l e c t s  the ignorance  of the well depth.)  The one-body reduced width,  2  n  , i s defined by,  By noting the boundary condition, Eq. (2.30), Eq. (2.29) can be written i n the form, 2 (2.34) 1  • Z  ^  »ot*£„- .> °a t t  -,C„„  y  n,n»  Defining the one-body decay constants, (2.35)  2P (e ,R )l^nJ c  n o  2  0  and the spectroscopic factors, (2.36)  S  c  J ^ C  n  c  ^ |  2  Eq. (2.34) can be written i n the form,  p  nc^n'c C  n n' |r£j2 0  - 19 (2.37) Eq.  J  =  ^  S . c  12.37) i s convenient f o r c a l c u l a t i o n s of the decay  from the n u c l e a r  models.  I t i s seen from Eq. (2.25) that we may parent  nucleus wave f u n c t i o n , b y  X, 0  HX  = E X  0  0  It i s a r e s u l t  approximate the  where X  f a m i l i a r resonant state of r e a c t i o n theory (2.38)  constant  0  i s the  d e f i n e d by,  .  0  of r e a c t i o n theory  that the width of the e l a s t i c  s c a t t e r i n g c r o s s - s e c t i o n (as determined "by the Breit-V/igner • s i n g l e - l e v e l resonance formula) i s , (2.39) where S  c  P oc -  fc  o c  S, ~K° oc ' '"-"c. 0 c  b  i s the s p e c t r o s c o p i c  f a c t o r of E q . (2.36) and  "AQC^*  i s the one-body width, (2.40)  r°  b c  *  = 2P (^ ,R )|^°J c  o  Comparing E q . (2.40) and E q . T  (2.41)  o c  .  0  (2.35),  = "Xh,  which i s an e x p r e s s i o n  of the U n c e r t a i n t y  Principle.  I f the r e s i d u a l i n t e r a c t i o n i s s u f f i c i e n t l y only the one-body wave f u n c t i o n s , u a b l y i n the parent  n Q  small,  , are contained  nucleus wave f u n c t i o n .  then  appreci-  Eq. (2.36) then  reduces t o , (2.42) and Eq. (2.43)  S  c  =  n J !  Ci i Q  (2.37).to, -X = ? * 8 - H c  n o 0  To t e s t the v a l i d i t y  |  2  •  of the one-body model, we note that  many of the e f f e c t s of the r e s i d u a l i n t e r a c t i o n can be accounted f o r by the o p t i c a l model p o t e n t i a l (Eq. (2.11)).  The absorp-  - 20 t i o n c r o s s - s e c t i o n f o r such a p o t e n t i a l i s of l 2  '  4 4 )  6  abs = J "  L ScT ;  (k  c  c  =  -  the'form, J W )  Tjj i s the o p t i c a l model t r a n s m i s s i o n f u n c t i o n i n the  channel,  c; L i s the r e l a t i v e angular momentum i n the channel, c;  g  c  i s a numerical f a c t o r depending on the angular momenta of the parent n u c l e u s , the daughter If W  nucleus, and  the a l p h a - p a r t i c l e .  i s taken as the order of magnitude of the depth of the  complex p a r t of the o p t i c a l model p o t e n t i a l , body approximation  about E , 0  then, i n the  one-  the t r a n s m i s s i o n f u n c t i o n i s of  the form, (2.45) c  L  T (E) C  r 4P (£ ,R ) c L  L  0  being a n u m e r i c a l f a c t o r .  a c t i o n spreads  L  the one-body statey-U  g r e a t e r than f i f t e e n MEV  and  + W§  ,  Thus the r e l a t i v e r e s i d u a l  Since i n t y p i c a l heavy n u c l e i ,  than ten MEV,  (E -  jrxf  inter-  , through a width,  r  the s e p a r a t i o n s , ! ^ -£^«l  reasonable v a l u e s f o r W  i t would seem reasonable  t h a t the parent  W. Q  f  are  are less  nucleus  wave f u n c t i o n should c o n t a i n a p p r e c i a b l y only the one-body Q  wave f u n c t i o n , H (2.42) and The  n o  , and  i t s n e a r e s t neighbours.  (2.43) should be a moderately  s t a t i s t i c a l f a c t o r s ,j 'C J l  n  2  good  Hence, Eqs.  approximation.  , are reasonably  inter-  p r e t e d as measuring the p r o b a b i l i t y that the resonant  state  can be represented by the a p p r o p r i a t e decay p r o d u c t s .  The  extent to which the i n d e p e n d e n t - p a r t i c l e model c o r r e c t l y accounts  f o r these c o r r e l a t i o n s w i l l be shown to provide a  a t e s t of i t s v a l i d i t y i n the n u c l e a r s u r f a c e .  - 21 -  CHAPTER 3  EXPERIMENTAL REDUCED WIDTHS  We have contended  i n the i n t r o d u c t i o n that the c a l c u l a -  t i o n s upon a l p h a - p a r t i c l e decay r a t e s performed by previous authors have, e s s e n t i a l l y , used the e q u i v a l e n t nucleus model of Vogt.  square-edge  They have c u s t o m a r i l y w r i t t e n the  decay constant as a product of tv/o f a c t o r s :  a) a n u c l e a r  reduced width, which depends only upon the i n t e r n a l a s p e c t s of the nucleus and which i s i n t e r p r e t e d as measuring the frequency a t which a l p h a - p a r t i c l e s appear at the n u c l e a r s u r f a c e ; b) a p e n e t r a b i l i t y , which depends only upon the p o t e n t i a l b a r r i e r and which i s i n t e r p r e t e d as measuring the ease w i t h which an a l p h a - p a r t i c l e can p e n e t r a t e the b a r r i e r and appear on the o u t s i d e .  The reduced widths have g e n e r a l l y  been c a l c u l a t e d u s i n g square-edge n u c l e i , while - the . b a r r i e r has g e n e r a l l y been rounded-off w i t h the n u c l e a r p o t e n t i a l of Igo (1959).  Hence, the anomalous r e f l e c t i o n of the square-  edge nucleus has i m p l i c i t l y been removed; i n f a c t , responds to a c a l c u l a t i o n w i t h the e q u i v a l e n t model.  In the present s e c t i o n , we w i l l  this  cor-  square-edge  check that p r e v i o u s  J.W.K.B. estimates of the p e n e t r a b i l i t i e s are r e a s o n a b l e ; i n the language of Vogt, t h i s i s e q u i v a l e n t to checking that the r e f l e c t i o n f a c t o r s have been c a l c u l a t e d a c c u r a t e l y .  We  have a l s o i n c l u d e d the p e n e t r a b i l i t y estimates f o r the unmodi f i e d Coulomb b a r r i e r .  F o r completeness, we have checked the  reduced d e r i v a t i v e widths which have been used by some previous authors. i s defined  i n terms of  - 22 the  p r o b a b i l i t y amplitude,  In the channels of alpha-decay, the channel wave are  functions  of the form,  ^•  ^ c  2 )  i o  =  X  l  L  ' L  m ^ M ^ ^ i ^ j m j ^ . B j Y ^ ^ ) ,  ]  where the s i n g l e bound-state of the a l p h a - p a r t i c l e , which has zero angular momentum, has been denoted by .X (L» ft*)  a n c  the bound s t a t e s of the daughter f u n c t i o n s , have been denoted  by  i|7 j ^  m < (  ^ where  . (x^., s_ ) , tf  l|/j lx<r,s^) . m  I t f o l l o w s from E q . [2.20) that, (3.3) Mi ••• Y  dfid£dx«.d.s,  L  whence, 2  13.4)  Sic n'c  n.n'  n  n*  I t was noted i n Chapter 2 that a l l the c o e f f i c i e n t s , C be n e g l e c t e d except C . 2 ..|Y, c l' 13.5) 'n c n  Then,  c  P  nc' may  2  0  and, from Eqs. (3  .6)  \  (2.35) and (2.43),  = ^  2  P r - ^ n , »n? 0  *  The reduced d e r i v a t i v e width has a l s o been used by some a u t h o r s , i n p a r t i c u l a r , by Mang (1960). defined  the reduced d e r i v a t i v e width as,  (3.7) where S 3-1:  Thomas [1954) has  lcT | c  c  2  =  S,  i s the s h i f t THE REACTION  In the subsequent  2 f u n c t i o n of E q . (2.31). Po 212 d i s c u s s i o n , only the p a r t i c u l a r l y simple  219 decay from the ground s t a t e of Po to the ground s t a t e of 6  203 Pb  p l u s an a l p h a - p a r t i c l e has been c o n s i d e r e d .  Since the  parent nucleus and both of the decay products have s p i n zero, it  i s seen that the o r b i t a l angular momentum of the a l p h a 208  particle relative Eq.  to the Fb  ° nucleus must be z e r o .  (3.6), the decay constant i s then  From  simply,  (3.8) \ : c , where the s u b s c r i p t r e f e r s to the r e l a t i v e angular momentum of the channel. 2  The l 3  '  9 )  P  2  r a d i a l Schroedinger Equation,. Eq. (2.14), then becomes, 2 2 "IVdR* l V R ) + V j , l R ) ) u = fr u +  0  f  whex-e we have n e g l e c t e d the imaginary part of the o p t i c a l model p o t e n t i a l and where we have s e t the r e l a t i v e momentum equal to zero.  orbital  angular  Here V ( R ) i s the e l e c t r o s t a t i c C  t i a l as c a l c u l a t e d from the P b  2 0 8  charge  poten-  d i s t r i b u t i o n ; V^(R)  is  the n u c l e a r one-body p o t e n t i a l ; £  of  the decay fragments.  of  8.81 MEV f o r the decay energy i n the l a b o r a t o r y frame y i e l d -  ing  q  i s the r e l a t i v e  energy  Rasmussen (1959) has stated a v a l u e  a r e l a t i v e decay energy of 8.98 MEV. In  the present c a l c u l a t i o n , the e l e c t r o s t a t i c  potential  has been taken as the Coulomb p o t e n t i a l and the n u c l e a r t i a l as that d e r i v e d by Igo (1959) from the e l a s t i c  poten-  scattering  data: (3.10)  V ( R ) = -1100 E X P i ' 1  N  1  q  > 5 ? 4  )  MEV. (  |v (R)|£lO MEV) N  The n u c l e a r p o t e n t i a l i s somewhat u n c e r t a i n .  Firstly,  i t has been d e r i v e d a t h i g h e r energies (~40 MEV) and i t i s  p r o b a b l y energy dependent. inward a l i t t l e the  be extended  beyond the range of v a l i d i t y determined from  40 MEV s c a t t e r i n g  part  Secondly, i t w i l l  experiments.  Thirdly,  only the r e a l  of the o p t i c a l model p o t e n t i a l has been used; the s c a t -  t e r i n g data can,  i n f a c t , only be f i t t e d w i t h a f u l l  model p o t e n t i a l .  optical  I t i s the imaginary term i n the o p t i c a l  model which b r i n g s i n the many-body a s p e c t s of the problem in  a phenomenological way; i t s n e g l e c t i n the p e n e t r a b i l i t y .  is  justified  i f the imaginary p o t e n t i a l  i s ascribed  to the  nuclear  interior.  5-2:  COMPARISONS WITH J.W.K.B. AMD SQUARE-WELL ESTIMATES The  from Eq. GQ  p e n e t r a b i l i t y and s h i f t f u n c t i o n (2.31) u s i n g Eqs.  i s much g r e a t e r than P  Eq.  (3.9) Q  have been  and (3.10).  In heavy n u c l e i ,  a t the n u c l e a r s u r f a c e so that  (2.31) e s s e n t i a l l y reduces t o ,  (3.ii) P U , R ) 0  0  0  = _ G  _  ° , ; s ( e , R ) = R dR_ o( o) G (R) 0  0  0  R  0  We need t h e r e f o r e only evaluate the i r r e g u l a r The  calculated  numerical c a l c u l a t i o n  R=R  0  function.  of the s h i f t f u n c t i o n  and the  p e n e t r a b i l i t y has been performed by e v a l u a t i n g the Coulomb f u n c t i o n s a t 24 f e r m i s w i t h an A i r y then i n t e g r a t i n g  the i r r e g u l a r s o l u t i o n  wards to the n u c l e a r s u r f a c e . by use  function  expansion and  of Eq.  (3.9)  back-  This c a l c u l a t i o n was performed  employing the Runga-Kutta method of order f o u r w i t h the of the U.B.C. IBM 7040 computer. In F i g u r e 1, the p o t e n t i a l s  defined  i n E q . (3.9)  have  - 25 "been p l o t t e d . The reduced width and the reduced d e r i v a t i v e width have been c a l c u l a t e d from Eq. (3.8) and (3.7), r e s p e c t i v e l y ; the r e s u l t s of the c a l c u l a t i o n s a r e tabulated i n Table 1. TABLE 1 EXPERIMENTAL REDUCED- WIDTHS ' o (fermis) R  S  o  P  r  o  2  0  .  J.'  (e.v.)  (k.e.v.) 5.9  10.5  -15.4  3.02 x 1 0 "  1 1  24  10.0  -14.1  6.76 x 1 0 "  1 2  111  9.5  -11.2  1 2 1.72 x l e "  436  55.2  9.0  - 4.0  6.71 x 1 0 "  1119  17.8  1 3  212 The l i f e t i m e (Rasmussen  of Po  ,  22.1  -7 has been taken as 3.04 x 10  sec.  (1959)).  The square-we 11 approximation 'of the p e n e t r a b i l i t y and the  s h i f t f u n c t i o n are g i v e n by,  vt? d G (R) (3.12) P ( * . R ) = J S * - , I S ( f , R ) = R d R where G i s the i r r e g u ^ l aor^ oCoulomb function. ) D  0  0  0  0  0  I  0  R=R,  0  The J.W.K.B. estimate i s given by, (3.13)  where R  Q  P ( ^ , H ) = qR EXP( -2 J R, 0  0  0  Q  i t h e .nnuucclle a r ]r a d i u s , r Iss the.  Q  q(R')dR« ;  i s the outer  classical  t u r n i n g p o i n t , and where q(R) «  2>;(Vu(R) + V ( R ) N  e) 0  In F i g u r e 2, the p e n e t r a b i l i t y , s h i f t f u n c t i o n ,  reduced  width, and reduced d e r i v a t i v e width have, been p l o t t e d as  - 26  9,0  9,5  J 0.0  10.5  FERMIS  I 1.0  11.5  FIGURE  2  PEI NET R A EM LITY  SHIFT  T. P. pgZ08  FUNCTION  •_  4  PARAMETERS  — , w  1  ACTUAL SQUARE  ^  _  _  _  _  _  _  1.  3  REDUCED DERI V A T I VE WIDTH  T.R  ACTUAL J.W.K,B 0  \  - 28  -  c a l c u l a t e d "by each of the above methods. It would seem from F i g u r e e m p i r i c a l reduced widths u s i n g not  classical  inner  ing  the J.V/.K.B. approxima-  quite good to w i t h i n a tenth of a fermi turning point.  error incurred i n using than an order  estimates of th  the 'J.V/.K.B. approximation, have  incurred serious error; in f a c t ,  t i o n i s seen to be the  2 that previous  I t i s a l s o seen that  of the c l a s s i c a l  turn  Sandelescu (1966) have r e c e n t l y i n v e s t i g a t e d  of the J.W.K.B. estimate of the p e n e t r a b i l i t y i n  the r e a c t i o n P u  2 3 8  >  U  2 3 4  f- o< .  They f i n d that  J.V/.K.B. estimate i s low by a f a c t o r of from two  the  to f i v e near  inner t u r n i n g p o i n t , i n agreement w i t h the r e s u l t s p r e -  sented i n t h i s chapter.  In f a c t , they c l a i m a much deeper  one-body p o t e n t i a l than i s c u s t o m a r i l y  believed  i n t h i s manner, obtain a f u r t h e r increase  trabilities. face  inner  point.  the v a l i d i t y  and,  the  the unmodified Coulomb b a r r i e r i s l e s s  of magnitude outside  Bencze and  the  of  In Chapter 4, we  (-^  231  MEV)  i n the pene-  show t h a t , . i f the n u c l e a r  sur-  i s t r e a t e d i n a d i r e c t manner, i t i s not necessary to r e -  s o r t to such extreme w e l l depths to obtain a reasonable agreement w i t h the  e m p i r i c a l decay r a t e s .  - 29 CHATTER 4  INDEPENDENT-PARTICIE' MODEL DECAY RATES OF  In the the  present chapter, we  square-well c a l c u l a t i o n  of P o  2 1 2  calculated  of the  m i x i n g , Harada has  ment with the  empirical  nucleus.  alpha-particle w i l l re-examine  With the  decay r a t e s ,  but has  to be  sponds to a c a l c u l a t i o n w i t h the will  i n t h i s model i s much l a r g e r  calculation  edge into the  had  The the we  e f f e c t of the  (2.37)  i n Eq.  "X =  f  t y p i c a l of a heavy  show that  the  of the  shape and  radius  also  diffuse-  nucleus upon By  this  (2.37), S  c  ,  s i z e and  nature of the  spectroscopic factors,  have c a l c u l a t e d  hence, the many-body e f f e c t s )  the  quite s e n s i t i v e  surface of the S, c  to these c h a r a c t e r i s t i c s . • I t i s our  p r e t e d the  V/e w i l l  a one-body e f f e c t .  the  calculations  involved  conventional  surface of the  one-body decay constants, "X ' * , are  whereas the  corre-  i n a d i r e c t manner.  the  to the  to  e q u i v a l e n t square-edge nucleus  than that  decay r a t e s i s , p r i m a r i l y , mean that  of  to r e s o r t  of Harada, i n t r o d u c i n g the  calculation  rate  inclusion  d i f f u s e - e d g e nucleus to which i t corresponds. repeat the  decay  c o n t e n t i o n that h i s c a l c u l a t i o n  model of Vogt; i n f a c t , we  of  obtained reasonable agree-  than those b e l i e v e d  I t i s our  effect  alpha-particle  performed "by Harada (1961).  radii  2 1 2  In p a r t i c u l a r , we  configuration  larger  Po  w i l l examine the  d i f f u s e n u c l e a r edge upon the  decay r a t e s of heavy n u c l e i . the  -  are  rather  r a d i u s i n v o l v e d i n the  nucleus insensitive  c o n t e n t i o n that  previous  spectroscopic factors  c o r r e c t l y , but calculation  to  (and,  have m i s i n t e r of the  one-body  - 30 decay  constant. I t has "been customary i n previous  c a l c u l a t i o n s to c a l -  c u l a t e the a l p h a - p a r t i c l e decay r a t e s from E q . (3.6): ( 3  .  }  6 )  £  =  e The  2r fcS c  f  0 t  R  0  )  t  y i a c'  •  p e n e t r a b i l i t y has, 'generally, been c a l c u l a t e d f o r a d i f -  fuse-edge b a r r i e r , as has been d i s c u s s e d the  other hand, the n u c l e a r  c a l c u l a t e d by u s i n g  i n Chapter 3.  On  reduced width has, g e n e r a l l y , been  the resonant one-body wave f u n c t i o n of a  square-edge w e l l i n the f o l l o w i n g sense:  one sets the ampli-  tude of that mode of the n u c l e a r wave f u n c t i o n which  describes  the r e l a t i v e motion of the decay products equal to the a m p l i tude of the resonant one-body wave f u n c t i o n a t the n u c l e a r radius.  I t i s i n the sense of the l a t t e r procedure that the  nucleus i s taken to be square.  We have noted i n Chapter 3  t h a t these procedures f o r c a l c u l a t i n g the decay r a t e s respond to a c a l c u l a t i o n with the e q u i v a l e n t nucleus of Vogt  square-edge  (1962).  I t i s o f h e u r i s t i c value  to consider  the freedom w i t h  which an a l p h a - p a r t i c l e can move w i t h i n the nucleus.  To see  how f a r an a l p h a - p a r t i c l e can t r a v e l i n the nuclear before  being absorbed, consider a square-well  i n g t y p i c a l r e a l and imaginary p a r t s , V V  o>> o'^o W  Q  and W , Q  where,  = sin  K R ,  such that  .  s o l u t i o n of E q . (3.9) i s of the form, U ( R )  volume  p o t e n t i a l hav-  Assuming the Coulomb e f f e c t s to be i n c o r p o r a t e d regular  cor-  i n - V , the Q  - 31 -  ~  J  T  2JV 6 0  . (k =  0  J^±L-~  1.3 )  H  For a t y p i c a l heavy nucleus., V 100 MEV and W  Q  Q  may he taken to be about  to be from 5 MEV to 10. MEV.  path of an a l p h a - p a r t i c l e w i t h i n the nuclear (4.1)  The mean f r e e volume,  I ~/3SZP_ kW n  N  0  then ranges from about f i v e to about two f e r m i s ( r e s p e c t i v e l y ) . There i s some evidence that the imaginary p o t e n t i a l i s l a r g e only i n the surface r e g i o n , so that the a l p h a - p a r t i c l e might have even g r e a t e r freedom w i t h i n the n u c l e a r volume; i n gene r a l , V/ i s then l a r g e r i n the surface r e g i o n so that here the a l p h a - p a r t i c l e has l e s s freedom.  Thus, to the extent  that  the-one-body model i s v a l i d , the a l p h a - p a r t i c l e moves r a t h e r f r e e l y within, the nucleus  except i n the v i c i n i t y of the n u c l e a r  surface. I t can e a s i l y be shown from the J.V/.K.B. approximation t h a t the mean f r e e path of an a l p h a - p a r t i c l e i n the b a r r i e r r e g i o n i s about,  where V i s the h e i g h t many-body a s p e c t s may take  I f we a t t r i b u t e the  of the problem to the n u c l e a r i n t e r i o r , we  the imaginary p o t e n t i a l (which accounts f o r the many-  body aspects radius,  of the b a r r i e r .  of the problem) as v a n i s h i n g outside the n u c l e a r  This i§ the assumption ma.d§ i n Chapter 3 , where we  have a t t r i b u t e d a l l the p e n e t r a t i o n e f f e c t s of the b a r r i e r to the p e n e t r a b i l i t i e s , and i s probably  extreme.  With t h i s  assumption, i t i s seen from F i g u r e 1 and E q . ( 4 . 2 ) that the  mean f r e e path i n the h a r r i e r and near the surface square-edge  well i s considerably  fuse-edge w e l l .  for a  l e s s than that f o r the d i f -  I t i s i n t h i s sense that we expect a d i f f u s e - •  edge w e l l t o behave l i k e a square-edge  w e l l of l a r g e r r a d i u s .  In the d i s c u s s i o n to f o l l o w , t h i s e f f e c t i s developed from the f o l l o w i n g point of view.  The e f f e c t of the d i f f u s e -  edge on the one-body resonant wave f u n c t i o n s it  i s found t h a t the d i f f u s e - e d g e  i s examined and  enhances the resonant wave  f u n c t i o n a t and beyond the n u c l e a r  radius.  Hence, the one-  body reduced widths are enhanced i n a s i m i l a r manner. Eq.  From  (2.40), one would expect to observe t h i s enhancement i n  the e l a s t i c s c a t t e r i n g c r o s s - s e c t i o n s  (the width of the e l a s -  t i c - s c a t t e r i n g c r o s s - s e c t i o n i s p r o p o r t i o n a l to the decay constant).  We have demonstrated  t h i s i n a d i r e c t manner by  c a l c u l a t i n g the d i f f e r e n t i a l e l a s t i c of P b  2 0 8  at 9 0 °  by r e p e a t i n g  C M  . We have i n c l u d e d  the i n d e p e n d e n t - p a r t i c l e  the decay constant of P o ^ 2  the d i f f u s e - e d g e  2  performed  scattering cross-section the many-body e f f e c t s model c a l c u l a t i o n of by Harada  (1961) w i t h  of the nucleus.now being taken i n t o account  i n a d i r e c t manner. 4-1:'  ONE-BODY REDUCED WIDTHS AND EXPERIMENTAL SPECTROSCOPIC FACTORS  In the x ^esent s e c t i o n , we w i l l examine the e f f e c t s of 3  diffuse-edge  of the nucleus upon the one-body reduced w i d t h s .  We a l s o p r o v i a a an e s t i m a t e of the c o r r e c t i o n mental s p e c t r o s c o p i c  f a c t o r s due to the d i f f u s e - e d g e .  The reduced mass i n E q . grams.  i n the e x p e r i -  (3.9) has the value 5.9031 x 1 0 ~  2 4  - 33 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 ) , has heen c a l c u l a t e d C  from the P b  2 0 8  (4.3)  charge d i s t r i b u t i o n of H i l l Cl - £ EXP(R/6.7 - 10) •/  J>(R)  . • V ( R ) = 82 2 x e x  C  V 2  x  i f R>6.7 f .  The  nuclear  .  R  £ Jj> (R')R« dR' + • •• 2  -1  RQ  + J R  (1955) :  i f R^-6.7 f .  \-k EXP(10 - R/6.7) Then, (4.4)  and Ford  .P (R' )R'dR'  f °J (R* )R dR* :>  ,2  p o t e n t i a l s , V ( R ) , have been chosen to be of the N  Saxon-Woods form, (4.5)  V (R) = V N  In F i g u r e  Saxon-Woods shape of t h i c k n e s s  . corresond-  (a) 0.5 f e r m i s ,  ( r ) 9.0 f e r m i s , and depth ( V ) 105 MEV; we have a l s o Q  p l o t t e d a square-well depth.  1  Q  3, the p o t e n t i a l s have been p l o t t e d  i n g to a n u c l e a r radius  x [ l + EXP( (R - r ) / a )]  -  Q  of the same r a d i u s and of a s i m i l a r  The depths of the w e l l s were chosen so that the regu-  l a r s o l u t i o n s of E q . (3.9) were resonant and had the same number of nodes; these resonant v/ave f u n c t i o n s have been p l o t ted i n F i g u r e  4.  The a n a l y s i s of decay r a t e s and s c a t t e r i n g c r o s s - s e c t i o n s a s s i g n s a l l o f the e f f e c t s of the i n t e r i o r of the nucleus to the reducd widths, these being face.  evaluated  i n the n u c l e a r  I t i s e x a c t l y i n t h i s sense that the d i f f u s e - e d g e  can be r e p l a c e d by a square-edge w e l l of g r e a t e r r a d i u s . the d i f f u s e - e d g e having  surwell If  w e l l i s r e p l a c e d by that square-edge w e l l  a r a d i u s such that t h e i r reduced widths are equal, the  w e l l s a r e i n d i s t i n g u i s h a b l e v/ithin the n u c l e a r volume; the d i f f e r e n c e i n the r e f l e c t i o n of the w e l l s outside  of the  - 34 -  FIGURE 3  FERMIS  - 36 nuclear  volume i s accounted f o r by a t y p i c a l l y s m a l l  tion factor.  In f a c t ,  t h i s i s e s s e n t i a l l y the  square-edge nucleus model of Vogt  reflec-  equivalent  (1962).  I t i s seen from F i g u r e 4 t h a t the r a d i u s of t h i s alent  square-well"  should  be  r a d i u s of the d i f f u s e - e d g e contention larger  "equiv-  s i g n i f i c a n t l y l a r g e r than  well.  that a d i f f u s e - e d g e  the  T h i s i s the b a s i s of  w e l l should  our  behave l i k e a  square-edge w e l l i n the a n a l y s i s of decay r a t e s .  e x h i b i t t h i s c l e a r l y , we  will  widths.with.a d i f f u s e - e d g e  -  c a l c u l a t e the  w e l l and  square-edge w e l l estimate of Harada  To  one-body reduced  compare them w i t h  the  (1961).  A s i n g l e nucleon i n a heavy nucleus moves i n a p o t e n t i a l h a v i n g a depth of about 50 MEV.  I t would, t h e r e f o r e ,  t h a t reasonable depths f o r the p o t e n t i a l w e l l should tween 50 MEV being Igo  and  200 MEV,  fashionable.  values  of 100-150 MEV  I f i s a l s o seen from Eq.  i l y within  this radius.  l i e be-  currently  (3.10) that  p o t e n t i a l i s known inwards only to about 9.7  t h a t the shape of the w e l l edge may  seem  fermis  be chosen r a t h e r  the so  arbitrar-  I t , t h e r e f o r e , seemed reasonable to  choose the p o t e n t i a l s of the  type of Eq.  (4.5)  i n the  fol-  lowing manner. The values R, 0  thickness  tabulated  s u r f a c e , a, was  i n Table 2..  The  chosen to have  depth, V , Q  and  were then v a r i e d so that the p o t e n t i a l was  equal to the lar  of the  Igo p o t e n t i a l a t 10 f e r m i s and  s o l u t i o n of E q .  lomb f u n c t i o n  (3.9)  the  was  the  c o n d i t i o n f o r resonance.  radius,  simultaneously  so that  asymptotic to the  the  the  regu-  i r r e g u l a r Cou(There i s  - 37 only  one such p o t e n t i a l f o r a given number of nodes.)  the d i s c u s s i o n of Chapter 2, one of these p o t e n t i a l s be a good r e p r e s e n t a t i o n  of the a c t u a l p o t e n t i a l .  By should  Exactly  which one i s the a c t u a l p o t e n t i a l cannot be decided  until  some f u r t h e r c r i t e r i o n i s e s t a b l i s h e d f o r d e c i d i n g  the w e l l  depths more p r e c i s e l y . not  Neither  fundamental n u c l e a r  the a n a l y s i s of a l p h a - p a r t i c l e s c a t t e r i n g data  the w e l l depth very  theory defines  c l e a r l y w i t h i n the range 50 MEV to 150  MEV. Examples of the r e s u l t i n g p o t e n t i a l s a r e summarized i n Table 2.  The behaviour of the resonant wave f u n c t i o n s as a  f u n c t i o n of the w e l l depth has been i l l u s t r a t e d and  t h e i r behaviour as a f u n c t i o n of surface  Figure  i n Figure 5  thickness i n  6. TABLE 2 ONE-BODY POTENTIALS RADIUS  o  THICKNESS a (fermis)  DEPTH Vo (MEV)  (fermis)  A  4  0.5  - 49.1  8.94  B  6  0.5  - 64.5  8.79  C  8  0.5  - 86.3  8.63  POTENTIAL V N  NODES . n  D  10  0.5  -114.5  8.48  E  12  0.5  -149.6  8.34  F  10  0.3  -103.4  9.12  G  10  0.7  -128.9  7.79  The  one-body reduced widths have been c a l c u l a t e d from  the resonant wave f u n c t i o n s  (Eq.  (2.33)) and have been used  - 38 to c a l c u l a t e the experimental and  spectroscopic factors  (Eq.  (3.5)  Table 1).  TABLE 3 EXPKRIMENTAL S PEG TR OS COPIC FACTORS EVALUATED AT 9.0 f .  POTENTIAL  ^  "o (kev)  EVALUATED AT 10.0  C 2 n  2  exp (kev)  0  C 2 o  7f 2 n (kev)  exp (kev)  11.5  0.111  0.98xl0"  16.9  0.111  0.66  0  n  A  105  1.12  1.07xl0"  B ••"  150  1.12  0.75  C  188  1.12  0.59  21.8  0.111  0.51  D  223  1.12  0.50  26.3  0.111  0 .42  E  256  1.12  0.44  30.5  0.111  0.36  F  101  1.12  1.11  42.4  0 .111  0.26  G  170  1.12  0.65  15.5  0 .111  0.71  "  2  I f the p o t e n t i a l s chosen i n e v a l u a t i n g the reduced widths  and  pendent of the r a d i u s . cluded to i n d i c a t e  inde-  r a d i a l dependence has been i n -  the e f f e c t  of the nature  of the surface  chosen and, f o r reasonable  s u r f a c e t h i c k n e s s e s of 0.5  f e r m i s , t h i s i s seen  slight.  to be  "  been the  s p e c t r o s c o p i c f a c t o r s would be The  2  experimental  the one-body reduced widths had  same, the experimental  f.  - 0.7-  Harada, u s i n g a square-edge w e l l of r a d i u s 10 f e r m i s , has  obtained a value of 0.143  f o r the experimental  reduced  -2 width and a value of 0.08 scopic factor.  x 10  f o r the experimental s p e c t r o -  I t would appear from Table 3 that the s p e c t r o -  s c o p i c f a c t o r might be about an order of magnitude l a r g e r this value.  than  - 39 -  .0|  FIGURE  5  EFFECT  OF WELL DEPTH  E -49 A -150 CMEV)-  R  0.5h r  i.o  6 FERMIS  o  8  0  12  10  - 40 l.Oi  F I G U R E  .6  EFFECT OF SURFACE "THICKNESS  F 0 . 3 F, G 0 . 5 F.  •o  .0  "0  4  6 FERMIS  8  10  - 41 4-2:  -  ONE-BODY DIFFERENTIAL ELASTIC SCATTERING GROSS-SECTION OF P b FOR AN ALPHA-PARTICLE AT 9 0 ° 2 0 8  C M  One  of the most d i r e c t ways i n which to i l l u s t r a t e  e f f e c t s , of the i t s e f f e c t on (As has  d i f f u s e - e d g e on the  one-body e l a s t i c s c a t t e r i n g  previously  t i o n a l to the  the decay r a t e s  been noted, the  i s to c a l c u l a t e cross-sections.  one-body width i s propor-  one-body decay constant.)  one-body d i f f e r e n t i a l e l a s t i c  In t h i s s e c t i o n ,  scattering cross-section  s c a t t e r i n g of an a l p h a - p a r t i c l e from the ground s t a t e w i l l be one  c a l c u l a t e d by d i r e c t and  then be  calculated  procedure f o r c a l c u l a t i n g the niques i s d i s c u s s e d It mic  i s worth n o t i n g  that  cross-sections  the  required  i n f a c t , the  Pb ° 2  What  one-body  manner.  by these  to the very small  q u i r e d many hours of computer time on  tech-  widths  the  thesis  p o t e n t i a l s and IBM  re-  7040 computer  used i n these c a l c u l a t i o n s .  F o r . t h i s reason, such d i r e c t  c a l c u l a t i o n s have p r e v i o u s l y  been avoided by  The  Pb  2 0 8  ( °C ,  )Pb  2 0 8  c r o s s - s e c t i o n a t 9 0 ° q j has for  a square-well and  radius. tic  other a u t h o r s .  differential elastic  scattering  been p l o t t e d i n F i g u r e 7,  both  f o r a d i f f u s e - e d g e w e l l of the  same  I t i s seen t h a t the d i f f u s e - e d g e enhances the  scattering cross-section  The  c a l c u l a t i o n of the l o g a r i t h -  a c c u r a c y i n the  8  deriv-  c a l c u l a t i o n s performed i n t h i s  seventeen place  the  C.  d e r i v a t i v e i s very t e d i o u s due  involved;  resonance; the  in a straightforward  i n Appendix  for  well-known procedures.  a t i v e of. the wave f u n c t i o n about the  the  of  does i s to study the behaviour of the l o g a r i t h m i c  w i d t h can  the  elas-  (and, hence, the one-body decay  - 42 -  FIGURE 7 0.2 4 —  ONE-BODY SCATTERING CROSS-SECTION  PB 2 0 8  -0.04  6  IO"  8  L 3  MEV  o  - 43 constant) The  "by about an order of magnitude. d i f f u s e - e d g e w e l l has here been taken  Saxon-Woods type, E q . (4.5), with V  0  - - 48.9 MEV;  r  the parameters,  = 9.0 f . ;  Q  a = 0.5 f . ;  the parameters of the square-well are taken V The  s  - - 47.8 MEV;  r  to be of the  to be,  » 9.0 f . .  s  depths of the w e l l s have been chosen so that the wave  f u n c t i o n s are resonant  a t 8.9795 MEV and have the same number  of nodes. The behaviour  c r o s s - s e c t i o n s have been p l o t t e d by studying the of the l o g a r i t h m i c d e r i v a t i v e of the wave f u n c t i o n s  a t 24.0 f e r m i s . A width of 1.37 x 1 0 "  1 3  MEV was found  f o r the d i f f u s e -  edge w e l l and a width of 1.94 MEV f o r the square-edge w e l l ; hence, the enhancement i n the c r o s s - s e c t i o n (or decay cons t a n t ) due to the d i f f u s e - e d g e 4-5:  i s a f a c t o r of 7.1.  HARADA S FORMULA FOR INDEPENDENT-PARTICLE MODEL REDUCED . WIDTHS 1  A convenient c l e model reduced  technique  f o r evaluating  widths has been developed  independent-partiby Harada  (1961).  We provide here only an o u t l i n e of h i s r e s u l t f o r the simple case  of P o  2 1 2  ; a more complete d i s c u s s i o n i s g i v e n i n Appendix  A. I t w i l l be assumed that the z e r o t h order  approximation  of E q . (2,25) can be d e s c r i b e d by the ground s t a t e c o n f i g u r a t i o n of the i n d e p e n d e n t - p a r t i c l e model. t i a l antisymmetrization  Employing the par-  scheme.of Chapter 2, and n e g l e c t i n g  core e x c i t a t i o n s , the parent nucleus wave f u n c t i o n can be  - 44 written  i n the form,  (4.6) IP0(12...A) »  i ( l 2 3 4 ) lpo (56...A) , 0  v/here IJ^QQ i s the wave f u n c t i o n of the doubly magic core.  (The n o t a t i o n  and n o t i n g  i s that of Chapter 2.)  Prom Eq.  2 0 8  (3.3),  that the parent n u c l e u s , daughter n u c l e u s , and  a l p h a - p a r t i c l e have s p i n (4.7)  Pb  If  =  zero,  f  f(!234) X ( 1 2 3 4 ) Y ° ( ^ )  did*, .  Harada has chosen the a l p h a - p a r t i c l e wave f u n c t i o n to be o f . t h e Gaussian type: (4.8,  :X(  1 2 3  4) . (  | ^ )  i  3  /  a  KP(- | ( | !  ... (4TJ-3/2 - J ( 0  ( 1 2 )  i f  +  ?!) )  +  X°(34);  where >CQ denotes the s p i n s i n g l e t f u n c t i o n .  .  Here,  has  . been w r i t t e n i n terms of i t s r a d i a l and angular components, fj_ and  respectively.'^  Harada chooses the  that the r.m.s. r a d i u s of the charge d e n s i t y  parameterso  i s equal to the,  measured value (1.6 f e r m i s ) ; he f i n d s a value of 0.44  fermis**  2  f o r Jb . In Appendix A, i t i s shown that Eq. i n the form, (4.9)  X  Q  /h R 2  - .JWj  (4.7) can be w r i t t e n  r ' " [ ( ; J l O l ) ^(00; j j ) j T  0 0  3  •••7« ' n ^ n o ^ l ^ l ^ •^'^l'$2'^3'—* ' • 3 ... < N O n 0 ; 0 l V l V l ; Q > / v  s  0  3  ^ !  1  ^ !  1  ^ ^  n  2  •••  2  3  3  3  3  <  1 T 0 n 5 0 ; 0  ] ^ 0 3 ^ 0 ;0>  .^8(l2)^C°(34)]d£l^^^|d^ d^ ds^ 2  3  .  Here i t has been assumed that the one-nucleon wave•functions if The d e f i n i t i o n of \j_ d i f f e r s from Harada's. ( c . f . Appendix A)  . - 45 for  the two protons {neutrons) i n the u n f i l l e d  s c r i b e d by the quantum numbers ("LAIj-.m-. )  (  s h e l l a r e de( i C l j m ) ), 7  o o o o  where 1 ^ • ( V ) 3  f  ±  1  (l ),  (j ),  3  and m  3  1  (m ) denote 3  the p r i n -  c i p a l quantum number, the o r b i t a l angular momentum, the t o t a l a n g u l a r momentum, and the magnetic quantum numbers of the onep r o t o n (one-neutron) o r b i t a l s .  In Eq. (4.9), the f i r s t b r a c k -  eted term c o n t a i n s the T c o e f f i c i e n t s of Rose' (1957) f o r t r a n s forming the j - j c o u p l i n g scheme to an L - S c o u p l i n g scheme; the second bracketed term expands  the wave f u n c t i o n i n terms  of the r e l a t i v e c o - o r d i n a t e s ; the f i n a l b r a c k e t e d term contains  the s p i n f u n c t i o n s .  A statistical  constant and a dou-  b l e parentage c o e f f i c i e n t , both of which a r e u n i t y , have omitted from the d i s c u s s i o n .  ^n^o  ( 1  s  been  1»2,3 ) and  are one-nucleon o r b i t a l s  of zero a n g u l a r momenta w i t h p r i n -  c i p a l quantum numbers n  ( i = 1,2,3 ) and N, r e s p e c t i v e l y .  The  i  c o e f f i c i e n t s a p p e a r i n g i n the summation of the second  bracketed  term a r e the Talmi c o e f f i c i e n t s  formula f o r the Talmi c o e f f i c i e n t s mation of harmonic  (Talmi  incurred  (1952.)); a  i n the t r a n s f o r -  oscillator.wave functions i s derived i n  Appendix B. The r a d i a l harmonic  o s c i l l a t o r wave f u n c t i o n i s of the  form,  where,  .• L^Mhr ) 2  =  b being the harmonic  g\ Q  n-h )  oscillator  h'.  j  '  s i z e parameter.  convenient to w r i t e E q . (4.7) i n t h e form,  I t w i l l be  -  2T" lf = j p o  (4.11)  c  46  -  2^, f ^(R ). o  2^  and to assume that the one-nucleon wave f u n c t i o n s of the indep e n d e n t - p a r t i c l e model and the f u n c t i o n s l ^ and  IJ/^Q  tions. (4.12)  can he approximated by harmonic The o v e r l a p '  (R ). =  o  n i  ( *  =  1  » ,3) 2  o s c i l l a t o r wave f u n c -  integral, |^(R ),  Q  0  i s found by performing the i n t e g r a t i o n s ; the r e s u l t i s , (4.13)  C^j(R )  T(00;j j ) ^  '^(OOJJ-LJ-L)  A  0  •' •^ 0 n (i!)-  l  0  3 / 2  3  ;0 ) 1^ (n  1 +  ^N0n 0;0lK 0H 0;0>  n  3  1  2  1 ; > ^ N 0 n 0 ; 0 1 ^ 1 ^ 1 ; 0> x  0  2  2  i ) I ( n i ) I ( n ^ - ) f * l2JTb '.n :n I I ^ ^ b 2 +  n i  The r a d i a l  3  2  3  \  9/2-  3  functions, K0» d e f i n e the center-of-mass  dependence of the system  (1234).  I f most of the c o n t r i b u t i o n  comes from a p r i n c i p a l node, N , one would function, (jy^Q Eq.  free  (3.9)  expect the p r i n c i p a l  , to be s i m i l a r to the resonant s o l u t i o n of  having the same number of nodes.  parameter of the harmonic  Since the  only-  o s c i l l a t o r wave f u n c t i o n i s the  s i z e parameter, b, i t s c h o i c e must be somewhat a r b i t r a r y . Since o n l y the amplitudes enter the d e t e r m i n a t i o n of the reduced width, i t might seem reasonable to choose the amplitude of the p r i n c i p a l f u n c t i o n a t the n u c l e a r r a d i u s to be equal to the c o r r e s p o n d i n g amplitude of the one-body wave f u n c t i o n as ted from Table 2 . . Of course, such a s o l u t i o n w i l l , e r a l , neither y i e l d  selec-  i n gen-  the c o r r e c t energy nor s a t i s f y the boundary  - 47 c o n d i t i o n , Eq.  (2.30).  I t i s not c l e a r t h a t c o n n e c t i n g  the i n d e p e n d e n t - r ^ a r t i c l e  model c a l c u l a t i o n w i t h the one-body problem i n the above manner i n t r o d u c e s the s u r f a c e and  the s i z e e f f e c t s of the  body problem i n t o the c a l c u l a t i o n c o r r e c t l y . Eq.  (3.5)  one-  I t i s seen from  t h a t the s p e c t r o s c o p i c f a c t o r s s h o u l d be r a t h e r i n -  s e n s i t i v e to the s u r f a c e and  the s i z e of the n u c l e u s  so t h a t  these f e a t u r e s s h o u l d a f f e c t o n l y the one-body a s p e c t s of the c a l c u l a t i o n ; i n f a c t , i n t h i s c a l c u l a t i o n , the  spectroscopic  f a c t o r s are connected to the one-body problem o n l y through c h o i c e of the s i z e parameter.  Hence, i f the use  the  of the h a r -  monic o s c i l l a t o r one-nucleon f u n c t i o n s i s r e a s o n a b l e ,  the  cri-  t e r i o n f o r the v a l i d i t y of the above procedure becomes t h a t the s p e c t r o s c o p i c f a c t o r s must not depend s e n s i t i v e l y upon the c h o i c e of the s i z e parameter. 4-4:  NUMERICAL INDEPENDENT-PARTICLE MODEL REDUCED WIDTHS In  the p r e s e n t  s e c t i o n , we w i l l  c a l c u l a t e the  independent-  p1 p  p a r t i c l e model reduced w i d t h of Po  by the t e c h n i q u e  H a r a d a , but employing the one-body r e s o n a n t a d i f f u s e - e d g e w e l l r a t h e r than those In h i s c a l c u l a t i o n , Harada has  of  wave f u n c t i o n s of  of a square-edge w e l l .  assumed a square-edge w e l l of  r a d i u s ten f e r m i s ; w i t h the i n t r o d u c t i o n of c o n f i g u r a t i o n mixing  i n the p a r e n t n u c l e u s wave f u n c t i o n , he o b t a i n s a  a b l e v a l u e of about o n e - t w e n t i e t h  the e m p i r i c a l decay  A l a r g e r r a d i u s would produce a l a r g e r decay c o n s t a n t  reasonconstant. but,  f r o m F i g u r e 1, i t would appear t h a t over ten f e r m i s i s too large a nuclear r a d i u s .  In f a c t , as w i l l be shown i n the  sub-  - 43 -  sequent d i s c u s s i o n , t h i s l a r g e square-edge w e l l to a s m a l l e r d i f f u s e - e d g e  corresponds  well.  Harada has chosen the s i z e parameter so that the p r i n c i p a l f u n c t i o n should (ten fermis)  have an amplitude a t the nuclear  radius  equal to that of a square-v/ell resonant wave  f u n c t i o n ; the square-v/ell has been chosen so that t h i s wave f u n c t i o n has the same number of nodes as the p r i n c i p a l f u n c t i o n and s a t i s f i e s the boundary c o n d i t i o n , E q . (2.30), The  only aspect  i n which the present  exactly.  c a l c u l a t i o n d i f f e r s from  H a r a d a s i s that the s i z e parameter has been chosen so that 1  the amplitude of the p r i n c i p a l f u n c t i o n i s equal to the amplitude of the corresponding resonant f u n c t i o n of a w e l l as s e l e c t e d from Table 2. incorporate  diffuse-edge  This procedure i s intended to  the p r o p e r t i e s of the diffuse-edge  l a t e d reduced widths i n a more d i r e c t way.  i n t o the c a l c u -  I t makes much  c l e a r e r the r o l e of the p o t e n t i a l and the value  of the r a d i u s  of the resonant s t a t e . The  c o n t r i b u t i o n s to the overlap  O J J ( R Q ) . ^ e given a  has  i n Table 4.  i n t e g r a l from each node,  Here, the p r i n c i p a l f u n c t i o n  been chosen to have t e n nodes and the r e s u l t s are tabu-  lated f o r various  nuclear  s i z e s and surface  thicknesses.  The  r a t e s of c o n t r i b u t i o n were found to be r a t h e r i n s e n s i t i v e to the harmonic o s c i l l a t o r The  s i z e parameter.  c a l c u l a t e d values  i n Table 5.  of the'reduced widths are tabulated  A comparison v/ith the experimental values  (Table  l ) and, hence, of the decay r a t e s , has a l s o been i n c l u d e d i n the  tabulation.  - 49 TABLE 4 OVERLAP INTEGRALS a) Surface Thickness : 0.5 fermis R ( f 7  3  /  2  )  0  = 9.0  f.  //b - 0.116 f 7  R - 9.5 f . b = 0.095 f 7  R  0  2  5.68x10"  8  2.73xl0**  4.43xl0"  8  1.67xl0"  2  0  » 10.0 f .  b = 0.116 f r  7  1.7 6x10 "  9  7  1.83xl0"  9  4.89xl0" '  1.43xl0*"  6  2.75xl0"  8  7. 98x10"  6  1.75xl0"  5  6.35xl0"  7  6.06x10"°  9.62xl0~  5  7.18xl0"  6  1.31xl0"  4  1.39x10" • 4  2.46x10"  0~io  9.6lxl0"  5  5.4lxl0"  5  3.32x10'°  Oil  1.59xl0"  5  1.08xl0~  6  7  2  5  • 1.42xl0"  5  # This was the l a r g e s t p o s s i b l e amplitude and i s s l i g h t l y l e s s than the resonant amplitude. b) Surface Thickness: 0.7 f e r m i s R ( f r V 2 ) Oa  Q  ='. 9 . O f .  b = 0.110 f 7  R 2  0  =.9.5  R : =.10.0 f .  f.-.  b = 0.092 f 7  Q  2  b = 0.120 f r  1.26xl0"  7  3.93xl0  8.84xl0"  8  2.2.8xl0- .  l.OlxlO"  9  8.75x10"  7  1.84X10.**  1.62xl0"  8  1.27xl0~  5  2.10xl0"  5  3.99xl0"  7  8.36xl0"  5  1.06xl0"  4  4.82xl0~  6  #9  1.51x1b"  4  1.36xl0"  4  1.78xl0"  5  <?io  8.52x10"  5  4.05xl0 ;  2.61xl0"  5  Oil  6.81xl0~  6  4 .49xl0"  1.23x10"  - 7  9.27xl0"  7  6  -5  6  5  10  2  - 50 TABLE 5 COMPARISON OF DECAY RATES a  ^£heor  exp  ~X theor  (ev)  (kev)  A exp  0.5  5.20  1.119  1/220  9.5  0.5  5.19  10.0  0.5  0.36  0.111  1/310  9.0  0.7  6.16  1.119  1/180  9.5  0.7  5.18  10.0  0.7  0.22.  0.111  1/500  10.0  0.0  1.3  0.143  1/110  S i z e of Nucleus  Ro (f.)  (f.)  ~8.5 f .  9.0  ~8 0 #  //Harada: ~10.0  u  # (Harada (1961)) We have p r e v i o u s l y noted should be r a t h e r i n s e n s i t i v e the n u c l e u s .  that the. s p e c t r o s c o p i c f a c t o r s to the s i z e and the surface of  In Table 6, the s p e c t r o s c o p i c f a c t o r s found i n  the present c a l c u l a t i o n have been compared w i t h those of Harada. TABLE 6 CALCULATED SPECTROSCOPIC FACTORS E v a l u a t e d a t 9.0 f . c  E v a l u a t e d a t 10.0 1 . C o  S i z e of Nucleus  Surface Thickness  ~8.5 f .  0.5 f .  2.4x10"  5  1.3xl0"  5  -8.0 f .  0.7' f .  3.6x10"  5  1.4xl0"  5  #Harada: ~10.0 f .  0.0 f .  0.9xl0~  5  # (Harada  (1961))  2  "O  2  n  - 51 I t Is' seen from Table 5 t h a t , by t r e a t i n g the a s p e c t s of the problem w i t h a d i f f u s e - e d g e  one-body  w e l l , an agreement  w i t h the e m p i r i c a l decay r a t e s can be o b t a i n e d as  -  t h a t i s as good  t h a t found by Harada employing an anomalously l a r g e r a d i u s ;  i n f a c t , the v a l u e s  suggested f o r the n u c l e a r  a r e r a t h e r modest s i z e s f o r heavy n u c l e i .  s i z e s i n Table 5  Harada has  found  t h a t the i n t r o d u c t i o n of c o n f i g u r a t i o n m i x i n g i n c r e a s e s c a l c u l a t e d decay r a t e s by a f a c t o r of between f i v e and I t should  ten.  be p o s s i b l e to remove much of the r e m a i n i n g d i s c r e p -  ancy -by c h o o s i n g a l a r g e r , but nucleus.  the  s t i l l reasonable,  size for  the  - 52 CHAPTER 5 In  THE  EQUIVALENT SQ.UARE-EDGE NUCLEUS MODEL  the p r e s e n t c h a p t e r , we w i l l d e f i n e , f o r a conven-  t i o n a l ' d i f f u s e - e d g e n u c l e u s , a square-edge n u c l e u s which, f o r the purposes of s t u d y i n g decay r a t e s , s c a t t e r i n g d a t a , and a b s o r p t i o n p r o c e s s e s , e x h i b i t s many of the p r o p e r t i e s of the diffuse-edge nucleus  to which i t c o r r e s p o n d s .  In f a c t , the  " e q u i v a l e n t square-edge n u c l e u s " w h i c h v/e w i l l d e f i n e i s , e s s e n t i a l l y , t h a t which has been used by p r e v i o u s a u t h o r s , i n p a r t i c u l a r Harada (1961), i n the c a l c u l a t i o n of the a l p h a p a r t i c l e decay r a t e s of' heavy n u c l e i . The  u s e f u l n e s s of r e p l a c i n g a d i f f u s e - e d g e n u c l e u s w i t h ,  an " e q u i v a l e n t " square-edge n u c l e u s has been noted p r e v i o u s l y by Vogt (1962); he has d e f i n e d and employed the " e q u i v a l e n t square-edge n u c l e u s " i n the a n a l y s i s of the s c a t t e r i n g a b s o r p t i o n of n e u t r o n s .  and  In p a r t i c u l a r , he has found t h a t the  parameters' of t h i s model are i n s e n s i t i v e to the i n c i d e n t energy and  channel and a l s o to the many-body a s p e c t s of the n u c l e a r  problem; hence, the e q u i v a l e n t square-edge n u c l e u s i s d e f i n e d i n a r e a s o n a b l y unique manner f o r a g i v e n d i f f u s e - e d g e n u c l e u s . V o g t , Michaud, and Reeves (1965) have found for  similar  a l p h a - p a r t i c l e s c a t t e r i n g from l i g h t n u c l e i .  results  The  useful-  ness of the e q u i v a l e n t square-edge n u c l e u s i s then t w o - f o l d : a) once h a v i n g determined lows one  the parameters of the model, i t a l -  to e x p l o i t the s i m p l e a n a l y t i c p r o p e r t i e s of  w e l l s i n subsequent c a l c u l a t i o n s ; b) i t p r o v i d e s a set  square-  convenient  of parameters f o r s t u d y i n g the e f f e c t of the d i f f u s e  nu-  c l e a r edge on decay r a t e s , s c a t t e r i n g c r o s s - s e c t i o n s , and'ab-  - 53  sorption cross-sections.  -  T h i s i s p a r t i c u l a r l y h e l p f u l i n the  study of a b s o r p t i o n c r o s s - s e c t i o n s a t the low e n e r g i e s t e r e s t to a s t r o p h y s i c s where the c a l c u l a t i o n s are  of i n -  otherwise  quite tedious. In the d i s c u s s i o n to f o l l o w , we w i l l p r o v i d e  the  t i o n of the e q u i v a l e n t square-edge nucleus model and  definiwill  check the e x t e n t to w h i c h i t a p p l i e s to a l p h a - p a r t i c l e 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 . 5-1:  THE The  nucleus  EQUIVALENT SQ.UAIiE-EDGE NUCLEUS MODEL  e q u i v a l e n t square-edge n u c l e u s  of a  i s d e f i n e d i n the f o l l o w i n g manner.  diffuse-edge The r a d i u s  and  depth of a r e a l s q u a r e - w e l l p o t e n t i a l are chosen so t h a t the f o l l o w i n g c o n d i t i o n s are s a t i s f i e d :  a) i t e x h i b i t s a r e s o -  nant wave f u n c t i o n i n the c h a n n e l , c, a t the resonance energy, £c, a p p r o p r i a t e to the decay, s c a t t e r i n g , or a b s o r p t i o n problem of i n t e r e s t ; b)the resonant w e l l has ing  wave f u n c t i o n of the square -  the same number of nodes as t h a t of the  resonant  correspond-  f u n c t i o n of the d i f f u s e - e d g e . w e l l which d e s c r i b e s  the average i n t e r a c t i o n between the i n c i d e n t p a r t i c l e and t a r g e t nucleus £c;  i n the c h a n n e l , c, and a t the resonance energy,  c) the reduced w i d t h s  of the two w e l l s , e v a l u a t e d a t  r a d i u s of the s q u a r e - w e l l , are e q u a l i n t h i s channel and t h i s energy.  the  One  the at  d e f i n e s the " r e f l e c t i o n f a c t o r " , w h i c h a c -  counts f o r the anomalous r e f l e c t i o n of the s q u a r e - w e l l ,  as  the r a t i o of the p e n e t r a b i l i t y of the d i f f u s e - e d g e w e l l to t h a t of the s q u a r e - w e l l a t the s q u a r e - w e l l r a d i u s . many-body a s p e c t s  I f the .  of the n u c l e a r problem have been accounted  - 54  for  -  by c h o o s i n g the d i f f u s e - e d g e w e l l to be an o p t i c a l model  p o t e n t i a l , the imaginary  term i n the s q u a r e - w e l l p o t e n t i a l i s  chosen to have the r a d i u s of the square-we 11 and the depth the d i f f u s e - e d g e w e l l .  The  of  e l e c t r o s t a t i c p o t e n t i a l i s chosen  to be the same f o r b o t h ' w e l l s .  The  square-well defined i n  t h i s manner i s c a l l e d the " e q u i v a l e n t s q u a r e - w e l l " (ESW). In g e n e r a l , the parameters•of sitive  the ESW  to the channel and the energy.  corresponding etrabilities  are r a t h e r i n s e n -  S i n c e the ESW  and  the  one-body d i f f u s e - e d g e v / e l l have the same penand reduced  the one-body' problem.  w i d t h s , they are i n t e r c h a n g e a b l e i n  I t i s seen from Eq.  (3.5)  t h a t the  s p e c t r o s c o p i c f a c t o r s should be r a t h e r i n s e n s i t i v e to the c l e a r s i z e and  s u r f a c e s i n c e the n u c l e a r reduced  width  nu-  and  the one-body reduced w i d t h depend upon these a s p e c t s of the n u c l e u s i n a s i m i l a r manner; i n f a c t , we have checked t h i s a s s e r t i o n i n " T a b l e 6. account  Thus the s p e c t r o s c o p i c f a c t o r s , w h i c h  f o r the many-body a s p e c t s of the n u c l e a r problem,  s h o u l d be about the same whether the nucleus  i s chosen to  have the s i z e and s u r f a c e of the d i f f u s e - e d g e v / e l l or of the c o r r e s p o n d i n g ESW.  Thus we  can i n a m e a n i n g f u l manner r e -  p l a c e the c o n v e n t i o n a l d i f f u s e - e d g e nucleus w i t h the e q u i v a l e n t square-edge nucleus f o r the purposes of a n a l y z i n g decay r a t e , s c a t t e r i n g , and a b s o r p t i o n d a t a . 5-2:  APPLICATIONS TO HEAVY NUCLEI In  the p r e s e n t s e c t i o n , we w i l l examine the v a l i d i t y  the e q u i v a l e n t square-edge n u c l e u s model i n heavy n u c l e i c o n s i d e r i n g the a b s o r p t i o n of a l p h a - p a r t i c l e s .  The  of by  technique  - 00 -  employed i s t o e v a l u a t e  the ESW f o r a f a s h i o n a b l e  optical  model p o t e n t i a l and then to study the dependence of the r e f l e c t i o n f a c t o r upon the parameters of the problem by cons i d e r i n g the t r a n s m i s s i o n  f u n c t i o n s o f the two one-body poten-  tial. The one-body r a d i a l S c h r o e d i n g e r E q u a t i o n ,  w i t h an o p t i c a l  model p o t e n t i a l , i s o f the form, ( 5 . 1 )  "W  :  (  +  V  C ^ )  % ( K )  +  i % ( R )  +  +  | ^ ^ J . ) = cuc £  D e f i n i n g the incoming wave i n the c h a n n e l , c, by, (5.2) : I ( R ) = e- c ( G iW  C  c  t-if ), c  one h a s , a t s u f f i c i e n t l y l a r g e r a d i u s , u»(r)  (5.3)  { r  - jTy  = G (R)+  I  C  =  c  and D  C  Eq.  c  -  I (R)  -  i*c) i t ( R )  e^c* e  2i(-  +iJ ) I (R)  c  c  C  a r e the r e a l and i m a g i n a r y 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 of u In g e n e r a l , J>  iD (R)  Il.(R) C  where C  that  c  and where <*+- i$ i s the phase s h i f t , c  c  i s very s m a l l v/hence i t i s e a s i l y shown from  (5.3) that,  (5.4)  •' T  c  - 4D  #  ? ^c = 4  (c G C  C  - G  C  - D  C  C  F  C  ) 2  + (F« - C E C  C  -  D G ) C  C  R=  v/here T R  m  i s the o p t i c a l model t r a n s m i s s i o n f u n c t i o n and where  i s s u f f i c i e n t l y l a r g e t h a t there  i s no f u r t h e r  absorption  due t o the o p t i c a l model p o t e n t i a l . If  there  i s no a b s o r p t i o n  due t o the o p t i c a l model poten-  t i a l beyond the ESW r a d i u s , one can d e f i n e ,  - 56 -  •to  (5.5) a)  Pi s-HkwD . e  b)  s  l  >, ^  0  (  =  j  = -•^•• c ,  c  li  ^  -  *  0  and w r i t e the o p t i p a l model t r a n s m i s s i o n f u n c t i o n i n the form (Preston  (1962)),  (5.6)  4P T  n  s  Tq ( S _- o. S ) c  i  (•*>_ (PC  2  c  MR  C  +  .+ >. i P \2 )2 iT  i c  f° 2 JJ K ( ) 0  W  h  R  2  u (R) c  c  Assuming u  to he normalized  c  dR.  u (R J 0  i n R , and n o t i n g that the s u r 0  face, t h i c k n e s s i s s m a l l , one can w r i t e E q . (5.6) i n the form, (5.7)  4Pc  W  0  ' ( s - s ) + (P C -+ i P i J . f T i ^ y ••'  c  a  c  5 4  i p  where, (5.8)  c  •• i n " \ \  ]j£  f (R ) = Q  u (R ) c  0  0  Choosing the n u c l e a r r a d i u s , R ing  t h a t , a t resonance,  one  might expect i n g e n e r a l  (5.9) .  fESw(%Sw) f  2  ^  f(RJJSW^  to he the E S W r a d i u s , and not-  Q  i  s  t  h  e  one-hody reduced width,  that,  1.  d i f f ( ESW) R  2 Numerical c a l c u l a t i o n s show that the denominator, ( S +  ( P +• P i ) , c  the w e l l to w i t h i n a few p e r c e n t . (5.10)  T diff ' c  . ESW  *  T  C  c  Thus,  P diff  t  c  P  cESW  I t w i l l he convenient ficient,  - Si )  i s independent of the i n t e r n a l f e a t u r e s of  2  c  c  to regard  the t r a n s m i s s i o n  coef-  T , of E q . (5.4), a s a f u n c t i o n of the r a d i u s o f c  evaluation, R  M  .  To study  the e f f e c t s of the imaginary  poten-  t i a l w i t h i n the b a r r i e r , one need then only c o n s i d e r the r a t i o of the t r a n s m i s s i o n f u n c t i o n of the d i f f u s e - e d g e w e l l to that  - 57 of the ESW.  W i t h i n the b a r r i e r ,  o>^ o  G  . .  F  D < < 1 s o that, (5.11) 'g(R) =  T  diff( )  Pdiff(R)  ESW^ '  PESW^^T  R  T  R  #  One may then c o n s i d e r that the a b s o r p t i o n due to the d i f f u s e imaginary p o t e n t i a l ceases where g(R) becomes asymptotic to the r e f l e c t i o n f a c t o r , f . From E q . (5.1), (5.12) a) d C l R l I b) d D ( R l •  d  R  z  k  k  ( *c  2  2  %  +  .  l  _ 2  }  c  ( r )  _ 2 D  _ C(R).D(R) .  %  (L = 0 )  2  .  ,  .  ( R ) f  -  The s o l u t i o n of Eq. (5.12) b) i n the b a r r i e r i s of the form, (5.13)  D  (  R  E%_  conatant-e-^/Zr^-R  _ fr  ,  •-  s i n c e , i n the b a r r i e r , P i s very s m a l l and '(J(R)'N-'k^p. I f V /e c  -  i s s u f f i c i e n t l y l a r g e , the f i r s t  term dominates the  second so that the a b s o r p t i o n from the o p t i c a l model potential  exceeds that from the b a r r i e r . The n u m e r i c a l example considered was that of s-wave s c a t O AO  terin-g. of an a l p h a - p a r t i c l e from a Pb tial  u  target.  The poten-  208 of the Pb - t a r g e t was taken to be of the form,  (5.14)  V (R) '+ iW (R) = (V + i W ) - ( l N  H  Q  0  4;EXP( -^a R  where the parameters where s e l e c t e d to be, V  0  = -99.58 MEV;  r  0  = 8.556 f . ; a = 0.5 f . ;  ) J" , 1  W  = -10'  Q  The  MEV.  c o r r e s p o n d i n g ESW V  E S W  was  found to have the. parameters,  = -99.99 MEV;  R  E3V  , « 9.39  f. ; W  E 3 W  = -10  f = 1.9; a t an  incident  energy) .  The  The  the  r e l a t i v e energy of 8.9795 MEV  e q u a t i o n s , Eqs.  effects  illustrated  ^  technique employed was  differential  of the  i n F i g u r e 8.  the b a r r i e r  model c a l c u l a t i o n s ,  " of 8.9795 MEV,  the  to  barrier.  not  D.  are  the This  have a good phys-  imaginary p a r t of the  of 1.9  of the  the  same as  n o t i n g that  o p t i c a l model  to the  potential.  A  ana-  d i s c u s s e d i n Chapfound to  be  Coulomb b a r r i e r ;  from  depend s e n s i t i v e l y upon  parameters should be  to the decay channel and  insensitive  of the ESW.  obtained by  r e f l e c t i o n f a c t o r was  the ESY/  po-  alpha-decay energy  p e n e t r a b i l i t i e s as  i t does not  the  considered.  diffuse-edge  that  i s obtained at the  to the h e i g h t of the  energy, i t i s seen that  to be  f o r C and  (to remove i t s t a i l ) s h a l l be  value of the  rather i n s e n s i t i v e  insensitive  coupled  f o r heavy n u c l e i ,  comparing w i t h a value of 2.0  l y t i c calculation  and  decay-  i n t r i n s i c p r o p e r t y of a l l o p t i c a l  been chosen to be  r e f l e c t i o n factor  this,  _  Where i t becomes dominant, a m o d i f i c a t i o n of  In Figure, 9,  The  b),  i s very s e n s i t i v e  even though i t may  absorptive potential  t e r 3.  (5.12) a) and  I t i s seen t h a t ,  long-range a b s o r p t i o n i s an  t e n t i a l has  to s o l v e the  imaginary p o t e n t i a l w i t h i n the  i c a l basis.  (Po  of the h e i g h t of the Coulomb b a r r i e r  t r a n s m i s s i o n of  tail  MEV;  energy.  s t r e n g t h of the  I t was  the  rather also  imaginary p a r t of  found the  106,  104-  BARRIER ABSORPTION  FIGURE 8  TDIFF/  PB 208  TESW  10' I0  ;  102  DIFFUSE IMAGINARY  10 SQUARE IMAGINARY  10 M E V "it  13  14  15  16  FERMIS  17  18  19  20  ;  In the c a l c u l a t i o n of Vogt, Michaud, and Reeves the d i f f u s e - e d g e a diffuse  6 1  -  (1965),  o p t i c a l model p o t e n t i a l was chosen to have  complex p a r t .  They o b t a i n a value  of 4.62 f o r the  r e f l e c t i o n f a c t o r ; r e p l a c i n g the d i f f u s e - e d g e  complex p a r t  by the complex p a r t of the a p p r o p r i a t e ESW y i e l d s a value of 2.8.  I t might, t h e r e f o r e , seem that the r e f l e c t i o n  has been overestimated  by a f a c t o r of two.  factor  They a l s o s t a t e  a r e f l e c t i o n f a c t o r of 100 when the ESW i s r e p l a c e w i t h a s q u a r e - w e l l having In the present  the same r a d i u s as the d i f f u s e - e d g e w e l l .  c a l c u l a t i o n , the r a t i o of the p e n e t r a b i l i t y  of the d i f f u s e - e d g e w e l l to a square-well was found  to vary  r a t h e r s l o w l y with the r a d i u s s e l e c t e d . It  i s seen.from the example s t u d i e d i n t h i s Chapter t h a t  the r a d i u s of the ESW i s about a ferrai l a r g e r than the corresponding  diffuse-edge w e l l .  that of  I t i s apparent from  Harada's c a l c u l a t i o n t h a t he has, e s s e n t i a l l y , used such an ESW f o r c a l c u l a t i n g the i n d e p e n d e n t - p a r t i c l e model widths.  reduced  Thus the l a r g e r a d i u s i n h i s c a l c u l a t i o n i s c o n s i s t -  ent w i t h c o n v e n t i o n a l n u c l e a r s i z e s , p r o v i d e d  that i t i s i n -  t e r p r e t e d as the r a d i u s of. an e q u i v a l e n t square-edge  nucleus.  - 62 CHAPTER 6  CONCLUSIONS  In t h i s t h e s i s we have shown that much of the d i s c r e p a n c y ' between the e m p i r i c a l a l p h a - p a r t i c l e decay r a t e s of heavy n u c l e i and those estimated from n u c l e a r s h e l l - m o d e l c a l c u l a t i o n s can be removed by a more d i r e c t treatment  of the  nuclear surface.  calculations  We have contended  that p r e v i o u s  on a l p h a - p a r t i c l e decay r a t e s have e s s e n t i a l l y used  the e q u i v -  a l e n t square-edge nucleus model of Vogt; we have shown by cons i d e r i n g the one-body problem that the e q u i v a l e n t square-edge nucleus has a c o n s i d e r a b l y l a r g e r r a d i u s than the d i f f u s e - e d g e nucleus  to which i t corresponds.  We have concluded  l a r g e r a d i i which were found necessary i n p r e v i o u s  that the calcula-  t i o n s on a l p h a - p a r t i c l e decay r a t e s to o b t a i n agreement w i t h the e m p i r i c a l v a l u e s were the r a d i i of t h e . e q u i v a l e n t  square-  edge nucleus model r a t h e r than the a c t u a l r a d i i of the decayi n g systems being c o n s i d e r e d .  In f a c t , we have shown that  the r a d i i o f the corresponding d i f f u s e - e d g e n u c l e i agree  with  the c o n v e n t i o n a l r a d i i b e l i e v e d to be t y p i c a l o f a heavy nucleus. - In our c a l c u l a t i o n s , . we have checked and  that the J.W.K.B.  the square-well estimates of the n u c l e a r p e n e t r a b i l i t y  used by. previous authors are i n reasonable agreement with the a n a l y t i c v a l u e s .  In f a c t , we have found i n Chapter 3  that the -J.W.K.B. estimate i s q u i t e good to w i t h i n about onetenth -of a f e r m i of the c l a s s i c a l i n n e r - t u r n i n g p o i n t and that the square-well estimate d i f f e r s by l e s s than an order of magnitude a t the c l a s s i c a l i n n e r t u r n i n g p o i n t .  - 63 We have demonstrated i n Chapter 4 that the one-body :  21P reduced  widths a p p r o p r i a t e  to the decay of Po  c o n s i d e r a b l y by a d i f f u s e n u c l e a r edge.  are enhanced  In f a c t , we have  checked the e f f e c t on the one-body decay constant by c a l c u l a t i n g the d i f f e r e n t i a l e l a s t i c  directly  scattering cross-  s e c t i o n f o r the one-body s c a t t e r i n g of an a l p h a - p a r t i c l e from P b " ^  8  at 90°Q^.  We have i n c l u d e d the many-body a s p e c t s  of the problem by r e p e a t i n g the c a l c u l a t i o n of Harada with the square-well  one-body wave f u n c t i o n s of h i s c a l c u l a t i o n  being r e p l a c e d with the resonant  wave f u n c t i o n s of a d i f f u s e -  edge w e l l .  We have found  that h i s square-well  corresponds  to a d i f f u s e - e d g e w e l l c a l c u l a t i o n of a s m a l l e r  and more c o n v e n t i o n a l n u c l e a r r a d i u s . tended t h a t h i s c a l c u l a t i o n corresponds square-edge nucleus  calculation  In f a c t , we have conto the e q u i v a l e n t  c a l c u l a t i o n f o r such a d i f f u s e - e d g e w e l l .  We have examined the dependence of the e q u i v a l e n t  square-  edge n u c l e u s model parameters i n Chapter 5 by c a l c u l a t i n g the one-body t r a n s m i s s i o n f u n c t i o n s f o r the a b s o r p t i o n of an a l p h a 208 p a r t i c l e by a Pb tail  nucleus.  of the imaginary  We have found  t h a t , provided the  p a r t of the o p t i c a l model p o t e n t i a l i s  t r u n c a t e d a t the n u c l e a r r a d i u s , the e q u i v a l e n t square-edge nucleus.model parameters are r a t h e r i n s e n s i t i v e  to the nature  of the r e a c t i o n and to the many-body a s p e c t s of the problem. We have from t h i s concluded  that the e q u i v a l e n t square-edge  nucleus model should have some v a l i d i t y  i n heavy n u c l e i .  We have i n t e r p r e t e d the r e s u l t s of our c a l c u l a t i o n s as suggesting  that the i n d e p e n d e n t - p a r t i c l e model, with  only  - 64 those c o r r e l a t i o n s i n t r o d u c e d  by c o n f i g u r a t i o n mixing,' can  s a t i s f a c t o r i l y account f o r the c l u s t e r i n g i n t o complex particles  i n the n u c l e a r  surface;  i n p a r t i c u l a r , we b e l i e v e  such a model can p r e d i c t reasonable v a l u e s p a r t i c l e decay r a t e s of heavy n u c l e i .  f o r the a l p h a -  that  - 65 BIBLIOGRAPHY Arima, A., and Terasawa, T., "Progress o f T h e o r e t i c a l P h y s i c s 23, 115" (I960). Bencze, Gy., and Sandelescu, A., " P h y s i c s L e t t e r s 22, 473" (1966), F o r d , K.W., and H i l l , D.L., 5_, 25" (1955).  "Annual Review o f Nuclear Science  Harada, K, , " P r o g r e s s of T h e o r e t i c a l P h y s i c s 26,. 667" Igo, G., " P h y s i c a l Review 115, 1665"  (1961).  (1959),  Mang, H.J.,  " P h y s i c a l Review 119. 1069" ( i 9 6 0 ) .  Mang, H.J.,  "Annual Review of N u c l e a r Science 14_, 1" (1964)  M e s s i a h , A., "Quantum Mechanics, Volume I" (1962), (John Wiley & Sons, Inc., New York) Noya, H., Arima, A., and H o r i e , H., "Progress of T h e o r e t i c a l P h y s i c s , Supplement No. 8_, 33" (1959). P r e s t o n , M.A., " P h y s i c s o f the Nucleus" (Addison-Wesley, Reading, Mass.) Rasmussen, J.O.,  " P h y s i c a l Review 113.  (1962). 1593" (1959). .  Rose, M.E., "Elementary'Theory o f Angular Momentum" (1957). (John W i l e y & Sons, Inc., New York) Talmi, I . , " H e l v e t i c a P h y s i c a A c t a 25, 185" (1952).. Thomas, R.G., "Progress of T h e o r e t i c a l P h y s i c s 12, 253" (1954). Vogt, E.W., "Reviews of Modern P h y s i c s 34, 723" (1962). Vogt, E.W. 'Michaud, 570" (1965).  G., and Reeves, H., " P h y s i c s L e t t e r s 19,  W i l k i n s o n , D.H., "Proceedings of the R u t h e r f o r d Conference on N u c l e a r S t r u c t u r e , 339" (1961). Zeh, H.-D., and Mang, H.J., Zeh, H.-D., " Z e i t s c h r i f t  "Nuclear P h y s i c s 29, 529" (1962).  f u r P h y s i k 175, 490" (1963).  - 66 APPENDIX A  HARADA'S FORMULA FOR THE REDUCED WIDTHS FOR ALPHA-PARTICLE DECAY IN EVEN-EVEN NUCLEI  Harada (1961) has u a t i n g the  d e r i v e d a convenient  formula  for eval-  i n d e p e n d e n t - p a r t i c l e model reduced widths f o r the  ground s t a t e a l p h a - p a r t i c l e decay t r a n s i t i o n i n even-even n u c l e i with  the use  of harmonic o s c i l l a t o r one-nucleon wave  functions.  In t h i s appendix, we  present an o u t l i n e of  the  d e r i v a t i o n of h i s r e s u l t . The  reduced w i d t h amplitude f o r a l p h a - p a r t i c l e decay has  been d e f i n e d i n Eq,  (3.3),  • • • Y^ where:  L  dAdldx^ds.  i s the parent  , ^  nucleus wave f u n c t i o n ; X  a l p h a - p a r t i c l e wave f u n c t i o n ; , IL/T  i s the  i s the daughter  nucleus  wave f u n c t i o n ; Yj-^ i s the s p h e r i c a l harmonic d e s c r i b i n g the r e l a t i v e motion of the decay fragments.  Here, "iX"and ^ j  are assumed to be p r o p e r l y antisymrnetrized and I|J  0  to be ter  p a r t i a l l y antisymrnetrized  j  i s assumed  i n the sense d e f i n e d i n Chap-  2. We  assume the parent  even-even n u c l e i and  nucleus  and  daughter nucleus  to be represented  model wave f u n c t i o n s of s e n i o r i t y zero. ground s t a t e t r a n s i t i o n s , nucleus,  and  to be  independent-particle We  consider nucleus,  only  the  daughter  the a l p h a - p a r t i c l e have zero angular momentum; o r b i t a l angular  i s then a l s o  zero.  The  by  so that the parent  the r e l a t i v e  w  m  momentum of the decay fragments  n o t a t i o n i s t h a t of Chapter 2 and  Chapter  3.  - 67 It tion  i s convenient  to expand the parent nucleus wave f u n c -  i n the two-proton and the two-neutron wave  which can be formed from the u n f i l l e d nucleus c o n f i g u r a t i o n ; to c o n t r i b u t e  subshell  j  m  of the parent  i t i s these nucleons which we  parentage  coefficients,  we r e q u i r e the  ( j " ( s j ) j ( J ) jfj j S J ) . m  2  2  1  m  denotes the angular momenta of the m one-nucleon  t i o n s of the u n f i l l e d  Here  wave func  s u b s h e l l of the parent c o n f i g u r a t i o n ; j  denotes the angular momenta of the two one-nucleon tions  expect  to alpha-decay.  In order to make the above expansion, double  functions  of the u n f i l l e d  wave func-  s u b s h e l l which are taken to c o n s t i t u t e m-2  the two-nucleon wave f u n c t i o n ; j menta of the remaining m-2 daughter  configuration.  denotes the angular  one-nucleon  wave f u n c t i o n s  mo-  of the  S and J are the s e n i o r i t y and the  t o t a l a n g u l a r momentum of the parent nucleus,' r e s p e c t i v e l y ; s and j are the s e n i o r i t y and the t o t a l angular momentum of the daughter  nucleus, r e s p e c t i v e l y ; J ' i s the t o t a l  angular  momentum of the two-nucleon system formed from the u n f i l l e d subshell. N o t i n g the p a r t i a l a n t i s y m m e t r i z a t i o n of VJ , 0  subscripts  u s i n g the  1 and 3 to denote proton and neutron quantum num-  b e r s , r e s p e c t i v e l y , and Invoking (A.l)  our p r e v i o u s  assumptions,  - 68 -  dx ds AT-2,  (^^(OOjj^OjolBj^OOHJn^lOOjj^oJol^j^OO)  M/ 12  A  3l  p  2J  —  (2j  1  »-p  3 - A )  +  (2j t  x  3  L  H e r e , (J)^ ( ( | ) N ) d e p e n d s nates,  (x]_, 2^2^  protons j  ( j  the (0  o  n  ).  3  (A -l)(2j tlJ 3  3  o n l y upon t h e p r o t o n  (neutron)  ^ (2^.3. 2L4) ) » A]_ ( A 3 ) d e n o t e s  (neutrons)  i n the u n f i l l e d  ) , o f the p a r e n t  protons  3 - A )  (neutrons)  In the f i r s t  proton  co-ordi-  t h e nurnher o f  (neutron)  subshell,  c o n f i g u r a t i o n , the a n g u l a r momenta o f in this  s u b s h e l l being  denoted  by j"  s t e p o f E q . ( A . l ) , we have e x p a n d e d t h e  wave n a r e n tf u n uc ctlieounss wave o f t hfeu nu cn tf ii olnl'e di n stu h be sht ew lo l;proton  ( t w o - inse ua t rnoonr )-  m(anleiuzt ar to in s o )n may f a c t obe r as ce cl o o) ns eu c tn et di n gf r ofmo rt t hh e eu nf af ci tl l etdh a t p r othe t o n two ( n e pu rt o rt on subshell  in  ( (A^) ) ways.  t e g r a t i o n has been performed; over  t h e ways o f s e l e c t i n g  unfilled  proton  have noted (A.2)  (neutron)  the r e s u l t  In the second  i n the t h i r d  two p r o t o n s subshell.  m  2  2  s t e p , we have  m  insumme  (neutrons)  from the  In the f i n a l  s t e p , we  of Noya, A r i m a , and H p r i e  (j - (00)j (0)0|3j 00) =  s t e p , the  [ [ ^ ( g -  (1959),  - 69 Noting,  and E q . ( A . l ) , and w r i t i n g , (2^-f  3 - A ) 1  U -I)(2j 1  Eq.  1 +  iFTsj + 3 - A ) 3  12  3  L_1_A -I)(2j +1) 3  3  (3.3) becomes, d-ad|_d_s . 0<  We have d e f i n e d  the a l p h a - p a r t i c l e wave f u n c t i o n ,  in  Chapter 4:  (4 .8) X(1234) = ••(4T)- /^ (l2)^(34). 3  0  To perform the i n t e g r a t i o n i n E q . (A.4) i t i s , t h e r e f o r e , convenient  to transform  the proton and the neutron wave f u n c t i o n s  to the L - S r e p r e s e n t a t i o n and to then transform and  neutron c o - o r d i n a t e s  dinates To  the proton  to the i n t e r n a l and r e l a t i v e  co-or-  of the a l p h a - p a r t i c l e . see the manner i n v/hich to transform  the two-nucleon  wave f u n c t i o n s to the L - S r e p r e s e n t a t i o n , we c o n s i d e r the  j - j coupled  only  two-proton wave f u n c t i o n , (R,(j ,0). (The 2  neutron case i s analogous.) ,i - ,i r e p r e s e n t a t i o n :  i{  2 )  = il  2 )  +s<  2 )  :  L - S representation  ' S = s j J  the  1  Both the ,1 - ,j coupled  #Harada employs the c o - o r d i n a t e s  2  = L +- S = 0;  s u p e r s c r i p t ( l ) or (2) denotes the f i r s t  respectively.  ^si ) .  =  or second  proton,  wave f u n c t i o n s and the  ~2'i-i '• 12H ~i-2» -£3H i-3' 3-  =  - 70 L - S coupled wave f u n c t i o n s form complete sets so that we may make the expansion,  •••|(l< »l« >)L.(.i >.« 's 1  2  1  . '  a  :  V/e adopt the n o t a t i o n of Hose (1957):  <(ltl)l« ))L.(4l).« ))S ™|(ltl).(l))iU).(l<8).(2))42) jll>; 2  2  :  :  From E q . ( 4 . 8 ) , only that m a t r i x element f o r which S - 0 (and, hence, L = 0) can c o n t r i b u t e to the i n t e g r a l i n Eq. (A.4).  We need, t h e r e f o r e , only c o n s i d e r c o e f f i c i e n t s of the  form, T(00;jj").  We evaluate t h i s c o e f f i c i e n t b y , f i r s t  express-  i n g i t i n terms of a 9-j symbol: /l T(00; oj) =  (2-J+-1)  1 0 £ Oi s  the 9-j symbol can be reduced to a W c o e f f i c i e n t symbol) which i s r e a d i l y e v a l u a t e d .  (or a 6-j  Conforming w i t h Harada,  we w r i t e the s o l u t i o n ' i n the form, (A.5)  T(00;oj) =  ^  (2j+-l)W( jljl;£0)  #  - i / 2.1+1 ' " J 2 J 21+1 . We w r i t e E q . (A.4) i n the L - S coupled scheme as, #The value of the matrix elements, T ( 0 0 ; j j ) , stated by Harada c o n t a i n s a m i s p r i n t (being i n e r r o r by a f a c t o r of 1 / ^ 2 ) . T h i s e r r o r does not appear to have been c a r r i e d through i n • h i s subsequent c a l c u l a t i o n s .  - 71 = y w / ^ S  A  l  3  A  T ( p O ; J 3 ) T { 0 0 ; J J ) / ' fp(L=0,S=0) 1  1  3  3  ••• ^ (L=0,S=0)5CdJid|ds. . n  The  K  one-nucleon wave f u n c t i o n s are taken  to he harmonic  o s c i l l a t o r wave f u n c t i o n s :  (A.7)  f^jlr.s)  =(J4 (r)|:<li«.-*|J^(|)^  .  1  where,  n  '  h=0 \ n-n  ("b i s the harmonic o s c i l l a t o r (A.9)  hi  J  s i z e parameter.)  Then,  ^ l (x )(/yi/ l (x )g<l l ^ -^ joq>  i ) (J) (L = 0,S = 0) =  1  p  1  1  1  1  1  1  1  X  J  X  2  1  Y ( £):X°(l2); 1 x  •••Y.ia -l  and  '  1  l  2  1^ (V* & d 1 ) are the p r i n c i p a l and o r b i t a l quantum n  3  3  numbers, r e s p e c t i v e l y , of the u n f i l l e d proton  (neutron) sub-  shell. To evaluate  the i n t e g r a l s i n Eq. (A.6), i t i s convenient  to w r i t e the c o - o r d i n a t e s , angular  , i n terms of t h e i r r a d i a l and  components, 3^ and-ft-^, r e s p e c t i v e l y .  I t i s a l s o con-  v e n i e n t to d e f i n e the c o - o r d i n a t e s , £12  =  Kx-L-t-Xr,) ;  Expressing  r  3  4  =  Kx tx ). 3  4  the two-proton and the two-neutron wave f u n c -  t i o n s i n terms of these  co-ordinates,  -  -  72  -Pi  (A.10) i)lpT5_l (x )lpV l (x ) S <l l ^ -^il00>Y^ (^)Y (^2) 1  1  1  1  1  1  1  1  1  1  1  N-  i i ) l ^ l ( x ) ^ l ( x ) l <l V -^|00>Yf (|i) j!|i) 3  3  3  4  3  3  3  3  Y  3  4  6 <L  2' 2  n  X  •• •  I<W 1  0 0  2  >  r  Y  2  Y  T  P  34 ^  (-^2) •  S u b s t i t u t i n g E q s . (A.9) and {A.10) i n t o E q . (A.6), and i n t e g r a t i n g over (A.ll)  and  =^TFM^o  c>lA  TtOQjJiJjKOO;^)  3  J1J3 ... f  <N 0n 0;0|'V- l V l ;0> 1  1  1  1  1  1  ' l'^2'-3'-^  '' ' ^ 2 °  ( r3 4}  ••• l p n b ^ ) X 2  d A  2  (  ^  ( M}  ^d5 'S|<i5 di ds 1  a  3  ^1°  ( ? 1 }  •"  .  K  Noting, (A.12)  l[7 (r )^(^|)l|; (r )Y°(Si) =... El0  H20  12  34  ••• = I<»0n 0 0] 0N 0 0>l|| (R )Y (A)i|; (? ) 3 0  3  :  Ml  2  ;  I0  o  0  n3o  3  rfThe Talmi t r a n s f o r m a t i o n c o e f f i c i e n t s , N L N L ; L n ^ l ^ n g ^ ^ » are d e f i n e d and evaluated i n Appendix B. 1  1  2  2  - 73 s u b s t i t u t i n g E q . (4.8) and E q . (A.12.) i n t o E q . (A.11), and integrating (A.13)  over  and  3  ^ = Jni ° A A  (J  2 R  S  X  3  ^  JlJ3  where, = T ( 0 0 ; J J ) T ( 0 0 ; J J ) (Jj^j  (A.14) O(H ) 0  1  1  3  /  3  <N0n 0 ; 0\ U-jONgO \ O^N-jOn-jO ; 0 | " ^ l ] y l ; 0j> 3  n  l f  n ,n 2  1  3  • ' -<N 0n 0 ; 2  i  2  0|  ^^^fj\)  •• /H n o(^)EXP(-|||)5|d5  ( \)EXP( - f  ^5  x  J  :  2  2  n (^)EXP(-|5|)^dS3-lV 3  P e r f o r m i n g the i n t e g r a t i o n s  K 0  (R ) o  .  i n Eq. (A.14), we can w r i t e the  o v e r l a p i n t e g r a l i n the form, (A.15)  = I  <y s ) (  ,  o  where, (A.16)  C (R ) H  0  = T ( 0 0 ; J J ) T { 0 0 ; J J ) ^> < 1  1  3  3  n  N 0  n 0 ; 0| 1^01^0 ; 0> 3  l' 2. n  '•••<N On 0 ; 0 | V l V ~ l ;6>/N On 0 ; 0| ir 1 "V 1 ; 0> 1 1 ' 1 1 1 1 ' 2 2 ' 3 3 3 3' / • N  1  /  s  1  -a  7* ( n ^ t ) t(n +t) ' . ( n 3 H )  ^ / 2i/£b\ 9/2.  2  n !n «.n i 1  2  .. . /JL=_k\ 11-,+ '1' np+n, "2' " 3 .3 + b  /S  3  NO ^'0 ^  The reduced width can now be c a l c u l a t e d Eq.  (A.16).  +bJ  '  f r o m S q . (A.13) and  - 74 APPENDIX B  TALMI TRANSFORMATION  COEFFICIENTS  The Talrni t r a n s f o r m a t i o n c o e f f i c i e n t s are the t r a n s f o r mation c o e f f i c i e n t s f o r expanding the s h e l l model wave f u n c t i o n of a t w o - p a r t i c l e system i n terms of the wave f u n c t i o n of t h e i r r e l a t i v e and center-of-mass c o - o r d i n a t e s . I f the average f i e l d  i s taken to be an harmonic o s c i l l a t o r  well,  these c o e f f i c i e n t s can be c a l c u l a t e d i n a simple manner.  In  t h i s appendix, a r e c u r s i v e r e l a t i o n i s d e r i v e d which i s conv e n i e n t f o r the computer  e v a l u a t i o n of those Talmi  coeffi-  c i e n t s which have been used i n Chapter 4 and Appendix A. more g e n e r a l r e c u r s i v e r e l a t i o n f o r the Talmi  A  coefficients  a p p r o p r i a t e to an harmonic o s c i l l a t o r v/ell has been d e r i v e d by Arima and Terasawa  (1959), and the technique employed i n  the present d i s c u s s i o n i s based upon t h e i r  calculation.  In the d i s c u s s i o n to f o l l o w , the wave f u n c t i o n s  discus-  sed w i l l be assumed to be harmonic o s c i l l a t o r f u n c t i o n s .  The  s p a t i a l harmonic o s c i l l a t o r f u n c t i o n s a r e of the form,  where:  r and-^- are the r a d i a l and angular components  respectively; Y  1  of r ,  i s a s p h e r i c a l harmonic; b i s the harmonic r a d i a l harmonic  oscillator o s c i l l a t o r function defined  i n Eq. (A.8).  The s p a t i a l s i n g l e - p a r t i c l e s t a t e s w i l l be taken to be The s p a t i a l wave f u n c t i o n of the two pa_rticles i s then,  The r e l a t i v e c o - o r d i n a t e nate w i l l  and the center-of-mass c o - o r d i -  he denoted by, £  respectively. co-ordinate  s  r  x  R- = i ( r -+• r2) ;  - r ;  x  2  The o n e - p a r t i c l e wave f u n c t i o n i n the r e l a t i v e  w i l l be denoted by 4^nl^» )» c  a  n  d  °^e-particle  t 1 n e  wave f u n c t i o n i n the center-of-mass by 4^Pi/!L> ^ •  ^  i s  e a s i l  y  shown t h a t , c = £b; One can c o n s t r u c t  C s 2b.  the t w o - p a r t i c l e wave f u n c t i o n s f r o m the  r e l a t i v e and center-of-mass wave f u n c t i o n s :  S i n c e b o t h the s e t of t w o - p a r t i c l e s t a t e s d e f i n e d  by E q . (B.2)  and t h a t d e f i n e d by E q . (B.3) are complete, we can make the expansion,. =  (B.4) ;  YT^l^lg  ( f ) ™  ii)  T ILnl  2  <KLnl;L-M*|TTl - y i  jj L t  =  f I 1  1  ^  .  1  ';m);  T ITLnl '  1-1 2-2  ^ l ^ l ^ L - M ' l NLnl;LM>4Vl,Vll -1 1 2 2  :  ' 1 1 2 ^  L» ,M« where  <NLnl  ;L"M»1 V ^ l ^ l ^ I ^ i n d ^ j l ^ ^ I^'M" (  K  expansion c o e f f i c i e n t s . were f i r s t name (Talmi  L  N  L  ""W^  a r  e  The p r o p e r t i e s of these c o e f f i c i e n t s  s t u d i e d by T a l m i , and the c o e f f i c i e n t s bear h i s (1952)).  To d e r i v e a f o r m u l a f o r the T a l m i c o e f f i c i e n t s , the procedure of A r i m a and Terasawa (1959). t h a t , from conservation  of energy,  We f i r s t  we f o l l o w note  - 76 (B.5)  (27^ + l + 2 ^ 4-1 ) x  = (2N + L f 2n +1).  2  Moreover, from the o r t h o g o n a l i t y eigenstates, L  = L ; M*  1  = M.  of the angular momentum  In f a c t , we a s s e r t that the  Talmi c o e f f i c i e n t s are independent of the magnetic quantum number, M. To see the l a t t e r p o i n t , i t i s convenient to choose _r and r_^ to be p a r a l l e l .  Making t h i s assumption, and  noting  tha t , m , ^ , THr 2-. ^ l o m ^ l I M ? ^ ! ^ ) 2 ( i l ) om mo)UI/r l ' 2 < l T l"* n  Y  m  m  1  n  / ( 2 ^ + 1)121 +- 1) 2  J  =  2  <lll 00|L0> Y r ^ )  4T (2U+-1)  2  ,  E q . (B.2) can be w r i t t e n as,  / (21, f 1) ( 2 1 f 1) ' • F .../ _ ^ < i l 00/L0> Y -^-) , ^ 4T (2L + 1) ^ ?  1  2  1  r  L  and Eq. (B.3) can be w r i t t e n as,  • • • -<L100| L0> Y^ (-O )  .  N o t i n g the l i n e a r independence of the s p h e r i c a l harmonics, Eq. (B.4) can be w r i t t e n as, (B.8)  i ) l j > i ( r , b ) l ^ i r l ( r , b ) x / (21-L + l ) ( 2 l + - l ) 1  1  1  2  <1 1 00| L0> 1  2  2  2  2  =  X^NLnljIlIl^l^ig.lJI^^lR.C) N ,L • -  n  '  1  ••• v/ (2L+ 1)121+1) <L1001L0>  ;  lp (r,c) nl  - 77 I I )  l|J (r,c)v/ (2L-t 1)(21+ 1) <L100|L0? =  U J L ^ . C )  nl  ^1', 1 2' 2  ar  ••• ^ 2 2 ( 2 » ) v i r  1  r  l 3  1)(21 +-D  ( 1 2 1  /  +  OilgOOlLC^  2  It  i s seen that the Talmi c o e f f i c i e n t s a r e , indeed, independ-  ent  of the magnetic quantum number,M; we w i l l h e n c e f o r t h deno  them'by <NLnl ;L l ^ l ^ l g ;L> and O ^ l i 1 ; L ! N L n l ; L > . 2  For  2  the purposes of t h i s t h e s i s , only the Talmi  coeffi-  c i e n t s of the form, < N 0 n 0 ; 0 ] i l i ^ i l i ; 0 ^ w i l l be r e q u i r e d . v  Taking L to be zero i n Eq. (B.8) i ) , (B.9)  l ^ ' V - i ( r , b ) l | v l ( r . b ) ( 2 1 ^ 1) O - ^ O O f 0 0 >  -  /  i  i  1  i  1  2  i  <NLnL;0['V l -V- i ;0> 4 V N,n., 1  1  1  1  R  '  C  ) ^no^ . ) c  L •  (2L + 1) <1L00(00>  .  Nov/, (B.10).\j/v  l l l  (r ,b)l(iv i (r h) 1  1  1  (21 + 1) ^1 1 _00 100>  2 i  x  3/  ...  1  =  ]  2  -  1^ (V^l^i)'.  g  **'  1  2 1  t  1  +  n  1  j=0 k=0 p=0 q=0 ...  (  i\ q /  2 k K L  l  .  - ^(2R + r /2 ) 2  2 V  J  R  2  6  / l^l-k /  Wl-J  MjL^ " k  q  b  j  +  k  +  1  2  i  \  P ^ * ^ ^ 2  3  1  P  -  J (P+I).  3 Ik . 1  P+q  and, (B.ll)  X  N,n, L. •••  < N L n L ; 0 | V l v- 1  (2L t 1 )  ;0  ^(j/  <LL00\00>  (.R t 2b)  =  tiA - (r ,£b) 1  n  0  - 78 , / 41\ lnl(2b-.Vb)' • . . = 2_ <NLnL; 0) ^ i l i ^ l l ! ; 0>7 2L + 1' J { i ) . (f+L+i) « N,n, L T  w  Nt L  - I2bl e ^  . iibl e ^  2 R  r  2 s  s t  whence ( e l i m i n a t i n g  n-t  = 0 t=0 ^  . . , t + £L 2t+L _2s+L (ib) r K  (2b)  sIti  +  common f a c t o r s )  Eq. (B.9) can be w r i t t e n  i n the form,  j^jyr  (B.12) V - 2 T ^ T T  #  _  T>  Jo Jo  ^o  q=0  1  q  r  n  2:  t ( +Lt^)». =o f-l 1  t=o  s  n  /n^Tr-iA  can  i 2  R  2  JZLTTJ  A recursive  T x  2(j^-k-tl ) - ( p * ) p t q 4N *.n I  ~  in  s+t+L  s-t  2t-f-L  formula f o r the r e q u i r e d Talmi  2s+-L coefficients  be found by equating the c o e f f i c i e n t s of the monomials,  r°R , 2 s  i n Eq. (B.12).  L = t=0; and we  For these c o e f f i c i e n t s :  p q=s;  j + k+l-^s;  p=2j + l ;  q=2k+l-  1  L  o b t a i n the r e l a t i o n , 2V I 1  ( B  *  1 3 )  ^ V "  ^  T ^ P W T  3  =  <  iT*S  n  H  2 j s 1  O  n  o  ^  :  k  ^ i  o ,  =  V  s  T  l  W  "1  l  , g r *  ;  79 where, N  m  =  2^  i ^  . V 2  t 1  X  2  From E q . (B.13), i t i s s t r a i g h t f o r w a r d (B.14)  to show t h a t ,  <sO(N -s)0;0|^ l 1r i ;0> m  1  1  1  1  2~ V S  '  *  2_  (V I 1 + t ) i  21 T 1  1L  N=s 1  <K0 (N -N) 0 ; 01 ^ I f ^ i l m  X  (T^ - j ) 1 (^i  ( j + I ^ i ) : (k + i + i ) : j i k i  ^  V I  x  -l  ; 0>  -l• The Talmi c o e f f i c i e n t s evaluated  employed i n t h i s t h e s i s have been  from the r e c u r s i v e formula, Eq. (B.14).  - 80 -  APPENDIX C  CALCULATION OF THE ONE-BODY DIFFERENTIAL ' ELASTIC SCATTERING CROSS-SECTION  In t h i s appendix, we d e r i v e tic  scattering cross-section  the s-wave d i f f e r e n t i a l  f o r the s c a t t e r i n g of a p a r t i c l e  from a s p h e r i c a l one-body p o t e n t i a l . adopted arithmic  The approach which i s  i s to express the c r o s s - s e c t i o n  i n terms .of the l o g -  d e r i v a t i v e of the resonant one-body wave  T h i s g e n e r a l method of approach theory of n u c l e a r r e a c t i o n s , the simplest  elas-  function.  i s f a m i l i a r from the ©-matrix  the present case being, perhaps,  example.  We take the one-body Schroedinger equation to be,  -  (c  1}  [~%^ * ^£^n*)~\ 2  *(R)  =_ to) £  ,  where P i s the reduced mass' of the one-body system;  Ze and  Z'e a r e the charges of the two i n t e r a c t i n g p a r t i c l e s . (C.2)  Y(R)  =  - Z ^ § i  Y (R) C  +  V  H  ( R )  Here,  ,  where Vj-,(R) and V-^(R) are the one-body e l e c t r o s t a t i c and nuclear potentials, respectively.  Due to the short range of  the n u c l e a r f o r c e , V(R) i s t y p i c a l l y a short-range p o t e n t i a l . The  scattering cross-section  from such a p o t e n t i a l i s w e l l -  .known (Messiah (1962)): (C.3)  6(6)  -  |f (6)| , 2  where,  (C.4)  f(6) = •  .  7 1  CO  Here, i) k =  2  2ksinHQ  . .. -JL_ ^ 2ik 1=0  (C.5)  EXP(-i?Un(sin ie) + 2 i o ~ )  ;  (21+1) e  2 i  ^l  (e  j,  2 i  ••• •  1 - DP^cosO) . l ' ;  v  - 81  7i =  ii) iii) S and  /-^  i s  P (l  + 1 •+ iTj.) ; of the  = 71 l n ( s i n H e )  i) P  f  ii)  <2t  phase s h i f t  t 5 i e  (C.6)  ZZ'e'  - arg  1  =  —2^7-  .  sin^^e and  neglecting  2  1  t h  p a r t i a l wave.  "  ' to the  c r o s s - s e c t i o n from a l l  t h a n s-waves ( l = 0), Eq. (C.3) \ cosU + / )s.incT + l i ^ (C.7)  t h a t , to evaluate  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 , we the phase s h i f t , £ .  The  Q  phase s h i f t  the  i s defined  by  u (R) 0  —  A[(C-  - iE )  0  -  Q  e ^°(G  0  angular momentum; u ( R ) Q  d i a l Schroedinger -  0  s  U  °  ( R  2 2 +  zero parra-  L  'e R  2  +  V(R)  u  o  =  £  o  u 0  the CR f u n c t i o n to  be,  )  R ^Uoill dR  0  of  and  0  ( C . l ) , being a' s o l u t i o n of the  I t i s convenient to define <R  Here, E  equation,  djuo dR2  2P  ,  i s the r a d i a l part of the zeroth  t i a l wave s o l u t i o n of Eq.  where R  as-  Q  i r r e g u l a r Coulomb f u n c t i o n s  the r e g u l a r and  (C.10)  the  -Y iF _)]  2 i  G  (C.9)  evaluate  property,  i s a short-range f o r c e .  are  differ-  need only  which h o l d s s i n c e V(R) 0  becomes,  0  I t i s seen from Eq.  (C.8)  Defining,  ;  contributions  p a r t i a l waves other (C.7) tS(Q) = 4k  ymptotic  -  R=R,  i s a s u i t a b l y large radius.  I t i s then e a s i l y shown  that, Pu? (R,£«) 0  G (R, 0  £')/<R  -  £')Af  E (R, - RGo(R, £ • ) 0  R=R  f  N o t i n g the Wronskian r e l a t i o n , (C.12) Eq.  ]?«G  (C.ll)  (C.13)  = G*]?. -r k  0  ,  0  can he w r i t t e n cT(^t) =  tan"  as, kR  1  - Po R=R G. 6 e» J  G/ The  c r o s s - s e c t i o n i n s e c t i o n 4 - 2 of t h i s  p l o t t e d by n u m e r i c a l l y  evaluating  phase s h i f t was  s e c t i o n from Eq. The  thesis  was  the l o g a r i t h m i c d e r i v a t i v e  of the one-body wave f u n c t i o n of Eq. The  0  a  (C.9)  then found from Eq.  about resonance.  (C.13), and  the  cross-  (C.7).  v/idth of the c r o s s - s e c t i o n can be obtained  from  B r e i t - W i g n e r o n e - l e v e l formula; a d i s c u s s i o n of the  the  one-level  approximation can be found i n the book by P r e s t o n  (1962).  can be  cross-sec-  shown from the formulas f o r ' t h e  t i o n s t a t e d by P r e s t o n (G.14) <S(Q)  =  scattering  It  that,  ^ 4 k'2  *  2  cos(^D )sinD 0  0  -j-  s i n 2 p  k2  o  where, (C.15)  i)'sin D 2  ii)  B -  0  =  f/^  ( 6  1 - d£  ^  ^  6=6,  Here, P i s the width of the  scattering cross-section; 6  Q  i s the  resonance energy; (C.16) A : i s the body  level  -S  0  7T  shift  2 (S  reduced w i d t h ) .  being the s h i f t f u n c t i o n and 7f By  comparing Eqs.  (C.7)  and  the  one-  (C.ll), i t  - 83  i s seen t h a t we may i d e n t i f y D  0  w i t h the phase s h i f t , <^ . 0  -  The  c o e f f i c i e n t , B, can then be found from Eq. (C.15) i ) by maki n g a l e a s t squares f i t to the phase s h i f t , To e v a l u a t e the w i d t h of the s c a t t e r i n g  <^ , 0  cross-section  f r o m E q . (C.15) i i ) , we r e q u i r e the energy dependence of the level shift.  F o r heavy n u c l e i , where there a r e l a r g e Coulomb  b a r r i e r s , t h i s energy dependence i s g i v e n t o good approximat i o n by,  In t h i s t h e s i s , the e f f e c t of the energy dependence of the l e v e l s h i f t has been e s t i m a t e d from Eq. (C.17).by c a l c u l a t i n g the energy dependence of G , fl  

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