UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

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

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata

Download

Media
831-UBC_1969_A1 S34.pdf [ 10.93MB ]
Metadata
JSON: 831-1.0084757.json
JSON-LD: 831-1.0084757-ld.json
RDF/XML (Pretty): 831-1.0084757-rdf.xml
RDF/JSON: 831-1.0084757-rdf.json
Turtle: 831-1.0084757-turtle.txt
N-Triples: 831-1.0084757-rdf-ntriples.txt
Original Record: 831-1.0084757-source.json
Full Text
831-1.0084757-fulltext.txt
Citation
831-1.0084757.ris

Full Text

5 u ^ ON THE INTERACTION 03? LOW ENERGY PIONS WITH NUCLEI LEONARD RAYMOND SCHERX B. Sc.,' University of B r i t i s h Columbia, 1965 M. So. University of B r i t i s h Columbia. 1967 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n the Department of PHYSICS We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA July, 1969 In p r e s e n t i n g t h i s t h e s i s in p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an advanced d e g r e e at the U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and S t u d y . I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by the Head o f my Department or by h i s r e p r e s e n t a t i v e s . It i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s thes . is for . f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Department o f Physics The U n i v e r s i t y o f B r i t i s h Co lumbia V a n c o u v e r 8, Canada Date September B, 1969 i i ABSTRACT The o p t i c a l i n t e r a c t i o n of low energy f^30 MeV) pions with n u c l e i i s discussed. In p a r t i c u l a r , i t i s shown that, since the nuclear density enters the low energy pion-nucleus i n t e r a c t i o n i n a very d i r e c t manner, this i n t e r a c t i o n pro-vides a sensitive means of inve s t i g a t i n g such properties of the nuclear density.as the diffuseness of the nuclear surface. A.geometric discussion of the structure of the low energy pion-nucleus i n t e r a c t i o n i s given which emphasizes the analogy hetween adding the scattered pion waves i n the nuclear medium and adding e l e c t r i c potentials i n a c l a s s i c a l d i e l e c t r i c . The parameters of the o p t i c a l potential which represents the i n t e r a c t i o n are taken to he those calculated by e a r l i e r authors who have used a multiple-scattering formalism to deduce the d e t a i l s of the-optical i n t e r a c t i o n from a micro-scopic point of view. The i n t e r a c t i o n i s strongly momentum-dependent and the l o c a l part of the i n t e r a c t i o n i s repulsive. I t i s shown that i n o p t i c a l s c attering and absorption, the resonance aspects of the problem depend mainly only upon the height of the l o c a l p o t e n t i a l barrier (^15 MeV) because of the long wavelength of the pion inside the nucleus. The o p t i c a l absorption of low energy pions i s shown to be sensit i v e to the diffuseness of the nuclear surface through the strong suppression of momentum-dependent absorption near the top of the po t e n t i a l b a r r i e r . I t i s argued that low energy pions can therefore be used to resolve the c o n f l i c t s which presently e x i s t i n the information available from i i i several experiments concerning the d i s t r i b u t i o n of neutrons i n the nuclear surface. The ideas developed i n discussing the o p t i c a l absorption of pions are extended to the ex c i t a t i o n by pions of r o t a t i o n a l states i n strongly deformed nuclei. I t i s shown that, unlike the e x c i t a t i o n cross sections obtained with more conventional interactions (such as the Coulomb interact!on) , the exc i t a t i o n cross sections obtained from the pion-nucleus o p t i c a l i n t e r -action depend strongly upon such c h a r a c t e r i s t i c s of the nuc-lea r density as i t s surface thickness. The r o t a t i o n a l model of strongly deformed nuclei i s assumed and an analysis of the e x c i t a t i o n cross sections i s made i n the Distorted Wave Born Approximation. I t i s shown that the s e n s i t i v i t y to the nuclear surface thickness i n the e x c i t a t i o n cross sections arises from the suppression of the e x c i t a t i o n processes due to the momentum-dependent i n t e r -action near the top of the potential b a r r i e r . I t is' suggested that the e x c i t a t i o n of r o t a t i o n a l levels i n strongly deformed nuclei by pions can therefore be used to c a r e f u l l y examine the d i s t r i b u t i o n of neutrons i n the surfaces of strongly deformed n u c l e i . i v • TABLE OF CONTENTS Page CHAPTER 1 INTRODUCTION 1 CHAPTER 2 THE LOW ENERGY PION-NUCLEUS INTERACTION 12 4 2-1 Structure of the Interaction 17 2-2 A Microscopic Derivation of the 25 Interaction t 2- 3 Experimental V e r i f i c a t i o n 39 CHAPTER 3 THE OPTICAL PROPERTIES OP LOW ENERGY 43 • - PTONS IN SPHERICAL NUCLEI 3- 1 A Formalism For Analyzing Optical 50 Scattering and Absorption 3-2 Qualitative Features of Low Energy 57 Pion Optics i n Nuclei 3-3 Pion-Hucleus Optical Potential 60 3-4 Numerical Discussion 64 3- 5 Charge Exchange and other Exotic Terms . 90 i n the Ericsons' Potential CHAPTER 4 EXCITATION OF ROTATIONAL LEVELS IN 94 DEFORMED NUCLEI BY PIONS 4- 1 A Review of the Rotational Model and a 98 Discussion of Pion E x c i t a t i o n 4-2 DWBA Formulae for Pion Excitation 105 Cross Sections 4-3 Numerical Discussion 118 CHAPTER 5 CONCLUSIONS 136 BIBLIOGRAPHY . 144 APPENDIX A EQUATIONS FOR THE INTERIOR LOGARITHMIC 146 DERIVATIVES A„i Integral•Equation for Absorption 146 A-2 Uniform D i s t r i b u t i o n .148 V APPENDIX B B-l B - 2 APPENDIX C NOTES Oil EVALUATION OP PION EXCITATION CROSS SECTIONS Evaluation of DWBA Matrix Elements Evaluation of Electromagnetic Potential For a Deformed Charge D i s t r i b u t i o n Page 150 150 155 THE NUCLEAR OPTICAL .MODEL AND WAVE. i« 155 PROPERTIES: BARRIER PENETRATION, REFLECTION, ABSORPTION, AND RESONANCE by G, Michaud, L. Scherlc, and E. Vogt LIST OF FIGURES Nuclear Density and Square of Nuclear Densi ty C a 4 0 Calculated E l a s t i c Scattering Cross Sections Jt r 4 C a 4 0 Calculated Total Absorption Cross Sections W * c a 4 0 Dependence of Real Phase S h i f t s upon Surface Parameter TV* + C a 4 0 Dependence of Real Phase Sh i f t s upon Potential B a r r i e r Tt +• Ca Dependence of Real Phase S h i f t s upon 40 Radius 71 + Ca Real Logarithmic ^ Derivatives 3T +• Ca Imaginary Phase S h i f t s K"1" +• C a 4 0 Ratios of Imaginary Phase S h i f t s ^ + C a 4 0 s-wave Absorption 7Tf •* Ca 4^ s-wave Imaginary'Phase S h i f t s for > ~ f C a 4 0 and Kr + C a 4 0 s-wave Imaginary Phase Shift s for Tr- +- P b 2 0 8 and ^ - C a 4 0 s-wave Imaginary Phase Sh i f t s for K f + p b208 a n a _ 7T-+-Pb208 40 v i i Figure . Page 11 Imaginary Phase S h i f t s 714 + P h 2 0 8 and 87 13a Reduced Total E x c i t a t i o n Gross Sections 120 7X+ + A l 8 5 12b Reduced Total E x c i t a t i o n Cross Sections 121 7iY + n238 13a Reduced D i f f e r e n t i a l E x c i t a t i o n Gross 122 . Sections 7tf > A l 2 5 13b Reduced D i f f e r e n t i a l E x c i t a t i o n Cross 123 Sections nr ± T J 2 5 8 14 Relative Values of Radial Integrals 126 15a Radial Integrals Insensitive to Surface 127 Parameter. Kr +- A l 8 5 15b Radial Integrals Insensitive to Surface 128 Parameter X* *• U 2 3 8 16a Momentum-Dependent Radial Integrals 129 • • 7C + A l 2 5 16b Momentum-Dependent Radial Integrals 120 KT +• T J 2 3 8 ' v i i i ACMOWLEDGEkENTS -The author would l i k e to express his deep appreciation t o h i s Teacher and Research Supervisor, Professor E. W. Vogt. Only those who have had the good fortune to have personally known or have worked with Professor Vogt w i l l f u l l y appreciate the extent to which his immense patience and personal kind-ness, his thorough knowledge of a l l aspects of physics, and his p r o l i f i c and perceptive suggestions a s s i s t i n the soluti o n o f one's'problems. The author would also l i k e to thank Dr. P. C. Bhargava for many h e l p f u l discussions .and he would l i k e to thank Hr. C. T. Tindle for making available his computer program for spherical Coulomb functions. The author also expresses h i s appreciation to the National Research Council of Canada for providing f i n a n c i a l assistance through N. R. C. Scholarships. 1 CHAPTER 1 INTRODUCTION In this: thesis i t i s our purpose to describe the prop-e r t i e s of the low energy (^50 MeV) pion-nucleus o p t i c a l i n t e r a c t i o n and to suggest and examine some of the ways i n which this i n t e r a c t i o n can be used to investigate the struc-ture of nuc l e i . Because a transparent connection exists between the o p t i c a l i n t e r a c t i o n of a pion with a nucleus and the elementary interactions of the pion with the con-s t i t u e n t nucleons, the properties of this o p t i c a l i n t e r a c t i o n are related i n a very d i r e c t way to such macroscopic nuclear properties as the nuclear density. In this thesis we examine the role of the nuclear density both i n determining the o p t i c a l scattering and absorption of pions i n nuclei and also i n determining the e x c i t a t i o n by pions of r o t a t i o n a l states i n strongly deformed n u c l e i . We show that these processes depend strongly upon the detai l s of the density of nucleons i n the surface of the nucleus. We therefore suggest that pions provide an i d e a l probe for measuring such aspects of nuclear structure as the d i s t r i b u t i o n of nucleons, i n p a r t i c u l a r , neutrons i n the nuclear surface. Details of the neutron d i s t r i b u t i o n i n the nuclear sur-face are not e a s i l y available from more conventional i n t e r -actions (such as the nucleon-nucleus interaction) where the connection between the microscopic and macroscopic aspects of the problem i s less direct; i n fact, information about the surface d i s t r i b u t i o n of neutrons i n nuclei currently available from several experiments i s neither very well 8 determined nor are the results of these experiments mutually consistent, We suggest that the c o n f l i c t s i n this important aspect of nuclear structure can be la r g e l y resolved by means of experiments which invoke the o p t i c a l properties of pions, such as those to be discussed i n this thesis. The d i r e c t manner i n which the macroscopic aspects of the nucleus, such as the nuclear density, enter the pion-nucleus o p t i c a l i n t e r a c t i o n arises from the transparent con-nection between the o p t i c a l i n t e r a c t i o n and the scattering and absorption processes associated with the elementary nucleon (and two-nucleon) scatterers whose average over the sca t t e r i n g nucleus i t purports to represent. This singular directness i n the r e l a t i o n s h i p of the macroscopic to the microscopic in t e r a c t i o n s of pions i n nuclei makes the pion-nucleus o p t i c a l i n t e r a c t i o n of i n t e r e s t as one of the few examples of tractable many-body scattering problems as well as an excellent t o o l for studying the macroscopic properties of the nucleus. The s i m p l i c i t y i n the case of low energy pion scattering arises from the short pion-nucleon scattering lengths (~0,1 fm. ) combined with the small mass of the pion r e l a t i v e to the scatterers ( ~ l / 7 ) . In these circumstances, the pion interacts i n d i v i d u a l l y with each nucleon and, be-cause of the small energy transfers i n elementary c o l l i s i o n s , i t i s r e l a t i v e l y oblivious to the dynamics which govern the motions of the scat t e r i n g nucleons 0 This allows one to make the impulse approximation (making the,scattering matrix tr a c t -able) and greatly s i m p l i f i e s the Green's functions which describe the propagation of the pion i n the nuclear medium; the dynamics of the problem are thus taken care of through known one-body operators and the kinematics are described by using familiar techniques for adding multiply scattered waves. This i s to be contrasted with the other f a m i l i a r example of a strongly i n t e r a c t i n g p r o j e c t i l e , the nucleon, where contrary conditions hold and the connection between macroscopic and microscopic processes i s clouded by the i n t r i n s i c manner i n which the many-body aspects of the scat-terer manifest themselves i n the o p t i c a l properties of the nuclear medium. Apart from the simple connections which e x i s t between the low energy pion-nucleus i n t e r a c t i o n and i t s elementary o r i g i n s , the i n t e r a c t i o n i t s e l f possesses a structure and form which make i t of i n t e r e s t i n i t s own r i g h t . The basic structure of the i n t e r a c t i o n i s r e a d i l y seen by noting that, at the energies of i n t e r e s t , only s- and p-waves enter the elementary scattering processes. The geometric problem of adding these waves i s analogous to adding the potentials a r i s i n g from the charges and dipoles i n a c l a s s i c a l d i e l e c -t r i c . The s-waves add l i k e the potentials due to the charges i n the d i e l e c t r i c and lead to a l o c a l contribution to the pion-nucleus o p t i c a l p o t e n t i a l . The p-waves modify the i n t r a -nuclear momentum f i e l d of the pion i n analogy to the modifica-t i o n of the e l e c t r i c f i e l d due to the dipoles i n a d i e l e c t r i c ; the r e s u l t of the p~waves i s therefore to contribute a momentum-dependent term to the pion-nucleus o p t i c a l p o t ential. In fact, i f we include the effects of the short-range correlations of the nucleons within the nucleus, we find that the mod-i f i c a t i o n of the intra-nuclear momentum f i e l d due to the p-waves i s non-linear i n the nuclear density, i n analogy to • the well-known Lorenz-Lorentz e f f e c t i n dense o p t i c a l media. Since the Lorenz-Lorentz e f f e c t a f f e c t s the structure of the o p t i c a l i n t e r a c t i o n , i t raises hopes that pion optics might provide a tool for i n v e s t i g a t i n g the elusive nature of the. short-range correlations of the nucleons within the nucleus. In fact, these hopes are considerably minimi zed since the wavelength of a low energy pion i s much longer than the range of the correlations so that i t i s an insen-s i t i v e t o ol for measuring th e i r structure. Thus, while the pion-nucleus i n t e r a c t i o n may be strongly affected even i n i t s structure by the anticorrelations of nucleons at short distances (which removes the s e l f - e x c i t a t i o n of the scat-t e r e r s ) , i t i s probably i n s e n s i t i v e to their d e t a i l s . IFever theless, careful measurements of o p t i c a l l y scattered pions might provide information about these correlations which can presently be obtained only i n very i n d i r e c t ways (for i n -stance, from the studies of nuclear matter). While the analogy with electromagnetic theory provides the basic structure of the pion-nucleus i n t e r a c t i o n , a quan-t i t a t i v e estimate of the macroscopic parameters from the microscopic dynamics requires a considerably more detailed c a l c u l a t i o n . A thorough study of this problem has been given by Ericson and Ericson (1966). They have used the techniques of multiple scattering theory to calculate the macroscopic form of the i n t e r a c t i o n taking into account kinematic cor-rections associated with the motion of the scattering nucleons and inc l u d i n g terms which simulate o p t i c a l absorption. The o r i g i n of the absorptive terms i n the Ericsons' potential l i e s i n r e a l pion absorption on "elementary" two-nucleon scatterers (pions, being bosons, need not be con-served); kinematics i n h i b i t pion absorption by a single nuc-leon and the small mass of the pion i n h i b i t s nuclear exc i t a -t i o n through k i n e t i c exchanges. The absorptive aspects of the pion-nucleus o p t i c a l i n t e r a c t i o n are thus connected to elementary processes as d i r e c t l y as the r e f r a c t i v e aspects; this i s an enormous s i m p l i f i c a t i o n over the absorptive aspects of the nucleon-nucleus o p t i c a l i n t e r a c t i o n which arise s o l e l y from k i n e t i c exchanges between the p r o j e c t i l e and the scatterers. (and, hence, i n t r i n s i c l y involve the many-body aspects of the s c a t t e r e r ) . In addition to t h e i r contribution to absorption, the Ericsons find that these two-nucleon scat-terers also make important contributions to the r e f r a c t i v e properties of the o p t i c a l i n t e r a c t i o n , p a r t i c u l a r l y to the l o c a l i n t e r a c t i o n where a strong c a n c e l l a t i o n occurs i n the s-wave pion-nucleon s c a t t e r i n g lengths. With these e f f e c t s taken into account, the Ericsons find that the o p t i c a l s c a t t e r i n g of pions from nuclei can be sum-marized by a Schroedinger equation which has the structure predicted by the electromagnetic analogy (including the Lorens-Lorentz' effect) and whose parameters are transparently connected 6 to pion s c a t t e r i n g lengths from nucleons and deuterons. In addition, their o p t i c a l i n t e r a c t i o n includes o p t i c a l i s o s p i n terms which account for quasi-elastic charge exchange pro-cesses and includes a hyperfine term which takes into account the spin of ..the target nucleus. They find that the r e s u l t i n g i n t e r a c t i o n i s consistent with the ( e s s e n t i a l l y zero energy) 7t-mesic x-ray data hut, unfortunately, the i n t e r a c t i o n can-not he tested at higher energies u n t i l the advent of more intense-^pion beams (such as the Triumf F a c i l i t y proposed for U.B.C. ) . I t i s reasonable, however, to expect that the Ericsons' p o t e n t i a l should adequately describe the pion-nucleus o p t i c a l i n t e r a c t i o n up to 30 or 40 MeV where the zero-energy descrip-t i o n of the elementary s c a t t e r i n g processes begins to f a i l and where the effects of the 3 - 3 pion-nucleon resonance make the naive treatment of the multiple scattering problem i n v a l i d This l i m i t i s to some extent set by the Fermi motion of the nucleons and the parameters of the pot e n t i a l may be somewhat energy-dependent at the higher energies. Nevertheless, one would expect that the basic structure of the i n t e r a c t i o n should be correct and i n our calculations we have used the zero-energy parameters suggested by the Ericsons' analysis. This seems a reasonable procedure since, i n any event, the pote n t i a l has not been tested at higher energies and i t would not seem worthwhile without some reasonable empirical basis to attempt to untangle the various second order effects which determine i t s exact d e t a i l s . In the present thesis we have taken the Ericsons 1 form for the o p t i c a l pion-nucleus i n t e r a c t i o n and we have attempted both to describe pion optics i n the low energy region and to •'I elucidate those features of nuclear structtire which are sus-ceptible to i n v e s t i g a t i o n with the pion probe, i n p a r t i c u l a r , the density of neutrons i n the nuclear surface. The basic character of pion optics resides i n the mildly repulsive l o c a l p o t e n t i a l , i n the strong momentum-dependence of the i n t e r a c t i o n , and i n the fact that pions are very strongly absorbed by nu c l e i . These c h a r a c t e r i s t i c s lead to o p t i c a l phenomena which are q u a l i t a t i v e l y quite d i f f e r e n t from those encountered with more conventional p r o j e c t i l e s . In p a r t i c u l a r , the resonance aspects of the problem at the energies of i n t e r e s t (£z30 MeV) are governed by the presence of a potential b a r r i e r f^-15 MeV) which, combined with the small pion mass, yields a pion wavelength within the nucleus which i s considerably greater than the nuclear dimen-sions; the resonance aspects of the problem are, therefore, rather i n s e n s i t i v e to the detai l s of the i n t e r a c t i o n and are governed mainly by the size of the nucleus and by the proper-t i e s of the e l e c t r o s t a t i c b a r r i e r which surrounds i t . This contrasts with the strong resonances c h a r a c t e r i s t i c of the att r a c t i v e potentials encountered i n nuclear optics. The absorption of pions also presents some unusual features: pions are absorbed not only through a l o c a l i n t e r a c t i o n (which depends primarily upon the quantity of matter present) but also through a. momentum-dependent i n t e r a c t i o n (which depends s e n s i t i v e l y 8 upon the d i s t r i b u t i o n of t h i s matter). Because pions are absorbed primarily i n the nuclear surface, the energy depend-ence of pion absorption cross sections yi e l d s detailed i n -formation about the peripheral' nuclear density. The density of nucleons i n the nuclear surface has long been an elusive quantity so that the r e l a t i v e l y d i r e c t and simple technique of measurement provided by pion absorption represents one of the most useful aspects of pion optics. Of course, the issue i s somewhat complicated by the fact that pion absorption takes place primarily on two-nucleon scat-terers. However, provided adequate account i s taken of the long-range correlations of nucleons i n the nucleus (which are quite well understood), the connection between the density and the absorption rates can be obtained with considerable precision. In f a c t , the Ericsons have shown i n their calcu-latio n s that the absorption rates are simply related to the square of the density and they have found these absorption rates to be i n good agreement with the absorption rates re-quired to explain the widths of 7T-niesic x-rays. At any rate, o p t i c a l pion measurements of the nuclear density should be a considerable improvement over measurements with nucleons (which strongly polarize the target) and measurements based on the branching r a t i o s associated with kaonic atoms (where the analysis i s confused by complicated atomic physics). Of course, the i n t e r e s t i n finding the nuclear surface density i s to obtain separately the surface d i s t r i b u t i o n of protons and neutrons. This i s immediately accomplished since the charge density (or d i s t r i b u t i o n of protons) i s well-known 9 both from electron s c a t t e r i n g experiments and, more recently, from >/-mesic x-ray analysis. The information about proton and neutron d i s t r i b u t i o n s then provides a new dimension i n the tests of nuclear models (such as the s h e l l model) and i s also necessary information i n obtaining precise rates for var-ious processes associated with the -nucleus (such as pion pro-duction rates from high energy protons). In addition to o p t i c a l pion absorption, a second tech-nique can be used to u t i l i z e the momentum-dependence of the pion-nucleus i n t e r a c t i o n i n obtaining the surface features of deformed n u c l e i . In the strong-coupling model, a deformed nucleus i s described as the product of a r o t a t i o n a l and i n -t r i n s i c wave function; the pion f i e l d can be used to excite the r o t a t i o n a l states of such a nucleus i n the same way as both the e l e c t r o s t a t i c f i e l d (Coulomb excitation) and neutron f i e l d have been used. In fact, i n the Distorted Wave Born Approximation, the pion e x c i t a t i o n amplitudes are e s s e n t i a l l y sums of the Coulomb e x c i t a t i o n and l o c a l e x c i t a t i o n amplitudes along with a moment urn-dependent e x c i t a t i o n amplitude a r i s i n g from the momentum-dependent i n t e r a c t i o n . The simple DWBA i n t e g r a l s which r e l a t e the e x c i t a t i o n cross sections to the Interaction (and, hence, the density) then allow a d i r e c t connection between the cross sections and the properties of the i n t r i n s i c wave function of the nucleus. The Coulomb and l o c a l e x c i t a t i o n amplitudes, being i n t e g r a l s over the nucleus, are only moderately sensitive to the d e t a i l s of the nuclear surface; the momentum-dependent ex c i t a t i o n amplitude, however, i s very sensitive to these d e t a i l s since, 10 ! near the top of the pot e n t i a l h a r r i e r , the momentum varies r a p i d l y both with energy and across the nuclear surface. Thus the analysis of the energy dependence of the e x c i t a t i o n by pions of r o t a t i o n a l states i n deformed nuclei y i e l d s de-t a i l e d information about the surface features of the i n t r i n -s i c wave function. A more complete discussion of the topics i n this Intro-duction i s given i n the remaining chapters of the thesis. A review and i n t e r p r e t i v e discussion of the construction of the pion-nucleus o p t i c a l p o t e n t i a l i s given i n Chapter 2 . In Chapter 3 this i n t e r a c t i o n i s used to analyze the e l a s t i c s c a t t e r i n g and t o t a l absorption, cross sections of pions from various spherical nuclei with p a r t i c u l a r emphasis on the role of the nuclear surface. In Chapter 4 the e x c i t a t i o n of rota-t i o n a l l e v e l s i n deformed nuclei by pions i s discussed i n DWBA as a technique for in v e s t i g a t i n g the surface features of the i n t r i n s i c wave functions of deformed n u c l e i . Chapter 5 presents a review of the thesis and a summary of the conclu-sions reached i n the e a r l i e r chapters. This thesis also includes, as Appendix C, a preprint of a paper now submitted for publication of which the author of t h i s thesis i s a co-author and which represents a separate problem i n which the author became interested during the per-iod of his i n t e r e s t i n the pion-nucleus problem presented i n the body of this thesis. This problem was prepared i n c o l -laboration with Professor E, t w. Vogt, the author's Research Supervisor, and with Mr. Georges Michaud, a v i s i t i n g student .1 1 ' t i ' 11 I In astrophysics from the C a l i f o r n i a Institute of Technology. This paper investigates the o p t i c a l interactions between nuclei and heavy ions and p a r t i c u l a r l y examines the e f f e c t s of the - . ' i diffuse nuclear surface upon the s c a t t e r i n g and absorption cross sections. These cross sections are of considerable i n t e r e s t i n determining the rates of various processes en-countered i n astrophysics. The author's i n t e r e s t i n this problem arose from his e a r l i e r i n t e r e s t i n alpha-decay rates and his main c o n t r i -butions are to the discussion of the resonance properties of r e a l potentials i n the presence of large Coulomb b a r r i e r s , these being contained i n Sec. 4 of the paper. CHAPTER S THE LOW ENERGY PION - NUCLEUS INTERACTION In the present chapter we discuss the construction of the low energy (^50 MeV) pion-nucleus o p t i c a l i n t e r a c t i o n » 4 from the elementary processes of pion-nucleon and pion-deutron scattering. A comprehensive study of th i s problem and a review of the e a r l i e r work by other authors has been given i n an a r t i c l e by M. Ericson and T. E. 0. Ericson (1966) and our discussion i s to a large extent a summary of the i r discussion and r e s u l t s . A more general discussion of the i n t e r a c t i o n of pions with nuclei can be found i n a recent review a r t i c l e by Koltun (1969), The pion-nucleus i n t e r a c t i o n i s distinguished from most other complex interactions which admit an o p t i c a l descrip-t i o n such as the nucleon-nucleus i n t e r a c t i o n i n that i t can e a s i l y be constructed from the microscopic pion-nucleon and pion-deuteron scatterings (whose average i t represents). This s i m p l i f i c a t i o n arises i n part i n that, because the 4 , pion-nucleon scatterings lengths are small (^0.1 fm. ) com-pared to the inter-nucleon separation (^1.4 fm.) and because the pion mass i s small ( ~ l / 7 the nucleon mass) , one can assume that a pion scatters from a nucleon inside the nuc-leus i n the same way i n which i t scatters from a nucleon i n free space; that i s , one can make the impulse approximation. In addition, the small mass of the pion allows a consider-able s i m p l i f i c a t i o n of the Green's function which describes pion propagation within the nucleus since one may then largely neglect the effects of intermediate nuclear excitations. The problem i s thus e s s e n t i a l l y reduced to the c l a s s i c a l problem of c a l c u l a t i n g the t o t a l scattered wave from a system of massive scatterers. In fact, since at low energies scat-terings are mainly s-wave scatterings (resembling e l e c t r i c charges) and p-wave scatterings (resembling e l e c t r i c d i p oles), one can calculate the t o t a l scattered wave i n exactly the same way i n which one calculates the t o t a l e l e c t r i c p o t e n t i a l i n a d i e l e c t r i c medium. I t i s the purpose of t h i s chapter to present a derivation of the low energy pion-nucleus o p t i c a l p o t e n t i a l from this point of view and to demonstrate the d i r e c t manner i n which i t involves the density of the scat-t e r i n g nucleus. I t was r e a l i z e d by Peaslee (1952) that the scattering of pions from complex nuclei could be at l e a s t q u a l i t a t i v e l y obtained simply by making the impulse approximation and by summing the s i n g l y scattered pion waves from the elementary scatterers. In f a c t , by applying his analysis to the scat-t e r i n g of 62 MeV pions by carbon, he i n f e r r e d the 3-3 pion-nucleon resonance at 200 MeV. This led him to conclude that below this energy both s- and p-waves must be taken into •account i n the elementary scatterings while higher p a r t i a l waves can be neglected. His r e s u l t s were extended to include multiple scatter-ings by K i s s l i n g e r (1955) who used the multiple scattering formalism of Watson (1953) and Francis and Watson (1953) to write the pion-nucleus o p t i c a l i n t e r a c t i o n i n terms of a 14 pseudopotential„ While K i s s l l n g e r neglected absorption and the e f f e c t s of nucleon-nucleon co r r e l a t i o n s , his treatment showed the di r e c t manner i n which the nuclear density enters the i n t e r a c t i o n and c o r r e c t l y predicted the structure of the i n t e r a c t i o n showing i t to consist of a l o c a l i n t e r a c t i o n a r i s i n g from the elementary s-wave scatterings --- and a momentum dependent i n t e r a c t i o n a r i s i n g from the elemen-* tary p-wave scatterings. The strong momentum dependence of the i n t e r a c t i o n suggested by Kisslinger was also shown by Frank et a l , (1956) who evaluated both the r e a l and absorp-tive terms of the potential i n an i n f i n i t e medium of uncor-related nucleons. The basic structure of the pion-nucleus i n t e r a c t i o n i s elucidated by considering the analogy between adding the elementary s- and p-wave scatterings i n the nuclear medium and adding e l e c t r i c potentials and f i e l d s i n a d i e l e c t r i c with free charges • — the. geometry of the two problems i s e s s e n t i a l l y the same. Baker et al„ (1958) used c l a s s i c a l arguments e s s e n t i a l l y of this kind to present a s i m p l i f i e d derivation of the Ki s s l i n g e r p o t e n t i a l and K r o l l (1961), r e a l i z i n g the electromagnetic analogy, suggested that there should be a modification of the intra-nuclear pion momentum f i e l d since such an e f f e c t exists i n the r e f r a c t i v e proper-t i e s of dense o p t i c a l media (the Lorenz-Lorentz e f f e c t ) . However, the electromagnetic analogy does not seem to be written i n d e t a i l i n the l i t e r a t u r e and -~- because of i t s h e u r i s t i c value i n revealing the structure of the pion-nucleus 15 i n t e r a c t i o n we discuss i t i n some d e t a i l . A derivation of the low energy pion-nucleus i n t e r a c t i o n has recently been given by M. Ericson and T. E. 0. Ericson (1966) which takes into account both absorption and nuclear c o r r e l a t i o n s . They derive the multiple scattering equations for a pion i n the nucleus by a method s i m i l a r to Lax's treat-metn (1951) of multiple scattering i n c l a s s i c a l media. Their basic idea i s to use a systematic expansion into higher order c o r r e l a t i o n functions between the scatterers which they have truncated at pair c o r r e l a t i o n s . They find i n this way that the short-range antic orrelations of nucleons lead to a mod-i f i c a t i o n of the intra-nuclear momentum f i e l d i n analogy to the Lorenz-Lorentz e f f e c t i n a dense o p t i c a l medium. They make the impulse approximation taking the s- (p-)wave scat-t e r i n g lengths (volumes) to be those obtained from pion-. nucleon scattering; they then include kinematic corrections for the binding' and Fermi motion of the nucleons and also extend their calculations to take account of the eff e c t s of spin and iso s p i n . Their c a l c u l a t i o n i s further distinguished by the fact that, i n addition to accounting for the pion s c a t t e r i n g from nucleon scatterers, they also take i n t o account pion scattering from two-nucleon (or quasi-deuteron) scatterers. In this way they are able to describe pion o p t i c a l absorption which i s dominated by the absorption of a pion on two nucleons. In addition, they find that the two-nucleon scatterers make an important contribution to the l o c a l i n t e r a c t i o n where a strong 16 c a n c e l l a t i o n occurs i n the s-wave scat t e r i n g lengths. Their resulting p o t e n t i a l which contains no free parameters . — i s found to he i n good empirical agreement with the s h i f t s and widths of x-rays associated with 7C -mesic atoms. This transparent connection which exists between the elem entary scattering processes and the pion-nucleus o p t i c a l i n t e r action i s to be contrasted with the nucleon-nucleus o p t i c a l i n t e r a c t i o n . In the l a t t e r case, the interactions between the scattered nucleon and target nucleons within the nucleus depend strongly upon the properties of the nuclear medium, both because of the long nucleon-nucleon scattering lengths and because of the larger nucleon mass; hence, their i n t e r -actions can no longer be expected to be described i n such a simple way as was provided to us i n the pion-nucleus case by the impulse approximation. In addition, i n nucleon- , nucleus s c a t t e r i n g , we can no longer introduce the technical s i m p l i f i c a t i o n s Which were allowed us i n pion-nucleus scat-t e r i n g by the small pion mass and, i n general, both i n t e r -mediate nuclear excitations and kinematic corrections for r e c o i l of the nucleon target must be taken into account. The pion-nucleus o p t i c a l i n t e r a c t i o n therefore involves the m&croscopic properties of the s c a t t e r i n g nucleus i n a much more e x p l i c i t manner than the nucleon-nucleus o p t i c a l i n t e r -action so that pions are much better tools for in v e s t i g a t i n g such properties of the nucleiis as the nuclear density. In the present chapter our purposes w i l l be two-fold. F i r s t l y , we w i l l show that the structure of the pion-nucleus 1 7 i n t e r a c t i o n can be s i m p l y u n d e r s t o o d from the f a m i l i a r p r o b l e m o f a d d i n g p o t e n t i a l s and f i e l d s i n a c l a s s i c a l d i e l e c t r i c S e c o n d l y , we w i l l d i s c u s s the E r i c s o n s ' c a l -c u l a t i o n i n a manner whi c h emphasizes the m i c r o s c o p i c a s p e c t s o f the problem. The i n t e r a c t i o n w i l l t h e n be i n v e s t i g a t e d i n l a t e r c h a p t e r s w i t h emphasis on how i t can be used as a t o o l t o i n v e s t i g a t e m a c r o s c o p i c p r o p e r t i e s o f the n u c l e u s , p a r t i c u l a r l y the n u c l e a r d e n s i t y . 2 . 1 S t r u c t u r e o f the I n t e r a c t i o n I n the p r e s e n t s e c t i o n we p r e s e n t a s i m p l i f i e d c l a s -s i c a l d e r i v a t i o n o f the p i o n - n u c l e u s o p t i c a l i n t e r a c t i o n w h i c h emphasizes the s t r u c t u r e o f the i n t e r a c t i o n and which i s based on the a n a l o g y between a d d i n g s- and p-waves i n the n u c l e a r medium and a d d i n g e l e c t r i c p o t e n t i a l s and f i e l d s i n a dense d i e l e c t r i c . The i d e a i s t h a t s-wave s c a t t e r i n g s , b e i n g i s o -t r o p i c , l o o k l i k e f r e e c harges i n a d i e l e c t r i c medium; the p-wave s c a t t e r i n g s , b e i n g d i p o l e i n c h a r a c t e r , l o o k l i k e e l e c t r i c d i p o l e s . The g e o m e t r i c problem o f a d d i n g the s c a t -t e r e d p i o n wave from the n u c l e a r medium i s t h e n the same as a d d i n g the t o t a l e l e c t r o s t a t i c p o t e n t i a l from a homogeneous and i s o t r o p i c d i e l e c t r i c medium immersed i n an e l e c t r i c f i e l d . Of c o u r s e , a d e r i v a t i o n o f t h i s t y pe p o s s e s s e s a l l the n a f v e t ? o f g e o m e t r i c a l s o l u t i o n s and cannot be ex p e c t e d to q u a n t i t a t i v e l y i n c l u d e the dynamics or quantum a s p e c t s o f the problem; i n a d d i t i o n , the t e c h n i q u e o f s o l u t i o n was r e a l i z e d (though n ot s p e c i f i c a l l y s t a t e d ) by o t h e r a u t h o r s and, hence, we p r e s e n t n o t h i n g r e a l l y new. N e v e r t h e l e s s , a 18 s o l u t i o n of t h i s type which s p e c i f i c a l l y invokes a know-ledge of elementary electromagnetic theory - — i s of consid-erable value i n understanding-the structure of the i n t e r a c t i o n . We s h a l l f i r s t review the c a l c u l a t i o n of the e l e c t r o -s t a t i c p o t e n t i a l from a homogeneous and i s o t r o p i c d i e l e c t r i c immersed i n a constant e l e c t r i c f i e l d . We s h a l l then simply i d e n t i f y t h i s potential and the corresponding e l e c t r i c f i e l d with the pion o p t i c a l wave function and the pion o p t i c a l momentum, f i e l d , respectively, l e t us define the macroscopic properties of the d i e l e c -t r i c by defining i t to have a free charge density, ef°(r_) , and a p o l a r i z a t i o n J _ E ^ i ® t o t a l p o t e n t i a l at a point r , U(r), i s then the sum of the p o t e n t i a l . U ( r ) . associated with the external e l e c t r i c , f i e l d plus the p o t e n t i a l , U ^ f r ) , due to the d i e l e c t r i c : (2-la) Ufr) - U 0 ( r ) + U a(r) . The l a t t e r term i s well-known from elementary electromagnetic theory:^ (2-lb) U d ( r ) - U P - ep(r') .. / i JL-ZL 4- p ( r M ' V r - r j ^ T l J l f [ e p f f t ) g o (_r_p ») + P E ( £ . j V g 0 ( r - r «)J where we have defined the free p a r t i c l e Green's function, (2-lc) g]Jr_-r T) = 1 e i k l^ ' f £ 7 T Ir - r r A discussion of the macroscopic properties of d i e l e c t r i c s can be found i n most textbooks on c l a s s i c a l electromagnetic theory (for instance, c.f. Jackson (1962)). 1 9 Now, i f the d i e l e c t r i c were homogeneous and i s o t r o p i c on the microscopic scale as well as on the macroscopic scale, we would have that the e f f e c t i v e f i e l d at a dipole, E g f f ( r ) , was simply the macroscopic e l e c t r i c f i e l d i n the d i e l e c t r i c , E ( r ) : ( 2 - 2 ) l ef f(£) = E(r) = y U(r) . (no c o r r e l a t i o n s ) . However, i f the dipoles are anticorrelated i n the d i e l e c t r i c — - that i s , i f two dipoles cannot ex i s t at the same point the e f f e c t i v e microscopic f i e l d w i l l i n general d i f f e r from E ( r ) . This e f f e c t i s e s s e n t i a l l y the well-known Lorenz-Lorentz e f f e c t (although, s t r i c t l y , t h i s term i s used to r e f e r to the resultant e f f e c t on the r e f r a c t i v e index: of the medium)„ To ca l c u l a t e the Lorenz-Lorentz e f f e c t , l e t us f i r s t note that the p o l a r i z a t i o n i s simply related to the e f f e c t i v e f i e l d : (2-3) P ?(£) - * > d ( r ) 2 e f f(r_) # where Tf. i s the molecular p o l a r i z a h i l i ty and -^P^(r) i s the dipole density. Now the e f f e c t i v e f i e l d i s the sum of the macroscopic f i e l d plus the f i e l d a r i s i n g from the p o l a r i -zation: . (2-4> £ e f f ( r ) = E(r) + E^fr) . To calculate the l a t t e r , l e t us choose a sphere of radius, a, surrounding the dipole so that no other dipoles are i n the sphere (which we can do since the dipoles are anticorrelated) and assume that outside the sphere the medium i s homogeneous 20 and i s o t r o p i c . The e l e c t r i c f i e l d a r i s i n g from the po l a r i . zation i s then e a s i l y calculated. n Spherical Cavity Surrounding a Dipole i n a D i e l e c t r i c Referring to the above sketch and noting that -P/n i s the charge on the surface of the cavity (n i s the outward normal), we have that (2-5) E -P (-P.n) (-ndS) p cose n an. = ~? p Thus, from Eqs. 2-3 and 2-4, (2-6a) (2-6b) — e f f — — rn[ — E(r) 3 Q ~ P(r) -1 . L r ; ( r ) 3 d ~ E(r) (zero-range anticorrelations) Substituting these r e l a t i o n s i n - E g . . 2-la, we have an i n t e g r a l equation for the pot e n t i a l : o (2-7) U(r) = U Q(r) — 47T ]dr 5 21 e / ( r j ) g 0 ( r l r ' ) 1 _ -Mr/ d(r5 2 u ( ^ ) V ' g 0 ( r - r ' ) Eqs. 2-6 are often used i n elementary optics to explain the.non-linear dependence of the re f r a c t i v e index of an op-t i c a l medium upon i t s density the c l a s s i c a l Lorenz-Lorentz e f f e c t . This i s e a s i l y seen by remembering from elementary electromagnetic theory that the molecular p o l a r i z a b i l i t y , J£ , i s defined by the equation, (2-8a) P-'JCE, and that the r e f r a c t i v e index of the medium, n, i s given i n terras of X by the r e l a t i o n , (2-8b) n 8 = 1+ 47TX , so that (from Eq. 2-6b), (2-8c) o 4 7 r 2r P n^ = 1 + a " - c l 1 - B*J>d For our purposes, we w i l l be interested i n the geometric modification of the intra-nuclear pion momentum f i e l d sug-gested by Eqs, 2-6; the "r e f r a c t i v e index" of the pion i n the nucleus does not bear the same analogy to the electromagnetic case because the pion o p t i c a l wave and the e l e c t r i c potential are not described by the same equations. We w i l l now show that the o p t i c a l l y scattered pion wave from a nucleus i s described by an equation l i k e Eq, 2-7 provided' 22 we i d e n t i f y the o p t i c a l wave with the e l e c t r o s t a t i c p o t e n t i a l and provided we i d e n t i f y the s-wave sc a t t e r i n g with the e l e c t r i c charge and the p-wave sca t t e r i n g with the molecular, p o l a r i z a b i l i t y (for s i m p l i c i t y , we w i l l neglect the pion charge). To see the analogy, l e t us f i r s t note that the sc a t t e r i n g amplitude of a pion from a nucleon has the general form (neglecting e x p l i c i t reference to the i n t e r n a l pion and nucleon co-ordinates and talcing Ti s c = 1), (2-9) f = b + cK»._k , where the f i r s t term i s the s-wave sc a t t e r i n g amplitude and the second term i s the p-wave scattering amplitude; k i s the i n i t i a l pion momentum and It1 the f i n a l pion momentum. Now the wave, d $ s c ( r ) , scattered from an element of nuclear densi ty, j> (r') d r 1 , at the point, r', i s the product . of the amplitude for finding a pion at r T , described by the pion o p t i c a l wave, $ f r ' ) , with the strength of the scattering, f p(r')dr', and the sp h e r i c a l l y outgoing wave from the scat-terer, g j j E - r ' ) , where g^ i s the fr e e - p a r t i c l e Green's func-t i o n of Eq. 2-lc: (2-10) . i k j r - r ' l = -^Tlg^ r - r M b $(r» ) f ( r ')dr 1 • - i J t g ^ f r - r M o k ' - k J ( r f ) ? (£ ' ) dr 1 „ Nov? (2-11) Z'gij£-£') •= - i i 'Sic(£-£') and we may conveniently define the e f f e c t i v e intra-nuclear 23 momentum density ar r T (2-12) E ^ f r ' ) - -ik_$(r') . Ee_-_-(r') i s related to the macroscopic momentum density, (2-13) E(r) 5 i 2 $(r) , through Eqs w 2-2 and 2-6. Eq. 2-10 can then be rewritten i n the form of Eq. 2-lb: (2-lb 1) $ s c ^ ) s -4^ * v[b § ( r M / ( r M g i J r - r M with " p o l a r i z a t i o n " (2-3') P(r) = -c/(r) E_ e f f ( r ) . Assuming an incident plane wave, (2-14) f 0 ( r ) - e 1 ^ , the t o t a l wave becomes (2-15) £(r) = e 1 ^ - 47rjdr [bf ( r ' )f(r M g ^ r - r ' ) Thus to the extent that we can neglect the structure and quantum aspects of the scatterer the pion s c a t t e r i n g within the nuclear medium i s equivalent to the c l a s s i c a l o p t i c a l s c a t t e r i n g within a d i e l e c t r i c medium, provided we i d e n t i f y the s-wave s c a t t e r i n g with the e l e c t r i c charge, (2-16a) e ) b|(r ' ) , and provided we i d e n t i f y the p-wave sca t t e r i n g with the molecular p o l a r i z a b i l i t y , (2-16b) ^ -c We can write the i n t e g r a l Schroedinger Equation 2-15 i n conventional d i f f e r e n t i a l form by operating on both sides 24 wi th the operator 0 1 V 2 + k2 . Then (2-17) ( V 2 + k 2) J(r) = -47lb^(r • )$(r ) + 2 »4*c/(r < ) E e f f ( r : ) In an uncorrelated nuclear medium, we have from Eq„ 2-2 that i 4 — i (2-2') ' ^Q±f^S) ~ ^ n 0 correlations) so that (2-18a) ( V 2 + k ) J ( r ) = -4nb/>(r )<j (r) f ^  .4Xc/(rJ V $ ( r ) , (no correlations) which i s the form of the pion-nucleus i n t e r a c t i o n derived by K i s s l i n g e r (1955); i n a nuclear medium with short-range ant i c o r r e l a t i o n s we have the Lorenz-Lorentz e f f e c t , (2-6«) E f f ( r ) = \ n P/ T E r r ) , <zero-range -exx — l + , - - — • c / l r ) • . » a n t i c o r r e l a t i ons) «-> so that (2-18b) ( V a + k 3 ) $(r) z -47ibp(r)|(r) , . 4Jic.p(r] _ ^. . I f f cj>(r) (zero-range anticorrelations) which i s the modification to the Kisslinger equation:";sug-gested by K r o l l (1961) and shown i n d e t a i l by the Ericsons (1966). In the present section we have demonstrated the struc-ture of the pion-nucleus i n t e r a c t i o n by a c l a s s i c a l and geometric technique for adding the elementary scattered waves from a (macroscopically) homogeneous and i s o t r o p i c nuclear 25 medium. T. E. 0. Eri c s on (1967) has also considered the c l a s s i c a l problem hut from a point of view which emphasizes the role of the microscopic correlations of the scatte r e r s . His tech-nique i s to expand the o p t i c a l wave i n terms of the higher order c o r r e l a t i o n functions between the sca t t e r e r s , trun-cating the r e u l t i n g hierarchy of equations at pair c o r r e l a -tions. He then finds the U s s l i n g e r equation, 2-18a, i n the absence-of correlations and the lorenz-Loreritz e f f e c t , Eq. 2-18,; i n the case of zero-range a n t i c o r r e l a t i o n s . The advantage of EricsonJs technique i s that i t generalizes i n an obvious way to include the structure of the scatterer ana the quantum aspects o f the problem. This generalization has ' been made by the Ericsons (1966) and we discuss t h e i r c a l -c u lation i n the next section. 2.2 A Microscopic Derivation of the Interaction; In the preceding section, we have derived the struc-ture of the pion-nucleus i n t e r a c t i o n using arguments which involve only the geometry of adding c l a s s i c a l waves. In the present section we w i l l discuss a microscopic and quantum mechanical treatment of the problem which has been given by T. E. 0. Ericson and (his wife) M. Ericson (1966). The impor-tant feature of t h e i r c a l c u l a t i o n i s that i t enables one to introduce the structure of the scatterer and the mechanism for pion absorption. We s h a l l discuss t h e i r c a l c u l a t i o n from a point of view which emphasizes the assumptions which must 'be made i n order to obtain the s i m p l i f i e d p o t e n t i a l of the previous section and which points out the corrections by which i t can be improved. For the purposes of i t s interactions with nucleons, a low energy pion may be regarded as a l i g h t boson with zero the nucleon-nucleon i n t e r a c t i o n , the low energy pion-nucleon i n t e r a c t i o n i s described by very short scattering lengths; for instance, the NN s i n g l e t s c a t t e r i n g length i s about -24 fm. and the NU t r i p l e t s c a t t e r i n g length i s about 5 fm. while the IT i s o s p i n doublet and quadruplet scattering lengths are both of the order of 0,2 fm. I t i s mainly this property of the pion, combined with the small pion mass ("^1/7 the nucleon mass), which allows us to simply construct the aver-age pion-nucleus i n t e r a c t i o n from the elrnentary pion-nucleon sca t t e r i n g data; on the other hand, the corresponding problem for the nucleon o p t i c a l p o t e n t i a l i s almost untractable. spin and three charge states ( JI+, 71°, X ). Moreover, unlike Nucleons's view of a nucleus Pion's view of a nucleus (zero energy) To see how the small pion-nucleus scattering lengths s i m p l i f y the problem, i t i s h e l p f u l to consider the zero-energy scattering; we can then replace the actual interactions by equivalent hard sphere interactions having the radius of the s c a t t e r i n g length (above sketch). For a nucleon pro-j e c t i l e , the spheres are larger than the inter-nucleon sep-aration within the nucleus (-^1.4 fm.) , while for a pion they are much smaller. We'can i n t e r p r e t t h i s as meaning that a nucleon sim-ultaneously i n t e r a c t s with many nucleons within a nuclear medium so that the nucleon-nucleon s c a t t e r i n g operator i s a many-body operator depending upon the positions and motions of many other nucleons. On the other hand, a low energy pion i n t e r a c t s with only one nucleon at a time, and i t i s reason-able to assume that the pion-nucleon scattering operator within a nuclear medium i s the same as i n free space, par-t i c u l a r l y so since the pion i s l i g h t and does not f e e l the dynamics of the nucleus. This assumption i s commonly known as the impulse approximation (and i s a l s o . v a l i d at very high energies when the p a r t i c l e wavelength becomes smaller than the i n t e r - p a r t i c l e spacing). The small mass of the pion also s i m p l i f i e s the c a l c u l a -t i o n i n the following two ways. F i r s t l y , i n i t s elementary c o l l i s i o n s with the nucleon scatterers, the kin e t i c exchanges of the pion with the scatterers are small so that we can e s s e n t i a l l y neglect the ef f e c t s of intermediate nuclear excitations. In fact, i n this approximation we consider the 28 elementary c o l l i s i o n s to be e s s e n t i a l l y e l a s t i c , "which r e s u l t s i n an enormous s i m p l i f i c a t i o n of the Green's function describ-ing the propagation of the pion within the nucleus. The Ericsons have estimated the corrections from intermediate excitations and find them to be n e g l i g i b l e . A second s i m p l i f i c a t i o n results from considering the scatterers to be massive. This greatly s i m p l i f i e s the mul-t i p l e s c a t t e r i n g equations since i n the sc a t t e r i n g processes we do not have to account for the motion of the nucleons. This assumption can be improved by making the following two kinematical corrections: f i r s t l y , we must correct for the fact that the target nucleons are i n motion having a k i n e t i c energy of about 25 MeV (Fermi -motion); secondly, we must correct for the r e c o i l of the bound nucleons. The Ericsons have estimated both corrections, finding them to be of the order of 10$. . s ' To make these ideas more e x p l i c i t , we w i l l describe the Ericsons' technique for so l v i n g the multiple scattering problem. We w i l l not give a detailed exposition of the derivations, but w i l l content ourselves with s t a t i n g the highlights of the c a l c u l a t i o n . We begin by noting that the pion wave,(|/^. , (which, of course, i s an operator i n the space of the target nucleons) s a t i s f i e s the equation (Goldberger & Watson (1964)), (2'-19a) U/T = X , 4. 1 lira 1 T , \jf. where the " e f f e c t i v e wave'*, i s the. sum of the incident 29 J • • . • . i "wave.Xj^ /and the scattered waves from a l l scatterers except the i t n : • J(=i)=l ~ ' : HJJ i s the nuclear Hamlltonian and the pion k i n e t i c energy; A i s the atomic number of the scatterer; k i s the.pi on wave . number E^ the. t o t a l energy of the system; Tj[ i s the scattering operator from the i t h nucleon 0 We w i l l assume the nucleus to i n i t i a l l y be i n the ground state, /0> , and w i l l apply the operators, Eq„ 2-19, to the system i n the configuration space of the pion. Eqs. 2-19 are then equivalent to the hierarchy of equations, (2-20a) (/£(r)|0?= X Q (*) \0? t fjjg"(r ,r') f± (r' , r 1 ' W± (r ") '" x /07 dr'dr"; (2-20b) (jA(r)J07= X 0 ( r ) I O > + £1 (S(r,r')f J r T , r ' ') W. . (r' M /0>dr'drti (2-20c) l ^ ; i ( r ) = X Q ( r ) |0> -f-.(r ' M j O>dr'dr" where we have introduced an obvious notation to keep track of 4 the nucleon co-ordinates. Here (2-21a) ; . g ( r , r f ) = lim ^ r | - — s 2 /n><n| I r 1 > 1 l i m n < r | L — , JrT7-|n><nl Jk aM so i s the t o t a l pion-nuoleus Green's function, where we have defined the one-particle Green's functions, 4 (2-21b) • < g n f r , r ' ) s lim <r j JL_ i _ — — / r * > and where the sum i s oyer a l l nuclear states, / n>. For s i m p l i c i t y , we have ignored the pion charge i n the one-p a r t i c l e Green's function, Eq. 2-21b (and, for notational convenience, have suppressed the i n t e r n a l co-ordinates of the pion). In Eq. 3-20 we have also defined the matrix elements of the scattering operator, (2-22) f i ( r , r ' ) = < r | T i / £ > We have mentioned i n e a r l i e r paragraphs that we can s i m p l i f y the Green's function, Eq. 2-21, because of the l i g h t mass "of the pion and that we can s i m p l i f y the scattering operator, Eq„ 2-22, mainly because of the small s c a t t e r i n g lengths which describe the pion-nucleon i n t e r a c t i o n . Because of the small value of the pion mass, elementary ki n e t i c exchanges are small so that the propagation of the pion i s not s e n s i t i v e to the intermediate states of nuclear e x c i t a t i o n . Thus we may choose the elementary c o l l i s i o n s to be e l a s t i c , (2-23) ' g n ( r , r ' ) ^ g Q ( r , r ' ) whence the Green's function, Eq. 2-21, simply becomes (2-24) g f r . r 1 ) = g Q l.r ,r') X| n>&| = g 0 ( r , r * ) I N , where i s the i d e n t i t y operator i n the space of the nucleus. This approximation, of course, represents an enormous s i m p l i f i c a t i o n since i t lar g e l y decouples the structure of the scatterer from the o p t i c a l scattering. To see how the impulse approximation s i m p l i f i e s the sc a t t e r i n g operator, %(£.,£.')» w © w i l l write i t s matrix elements i n the configuration space of the nucleons (suppres-sing the i n t r i n s i c nucleon co-ordinates), From the impulse approximation we have that, (2-25) </£; R i . R E ^ o . R A l T i l ^ ; c ^ W ^ i - ^ V * % - ^ V ° ° B ^ ^ i - i - S i - i 1 where heing the free space s c a t t e r i n g operator of the pion from the i ^ k nucleon. Hence, the many-body scat t e r i n g operator i s reduced to a known two-body s c a t t e r i n g operator. The sc a t t e r i n g matrix i s further s i m p l i f i e d by the as sump ti o n that the nucleons are massive: (2-26) f i ^ , R i . ; r , t R ^ ) 5 f. (r ; -R^ ; r \£± ) / ( ^ - E f ) . • Since i n the impulse approximation the scattering oper-ator i s simply the free space sc a t t e r i n g operator of a pion from a nucleon, i t i s e a s i l y evaluated because i t s matrix elements i n momentum space are (at low energies) simply the s-wave and p-wave scat t e r i n g amplitudes: 32 (2-27) f . ^ ) S <t'|T i|k> - b ^ b ^ J T - j ^ (s-wave) (p-wave) Here b Q , "b are related to the s-wave pion-nucleon scattering lengths, and c . o„. d , a, are related to the p-wave scat-to ' * :. O* 1 0' 1 ter i n g volumes (c.f. Ericson and Ericson.(1966)); the spin-f l i p term has no s-wave contribution since the pion has spin zero. In Eq„ 2-27, T and are the isospi.n operators of the pion and the i t J l nucleon, respectively. As we have often noted, i t i s important to r e t a i n the p-wave c o n t r i -butions, even at low energies, because of the l i n g e r i n g e f f e c t s of the 3-3 resonance;the higher p a r t i a l waves make ne g l i g i b l e contributions. Employing the assumptions i n Eqs. 2-23 to 2-27, the o p t i c a l pion wave, (2-28) f ( r ) = <0|||£(r) ) 0^ , can be evaluated i n a straightforward manner from Eqs. 2-20. The Ericsons (1966) have performed the nuclear averages and find multiple scattering equations which are e s s e n t i a l l y those for multiple s c a t t e r i n g i n a c l a s s i c a l media (Lax (1951)): (2-29a) J(r) = X j r ) ^J'^Li^Lijj BQin-L,)fiL\-Li> L*-L^ x • J r ( r " ) d r ' d r n • (2-29b) (r) = XJx)/^^8;r1)dr^g0(r-r1') x f(r.?-r„; r " - r ' J (r" )dr*dr"; 33 (2-290) J r g . ^ W s ^o^+^^s^l^^sjTgo^^ x , f ( r T - r ; r"-r_„) J # (r")dr'dr" * Here >^ (r^) i s the average density of the nucleus ;/> ( r ^ j r ^ ) i s the p r o b a b i l i t y density for finding a p a r t i c l e at when there i s a p a r t i c l e at r ? ; 5 (r"0 i s the sum at r " of the incident wave and the scattered waves from a l l scatterers except, the one at r_g, subject to the knowledge that there i s a scatterer at r ^ . The other terms have a similar meaning. These equations describe e s s e n t i a l l y the same problem that we gave a geometric solution for i n Sec. 2-1 and they have been discussed i n various approximations by Ericson (1967). We s h a l l now wish to truncate the hierarchy of Eqs. £-29 at some point and solve the coupled equations. For instance i f we took -(2-30) $ (r) = X (r) Ll ~ o -i n Eq„ 2-29a, i t i s seen that we would simply have made the f i r s t Born approximation. That this i s not a good approx-imation can be seen from the dominance of the s- and p-waves i n the pion-nucleon s c a t t e r i n g amplitudes. As a second attempt - — one which includes multiple s c a t t e r i n g — . we might take 4 \ (2-31) § (r) = J ( r ) -1 This i s i d e n t i c a l to our geometric derivation i n the f i r s t section i n which we neglected nuclear correlations; i n fact. 34 this i s seen e x p l i c i t l y to be the case since terminating the hierarchy at Eq„ 2-29a i s equivalent to neglecting the cor-r e l a t i o n s ; Of course, t h i s approximation leads to the Ki s s l i n g e r Eq„ 2-18a, From our present point of view, we note that the primary handicap' i n this approximation i s that ? r (r) d i f f e r s from J ( r ) at r by a wave which i s singular at r. To avoid this s e l f - e x c i t a t i o n of the scatterer, we go to the -next approximation and set (2-32) I (r) = \ (r) - 2 ' - l . ~2 i n Eq. 2-29b„ Since r ^ and r ^ are generally far apart i n the nucleus, we do not expect the wave <F (r) to depend £-2;~l ~ very s e n s i t i v e l y on the exact po s i t i o n of r_ l 0 In fact, i f we write the j o i n t p r o b a b i l i t y density, 2) * i n t e r m s of the pair c o r r e l a t i o n function, G(r.,r P) , ( 2 - 3 3 ) \P(rQ;^i) = / >(£a ) ( 1 + G f£i»£a ) } * we can remove the unphysical divergence at r = v0 by i n t r o -"—1 ~<d ducing the zero-range a n t i c o r r e l a t i o n of two nucleons, (2-34) G(r ,r 2) = - 1 „ In f a c t , i t i s this choice of the pair c o r r e l a t i o n function 4 which leads to the Lorenz-Lorentz e f f e c t , as we foiznd i n our geometric derivation (Eq. 2-18'b) . The Ericsons have shown that replacing the zero-range c o r r e l a t i o n function, Eq. 2-34, with a function of f i n i t e 2 range, leads to effects of order (§ K) where ^ i s the range of the c o r r e l a t i o n function and K i s the wave number of the 35 pion i n the nuclear medium. The structure of the c o r r e l a t i o n region i s thus n e g l i g i b l e for wavelengths appreciably larger than one fermi; on the other hand, t y p i c a l wavelengths of low energy pions are several times the nuclear dimensions,, Hence, we expect to l e a m l i t t l e about the structure of the c o r r e l a t i o n function from low energy pion scattering,, Tn fa c t , the Ericsons have estimated these range corrections and find them to be of the order of 10%, This i s i n contrast to the measurements of molecular correlations through the scat-t e r i n g of slow neutrons by a c r y s t a l l a t t i c e ; here the neutron-nuclear s c a t t e r i n g lengths are also much smaller than the inter-molecular spacing, but the neutron wavelength i s of a comparable value (Van Hove (1954)), We have now discussed the connections between the micro-scopic quantum description of the pion-nucleus i n t e r a c t i o n and the geometric derivation given i n the previous section (Sec, 2-1), In addition, we have discussed the kinematic and range corrections which must be made to the pion-nucleus i n t e r a c t i o n , Eqs, 2-18, I t now remains to describe the Ericsons' technique for describing o p t i c a l pion absorption. There are e s s e n t i a l l y three mechanisms by which a low energy o p t i c a l pion can be absorbed: f i r s t l y , i t can be i n e l a s t i c a l l y scattered by the nucleus; secondly, being a boson, i t can be absorbed by a single nucleon, i t s rest mass being transformed into the k i n e t i c energy of the scatterer; t h i r d l y , i t can be absorbed by two nucleons. E x c i t a t i o n of the nucleus i s suppressed due to the small mass of the pion, so that effects of the f i r s t mechanism may be neglected. 36 In addition, the absorption by one nucleon requires a nuc-l e a r momentum of about 500 MeV/c from kinematic considerations while the Fermi momentum of the nucleons i s of the order of 850 MeV/c; hence, one-nucleon absorption i s also strongly 4 i suppressed. Thus only the process of two-nucleon absorp-t i o n contributes s i g n i f i c a n t l y to the o p t i c a l absorption. To s t r i c t l y account for two-nucleon pion absorption, we should use f i e l d theoretic techniques to describe the crea-t i o n and destruction of pions. In addition^ we would need a detailed model of the nucleus (such as the s h e l l model) to describe the kineriiatics of the absorption, which obviates the simple macroscopic description which we obtained for the r e f r a c t i v e processes. As discussed by the Ericsons, t h i s method of approach has not met with much success; i n fact, proceeding i n this manner, Spec tor (1964) .found absorption rates nearly two orders of~~magnitude smaller than those observed. This then would seem s u f f i c i e n t j u s t i f i c a t i o n to use a more phenomenological approach, and the Ericsons have done this by extending and reformulating the quasi-deuteron model of Brueckner et a l . (1951), In the Brueckner model, absorption i s considered to take place at zero energy with the two absorbing nucleons i n a r e l a t i v e s-state* the two-nucleon absorption amplitude i s then introduced as a phenomenological quantity accounting for a l l the short-range absorption processes. Hence, the problem of obtaining a description which i s s a t i s f a c t o r y from a f i e l d theoretic point of view i s l a r g e l y transferred to a j u s t i f i c a t i o n of these phenomenological quantities. 3 7 The Ericsons have generalized the Brueckner model (and i t s reformulation by Eckstein ( 1 9 6 3 ) ) i n two ways. F i r s t l y , they introduce an e l a s t i c s c a t t e r i n g amplitude to describe pion s c a t t e r i n g by a two-nucleon scatterer; following Eckstein ( 1 9 6 3 ) , they measure the parameters of this amp-li t u d e from considerations of pion production i n M c o l -l i s i o n s . Secondly, they extend the Brueckner model, which included only s-wave pion absorption, to include p~wave pion absorption. The r e s u l t i s that the two-nucleon amplitude can be written i n the form, ( 2 - 3 5 ) f. . = <T(r,-r.) J(r -i(r,+r,)) (B + C k'-k -+ (B c +- C k'. k)j~(T» t. ) (T't .)-f (T-t^) (T»t. A ) , 5 5— — L a -J J 1.J where B.. an C are li n e a r combinations of the various m m amplitudes of the angular momentum and i s o s p i n states of the (7I2W) system. Since Eq. 2 - 3 5 i s of the form of the (KW) amplitude, Eq. 2 - 2 7 , we can simply account for pion o p t i c a l absorption by adding the two-nucleon amplitude of Eq. 2 - 3 5 to the one-nucleon amplitude of Eq. 2 - 2 7 . Of course, the two-nucleon amplitudes w i l l also make contributions to the r e f r a c t i v e aspects of the problem through the r e a l parts of the parameters B^ and C ; o .r m m , these re a l parts are to some extent induced by the absorp-t i o n , and are not very well determined. This i s of par-t i c u l a r i n t e r e s t to the s_wave scattering, where a strong c a n c e l l a t i o n occurs i n the one-nucleon amplitude, b Q Of course, for the two-nucleon processes, the two-38 nucleon density replaces the one-nucleon density. To a good approximation the two-nucleon density can simply he replaced by the square of the density; however, i n the case of s-waves, where the s c a t t e r i n g lengths strongly cancel, the c o r r e c t i o n for long-range correlations should be included. The Ericsons show that the correction term i s also proportional to the square of the density; for our l a t e r purposes we w i l l only require the fact that absorp-ti v e processes are proportional to the square of the density and we w i l l not include t h i s correction i n our subsequent discussion. t We s h a l l now simply state the r e s u l t which the E r i c -sons obtain for the pion-nucleus i n t e r a c t i o n . Their r e s u l t i s e s s e n t i a l l y that the pion wave s a t i s f i e s the o p t i c a l equation, (2-36a) ( V S-f k a) i f f ) ' S 2^  T i -uil(r) -m 2(r) t 2" n l ( £ J + n 2 ^ -i47fd 0tf(r) X 1 + i d i ^ r ) + n 2 ( r ) ) 2 I<£) , where (2-36b) m, (r) = 4 j f ( l ) Mr) + '=1= pfr) ) M„ 0 A +. 2 r ( £) ( 1 + i ( n (r) + n 8 ( r ) ) I ' V P ^ " (2-36c) m (r) = 4n(l +- J^Z ) 8 - . . 2MH B P 2 ( r ) 8B 5(T.t)fT- t ) / ( r ) -° ~ + * ~ ~ 2M-X-ri.(n n(r) n Q ( r ) ) sr1 Q 1 — 2 — _ IT -1 2 < P 2 ^ 39 (2-36d) n, (r) = 47i(l + ^ ) 1 ^ % (2-36e) n 0 ( r ) = ""4*(1+3L) • ~ • 2M T • A C 0 / ( r ) + 8C 5(T t ) ( T t ) / ( r ) A * The e l e c t r o s t a t i c p o t e n t i a l , V c (?) * w h i c h w © n a v e n o t included i n Eqs. 2-36, i s to he evaluated from the nuclear charge d i s t r i b u t i o n ; we discuss the e l e c t r o s t a t i c potential i n Sec. 3-3 of the next chapter. -Comparing Eqs. 2-36 and Eqs. 2-18, we see that the i n t e r a c t i o n retains the simple form predicted by the geo-metric arguments of the previous section (Sec. 2-1). 2-3 Experimental V e r i f i c a t i o n In the present section we w i l l b r i e f l y review the experimental corroboration of the Ericsons' p o t e n t i a l , Eq. 2-36. V i r t u a l l y the only data available i n the low energy region i s that from the 7T -mesic x-ray experiments (which are at e s s e n t i a l l y zero energy) „ The technique i s to analyze the s h i f t s and the widths of these x-rays with the p o t e n t i a l , Eq. 2-36, and to compare the r e s u l t i n g parameters with the t h e o r e t i c a l l y predicted values. A general remark might be that the Ericsons (1966) and Ericson (1967) find theory to be i n s a t i s f a c t o r y agreement with ex-periment, the t h e o r e t i c a l and empirical parameters generally agreeing to better than 50%, We s h a l l only c u r s o r i l y re-view th e i r discussion here. The essential i de as" involved i n analj/zing the TT -mesic x-ray data for the parameters of the pion-nucleus potential 40 are as follows. Since a pion i s much heavier than an electron, a pion i n a 7T-mesic atom spends much more time i n the nucleus, than does an electron of corresponding quantum numbers i n a conventional atom. The energy levels of a 7C -mesic atom are thus considerably modified from those which we would expect from the purely electromag-netic forces. The e f f e c t of the r e a l terms i n the o p t i c a l potential i s to s h i f t the energy l e v e l s from those values calculated from the electromagnetic i n t e r a c t i o n and the e f f e c t of the absorptive terms i s to broaden the l e v e l s . Thus we can measure the r e a l terms i n the o p t i c a l potential by analyzing the l e v e l s h i f t s of the ir-mesic x-rays and we can measure the imaginary terms i n the o p t i c a l potential by analyzing t h e i r widths. A simple way to analyze the r e a l l e v e l s h i f t s and the widths, or imaginary l e v e l s h i f t s , i s to calculate the com-plex l e v e l s h i f t s , A E n l , of the l e v e l ( n , l ) , i n f i r s t - o r d e r perturbation theory: here <j>nl fr) i s the wave function of the pion i n the o r b i t with p r i n c i p a l quantum number, n, and angular momentum, 1, calculated from the electromagnetic i n t e r a c t i o n ; m(r) and n(r) schematically represent the l o c a l and momentum-dependent potentials i n Eq. 2-36 (neglecting the hyper fine term). How at the x-ray energies (^ -1 MeV) the intra-nuclear pion wavelength i s long compared to the nuclear dimensions 41 ana the s rwave function (1=0)' varies slowly with radius. The complex s-wave l e v e l s h i f t s are thus dominated by the l o c a l i n t e r a c t i o n , m(r) , i n Eq. 2-37, since the gradient term which enters the momentum-dependent i n t e r a c t i o n i s small. By analyzing the s-wave s h i f t s we therefore obtain the l o c a l i n t e r a c t i o n , m(r) , and, hence, the microscopic s-wave pion-nucleon and pion-deuteron parameters, "D and B , i n Eq. 2-36. For higher p a r t i a l waves, $n]_(£), i s a much stronger function of radius (since i t looks, l i k e a spherical Bessel function of order,1) so that, from the complex l e v e l s h i f t s of higher angular momentum states, we can obtain the momentum-dependent i n t e r a c t i o n , n(r) , and, hence, the microscopic p-wave pion-nucleon and pi on-deuteron parameters, c m and CXfl, of Eq. 2-36. In actual p r a c t i c e , the Ericsons have not used the perturbation formula, Eq. 2-37, to analyze the 7T-mesic x-ray data but rather have obtained analytic formulae for the l e v e l s h i f t s (for a uniform potential) by matching wave functions calculated inside the potential with the well-known Whittaker functions (bound-state spherical Coulomb functions) which describe the pion wave outside the pot e n t i a l . The v a l i d i t y of the Ericsons' potential i s tested by comparing the pion-nucleon and pion-deuteron sc a t t e r i n g lengths (or volumes) obtained from analyzing the TT-mesic x-ray data with Eq. 2-36 with those obtained from the elemen tary experiments. The most important uncertainty l i e s i n the r e a l part of the l o c a l i n t e r a c t i o n , where a large can-• 4a , c e l l a t i o n fl:10) occurs i n the s-wave scattering lengths. For t h i s reason terras o r d i n a r i l y of order 10'fo the Fermi motion corrections, the f i n i t e c o r r e l a t i o n length corrections, %he r e a l part of the two-nucleon scattering amplitude become of f i r s t order importance. Nevertheless, making these corrections, the Ericsons find remarkable agreement with the empirical values i^ZOfo) as calculated from the Is l e v e l s h i f t s i n L i and F, although this may be somewhat fortuitous. The l o c a l isotopic spin term i n the l o c a l i n t e r -action i s found to have, a s i m i l a r empirical v a l i d i t y . ' The l o c a l absorption term i s measured from the Is l e v e l widths and i s found to be i n agreement to about 50^. The non-l o c a l terms are measured from the Sp and 3d l e v e l s h i f t s and widths (where momentum-dependent ef f e c t s predominate) and are found i n si m i l a r agreement to the l o c a l terms. A good deal of this-imprecision may l i e i n the rather crude data obtained from the x-ray measurements and i n the only poorly determined empirical input scattering lengths rather than i n the p r e c i s i o n of the technique of c a l c u l a -t i o n . At any rate, the potential would seem to be suf-f i c i e n t l y well v e r i f i e d (at least at zero energy) to study i t s properties further. In the subsequent chapters we w i l l investigate the properties of the o p t i c a l i n t e r a c t i o n and suggest how i t may be used to measure the diffuseness of the nuclear surface. 4 3 CHAPTER 3 THE OPTICAL PROPERTIES 01 LOW ENERGY, PIONS IN SPHERICAL NUCLEI In the previous chapter we have shown that the struc-ture of the pion-nucleus o p t i c a l i n t e r a c t i o n i s quite well understood and we have suggested that i n the low energy region (^30 MeV) at least the qu a l i t a t i v e features of the i n t e r a c t i o n are known with reasonable certainty. Our pur-poses i n the present chapter are two-fold: f i r s t l y , we wish to- use this i n t e r a c t i o n to study the o p t i c a l properties of low energy pions i n spherical nuclei; secondly, we wish to study the effects of the diffuseness of the nuclear sur-face on s c a t t e r i n g and absorption cross sections. In a l a t e r chapter we w i l l use these results to i n t e r p r e t the behaviour of deformed nuclei i n the presence of the o p t i c a l pion f i e l d . In this chapter our f i r s t objective i s to display the various o p t i c a l processes associated with the low energy pion-nucleus i n t e r a c t i o n . We s h a l l be p a r t i c u l a r l y concerned with those processes which concern the wave properties of the pion inside the nucleus, namely, resonance and absorption. Since we are at low energies, i t i s convenient to demonstrate these properties by making a p a r t i a l wave analysis. We then follow the fam i l i a r technique of defining an interface i n the region of the nuclear surface and match wave functions calculated inside the interface from the o p t i c a l p o t e n t i a l with exterior wave functions which depend only upon the e l e c t r o s t a t i c and angular momentum barriers surrounding the 4 4 nucleus (spherical Coulomb functions). This allows us to state the p a r t i a l wave phase s h i f t s i n terms of the i n t e r i o r logarithmic derivatives which, for the low energy pion-nucleus i n t e r a c t i o n , have a simple and i n s t r u c t i v e form: the r e a l parts of the i n t e r i o r logar-ithmic derivatives, which describe the resonance properties of the o p t i c a l i n t e r a c t i o n , are (in the low energy region) i n s e n s i t i v e to the details of the o p t i c a l potential because of the long pion wavelength inside of the nucleus; the imaginary parts of the i n t e r i o r logarithmic derivatives, which describe absorption, can be written i n the form of a sum over l o c a l and momentum-dependent absorption so that we can e x p l i c i t l y state and examine the r e l a t i v e effects of these processes near the top of the l o c a l potential b a r r i e r where the pion momentum i s small. In Sec. 3-2 we present a q u a l i t a t i v e discussion of the pion-nucleus i n t e r a c t i o n treating the nucleus as having a uniform density. In the low energy region (^r30 MeV) scat-tering and absorption are governed by the potential b a r r i e r ("•^ 15 MeV) a r i s i n g from the repulsive l o c a l i n t e r a c t i o n . This b a r r i e r results i n . the pion wavelength i n the nucleus being very long so that the resonance properties are deter-mined mainly by the s i z e .of the scattering nucleus rather than by the d e t a i l s of the interaction, (in contrast to nucleon-nucleus s c a t t e r i n g where we have an a t t r a c t i v e pot-e n t i a l and a short nucleon wavelength inside the nucleus). In fact, the resonance properties of s-waves are quite 45 i n s e n s i t i v e to the i n t e r a c t i o n while the resonance properties of higher p a r t i a l waves depend mainly upon the e f f e c t i v e mass of the pion inside the nucleus a r i s i n g from the momentum-dependent i n t e r a c t i o n . This l a t t e r e f f e c t i s expected since angular momentum must "be conserved i n crossing the nuclear interface while the mass of the pion apparently changes. Nevertheless, because of the long pion wavelength, we learn l i t t l e about the structure of the nuclear interface for instance', the diffuseness of the nuclear surface from the. effects of resonance upon o p t i c a l scattering experiments. Of course, we can go to higher energies where the wavelength of the pion i s shorter and resonance ef f e c t s are more dependent upon the d e t a i l s of the p o t e n t i a l , but we then encounter two d i f f i c u l t i e s : f i r s t l y , except i n very l i g h t n u c l e i , several p a r t i a l waves enter the problem so that i t i s d i f f i c u l t to untangle these various p a r t i a l waves i n a convincing manner; secondly, the simple connection between microscopic processes and o p t i c a l phenomena which we have discussed i n Sec. 2-2 i s no longer v a l i d . However, from our discussion of the structure of the pion-nucleus i n t e r a c t i o n i n Sec, 2-1, we might expect that the only e f f e c t of the l a t t e r d i f f i c u l t y i s to make the parameters of the potential energy dependent. In f a c t , Kisslinger (1955) was able to show that 12 the strong back scattering of 62 MeV pions i n C could only be explained when proper account was taken of the nuclear surface. Nevertheless, as we w i l l now show5 low energy o p t i -c a l pion absorption cross sections should provide a much more 46 d i r e c t method for measuring the nuclear surface and one which can he used even i n heavy n u c l e i . The basic idea of using absorption cross sections to measure the nuclear dxffuseness i s the following. The o p t i c a l absorption of pions arises both from the l o c a l i n t e r a c t i o n and the momentum-dependent i n t e r a c t i o n . The absorption due to the l o c a l i n t e r a c t i o n i s s e n s i t i v e only to the t o t a l amount of matter present and i s rather i n s e n s i t i v e to i t s d i s t r i -bution. -.On the other hand, as the pion enters the nucleus and encounters the repulsion of the l o c a l p o t e n t i a l , the i n -ter n a l pion momentum becomes small so that momentum-dependent processes are both suppressed, and, at the same time, very energy dependent; i n fact, the. pion momentum varies r a p i d l y over the nuclear surface so that the i n t e r a c t i o n is-very s e n s i t i v e to the nuclear diffuseness. Since the momentum-dependent processes are comparable i n magnitude to the l o c a l processes, absorption cross sections have considerable struc-ture near the top of the pot e n t i a l barrier; i n fa c t , t h i s structure c l e a r l y mirrors the diffuseness of the nuclear sur-face. Of course, our arguments s t r i c t l y apply only to s-waves which have no angular momentum; however, the effects of angu-l a r momentum for higher p a r t i a l waves i s simply to " r a i s e " the b a r r i e r y i e l d i n g a s i m i l a r sort of structure only at higher energy. Since we are at low energies, only a few par-t i a l waves enter the problem and th e i r untanglement i s r e l a -t i v e l y straightforward. Optical pion absorption should there-fore be a good tool for measuring the nuclear diffuseness i n • 4 7 a large v a r i e t y of nu c l e i . -One of the major reasons for wishing to know the d i f f u s e -ness of the nuclear surface i s that i t allows us to i n f e r the neutron density i n the nuclear surface. In fact, the nuclear density i s simply the sum of the proton and neutron densities and, since the proton density i s well-known from electron s c a t t e r i n g and JJ -mesic x-ray experiments (Devons and Duerdoth (1969), Wu (1967)), we can e a s i l y i n f e r the l a t t e r . This e l u -sive quantity i s of considerable i n t e r e s t to the theory of nuclear structure (for instance, the shell-model) and res u l t s currently available from various experiments are not mut-u a l l y consistent, as discussed below. One technique for measuring the surface d i s t r i b u t i o n of neutrons has been to observe the branching r a t i o s for events involving various elementary p a r t i c l e s . Auerbach et a l . (1968) have reanalyzed the 700 MeV pion charge-exchange data of Abashian et a l . (1956) and find that the rms neutron radius agrees with the rms proton radius to within a few percent (generally being smaller) unless one admits sharply peaked neutron dist r i b u t i o n s i n the nuclear surface, Burhop (1967) has analysed the decay products r e s u l t i n g from the capture of kaons i n atoms and finds that the res u l t s can be accommodated by choosing the rms radius of the proton and neutron d i s t r i b u t i o n s to be the same but with the neutron surface thickness 50$ greater than that of the proton dis-t r i b u t i o n ; however, as Bugg et a l , (1969) have pointed out,' these results are somewhat clouded by the complicated atomic physics which'-'enters' the, problem. 48 Temmer (1966) suggested that the observed Coulomb displacement energies between isobaric analog states i n lead isotopes was evidence for a neutron excess i n heavy nuclei; ITolen et al„ (1968) extended these re s u l t s and found the rms neutron radius to be about a percent larger than that of the. charge radius. .This i s i n agreement with results using o p t i c a l scattering of nucleons: Greenlees et a l . (1966) measured the rms radius of the neutron d i s t r i b u t i o n by choos-ing the'rms radius of the mass d i s t r i b u t i o n to be given.by the s p i n - o r b i t term i n the o p t i c a l potential; Elton (1968) deduced the di s t r i b u t i o n s from the shell-model potentials obtained from analysis of the s i n g l e - p a r t i e l e and s i n g l e -S 0 8 hole states i n Pb . Both authors found that the neutron d i s t r i b u t i o n i s up to ten percent greater than the proton d i s t r i b u t i o n . We f e e l that these disagreements can be l a r g e l y resolved simply by measuring the o p t i c a l absorption of low energy pions i n heavy nuclei where the nuclear density enters the i n t e r a c t i o n i n a much more di r e c t manner. Unfortunately, at the present time there i s e s s e n t i a l l y no o p t i c a l s c a t t e r i n g data available for pions i n the low energy region (< 50 MeV). We therefore though i t useful to discuss one example (the' s c a t t e r i n g of positive pions from 40v Ca ) from the point of view from which an experiment might be analyzed. We f i r s t give a numerical discussion of results for the Ericsons' p o t e n t i a l described i n Sec. 2-2''taking into account the e l e c t r o s t a t i c potential a r i s i n g from the nuclear charge d i s t r i b u t i o n and noting r e l a t i v i s t i c corrections which 49 may become important at higher energies; this discussion i s contained i n Sec. 3-3. While th i s zero-energy potential may neglect some energy dependence of the parameters of the potential at higher energies, i t should provide a reason-able description of empirical processes. In Sec. 3-4 we use this potential to construct the e l a s t i c s c attering and absorption cross sections of pions from a l i g h t nucleus such as C a ^ , We discuss these cross sections, i n terms of a p a r t i a l wave analysis and we use the formalism developed i n Sec. 3-1 to provide an explanation of the p a r t i a l wave phase s h i f t s i n terms of the i n t e r n a l log-arithmic derivatives. We examine the op t i c a l properties of the p o t e n t i a l for this quantitative example i n terms of the qu a l i t a t i v e features which we describe for a uniform d i s t r i -bution i n Sec. 3-2. In addition, we suggest a procedure for obtaining the empiricar-parameters of the poten t i a l from the experimental, cross sections. We also extend our d i s -cussion to examine the ef f e c t s of the diffuseness of the nuclear surface on the o p t i c a l cross sections, p a r t i c u l a r l y the absorption cross sections. We then extend our discussion to the properties of pion s c a t t e r i n g and absorption i n heavy n u c l e i , choosing as an example the nucleus Pb 2^ 8. In Sec. 3-5 we b r i e f l y mention some of the terms of order A""*" and higher which enter the Ericsons 1 potential described i n Eq, 2-36. These include i s o s p i n terms, which macroscopically account for single and double charge exchange, and terms which depend on the nuclear spin and lead to 50 h y p e r f i n e s p l i t t i n g i n t h e 7 i - m e s i c x - r a y s p e c t r u m . B e c a u s e t h e s e t e r m s a r e s m a l l a n d p l a y a s u b o r d i n a t e r o l e i n d e t e r -m i n i n g t h e o p t i c a l p r o p e r t i e s o f p i o n s i n n u c l e i , t h e y h a v e b e e n o m i t t e d f r o m t h e r e m a i n i n g d i s c u s s i o n o f t h e t h e s i s . 3-1 A F o r m a l i s m f o r A n a l y z i n g O p t i c a l S c a t t e r i n g a n d  A b s o r p t i o n • I n t h e p r e s e n t s e c t i o n we w i l l d e s c r i b e a f o r m a l i s m f o r a n a l y z i n g o p t i c a l s c a t t e r i n g a n d a b s o r p t i o n c r o s s s e c t i o n s b y a p a r t i a l w a v e a n a l y s i s . O u r t e c h n i q u e i s t o f i r s t c h o o s e a n i n t e r f a c e i n t h e r e g i o n o f t h e n u c l e a r s u r f a c e . T h e n , f o r e a c h p a r t i a l w a v e , we c a l c u l a t e t h e r a d i a l w a v e f u n c t i o n s i n s i d e t h e i n t e r f a c e f r o m t h e p i o n - n u c l e u s o p t i c a l p o t e n t i a l ; t h e p a r t i a l w a v e p h a s e s h i f t s a r e t h e n d e f i n e d b y m a t c h i n g t h e s e i n t e r i o r w a v e f u n c t i o n s t o t h e i n c o m i n g a n d o u t g o i n g s p h e r i c a l C o u l o m b f u n c t i o n s w h i c h d e s c r i b e t h e i n t e r a c t i o n o u t s i d e t h e i n t e r f a c e . T h i s p r o c e d u r e a l l o w s u s t o e x p r e s s t h e p h a s e s h i f t s i n t e r m s o f q u a n t i t i e s w h i c h a r e s p e c i f i c t o t h e b a r r i e r - - ~ t h e p e n e t r a t i o n f a c t o r a n d s h i f t f u n c t i o n a n d q u a n t i t i e s w h i c h a r e s p e c i f i c t o t h e o p t i c a l p o t e n t i a l t h e i n t e r i o r l o g a r i t h m i c d e r i v a t i v e s . T h e f o r m u l a e we d e r i v e a r e w e l l - k n o w n f o r l o c a l p o t e n t i a l s a n d , , w h e r e n e c e s s a r y , w e e x t e n d t h e m t o m o m e n t u m - d e p e n d e n t p o t e n t i a l s . T h e a d v a n t a g e o f o u r f o r m u l a e i s t h a t t h e y a l l o w u s d i r e c t a c c e s s t o t h e i n t e r i o r l o g a r i t h m i c d e r i v a t i v e s . F o r t h e p i o n - n u c l e u s i n t e r a c t i o n t h e s e q u a n t i t i e s a r e b o t h s i m p l e a n d i n s t r u c t i v e : a s we s h o w i n t h e n e x t s e c t i o n , t h e r e a l p a r t s o f t h e l o g a r i t h m i c d e r i v a t i v e s , w h i c h d e s c r i b e t h e 51 resonance aspects of the problem, are roughly constant because of the long pion wavelength inside the nucleus; the imaginary parts are related i n a very dir e c t way to absorption. Hence, i n the discussion of the l a t e r sections, we w i l l emphasize the role of the logarithmic derivatives i n understanding the p a r t i a l wave phase s h i f t s ; the connection between these con-cepts and the empirical data i s the subject of the present section. A s p h e r i c a l l y symmetric and momentum-dependent p o t e n t i a l , such as that which describes the pion-nucleus o p t i c a l i n t e r -action i n spherical n u c l e i , has the general form, (3-la) VCr) B -^Y^(r) V + V (r) -f V(r) + IW(r) . Here, (3-lb) • * e(r)= •= *°(r) -t-irt^r) 1 _  0 1 ' r i 3 i s the e f f e c t i v e strength of the. momentum-dependent i n t e r a c t i o n , including the term ( 1 - )~^~ a r i s i n g from the Lorenz-• o Lorentz e f f e c t , where <*(r) represents terms i n Eq„ 2-36, n^(r) and n p ( r ) , which are proportional to the density (or square of.the density); V"c(r) i s the e l e c t r o s t a t i c p o t e n t i a l , V(r) i s the r e a l l o c a l p o t e n t i a l , and W(r) i s the absorptive l o c a l p o t e n t i a l . The n o n - r e l a t i v i s t i c Schroedinger equation which describes the pion-nucleus i n t e r a c t i o n then has the form (1 2) - K 2 2 ' ~ V ^)(r) -f o r( r)|(r) s Ej>(r_) ; E i s the r e l a t i v e energy (andjJ' the reduced mass) of the pion and the target nucleus. Eq. 3 - 3 can be reduced to a set of r a d i a l equations by making the p a r t i a l wave expansion, ( 3 - 3 ) Mr) = gkrtZJl + 1 ) i J * t ± Y j f e ) , 4=0 r where Jt i s the o r b i t a l angular momentum. Substituting Eq. 3 - 3 i n E q . 3 - 3 , we deduce the r a d i a l Schroedinger equations, f o r each angular momentum value, i ? , for the r a d i a l pion . waves, U J J (r) (3-4'' - \ (l-fcf(r)lai'fr) '. "L. f^rlu,' (r) 2.V r ^ 2 ^ r + [y c( r)+V(r)+iWfr) u.fr) * Eu,(r) f J5 ^  0, 1, 2, ... ) To begin our c a l c u l a t i o n , we w i l l f i r s t define the bound dary conditions on E q . 3-4Tby assuming the usual experimental s i t u a t i o n of a beam of p a r t i c l e s described by an incident distorted plane wave, <J>Q: ( 3 - 5 ) f c f ^ ) = \ ^hn(ZSl + 1 ) i V ^ ( V ) Y j ( e ) ; P JL = 0 here Pi>(/,?l) i s the regular sp h e r i c a l Coulomb wave function, i • j? <3AC = arg V + 1-f 171) 2 . 6 " c -f X t a n - 1 £ 0 s a l s i s the usual Coulomb phase s h i f t , and jp and 7^ are the con-ventional Coulomb parameters, p = kr; n\ = z Z e 2 n vj 2S where k i s the free space wave-number, k - s T 3 ^ 53 % i s the charge of the target nucleus and z the charge of the pion (1,0, or -1). The technique i s to now solve Eq. 3-4 inside some match-ing radius, RQ, which i s chosen s u f f i c i e n t l y close to the nucleus so as to exclude b a r r i e r properties but s u f f i c i e n t l y far removed so as to e s s e n t i a l l y include a l l aspects of the po t e n t i a l . Subject to Eq. 3-5, we then obtain the familiar continuity conditions at the matching radius for each p a r t i a l wave and i t s derivative i n terms of the (complex) phase s h i f t s (3-6a) MV (3-6b) i ; ( ^ 0 ) where (3-7a) M )^ (3-7b) 0^(/ Q) are the exterior spherical incoming and outgoing Coulomb waves, respectively: G^{P^) i s the i r r e g u l a r spherical Coulomb wave function. Of course, Eqs. 3-6 are t r i v i a l l y soluble for the t o t a l phase s h i f t s , oty, The problem i s to re-express the solution i n a form which e x p l i c i t l y states the optic a l properties of the pion-nucleus i n t e r a c t i o n . To do this we need only note that the p a r t i a l wave phase s h i f t , ^ , i s the sum of two terms (3_8) . o ( j s < r / + ( / / H ^ ) , where <f^ c i s the Coulomb phase s h i f t associated with the long-e 2 i < < e 0 , ( / 0 ) r u ^ ) ; (/0 g kR o' e 8 1 ^ 0 ; ( / n ) = U ' ( 7 ) ; I _ = d d/ e ^ M G ^ ) 1 I F , f/ o)) . e- 1 , rMGj(/ 0) 4 i ^ f ^ ) ) J 54 range Coulomb po t e n t i a l and where (^i+±J^) i s the phase s h i f t which arises from the short-range pion-nucleus poten-t i a l . I t i s the l a t t e r phase s h i f t which contains the op-t i c a l properties of the po t e n t i a l and i t i s this phase s h i f t which we wish to evaluate. To perform this evaluation, i t - i s convenient to f i r s t account for the Coulomb and angular momentum barriers which surround the target nucleus. These barri e r s have essen-t i a l l y two effects which are s p e c i f i c to the short-range i n t e r a c t i o n . F i r s t l y , we must, for each p a r t i a l wave, account for the quantum mechanical penetration of the pion through these barriers which we do by introducing the conventional penetration factors, (3-9a) -p P „ f / ) = ±o Secondly, we must account for the phase of the waves i n going through the b a r r i e r s , which i s taken care of by the # s h i f t functions , ( 3 * 9 l 3 ) s ( / > ) - / o f e ^ W ^ o ) + G J C P Q I G , ( / » 0 ) ] I t i s convenient to note that these quantities are summarized i n the exterior derivative quantity, The term " s h i f t function" i s used for a s l i g h t l y d i f f e r e n t quantity i n the -matrix theory of nuclear reactions (Vogt (1962)), I t d i f f e r s from our s h i f t function mainly by the addition of a boundary condition number. 55 (3-10) V -^ ) ' ~ fo -J~-2-~ - g ty> \ . iv.lp ) * 0 The properties of the short-range potential are en-t i r e l y contained i n the i n t e r i o r logarithmic derivatives, the r e a l part, cfg , contains the resonance properties of the i n t e r a c t i o n and the imaginary part, 7Cg , the absorptive loroperties. The match of the i n t e r i o r and exterior wave functions i s provided by Eqs„ 3-6 which can now be conveniently rewrit-ten i n the form, (3 12) " ' 0gi«feo <0' 0) :_ frlrp) - M A ) ) ' Now, as i s e a s i l y shown from Eqs. 3-7, (3ll3) M T o l !. ' - A I C ' _ 2 I N J E where <5^cis the Coulomb phase s h i f t (Eq. 3-5 f f . ) and < 3" 1 4 ) J l / = t a n - ^ ' i ) ' ' i s the well-known hard-sphere phase s h i f t which accounts for the position of the matching radius with respect to the origin . Defining the p a r t i a l wave transmission functions, (3-15) Ts = 1 - e " 4 ^ , and using Eqs. 3-10, 3-12, and 3-13, we see that (3-l6a) h . „ ! aP»(«5 - Sg) O < 2 . (<5} _ ) 2 + x 2 „ p 2 56 (3-16b) The r e a l part, ^ , of the p a r t i a l wave phase s h i f t s accounts for the r e f l e c t i v i t y and resonance properties of the o p t i c a l the net flux of p a r t i c l e s through the nuclear interface. We w i l l show i n subsequent sections that the r e a l part of the i n t e r i o r logarithmic derivative, <Xe , which contains the resonance aspects of the problem, has rather simple properties: for s-waves i t i s nearly unity and for higher p a r t i a l waves i t depends s e n s i t i v e l y only upon the ef f e c t i v e mass of the pion inside the nucleus. The absorptive term, 7Q , deserves further comment. We show i n Appendix A, Sec. A-1, that 7fy can, for each p a r t i a l wave, be written i n terms of an i n t e g r a l over the wave functions: A\ 0 C l e a r l y , the f i r s t term i s the l o c a l absorption and the second term i s the momentum-dependent absorption. Hence, we can e a s i l y segregate and display those absorptive effects a r i s i n g from the momentum-dependent;'interaction. potent i a l ; the imaginary part, P $ , or, a l t e r n a t i v e l y , the transmission function, T^, accounts for absorption, s t a t i n g 57 3-2 Qualitative Features of Low Energy Pion Optics i n Nuclei In the previous section we have derived a formalism for describing the o p t i c a l properties of pions i n n u c l e i . In the present section we w i l l use th i s formalism to investigate the q u a l i t a t i v e features of the pion-nucleus i n t e r a c t i o n . These features are mainly governed by the long pion wave-length within the nucleus a r i s i n g from the repulsive l o c a l p o t e n t i a l (and the small mass of the pion) and the strong energy dependence of processes a r i s i n g from the momentum-dependent i n t e r a c t i o n when we are near the top of the poten-t i a l b a r r i e r . Because of the long pion wavelength, o p t i c a l resonance properties tend to depend mainly upon the size of the scat-terer and the e f f e c t i v e mass of the pion inside the nucleus rather than upon the detailed structure of the i n t e r a c t i o n i n the nuclear surface. On the other hand, o p t i c a l absorp-tion exhibits considerable structure through the momentum-dependent i n t e r a c t i o n since the pion momentum i n the nucleus varies r a p i d l y near the top of the poten t i a l b a r r i e r . To make these points clearer we w i l l i n this section consider the nucleus to have_ a uniform density, i n which case we can obtain simple analytic formulae for the scat t e r i n g parameters. We will extend our r e s u l t s to diffuse nuclei i n our numerical discussion i n Sec. 3-4. 3Por a uniform nuclear density, of radius, RQ, the poten-t i a l parameters of Eqs. 3-1 have the r a d i a l dependence, 58 (3-18a) (3-18b) (3-18c) c* e(r) V(r) r W(r) = < = ~«Ro 0 , 0 . r > R „ ; -V/ , r ^ R • o ' o * 0 e r ^ R, l o , - -"o r ^ R 0 ; anfl W have been chosen to be p o s i t i v e . Ro' l o » o- p We then have from Eq. 3-4 that the wave function for the pion inside the nucleus i s (3-19) u^(r) = Ajgrj^(br), r < R 0 , where the i n t e r n a l wave number, b, i s given by the r e l a t i o n , (3-20) (E V - V 4 iW ) vc 0^ x 0' (V being the average value of the e l e c t r o s t a t i c potential c within the nucleus). As we show i n Appendix A, Sec, A-2, the logarithmic derivative at R Q i s given by the r e l a t i o n , (3-21) MH0) - ( 1 4 « * ) r f p h<**) r=R, 04(br) Now, near the top of the p o t e n t i a l b a r r i e r , E ^ ( V _ + V ) , ^ 0 so that bR 0 i s small (W0 i s small). Thus, we may make an asymptotic expansion of the spherical Bessel functions: (3-22) 3 ( b R j (an-i):: (bR 0) 4+2 I T o 0 Tn this case, Eq. 3-21 has the approximate form (to order ( b R 0 ) 2 ), (3-23) _ :-[ • ' • /, R sZ * 0 2-*+3 ' or, from Eqs. 3-11, 3-18, and 3-20, (3-24a) '• n i r v ) R 3 ' (a-t + 3) n (3-24b) 7Tj(R0)'= ^ f 0 + 2^W0 ( 2 4 - f - 3 ) h 2 From Eq. 3-24a, i t i s seen that the resonance aspects of the problem, which are contained i n the logarithmic deriv-atives, <5~JI , are i n s e n s i t i v e to the detail s of the. potential ' for s-waves and, for higher p a r t i a l waves, depend only upon the r e a l part of the ef f e c t i v e strength of the momentum-dependent i n t e r a c t i o n , °^^0» inside the nucleus. We therefore do not expect to find i n the case of a diffuse-edge nucleus that these quantities depend very s e n s i t i v e l y upon the diffuseness of the nuclear surface; i n ' f a c t , we w i l l show this to be the case i n Sec. 3-4. I t i s also seen from Eq. 3-24b that s-wave absorption, which i s described by the logarithmic derivative, 7tQ, i s , near the top of the potential b a r r i e r , dominated by the l o c a l i n t e r a c t i o n . This arises since, near the top of the ba r r i e r , the pion momentum i s small. The momentum-dependent processes are therefore strongly energy dependent i n the v i c i n i t y of the b a r r i e r and, as we w i l l show i n Sec. 3-4, the det a i l s of 60 t h i s energy dependence provide a sensitive tool for measuring the diffuseness of the nuclear surface. Since the e f f e c t of increasing the angular momentum i s simply to rai s e the pot e n t i a l h a r r i e r , we expect to find that for absorption involving higher p a r t i a l waves there i s a s i m i l a r structure to that demonstrated here for s-waves, only now at a higher energy. Unfortunately, we cannot so e a s i l y demonstrate this a n a l y t i c a l l y since we can no longer use simple expansions for the spherical Bessel functions. We w i l l therefore s a t i s f y ourselves with demonstrating this numerically i n a subsequent section, Sec. 3-4. To do t h i s we w i l l f i r s t require numerical values for the o p t i c a l p o t e n t i a l . 3-3 Pion-Nucleus Optical Potential In the f i n a l sections of this chapter, we w i l l demon-strate the o p t i c a l properties of the pion-nucleus i n t e r a c t i o n i n a more quantitative way. To this purpose we w i l l require a detailed o p t i c a l potential which we w i l l take to be the Ericsons' po t e n t i a l of Sec. 2-2. This pot e n t i a l would seem I 4 from the XT- mesic x-ray data to be quite good at low energies (Sec. 2-3) and at higher energies (^30 MeV) i t should s t i l l reasonably represent the o p t i c a l i n t e r a c t i o n , at l e a s t i f one allows for some energy dependence i n i t s parameters. In addition, we must include the e l e c t r o s t a t i c p o t e n t i a l a r i s i n g from the nuclear charge d i s t r i b u t i o n and we also mention r e l -a t i v i s t i c e f f e c ts of the pion which become important at'higher energies. 61 To consider the r e l a t i v i s t i c e f f e c t s , we f i r s t note that, since the pion i s s p i n l e s s , i t s o p t i c a l wave function should s a t i s f y a Klein-Gordon equation of the form, (3-25) ! (v. •hS c2y2 -/2C2 + [(E+^c 2) - V 0 ( r ) - I f f : 4 <j>(r) = 0 Here E i s the k i n e t i c energy of the pion, jJ i s i t s (reduced) r e s t mass, V (r) i s the e l e c t r o s t a t i c p o t e n t i a l a r i s i n g from the nuclear charge d i s t r i b u t i o n , andlTfr) i s the momentum-dependent short-range i n t e r a c t i o n of the pion with the nuc-leus discussed i n Chapter 2 and Eq. 3-1. Eq. 3-25 can be rewritten i n the convenient form. (3-26) iii vs 2^ + (E - V o ( r ) - V ( r ) ) i - H E - Vc(r) - i r ( r ) x <j> (r) = 0 From Eq. 3-26 we see that to order 4 4 •• E - Y c ( r ) -'Vffr) 2JJGC ^ 20:MeV ^ 2 8 0 MeV 0 1 the Klein-Gordon equation reduces to the non-relati v i s t i c Sehroedinger equation, (3 ?7) ' -~2 2 " ~ V . <j)(r) 4- ( V c ( r ) 4 V ( r ) ) | ( r ) « E j . f r ) . Hence, neglecting r e l a t i v i s t i c corrections w i l l make parameters of the po t e n t i a l energy-dependent to the order of 10$. Since the parameters are probably somewhat energy dependent i n any event, we have simply neglected these corrections. The e l e c t r o s t a t i c p o t e n t i a l , v" cfr), for a uniform d i s -t r i b u t i o n i s well-known and, as we show i n Appendix B, Sec. B-2 62 i s given to good approximation (even for diffuse-edge nuclei) by the formula, (3-28) V c ( r ) zZe 8 R c zZe 8 L r , r?R, where R c i s the rms radius of the charge d i s t r i b u t i o n , Z i s the nuclear charge, and z i s the charge of the pion. As discussed i n Chapter 2, we w i l l take the short-range pion-nucleus o p t i c a l potenti a l ,*\T(r) , to be the Ericsons' p o t e n t i a l given i n Eq. 2-36 and Eq. 3-1. We w i l l only consider spherical nuclei having zero spin and isospin; i n any event, corrections for these e f f e c t s are small (of order A"*"*") except i n very l i g h t n u c l e i . Eq. 2-36 can then be rewritten i n the form o.f Eq. 3-2: 3 -y |Vc(r) +V(r)+ iW(r)| $>(r) = E^(r) where (3-29b) V(r) = VQj>(r) 2 o - M U o ; e f f B (3-29c) W(r) r -W^ 8(r) , W0 r | | 47T(l +__^ ) I m ( B 0 ) ( ^ ) - 4 (3-29d) <*(r) = -K&g(r) -io<Ioj!>lr) 63 ^ I o z 47T(l.+ ^ L . ) l m(G 0 ) ( JL . ) 0 81% m * ° 6 We have here included the conversion factors from the units IT = C = 1 used i n the Er i c s o n s 1 paper to e.g. s. units. In calculations with Eq. 3-29 we have used the experi-mental values for the parameters given by the Ericsons (1966) (3-30a) C b 0)eff ~ -0.0Z8; (3-30b) Im(B 0) = 0.028; (3-30c) . ( c 0 ) e f f = 0.208; (3-30d) Im(G0) = 0.134. We have chosen the nuclear density i n Eq. 3-89 to have a Saxon-Woods d i s t r i b u t i o n : (3-31) Pn • 1 4- e- a l / 3 where 'a 1 i s the diffuseness parameter, R 0 = 1.05 A '* fm. , and J> i s normalized to the atomic number. With these values for the parameters, we find that t y p i c a l values of the poten t i a l inside a nucleus are, (3-32a) V 0 ^ 15 MeV ; (3-32b) WQ ^ 7 MeV ; (3-32c) <* R q ^ -1.5 ; 4 , (3-32d) . * T ^ '_o 5 Of course, the exact values depend somewhat on the p a r t i c u l a r nucleus through the choice o'"f the density function, y ( r ) . 64 3-4 lamerical Discussion . In the previous sections we have q u a l i t a t i v e l y described the o p t i c a l s c a t t e r i n g and absorption of low energy pions i n n u c l e i . In p a r t i c u l a r , we have noted that resonance proper-t i e s tend to be rather i n s e n s i t i v e to the d e t a i l s of the pot-e n t i a l near the top of the potential b a r r i e r because of the long pion wavelength. On the other hand, we have suggested that opt i c a l absorption exhibits considerable structure near the top- of the potential b a r r i e r through the momentum-depend-ent i n t e r a c t i o n ; i n f a c t , we have suggested that t h i s l a t -ter e f f e c t i s strongly dependent on the details of the nuclear surface and, hence , provides.a sensitive tool for measuring the diffuseness of the nuclear surface. In the present section we w i l l make these concepts more quantitative by employing the pion-nucleus o p t i c a l potential discussed i n Sec. 3-3. We w i l l f i r s t provide a rather detailed analysis of one example the interactions of pions with a l i g h t nucleus, such as Ca 4^ and from the point of view from which an experiment might be analyzed. (This seems worthwhile for descriptive purposes since there i s presently e s s e n t i a l l y no experimental data i n the energy region of i n t e r e s t (^30 MeV).) We w i l l then extend these ideas to the interactions of pions with a heavy nucleus, such as Pb 2^ 8. Prom the experimental point of view, the raw data to be analyzed i s provided by the d i f f e r e n t i a l e l a s t i c s cattering cross sections and the t o t a l absorption cross sections. We have calculated these cross sections for the example of 65 40 pos i t i v e pions on Ca employing the potential of Eqs. 3-S8 ana 3-29 i n Eq. 3-4 to calculate the r a d i a l p a r t i a l wave functions. The p a r t i a l wave phase s h i f t s , <^  + i/fy , are then calculated from Eqs. 3-6 fat an asymptotic matching radius) and the d i f f e r e n t i a l e l a s t i c s cattering cross section and absorption cross section are calculated, respectively, from the formulae: f3-33a) d_6elf0) d-n. 2 6 71 -iTllogisin 2) 2 k s i n * f 6 1 V / . ,\ i f f a ^ K i ^ ) ' _ , 1 (2*+l)e s i n ( « 5 + i ^ e ) ^ k i=0 x P^fcose) 2 €*£> (s-33b) <5abs = f a m ) ^ . Of course, i n practice the experimental phase s h i f t s are to be deduced by analyzing the empirical cross section data with Eqs. 3-33. Since one of our aims i s to display the dependence of opt i c a l cross sections on the diffuseness of the nuclear surface, we have performed our calculations for both a sharp-edge fdiffuseness parameter, a=0.1 fm.) nucleus and for a "more conventional fa=0.5 fm. ) diffuse-edge nucleus, A plot of these two densities for a Saxon-V/oods density d i s t r i b u t i o n fEq, 3-31') with the above surface parameters, and a radius appropriate to C a 4 0 fR 0= 1.05 A 1 ^ 3 fm. = 3.59 fm.) i s given i n Pig. 1. V/e have i n this*'figure also plotted the squares of the densities upon which the o p t i c a l absorption depends. 67 Substituting these densities i n the potential of Eq. 3-29, we have calculated the d i f f e r e n t i a l e l a s t i c scattering cross section and .total absorption cross section for a var-i e t y of energies. The res u l t s (for the two densities of Pig. 1 ) are presented i n Fig . 2a ( d i f f e r e n t i a l e l a s t i c scat-t e r i n g cross section) and F i g . 2b (t o t a l absorption cross section) „ Tt i s seen from F i g . 2a that the d i f f e r e n t i a l e l a s t i c s c a t t e r i n g cross sections possess considerable structure and are quite s e n s i t i v e to the diffuseness of the nuclear surface. The strong destructive interference at higher energies between the Rutherford scattering and the s c a t t e r i n g due to the strong part of the pion-nucleus i n t e r a c t i o n (middle and lower figures) arises from the interference between the p-wave (and higher p a r t i a l wave) scattering which i s dominated by the a t t r a c t i v e momentum-dependent potential (c,f. Eq, 3-24) with the sca t t e r i n g due to the Coulomb po t e n t i a l - — which i s repulsive. At lower energies the interference i s constructive (top figure) since, l i k e the Coulomb i n t e r a c t i o n , the s-wave'interaction i s repulsive (c.f. Sq. 3-24). These properties of the strong pion-nucleus i n t e r a c t i o n are demonstrated more e x p l i c i t l y i n Figs. 3 where we have plotted the r e a l p a r t i a l wave phase s h i f t s , • the s-wave phase s h i f t s are seen to be negative (corresponding to a repulsive Interaction) and the p-wave phase s h i f t s (and d-wave phase s h i f t s ) are seen to be positive (corresponding tp an at t r a c t i v e i n t e r a c t i o n ) . I t i s seen that at large 6 8 2 0 10 0 Fl G. 2 a E L A S T I C S C A T T E R I N G CROSS SECT IONS 7T* + Ca TOTAL R U T H E R F O R D E " 15-MeV a = 0.5 Fm 180 , ^ 4 0 -oho 0 400 2 0 0 °0° JL=* J L 30° 60° 90° e E = 2 3 M * V 0.1 F E = 31 MeV 0 . 1 f m 120° 150° I8CP 69 6 0 0 5 0 0 •N.400 - a b 3 0 0 200 100 0 FIG. 2b T O T A L ABSORPTION CROSS SECTIONS r 4 0 s s — s -/ y / _ y / s / y / y / y / s / — s / / / / / / ' / 1 1 a = 0.5 fm — ~ a. = 0 . | F m 1 / 1/ • 1 i i l i 0 10 E 20 MeV 25 3 0 7 2 FIG, 4 R E A L LOGARITHMIC DERIVAT IVES * + Ca 40 EQ 3 - 2 4 a 0 E 5 2 0 2 5 3 0 M eV 74 75 FIG. 6 . RATIOS OF I MAG I N A R Y P H A S E SH IFTS o Ol 0 0 10 15 E 2 0 MeV 2 5 3 0 76 angles there i s also (as expected) interference between the s-wave s c a t t e r i n g and the scatterings of the higher par-t i a l waves. The dependence of the e l a s t i c s c a t t e r i n g on the d i f -fuseness of the nuclear surface i n F i g , 2a does not depend on the resonance properties of the p o t e n t i a l but rather i t depends on the difference i n the d i f f r a c t i o n associated with the difference i n the absorption between the diffuse-edge and sharp-edge choices for the nuclear density. This i s shown i n F i g . 3a, where we see that the r e a l phase s h i f t s associated with the two potentials are nearly the same, and i n F i g . 5, where we see that the corresponding imaginary phase s h i f t s are quite d i f f e r e n t . The strong dependence of absorption on the diffuseness parameter i s also demonstrated i n F i g . Eb, where we have plotted the t o t a l absorption cross sections. We w i l l now address ourselves to the problem of obtaining the empirical parameters of the pion-nucleus p o t e n t i a l , Eq. 3-29, from the phase s h i f t s which describe the e l a s t i c s c a t t e r i n g and absorption cross sections.. Let us f i r s t . enumerate the e s s e n t i a l s i x parameters of the potential assuming the Saxon-Woods density d i s t r i b u t i o n , Eq. 3-31: (1) The r e a l l o c a l i n t e r a c t i o n , V 0; (2) The absorptive l o c a l i n t e r a c t i o n , WQ; (3) The r e a l momentum-dependent strength, °^Ro* ^ '^ie absorptive momentum-dependent strength, » . (5) The nuclear radius parameter, RQ; (6) The nuclear surface parameter, a. Before proceeding, i t w i l l be convenient to r e c a l l Eq, 3-i n which we stated the properties of the i n t e r i o r logarithmic derivatives near the top of the l o c a l potential "barrier for a uniform density d i s t r i b u t i o n : (3-24a) c^(R 0) '= 1 + L * § 0 ) - V Q- V Q)R 2 ^ . (2-0 + 3)ft 2 (3-241)') X j f H j L j f * ? u 2 > o R Q ( 2 ^ + 3 ) ^ In Eqs.. 3-24, the e f f e c t i v e strengths of the momentum-depend-ent i n t e r a c t i o n inside the nucleus, oZgQ and <*L® d i f f e r from the strengths c < ^ 0 and < -^j-0 (which are proportional to the density) through the Lorenz-Lorentz e f f e c t . I t i s the e f f e c t i v e strengths which are important i n determining the scattering, as can he seen from Eq. 3-24, Of course, the e f f e c t i v e strengths vary i n the surface through the Lorenz-Lorentz e f f e c t (Eq. 3-lb) but, to f i r s t order, the o p t i c a l properties are determined by their values inside the nucleus where the density i s constant (as can be seen from Eq. 3-24 and as we s h a l l presently show). That i s , the o p t i c a l prop-e r t i e s are sensitive to the t o t a l change i n the e f f e c t i v e mass of the pion i n entering the nucleus which involves an i n t e g r a l over the. nuclear surface. i t * For s-waves (4=0) i t i s seen from Eq. 3-24a that <?Q i s almost unity and, to f i r s t order, depends only on V Q and'R (V i s fixed bv the nuclear charge d i s t r i b u t i o n ) . ^ Vv'e c . if 'IMs r e s u l t i s to be contrasted with the behaviour of logar-ithmic derivatives for.interactions involving strongly a t t r a c t i v e potentials (such as the nucleon-nucleus i n t e r -• action) where the i n t e r i o r logarithmic derivatives are rapi d l y varying functions of the energy. Tn fact, i t i s this property of the derivatives which gives r i s e to the strong resonance properties c h a r a c t e r i s t i c of such potentials 78 have plotted <T0 for "both the diffuse-edge and sharp-edge densities of Fig. 1 i n F i g . 4 (upper figur e ) . I t i s seen from F i g . 4 that CT 0 i s rather i n s e n s i t i v e to the d i f f u s e -ness of the nuclear surface; i n fact, i n both cases the logarithmic derivative i s seen to he i n close agreement with the approximate value calculated from Eq. 3-24a (dashed). How the r e a l part of the s-wave phase s h i f t , <P0. depends mainly on C5~0 (and the h a r r i e r properties) since, i n general, i s small (c.f. Eq. 3-l6a). We therefore expect that fQt l i k e <5"0, i s rather i n s e n s i t i v e to the diffuseness of the nuclear edge; t h i s i s shown for our present example i n F i g . 3a (upper fi g u r e ) . On the other hand we expect J*0 to depend s e n s i t i v e l y on V (which determines the height of the poten-t i a l harrier) and R 0 (which determines the penetration); th i s i s demonstrated i n Figs. 3b and. 3c (upper f i g u r e s ) , respectively. Hence, <FQ depends s e n s i t i v e l y only on VQ and RQ. For higher p a r t i a l waves, we see from Eq. 3-24a that depends s e n s i t i v e l y only <x'j| (through the i n t e r i o r logarithmic derivative, cTji) and R Q (through-barrier penetra-t i o n ) . The i n s e n s i t i v i t y of JJI to diffuseness parameter, a, and the r e a l p o t e n t i a l , V , i s demonstrated i n Figs. 3a and 3b (lower figures); the s e n s i t i v i t y to R 0 i s demonstrated i n F i g . 3c (lower f i g u r e s ) . We have demonstrated the s e n s i t i v i t y of <5~2_ (and, hence, /]_) to c * ^ i n Fig. 4 (lower figure); i t i s seen that <5^  i s i n close agreement with the approximate value (dashed) calculated from Eq. 3-24a for both the sharp-7 9 edge and diffuse-edge potentials. I t i s seen from the above discussion that an analysis of the r e a l parts of the phase s h i f t s , , yields the para-meters V - . , R , and cxcfh of the pion nucleus potential o» o* Ro r e l a t i v e l y independent of the other parameters, i n p a r t i c u l a r , independent of the diffuseness parameter, a, Hear the top of the potential b a r r i e r (V + V ^ 2 6 MeV); momentum-dependent absorption processes are suppressed. Hence, near the top of the b a r r i e r , the imaginary logarithmic derivative, 7T0 and, therefore, the imaginary phase s h i f t s , J*0, (Eq. 3-16), are sensitive only to the l o c a l absorption parameter, W0. This i s demonstrated i n Fig. 6 (upper figure) where wehave plotted the ratios of the imaginary phase s h i f t s appropriate to the sharp-edge and diffuse-edge densities of Fi g , 1; i t i s seen that, near the top of the poten t i a l barrier (^26 MeV) , ' i f . ^ l . Hence, 1SL can be obtained from the absorption cross sections near the top of the b a r r i e r (which i s known from analysis of the r e a l parts of the phase s h i f t s , I t i s therefore possible to obtain values for the para-meters V Q, R Q , p^fj0» a n < 3 ^ 0 r e l a t i v e l y independent of the nature of the nuclear surface. Now, as we s h a l l shortly discuss i n more d e t a i l and as can be seen from F i g , 5, the energy dependence of the imaginary parts of the phase s h i f t s , Pj( , i s a strong 'function of the diffuseness parameter, a, through the momentum-dependent absorption. Hence, using the values which we have previously obtained for the other para-80 meters o f the i n t e r a c t i o n , we can a n a l y z e the phase s h i f t s , J ^ , f o r the d i f f u s e n e s s parameter, a, and the a b s o r p t i v e term i n the momentum-dependent i n t e r a c t i o n , ^ f 0 . Of c o u r s e , by f o l l o w i n g t h i s p r o c e d u r e w h i c h we have o u t l i n e d o n l y v e r y approximate v a l u e s a re o b t a i n e d f o r the parameters o f the p o t e n t i a l . Of c o u r s e , b e t t e r v a l u e s can be found by r e p e a t i n g the a n a l y s i s w i t h the v a l u e s found from the f i r s t attempt. I n any e v e n t , a t l e a s t a p r e l i m i n a r y use o f these p r o c e d u r e s would seem to be b o t h more p h y s i c a l l y i n s t r u c t i v e and more e f f i c i e n t t h e n s i m p l y a t t e m p t i n g to s i m u l t a n e o u s l y vary the s e v e r a l ( s i x ) p a r a -meters which e n t e r the i n t e r a c t i o n t o o b t a i n a b e s t f i t t o e x p e r i m e n t a l c r o s s s e c t i o n s (when such c r o s s s e c t i o n s become a v a i l a b l e , ) , , - As we have noted i n P i g s , 5 and 6 , the o p t i c a l absorp-t i o n d e s c r i b e d by the i m a g i n a r y phase s h i f t s , ^ , s h o u l d depend s t r o n g l y on the t h i c k n e s s o f the n u c l e a r s u r f a c e . T h i s r e s u l t s s i n c e , near the t o p o f the p o t e n t i a l b a r r i e r , the momentum-dependent a b s o r p t i o n i s s t r o n g l y s u p p r e s s e d . Now, because a d i f f u s e - e d g e n u c l e u s c o n t a i n s a g r e a t e r amount o f m a t t e r a t lower d e n s i t i e s t h a n a square-edge n u c l e u s , the b a r r i e r appears lower f o r the former than the l a t t e r . a n d , ; i n f a c t , the s t r u c t u r e of the i m a g i n a r y phase s h i f t s ( o r imag-i n a r y i n t e r i o r l o g a r i t h m i c d e r i v a t i v e s ) c l e a r l y m i r r o r s the d i f f u s e n e s s o f the n u c l e a r s u r f a c e . We have a l r e a d y seen t h i s e f f e c t i n P i g , 5 where we have p l o t t e d the i m a g i n a r y phase s h i f t s , f o r two p o s s i b l e c h o i c e s f o r the n u c l e a r s u r f a c e t h i c k n e s s g i v e n i n P i g , 1„ We now e x h i b i t the 81 82 • \ i s p e c i f i c e f f e c t of the momentum-dependence more c l e a r l y "by making use of Eq„ 3-17 which allows us to segregate the effects of the l o c a l and momentum-dependent absorption: ( 3 " 1 7 ' 7t,(R ) '. - ^ 2 . o f WW h(")\8 dr h & • l m f H 0 H E , 0 (Local) ( M o m e n t u m -d e p e n d e n t ) I n F i g . 7 we have used Eq_. 3-17 t o p l o t the l o c a l (solid) and momentum-dependent (dashed) absorption for the two den-s i t i e s o f F i g . 1 ( a = 0.5 f m . , left-hand side; a = 0.1 f m . , right-hand s i d e ) . For reference purposes, we have included t h e t o t a l l o c a l potential b a r r i e r , V c (r) •+ V 0 (r) , and the incident energy of the pion (marked T E T ) . I n Pig. 7a and 7b, we have compared s-wave absorption a t an energy below the ba r r i e r for the diffuse-edge and sharp-edge nuclear densities of F i g . 1, I t i s seen that, for the diffuse-edge density, most pions are absorbed at nearly zero momentum so that momentum-dependent absorption i s strongly ' • • .2 suppressed (the momentum i s roughly proportional to JE-Vc (r)-V(r |)J o n the other hand, for the sharp-edge density, pions are being absorbed inside the ba r r i e r at higher momentum so that momentum-dependent absorption i s considerably less suppressed. We see from Fig, 7c and 7d that, by,similar arguments, momentum-dependent absorption i s suppressed for both choices of the 83 density near the top of the ha r r i e r . In Pig. 7e and 7f we see that, for energies above the b a r r i e r , momentum-dependent absorption i s more suppressed for the sharp-edge density (where absorption takes place inside the barrier) than for the diffuse-edge density (where, because there i s more sur-face, absorption takes place at lower density and higher momentum). Por higher p a r t i a l waves, this simple analysis becomes somewhat more complicated through the presence of angular momentum b a r r i e r s . F i r s t l y , i t i s seen from Eq. 3-17 that, f o r ^ ^ 0, there i s a contribution to the absorption from the angular terms which enter the momentum-dependent i n t e r -action as well as from the r a d i a l terms; l i k e the l o c a l absorption, these terms are i n s e n s i t i v e to the diffuseness of the nuclear surface. In addition, the r a d i a l terms which enter the momentum-dependent i n t e r a c t i o n for the higher p a r t i a l waves see a higher p o t e n t i a l b a r r i e r through the appearance of the c e n t r i f u g a l potential; we therefore expect the contribution to the momentum-dependent absorption from the r a d i a l terms to exhibit a sim i l a r dependence on the potent i a l b a r r i e r to that shown i n Pig. 7 for s-waves, only now at higher energies. In fact, this i s seen to be the case from Pig. 5 . We w i l l discuss these effects more exp l i c -i t l y i n Sec. 4-3 where si m i l a r e f f e c t s are found i n the cross sections which describe the. excl tation of ro t a t i o n a l l e v e l s i n deformed n u c l e i . Now, as we have previously discussed, we know the height of the p o t e n t i a l b a r r i e r inside of the nucleus from 84 FIG. 8 S - W A V E IMAGINARY P H A S E SHIFTS F I G . 9 S - WAVE IMAGINARY ' P H A S E SHIFTS 0.15 A 0.1 0 0.05 0.15H O.IOh 0.0 5 h jC + P l o 87 80 our analysis of the r e a l part of the phase s h i f t s , Jj;, which describe the resonance aspects of the problem. On the other hand- we see here .that the imaginary part of the phase s h i f t s , ^ , are determined by the behaviour of the p o t e n t i a l b a r r i e r (and, hence, the nuclear density) i n the region of the nuclear surface. It.should therefore-be pos-s i b l e from o p t i c a l scattering and absorption experiments with pions to very accurately determine the diffuseness of the. nuclear surface. In the preceding discussion, we have noted that the main ef f e c t of the electromagnetic i n t e r a c t i o n upon our analysis was i n adjusting the e f f e c t i v e height of the potential bar-r i e r . In the case of C a 4 ( \ the e l e c t r o s t a t i c potential has . an average value inside the nucleus of about 10 MeV. We therefore would expect i f we were to use negative pions rather than positive pions that the only important e f f e c t on our analysis would be that the b a r r i e r would appear 20 MeV lower. This e f f e c t i s shown i n F i g . 8, where we have compared the s-wave imaginary phase s h i f t s , J^0, for the two reactions 7T + + Ca 4 0.and TT*" +• C a 4 0 employing both the' sharp-edge (dashed) and diffuse-edge (solid) nuclear densities of our previous calculations (Fig. 1). In a si m i l a r manner, we expect that i f we go to heavier n u c l e i , the only major changes i n our analysis w i l l be due to the large e l e c t r o s t a t i c barriers (as seen from Eq. 3-29, the short-range pion-nucleus i n t e r a c t i o n i s insen s i t i v e to atomic number). We have demonstrated this e f f e c t i n Fig. 9 89 where we compare the s-wave imaginary phase s h i f t s , ^ , for the reactions TT1" t C a 4 0 and TT* •+• P b 2 ° 8 . Both calculations are for Saxon-Woods densities (Eq. 5-31) with two choices for the diffuseness parameter, a = 0.1 fm. (dashed) and a - 0.5 fm. ( s o l i d ) . It i s seen that, apart from the difference i n penetration, the only e f f e c t of going to a heavier nucleus i s to raise the e f f e c t i v e repul-sive b a r r i e r through the difference i n the e l e c t r o s t a t i c potentials. In fact, in. heavy nuclei the e l e c t r o s t a t i c potentials are so large that, for negative pions, i t i s possible that the net potential might be a t t r a c t i v e . This e f f e c t i s demon-strated i n F i g . 10 where we compare the reactions 17"*"+ P b 2 ^ 8 — p o ft and Tf + Pb . The average e l e c t r o s t a t i c p o t e n t i a l inside the nucleus i s s i m i l a r to 31 Mev, whereas.our choice for the short-range potential b a r r i e r i s s i m i l a r to 16 MeV. For the net depth of the p o t e n t i a l suggested by these estimates (roughly, - 5MeV), i t would seem that the pion mass i s too l i g h t for there to be strong resonance effects at low energies. Of course, as we have discussed i n Chapter 3,.the parameters of the pion-nucleus p o t e n t i a l , i n p a r t i c u l a r , the height of the p o t e n t i a l b a r r i e r , are at present not very well deter-mined. Another e f f e c t of the e l e c t r o s t a t i c potential i n the case of negative pions i s that i t s u f f i c i e n t l y reduces the angular momentum barrie r s so .that many p a r t i a l waves enter the problem, V/e have demonstrated this e f f e c t i n Pig, 11. 90 for the reaction TJ P b 2 0 8 „ flower figure). The untang-l i n g of the p a r t i a l waves i s then a formidable problem. Of course, i n the case of positive pions, the converse e f f e c t applies since the e f f e c t of the e l e c t r o s t a t i c pot-e n t i a l i s to increase the penetration factors (Pig. 11, upper figure) Prom the analysis presented i n t h i s chapter, we believe that we have c l e a r l y demonstrated that the nuclear surface thickness plays an important r o l e i n determining the absor-ption rates of pions i n nuclei, Tn fact, we would suggest that, on the basis of these r e s u l t s , i t i s of great i n t e r e s t to measure low energy pion o p t i c a l data, p a r t i c u l a r l y pion absorption, as a means to measuring the density of nucleons i n the nuclear surface. Accurate measurements of this nature should f i n a l l y resolve the considerable controversy over the surface d i s t r i b u t i o n of neutrons i n heavy nuclei which we have discussed i n the introduction to this chapter. 3-5 Charge Exchange and other Exotic Terms i n the Ericsons'  Potential' " In the present section we w i l l b r i e f l y mention some i terms of order A" and higher which enter the'Ericsons' p o t e n t i a l , Eq;. 2-36, and which we have u n t i l now omitted from our-discussion. These terms consist of i s o s p i n terms, which macroscopically account for single and double charge exchange between nuclear analog states, and terms a r i s i n g from the spin of the nucleus, which lead to hyper fine s p l i t -t i n g i n the TT-mesic x-ray spectrum. Hence, while these terms make small contributions to the pion-nucleus o p t i c a l 91 i n t e r a c t i o n , they lead, to q u a l i t a t i v e l y d i f f e r e n t sorts of ef f e c t s which are both i n t e r e s t i n g and e a s i l y measured. The problem of charge exchange has long been of i n t e r -est since the pion, unlike most other familiar elementary p a r t i c l e s , has three charge states ~_- jfi jt* 71". Hence, i n addition to single charge exchange, there are the i n t e r -esting processes of double charge exchange by which, for example, one can reach nuclear isobars of greater (or lesser) charge than have heretofore been available. The f i r s t approaches to charge exchange were microscopic i n nature i n which, so to speak,, one traces the path of the pion through the nucleus and attempts to account for the p a r t i c u l a r nucleon where charge exchange occurred. For these purposes one requires a microscopic description of the nucleus which i s usually taken to be that given by the s h e l l model. The pion-nucleon scattering operators are often taken to be the free space scattering operators (the impulse approx-imation) and the propagation of the pion within the nucleus i s often treated by making some average of the intermediate e x c i t a t i o n energies so that one can reduce the many-particle Green's function to a one-body Green's function for the pion (the "closure approximation"; c.f. Eq, 2-Sla). Calcu-latio n s of this type have been attempted;by several authors; a summary of the current status o f both single and double charge exchange reactions, along with a review of the e a r l i e r calculations and experiments, can be found i n a recent review a r t i c l e by Koltun (1969) A second manner i n which one can account for charge, exchange i s through an o p t i c a l p o t e n t i a l , such as the E r i c -sons' o p t i c a l potential of Eq„. 3-36. Such an o p t i c a l descrip-t i o n states that charge exchange can only occur through those nucleons which contribute to the macroscopic i s o s p i n of the nucleus, that i s , between i s o t o p i c analog states; charge exchange which takes place i n this manner i s often referred to as quasi-elastic scattering since only the charge 'of the nucleus i s changed i n the scattering process. As an example of quasi-elastic scattering, l e t us con-sider the A = 14 iso s p i n t r i p l e t , ( C 1 4 , I 1 4 , 0 1 4 ) . A l l these nuclei have closed s-,/r> orbits and p , orbits; the 3/2 remaining two nucleons are i n p ^ g orbits and are coupled to spin, J = 0 , and i s o s p i n , T ~ 1, with M T - -1 for C 1 4 , M T = 0 for N 1 4 , and % = 1 for 0 1 4. Quasi-elastic scat-ter i n g can only account for changes i n 1A<% and, hence, only involves the VxfZ n u c l e ° n s ; o n "the other hand, we might well expect charge exchange to occur on the p _ / o nucleons or 3/ « even the s-^g nucleons (of course, charge exchange i s more l i k e l y i n open then closed s h e l l s from energetic considera-ti o n s ) . In fa c t , from considerations only of quasi-elastic s c a t t e r i n g , we would conclude that charge exchange i s for-bidden i n nuclear states of is o s p i n , T = 0, such as the 13 ground state of C „ An o p t i c a l description of charge ex-change, such as that given by the Ericsons, i s therefore only of value i n considering quasi-elastic scattering bet-ween nuclear analog states. 93 As can be seen from Eq. 2-36, the Eric s o n s 1 potential accounts for optic a l charge exchange processes a r i s i n g i n a variety of ways. F i r s t l y , there are single charge exchange 4 processes a r i s i n g from "both the s-wave (terms i n b-j_) and p-wave (terras, i n c^) interactions (including a charge ex-4 change terra through the Lorenz '-Lorentz effect). Double charge exchange can occur either through repeated application of these vector couplings or d i r e c t l y through the tensor coupling (terms i n Bg and Cg) which arise from the two-nucleon processes. Calculations with charge exchange i n t e r -actions being described by an o p t i c a l p o t e n t i a l , such as the Ericsons', are found to be i n moderate agreement with experiment (Koltun (1969)). In addition to terms accounting for i s o s p i n , the E r i c -sons' potential also takes into account the coupling between the angular momentum of the incident pion and the nuclear spin (terms i n d Q) the hyperfine i n t e r a c t i o n . At present pion i n t e n s i t i e s are too low to observe hyper fine s p l i t t i n g i n the TT-mesic x-ray spectra, but i n t e r e s t i n g information about these terms should be obtained once intense pion beams (such as those provided by the proposed Triumf f a c i l i t y ) bee ome av ai1able„ 94 CHAPTER 4- ' THE EXCITATION 0? ROTATIONAL LEVELS IN DEFORMED NUCLEI BY PIONS In the l a s t chapter'we have examined the o p t i c a l proper-t i e s of pions i n nuclei and, i n p a r t i c u l a r , we have shown that the o p t i c a l absorption of pions i n nuclei depends strong-l y upon the d e t a i l s of the nuclear surface. In the present chapter we w i l l discuss a second way i n which the pion o p t i c a l f i e l d may be used to examine nuclear surface features, namely, by the e x c i t a t i o n of r o t a t i o n a l states i n strongly deformed n u c l e i . Conventional techniques for studying deformed nuclei the e x c i t a t i o n of r o t a t i o n a l levels by electromagnetic or nucleon f i e l d s ; the electromagnetic decay of these excited l e v e l s e s s e n t i a l l y involve i n t e g r a l s over the nucleus which are sensitive only to the average nuclear deformation and which are i n s e n s i t i v e to the diffuseness of the nuclear edge. From such experiments on deformed n u c l e i , we learn detailed information about the quadropole "and higher multi-pole deformations of the nuclear density, but we learn l i t t l e about the details of i t s shape, such as the nuclear surface thickness. Our purpose i n this chapter i s to show that the e x c i t a t i o n of r o t a t i o n a l l e v e l s i n deformed nuclei by pions --- which we w i l l henceforward (and somewhat loosely) c a l l "pion e x c i t a t i o n " ( i n analogy with Coulomb excitation) does i n t r i n s i c l y involve the surface features of the nuclear density because of the momentum-dependent part of the pion-nucleus i n t e r a c t i o n . Our technique for demonstrating this '95 .] ' Is to segregate the Coulomb and l o c a l excitation processes i n "pion excitation'* from the momentum-dependent ex c i t a t i o n processes by. employing the Distorted 17ave Born Approximati on. The information obtained from pion e x c i t a t i o n , when ! combined with the analysis of o p t i c a l absorption experiments fas discussed i n Chapter 3 ), should allow a comprehensive study of the nuclear surface i n deformed n u c l e i . Since the charge d i s t r i b u t i o n i n deformed nuclei i s becoming increas-i n g l y well-known from jJ-mesic x-ray experiments, t h i s should y i e l d quantitative information about the surface d i s t r i b u -tions of protons and neutrons i n such nuclei including, possibly, the angular dependence of these d i s t r i b u t i o n s . This information should be of considerable value i n r e f i n i n g the microscopic descriptions of deformed nuclei and i n com-paring the extent to which these descriptions correspond to the microscopic descriptions of spherical n u c l e i . Our picture of a strongly deformed nucleus i s that of the strpng-.couplingor r o t a t i o n a l model i n which the nucleus i s viewed as an falmost) spheroidal body rota t i n g i n space f c . f . Sec. 4-1). The e s s e n t i a l idea i s that when the nucleus i s immersed i n - a pot e n t i a l f i e l d the speed of rotation i n -creases i . e . we excite a r o t a t i o n a l degree of freedom of the nucleus; i n general, as we w i l l discuss i n Sec. 4-1, we may neglect excitations of other (vibrational and i n t r i n s i c ) degrees of freedom, so that we can d i r e c t l y connect the cross sections for e x c i t i n g r o t a t i o n a l states to such macroscopic aspects of the nucleus as the nuclear density. 96 For instance, the e x c i t a t i o n of r o t a t i o n a l l e v e l s due to the e l e c t r i c f i e l d "between the p r o j e c t i l e and the deformed target nucleus i s the well-known phenomena of Coulomb e x c i t -ation (Alder and Winther (1966) ), From Coulomb e x c i t a t i o n of r o t a t i o n a l levels we can i n f e r the s t a t i c multipole moments of the deformed nucleus and, hence, the average deformation of the nuclear charge d i s t r i b u t i o n (which we expect to be s i m i l a r to the average deformation of the mass d i s t r i b u t i o n ) ; i n fact", recent experiments on i n e l a s t i c alpha p a r t i c l e scat-t e r i n g have allowed the evaluation of even, the higher multi-pole moments with considerable p r e c i s i o n (Bernstein (1969)). The e x c i t a t i o n of r o t a t i o n a l l e v e l s due to a l o c a l strong-i n t e r a c t i o n has also been investigated by bombarding deformed nuclei with neutrons (Chase, Wilets, and Edmonds (1958)), However, the excitations of r o t a t i o n a l levels with either the Coulomb or neturon f i e l d s e s s e n t i a l l y involve i n t e g r a l s over the nucleus and, hence, are rather i n s e n s i t i v e to the d e t a i l s of the nuclear surface. In the case of the nuclear charge density this i s not too disappointing since the surface d e t a i l s can be obtained moderately well from other sources, such as the p-mesic x-ray experiments (Devons and Duerdoth (1969); Wu (1967)), The d e t a i l s of the nuclear mass density, however, are not e a s i l y obtained from current experiments; i n f a c t , as we discussed i n Chapter 3, the surface d i s t r i b u t i o n of neutrons i s a controversial issue even i n spherical nucleic The purpose'of this chapter i s to demonstrate that the surface 97 features of the mass density (and, hence of the neutron di s t r i b u t i o n ) are susceptible to in v e s t i g a t i o n by pion e x c i t -ation (as well as by o p t i c a l absorption) because of the mo-mentum dependent pion-nucleus i n t e r a c t i o n . Our basic idea i s the same as i n our study of pion o p t i c a l absorption i n Chapter 3: near the top of the poten-t i a l b a r r i e r a r i s i n g from the l o c a l pion-nucleus i n t e r a c t i o n the momentum-dependent processes are strongly suppressed; i n fact, t h e i r v a r i a t i o n with energy i n t h i s region should strongly r e f l e c t the details of the nuclear surface because of the va r i a t i o n of momentum across the nuclear surface. Of course, the "pion e x c i t a t i o n " cross sections also include Coulomb e x c i t a t i o n , due to the pion charge, and l o c a l e x c i t a t i o n , due to the l o c a l pion-nucleus i n t e r a c t i o n . A graphic technique for segregating these processes from the in t e r e s t i n g momentum-dependent processes i s provided by em-ploying the Distorted Wave Born Approximation. As we discuss i n Sec. 4-1, we expect the DWBA to be reasonably good since the e x c i t a t i o n cross sections are much less than the geomet-r i c cross section of the nucleus. In DWBA the amplitude for e x c i t a t i o n i s proportional to int e g r a l s over the pion o p t i c a l p o t e n t i a l and the pion o p t i c a l wave functions. The t o t a l amplitude i s then e s s e n t i a l l y a sum of three terms a r i s i n g from the three terms i n the o p t i c a l p o t e n t i a l : a Coulomb ex c i t a t i o n term, a r i s i n g from the electo s t a t i c potential; a l o c a l e x c i t a t i o n term, a r i s i n g from the l o c a l pion-nucleus i n t e r a c t i o n ; a momentum-dependent term, 98 a r i s i n g from the momentum-dependent -.pion-nucleus i n t e r a c t i o n . We can therefore c l e a r l y demonstrate the e f f e c t s of the momen turn-dependent i n t e r a c t i o n by examination of the corresponding DWBA amplitude.. Tn Sec.. 4-1 we review the r o t a t i o n a l model of strongly deformed nuclei and we discuss the'excitation of r o t a t i o n a l states i n such nuclei by pions. In Sec.. 4-2 we obtain the cross sections for these e x c i t a t i o n processes i n DWBA, In Sec. 4-.3 we employ the Ericsons' potential of Chapter 2 and Sec. 3-3 to evaluate these formulae for a r e a l i s t i c pion-nucleus i n t e r a c t i o n and, hence, to demonstrate the strong dependence of the pion e x c i t a t i o n cross sections on the nuclear surface parameters. '4-1 A Review of the Rotational Model and a Discussion of  Pion E x c i t a t i o n In the present section we f i r s t b r i e f l y review the rota-t i o n a l model of strongly deformed n u c l e i . We then discuss th exc i t a t i o n of r o t a t i o n a l states i n such nuclei by pions and we suggest that these processes can be well described by the techniques of DWBA. To understand the strong-coupling or rota t i o n a l model, i t i s worthwhile to review our basic knowledge of the i n t e r -actions of a constituent nucleon i n i t s . nuclear environment. Prom studies of nuclear matter we know that, because of the Pauli exclusion p r i n c i p l e , a nucleon i n nuclear matter moves i n the average potential of the surrounding nucleons and i s rather i n s e n s i t i v e to the d e t a i l s of their positions and motions. Hence, i n a f i n i t e nucleus we expect each nucleon to move e s s e n t i a l l y i n a s i n g l e - p a r t i c l e o r b i t . On the other hand, the average pote n t i a l i n which the nucleons move i s determined by the nuclear density which i s simply the sum over the nucleus of the nucleon densities obtained from the s i n g l e - p a r t i c l e orbits. Hence, the problem of fin d i n g the average potential and the problem of finding the si n g l e - p a r t i c l e orbits are coupled. The way out of this apparent "chicken or egg" conundrum i s provided by the var-i a t i o n a l principle- which says that the resultant nuclear wave function must represent a state of minimum energy. In a nucleus with many nucleons i n an u n f i l l e d s h e l l , i t turns out that because of the i n t e r a c t i o n between the extra-core nucleons, which tend to polarize the nuclear core, and because the nuclear core which, l i k e a closed s h e l l nucleus, prefers to be sp h e r i c a l , the equilibrium shape of the nucleus i s strongly deformed. Such a deformed nucleus exhibits c o l -l e c t i v e modes of o s c i l l a t i o n , rotations and vibrations, i n addition to the s i n g l e - p a r t i c l e excitations a r i s i n g from the basic s h e l l structure of the nucleus. The s i n g l e - p a r t i c l e aspects and the rota t i o n a l aspects of the problem are strongly coupled since a rot a t i o n of the nucleus corresponds to a rota t i o n of the average potential which determines the s i n g l e - p a r t i c l e o r b i t s . The problem i s considerably s i m p l i f i e d by noting that the nucleons o r b i t s u f f i c i e n t l y quickly compared- to the r o t a t i o n a l motion that they respond a d l a b a t i c a l l y to the changing average potential. 100 j Thus the t o t a l wave function of the nucleus may be treated a d i a b a t i c a l l y and taken to include the product of an i n t r i n s i c wave function, 2C, describing the shape and structure of the nucleus, and a r o t a t i o n a l wave function, D, describing i t s o r i e n t a t i o n i n space. In addition to rotation, the nucleus also exhibits small c o l l e c t i v e vibrations about i t s equilibrium shape. Since these vibrations only s l i g h t l y modify the aver-age p o t e n t i a l seen by the nucleons, they are only weakly couple to the s i n g l e - p a r t i c l e and ro t a t i o n a l aspects of the problem and we can account for them separately with, say, a v i b r a t i o n a l wave function, ^»viij. Thus for the moment we can take the t o t a l nuclear Wave function to have the schematic form, ' -i n fact, v i b r a t i o n a l excitations generally occur at consid-erably higher energies then the r o t a t i o n a l excitations i n which we s h a l l be interested and we w i l l simply omit them from our discussion ftaking the nucleus to always be i n i t s v i b r a t i o n a l ground, st a t e ) . The motivation of our ca l c u l a t i o n i s now clear. Our purpose i s to investigate the exc i t a t i o n of r o t a t i o n a l states, D, through the o p t i c a l pion-nucleus i n t e r a c t i o n which, from our discussion i n Chapter 2, involves the nuclear mass density, _?(r) , i n a very d i r e c t manner. In fact, what we w i l l show i s that the e x c i t a t i o n of these r o t a t i o n a l states depends s e n s i t i v e l y on the surface d e t a i l s of„P(r) through the momentum-dependent pion-nucleus i n t e r a c t i o n . Since J>{v) 101 . i s simply the sum of the s i n g l e - p a r t i d e densities obtained from the i n t r i n s i c wave function, OC , and since we know the surface d i s t r i b u t i o n of protons i n 3C from other experiments (such as the JJ -mesic x-ray experiments) , this allows us to state separately the surface d i s t r i b u t i o n of neutrons and protons. When combined with the results of o p t i c a l absorption experiments, as described i n Chapter 3, t h i s should allow a comprehensive i n v e s t i g a t i o n of the nuclear surface i n deformed nuclei and an accompanying increase i n our under-standing of the microscopic aspects.of the problem accounted for i n the i n t r i n s i c wave function, X . Before proceeding, we require a more complete descrip-tion of the nuclear wave function which we have schematically written i n Eq, 4-1. A detailed derivation of the r o t a t i o n a l model can be found i n most elementary textbooks on nuclear physics (Preston (1962)) ""and we w i l l s a t i s y ourselves with a b r i e f description. The basic postulate of the adiabatic assumption Is that the t o t a l nuclear Hamilton!an, h, can be written i n the form, (4-2) h = H i n t l,(x') -f T r o t , ' 4 where % n t r ^ x T ^ i s » s a y » a shell-model Hamiltonian appropriate to a deformed average potential and depending upon the i n t r i n -s i c co-ordinates, x T, and where T r o t i s the k i n e t i c energy of the rotational motion. For our purposes, we w i l l be interested i n the (large) class of deformed nuclei which are a x i a l l y symmetric, these e x h i b i t i n g primarily spheroidal deformations. The rotation' ICS of such a body i s analogous to the c l a s s i c a l j>recession of a symmetric top (Goldstein (1950)). We therefore might expect the ki n e t i c energy operator to have the form, m _ h _ ^ 2 r o t " ?J(J) v 'where i s the "moment of i n e r t i a " of the nucleus, depend-' **** ing upon the average deformation, , and where I i s the t o t a l angular momentum operator of the nucleus. The norm-al i z e d eigenstates , D, of T r o t are then e a s i l y shown to have the form, ( 4 - 4 ) . = Tor , . - , !* n I where <=< = ( <* , J 5, ^  ) are the Euler angles (Goldstein (1950)) K T . . . . which rotate the space-fixed axes, (x,y,z), z being i n the di r e c t i o n of the axis of quantization, into the body-fixed axes, (1,8,3), 3 being the symmetry axis of the nucleus. oDcKjj.) i s the matrix element of the rotation matrix, (Preston (1962)) corresponding to the eigenvalues I, M, and K of the operators I, I z , and I respectively. C l e a r l y , these are the quantities which we-expect to be constants of the motion from the analogy with the c l a s s i c a l symmetric top. In addition, since the nucleus i s a x i a l l y symmetric, we expect i t s spin i n the 3 - d i r e c t i o n , say, _Q. , to also be a constant of the motion so that the eigenstates of the i n t r i n s i c Hamilton!an, H j _ n ^ : r , ^ x I ^  » have the form " X ^ x 1 ) , where T are the remaining quantum numbers required to specify 103 the i n t r i n s i c nuclear state. I t can-easily be shown that a x i a l symmetry requires t h a t X l = K. . The nuclear wave func-t i o n therefore has the form, - ( 4 " 5 a > % = f i g * : T * ' ° . o ) . A c t u a l l y , Eq. 4-5a c o r r e c t l y represents the nuclear state only i f K = 0; for higher values of K we require linear combinations of states l i k e Eq. 4-5a i n order to have states of definite p a r i t y (Preston (1962)). The r e s u l t i s then of the form, 4 (4-5b) L16J12. m , — ii . j . ( K k 0 ) ( -f or — depending on whether the state has positive or neg-ative p a r i t y , respectively) where we have expanded 'JC'K. i n terms of i t s angular momentum sub states, "3C.. ^ c Prom Eqs. 4-5 i t i s seen that a given i n t r i n s i c state, " ^ CJJ , may e x i s t i n various states of r o t a t i o n as prescribed by the r o t a t i o n a l wave functions, S)T^r „ Such a set of rota-Mil t i o n a l states i s ca l l e d a " r o t a t i o n a l band". I t i s e a s i l y shown that, i f K - 0, I*increases i n steps of two eg. l u - 0* 2"^  4*", ... while i f K § 0, locincreases i n steps of one Ioj = K, K>1, E-+2, ... . This band structure i s the signature of the energy l e v e l spectrum of a strongly deformed nucleus. 104 Our technique for evaluating the r o t a t i o n a l e x c i t a t i o n . cross sections w i l l he to use the Distorted Wave Born Approximation (c.f. Kikuchi and Kawai (1968), Messiah (1963)). In DWBA we see the nucleus i n i t s ground state as being im-mersed i n the e l a s t i c a l l y scattered irion f i e l d which we c a l -culate from the pion-nucleus i n t e r a c t i o n , say that described by the Ericsons' potential discussed i n Chapter 2 and Sec. 3 - 3 . . The pion f i e l d then causes the nucleus to increase i t s rota-t i o n , e x c i t i n g i t to some higher state i n the r o t a t i o n a l band. Before proceeding to evaluate the exci t a t i o n cross sections i n DWBA we must f i r s t ask ourselves about the v a l i d i t y of this approximation. We argued i n Chapter 2 that the absorptive e f f e c t s due • to e x c i t a t i o n within the i n t r i n s i c wave function were small compared to r e a l pion absorption (which i s taken into account i n the Ericsons' p o t e n t i a l ) . In addition,, we do not expect absorptive effects of v i b r a t i onal states to be important since they occur at considerably higher energies then r o t a t i o n a l states. The question of the v a l i d i t y of DWBA can therefore to .a large extent be rephrased by enquiring what the absorptive e f f e c t s of r o t a t i o n a l e x c i t a t i o n are on the ground state wave function that i s , we wish to determine the coupling of the r o t a t i o n a l channels. I f i t i s small, we may simply use DWBA;-if i t i s large (as i t i s i n the excitation of ro t a t i o n a l states by neutrons), the distorted wave calculated from the op t i c a l pion-nucleus i n t e r a c t i o n (which does not include the eff e c t s of -channel-coupling) does not accurately describe the 105 pion f i e l d seen by the nucleus i n i t s ground state and we must resort to some coupled channel approximation. . The es s e n t i a l condition for the v a l i d i t y of DWBA i s therefore that the e x c i t a t i o n cross sections be much smaller than the geometric cross section 4JIR , where R i s the radius of the target nucleus. • As we w i l l show i n the numerical discussion '(Sec. 4-2), the e x c i t a t i o n cross sections are a few millibarns while the geometric cross section i s a few barns. This condition would therefore seem to be well s a t i s -fied and DWBA should be good at least to f i r s t order. This i s to be contrasted with the e x c i t a t i o n of r o t a t i o n a l states by neutrons where i t i s found.that the geometric cross section i s of the same order as the e x c i t a t i o n cross sections so that, one must resort to some coupled-channel approximation (Chase, Wilets, and Edmonds (1958)). Even i f i t should turn out that one might l a t e r wish to take more ca r e f u l account of the coupling of the r o t a t i o n a l channels, the DWBA calculations are of considerable i n t e r e s t since they allow us to separately state the e x c i t a t i o n pro-cesses due to the Coulomb, l o c a l , and momentum-dependent pion-nucleus interactions; these effects are not e a s i l y sep-arated i n , say a coupled-channel approximation. 4-2 DWBA Formulae for Pion E x c i t a t i o n Cross Sections In the present section we derive i n the Distorted Wave Born Approximation -the formulae for the e x c i t a t i o n of rota-t i o n a l states i n strongly deformed nuclei by pions. A gen-eral discussion of DWBA can be found i n many text books on 106 nuclear physics (KLkuchi and Kawai (1968)-, Preston (1963)) and quantum mechanics (Messiah (1963)), For our purposes, we only c u r s o r i l y review the-subject and from the point of view of the problem at hand, We then use the DWBA expressions to present formulae suitable for evaluating and i n t e r p r e t i n g the pion e x c i t a t i o n cross sections. In Sec. 4-3 we w i l l evaluate these formulae numerically for a r e a l i s t i c pion-nucleus p o t e n t i a l (the Ericsons' p o t e n t i a l of Chapter 3 and Sec. 3-.3) with p a r t i c u l a r emphasis upon the role of the nuc-le a r surface thickness. To begin with, l e t us consider a pion, described by a k i n e t i c energy operator, T^, to be incident on a deformed target nucleus described by the Hamiltonian, h, of Eq, 4-3, •• We w i l l assume that the pion i n t e r a c t s with the nucleus through and average potential,"\f(r) , such as the momentum-dependent Ericsons' potential discussed i n Chapter 3. We w i l l assume that r e a l pion absorption and absorptive effects due to the many-body structure of the nucleus are accounted for i n the imaginary part of this potential; the only channels a v a i l -able to the problem are then through those c o l l e c t i v e modes of o s c i l l a t i o n which can be excited by the average i n t e r a c t i o n , i n p a r t i c u l a r , the r o t a t i o n a l channels. In our calculations we w i l l assume the deformed nuc-leus to be a x i a l l y symmetric. To evaluate the e x c i t a t i o n cross sections i t i s then convenient to expand ~[f(r) i n the body-fixed frame i n a series of spherical harmonics 8 (4-6) 107 where the body-fixed co-ordinate system (r-^ , 0^, ^ ) is chosen to have its origin at the center of the nucleus and its polar axis along the nuclear axis of symmetry. We can then divide ~V(r) into the spherically symmetric potential, 1/"0(r), and the perturbing potential, (4-7) -V'(r) = £l#r b)Y£(e b ) , which is responsible for the nuclear excitations. The total Hamiltonian of the system can now be written in the 'form, ( 4 - 8 a ) H Ex +--V* (r) , where the unperturbed Hamiltonian is ( 4 - 8 b ) H X = H 0 + V^(r) and where the Hamiltonian in the absence of interaction is ( 4 - 8 c ) H = T r 4-h. Before proceeding, it will be convenient to define 'the ei gens tat es of the Hamiltonians of Eqs. 4 - 8 . Let us first define the eigenstates, J j a , of E , the Hamiltonian in the absence of interaction: ( 4 - 9 ) J a = ^e 1^"-where the nuclear states, , are given by Eq. 4-5. , and ik • r e _a — is the pion wave function, ka being the incident pion momentum (for notational convenience, we will treat the incident pion as a plane wave and neglect terms in log(3k ar ) which arise from the long-range effects of the pion-nucleus interaction; the generalized discussion is essentially the same) „ 108 The t o t a l state of the system i s represented "by the wave function, ,\u , which i s that eigenstate of the t o t a l Hamiltonian, H, of t o t a l energy, E, which asymptotically contains an incoming plane wave i n the incident channel, , with pion momentum, k , and outgoing waves i n : a l l other channels, say ^  , with corresponding pion momentum, say k : U-lOa) H ty1*' = E ; (4-10D) M / W s-^, Ta r —>•<> % -ik^r^T ik°r„, „ (+•;, . e a' a * r * r r Here fil'f-Ap) i s the scattering amplitude from the incident channel, , into the channel,^3 . We can define states, OCa, for the unperturbed Hamil-tonian, Hj_, s i m i l a r to the t o t a l state, •, of Eqs. 4-10, except that JQ^' contains only an outgoing wave i n the incident . EL channel, «*• ^  since H-j_, having sp h e r i c a l symmetry, contains no mechanism for e x c i t i n g a r o t a t i o n a l state. We can then write (4 - l la) a " = . where h^'ir^) i s the o p t i c a l wave function of the pion s a t i s -i a fying the equation, (4-llb) (T^ + ^ (rJJ^ Vr*) = (E - 6* , and having the asymptotic form, ( 4 - l l c ) i , , iK^r, ^C4; . A . e " i k a r -Ta — <  r., -><» J-a^ . * . .r* » • 109 here the pion wave number i s given by the r e l a t i o n , (4-iia) < , ' a h where ^  i s the reduced mass of the pion. The DWBA .now says the following:. I f the amplitudes f^yf-M {f $*) i n Eq. 4-10 b are small _ — i . e . , i f the ex c i t a t i o n cross sections are small we may omit from [|/a those terms which lead to asymptotically outgoing waves i n an excited channel: i t i s then reasonable also to neglect the absorptive effects on fa^ f-^ *<) of the coupling between the incident channel, , and the excited channels, V t i n it) . Since the perturbation term, i f M r ) , i s small com-pared to IT ( r ) f we may also neglect i t s e f f e c t s on the e l a s t i c s c a t t e r i n g and thus take (4-13) t£(-"*) = -Hence, DWBA says that, to good approximation, * •a (4-13) ( J/^ = We c a l l X l ^ t h e "distorted .wave", a. I t i s to be noted that "X^contains almost a l l the effects / a of the pion-nucleus i n t e r a c t i o n . We therefore expect DWBA to be a much better approximation than plane wave Born Approximation, where we take - ^ , the incident plane wave; i n fact, i n pion-nucleus scattering, the e l a s t i c a l l y scattered wave i s strongly distorted and we do not expect the l a t t e r approximation to be good at a l l . I t remains now only to evaluate the ex c i t a t i o n cross sections. From•elementary quantum mechanics (Messiah (1963)), 110 the exc i t a t i o n cross section i s given by / ( 4 - 1 4 ) aera-*b ' ^ , ... 12 Gl k n W i ^ ..|-a-*b| a \ where the matrix elements, Ta_yt,» o f the scattering matrix, T, are given by the expressions, T a ^ b = < 5 b l T l i a > = < $ „ | ( T T 0 f r ) + T P ( r ) ) j | | / B ^ . I t i s then a well-known r e s u l t (Messiah (1968)) that, '4-16) T a ^ '. < - J „ |TT0-fr) | '^> + <X' |V' (r)|l|/aW> The distorted wave, " X ^ , i n E3. 4-16 s a t i s f i e s equa-tions analogous to Eqs. 4-11: (4-17a> ah' = I (4-17b) (T, +ir ( r ^ ) ) ^ ! ^ ) = (E - ^ ^ ( r ^ ; f 4 ~ 1 7 a ) v - ay(E'- e,) b -h ' except that i t i s asymptotic to incoming rather than out-going scattered waves: ' •>V e ~* ~p + -77- i here k, i s the momentum i n the di r e c t i o n of the scattered pion wave. We w i l l defer giving e x p l i c i t expressions for ,<j>1 '^ and ffl u n t i l we are ready to evaluate the matrix elem-ents of Sq, 4-16 (c.f. Eq, 4-30 f f. ). I l l Now, i f ~i> £ , we have from-the sph e r i c a l symmetry of if (r) and from Eqs. 4-5 and 4-11 that i n Eq. 4-16, ( 4 - 1 8 ) < J 0 K W | y a ' W > V 0 ' . . I f we further make the DWBA, Eq. 4-13, we have from Eqs. 4-14 and 4-16 that (4-19) d£a->c k x 2. (-, *a Eq. 4-19 can be evaluated i n a straightforward manner and the details of the evaluation have been relegated to Appendix B of this thesis. As i s shown i n Appendix B, Sec. B - l , the r e s u l t for an unpolarized beam of pions (where we average over i n i t i a l states and. sum over f i n a l states) i s that (4-20) d£-a*c „ \ 2 * f 7 . £ I fl^2 d X i y : -2.1 o\ £ ^ 1 ex a c where A ^ are s t a t i s t i c a l c o e f f i c i e n t s describing the rota-t i onal band, (4-21a) = — r r — : ( J even) ; Q K + 1 (4-21b) A** = 0 . (7- odd) ; and where' G^ac are matrix elements over the pion co-ordinates, Before proceeding to evaluate E q . 4-22 for the matrix I I S elements ^ we s h a l l need to describe the potentials ac o r fr) and I T ' ( r ) of Has. 4-6 and 4-7 i n more d e t a i l . We w i l l take the pion-nucleus i n t e r a c t i o n to be described by the Ericsons' potential of Eq. 2-36. • Now we know from electromagnetic experiments, such as Coulomb e x c i t a t i o n , that a strongly deformed nucleus i s nearly spheroidal so that we can write the nuclear density as (4-23.) / ( r ) = J> ( r ( ) / x ^ where J> (r) I s , say, the Saxon-Woods density of Eq. 3-31. The average deformation parameter, J? , i s t y p i c a l l y small (^.0.3) so that we can take (4-24) As can be seen from Pig. 1 , Sec. 3-4, J> (r)-'-\-/p (r) , and we w i l l simply make thi s replacement i n Eq. 3-29. We can then take the pion-nucleus p o t e n t i a l to be given by (4-25) -V (r) = - - V' ~^[7T 2 + Vc (r).+V(r)+iW(r) where (4-26a) c<(r) : : ac.(r) o (4-26b) V ( r ) : V (r) 0 (4_26c) W(r) = W o(r) v M r ) Yg(6 b) ; - Jfr W«(r) Y°(e b) ; here <^ 0(r), V ( r ) , and W Q(r) are the potentials (r),'V(r) , 113 and Wfr) i n Eqs. 3-29, The Coulomb p o t e n t i a l . V (r) , i s evaluated i n Appendix B, Sec. B-2, where we find that Ze 2 p3 1 /r\ 2 (4-27b) V c(r) = — " + " Y g . f e b ) , . r - R ; (here we have treated the charge density as uniform and of radius,-. R). We therefore have that - , > • ' 2 ^ ^ o f r ) (4-28) i T f r ) r - 1 - V • ' ~TT 1 o - 1 - ^ 0 ( r ) 1 .'. 3 + V c(-r) 4- V Q ( r ) +- iW Q(r) , i s the o p t i c a l potential of Eq, 3-29 and that the perturbing p o t e n t i a l i s (4-29a) -V'(r) = fL(r)Yg(eb) + V - % D ( r ) Y°(^ ) 2 4-fc(r)YX) where (4-29b) f (r) = ~rV*(r) - irW'(r) C O o i s the l o c a l i n t e r a c t i o n , where , 2 r o < 0 ^ ) (4-29c) „ , . _ Tia . = T-fTtT-F i s the momentum-dependent strength, and where r > R ; f ( r ) = - j W 4 c 5 K 3 , r < R ; 114 i s the electromagnetic i n t e r a c t i o n . Of course, we should s t r i c t l y have calculated the poten t i a l harmonics, l / ^ f r ) , from Eq. 4-6. For instance, such a c a l c u l a t i o n has been performed by Chase, Wilets, and Edmonds (1958) for the e x c i t a t i o n of r o t a t i o n a l states i n deformed nuclei by neutrons; they find that'estimates such as those made i n deriving Eqs. 4-28 and 4-29 are of only moderate accuracy, nevertheless, for our purposes we simply wish to demonstrate the e f f e c t of the nuclear surface on the exc i t a -tion cross sections and for this purpose these estimates should be quite s a t i s f a c t o r y . A more accurate c a l c u l a t i o n would, of course, take a more careful account of the poten-t i a l s . Since we are interested i n pion ex c i t a t i o n at low ener-gies (^30 MeV) , only a few p a r t i a l waves enter the problem (as can be seen from Figs. 3, 5, and 11 of Sec. 3-4) and i t w i l l be convenient to make a p a r t i a l wave expansion of the o p t i c a l waves, ^a^sJ » a n < 3 tfc^?) • Prom Eqs. 4-11 and 4-17 i t can be shown (Messiah (1962)) that (4-30) = g 2 ^ - A , ^ tff})Y?(£)I> ( k : r ) , where ^ + iJlf are the p a r t i a l wave phase s h i f t s ( and F^(k;r) i s the regular spherical Coulomb function). Hence, (4-31) tf"'(r) = Art*('-r) . .: 'k - I t At the energies which we w i l l consider (several MeV) we can neglect the effects of nuclear ex c i t a t i o n ( C v ^ ^ l Me .and take k.G = k In addition, we w i l l take k a, the beam 115 d i r e c t i o n , to be along the (space-fixed) n-axis and we w i l l henceforward suppress the indices a, c, for example, taking A = k c to (4-32a) /  k = k c to be the d i r e c t i o n of the scattered wave. Then, ^ ( r ) = j>(r) = F 1 ~ * 0 ( r ) ; ( 4 " 3 2 b ) f ( r ) = f(lr) = - 277 ^ V + l u*(r) , , where the p a r t i a l waves, u ^ ( r ) , are solutions of the r a d i a l Schroedinger Eqs. 3-4. Considering Eqs. 4-29 and 4-32, we can rewrite Eq. 4-20 i n the convenient form, (4-33a) acr0,->y ;_ j 3 2 A ^ ^ ? r e < 3 d X L a where (4-33b) d6"red * f JJ \ Z W A Z here, (4-34) I* = Jar $ ( - r J ^ n " Y f^e.^ r^) . dCT red As can be seen from Eq. 4-34', —- i s independent of the r o t a t i o n a l band (to the extent to which we can neglect the energies associated with nuclear excitations) and of the nuclear deformation (to the extent Eq. 4-29 correc t l y do" red represents the perturbing potential) and i t i s ' which we s h a l l consider i n our ca l c u l a t i o n s . The evaluation of simply involves the sub s t i t u t i o n ,of the p a r t i a l wave expansion, Eq. 4-32, for ij> (r) and the 116 s u b s t i t u t i o n o f t h e p e r t u r b i n g p o t e n t i a l , Eq. 4--S9, i n Eq. 4-34. The i n t e g r a t i o n o v e r t h e a n g u l a r v a r i a b l e s i s t h e n s t r a i g h t f o r w a r d a n d we c a n c o n v e n i e n t l y s t a t e t h e r e s u l t o f t h e c a l c u l a t i o n b y d e f i n i n g t h e f o l l o w i n g r a d i a l i n t e g r a l s . F i r s t l y , t h e r e a r e r a d i a l i n t e g r a l s (4-35a) ,«« f 4 . Fj* = j d r i y ( r ) f L ( r ) u ^ f r ) / a r i s i n g f r o m t h e l o c a l i n t e r a c t i o n ; t h e r e a r e s i m i l a r r a d i a l i n t e g r a l s , ., (4-35b) F ^ •= J d r l y f r ) , f c f r ) u ^ f r ) , a r i s i n g f r o m t h e e l e c t r o m a g n e t i c i n t e r a c t i o n . T l i e r e a r e e s s e n t i a l l y two k i n d s o f r a d i a l i n t e g r a l s a r i s i n g f r o m t h e m o m e n t u m - d e p e n d e n t i n t e r a c t i o n : f i r s t l y , t h e r e a r e r a d i a l i n t e g r a l s r i'u; • ^ r a r i s i n g f r o m t h e r a d i a l t e r m i n t h e momentum o p e r a t o r ; s e c o n d l y , t h e r e a r e r a d i a l i n t e g r a l s (4-35d) j o ' f „ f x IP.** = jdr u « f r ) f ( r ) ^ ( r ) a r i s i n g f r o m t h e a n g u l a r t e r m s i n t h e momentum o p e r a t o r W i t h t h e s e d e f i n i t i o n s i t i s f o u n d t h a t t h e d i f f e r e n t i a l c r o s s s e c t i o n s , E q . 4-33, a r e g i v e n i n t e r m s o f I " " b y t h e e q u a t i o n # N o t e t h a t , f r o m E q . 4-34-7 t h e i n t e g r a l s are o v e r u v f r K n o t 117 (4-36) (-) n /2^ n isTTi M)\ Y;(k) k 2 * P " ' 1 J I ( i + i ) MD,9 '. * L J ( V i ' W ' + l ) - ^ - ! ) < 2 J ? - / ^ / - l | i - l ^ f J (/+!)'->/C"+l) ^ 24 IjiM+l [ Si 1 ^ ) J ' Eq. 4-33 i s e a s i l y i n t e g r a t e d to give the t o t a l . e x c i t -a t i o n cross s e c t i o n , the r e s u l t b e i n g (4-37a) (4-37b) where (4-38a) * JaMi</i00120/<tf i.00120? ' 0 ' / M » = ^ / / ( i ' + i ) - ^ l ) < 2 i - H * 0 > * ^JY(-*'-»-l) - -v(^ + l ) < - 8 i - ^ l H . 1~7 + J iV ' -H) - W ^ - l ) < ?J^X- l | - f ( -1>]. I t i s seen i n Eqs„ 4-36 and 4-37 that we have succeeded 118 In separating those e x c i t a t i o n processes which arise from the momentum-dependent i n t e r a c t i o n from those which arise from * the l o c a l and electromagnetic i n t e r a c t i o n ; i n fact, the mom-entum-dependence is contained i n the i n t e g r a l s , F^ -p p . In the next section we w i l l provide numerical estimates of the integrals,.Eq. 4-35, and the cross sections, Eqs. 4-33, 4-36 and 4-37, "based on the Ericsons' p o t e n t i a l of Chapter 3 and Sec. 3-3. 4-3 Numerical Discussion In the present section we present a numerical discussion of pion ex c i t a t i o n "based on the DWBA cross section formulae and the r a d i a l integrals which we have derived i n the l a s t . section. In these c a l c u l a t i o n s , our main objective i s to demonstrate the s e n s i t i v i t y of the e x c i t a t i o n cross sections to the diffuseness of the nuclear surface. As discussed i n the previous section, this s e n s i t i v i t y i s contained i n the r a d i a l i n t e g r a l s , F ^ r which arise from the r a d i a l terms i n the momentum-dependent part of the pion-nucleus i n t e r a c t i o n To show that the experimental data can he analyzed for these quantities, we first construct the t o t a l and d i f f e r e n t i a l excita.tion cross sections and demonstrate that these cross sections are sensitive to the de t a i l s of the nuclear surface. We then show that this s e n s i t i v i t y i s contained i n the r a d i a l i n t e g r a l s , F^p r and that i t can be understood i n terms of the suppression of momentum-dependent processes near the top of the potential b a r r i e r f i n analogy with our discussion i n -Chapter 3). 119 For the purposes of the present discussion, we have chosen the potential to be that, given by Eqs. 4-28 and 4-29 of Sec. 4-2 where we have expanded the actual potential i n terms of the average nuclear deformation parameter,^ 9 r e t a i n i n g only terms of order f . For the parameters of the i n t e r a c t i o n , we have used the values obtained from the Ericsons' c a l c u l a -t i o n (1966) which we have discussed i n Sec. 3-3 of the prev-ious chapter. We have then introduced the diffuseness of the nuclear surface by employing a Saxon-Woods d i s t r i b u t i o n (Eq. 3-31) for the nuclear density. In the Saxon-Woods den-' 1/3 s i t y we have chosen R 0 = 1.05 A ' fm. and we have performed our calculations for two choices of the surface parameter, a = 0.5 fm. ('which i n our figures we denote by s o l i d l i n e s ) and a = 0.1 fm. (which i n our figures we denote by dashed l i n e s ) . We have performed our calculations for both a l i g h t O K nucleus, where we have chosen parameters; appropriate to A l , and for a heavy nucleus, where we have chosen parameters 2 38 appropriate to IT In Figs. 12a and 12b we have plotted the reduced t o t a l e x c i t a t i o n cross sections, C 5 ^ e ( j ? for A l and 11"° , respec-t i v e l y , ^ r e d ^ s r e l a t e d to the actual e x c i t a t i o n cross sections, C5£MT, by Eq. 4~37b of Sec. 4-2: (4-37b) = ^ 2 A ^ ^ " e d 5 where J$ i s the average nuclear deformation parameter and A^< i s a s t a t i s t i c a l c o e f f i c i e n t (Eq. 4-21) describing the i n i t i a l and f i n a l r o t a t i o n a l states involved i n the excitation. 2 Ty p i c a l l y J 3 ^ o . 3 and AVe< i s a number always less than unity 120 FIG. 12a REDUCED T O T A L EXCITATION CROSS SECTIONS 0.20 ; f + A I 2 5 c 0.15 4. 0.10 0.05 0 0 / / a = 0.5 Fm a = Oo I Fm 2 0 MeV 35 P I G 1 2 b REDUCED TOTAL ' EXC ITAT ION CROSS SECTIONS 71^ U 2 3 @ E , MeV i2a F I G . 13a REDUCED D I F F E R E N T I A L EXCITATION CROSS S E C T I O N S > + Al 25 3 0 2 0 i IOI O ot 75 5 0 a = 0.5 Fm a = 0.1 F m 25 E = 35 MeV 01 0 ( 3 0 ( 60° 90° 120° 15 0° I80 c e 123 F I G . 13b REDUCED DIFFERENTIAL EXCITATION CROSS SECTIONS i j I I J I ! _ I 0° 30° 60° 9 0° 120° 150° 18 0° . 9 (for example, A6+ 2^+ =' i . ). Tt i s therefore seen from Figs. 12 that, t y p i c a l l y , <5^->~s <: 0.03 barns, considerably less than the geometric cross section of a few barns. From our previous discussion, t h i s would seem s u f f i c i e n t j u s t i -f i c a t i o n to.employ the DWBA. I t i s seen from Figs. 12.that, i n both l i g h t and heavy n u c l e i , even the t o t a l e x c i t a t i o n cross sections are quite sensitive to the diffuseness of the nuclear surface. In p r a c t i c e , the t o t a l cross sections can be measured very accurately by measuring the de-excitation "tf-rays when the excited r o t a t i o n a l state decays so that these effects can be measured with considerable p r e c i s i o n . Although the effects of the diffuseness of the surface become greater, at higher' energies, the simple connection between the density and the pion-nucleus i n t e r a c t i o n many then break down, as was d i s -cussed i n Chapter 2. Nevertheless, i f our present understand-ing of the i n t e r a c t i o n i s even q u a l i t a t i v e l y correct, the e x c i t a t i o n cross sections should provide a powerful t o o l for i n v e s t i g a t i n g the d e t a i l s of the nuclear surface. In Figs. 13a and 13b we have plotted the reduced d i f f e r -e n t i a l cross sections (for the i n e l a s t i c a l l y scattered pions) for A l and u corresponding to the t o t a l cross sections i n Figs.. 12a and 12b, respectively. Measurement of the di f-f e r e n t i a l cross sections involves measuring the inelas t i c a l l y scattered pions and i s probably somewhat more d i f f i c u l t than measuring the t o t a l cross sections. I t i s seen, however, that the d i f f e r e n t i a l cross sections have a structure which i s 125 strongly dependent upon the diffuseness of the nuclear surface; i n Pig. 13a (upper figure) i t i s seen that this may-be the case even when the t o t a l cross sections exhibit a much smaller s e n s i t i v i t y . The large back sca t t e r i n g arises since i t i s more probable for the pion to excite the nucleus i n a head-on c o l l i s i o n ( 6 = 180° ) than i n a grazing c o l -l i s i o n ( 0 = 0 ° ) . Before we proceed to analyze the cross sections i n more d e t a i l , i t w i l l be useful to remember from our e a r l i e r d i s -cussion (Eqs. 4-35) that pion e x c i t a t i o n arises from essen-t i a l l y four separate processes; f i r s t l y , there i s e x c i t a t i o n due to the loe a l pion-nucleus i n t e r a c t i o n , accounted for i n the i n t e g r a l s , F^; secondly, there i s "Coulomb e x c i t a t i o n " due to the e l e c t r o s t a t i c p o t e n t i a l , accounted for i n the i n t e g r a l s , F ; t h i r d l y , there i s e x c i t a t i o n due to the angular terms i n the momentum-dependent i n t e r a c t i o n , accounted for i n the i n t e g r a l s , P ^ ^; f i n a l l y , there i s exci t a t i o n due to the r a d i a l terms i n the momentum-dependent i n t e r a c t i o n , accounted for i n the i n t e g r a l s , 3?. _ „ In the subsequent MD9r A discussion we w i l l show that the f i r s t three integrals are i n s e n s i t i v e to the de t a i l s of the nuclear surface and that the s e n s i t i v i t y demonstrated i n Pigs. 12 and 13 i s contained mainly i n the i n t e g r a l s , r „ Tt w i l l then be our objec-tive to show that this s e n s i t i v i t y can e a s i l y be understood i n terms of the suppression of pion momentum near the top of the potential barrier. In Pig. 14 we have plotted the absolute values of the r a d i a l i n t e g r a l s for the';local i n t e r a c t i o n , j F-^  j, at 35 MeV 126 F l r - 14 R E L A T I V E VALUES O F RADIAL I N T E G R A L S E = 35 MeV TC+ Al 25 E = 3 5 MeV U 2 3 8 0 - 2 2 - 2 1-3 3 - 3 129 F I G . 1 6 . R A D I A L I N T E G R A L S 130 F I G . 16b R A D I A L I N T E G R A L S for the allowed angular momentum tran s i t i o n s between the i n i t i a l state p a r t i a l wave of angular momentum, J[ , and the f i n a l state p a r t i a l wave of angular momentum, |'„ We have r e s t r i c t e d our c a l c u l a t i o n to X, |' ^ 3 (although a more j precise c a l c u l a t i o n would take into account the higher p a r t i a l waves). I t i s seen that-the main contribution to the e x c i t a t i o n arises from the. i n t e g r a l F-^ and the secondary L contribution from the i n t e g r a l F ^ (or, equivalently, F ^ ) , JJ 1/ Hence, even at moderately high energies, only a few p a r t i a l waves enter the problem; of course, as we go to lower energies the r e l a t i v e dominance of the i n t e g r a l s , F-*-^ , increases. We w i l l therefore here r e s t r i c t ourselves to integrals of the form F"1"^  and 3?^; i n any event, the generalization to integ-r a l s i n v o l v i n g p a r t i a l waves of higher angular momentum w i l l be apparent from our discussion. In F i g . 15a and 15b we have plotted the r e l a t i v e mag--11' 1 1 nitudes of the absolute values of the i n t e g r a l s , 3?^ , 9.-.3? , 11 p-R ? 38 and 3? ' , for A l 4 0 and U" , respectively. I t i s seen X l i s p \J ' t from Figs. 15 that these r a d i a l integrals are quite i n s e n s i t -ive to the diffuseness parameter, a, even i n the case of 25 A l , which i s mostly surface. This arises since these quantities involve i n t e g r a l s over the nucleus and are sen-s i t i v e only to the amount of nuclear matter present rather than to i t s d i s t r i b u t i o n . A s i m i l a r i n s e n s i t i v i t y to the e'Jt ' ' 0" • ' surface i s found i n general for the int e g r a l s F T , P , a n a V . e • In F i g . 16a and 16b "(upper figures) we have plotted (on the same scale) the corresponding values of the int e g r a l s a r i s i n g from the r a d i a l term i n the momentum-dependent i n t e r -action,, I' l l ^ I t i s seen that, unlike the other i n t e g r a l s , these inte g r a l s depend very strongly on the diffuseness para-meter, a. In fact, as we w i l l now show, the behaviour of these i n t e g r a l s can be understood i n an analogous manner to the behaviour of momentum-dependent absorption discussed i n Chapter 3. •In F i g . 16a we have plotted the r a d i a l i n t e g r a l s | F ! 1 (upper figure) and J F ^ 2 ' I (lower figure) appropriate to A l .. Since the parameters of the pion-nucleus potential for A l 2 5 are si m i l a r to those of Ca, 4 0, which we discussed i n connection with o p t i c a l absorption i n Sec. 3-4, we w i l l . int e r p r e t the effects found here i n terms of the explanations given there. For instance, the behaviour of P ? 2 depends on the ' , MD,r properties of the r a d i a l s-wave function. Now, near the top of the potential b a r r i e r , the r a d i a l momentum associated with the s-wave becomes small. For the diffuse-edge poten-t i a l , most of the i n t e r a c t i o n takes place i n the surface of the nucleus where the nuclear density i s lower; hence, for the diffuse-edge p o t e n t i a l , the e f f e c t i v e b a r r i e r appears lower. Thus, we expect at lower .-energies to find that the value of |FT9-2 I i s less i f we assume a diffuse-edge poten-I MD,rl , t i a l than i f we assume a sharp-edge potential and, at higher energies, we expect to find that i t i s greater. This i s i n analogy with our discussion of momentum-dependent absorption 133 ! i i n C a 4 0 which, we have discussed i n Fig. 7. This e f f e c t i s shown here i n F i g . 16a (lower figure) and i s to he compared with the imaginary s-wave phase s h i f t for o p t i c a l absorption i n C a 4 0 (Fig. 5). ! As we discussed i n Sec. 3-4, the e f f e c t ' of increasing the angular momentum i s to increase the p o t e n t i a l b a r r i e r . We therefore expect at higher energies to see a s i m i l a r behaviour for the i n t e g r a l . F H , as was found at lower * MD,r * 02 energies for the i n t e g r a l • F,!: . This e f f e c t i s shown ° * MD,r i n F i g . 16a (upper fi g u r e ) . As we discussed for P b ^ 0 8 i n Sec. 3-4 (Fig. 8), the major e f f e c t i n going to a heavy nucleus i s an increase i n the e f f e c t i v e b a r r i e r due to an increase i n the e l e c t r o s t a t i c p o t e n t i a l . This e f f e c t i s seen i n F i g . 16b where i t i s shown that even at higher energies both the i n t e g r a l s and 3?0S ^ are more strongly suppressed for a diffuse-edge potential than for a sharp-edge p o t e n t i a l . The attenuation at lower energies which i s seen In the diagram i s due, of course, to the e f f e c t s of penetration. We might now return b r i e f l y to the cross sections cor-responding to these r a d i a l i n t e g r a l s which we have given i n Figs. 12 and 13. We believe that i n the discussion which we have presented here we have c l e a r l y shown that the dependence of these cross sections on the nuclear surface arises i n an e a s i l y understandable way through the momentum-dependent in t e r a c t i o n . In fact, our discussion depended only upon the q u a l i t a t i v e features of the pion-nucleus i n t e r a c t i o n ' - - - the 134 presence o f a p o t e n t i a l b a r r i e r and a s t r o n g momentum-depend-e n t i n t e r a c t i o n . We have n o t i n v e s t i g a t e d the s t r u c t u r e o f the c r o s s s e c t i o n s i n d e t a i l s i n c e , f i r s t l y , the parameters o f the p o t e n t i a l a re n o t v e r y w e l l determined (and are prob-a b l y somewhat energy-dependent) and, s e c o n d l y , our rough t r e a t m e n t o f t h e p o t e n t i a l (Eqs. 4-28 and 4-29) i s o n l y a p p r o x i m a t e l y c o r r e c t . F u t u r e c a l c u l a t i o n s o f t h i s type w i l l p r o b a b l y employ p o t e n t i a l s o b t a i n e d from s c a t t e r i n g e x per-i m e n t s ( s u c h as those suggested i n Chapter 3) and w i l l t r e a t the p e r t u r b i n g p o t e n t i a l i n a more p r e c i s e manner (such as t h a t s u g g e s t e d by Chase, W i l e t s , and Edmonds (1958)). I t i s w o r t h w h i l e , however, to mention one f u r t h e r e f f e c t which i s seen i n the s e c r o s s s e c t i o n s . Tt i s seen i n F i g . 12 t h a t , i n the case o f A l 2 5 , i n c r e a s -i n g the d i f f u s e n e s s parameter,-a, d e c r e a s e s the t o t a l c r o s s ? ^8 s e c t i o n w h i l e , i n the case o f U , the t o t a l c r o s s s e c t i o n i s enhanced. On the o t h e r hand, we know from F i g . 16 t h a t the r a d i a l i n t e g r a l s , F ^ r the o n l y a s p e c t s o f the prob-lem w h i c h are s e n s i t i v e to the d i f f u s e n e s s parameter are g e n e r a l l y s u p p r e s s e d a t low e n e r g i e s f o r a" d i f f u s e - e d g e n02 . no p o t e n t i a l ( w i t h the minor e x c e p t i o n o f F ^ r i n the case o f A l 8 5 ) o The e x p l a n a t i o n o f t h i s b e h a v i o u r o f the c r o s s s e c t i o n s i s u n d e r s t o o d from the f o r m u l a f o r the t o t a l c r o s s s e c t i o n . Eq. 4-37, i n w h i c h i t i s seen t h a t the c r o s s s e c t i o n i n v o l v e s p»o • i n t e r f e r e n c e between the a m p l i t u d e s . F, j ; and the o t h e r ' IVLD , r a m p l i t u d e s which e n t e r the e x c i t a t i o n p r o c e s s ( F ^ , 3?f^ , and 135 i M^D 0*°' ^ n ^ a c^» by comparing the absolute values of these amplitudes on Figs, 15a and 15b and on Figs, 16a and 16b, respectively, (which have been drawn to the same scale) i t i'j i i s clear that the e f f e c t s of F,,^  are seen primarily through' MD,r . * J to interferences with the more dominant terms, / 25*. $ Hence, i n Fig, 12a (Al ) the decrease i n F j ^ r with increasing diffuseness parameter repre-sents a decrease i n constructive interference; i n F i g , 12b (U ) i t represents a decrease i n destructive interference and, hence, an en-hancement of the cross section,, I t i s d i f f i c u l t to see through the complex algebra i n Eq„ 4-37 and to present these argu-ments i n a more s a t i s f a c t o r y form. However, a more careful analysis of e f f e c t s such as-these w i l l be required i n any event when more sa t i s f a c t o r y values for the pion-nucleus pot-e n t i a l become available. 136 CHAT^TSR 5 CONCLUSIONS In this thesis we have attempted to delineate the impor-tant features of the low energy pion-nucleus i n t e r a c t i o n and, i n p a r t i c u l a r , we have emphasized the usefulness of this i n t e r -action as a tool for i n v e s t i g a t i n g the structure of the nuclear surface. In Chapter 2 we have investigated the construction of the pion-nucleus o p t i c a l p o t e n t i a l from the elementary pion-nucleon and pion-deuteron processes; we have shown that a transparent connection exists between the macroscopic and microscopic descriptions of pion interactions i n nuclei and, i n f a c t , that the potential which describes the macroscopic i n t e r a c t i o n involves the nuclear density i n a very direct manner. In Chapter 3 we have used this potential to .investigate the o p t i c a l properties of pions i n nuclei and we have shown that the e l a s t i c s c a t t e r i n g and absorption cross sections depend strongly upon the d e t a i l s of the nuclear•surface through the momentum-dependent absorption. In Chapter 4 we have shown that the cross sections for the e x c i t a t i o n of r o t a t i o n a l states i n deformed nuclei through the opti c a l pio.n-nucleus interaction, depends strongly on the nuclear surface through momentum-dependent e x c i t a t i o n for analogous reasons to those which were discussed i n Chapter 3 for o p t i c a l absorption. We have con-cluded from our calculations and:discussion that low energy pions provide an i n t e r e s t i n g new probe of nuclear structure; i n p a r t i c u l a r , they provide a d i r e c t means of measuring the 137 1 d i s t r i b u t i o n of nucleons i n the nuclear surface. In Chapter 2 we have discussed the construction of an o p t i c a l p o t e n t i a l which describes the low energy pion-nucleus o p t i c a l i n t e r a c t i o n i n terms of elementary pion-nucleon and pion-deuteron sc a t t e r i n g and absorption processes. In par-t i c u l a r , we have shown that this i n t e r a c t i o n involves the density of the nucleus i n a very dir e c t manner. In Sec. 2-1 we have discussed a geometric technique for exh i b i t i n g the structure of the low energy'pion-nucleus op-t i c a l potential; we have argued that, since the elementary pion-nucleon and pion-deuteron scatterings are s-wave and p-wave scatterings, the elementary scattered waves should add to form the t o t a l scattered wave i n an analogous way to that i n which one adds the e l e c t r i c potentials of dipoles and charges i n a c l a s s i c a l d i e l e c t r i c . We have shown that the s-wave scatterings add l i k e the potentials from free charges i n a d i e l e c t r i c and lead to a l o c a l term i n the pion-nucleus poten t i a l ; the p-wave scatterings modify the pion momentum-f i e l d i n the nucleus, i n analogy to the manner i n which the dipoles modify the e l e c t r i c f i e l d i n a d i e l e c t r i c , and lead to a momentum-dependent term i n the pion-nucleus i n t e r a c t i o n which i s analogous to the Lorenz-Lorentz e f f e c t i n a dense c l a s s i c a l medium. We have discussed the derivation of the o p t i c a l potential from a microscopic and quantum mechanical point of view i n Sec. 2-2 where'we have reviewed an e a r l i e r discussion due to Ericson and Ericson (1966). I've have, - shown that the many-138 body aspects of the problem are reduced to tractable propor-tions mainly through the short pion-nucleon scattering lengths and the small mass of the pion. We have shown that these properties allow us to make the impulse approximation, so that we may use the known free space pion-nucleon scattering amp-l i t u d e s , and allow us to simplify the Green's functions which describe the propagation of the pion i n the nucleus, since the elementary pion c o l l i s i o n s may be treated as being e s s e n t i a l l y e l a s t i c . Prom these approximations, and the assumption that the nucleoli scatterers are massive, we were able to perform the nuclear averages and to obtain multiple scattering equations for the pion analogous to those describing multiple scattering i n a c l a s s i c a l medium. In fact, we were able to show that these equations could be reduced to conventional Schroedinger equations for the pion in'which the o p t i c a l p o t e n t i a l involved the density of the nucleus'in a very direOt manner and which had the structure predicted by the electromagnetic analogy. In Sec, 2 - 3 we presented the rather sparse experimental v e r i -f i c a t i o n of the parameters of t h i s potential provided by the JC - mesic x-ray experiments. In Chapter 3 we used t h i s potential to examine.the op-t i c a l properties of pions i n nuclei and to investigate the ro l e of the nuclear surface i n determining e l a s t i c scattering and absorption cross sections, Tn Sec, 3-1 we have made a p a r t i a l wave analysis of the cross sections and we have re-viewed the derivation of well-known formulae which express the 139 p a r t i a l wave phase s h i f t s i n terms of the i n t e r i o r l o g a r i t h -mic derivatives; where necessary, we have extended these for-mulae (which are conventionally written for l o c a l interactions) to momentum-dependent interactions. In Sec.- 3-2 we have used the formalism developed i n Sec. 3-1 and the assumption of a uniform nuclear density to obtain analytic formulae which describe the op t i c a l properties .of the pion-nucleus i n t e r a c t i o n . We have shown that the res-onance -aspects of the problem, which are contained i n the r e a l parts of the i n t e r i o r logarithmic derivatives, are i n -sensitive to the details of the i n t e r a c t i o n because of the long pion wavelength near the top of the poten t i a l b a r r i e r . In fact, we have shown that the resonance aspects of the prob-lem depend only upon the height of the potential ba r r i e r and, for higher p a r t i a l waves, the rea l term i n the momentum-depen-dent i n t e r a c t i o n . We have also shown that, near the top of the potential b a r r i e r , the momentum-dependent absorption pro-cesses, which are described by the imaginary parts of the log-arithmic derivatives, are strongly suppressed due to the small momentum of the pion. In Sec. 3-3 we have stated numerical values for the par-ameters of the pion-nucleus o p t i c a l potential which we have taken to be the zero-energy values given by Ericson and Ericson (1966). In Sec. 3-4 we have used this potential assuming a diffuse-edge nuclear density to give a more quan-t i t a t i v e discussion of the ideas developed i n Sec. 3-2. We have shown that the r e a l parts of the p a r t i a l wave phase s h i f t s 140 d e p e n d s e n s i t i v e l y o n l y u p o n t h e h e i g h t o f t h e p o t e n t i a l b a r -r i e r ( t h r o u g h t h e r e s o n a n c e a s p e c t s . o f t h e p r o b l e m ) a n d u p o n t h e r a d i u s - c h o s e n f o r t h e n u c l e u s ( t h r o u g h t h e p e n e t r a t i o n a s p e c t s o f t h e p r o b l e m ) . We h a v e u s e d t h e s e r e s u l t s t o s u g -g e s t a p r o c e d u r e f o r a n a l y z i n g e m p i r i c a l d a t a . We h a v e m a d e a t h o r o u g h i n v e s t i g a t i o n o f t h e a b s o r p t i v e a s p e c t s o f t h e p r o b l e m a n d we h a v e s h o w n t h a t t h e i m a g i n a r y p a r t o f t h e p a r t i a l w a v e p h a s e s h i f t s , w h i c h d e t e r m i n e s a b s o r p t i o n a n d d i f f r a c t i o n , d e p e n d s v e r y s e n s i t i v e l y u p o n t h e d i f f u s e n e s s o f t h e n u c l e a r s u r f a c e . We. h a v e p r o v i d e d a n e x p l a n a t i o n o f t h i s e f f e c t b y s e g r e g a t i n g t h e l o c a l a b s o r p -t i o n f r o m t h e m o m e n t u m - d e p e n d e n t a b s o r p t i o n i n t h e i m a g i n a r y ; p a r t o f t h e s - w a v e i n t e r i o r l o g a r i t h m i c d e r i v a t i v e . We h a v e s h o w n , b y c o m p a r i n g t h e a b s o r p t i o n f o u n d w i t h a s h a r p - e d g e n u c l e a r d e n s i t y a n d a m o r e c o n v e n t i o n a l d i f f u s e - e d g e n u c l e a r d e n s i t y , t h a t t h e e f f e c t o f t h e d i f f u s e - e d g e i s t o l o w e r t h e e f f e c t i v e p o t e n t i a l b a r r i e r ( w h i c h i s p r o p o r t i o n a l t o t h e n u c l e a r d e n s i t y ) s e e n b y t h e a b s o r b e d p i o n s . T h e s t r o n g s u p -p r e s s i o n o f m o m e n t u m - d e p e n d e n t a b s o r p t i o n t h e r e f o r e o c c u r s a t a l o w e r e n e r g y i n a d i f f u s e - e d g e ' n u c l e u s t h a n a s h a r p -e d g e n u c l e u s . O n t h e o t h e r h a n d , w e k n o w t h e h e i g h t o f t h e p o t e n t i a l b a r r i e r ( a n d , h e n c e , t h e n u c l e a r d e n s i t y ) i n s i d e " t h e n u c l e u s f r o m t h e r e a l p a r t o f t h e p a r t i a l w a v e p h a s e s h i f t s w h i c h c o n t a i n t h e r e s o n a n c e a s p e c t s o f t h e p r o b l e m . We t h e r e - ' f o r h a v e c o n c l u d e d t h a t t h e t h i c k n e s s o f t h e n u c l e a r s u r f a c e c a n b e a c c u r a t e l y d e t e r m i n e d f r o m m e a s u r e m e n t s o f p i o n - n u c l e u s e l a s t i c s c a t t e r i n g a n d a b s o r p t i o n c r o s s s e c t i o n s . We h a v e ' - 141 shown that the main e f f e c t on the analysis of pion-nucleus cross sections of using positive or negative pions, or of using d i f f e r e n t n u c l e i , i s simply to adjust the e f f e c t i v e height of the potential h a r r i e r seen by the pion through the e l e c t r o s t a t i c interaction.. In Sec. 3-5 we have b r i e f l y discussed the isospin and hyperfine terms which enter the pion-nucleus i n t e r a c t i o n . In Chapter 4 we have examined the ex c i t a t i o n of r o t a t i o n a l l e v e l s .in deformed nu c l e i by pions and we have shown that, unlike e x c i t a t i o n processes a r i s i n g from more conventional i n t e r a c t i o n s , the exci t a t i o n process found here depends sen-s i t i v e l y upon the structure of the nuclear surface. In Sec. 4-1 we have b r i e f l y reviewed the description of strongly deformed nuclei provided by the r o t a t i o n a l model and we have discussed the v a l i d i t y of the Distorted Wave^'Born Approxima-tio n description of pion ex c i t a t i o n processes. In Sec. 4-3 we have employed the DWBA to write the pion ex c i t a t i o n amp-li t u d e s i n a form which separately segregates the effects of Coulomb e x c i t a t i o n , l o c a l e x c i t a t i o n , and momentum-dependent excit a t i o n . • - • In Sec. 4-3 we have used the Ericsons' p o t e n t i a l of Sec. 3-3 to calculate the ex c i t a t i o n cross sections for a r e a l i s t i c diffuse-edge potential and for a sharp-edge pot-e n t i a l . We have shown that the structure of the cross sections i s strongly affected by the thickness of the nuclear surface and that this structure i s mainly contained i n the r a d i a l i n t e g r a l s which describe.the momentum-dependent excitation. 142 We have shown that the e f f e c t of the diffuse nuclear surface • i s to lower the e f f e c t i v e potential b a r r i e r so that the strong suppression of momentum-dependent ex c i t a t i o n occurs at a lower energy for a diffuse-edge nucleus than for a sharp-edge nucleu In f a c t , this i s ' i n analogy to our discussion of momentum-dependent absorption i n Sec. 3-4 and we have found here, as we found there, that the main e f f e c t of considering higher p a r t i a l waves i s to ra i s e the e f f e c t i v e b a r r i e r seen by the pion through the c e n t r i f u g a l potential and that the main e f f e c t of considering d i f f e r e n t nuclei or pions of d i f f e r e n t charges i s to change the ef f e c t i v e b a r r i e r through the electro s t a t i c p o t e ntial. We have concluded from these results that pion e x c i t a t i o n cross sections provide a sensitive technique for i n v e s t i g a t i n g the structure of the nuclear surface i n deformed n u c l e i , • • , ' . . The general conclusions of this thesis are that low energy pions have Optical properties i n nuclei which are e a s i l y understood and which are q u a l i t a t i v e l y quite d i f f e r e n t from those encountered with more conventional interactions because of the appearance of a short-range potential b a r r i e r and a strongly momentum-dependent contribution to the i n t e r -action. These properties of the i n t e r a c t i o n , and the fact that the nuclear density enters the pion-nucleus o p t i c a l pot-e n t ! el i n a very d i r e c t manner, allow one to investigate aspects of nuclear structure which have previously not been e a s i l y accessible,, Tn p a r t i c u l a r , an analysis of pion-nucleus e l a s t i c scattering and absorption cross sections and of pion 143 e x c i t a t i o n cross sections i n deformed nuclei should reveal detailed information ah out the d i s t r i b u t i o n of nucleons i n the nuclear surface, information c r u c i a l to a refinement of the detailed models of nuclear structure. BIBLIOGRAPHY" 144 Abashian, A., Cool, R„ , ana Cronin, J.W., Phys. Rev, 104. 855 (1956). Alaer, £. , and Winther, A., "Coulomb Excitation", (Academic Press, Hew York, 1966) Auerbach, E,H. , Qureshi , H.M. , and. Sternheim, M.H. , Phys. Rev. Letters 21, 162 (1968). Baker, W. P. , By f i e l d , H. , and Rainwater', J. , Phys. Rev. 112, 1773 (1958). Bernstein, A., "Advances i n Nuclear Physics, Volume 3", eds. M, Baranger and E, Vogt. (Plenum Press, New York, 1969) Brueckner, K.A. , Serber, R. , and Watson, K.M. , Phys. Rev, 84, 258 (1951). Bugg, W.M. , Condo, G. T. , Cohn, H.O., and H i l l , R. D. , Nucl. Phys. A124, 212 (1969). Burhop^, 'E.H.S. , Nucl. Phys. B l , 438 (1967). ' Chase, D.M. , Wilets, L. , and Edmonds, A. R„ s Phys. Rev. 110, 1080 (1958). Devons, S., and Duerdoth, I . , "Advances i n Nuclear Physics, Volume 2", eds. M„ Baranger and E. Vogt. (Plenum Press., New York, 1969) Eckstein, S.G. , Phys Rev. 129, 413 (1963). Elton, I.R.B., Phys. Letters 26B, 689 (1968). Ericson, M. , and Ericson, T.E.O. , Ann. Phys. (N.Y. ) -36, 323 (1966). Ericson, T.E.O., Lectures given at the International School of Physics, "Enrico Permi", Course XXXVIII, Varenna sul Lago ...di Como, 67/164/5 - TH. 746 (Feb. 9, 1967) Francis, N.C. , and Watson, K.H, , Phys. Rev. 92_, 291 (1953). Frank, R.M, Gammel, J. I. , and Watson, K.M„ , Phys. Rev. 101, 891 (1956). Goldberger, M.L., and Watson, Z.M " C o l l i s i o n Theory". (John Wiley & Sons, New York, 1964) 145 Goldstein, H., " C l a s s i c a l Mechanics". (Addison-Wesley, Reading, Mass., 1950) Greenlees, GJ.V. , Pyle, G.J., and Tang, Y.C., Phys. Rev. Letters 17, 33 (1966) Jackson, J.L., " C l a s s i c a l Electrodynamics", (John.Wiley & Sons, New York, 1962) Eikuchi, X. , and Sawai , M. , "Nuclear Matter and Nuclear Reactions". (John Wiley & Sons,. New York, 1968) K i s s l i n g e r , L., Phys. Rev. 98, 761 (1955). Koltun, L.S., "Advances i n Nuclear Physics, Volume 3", eds. H s Baranger and E, Vogt. (Plenum Press, New York, 1969) K r o l l , N.M. , c i t e d i n Edelstein, R . M . , Baker, P.W. , and Rainwater, J., Phys. Rev. 122, 252 (1961). Lax, M. , Revs. Mod.. Phys. 23, 287 (1951). Messiah,'A., "Quantum Mechanics, Volume 2". (Addison-Wesley,. Reading, Mass., 1962) Nolen, -J.A. Jr. , S c h i f f e r , J.P. , and Williams, N. ,-Phys, Letters 27B, 1 (1968). Peaslee, D. C. , Phys. Rev. E57, 862 (1952). Preston,-M o A. , "Physics of the Nucleus". (Addison-Wesley, Reading, Mass., 1962) Spector, R.M., Phys. Rev. 129, 413 (1964). Temmer, G.M., "International Nuclear Physics Conference, Gatlinhurg, Tennessee, Sept. 12-17,.1966, 223." (Academic Press, New York, 1967) Van Hove, L. , Phys. Rev. 95_, 249 (1954). Vogt, E.W., Revs. Mod. Phys. 34, 723 (1962). Watson, K.M. , Phys. Rev. 89, 575 (1953). Wu, C.S., "International Nuclear Physics Conference, Gatlinhurg, Tennessee, Sept. 18-17, 1966, 409" (Academic Press, New York, 1967) 146 APPENDIX A EQUATIONS FOR THE INTERIOR LOGARITHMIC DERIVATIVES In the present appendix, we w i l l be concerned with eval-uating the i n t e r i o r logarithmic derivatives, given i n Eq. 3-13. F i r s t l y , we w i l l find an i n t e g r a l solution for their imaginary parts, T t y , i n terms of the p a r l a l wave functions, u^(r). Secondly, we w i l l evaluate 7^ for a uni form nucl ear density i n terms of the i n t e r i o r wave func-tions.. A-1 Integral Equation for Absorption In Eq. 3-4 we have written r a d i a l Sehroedinger equa-tions for the p a r t i a l waves, u ( r ) : Zp ••* 2^ + M±H (1 + ^ e ( r ) ) u , ( r ) r 2 4- tt. u ( r ) + [Y f r ) + V ( r ) + - i W ( r l u«(r) Zp r * U c J = Euj,(r) ( J? = 0, 1, S, ...) . We w i l l now use this equation to obtain an i n t e g r a l equation for the absorptive term, 71^ , of the logarithmic derivative, Jj{ ° Le t us f i r s t de fine the quanti ty, ( A - i ) f t ( , ) . : i i ^ M ) ^ _ I t i s then e a s i l y shown from Eq. 3-4 that 147 (A-8) f ' f r ) = ( l + * e ( r ) ) ,i2 ^ [ y c ( r ) + v ( r ) + l W ( r ) | -. r H 2 L J k< (r) Let us aivide fjj(r) into i t s r e a l and imaginary parts, f^(r) , and f ^ f r ) : ( A-3) f 4 f r ) = fff.r) + i f ^ ( r ) # Then, from Eq. A - 2 and remembering Eq. 3-1*0, (A'-4) . a f J( r ) = <^(r) M i l l + * f W(r) dr r • r 1 1 1 21 # 2 . uj{(r) ^ Now, from Eq. A-1 (and suppressing r ) , u<> v* !k + ( i ) JL 81 , P ' o*^ t # 2 ( 1 4 ^ e ) ( 5 L ) 2 - ^ ^ - e u. UJ0 Substituting E q . A-5 i n E q . A-4, we have that (suppressing r) 2 f A - 6 ) d f J e M + l ) , . dr r 4 u,' h u^ dr Ju{l2 As i s e a s i l y checked, a s o l u t i o n to Eq , A-6 i s given by the i n t e g r a l equation 148 (A-7) f ^ W ( r ' ) 0 Tf dr L 1(1+-1) dr -r u 0 ( r ) Now outside the i n t e r a c t i o n region, say at the matching e radius, R . oC (r) vanishes. In addition, we have from * o , Eq. A-1 that f (r) i s then equal to  JJ± (Eq, 3-11). Thus Ro 7fy i s given hy the r e l a t i o n , (A-8) r , ( R ) = - 0/°W(r) {u,(r)| 3dr Ju«(Ro)|2 0 ) ° ^ ? ( r ) a u^fr) ar — ^ — u^(r) p r dr M M 2 A-2 Uniform D i s t r i b u t i o n We w i l l here derive the logarithmic derivative associated with. Eq. 3-4 for a uniform d i s t r i b u t i o n , fc. f, Eq. 3-18). Let us f i r s t define the quantity (A-9) d u j r ) g^(r) = (1 4-<e(r) ) r u jjfr)/r I t i s then e a s i l y shown that Eq. • 3-4 cam be rewritten i n terms of the R i c a t t i equat'on,. 149 (A-10) g j ( r ) = ( l + * e f r ) ) H i t H . . + l£ jV f r ) ! V ( r ) f i W ( r r r f -h2 L c k 2 _ 2 g$(r) (r) 1-f- oce(r) Now, from. Eq,. 3-4, i t i s seen that inside R0, ( A - l l ) ^ ( r ) = A j p r j ^ b r ) , where A^ i s a normalization constant and. where b r J H- •< g 0 o o 7 From Eq. A-10 i t i s seen that the c o e f f i c i e n t functions of the d i f f e r e n t i a l equation contain no s i n g u l a r i t i e s so that l i r a r -» R Thus, (A-13) d dr Ujj f r ) /r r=R. 0 0 ,d fo(br) fy.(br) r=R o » or, rewriting Eq. A-13 in/teimis of fA-14) >^(R0) « (1 4- *|) r d _ ^ ( b r ) dr_ TbrT' r=R, 150 APPENDIX B NOTES ON EVALUATION OF PION EXCITATION CROSS SECTIONS In the present Appendix we w i l l evaluate some quant-i t i e s which are required i n the evaluation of the pion excitation, cross sections given i n Sec. 4-2. In Sec. B - l , we provide d e t a i l s of the evaluation of the matrix elements which enter the pion e x c i t a t i o n cross section formulae. In Sec. B-2, we evaluate the e l e c t r o s t a t i c p o t e n t i a l for a uni-form deformed charge d i s t r i b u t i o n . B - l Evaluation of DWBA Matrix Elements In the text, Sec. 4-2, we have shown that i n DWBA the cross sections for the e x c i t a t i o n of a r o t a t i o n a l state i n a strongly deformed nucleus are given by the expression, (4-19) a j X V k . N2 ' , M- v i 2 . (4-5a) 23^ + 1 en2 _ where, taking the example of a K - 0 band (c.f. Eq. 4-5), here = ( , P, V ) are the Euler angles (in the conventions of Goldstein. (1950)) which rotate the space-fixed axes (x,y,z) into the body-fixed axes (1,2,3) (see text); and $^(r) are solutions of Eqs. 4 - l i b and 4-17b, 'respect-i v e l y ; and 1T'(r) i s the perturbing potential:' (4-7) . ^ We begin o\ir evaluation by rewriting Eq. 4-7 i n the 151 space-fixed co-ordinate system,, To do this l e t us f i r s t note that (B-1) ^ e b , * b ) = l ^ ^ f ^ ) <(e,*) where f r , 6, (J>) are the spherical co-ordinates i n the space-fixed system and f r ^ , 0^, ^ ) are the spherical co-ordinates i n the body_fixed system. Both systems are chosen to have th e i r origins at the center of the nucleus so that r = r^. Using Eg., B-1, we can write Eq, 4-7 i n the form, We s h a l l now assume that we have an unpolarized beam i n which case we average over i n i t i a l states and sum over f i n a l states. To do this we see from Eq, 4-21 that we f i r s t require matrix elements of the form (taking K .= 0): (B-3a) -tt< drd^dx' $>0(r) 2Iy +1 SI**1" . 8 1 ' where (B_3b) "Vrf ; M f M x 4? ^ r ^ a*' 2. ^ - ^ T f c t j M ' M f t K ac 8I*+1 8/i s J L en* i 81*+1 ^ k Jwo'(V ^ * o ( o < k > $ 152 Remembering the r e l a t i o n (c.f. Preston (1962)), o $ . J 3 V $ l Z r r <PiSv " — ^ J i JpMiMpJ J J , > k l E l to2K2 M 3 K 3 k 2 J 3 + 1 1 2 1 2 2 3 we find that (after some elementary manipulations), x <:^ioo|i,o> Now, averaging over i n i t i a l states and summing over f i n a l states, we have that i ^ U h 2 J 2im M.M* ' k a U*h S/ -3I.C-+ 1 ^acWac M M . ° W . M . M C W ; M » M where (B1 6) A I < '= ^100/ L O * " ( z - o ) - • ' + 1 For hands where Z £ 0 we need only replace Eq_B 4-5a with Eq_0 4-5b of the test i n the above derivation. I t i s then e a s i l y shown that the generalization of Eq. B-6 i s given by the r e l a t i o n , ~ — ^ ( 7 Q V e n ) . (B-7b) At* ~ 0 , (7. odd) „ B-2 Evaluation of Electromagnetic Potential for a Deformed  Charge D i s t r i b u t i on .In the present section, we w i l l evaluate the e l e c t r o -magnetic p o t e n t i a l , V fr) , for a deformed charge d i s t r i b u t i o n of uniform charge density,^pfr)„ As i s known from elementary electromagnetic theory, (B-8) 1 r 1 T Y * t V fr) = f 7 f £ > a £ • C ~ J r ' - r so that, because of the i n t e g r a l , we make neg l i g i b l e errors by neglecting the diffuseness of the nuclear edge. We take the charge d i s t r i b u t i o n to have a spheroidal deformation, as i n the text (Eq. 4-23): (B-9a) • 1 + J»"*§(e 1 ) )_ ^ r _ 2 for a uniform d i s t r i b u t i o n , (B-9b) J" fr) - />0H(R-r) , where . (B'-9c) n '-- ' I f Xs)'1 and where H(R-r) i s the well-known. Heaviside function. Prom 154 Eqs„ B-9a and B-9b t (B-10) y(r) - P H ( R - r ) 4 - ^ B r / ^ f R - r ) whence, (B-ll) * § p ^ o ( V k > Y 2 ( e . ' 0 0^  y |r - P'J To evaluate the angular Integrals i n Eq, B - l l , i t i s convenient to note the i d e n t i t y , (B-12) $ Ji I^TTl = j lo ^ - f ^ m r j * (e»,At) where (r^. } i s the smaller (larger) of r and R„ Using t h i s i d e n t i t y , we find that (B-13a) d J 2 . r _ r 4c K 4Jt r r < r 1 : r y r 1 ; and that (B-13b) 4n r ' 2 5 " ^ VpJtA) , r > r » Substituting Eqs, B _ 1 3 i n Eq, B - l l , and noting Eq„ B-9, we find that APPENDIX C The Nuclear O p t i c a l Model and Wave Properties: Ba r r i e r Penetration, R e f l e c t i o n , Absorption and Resonance by Georges Michaud, Department of Astronomy C a l i f o r n i a I n s t i t u t e of Technology and Leonard Scherk and E r i c h Vogt, Physics Department University of B r i t i s h Columbia - 1 -ABSTRACT A d e t a i l e d s t u d y o f the wave p r o p e r t i e s o f the n u c l e a r o p t i c a l . m o d e l i s p r e s e n t e d to e l u c i d a t e the problem o f b a r r i e r p e n e t r a t i o n by c h a r g e d p a r t i c l e s and t o remove some o f the m y s t i q u e o f o p t i c a l model c a l c u l a t i o n s . The wave p r o p e r t i e s and the c o n c o m i t a n t p e n e t r a t i o n are most s t r a i g h t f o r w a r d , f o r square w e l l s f o r w h i c h the r e s o n a n c e , r e f l e c t i o n and p e n e t r a t i o n a r e e a s i l y a s c r i b e d t o s e p a r a t e f a c t o r s . We show t h a t the wave p r o p e r t i e s o f more g e n e r a l d i f f u s e - e d g e o p t i c a l p o t e n t i a l s a c h i e v e a s i m i l a r s i m p l i c i t y by the c o n s t r u c t i o n o f an e q u i v a l e n t square w e l l (ESW'J w h i c h has the same r e s o n a n c e , p e n e t r a t i o n and a b s o r p t i o n f a c t o r s as the o p t i c a l p o t e n t i a l b u t w h i c h d i f f e r s i n i t s r e f l e c t i o n f a c t o r . A g e n e r a l c o n s t r u c t i o n o f the ESW i s g i v e n and we a p p l y i t t o the f o l l o w i n g p r o b l e m s : (1) the v e r y narrow s i n g l e - p a r t i c l e r e s o n a n c e s o f r e a l o p t i c a l p o t e n t i a l s w h i c h o c c u r a t e n e r g i e s f a r below t h e Coulomb b a r r i e r ( 2 ) the n u c l e a r a b s o r p t i o n c r o s s s e c t i o n s i n the p r e s e n c e o f b a r r i e r s ; . (3) the c a l c u l a t i o n o f a b s o r p t i o n c r o s s s e c t i o n s a t a s t r o p h y s i c a l e n e r g i e s (extreme b a r r i e r p e n e t r a t i o n ) e m p l o y i n g o p t i c a l models f i t t e d to d a t a a t h i g h e r e n e r g i e s ; ('-I) the v a l u e o f the n u c l e a r r a d i u s and sum r u l e , l i m i t s a p p r o p r i a t e to the . a n a l y s i s o f n u c l e a r r e a c t i o n s . I n some cases o f extreme b a r r i e r p e n e t r a t i o n the e q u i v a l e n t s q uare w e l l f a i l s t o y i e l d a l l the p r o p e r t i e s . F o r example, c a s e s a r e d e s c r i b e d where the b u l k o f the a b s o r p t i o n may a t t a i n i n the d i s t a n t " t a i l " o f the i m a g i n a r y term i n the o p t i c a l p o t e n t i a l : the c o r r e s p o n d i n g r e a c t i o n r a t e s can y i e l d i n f o r m a t i o n about the b e h a v i o u r o f the n u c l e u s a t d i s t a n c e s much beyond.the normal n u c l e a r r a d i u s . - 2 -1. I n t r o d u c t i o n ' The b e h a v i o u r o f most n u c l e a r r e a c t i o n s a t low energy -p a r t i c u l a r l y t h o s e o f i n t e r e s t f o r a s t r o p b y s i c a l systems - i s dominated by Coulomb and a n g u l a r momentum b a r r i e r s . E a r l y treatments''"'"^ o f such r e a c t i o n s employed a s i m p l e p i c t u r e , the " b l a c k n u c l e u s " o r " b l a c k box" p i c t u r e . I n t h i s p i c t u r e a bombarding p a r t i c l e was v i e w e d as p a s s i n g t h r o u g h known Coulomb and a n g u l a r momentum b a r r i e r s up t o the n u c l e a r r a d i u s . A t the n u c l e a r r a d i u s i t was a b s o r b e d by the " b l a c k box". T h i s e a r l y p i c t u r e was based on the s h o r t range and g r e a t s t r e n g t h o f the n u c l e a r f o r c e s . D u r i n g the l a s t two decades a m i c r o s c o p i c model o f n u c l e a r s t r u c t u r e has emerged which has changed our view o f n u c l e a r r e a c t i o n s and o f b a r r i e r p e n e t r a t i o n . I n s p i t e o f the s h o r t range and g r e a t s t r e n g t h o f n u c l e a r f o r c e s i t has been f o u n d t h a t i n z e r o -o r d e r n u c l e i may be r e g a r d e d as composed o f n e u t r o n s and p r o t o n s moving i n o r b i t s o r s h e l l s and i n t e r a c t i n g w i t h moderate p o t e n t i a l s o f f i n i t e r a n g e . The s i n g l e - p a r t i c l e o r b i t s o r s h e l l s a r e those o f an a p p r o p r i a t e average p o t e n t i a l w e l l . The v e s t i g e s o f such a s i n g l e - p a r t i c l e p i c t u r e r e m a i n i n n u c l e a r r e a c t i o n s . The n e u t r o n s and p r o t o n s a t low energy e x h i b i t " g i a n t " resonances''^ a t the p o s i t i o n o f the " s i n g l e - p a r t i c l e l e v e l s o f the average p o t e n t i a l w e l l . The re s o n a n c e s are broadened by n u c l e o n - n u c l e o n i n t e r a c t i o n and, a t h i g h energy, the o b s e r v e d r e s o n a n c e s resemble t h o s e o f the Ramsauer-Townsend. e f f e c t i n the e l e c t r o n bombardment o f atoms. The o p t i c a l model o f n u c l e a r r e a c t i o n s a c c o u n t s f o r 7 the o b s e r v e d s i n g l e - p a r t i c l e e f f e c t s . I n the o p t i c a l model, the i n t e r a c t i o n o f a bombarding p a r t i c l e and a t a r g e t n u c l e u s i s d e p l e t e d i n terms o f a complex p o t e n t i a l w e l l . The r e a l p a r t o f the p o t e n t i a l r e f r a c t s the i n c o m i n g .waves and the i m a g i n a r y p a r t a b s o r b s them. The magnitude o f c r o s s s e c t i o n s i s d e t e r m i n e d by th e magnitude and shape o f the o p t i c a l p o t e n t i a l . When Coulomb and a n g u l a r momentum b a r r i e r s a r e p r e s e n t the whole r e a l p o t e n t i a l o f the system i s the sum o f the r e a l p a r t o f the o p t i c a l p o t e n t i a l w i t h the .Coulomb p o t e n t i a l and c e n t r i p e t a l p o t e n t i a l . Our aim i n the p r e s e n t paper i s t o u n d e r s t a n d the f o r m a t i o n o f the compound n u c l e u s i n the p r e s e n c e o f b a r r i e r s and o p t i c a l p o t e n t i a l s : the a b s o r p t i o n o f waves by the i m a g i n a r y p a r t o f the o p t i c a l p o t e n t i a l y i e l d s the compound n u c l e u s f o r m a t i o n c r o s s s e c t i o n ; the b a r r i e r s dominate t h e b e h a v i o u r o f the wave a m p l i t u d e s i n the a b s o r b i n g r e g i o n . The f o r m a t i o n o f the compound n u c l e u s has the f o l l o w i n g ' a s p e c t s w h i c h a r e t r e a t e d i n subsequent s e c t i o n s o f t h i s p a p e r . 1) On the one hand.the o p t i c a l model i s a wave model and d e a l s w i t h the s i m p l e p r o p e r t i e s o f wave p e n e t r a t i o n , r e f r a c t i o n , r e f l e c t i o n and a b s o r p t i o n . On the o t h e r hand, the o p t i c a l p o t e n t i a l 7 i n common usage have many p h e n o m e n o l o g i c a l p a r a m e t e r s . I n o r d e r t o u n d e r s t a n d how v a r i a t i o n s i n the para m e t e r s a f f e c t c r o s s s e c t i o n s i t .is u s e f u l t o d e s c r i b e the r e l a t i o n s h i p between the parameters and. the b a s i c wave p r o p e r t i e s . I 1 1 - II - -j I I 2) Many o f the r e a c t i o n s o f . i n t e r e s t i n a s t r o p h y s i c s o c c u r a t such low energy t h a t they a r e r a r e l y measured i n the l a b o r a t o r y . I n such cases i t i s t e m p t i n g t o use an o p t i c a l model w h i c h f i t s the more abundant d a t a a t h i g h e r e n e r g i e s t o ' c a l c u l a t e the d e s i r e d r a t e s . Thus the o p t i c a l model s e r v e s as an e x t r a p o l a t i o n f o r m u l a . Our a n a l y s i s o f the wave p r o p e r t i e s f o r the n u c l e a r o p t i c a l model a l l o w s us t o u n d e r s t a n d the f a c t o r s a f f e c t i n g the e x t r a p o l a t i o n . 3) The e a r l y b l a c k n u c l e u s models a r e s t i l l f r e q u e n t l y employed, p a r t i c u l a r l y i n a s t r o p h y s . i c a l c a l c u l a t i o n s where a v e r y l a r g e number o f r e a c t i o n r a t e s a r e r e q u i r e d i n a s i n g l e c a l c u l a t i o n . The o p t i c a l model i s then o f t e n too cumbersome. I t i s t h e r e f o r e u s e f u l t o know the r e l a t i o n between the c r o s s s e c t i o n o f the b l a c k n u c l e u s model and the c o r r e s p o n d i n g c r o s s s e c t i o n s o f the o p t i c a l model. We d e v e l o p such, r e l a t i o n s h i p s ( s e e . 2 and s e c . 3) as part- o f a g e n e r a l c o n n e c t i o n between d i f f u s e edge p o t e n t i a l s ' a n d square w e l l s . 8 9 'I) The t h e o r y o f resonance r e a c t i o n s i s a v e r y g e n e r a l and p o w e r f u l framework f o r d e s c r i b i n g n u c l e a r r e a c t i o n s : i t has been s u c c e s s f u l l y a p p l i e d t o a n a l y z i n g compound n u c l e u s r e sonances and t o the c r o s s s e c t i o n s o f the s t a t i s t i c a l t h e o r y o f n u c l e a r r e a c t i o n s 1 " " " i n w h i c h a v e r a g e s a r e made o v e r the compound n u c l e u s r e s o n a n c e s . I t p r o v i d e s a j j u s t i f i c a t i o n ' ' * 1 f o r the d e s c r i p t i o n o f the average c r o s s s e c t i o n s by an o p t i c a l model p o t e n t i a l . N o n e t h e l e s s , the resonance t h e o r y has some o f Hie u n d e s i r a b l e f e a t u r e s o f the e a r l y b l a c k box p i c t u r e : i t r e l i e s e x p l i c i t l y on. the use o f a d e f i n i t e n u c l e a r r a d i u s . Many o f the common r e s u l t s o f the resonance t h e o r y i m p l y a square edge t o the n u c l e a r s u r f a c e . I n an e a r l i e r 3 2 a r t i c l e one o i us t r i e d t o show how the d i f f u s e edge o f the o p t i c a l model c o u l d be accommodated i n the g e n e r a l resonance t h e o r y j f o r n e u t r o n r e a c t i o n s . That accommodation i s e x t e n d e d i n the p r e s e n t a r t i c l e ( s e c . 1) t o c o v e r r e a c t i o n s i n v o l v i n g Coulomb and a n g u l a r momentum b a r r i e r s . 5) I n s e e k i n g t o d e s c r i b e a l l n u c l e a r r e a c t i o n s w i t h the p h y s i c a l i d e a s o f the o p t i c a l p o t e n t i a l we need t o re-examine the v a l u e o f the n u c l e a r r a d i u s . I n the s e v e r a l decades d u r i n g w h i c h the s i m p l e b l a c k n u c l e u s p i c t u r e was'employed t o d e s c r i b e n u c l e a r 3 3 / 3 l / " * r e a c t i o n s the u s u a l c h o i c e o f r a d i u s was R = l . ' l (A^ + A^ ) f e r m i s . Here A.j i s the a t o m i c w e i g h t o f the t a r g e t n u c l e u s and A2 t h a t o f the bombarding p a r t i c l e . The n u c l e a r r a d i u s o f the o p t i c a l model i s determined, q u i t e s e n s i t i v e l y by the f i t s t o the d i f f r a c t i o n p a t t e r n s o f e l a s t i c s c a t t e r i n g d a t a and i t t u r n s out t o be much s m a l l e r , e.g. f o r n u c l e o n s i t s v a l u e i s a p p r o x i m a t e l y ] /3 R = 1.25 A£ f e r m i s . ( I n t u r n , the n u c l e a r charge r a d i u s as measured i n e l a s t i c s c a t t e r i n g i s s m a l l e r s t i l l ; the charge r a d i u s 1/3 i s R = 1.09 A^ f e r m i s . The s m a l l d i f t e r e n . e e o f t h i s . r a d i u s from the n o r m a l r a d i u s o f the n u c l e o n - n u c l e u s i n t e r a c t i o n l i e s i n a number o f e f f e c t s ; s u c h as c o r e p o l a r i z a t i o n , w h i c h are beyond the scope o f the p r e s e n t w o r k ) . There i s an overwhelming amount o f r e c e n t e v i d e n c e s u g g e s t i n g t h a t the s m a l l e r o p t i c a l model r a d i u s i s t h e ' r i g h t one. Why was the e a r l i e r model wrong f o r s e v e r a l decades, p a r t i c u l a r l y f o r r e a c t i o n r a t e s i n v o l v i n g b a r r i e r s where the c r o s s s e c t i o n , depends q u i t e s t r o n g l y on the c h o i c e , o f n u c l e a r -- 6 -r a d i u s ? Our e l u c i d a t i o n o f the o p t i c a l p o t e n t i a l s u g g e s t s a f a i r l y u n i v e r s a l e x p l a n a t i o n . The e a r l y b l a c k - n u c l e u s model i m p l i e d a square n u c l e a r edge and t h e r e f o r e had an u n r e a l i s t i c amount o f wave r e f l e c t i o n . I t compensated f o r t h i s a t t e n u a t i o n o f a b s o r p t i o n by an a p p r o p r i a t e i n c r e a s e i n the n u c l e a r r a d i u s . Some p r e l i m i n a r y r e s u l t s o f our p r e s e n t i n v e s t i g a t i o n ] 3 were d e s c r i b e d b r i e f l y i n an e a r l i e r p a p e r . I n our p r e s e n t p a p e r we t r y to" g i v e a complete a c c o u n t o f a l l a s p e c t s o f b a r r i e r p e n e t r a t i o n and the o t h e r wave p r o p e r t i e s w i t h i n t h e scope o f the o p t i c a l model. We b e g i n our a n a l y s i s (Sec. 2) w i t h a comparison o f the wave p r o p e r t i e s f o r d i f f u s e - e d g e o p t i c a l p o t e n t i a l s and f o r s i m i l a r s q u a r e w e l l s . The m o t i v e f o r b r i n g i n g s q uare w e l l s , i n t o the a n a l y s i s .is t w o - f o l d : f i r s t o f a l l , the v a r i o u s c r o s s s e c t i o n s f o r a square w e l l can be w r i t t e n i n the s i m p l e , f a m i l i a r forms w h i c h make p o s s i b l e easy e v a l u a t i o n o f the wave p r o p e r t i e s ; s e c o n d l y , f o r a square w e l l the b a s i c wave p r o p e r t i e s are e a s i l y s e p a r a t e d - b a r r i e r p e n e t r a t i o n o c c \ i r s o n l y beyond the square w e l l r a d i u s , r e f l e c t i o n o c c u r s a t the square w e l l r a d i u s and a b s o r p t i o n and resonance w i t h i n t h e square w e l l r a d i u s . The v e h i c l e f o r the co m p a r i s o n (See. 2) i s the a b s o r p t i o n c r o s s s e c t i o n . The comparison y i e l d s the dominant r e s u l t o f our work: i t i s f o u n d t h a t f o r each d i f f u s e - e d g e o p t i c a l p o t e n t i a l an e q u i v a l e n t square w e l l can be d e f i n e d u n i q u e l y . The c r o s s s e c t i o n s o f the d i f f u s e - e d g e p o t e n t i a l have the same s i m p l e form as tho s e o f i t s e q u i v a l e n t square w e l l w i t h a l l the b a s i c wave p r o p e r t i e s c l e a r l y s e p a r a t e d . The o n l y q u a n t i t a t i v e d i f f e r e n c e between the wave p r o p e r t i e s o r c r o s s s e c t i o n s o f the two w e l l s i s shown to r e s i d e i n the p e n e t r a t i o n and s h i f t f u n c t i o n s . The c o n v e n t i o n a l square w e l l p e n e t r a t i o n and s h i f t f u n c t i o n s a p p l y to the d i f f u s e - e d g e j ' w e l l when they are m u l t i p l i e d by a r e f l e e t i o . n f a c t o r w h i c h depends on the s u r f a c e t h i c k n e s s ' and reduced mass b u t n o t on the ch a r g e , energy o r a n g u l a r momentum o f . t h e bombarding p a r t i c l e . Thus the breakdown o f the o p t i c a l model i n t o b a s i c wave p r o p e r t i e s i s j a c h i e v e d by means o f e q u i v a l e n t square wells.. I n Sec. 3 we c a r r y o u t , f o r d i f f u s e - e d g e o p t i c a l p o t e n t i a l s , the e x p l i c i t c o n s t r u c t i o n o f e q u i v a l e n t square w e l l s and d e t e r m i n e the c o r r e s p o n d i n g r e f l e c t i o n f a c t o r s . I n Sec. 1 we show t h a t our a n a l y s i s o f wave p r o p e r t i e s based on the b e h a v i o u r o f a b s o r p t i o n c r o s s s e c t i o n s (Sec. 2) a l s o a p p l i e s t o p u r e l y r e a l w e l l s : the sharp r e s o n a n c e s o f a d i f f u s e - e d g e r e a l p o t e n t i a l a r e g i v e n a p p r o x i m a t e l y i n terms o f the sharp resonances o f the e q u i -v a l e n t square w e l l when the p e n e t r a t i o n f a c t o r and s h i f t f u n c t i o n o f the l a t t e r a r e m u l t i p l i e d by a known r e f l e c t i o n f a c t o r . I n Sec. 5 we d e s c r i b e the b e h a v i o u r o f g e n e r a l o p t i c a l model a b s o r p t i o n c r o s s s e c t i o n s and the c i r c u m s t a n c e s under w h i c h t h e r e a r e some d e p a r t u r e s o f t h e wave p r o p e r t i e s from those w h i c h may be d e a l t w i t h i n . terms o f e q u i v a l e n t square w e l l s . I n Sec. 6 we use our a n a l y s i s o f the b a s i c wave p r o p e r t i e s o f the o p t i c a l p o t e n t i a l t o d i s c u s s the u n c e r t a i n t i e s i n the c o n v e n t i o n a l approach o f a s t r o p h y s i c s to the problem o f extreme b a r r i e r p e n e t r a t i o n . I n Sec. 7 we d i s c u s s a number o f c o n c l u s i o n s r e s u l t i n g from our work i n c l u d i n g the p l a c e o f the n u c l e a r r a d i u s i n the a n a l y s i s o f n u c l e a r r e a c t i o n d a t a . The o p t i c a l model has a f i r m f o u n d a t i o n o n l y f o r n u c l e o n r e a c t i o n s b u t i t has a l s o been found to be a v e r y u s e f u l t o o l f o r d e s c r i b i n g heavy i o n r e a c t i o n s . I t i s more ambiguous f o r heavy i o n s the r e sonance e f f e c t s w h i c h a r e the s t r o n g e s t s i g n a t u r e o f the optica'' model are l a r g e l y m i s s i n g i n t h i s case b u t i t can accommodate the s t r o n g a b s o r p t i o n and the d i f f u s e n u c l e a r edge w h i c h ar e o f i m p o r t a n c e f o r heavy i o n s . T h e r e f o r e the o p t i c a l model i s perhaps the b e s t s i m p l e model, f o r d e a l i n g w i t h heavy i o n s . Because o f ' the mass and charge o f the heavy i o n s many o f the b a s i c wave p r o p e r t i e s a r e more complex than f o r n u c l e o n s . Such r e a c t i o n s s e r v e as a u s e f u l measure o f the s u c c e s s ' o f our approach. A l t h o u g h we s h a l l seele t o be g e n e r a l most o f our r e s u l t s w i l l be i l l u s t r a t e d w i t h a p a r t i c u l a r 'case: the i n t e r a c t i o n o f a l p h a p a r t i c l e s w i t h 32 S. T h i s example o f f e r s extreme b a r r i e r p e n e t r a t i o n and i s t y p i c a l o f the a s t r o p h y s i c a l r e a c t i o n r a t e s t o w h i c h our r e s u l t s might be a p p l i e d . R e c e n t l y a m o d i f i e d v e r s i o n o f our methods was a p p l i e d t o . an extreme case o f b a r r i e r p e n e t r a t i o n - t h a t o f a l p h a decay i n heavy ]' I n u c l e i . O t h e r a p p l i c a t i o n s a r e s u g g e s t e d i n the f o l l o w i n g s e c t i o n s . 2. Comparison o f the A b s o r p t i o n C r o s s S e c t i o n s o f D i f f u s e - E d g e O p t i c a l P o t e n t i a l s w i t h t h o s e of Square W e l l s I n o r d e r t o d i s p l a y the b a s i c wave p r o p e r t i e s o f d i f f u s e -edge o p t i c a l p o t e n t i a l s we s h a l l compare them w i t h square w e l l s f o r w h i c h the wave p r o p e r t i e s a r e much more p e r s p i c u o u s . We b e g i n by c h o o s i n g and p a r a m e t e r i z i n g those terms o f the modern n u c l e a r o p t i c a l . p o t e n t i a l w h i c h ar e o f i n t e r e s t t o u s . F o r most s c a t t e r i n g and a b s o r p t i o n , c r o s s s e c t i o n s the p r i n c i p a l terms o f the o p t i c a l p o t e n t i a l may be w r i t t e n V O ) •= - V (1 + e ^ r ~ R o ^ a ) " 3 1 W(r) (1) Here V q i s the " d e p t h " o f the r e a l p a r t o f the p o t e n t i a l and has a v a l u e i n the n e i g h b o u r h o o d o f 50 MeV f o r n u c l e o n s and c o n s i d e r a b l y more f o r heavy i o n s ; R i s the n u c l e a r r a d i u s whose v a l u e i s about o ] /3 i 1.25 A fm f o r n u c l e o n s (where A i s the at o m i c w e i g h t o f the t a r g e t n u c l e u s ) and a s l i g h t l y l a r g e r v a l u e f o r heavy i o n s ; a i s the " s u r f a c e t h i c k n e s s " w h i c h has a v a l u e o f about 0.5 fm f o r n u c l e o n s and heavy i o n s ; I W(r) i s the i m a g i n a r y term o f the o p t i c a l p o t e n t i a l w h i c h l e a d s t o a b s o r p t i o n . The i m a g i n a r y term i s u s u a l l y c h o s e n _ t o have a shape e i t h e r l i k e t h a t o f the r e a l term, .W(r)' = - W (1 + e ^ V ^ ) " 1 (2) w h i c h i s c a l l e d "volume a b s o r p t i o n " o r , a l t e r n a t i v e l y , W(r) i s chosen t o be l a r g e r i n the r e g i o n o f the n u c l e a r s u r f a c e ( t h i s c h o i c e i s c a l l e d " s u r f a c e a b s o r p t i o n " ) . I n the main p a r t o f our d i s c u s s i o n we w i l l choose volume a b s o r p t i o n i n o r d e r t o be s p e c i f i c : i n Sec. 5 we w i l l d i s c u s s i n some d e t a i l the. e f f e c t o f the shape o f W(r) on the wave p r o p e r t i e s . The dep t h p a r a m e t e r , W , t y p i c a l l y has a v a l u e o f 2-5 MeV f o r n u c l e o n s and somewhat l a r g e r v a l u e s f o r heavy i o n s . We have chosen t o c o n c e n t r a t e on the p r i n c i p a l terms, (1) , of the o p t i c a l p o t e n t i a l because the o t h e r terms, which we n e g l e c t , do n o t mo d i f y our main r e s u l t s about wave p r o p e r t i e s . F o r example, the s p i n - o r b i t c o u p l i n g term w h i c h i s p r o p o r t i o n a l t o JL . _s (where X i s the o r b i t a l a n g u l a r momentum and £ the s p i n ) makes the o p t i c a l . model phase s h i f t s ' depend on b o t h J and j Q = + s) . Our t r e a t m e n t below must then a l s o be c a r r i e d out s e p a r a t e l y f o r each v a l u e o f X. and j . - S i m i l a r l y the i s o t o p i c s p i n term when i m p o r t a n t makes i t n e c e s s a r y t o t r e a t n e u t r o n s and p r o t o n s on a common f o o t i n g and l e a d s t o u n u s u a l c r o s s s e c t i o n ; c o n t r i b u t i o n s s u c h as the " c j u a s . i c l a s . t i c (p, n) r e a c t i o n s . We c o u l d a l s o accommodate such terms i n our a n a l y s i s . However t o c l a r i f y the wave p r o p e r t i e s we w i l l i g n o r e such r e f i n e m e n t s and d e a l o n l y w i t h the dominant terms o f ( 1 ) . The o p t i c a l p o t e n t i a l o f (1) (or Saxon-Woods p o t e n t i a l w h i c h t h e p a r t i c u l a r w e l l shape o f (1) i s f r e q u e n t l y c a l l e d ) w i l l be shown t o e x h i b i t a l l o f the b a s t e wave p r o p e r t i e s : b a r r i e r p e n e t r a t i o n , r e s o n a n c e , r e f l e c t i o n and a b s o l u t i o n . The most s t r a i g h t f o r w a r d m a n i f e s t a t i o n o f thes e p r o p e r t i e s o c c u r s i n the a b s o r p t i o n c r o s s s e c t i o n and t h e r e f o r e we choose t o b e g i n our a n a l y s i s w i t h i t . F o r any r e a c t i o n c h a n n e l cl ( l a b e l s t h e p a i r o f r e a c t i o n p r o d u c t s and. t h e i r s t a t e o f e x c i t a t i o n ) d e s c r i b e d by an o p t i c a l p o t e n t i a l w i t h phase s h i f t s S the a b s o r p t i o n c r o s s s e c t i o n i s : ' CT.(abs) •= Or A 2 ) £ , ( 2 * + 1) T (oQ (3) where 2 i 6 2 T><) S i - e W (<!) The T f ' i ) a r e c a l l e d , n u c l e a r t r a n s m i s s i o n f u n c t i o n s . A l l o f the p r o p e r t i e s o f the a b s o r p t i o n c r o s s s e c t i o n a r e tho s e o f the t r a n s -m i s s i o n f u n c t i o n s o r , e q u i v a l e n t l y , those o f the phase s h i f t s . T h e r e f o r e the n u m e r i c a l c a l c u l a t i o n o f the a b s o r p t i o n c r o s s s e c t i o n i s s t r a i g h t f o r w a r d - one needs m e r e l y t o i n t e g r a t e the S c h r o e d i n g e r e q u a t i o n c o n t a i n i n g the o p t i c a l p o t e n t i a l t o a l a r g e d i s t a n c e where the decomposition, o f the wave f u n c t i o n , i n t o i n c o m i n g and o u t g o i n g waves y i e l d s the phase s h i f t . But the n u m e r i c a l s o l u t i o n does n o t d i s p l a y the wave p r o p e r t i e s . To see the s e we f i r s t examine the t r a n s m i s s i o n f u n c t i o n s o f a complex square w e l l p o t e n t i a l . The t r a n s m i s s i o n f u n c t i o n o f a complex square w e l l o f depth - ( V q H- i W ) , and r a d i u s R ^ , may be w r i t t e n * * F o r a d e r i v a t i o n o f t h i s r e s u l t and a d e t a i l e d d i s c u s s i o n o f n u c l e a r t r a n s m i s s i o n f u n c t i o n s see pages 281-29 5 o f r e f e r e n c e 10 'I P, 1" > • ^ ( O " = 1 _ 1 : (5) ( I - S ; f J e + P , f , J m ) 2 + O^T + ^ ^ ' V Here i s the u s u a l p e n e t r a t i o n f a c t o r used i n n u c l e a r r e a c t i o n s t u d i e s , where F, and G a r e , r e s p e c t i v e l y , the r e g u l a r and i r r e g u l a r Coulomb wave f u n c t i o n s o f the c h a n n e l <X . The a r e the u s u a l s h i f t f u n c t i o n s o f n u c l e a r r e a c t i o n t h e o r y / r F^ dE, / d r + r G., dG^ /dr j S. = - b R., ( 7 ) i n w h i c h t he b ^  a r e boundary c o n d i t i o n numbers determined, by the res o n a n c e p r o p e r t i e s o f the w e l l (see b e l o w ) . F i n a l l y , we have f ( f + i f r e l a t e d t o the l o g a r i t h m i c d e r i v a t i v e o f the wave f u n c t i o n i n s i d e t h e square w e l l , j (\-r) ' ^ n ^ e f o l l o w i n g way • • The f are complex numbers because the K . are comple) K = 2 m . . - i (E + V + i W J / * ( 9 ) . 2 • " 0 ' - " Q . The q u a n t i t i e s f. , } r e f e r to the p r o p e r t i e s of the square w e l l j '-\\ w h i l e the q u a n t i t i e s P and S, r e f e r to the e x t e r n a l p r o p e r t i e s . The t r a n s m i s s i o n f u n c t i o n s (5) c l e a r l y e x h i b i t a b s o r p t i o n and b a r r i e r p e n e t r a t i o n i n separate f a c t o r s . .The m a n i f e s t a t i o n of resonance and r e f l e c t i o n r e q u i r e s f u r t h e r a n a l y s i s . To e x h i b i t the "resonances" of 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 square w e l l we take the r e a l p a r t of the p o t e n t i a l and f ind. the normal modes or resonances . The normal modes are s o l u t i o n s of the Sehroed.inger e q u a t i o n , which vanish, a t the o r i g i n and s a t i s f y a s u i t a b l e boundary c o n d i t i o n a t the square w e l l r a d i u s . I f we w r i t e the s o l u t i o n s as A i , (K , r) where A . i s a n o r m a l i z i n g c o n s t a n t and K , ( ~ f 2 m . : h ~ (E „ + V ) ~ 2 ) i s the d i s c r e t e v a l u e px v L o- P'- o . i J of the wave number f o r which the f o l l o w i n g boundary c o n d i t i o n i s s a t i s f i e d : r d j x / d r = b ^ -}A J r = R , (10) The c o r r e c t va lue of the boundary c o n d i t i o n number b ^ i s t h a t which makes the s h i f t f u n c t i o n ( 7 ) v a n i s h at the energy of ] 2 i n t e r e s t . "" The normal modes have s i n g l e p a r t i c l e energ ies F ' = K 2 JpA 2 m ^ p i The f , were g i v e n q u i t e g e n e r a l l y i n terms of s p h e r i c a l B e s s e l funci: ions o f complex arguments: a l t e r n a t i v e l y we can g i v e them i n terms of the normal modes or . resonances of the r e a l p a r t of the p o t e n t i a l . By u s i n g the completeness of the resonances we f i n d : - 13 -2 y w f l m , v * . = Y — ' - l ' ° :r~ (11) - ' P (E, -E ) 2 -I W 1 v Ap o o f Re = ( E ^ E ) Y,2 , p f s.fEndE-( E P - E ) 2 -I- W 2 J E' - E v Xp 0 ~ oo where P s t a n d s f o r t h e p r i n c i p a l v a l u e o f t h e i m p r o p e r i n t e g r a l 2 a n d wher.e Y, i s t h e s i n g l e p a r t i c l e r e d u c e d w i d t h . I f we c h o o s e A P t h e b o u n d a r y c o n d i t i o n number t o be b t - - X - w h i c h w o u l d be c o r r e c t f o r n e u t r a l p a r t i c l e s i n the l o w e n e r g y l i m i t - t h e n 2 2 2 Yjjp = "11 /m , . The q u a n t i t y s^ d e f i n e d by (11) i s u s u a l l y r e f e r r e d , t o as t h e s t r e n g t h f u n c t i o n . The n u c l e a r s t r e n g t h f u n c t i o n o f (11) c l e a r l y e x h i b i t s a r e s o n a n c e s t r u c t u r e . U s i n g t h e c o r r e s p o n d i n g v a l u e o f f ^ ' n i n t h e t r a n s m i s s i o n f u n c t i o n (S) - o r i n t h e a b s o r p t i o n c r o s s s e c t i o n (3) - shows t h a t t h e s e r e s o n a n c e s a l s o a r e e x p e c t e d t o a p p e a r i n t h e d a t a . T h e r e i s much e v i d e n c e f o r them i n low e n e r g y n u c l e o n a b s o r p t i o n . F o r h e a v y i o n s t h e v a l u e s of-W a r e l a r g e r and t h e v a l u e s o f E c l o s e r t o g e t h e r so t h a t t h e d a t a does n o t u s u a l l y e x h i b i t r e s o n a n c e s . The r e f l e c t i v i t y o f t h e s q u a r e w e l l c a n be t r a c e d t o t h e p e n e t r a t i o n f a c t o r . F o r X ~ 0 a b s o r p t i o n o f n e u t r a l p a r t i c l e s we h a v e no Coulomb o r a n g u l a r momentum b a r r i e r s and t h e r e f o r e the p e n e t r a t i o n f a c t o r , (6) , becomes P ~ k R . N e a r t h r e s h o l d , t h e r v J o ^ ci 1 s t r e n g t h f u n c t i o n r e m a i n s f i n i t e b u t ' P v a n i s h e s as E 2. T h i s v a n i s h i n g o f P c a n be t r a c e d t o t o t a l r e f l e c t i o n o f t h e wave o a p p r o a c h i n g t h e s q u a r e w e l l r a d i u s f r o m e i t h e r s i d e . - 1.1-1 -H a v i n g d i s p l a y e d a b s o r p t i o n , b a r r i e r p e n e t r a t i o n , r e s o n a n c e and. r e f l e c t i o n f o r s q u a r e w e l l s we now seele t o compare t h e c o r r e s p o n d i n g q u a n t i t i e s f o r d i f f u s e - e d g e w e l l s w i t h t h e r e s u l t s a b o v e . I t m i g h t a p p e a r r e a s o n a b l e t o b e g i n w i t h a d e c o m p o s i t i o n o f t h e t r a n s m i s s i o n f u n c t i o n as i n ( 5 ) . T h e r e i s no f e a t u r e o f t h e d e c o m p o s i t i o n o f (S) w h i c h does n o t a p p l y t o an a r b i t r a r y w e l l s h a p e . We c a n a l w a y s c h o o s e a m a t c h i n g r a d i u s , e v a l u a t e P . and S, f o r t h a t r a d i u s ( t a k i n g a c c o u n t o f t h e m o d i f i c a t i o n o f P, a n d S, by t h e t a i l o f t h e K X c o m p l e x p o t e n t i a l ) , and e v a l u a t e t h e l o g a r i t h m i c d e r i v a t i v e , f , f o r t h e c o m p l e x p o t e n t i a l a t t h a t m a t c h i n g r a d i u s . I n d e e d , we c a n e v e n decompose t h e l o g a r i t h m i c d e r i v a t i v e o f any w e l l t o e x h i b i t t h e r e s o n a n c e s as a b o v e . B u t , how do we choose t h e m a t c h i n g r a d i u s ? How do t h e p e n e t r a t i o n f a c t o r s , s t r e n g t h f u n c t i o n s , r e s o n a n c e e n e r g i e s , e t c . depend on w e l l p r o p e r t i e s ? I n o u r a n s w e r s t o t h e s e q u e s t i o n s w i l l r e s i d e o u r a n a l y s i s o f wave p r o p e r t i e s . B e f o r e a d a p t i n g o u r e x p o s i t i o n t o d i f f u s e - e d g e p o t e n t i a l s we i l l u s t r a t e t h e d e c o m p o s i t i o n o f t h e t r a n s m i s s i o n f u n c t i o n as i n (5) by t h e " b l a c k n u c l e u s " m o d e l - w h i c h i s n o t a p o t e n t i a l m o d e l a t a l l . I n t h i s c a s e i t i s assumed t h a t t h e r e a r e o n l y i n c o m i n g - i K r waves f o r r < . R e p l a c i n g j . by e i n (8) and c h o o s i n g b = 0 we t h e n h a v e O U K R ^ - 15 -U s i n g t h i s v a l u e o f f c U xn (5) we o b t a i n ' the b l a c k n u c l e u s t r a n s m i s s 1on f u n c t i on s (1 -I- P f / K R J 2 + ( s ^ / K i y 2 (HI) Uk/K f o r s-wave n e u t r o n s . (1 + k/K) 2 A t low e n e r g i e s (k<< K) a l l 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 b l a c k n u c l e u s model (even those f o r s-wave ne u t r o n s ) are v e r y s m a l l because o f wave r e f l e c t i o n o r wave p e n e t r a t i o n . A t h i g h e r e n e r g i e s (P ~->kR ,. k — > K) the b l a c k n u c l e u s t r a n s m i s s i o n f u n c t i o n s approach t h e i r maximum v a l u e o f u n i t y . r e f l e c t i o n t h an a r e a l n u c l e u s . To u n d e r s t a n d the wave a n a l y s i s o f the b l a c k n u c l e u s we can compare i t t o a wave gu i d e problem"'""' i n c l a s s i c a l e l e c t r o m a g n e t i c t h e o r y . The bombarding wave approaches the n u c l e u s w h i c h i s t r e a t e d l i k e a r e s o n a t i n g c a v i t y . The c a v i t y i s t u n e d so t h a t waves p r o p a g a t e i n w a r d a t the c a v i t y e n t r a n c e , R. w i t h the g i v e n wave p r o p a g a t i o n number K. The t u n i n g to a c c o m p l i s h t h i s i s n o t e a s i l y d u p l i c a t e d by a p o t e n t i a l w e l l model. J u s t as i t i s d i f f i c u l t t o b u i l d r e s o n a t i n g c a v i t i e s w h i c h are tuned i n a c e r t a i n way f o r a l l w a v e l e n g t h s , i t i s i m p o s s i b l e t o make o p t i c a l - m o d e l p o t e n t i a l s f o r w h i c h the b l a c k n u c l e u s c o n d i t i o n s a p p l y a t a l l e n e r g i e s . The b l a c k n u c l e u s t r a n s m i s s i o n f u n c t i o n s c o m p l e t e l y l a c k the r e s o n a n c e s o f the s q u a r e - w e l l : the square w e l l t r a n s m i s s i o n f u n c t i o n s o s c i l l a t e about t he b l a c k ' n u c l e u s t r a n s m i s s i o n f u n c t i o n L i k e the square w e l l model, the b l a c k n u c l e u s has more ' - l.G - ! " ii i! i f we choose K t o have the same v a l u e f o r b o t h . The two t r a n s m i s s i o n f u n c t i o n s have the same mean v a l u e . We 'show below t h a t the t r a n s m i s s i o n f u n c t i o n s o f a r e a l i s t i c n u c l e a r o p t i c a l p o t e n t i a l have a l a r g e r mean v a l u e than b o t h o f the ' ' i above models because the d i f f u s e n u c l e a r edge g i v e s l e s s r e f l e c t i o n than a square edge. The b l a c k n u c l e u s s u f f e r s the same u n r e a l i s t i c r e f l e c t i o n because o f the' sudden change i n the wave number a t R,. Thus the " b l a c k n u c l e u s " model f a i l s t o a c o n s i d e r a b l e degree i n i t s main o b j e c t i v e o f o p t i m i z i n g n u c l e a r a b s o r p t i o n . Much o f the above wave a n a l y s i s can be adapted a t once t o a r e a l i s t i c n u c l e a r o p t i c a l p o t e n t i a l . F o r example, the d e c o m p o s i t i o n o f the t r a n s m i s s i o n ' f u n c t i o n as i n (5) s t i l l a p p l i e s e x c e p t than P,, S , and f < a r e m o d i f i e d , and the c h o i c e X x a A - . . o f a m a t c h i n g r a d i u s a t w h i c h these t h r e e q u a n t i t i e s are t o be e v a l u a t e d i s no l o n g e r o b v i o u s . The d e c o m p o s i t i o n i s v a l i d f o r an a r b i t r a r y c h o i c e o f m a t c h i n g r a d i u s b u t f o r our purp o s e s a w e l l - d e f i n e d c h o i c e w i l l t u r n out t o be u s e f u l . F o r an a r b i t r a r y r a d i u s we can f i n d P,, and S. from the l o g a r i t h m i c d e r i v a t i v e o f X X an o u t g o i n g wave Cl, , • !d[kZ? = rdF, /dr + I r dG, /dr (for ^  r) (15) = S - l b , -I- I P„ f a t the m a t c h i n g r a d i u s ) s i m p l y by making the a n a l y t i c c o n t i n u a t i o n o f the l o g a r i t h m i c d e r i v a t i v e t o t h e m a t c h i n g r a d i u s as' i n d i c a t e d . S i m i l a r l y , f w . i s s t i l l g i v e n by (8) i f , i.Ln (8) , we r e p l a c e j by the - 17 -a p p r o p r i a t e r e g u l a r wave f u n c t i o n o f the o p t i c a l p o t e n t i a l . There are two i m p o r t a n t f a c t o r s which narrow down the c h o i c e o f m a t c h i n g r a d i u s i n such a wave a n a l y s i s . F i r s t o f a l l , i the n u c l e a r s u r f a c e o f r e a l i s t i c p o t e n t i a l s i s s t i l l q u i t e s h a r p : t y p i c a l l y a/R i n (1) has a v a l u e o f 0.1. S e c o n d l y , the res o n a n c e c o n d i t i o n s t h e m s e l v e s do n o t a l l o w much freedom o f C h o i c e . . C o n s i d e r i n g the n u c l e u s as a wave gu i d e we t h i n k o f s e t t i n g up normal modes o r r e s o n a n c e s . I f we i n c l u d e i n our d e f i n i t i o n o f the wave gu i d e p a r t o f the t r a n s m i s s i o n c h a n n e l s w h i c h l e a d i n t o i t then the r e s u l t i n g l a r g e •- a r t i f i c a l l y l a r g e -d i m e n s i o n o f the gu i d e g i v e i t u n u s u a l p r o p e r t i e s . Even when our w a v e l e n g t h i s c l o s e t o t h a t o f one o f the a r t i f i c i a l normal modes the one-mode a p p r o x i m a t i o n i s p o o r . I n the n u c l e u s the o n e - l e v e l B r e i t - W i g n e r f o r m u l a b r e a k s down. We must choose the mat c h i n g r a d i u s t o l i e r e a s o n a b l y c l o s e t o R . 12 I n an e a r l i e r work one o f us showed t h a t f o r s-wave n e u t r o n s t h e r e e x i s t e d a v e r y s i m p l e c o r r e s p o n d e n c e between the o p t i c a l p o t e n t i a l (1) and. a square w e l l , o f v e r y n e a r l y the same r a d i u s . F i g . 1 shows the i m p o r t a n t f e a t u r e s o f the c o r r e s p o n d e n c e . W i t h the r a d i i the same, the depth o f the square w e l l i s a d j u s t e d so t h a t the re s o n a n c e s n e a r z e r o energy c o i n c i d e f o r the two w e l l s . W i t h t h i s c h o i c e i t t u r n s out t h a t a l l o f the resonance e n e r g i e s and a l l o f the r e d u c e d w i d t h s o f the two w e l l s a r e the same. The two w e l l s have e x a c t l y the same resonance p r o p e r t i e s . But the p e n e t r a t i o n f u n c t i o n s and s h i f t f u n c t i o n s o f the two w e l l s are d i f f e r e n t . F o r the square w e l l the b r a c k e t e d term o f (7) i s e q u a l t o X a t low energy and t h e r e f o r e i t i s n a t u r a l ' to choose b = 0 so t h a t S. = 0 . F o r the d i f f u s e - e d g e w e l l , one o b t a i n s - 18 -b, . by c a l c u l a t i n g the a n a l y t i c c o n t i n u a t i o n o f the b r a c k e t e d term and by c h o o s i n g b , so t h a t S .. v a n i s h e s a t low e n e r g y . The c o r r e s p o n d e n c e o f F i g . 1 i s o b t a i n e d w i t h t h i s c h o i c e . The p e n e t r a t i o n f a c t o r s o f the too w e l l s d i f f e r by a c o n s t a n t f a c t o r , 1(3 f , w h i c h i s n e a r l y energy i n d e p e n d e n t . As shown f i r s t by P e a s l e e , . f can be w e l l a p p r o x i m a t e d by f — TT K a c o t h (IT K a) (16) w h i c h has a v a l u e o f about 2.5 f o r the o p t i c a l p o t e n t i a l s i n common use f o r n u c l e o n s . The f a c t o r f a c c o u n t s f o r the d i f f e r e n c e i n r e f l e c t i o n o f the d i f f u s e - e d g e and square-edge. F o r s~wave n e u t r o n s the c o r r e s p o n d e n c e between the o p t i c a l p o t e n t i a l and i t s e q u i v a l e n t s q u a r e w e l l extends t o a l l c a s e s , from r e a l w e l l s w i t h narrow r e s o n a n c e s t o complex p o t e n t i a l s w i t h b r o a d peaks i n the t r a n s m i s s i o n f u n c t i o n s . I n a l l c a s e s the s q u a r e w e l l r e s u l t s a p p l y .to d i f f u s e w e l l s a l s o i f we m e r e l y m u l t i p l y the s q u a re w e l l p e n e t r a t i o n f u n c t i o n and s h i f t f u n c t i o n by the r e f l e c t i o n f a c t o r f . Thus the whole wave a n a l y s i s i s e s t a b l i s h e d f o r the d i f f u s e - e d g e w e l l s . The c o r r e s p o n d e n c e o f F i g . 1. was e s t a b l i s h e d f o r p u r e l y r e a l w e l l s . I t e x t e n d s t o complex p o t e n t i a l s too because we can i n c l u d e an i m a g i n a r y term o f square shape i n the r e a l l o g a r i t h m i c d e r i v a t i v e s i m p l y by r e p l a c i n g E by E -- i W . Thus (11) and (12) a p p l y t o the d i f f u s e - e d g e w e l l a l s o . The shape o f the i m a g i n a r y term i s o f l i t t l e consequence because t h e r e i s l i t t l e r e f l e c t i o n from the i m a g i n a r y term ( s e e , however, Sec. 5 b e l o w ) . Mien th e t r a n s m i s s i o n f u n c t i o n s a r e much s m a l l e r t h a n u n i t y they can be w e l l a p p r o x i m a t e d by T U 'ITT P, s., f (17) Here they c l e a r l y show the common resonance and p e n e t r a t i o n f a c t o r s o f d i f f u s e - e d g e and square w e l l s and the r e f l e c t i o n f a c t o r w h i c h d i s t i n g u i s h e s them. The e a r l i e r a n a l y s i s a p p l i e d to s-wave n e u t r o n s and . i t i s n o t a t a l l c l e a r t h a t i n the p r e s e n c e o f Coulomb b a r r i e r s i t i s p o s s i b l e t o decompose the t r a n s m i s s i o n f u n c t i o n s i n t o wave p r o p e r t i e s i n q u i t e the same s i m p l e way. I t t u r n s out t o be so f o r p r o t o n s as we show on F i g . 2. Here t r a n s m i s s i o n f u n c t i o n s a r e shown w e l l below the Coulomb b a r r i e r f o r d-wave p r o t o n s . The r a d i i o f a l l the w e l l s a r e the same b u t the w e l l depths a r e ad j u s t e d * , t o make res o n a n c e e n e r g i e s c o i n c i d e r o u g h l y . A l t h o u g h * The a d j u s t m e n t p r o c e d u r e i s the f o l l o w i n g : I f V ( r ) i s the r e a l p a r t o f the o p t i c a l p o t e n t i a l the J V ( r ) r d r i s t a k e n t o be o c o n s t a n t f o r a l l p o t e n t i a l s . t h e r e i s s t r o n g b a r r i e r p e n e t r a t i o n h e r e the r e l a t i o n between the v a r i o u s t r a n s m i s s i o n f u n c t i o n s i s s t i l l t h a t o f ( 1 7 ) . I t i s n o t d i f f i c u l t t o u n d e r s t a n d why the same p r e s c r i p t i o n works f o r p r o t o n s . F i g . 3 shows the sum o f the n u c l e a r and Coulomb p o t e n t i a l f o r the o p t i c a l model and i t s e q u i v a l e n t square w e l l . The p o i n t i s t h a t the Coulomb p o t e n t i a l v a r i e s l i t t l e i n the s u r f a c e r e g i o n ( e n c i r c l e d ) " . W i t h i n the s u r f a c e r e g i o n we mig h t approximate the Coulomb p o t e n t i a l as b e i n g c o n s t a n t . 'Then our r e f l e c t i o n problem i s e x a c t l y l i k e t h a t f o r s-wave n e u t r o n s o f n e g a t i v e energy E - B where B 'is the b a r r i e r h e i g h t . The e a r l i e r a n a l y s i s would a p p l y t o t h i s c a s e . - 20 - ' I I • I I t was s u r p r i s i n g t o us t h a t t h e s i m p l e r e l a t i o n s h i p between d i f f u s e - e d g e w e l l s and square w e l l s appeared t o b r e a k down c o m p l e t e l y f o r heavy i o n s . 3 ? i g . '1 shows a s i m i l a r r a t i o o f 32 '-I-t r a n s m i s s i o n f u n c t i o n s l o r S + He. The e x p e c t e d r e f l e c t i o n f a c t o r i n t h i s case i s f = '1.6 a c c o r d i n g t o (17) . But the • r a t i o o f t r a n s m i s s i o n f u n c t i o n s i s much l a r g e r . Moreover i t v a r i e s s t r o n g l y w i t h the energy and depends s t r o n g l y on the o r b i t a l a n g u l a r momentum. There i s . a s i m p l e way i n w h i c h the'wave a n a l y s i s f o r heavy i o n s i s r e s t o r e d t o the s i m p l i c i t y o f t h a t f o r n u c l e o n s . We e x p l a i n e d t h a t the m a t c h i n g r a d i u s f o r the o p t i c a l p o t e n t i a l had t o be chosen t o be r e a s o n a b l y c l o s e t o R , the m i d p o i n t o f th e s u r f a c e . But i t does n o t heed t o be e x a c t l y R . I n the o • n e x t s e c t i o n we show t h a t i f we choose R , = R + /JS R - w i t h cA o /S.R c l e a r l y d e f i n e d and e v a l u a t e d - we a g a i n g e t r e s u l t s as s i m p l e as t h o s e o f ( 1 7 ) . The r o l e o f /\R i s i m p o r t a n t f o r heavy Ions and n o t f o r n u c l e o n s because the more m a s s i v e heavy i o n s have wave . f u n c t i o n s which, o s c i l l a t e more r a p i d l y ~ the "wave g u i d e " commences e a r l i e r . ' 3- The C o n s t r u c t i o n o f Eqiafval.ent Square W e l l s . (ESW) F o r a r b i t r a r y o p t i c a l p o t e n t i a l s o f the k i n d (1) we w i s h t o c o n s t r u c t e q u i v a l e n t square w e l l s . (ESW) i n o r d e r t h a t we may d i s p l a y the b a s i c wave p r o p e r t i e s o f the p o t e n t i a l . A c c o r d i n g t o the a p p r o x i m a t e v i e w o f n u c l e a r r e a c t i o n s d e p i c t e d on F i g . 3, Coulomb b a r r i e r s have no g r e a t e f f e c t on the r e f l e c t i o n o f a d i f f u s e - e d g e o p t i c a l p o t e n t i a l . N e i t h e r , do a n g u l a r momentum b a r r i e r s as we s h a l l show below. The problem o f d i s p l a y i n g the wave p r o p e r t i e s i s then r e d u c e d t o the case where we have s-wave 1 2 n e u t r a l p a r t i c l e s . A c c o r d i n g t o e a r l i e r work the ESW f o r the i case o f s-wave n e u t r a l p a r t i c l e s i s i n d e p e n d e n t o f the energy o f the p a r t i c l e s o r o f the a b s o r p t i v e ( i m a g i n a r y ) term i n the o p t i c a l p o t e n t i a l . We can t h e r e f o r e c o n s t r u c t the ESW u s i n g the s c a t t e r i n g o f z e r o - e n e r g y s-wave n e u t r a l p a r t i c l e s i n o n l y the r e a l p a r t o f the o p t i c a l p o t e n t i a l . The d e p t h , r a d i u s and r e f l e c t i o n f a c t o r o f the ESW c o n s t r u c t e d i n t h i s way are e x p e c t e d t o a p p l y t o a l l the o t h e r , more c o m p l i c a t e d cases as w e l l . I n l a t e r s e c t i o n s we show t o what e x t e n t the ESW i s u n i v e r s a l and t o what e x t e n t i t e n a b l e s us t o d i s p l a y the b a s i c wave p r o p e r t i e s f o r a l l c a s e s . H a v i n g r e d u c e d the c o n s t r u c t i o n o f the ESW t o the case o f z e r o - e n e r g y s-wave n e u t r a l p a r t i c l e s , of.mass m, i n the r e a l p a r t o f the o p t i c a l p o t e n t i a l we p r o c e e d , f o r a l l masses, i n t h e way t h a t was shown f o r n e u t r o n s on F i g . 1. S i n c e the ESW i s p u r e l y r e a l i t has o n l y - t w o p a r a m e t e r s : i t s depth and i t s r a d i u s . On the o t h e r hand the r e a l p a r t o f the o p t i c a l p o t e n t i a l (1) has t h r e e p a r a m e t e r s : V Q , R and a. We need two c o n d i t i o n s t o f i x and. R . B o t h w e l l s can have r e s o n a n c e s a t z e r o - e n e r g y : f o r example, i f we keep the depth o f e i t h e r w e l l f i x e d and v a r y the r a d i u s (as we w o u l d do, i n p r o c e e d i n g t h r o u g h the p e r i o d i c t a b l e ) r e s o n a n c e s o c c u r a t z e r o - e n e r g y whenever the r a d i a l wave f u n c t i o n has z e r o d e r i v a t i v e beyond the w e l l . ' S i n c e the resonances g r e a t l y a f f e c t the s c a t t e r i n g c r o s s s e c t i o n we must choose the p o s i t i o n o f the r e s o n a n c e s i n the two w e l l s t o be the same i f we want the w e l l s t o have e q u i v a l e n t p r o p e r t i e s . Making the p o s i t i o n s o f the resonances - 22 -c o i n c i d e f i x e s one o f the two par a m e t e r s o f the ESW. A second c o n d i t i o n i s o b t a i n e d from the w i d t h s o f the s c a t t e r i n g r e s o n a n c e s . A c o n v e n t i o n a l , d i f f u s e - e d g e p o t e n t i a l has much l a r g e r w i d t h s than a square w e l l , a f a c t w h i c h we a s s o c i a t e w i t h r e f l e c t i o n . F o r each w e l l we can w r i t e the w i d t h w i t h A = 0 i n our c a s e , where n i s the number o f nodes o f th e s-wave r e s o n a n t wave f u n c t i o n ) as a p r o d u c t o f a p e n e t r a t i o n f a c t o r and a r e d u c e d w i d t h P o Y no 2 (18) where the two f a c t o r s a r e each e v a l u a t e d a t a s u i t a b l e m a t c h i n g r a d i u s , R. F o r the square w e l l i t i s n a t u r a l and n e c e s s a r y t o choose R = R, and f o r t h i s c h o i c e we have P = kR.. and Y = i i /mR-, . I . o 1 no ' 1 F o r the d i f f u s e - e d g e p o t e n t i a l we can e a s i l y e x h i b i t a resonant-wave f u n c t i o n - as on F i g . 5 - by v a r y i n g R u n t i l the wave f u n c t i o n has z e r o d e r i v a t i v e f a r beyond the w e l l , b u t t h e r e i s a r b i t r a r i n e s s i n the c h o i c e o f the. m a t c h i n g r a d i u s R . The w i d t h i t s e l f i s i n d e p e n d e n t o f the c h o i c e o f R b u t the p e n e t r a t i o n f a c t o r and reduced w i d t h a r e n o t . A t t h i s s t a g e we make a c h o i c e * . We w i s h t o a s s o c i a t e * The c h o i c e i s n o t e n t i r e l y an a r b i t r a r y one - no more than the c h o i c e o f the m a t c h i n g r a d i u s i n the R ~ m a t r i x -theory, f o r w h i c h the r a d i u s must be t h a t o f the a c t u a l n u c l e u s i f the resonance f o r m u l a e a r e t o have r e a s o n a b l e convergence p r o p e r t i e s . t h e d i f f e r e n c e i n w i d t h s - the r e f l e c t i o n - w i t h the d i f f e r e n c e i n p e n e t r a t i o n o f the two w e l l s and t h e r e f o r e we choose a m a t c h i n g r a d i u s , R, such t h a t the o p t i c a l p o t e n t i a l and i t s ESW have the same r e d u c e d w i d t h s , a t the common m a t c h i n g r a d i u s . F i x i n g the p o s i t i o n and reduced. w i d t h o f -the ESW d e t e r m i n e s V , and R., . • The a p p l i c a t i o n o f the above two c o n d i t i o n s t o the c o n s t r u c t i o n o f the ESW i s i l l u s t r a t e d on F i g . 5 . F o r a g i v e n o p t i c a l p o t e n t i a l we f i x V and a and t h e n v a r y R t o f i n d a l l J o J o o f the r e s o n a n c e s . We t h e n have a d i s c r e t e s e t o f v a l u e s o f R q and the c o r r e s p o n d i n g r e s o n a n t wave f u n c t i o n s as shown on the f i g u r e . N e x t , we b e g i n w i t h a square w e l l w i t h an i n i t i a l c h o i c e o f R^( ^''RQ) J and f i - x V . ^ so t h a t we a l s o g e t a z e r o - e n e r g y resonance i n the square w e l l h a v i n g the same number o f nodes as the d i f f u s e edge r e s o n a n c e . F o r t h i s i n i t i a l c h o i c e we compare r e d u c e d w i d t h s o f b o t h w e l l s a t the same m a t c h i n g r a d i u s , R(= Rj) . I f ^ / / N O ( R 0 i s the r a d i a l wave f u n c t i o n o f the re s o n a n c e the r e d u c e d w i d t h i s , 1 . 2 g i v e n by I t i s the square o f the a m p l i t u d e o f the n o r m a l i z e d wave f u n c t i o n a t the m a t c h i n g r a d i u s . I f the two w e l l s do n o t have the same re d u c e d w i d t h f o r the i n i t i a l c h o i c e o f R , we v a r y R^  (always a d j u s t i n g V . ^ so t h a t the resonance p o s i t i o n s o f the two w e l l s c o i n c i d e e x a c t l y ) u n t i l they do. I n t h i s way we a c h i e v e a uniqu e •value o f the dep t h and r a d i u s o f the o p t i c a l p o t e n t i a l . T h i s c o n s t r u c t i o n h o l d s o n l y f o r tho s e v a l u e s o f R w h i c h c o r r e s p o n d t o r e s o n a n c e , b u t we can i n t e r p o l a t e the c o n s t r u c t i o n i n between r e s o n a n c e s . The r e f l e c t i o n f a c t o r i s a d i r e c t r e s u l t o f the above c o n s t r u c t i o n o f the ESW. I t can be defined, by |~ o ( d i f f u s e w e l l ) = f f (square w e l l ) (20) - 2'-l -S i n c e the r e d u c e d w i d t h s o f the two w e l t s are- the same t h i s y i e l d s P O ( d i f f u s e w e l l ) - f P q (square w e l l ) - f K R. (21) The c o m parison o f (20) and (21) employed ' n a t u r a l * w i d t h s - n o t \ \ the o b s e r v e d w i d t h s o f narrow resonances. We g i v e a complete d i s c u s s i o n i n Sec. 1 o f the r e l a t i v e v a l u e s o f o b s e r v e d and n a t u r a l w i d t h s f o r v a r i o u s w e l l s . • We can e v a l u a t e the d i f f u s e - w e l l p e n e t r a t i o n f a c t o r d i r e c t l y from (6) i f . we r e p l a c e F^ and G , by F^ and where the l a t t e r are the- r a d i a l wave f u n c t i o n s o f the o p t i c a l p o t e n t i a l w h i c h behave a t l a r g e d i s t a n c e l i k e u s u a l r a d i a l wave f u n c t i o n s - F^ and Gj,, r e s p e c t i v e l y - i n the absence o f a p o t e n t i a l . I n our case ' F = s i n k r and G = cos k r so t h a t a t z e r o energy F = 0 and o o ^ o G = 1 . T h e r e f o r e we f i n d o f = [ G O (R) / G o (R)j 2 = ( G O / G O (R) ) 2 - C r W ^ ^ W 1 ^ 2 = ^ 7 d l f f ^ ^ V 4 u u a r e ^ ^ s i n c e G i s p r o p o r t i o n a l t o the d i f f u s e - w e l l r e s o n a n t wave f u n c t i o n o f F i g . 5 and the r e s o n a n t wave f u n c t i o n s o f the two w e l l s a r e n o r m a l i z e d t o the same a m p l i t u d e a t the m a t c h i n g r a d i u s on F i g . 5. Then f i s j u s t the square o f the a m p l i t u d e r a t i o o f the r e s o n a n t wave f u n c t i o n s a t l a r g e d i s t a n c e , as i n d i c a t e d on the f i g u r e . I f we remember the d e f i n i t i o n (6) o f t h e . p e n e t r a t i o n f a c t o r and employ the c o n v e n t i o n a l d e f i n i t i o n o f an i n c o m i n g wave, 12 (see ('-I-1) below) , we can g i v e a s i m i l a r p h y s i c a l meaning t o (22) 25 -the r e f l e c t i o n f a c t o r w h i c h i s v a l i d , a t f i n i t e energy. The c o n v e n t i o n a l p e n e t r a t i o n f a c t o r may be w r i t t e n lc R ( 1 J . J ) • 1 F o r a d i f f u s e - e d g e w e l l we r e p l a c e X y by i t s a n a l y t i c a l c o n t i n u a t i o n I ^ , y i e l d i n g )<R ( I , I /O " 1 = f kR ( I , I*) 1 wh'ere f I, I * / 1,1* X A A A1. The i n t e n s i t y o f the Incoming wave a t l a r g e d i s t a n c e needed t o y i e l d a g i v e n . i n t e n s i t y a t the m a t c h i n g r a d i u s i s f ti m e s b i g g e r f o r a square w e l l t han f o r a d i f f u s e w e l l . The r e f l e c t i o n f a c t o r t h a t we d e t e r m i n e i s r e l a t e d t o wave r e f l e c t i o n . I n t h e above c o n s t r u c t i o n o f an ESW f o r a p a r t i c l e o f mass m and an a r b i t r a r y o p t i c a l p o t e n t i a l o n l y two o f the f o u r p a r a m e t e r s (V > R , a, m) are i n d e p e n d e n t . T h i s can be seen from the r a d i a l e q u a t i o n w i t h t h e o p t i c a l p o t e n t i a l V, o f (1) • 11! ^ 3 / -2m d r 2 V , , (r-Rl7a 1 + e *• o (23) A t z e r o energy the r i g h t - h a n d , s i d e v a n i s h e s and we can w r i t e (23) i n terms o f too d i m e n s i o n l e s s p a r a m e t e r s ? d \\J (K R. ) v o o' dx 1 -.- e*x - ] - M V a ) (211) - 26 where x = r/R " (25) K 2 = 2m V / l i 2 (26) o o \ J Thus R Q d e t e r m i n e s o n l y the s c a l e o f the a b s c i s s a on F i g . 5. The two i m p o r t a n t p a r a m e t e r s are K R and R / a . I f we f i n d f L o o o and the ESW f o r a l l c h o i c e s o f these two p a r a m e t e r s then we have c o v e r e d a l l c a s e s . We c a r r i e d o u t the c o n s t r u c t i o n o f the ESW and the a s s o c i a t e d r e f l e c t i o n f a c t o r f o r a l a r g e range o f v a l u e s o f K q R and a/R . The d e p t h o f the ESW i s n o t i m p o r t a n t i n many a p p l i c a t i o n s •- i t must always be a d j u s t e d t o produce the r e s o n a n t p o s i t i o n s a t the r i g h t e n e r g i e s , (see Sec. M-) and i s always found 2 2 t o be g i v e n v e r y n e a r l y by Rj ^ V Q R.q . The ESW 3?adius i s u s u a l l y i m p o r t a n t and d i f f e r s a p p r e c i a b l y from R . We p r e s e n t the r e s u l t s i n terms o f the r a d i u s d i f f e r e n c e /SR/R R R v > o o • w h i c h i s a l s o d i m e n s i o n l e s s . F i g . 6 shows the v a l u e s o f f and AR/R w h i c h we o b t a i n e d from the z e r o energy r e s o n a n c e s and i n t e r -p o l a t i o n s between r e s o n a n c e s . F i g . 6 i s an enlargement o f a s m a l l 13 r e g i o n o f a. c o r r e s p o n d i n g map g i v e n by us i n a p r e l i m i n a r y r e p o r t the enlargement g i v e n h e r e c o n t a i n s the r e g i o n o f the map used i n most a p p l i c a t i o n s . A l t h o u g h our wave a n a l y s i s o f p o t e n t i a l w e l l s has t a k e n the r e f l e c t i o n f a c t o r t o be energy i n d e p e n d e n t , the a n a l y s i s o f wave p r o p e r t i e s o f p o t e n t i a l s i n terms o f the e q u i v a l e n t square w e l l can be g e n e r a l i z e d and improved by a l l o w i n g f t o be energy | dependent. I n F i g . (7) we see the e x t e n t t o w h i c h t h e . r e f l e c t i o n f a c t o r i s i n d e p e n d e n t o f energy. The r e f l e c t i o n f a c t o r was c a l c u l a t e d by s u p p o s i n g an i n c o m i n g wave a t i n f i n i t y , and c a l c u l a t i n g the r a t i o f / 1DW V , where I„ V 1 and. a r e e v a l u a t e d a t R = 0 .1 R^ ,,,,. DW ESW - ESW To e l i m i n a t e any p o s s i b l e e f f e c t o f a b s o r p t i o n i n the b a r r i e r , we made our c a l c u l a t i o n s by s u p p o s i n g t h a t , f o r b o t h the d i f f u s e and square w e l l s , a b s o r p t i o n o c c u r r e d only' a t R. - 0 and. t h a t the i n c o m i n g wave was c o m p l e t e l y absorbed t h e r e . The r e f l e c t i o n f a c t o r i s t h e n seen t o be m o d e r a t e l y dependent o f energy. The energy dependence i s a l s o a f u n c t i o n of. the r e d u c e d mass; i t i s l a r g e r f o r the h e a v i e r p a r t i c l e s . We have a l s o checked t h a t the energy dependene o f the r e f l e c t i o n f a c t o r was n o t a f u n c t i o n o f c h a r g e . I t i s the same w i t h i n 10% o r s o , w h a t e v e r the Coulomb b a r r i e r , f o r a g i v e n r e d u c e d mass. B e f o r e a p p l y i n g the ESW i n some d e t a i l t o o b t a i n the. b a s i c wave p r o p e r t i e s o f v a r i o u s problems we g i v e a s i m p l e example 32 and d i s c u s s some s p e c i a l , c a s e s . F o r our s t a n d a r d example o f S +<& we can r e s o l v e the d i f f i c u l t i e s .of F i g . '-I. F o r a known o p t i c a l p o t e n t i a l w i t h a d i f f u s e - e d g e we now c o n s t r u c t the ESW as above, w i t h Z\R no l o n g e r z e r o b u t f i x e d , by F i g . 6. The r e s u l t s are shown - 28 -on F i g - 8. Nov; the r a t i o o f t r a n s m i s s i o n " f u n c t i o n s , a t low e n e r g y , i s r o u g h l y e q u a l t o the r e f l e c t i o n f a c t o r as e x p e c t e d from ( 1 7 ) . There a r e c o r r e c t i o n s however, the most i m p o r t a n t o f w h i c h i s a b s o r p t i o n i n the b a r r i e r . I t l a r g e l y c a n c e l s the energy v a r i a t i o n o f the r e f l e c t i o n f a c t o r so t h a t the n o t i o n o f a c o n s t a n t r e f l e c t i o n f a c t o r becomes a good a p p r o x i m a t i o n . The c o r r e c t i o n s t o the r e f l e c t i o n f a c t o r w i l l be f u r t h e r d i s c u s s e d i n Sec. 5. The p r e s c r i p t i o n f o r the ESW i s o b v i o u s l y q u i t e s u c c e s s f u l i n t h i s " c a s e . 12 F o r p r o t o n s , a l p h a p a r t i c l e s and C n u c l e i we can a p p l y F i g . 6 t o c o n v e n t i o n a l o p t i c a l p o t e n t i a l s . T h i s i s done on F i g . 9 w h i c h shows the v a l u e o f f and /^R f o r t h e s e p a r t i c l e s as a f u n c t i o n o f the bombarding n u c l e u s . The r e s u l t s f o r . f a r e a l s o compared on the f i g u r e t o the a p p r o x i m a t e r e f l e c t i o n f a c t o r , (3.6), d e r i v e d by P e a s l e e f o r s-wave n e u t r o n s . I . S i n g l e P a r t i c l e Resonances F o r R e a l P o t e n t i a l s The most s t r a i g h t f o r w a r d case i n w h i ch the ESW c o n s t r u c t e d i n Sec. 3 can be a p p l i e d t o d i s p l a y the wave p r o p e r t i e s o f o p t i c a l p o t e n t i a l s i s f o r the e l a s t i c s c a t t e r i n g r e s o n a n c e s o f r e a l p o t e n t i a l w e l l s . The ESW was c o n s t r u c t e d from a m a t c h i n g o f the r e s o n a n c e s o f n e u t r a l , p a r t i c l e s a t z e r o energy. We now go t o f i n i t e energy and a l s o add Coulomb b a r r i e r s , much as one m i g h t do i n a n a l y z i n g the s c a t t e r i n g o f a l p h a p a r t i c l e s from L i g h t n u c l e i . I n l a t e r s e c t i o n s we go f u r t h e r s t i l l and add a b s o r p t i o n t o the p i c t u r e (Sec. S ) . The i n t e r e s t h e r e i s t o show how we'll a z e r o - e n e r g y n e u t r a l - p a r t i c l e p r e s c r i p t i o n a p p l i e s t o a r e a l i s t i c p o t e n t i a l w i t h b a r r i e r s . The r e s u l t s w i l l be u s e f u l i n g e n e r a l i z i n g the f o r m a l many-channel t h e o r y o f n u c l e a r r e a c t i o n s (Sec. 7) t o take i n t o a c c o u n t the f i n i t e s u r f a c e 29 t h i c k n e s s o f n u c l e i . We b e g i n w i t h a. r e a l n u c l e a r p o t e n t i a l w h i c h m i g h t be 32 s u i t a b l e f o r S -l- C\ . C h o o s i n g t h e Saxon-Woods f o r m (1) we 1 /3 e x p e c t V t o be a b o u t 75 MeV, R t o b e * 1.25 A _ / +1.6 and a L o o * T h i s v a l u e o f t h e r a d i u s i s common].y u s e d f o r o p t i c a l m o d e l a n a l y s e s o f a l p h a - p a r t i c l e r e a c t i o n s . . I t s d i f f e r e n c e f r o m 1.25 + A?"^ J) o r f r o m the more f u n d a m e n t a l r a d i u s o f 1.09 (A^ " H- A^ ) - w h i c h i s s u g g e s t e d by e l e c t r o n s c a t t e r i n g d a t a - i s n e i t h e r j u s t i f i e d by t h e d a t a n o r I m p o r t a n t f o r o u r d i s c u s s i o n . . t o be 0.5 ' fm. We f i x a a t t h i s , v a l u e and R a t 5.5685 f ni. The o Coulomb b a r r i e r h a s a h e i g h t o f 8.3 MeV a t R. . T h e r e f o r e any a l p h a - p a r t i c l e r e s o n a n c e s a t a few MeV e n e r g y w i l l be v e r y n a r r o w . F o r p u r p o s e s o f i l l u s t r a t i o n we chose a f i x e d r e s o n a n c e e n e r g y , E =3.0 MeV, and v a r i e d V i n the v i c i n i t y o f 75 MeV t o o b t a i n r e s o a s i n g l e - p a r t i c l e , s-wave r e s o n a n c e . F i g . 10a shows s u c h a r e s o n a n t wave f u n c t i o n o b t a i n e d f o r V = 61.9 MeV. A t b a c k w a r d o a n g l e s w h e r e • C o u l o m b s c a t t e r i n g i s - m i n i m u m the s c a t t e r i n g c r o s s s e c t i o n i s t h a t o f the B r e i t - W i g n e r f o r m u l a . r - 2 d a ' _ 1 no d.O- i l k 2 (E - E - A ) 2 + ~ P 2  v no n o ' 4 no where i s t h e l e v e l s h i f t . F i g . 10b shows the s c a t t e r i n g no c r o s s s e c t i o n n e a r t h e e n e r g y o f the r e s o n a n t wave f u n c t i o n o f F i g . 1 0 a . The s i n g l e - p a r t i c l e r e s o n a n c e o f the o p t i c a l m o d e l w e l l i s 1.5 keV w i d e f o r t h i s c a s e . - 30 -I n f i n d i n g t h e s i n g l e - p a r t i c l e resonance o f an o p t i c a l model p o t e n t i a l we have c o m p l e t e l y s p e c i f i e d the par a m e t e r s o f the p o t e n t i a l . T h e r e f o r e we can a t once use F i g . 6 t o c o n s t r u c t the ESW. In. the p r e s e n t case the ESW has a r a d i u s o f 6.25 fm and a de p t h = S6.3 MeV i n o r d e r t h a t i t y i e l d an a l p h a - p a r t i c l e r e s o n a n c e w i t h f i v e nodes a t 3.0 MeV. The ESW and i t s r e s o n a n t wave f u n c t i o n are shown on F i g . 10a and the c o r r e s p o n d i n g s c a t t e r i n g c r o s s s e c t i o n on F i g . 10b. C l e a r l y , ' b o t h the o p t i c a l p o t e n t i a l and i t s ESW e x h i b i t r e s o n a n c e s . The re s o n a n c e o f the ESW i s c o n s i d e r a b l y n a r r o w e r t h a n t h a t o f the o p t i c a l model, a f a c t w h i c h might be a s c r i b e d t o r e f l e c t i o n . We w i s h t o e n q u i r e i n some d e t a i l how the o p t i c a l model resonance p r o p e r t i e s can be d e s c r i b e d i n terms o f the ESW resonance p r o p e r t i e s a f t e r t a k i n g a c c o u n t o f r e f l e c t i o n . I n o r d e r t o do so we need to f a c t o r y . t h e r e sonance w i d t h s i n t o p e n e t r a t i o n f a c t o r s and'reduced w i d t h s and we need t o a n a l y z e the b e h a v i o u r o f the l e v e l s h i f t . I n such a f a c t o r i z a t i o n a p a r t o f the o p t i c a l p o t e n t i a l - the " t a i l " beyond the m a t c h i n g r a d i u s - m o d i f i e s the s h i f t and p e n e t r a t i o n f a c t o r s from t h e i r c o n v e n t i o n a l f o r ms, (7) and (6) r e s p e c t i v e l y . A r e - d e r i v a t i o n o f the B r e l t - W i g n e r f o r m u l a (see 3?ef. 12, f o r example) shows t h a t P and S f s t i l l a re g i v e n by (6) and (7) i f we r e p l a c e F and G by t h e i r a n a l y t i c c o n t i n u a t i o n s , F, and G, , t o the mach.ing r a d i u s . D e n o t i n g t h e s e by P, and S^ we have 1 • = 2 P y 2 (29) no o no v ' J u s t as i n Sec. 3, the m a t c h i n g r a d i u s o f b o t h w e l l s f o r the f a c t o r i z a t i o n I s the ESW r a d i u s R.j = G.25 fm. The r e d u c e d w i d t h s a r e o b t a i n e d from the r e s o n a n t wave f u n c t i o n s by ( 1 9 ) . I n t u r n , the r e s o n a n t wave f u n c t i o n i s f o u n d t o be t h a t wave f u n c t i o n w h i c h behaves l i k e the i r r e g u l a r s o l u t i o n G„ f a r beyond the p o t e n t i a l . The boundary c o n d i t i o n number, b„ , o f the l e v e l s h i f t f u n c t i o n ( 7 ) i s s t i l l t o be chosen t o make the l e v e l s h i f t v a n i s h a t E = E . However b o t h P,. and S, are f u n c t i o n s o f the r e s X A energy and t h e r e f o r e we need t o r e t a i n both f and ^ i n the faJ- no no B r e i t - W i g n e r f o r m u l a . • There i s no doubt a.bout the v a l i d i t y o f the o n e - l e v e l 32 a p p r o x i m a t i o n i n our c a s e . The S + 0\ r e s o n a n c e s a r e about 10 MeV a p a r t compared t o the w i d t h o f 1.5'keV. The energy dependence o f the l e v e l s h i f t can be t a k e n i n t o a c c o u n t a p p r o x i m a t e l y by u s i n g a T a y l o r " s e r i e s f o r the s ! i i f." 1 about the energy and r e t a i n i n g o n l y the l o w e s t non-zero term d A no dE x (E - E ) (31) E = E no Combining (31) w i t h (28) we can w r i t e the s i n g l e - l e v e l r e sonance f o r m u l a i n the f o l l o w i n g way (r~ ,)2 der 1 *• no d-o. = (i _ ^ n 0 ) ( E : E )'2 -i- fr 2 (32) no 1 no dE The e f f e c t o f the energy dependence o f the l e v e l s h i f t on ] 7 c r o s s s e c t i o n s was f i r s t n o t e d by Thomas and i s d e s c r i b e d i n d e t a i l 32 by B r e i t " 0 who has a l s o computed those cases where a p p r o x i m a t i o n 3 7 (31) f a i l s . Thomas li a s s u g g e s t e d t h a t i f (31) were a good a p p r o x i m a t i o n (as i t s h o u l d be f o r the e x t r e m e l y narrow resonances ii w i t h i n a Coulomb b a r r i e r ) t h a t the energy dependence of the l e v e l s h i f t s be a s s o c i a t e d w i t h the reduced w i d t h s (Y< ) 2 ~ Y 2 / [ Y - d A n o 1 : "1 no ; (33) I n f a c t , he then found t h a t the reduced w i d t h s o f h i s c a l c u l a t i o n f o r a square w e l l w i t h Coulomb b a r r i e r were i n c l o s e agreement w i t h 2 2 t h o s e a p p r o p r i a t e t o a square w e l l w i t h o u t Coulomb b a r r i e r ( i . e . - - ! ! /mR ) W i t h t i l l s a s s o c i a t i o n we may n a t u r a l l y d e f i n e the o b s e r v e d w i d t h s o f t h e c a l c u l a t i o n 2 _ r / r i " &-A -' 1 - no i ' 2 P (Y 1 ) 2 = / / f 'no — o v no^ no ' I; dE so t h a t the s i n g l e - l e v e l f o r m u l a a t t a i n s the s i m p l e form a p p r o p r i a t e i n the absence o f l e v e l s h i f t s ^ = -1 ' ' n o > 2 . (3.1) d.n » . E-- , 2 ; 1 ( p ~ 2 v noJ 4 v 1 noJ I n comparing the resonance p r o p e r t i e s o f the r e a l o p t i c a l p o t e n t i a l and i t s ESW t h e r e are a number o f r a t i o s w h i c h might be r e l a t e d t o the r e f l e c t i o n f a c t o r o f 5.2 ( o r more p r e c i s e l y M.'l .at 3 MeV from F i g . (7)) f o u n d f o r our problem from F i g . 6. The r a t i o s a r e : • *) f l •« E k / ! 3 w - Y V F A = I - 2 2 , i i ) f y 2 - (rcl;i;1-:r) 2 / ( Y E g w ) 2 1.93. - 33 • -.p p „ p / p _ p 31.1. . 1 i v ) f ,, 1 " ^ ^ n o 1 ' ^ / d l ' ^ ) ; •• E = 0.727 (35) ^ A - _____ no v d i f 3 ?/dE) I r ~ no ' M t = E no v) f ( y t ) 2 =r ( Y < l f f ) 2 / ( Y ^ / - 1-38 v l ) f - Q f f / Tp,,, - M..M6 32 'I where the numbers r e f e r t o our s p e c i a l case o f "S + He a t 3 MeV. The f i r s t r a t i o , f ' , o f the o b s e r v e d w i d t h s d i f f e r s by an a p p r e c i a b l e amount from the z e r o - e n e r g y , z e r o - b a r r i e r r e s u l t o f 5.2 w h i l e the r a t i o , f , o f the one-body w i d t h s / i s in. m o d e r a t e l y good agreement w i t h t h e s e r e s u l t s and i n e x c e l l e n t agreement w i t h the one o b t a i n e d a t E = 3.0 MeV (from F i g . (7)) . I t must be remembered, however, t h a t i n t r o d u c t i o n o f the s u r f a c e e f f e c t s . i n t o t h e c h a n n e l w i d t h s w i l l , i n g e n e r a l , i n v o l v e an a d j u s t m e n t f o r the energy dependence o f the l e v e l s h i f t ( f ' - f f) . I t i s i n t e r e s t i n g t o n o t e t h a t t h e f a c t o r f e n t e r s i n a s i m p l e way i n t o t h e l e v e l s h i f t * d A d : i f f / di; I r = f d A K S W / dE no IL = L no E = E . (36) . no no * To show t h i s l e t '"d , U ( K ) : - I dE R u n l V ) d u ^ / d ^ ^ , n ; ; , ( R ) ^ l T (KCT^R) dG,/dr)j ] w h e r e R i s an a r b i t r a r y m a t c h i n g r a d i u s , u n ^ ; ( r ) i - s t b e n o r m a l i z e d wave f u n c t i o n a p p r o p r i a t e t o the nth. i -wave r e s o n a n c e a t e n e r g y , E (j , o f t h e - o n e - b o d y p o t e n t i a l , and G^ i s t h e a n a l y t i c c o n t i n u a t i o n o f t h e i r r e g u l a r A -wave Coulomb f u n c t i o n . S i n c e t h e o n e - l e v e l a p p r o x i m a t i o n • • ~2 a p p l i e s we h a v e , n e a r r e s o n a n c e , t h a t a ^ (R) = 7^ ^  (R) . M o r e o v e r , ( s u p p r e s s i n g t h e i n d i c e s ( n , £ )) i t i s e a s i l y shown t h a t (a (R) /a (R ) ) - o° (R) / a ° (R ) where R q i s t h e r a d i u s o f t h e c a l c u l a t i o n , s i n c e a ^ F F (R^) - a ^ g ^ (R^,) we h a v e (36). f , - _ & = Y d i f f " ^ = y d i f f _ ( v a d i f f 2." ,„ . „, 2~ ^ S w " Y ESW \ S W ( V *ESW ( V 32 I n f a c t i n t h e example o f S -I- c< t h e r a t i o of (36) t u r n e d o u t t o be 3.26 w h i l e f T h a s t h e v a l u e 3.22 o f (35) . The o b s e r v e d w i d t h s of t h e r e a l d i f f u s . e - e d g e p o t e n t i a l c a n t h e n a g a i n be s t a t e d i n terms of t h e p a r a m e t e r s o f i t s e q u i v a l e n t s q u a r e w e l l p' f r M i f f : " ^ f r m r ; (37) 1 - f "dl J n o 1 E IE = E no 35 by the p o s i t i v e s o l u t i o n o f the s i m p l e q u a d r a t i c e q u a t i o n c f ' 2 - f + ( f - c f ) = 0 where (38) . d A E S W c ~ no dE E = E. no Hence we may s t a t e the w i d t h s (37) of the d i f f u s e - w e l l e n t i r e l y i n terms o f the pa r a m e t e r s o f the square i n t e r a c t i o n and o f the standard' r e f l e c t i o n f a c t o r , f , o b t a i n e d from the z e r o - e n e r g y z e r o -b a r r i e r case .*. ' 1 '-I * I n an e a r l i e r c a l c u l a t i o n on a l p h a - d e c a y r a t e s I n heavy n u c l e i i t was o f h e u r i s t i c v a l u e t o d e f i n e an ESW i n a s l i g h t l y d i f f e r e n t manner. To have chosen the ESW t o have the pa r a m e t e r s s u g g e s t e d • i n Sec. 3 would have y i e l d e d a r e f l e c t i o n f a c t o r c o n s i s t e n t w i t h . t h a t f o u n d f o r n e u t r a l p a r t i c l e s a t z e r o energy b u t (as we have shown i n the p r e s e n t a n a l y s i s ) t h i s r e f l e c t i o n w o u l d have been d i v i d e d between the i n t e r i o r a s p e c t s ( r e d u c e d w i d t h s ) and e x t e r i o r a s p e c t s ( p e n e t r a b i l i t i e s ) o f ' t h e a l p h a - d e c a y p r o b l e m . Our p o i n t was t h a t p r e v i o u s a u t h o r s had. t r e a t e d the p e n e t r a b i l i t i e s c o r r e c t l y b u t bad. n o t a c c o u n t e d f o r the enhancement o f the wave f u n c t i o n i n the n u c l e a r s u r f a c e e f f e c t e d by the d i f f u s e n u c l e a r edge; i n f a c t , t h i s enhancement e x p l a i n e d the anomalous r a d i i w h i c h had p l a g u e d e a r l i e r a l p h a - d e c a y r a t e c a l c u l a t i o n s (which had g e n e r a l l y t a k e n the n u c l e a r i n t e r i o r t o be square) . T h i s was most e a s i l y d i s p l a y e d by c o n s t r u c t i n g an "ESW" from f i r s t p r i n c i p l e s , t h a t i s , i n the manner o f Sec. 3: the r a d i u s and depth o f the ESW were chosen so / t h a t the r e s o n a n t wave f u n c t i o n s o f the d i f f u s e and square w e l l had the same number o f nodes and the same a m p l i t u d e a t the ESW r a d i u s . The r a d i u s o f t h i s ESW was found to be c o n s i d e r a b l y l a r g e r than t h a t o f the d i f f u s e - w e l l , e x p l a i n i n g ! much o f the anomaly i n .the p r e v i o u s c a l c u l a t i o n s . • Complex O p t i c a l P o t e n t i a l s We w i s h t o e x t e n d our t r e a t m e n t o f b a r r i e r p e n e t r a t i o n t o the case o f complex o p t i c a l p o t e n t i a l s . As i n the case o f b a r r i e r s and r e a l o p t i c a l p o t e n t i a l s (Sec. '1) the c o m b i n a t i o n o f b a r r i e r s w i t h complex p o t e n t i a l s w i l l be shown t o e x h i b i t many o f the s i m p l e wave p r o p e r t i e s p r e s e n t where t h e r e a r e no b a r r i e r s a t a l l . A g a i n t h e r e a r e some s t r a i g h t f o r w a r d m o d i f i c a t i o n s o f the s i m p l e wave p r o p e r t i e s b r o u g h t about by the i n t r o d u c t i o n o f b a r r i e r s . 32 As i n our e a r l i e r d i s c u s s i o n , we s h a l l use the example S + C\ t o i l l u s t r a t e the main p o i n t s . I n the absence o f b a r r i e r s the a d d i t i o n o f an i m a g i n a r y term t o a r e a l o p t i c a l model p o t e n t i a l i n t r o d u c e s a b s o r p t i o n , i t broadens the wave r e s o n a n c e s , b u t has an a l m o s t n e g l i g i b l e e f f e c t 12 on wave r e f l e c t i o n . The i m a g i n a r y term f o r n u c l e o n s and even f o r heavy i o n s i s n o r m a l l y chosen t o have a s m a l l magnitude compared t o th e r e a l term. T h e r e f o r e the r e f l e c t i o n from the i m a g i n a r y term i s s m a l l - i n d e e d , the a b s o r p t i o n and s c a t t e r i n g c r o s s s e c t i o n o f a complex p o t e n t i a l w e l l a r e r e l a t i v e l y i n s e n s i t i v e t o the shape o f the i m a g i n a r y term. F o r the same r e a s o n s we e x p e c t the same k i n d o f changes when, i n the p r e s e n c e o f b a r r i e r s , we add an i m a g i n a r y •term t o the o p t i c a l p o t e n t i a l : a b s o r p t i o n e n t e r s the p i c t u r e , the res o n a n c e s broaden b u t the wave r e f l e c t i o n s h o u l d remain r e l a t i v e l y u n c h a n g e d . A t lev? e n e r g y , t h e wave r e f l e c t i o n p r o p e r t i e s m a n i f e s t t h e m s e l v e s m o s t d i r e c t l y i n n u c l e a r t r a n s m i s s i o n f u n c t i o n s . On F i g . 8 we s h o w e d a s t r a i g h t f o r w a r d a p p l i c a t i o n o f t h e z e r o - e n e r g y , 32 z e r o - b a r r i e r r e s u l t s t o t h e t r a n s m i s s i o n f u n c t i o n s f o r "S + C< . The s u c c e s s o f t h e m o d e l o f S e c . 3 a p p e a r s t o be g r e a t e r f o r t h e t r a n s m i s s i o n f u n c t i o n s t h a n f o r t h e w i d t h s d i s c u s s e d i n S e c . 4. I n o r d e r t o e l u c i d a t e t o w h a t e x t e n t t h e r a t i o o f t h e t r a n s m i s s i o n f u n c t i o n ' s s h o u l d , be r e l a t e d t o t h e r e f l e c t i o n f a c t o r , we h a v e c a l c u l a t e d t h a t r a t i o o v e r a w i d e r a n g e o f e n e r g i e s a n d f o r r e a c t i o n s i n v o l v i n g w i d e l y d i f f e r e n t C oulomb b a r r i e r s . To a n a l y z e t h o s e r e s u l t s , we w i l l e x a m i n e t h e ' e f f e c t o f i n t r o d u c i n g a s q u a r e a n d t h e n a d i f f u s e a b s o r b i n g p a r t t o t h e p o t e n t i a l . We s h a l l , . f i n d t h a t t h e r e f l e c t i o n p r o p e r t i e s a r e n e g l i g i b l y a f f e c t e d b y t h e p r e s e n c e o f an i m a g i n a r y p o t e n t i a l b u t t h a t , i f t h e a b s o r b i n g p o t e n t i a l h a s a d i f f u s e e d g e , an i m p o r t a n t f r a c t i o n o f t h e a b s o r p t i o n may t a k e p l a c e much b e y o n d t h e n o r m a l n u c l e a r r a d i u s . The n u c l e a r t r a n s m i s s i o n f u n c t i o n s a r e c a l c u l a t e d e x a c t l y by means o f ('-I) f r o m t h e c o m p l e x p h a s e s h i f t s a t c o m p l e x p o t e n t i a l s . I n t u r n , t h e p h a s e s h i f t s a r e c a l c u l a t e d d i r e c t l y f r o m n u m e r i c a l i n t e g r a t i o n o f t h e wave e q u a t i o n t o a . p o i n t w e l l b e y o n d t h e n u c l e a r r a d i u s w h e r e t h e n u m e r i c a l s o l u t i o n i s m a t c h e d on t o s t a n d a r d C o u l o m b wave f u n c t i o n s . The e x a c t t r a n s m i s s i o n f u n c t i o n s f o u n d i n t h i s way c a n h a v e t h e i r wave p r o p e r t i e s e x h i b i t e d a s i n S e c . 2. I f we a d o p t t h e f o r m (5) f o r t h e t r a n s m i s s i o n f u n c t i o n -w i t h t h e a p p r o p r i a t e mo d i f i c a t i o n s made f o r d i f f u s e - e d g e p o t e n t i a l s , as discussed i n Sec. 2 - then a simple choice of the boundary condit.ion numbe3: i s one which makes the s h i f t function, S vanish. With t h i s choice we have - r I C I + T ; A D 2 (39) where TT I S defined by •. x • r ; = ti tr f p. s , A A Here P^ 'is the penetration function, s ^ the strength function, and f the r e f l e c t i o n f a c t o r discussed i n Sec. 2. explains the r e s u l t s of Pig. 8 , both the simple r a t i o of trans-mission functions at low e n e r g y and also the corrections pertaining at higher energy. At low energy, far below the b a r r i e r , 7.' /'I 'is A, much smaller than unity so that T, " V , . . Then the r a t i o of the diffuse-edge transmission function to that of the ESW should be equal to the r e f l e c t i o n f a c t o r . This i s found, to be so in. F i g . 8 . At energies approaching the top of the Coulomb b a r r i e r i t i s not the transmission functions themselves but rather the r a t i o of 1'^ which should equal f . Again t h i s i s found to be v e r i f i e d by F i g . 8 . These re s u l t s and many other s i m i l a r results which we.obtained but cannot show here j u s t i f y our claim that the ESW should be independent of both the Coulomb and the angulai? momentum barri e r s . A comparison of F i g . 5 and F i g . 8 shows the importance of /^R i i i the choice of the ESW f o r heavy ion reactions. The r e s u l t (39) for the form of the transmission function Although the dominant, wave properties of complex wells a3."'e given i n the discussion above, there are some minor and some - 39 -major c o r r e c t i o n s i n c e r t a i n c a s e s . Two m i n o r . c o r r e c t i o n s 32 (each l e s s than 10% i n our s t a n d a r d S + r e a c t i o n ) a r i s e from resonance e f f e c t s - and from r e f l e c t i o n by the i m a g i n a r y p a r t o f the p o t e n t i a l . The former o f t h e s e m a n i f e s t s i t s e l f i n a s l i g h t energy dependence o f t h e ' s t r e n g t h f u n c t i o n when (39) i s f i t t e d t o c a l c u l a t e d t r a n s m i s s i o n f u n c t i o n s . A l t h o u g h m i n o r f o r most heavy i o n r e a c t i o n s the resonance c o r r e c t i o n s can become v e r y l a r g e i f the v a l u e o f W , the depth o f the i m a g i n a r y term i n the p o t e n t i a l , becomes s m a l l e r than the s i n g l e - p a r t i c l e l e v e l - s p a c i n g i n the r e a l p a r t o f the w e l l . The r e f l e c t i o n i n d u c e d by W i s s m a l l i n a l l cases o f p r a c t i c a l . i n t e r e s t . I t can be shown t h a t a rough e s t i m a t e o f t h i s e f f e c t 2 2 -i s t o r e p l a c e V r by (V^ + W ') 2 i n Peasl.ee's f o r m u l a , (16), f o r t h e r e f l e c t i o n f a c t o r . A major c o r r e c t i o n t o o p t i c a l model a n a l y s e s c o n c e r n s a b s o r p t i o n deep w i t h i n the b a r r i e r w h i ch o c c u r s a t e n e r g i e s v e r y f a r below the Coulomb b a r r i e r . I t f o l l o w s from an a p p l i c a t i o n 1 9 of Green's theorem t o the wave e q u a t i o n t h a t the t r a n s m i s s i o n f u n c t i o n , Tj. , i s e x a c t l y p r o p o r t i o n a l t o the i n t e g r a l o f the i m a g i n a r y p a r t o f the p o t e n t i a l where c i s a c o n s t a n t and vj/ i s the r a d i a l wave f u n c t i o n , o b t a i n e d by n u m e r i c a l i n t e g r a t i o n o f the wave e q u a t i o n w i t h the o p t i c a l p o t e n t i a l . 'In c e r t a i n c a s e s i m p o r t a n t c o n t r i b u t i o n s t o the i n t e g r a l o f (Ml) o c c u r w e l l beyond the normal n u c l e a r r a d i u s . The poi.nl: i s T (4-1) I ~ MO s t r a i g h t f o r w a r d . VJl.thin the b a r r i e r the wave f u n c t i o n i n c r e a s e s e x p o n e n t i a l l y : c>.c jL^f{ l<A<) " ('12) ( 2m 1 - -K = ™ (B - E) 2 (43) • l l . J where B i s the h e i g h t o f the b a r r i e r and E i s the energy. The a b s o r b i n g p o t e n t i a l (2) , f o r r - R a, f a l l s o f f as e . T h e r e f o r e the c o n t r i b u t i o n t o the a b s o r p t i o n from the r e g i o n w i t h i n the b a r r i e r becomes i m p o r t a n t a t e n e r g i e s such t h a t 2K approaches the v a l u e , o f a I f we t a k e a s t a n d a r d v a l u e o f 0.5 fm f o r a we f i n d 2Ka = 0.23\/m (B--E) where m i s the r e d u c e d mass ( i n u n i t s o f the p r o t o n mass) and (B~E) i s i n MeV. The a b s o r p t i o n w i t h i n the b a r r i e r i s • o f g r e a t i m p o r t a n c e f o r many heavy i o n r e a c t i o n s . I t s magnitude can be e s t i m a t e d by 32 means o f (Ml) . I n our s t a n d a r d case o f S + c< a t 3 MeV 16% o f the a b s o r p t i o n o c c u r s beyond the ESW r a d i u s . T h i s a b s o r p t i o n d o e s n ' t obey the n o r m a l r u l e s : i t i s n o t s u b j e c t t o a c o r r e c t i o n due t o r e f l e c t i o n and i t tends t o i n c r e a s e l i n e a r l y w i t h the v a l u e o f W . I t i s i m p o r t a n t t o remember t h a t the e f f e c t on the t r a n s -m i s s i o n f u n c t i o n s o f a b s o r p t i o n i n the b a r r i e r i s o p p o s i t e t o t h a t o f the energy dependence o f the r e f l e c t i o n f a c t o r . As we go down i n energy from the top o f the b a r r i e r the a b s o r p t i o n i n the b a r r i e r tends t o i n c r e a s e T^ w h i l e the energy dependence o f f d e c r e a s e s T^. F o r many commonly u s e d o p t i c a ] , p o t e n t i a l s the two e f f e c t s n e a r l y c a n c e l . More i m p o r t a n t l y , i t s u g g e s t s t h a t some a s t r o p h y s i c a l - Ml -r e a c t i o n r a t e s w i l l be dominated by the t a i l o f the i m a g i n a r y p a r t o f the o p t i c a l p o t e n t i a l . . I n such cases the o p t i c a l mode]. l o s e s i t s u t i l i t y i n e x t r a p o l a t i n g from measurements a t h i g h e n e r g i e s t o a s t r o p h y s l c a l r a t e s f a r below the b a r r i e r . C o n v e r s e l y i t r a i s e s the p r o s p e c t t h a t a b s o r p t i o n c r o s s s e c t i o n measurements a t low e n e r g i e s can p r o v i d e i m p o r t a n t i n f o r m a t i o n about n u c l e a r d e n s i t i e s f a r beyond the n u c l e a r r a d i u s . . • An extreme example o f b a r r i e r a b s o r p t i o n i s shown on 208' F i g . 11 f o r the Pb -l-CX r e a c t i o n a t an energy o f 8 MeV a p p r o p r i a t e 212 t o t he i n v e r s e o f the a l p h a decay o f " "P • Here the a b s o r p t i o n i s i m p o r t a n t a t t w i c e the n u c l e a r r a d i u s , a t v a l u e s o f r comparable t o R -1- 15a. Many o p t i c a l model programs i n c u r r e n t use w o u l d -throw away the b a r r i e r a b s o r p t i o n because they match 'the n u m e r i c a l l y i n t e g r a t e d wave f u n c t i o n s t o Coulomb f u n c t i o n s a t too s m a l l a. r a d i u s . - i|2 -•6. The Use o f the O p t i c a l Model i n A s t r o p h y s i c s C u r r e n t e f f o r t s by a s t r o p h y s i c i s t s t o d e t e r m i n e the l a t e s t a g e s o f e v o l u t i o n o f the s t a r s and to de t e r m i n e the e a r l y h i s t o r y o f the u n i v e r s e a c c o r d i n g t o the " B i g Bang" t h e o r y r e q u i r e the knowledge o f a l a r g e number o f c h a r g e d p a r t i c l e r e a c t i o n r a t e s . 20 A c c o r d i n g t o the "Big.Bang" t h e o r y m a t t e r w o u l d have emerged from thermodynamic e q u i l i b r i u m a t v e r y h i g h t e m p e r a t u r e s (T ^ , 10^ °K) and w o u l d have c o o l e d down as i t expanded. The amount o f n u c l e a r t r a n s m u t a t i o n s o c c u r r i n g as the m a t t e r expanded, th e energy r e l e a s e d u r i n g t h a t t i m e , and the t e m p e r a t u r e a t which' t h e n u c l e a r r e a c t i o n r a t e s become too slow t o a l l o w any more n u c l e a r r e a c t i o n s t o o c c u r a l l depend s e n s i t i v e l y on c h a r g e d p a r t i c l e r e a c t i o n r a t e s . On the o t h e r hand, once the s t a r s , i n ' t h e i r deep i n t e r i o r s , have bur n e d a l l t h e i r hydrogen and a l l t h e i r ' h e l i u m , ' C and " 0 a r e e x p e c t e d t o burn, u n t i l , a f t e r t h e i r 28 e x h a u s t i o n , S i decomposes i t s e l f i n t o a l p h a p a r t i c l e s w h i c h , 28 21 a d d i n g t h e m s e l v es t o S i s e e d n u c l e i , form the i r o n peak The ^2C + "^2C and the ^0 + "^ 0 r e a c t i o n s have been 22 23 s t u d i e d i n the l a b o r a t o r y '" , b u t n o t a t q u i t e as low an energy as they a r e e x p e c t e d t o o c c u r a t i n the s t a r s . One needs an under-s t a n d i n g o f the p h y s i c s o f the r e a c t i o n s t o e x t r a p o l a t e downwards . w i t h a c c u r a c y , o r a t l e a s t , t o e v a l u a t e the u n c e r t a i n t i e s o f the e x t r a p o l a t i o n . Many o t h e r needed c h a r g e d p a r t i c l e r e a c t i o n s have n o t been measured, e s p e c i a l l y ( 0(, p) , (C* > n) ('X, y ) and ( p , y ) 28 r e a c t i o n s f o r t a r g e t n u c l e i w i t h masses around and above S i . Some o f those r e a c t i o n s p r o b a b l y w i l l n e v e r be measured s i n c e they . ' i n v o l v e u n s t a b l e t a r g e t n u c l e i . T h e o r e t i c a l d e t e r m i n a t i o n s o f tho s e r a t e s , based on the o p t i c a l model have been vised by a s t r o -p h y s i c i s t s . - '13 -I n p r i n c i p l e they could, have s o l v e d the S c h r o e d i n g e r e q u a t i o n o f the Woods-Saxon o p t i c a l model f o r each c h a n n e l and f o r each n u c l e u s . Due t o the l a r g e number o f r a t e s needed, they have r e l i e d upon square w e l l , b l a c k n u c l e u s c a l c u l a t i o n s . The a c c u r a c y w i t h w h i c h the e q u i v a l e n t square w e l l r e p l a c e s a d i f f u s e w e l l v i n d i c a t e s t h e i r e f f o r t s , b u t a l s o shows the r e l a t i o n s h i p between the r a d i u s o b t a i n e d by f i t t i n g s c a t t e r i n g d a t a t o an o p t i c a ] , model and the r a d i u s one s h o u l d use i n a b s o r p t i o n c r o s s s e c t i o n s . By f i t t i n g e x p e r i m e n t a l r e a c t i o n c r o s s s e c t i o n s f o r (n , p) r e a c t i o n s ( f o r 3 7 < ^ t a r ^ 60) and f o r (p ,-X. ) > ( P > y ) and ( c ^ j V ) r e a c t i o n s ( f o r A 35) t o b l a c k n u c l e i c r o s s s e c t i o n s , 24 ] /3 ] /3 T r u r a n e t a l . d e t e r m i n e d a r a d i u s R = 1.2 (A + A ' ) fm v o I  J where A i s the mass number o f the p r o j e c t i l e and A , t h a t o f the o • . 1 • t a r g e t . From e l e c t r o n and e l a s t i c s c a t t e r i n g d a t a , one r a t h e r 1/3 e x p e c t s R = 1.2 5 A f e r m i s f o r p r o t o n s and n e u t r o n s and. , R = 1.09 A " * ' ^ + 1.6 fm f o r a l p h a p a r t i c l e s . From F i g . 9 , one s h o u l d add t o o b t a i n the ESW r a d i u s / \ R = 0.1fm f o r p r o t o n s and n e u t r o n s and. £ R = 0.7fm f o r a l p h a p a r t i c l e s . The r a d i u s o b t a i n e d by T r u r a n e t a l . i s seen t o be l a r g e r t han the s c a t t e r i n g one f o r n u c l e o n c h a n n e l s and v e r y c l o s e l y the same f o r a l p h a c h a n n e l s . P a r t o f the d i s c r e p a n c y p r o b a b l y comes from the a r t i f l e i a l r e f l e c t i o n o c c u r r i n g i n the b l a c k n u c l e u s . I f the r e f l e c t i o n f a c t o r i s used t o m u l t i p l y the p e n e t r a b i l i t y , the r a d i u s needed to f i t the (n, p) r e a c t i o n measurements i s s m a l l e r . The a l p h a c h a n n e l r a d i u s i s a l s o r e d u c e d however. We have made c a l c u l a t i o n s •I'l - 11 u s i n g d i f f u s e w e l l , t r a n s m i s s i o n f u n c t i o n s , o f the r e a c t i o n s 27 2' I 31 28 A (]?/•'•) M£ a n ( l P (lV% '0 S i . Comparing w i t h t h e . e x p e r t -25 m e n t a l c r o s s s e c t i o n s p o i n t s t o an a l p h a c h a n n e l r a d i u s R = 1.09 A^^ J + 1.6 + .4 f i n good agreement w i t h the r a d i u s f o r p a r t i c l e s c a t t e r i n g . I t i s then p o s s i b l e t o r e l a t e the par a m e t e r s t o be us e d i n r e a c t i o n c r o s s s e c t i o n , c a l c u l a t i o n s t o those o b t a i n e d L| from e l e c t r o n , meson, p, n and He s c a t t e r i n g by n u c l e i . 2 (5 Formulas l i k e C~60 o f F o w l e r and Hoyle may be used i f one r e p l a c e s P by f P , t h a t i s , i f one m u l t i p l i e s the p e n e t r a b i l i t y , o r e c j u i v a l o n t l y the s t r e n g t h f u n c t i o n , by the r e f l e c t i o n f a c t o r . The r a d i u s t o be used i s then the one o b t a i n e d from s c a t t e r i n g e x p e r i m e n t s p l u s the J^R, from F i g . 6 o r F i g . 6 . One can use Eq. C-GO o f Ref. 26 t o e s t i m a t e the u n c e r t a i n t y i n the r e a c t i o n r a t e s c a u s e d by the u n c e r t a i n t y i n the parameters o f the o p t i c a l model. Changing t he d e p t h , V , o f the p o t e n t i a l has o n l y a second o r d e r e f f e c t : the f i r s t o r d e r e f f e c t s i n Eq. (C-GO) are due t o r e f l e c t i o n and they a r e c a n c e l l e d by the dependence o f f on V . The e f f e c t o f the s u r f a c e t h i c k n e s s , a, i s more p r o f o u n d . I n c r e a s i n g i t by 25% w i l l g e n e r a l l y i n c r e a s e the f f a c t o r by 25% 32 4 b u t , more i m p o r t a n t , w i l l , sometimes, as f o r " "S + He, i n c r e a s e / \ R by a f a c t o r o f 2 (see F i g . 6 ) . A l l i m p o r t a n t u n c e r t a i n t i e s can then.be r e l a t e d t o the r a d i u s , R , o f the d i f f u s e w e l l , and o 32 4 t o the /\R needed t o o b t a i n the r a d i u s o f the ESW. F o r " "S 1 He, q a t T - 3.0 x 10* °K, i.t can'" e a s i l y be c a l c u l a t e d , u s i n g eq. C-GO, t h a t c h a n g i n g the r a d i u s from R = 5.6 fm t o R = 6.6 fin i n c r e a s e s the r e a c t i o n r a t e by a f a c t o r o f 5.0. U n c e r t a i n t i e s i n the r a d i u s and the s u r f a c e t h i c k n e s s t h e n seem t o i n t r o d u c e u n c e r t a i n t i e s o f a f a c t o r o f 5.0 i n the r e a c t i o n r a t e s i n v o l v i n g L|. \ He c h a n n e l s . I n the p r e c e d i n g d i s c u s s i o n , i t was assumed t h a t the o p t i c a l model i s a p r o p e r r e p r e s e n t a t i o n of the He r e a c t i o n c h a n n e l s . We can a l s o s t u d y how s e n s i t i v e the v a l u e o f <Ccr' i s on the u n d e r l y i n g p h y s i c s . To t h i s end, we have used as our o p t i c a l p o t e n t i a l , M n , ( r - R ) / a .~1 ... , ( r - R ) / a . -1 V ( r ) = - V Q (1 -I- e v oJ v) - l W (1 + e v oJ ' w) and we have v a r i e d a... w h i l e k e e p i n g a„ c o n s t a n t . T h i s r e p r e s e n t s W . ° V one o f the ways i n w h i c h the commonly used o p t i c a l mode], can be m o d i f i e d t o r e p r e s e n t d i f f e r e n t h y p o t h e s e s as t o matte]? d i s t r i b u t i o n and the l o c a t i o n o f the compound n u c l e u s f o r m a t i o n . 32 N u m e r i c a l c a l c u l a t i o n s have been c a r r i e d out f o r S + CX , l|,|T +e\ and 1 2 C + 1 2 C w i t h a „ / a „ . The r a t i o s T„ ( a „ / a w ) /T , (ESW) V W A W V' A a r e p r e s e n t e d on F i g . 12 . Whereas the e q u i v a l e n t square w e l l i s a good a p p r o x i m a t i o n t o the d i f f u s e w e l l f o r a^ = a i t b r e a k s down c o m p l e t e l y i f a^; >" a^. The c r o s s s e c t i o n i s then dominated by ] 2 ] 2 a b s o r p t i o n i n the b a r r i e r as can be seen f o r C + C on F i g . 13 . The low energy c r o s s s e c t i o n o b t a i n e d w i t h the b l a c k n u c l e u s c o u l d t hen e a s i l y be an u n d e r e s t i m a t e by a n o t h e r f a c t o r o f 5. T h i s i s e x p e c t e d t o be i m p o r t a n t s p e c i a l l y f o r a l p h a p a r t i c l e s i n c i d e n t on n u c l e i w i t h l a r g e Z (Z > 20)'. However, the d a t a a v a i l a b l e f o r a l p h a p a r t i c l e c h a n n e l s i n r e a c t i o n c r o s s s e c t i o n s a t low energy i s f o r - '16 -31 P (p,o'.) S t , 27 S i n c e the Q v a l u e o f those r e a c t i o n s i s p o s i t i v e , the t r a n s m i s s i o n f u n c t i o n f o r p r o t o n s i s a p p r o x i m a t e l y e q u a l , a t a g i v e n energy i n the compound n u c l e u s , t o t h a t f o r a l p h a ! p a r t i c l e s . The a l p h a p a r t i c l e t r a n s m i s s i o n f u n c t i o n i s r e a s o n a b l y l a r g e . a n d i s n o t e x p e c t e d t o show a b s o r p t i o n i n the b a r r i e r , w i t h b l a c k n u c l e i c r o s s s e c t i o n s and then e x t r a p o l a t i n g t o h i g h e r mass numbers c o m p l e t e l y n e g l e c t s the p o s s i b i l i t y o f a b s o r p t i o n i n the b a r r i e r . .More e x p e r i m e n t a l d a t a i s needed t o p e r m i t r e l i a b l e e s t i m a t e s o f a l p h a p a r t i c l e c h a n n e l s a t A ft-W. ] 2 12 The C + C r e s u l t s i n d i c a t e how s e n s i t i v e the cross, ] 2 12 s e c t i o n o f the C 1 C r e a c t i o n i s on the d e t a i l e d ' s h a p e o f the o p t i c a l model chosen. A 20% i n c r e a s e i n the d i f f u s e n e s s parameter o f the i m a g i n a r y p o t e n t i a l i n c r e a s e s the c r o s s s e c t i o n a t G ^ 3 MeV by a f a c t o r o f 5. Only a model t h a t would r e p r e s e n t c l o s e l y the ] 2 J 2 p h y s i c s o f the C + ' C system c o u l d be hoped t o p e r m i t any e x t r a -p o l a t i o n s . The r e c e n t r e s u l t s o f R e f . 22 may i n d i c a t e t h a t the o p t i c a l model i s too crude a t o o l . ^• G e n e r a l C o n c l u s i o n s C o n c e r n i n g N u c l e a r R e a c t i o n s Our a n a l y s i s o f the r o l e o f the o p t i c a l p o t e n t i a l i n b a r r i e r p e n e t r a t i o n s u g g e s t s a number o f genera], c o n c l u s i o n s about n u c l e a r - r e a c t i o n s . They c o n c e r n the r o l e o f the r a d i u s i n resonance r e a c t i o n s , the v a l u e of. the n u c l e a r r a d i u s i n • r e a c t i o n s and the use o f a b s o r p t i o n c r o s s s e c t i o n s , f a r below the b a r r i e r t o probe the t a l l o f the n u c l e a r d e n s i t y d i s t r i b u t i o n . -The many-channel t h e o r y o f resonance r e a c t i o n s appears t o have a much more c o m p l i c a t e d geometry than the one-channel p o t e n t i a l - s c a t t e r i n g p r o b l e m whose wave p r o p e r t i e s we d e s c r i b e d 1 i n Sec. 4 . We s h a l l show t h a t i t i s r e a s o n a b l e t o decompose the • s c a t t e r i n g m a t r i x i n a manner such t h a t each r e a c t i o n c h a n n e l p o s s e s s e s the o n e - c h a n n e l p o t e n t i a l - s c a t t e r i n g p r o p e r t i e s . Then we can vise our e a r l i e r r e s u l t s f o r the one-channel case t o remove many o f the a r t i f i c i a l " " s q u a r e - w e l l " a s p e c t s o f the resonance t h e o r y r e s u l t s . A s i m i l a r t r e a t m e n t f o r n u c l e a r r e a c t i o n s w i t h o u t ] 2 b a r r i e r s 'was g i v e n i n an e a r l i e r paper by V o g t . The g e n e r a l t h e o r y o f n u c l e a r reactions'""""' p r o v i d e s a framework i n w h i c h a l l c r o s s s e c t i o n s can be d e s c r i b e d , i n terms o f l e v e l p a r a m e t e r s . The r e l a t i o n between c r o s s s e c t i o n s and l e v e l p a r a m e t e r s i s made i n two s t e p s . F i r s t the c r o s s s e c t i o n , XT' ,, cc' f o r an i n i t i a l c h a n n e l c and. a f i n a l c h a n n e l c' , i s w r i t t e n i n terms o f s t a t i s t i c a l s p i n f a c t o r s and c o l l i s i o n m a t r i x components. Next the c o l l i s i o n m a t r i x components, U ,, a r e w r i t t e n i n terms o f l e v e l p a rameters . " c c t . = e 1 ( ^ c + - n c ' ) ( g c c t + i £ X X l r | c r | c , A X X I ) - (15) "where the a r e phase s h i f t s and the A^ A, the components o f a m a t r i x whose i n v e r s e i s ( A _ L > X X T = ( E X - E ) S A A , + A A A , - (|)| X X l (-16) - MS -T h i s form o f the framework i s c o m p l e t e l y g e n e r a l : i t a p p l i e s t o a l l approximate forms from the Breit-W.igner Formula t o the Hauser-Feshbaeh t h e o r y . The q u e s t i o n i s , how do the l e v e l p a r a m e t e r s change when a r e a c t i o n c h a n n e l i s assumed t o i n c l u d e an average Saxon-Woods p o t e n t i a l ? The square w e l l i n t e r a c t i o n i s e a s i l y a dapted t o the o r d i n a r y r e a c t i o n t h e o r i e s . F o r a square w e l l , both the p a r t i a l w i d t h s and the l e v e l s h i f t can be f a c t o r e d i n a manner w h i c h s e p a r a t e s out the many-body f e a t u r e s , o f the probl e m and w h i c h c l e a r l y d i s p l a y s the wave p r o p e r t i e s - a s s o c i a t e d w i t h the average i n t e r a c t i o n : r;c= 8Ac» r'p - v »; <yV m A A = - I S X P S , ( Y * V . ' (.18) >c ^ ____ A I The s p e c t r o s c o p i c f a c t o r s , ^, a r e e s s e n t i a l l y s t a t i s t i c a l c o e f f i c i e n t s w h i c h measure the p r o b a b i l i t y f o r f i n d i n g the compound n u c l e u s i n the p a r t i c u l a r mode s p e c i f i e d by the c h a n n e l number, c. S i n c e they depend on averages over the n u c l e a r volume t h e y a r e i n s e n s i t i v e t o the d e t a i l s o f the n u c l e a r s u r f a c e . Thus the e f f e c t s o f the s u r f a c e i n f l u e n c e o n l y the one-body a s p e c t s o f the p r o b l e m , the s i n g l e p a r t i c l e w i d t h s o r the c o r r e s p o n d i n g s i n g l e - p a r t i c l e r e d u c e d w i d t h s (7 ) . T h e r e f o r e the t r a n s i t i o n f r om a square average i n t e r a c t i o n to a Saxon-Woods i n t e r a c t i o n a f f e c t s o n l y the s i n g l e - p a r t i c l e w i d t h . The s i n g l e - p a r t i c l e width, i s a f f e c t e d , by the d i f f u s e - e d g e i n the manner d i s c u s s e d , i n Sec. 2 and Sec. 3. Thus the i n s e n s i t i v i t y o f the s p e c t r o s c o p i c f a c t o r s t o the p r e c i s e v a l u e o f the m a t c h i n g r a d i u s makes our wave a n a l y s i s _ 4 9 _ i a p p l y to each r e a c t i o n c h a n n e l s e p a r a t e l y . : The c o n v e n t i o n a l t r e a t m e n t o f n u c l e a r r e a c t i o n s by the b l a c k - b o x o r resonance t h e o r i e s i s e s s e n t i a l l y a s q u a r e - w e l l t r e a t m e n t . Because o f t h i s f a c t the n u c l e a r r a d i u s r e q u i r e d i n 'j t h i s t r e a t m e n t t o f i t o b s e r v e d r e a c t i o n r a t e s was a r t i f i c i a l l y 3 l a r g e . F o r s e v e r a l decades the u s u a l v a l u e o f the n u c l e a r r a d i u s f o r n u c l e a r r e a c t i o n s was ' • R = l . ' l - ( A ^ 3 + A ^ ) F M where A ^ i s a t o m i c w e i g h t o f the t a r g e t n u c l e u s and A ^ t h a t o f the b o m b a r d i n g - p a r t i c l e . On the o t h e r hand the u s u a l o p t i c a l model r a d i u s f o r n u c l e o n s i s R = 1.25 A ] / 3 fm (50). and f o r heavy i o n s R - 1.25 ( A * / 3 + A ? / 3 ) fm (51) o r perhaps R = 1.09 ( A ^ 3 + A ^ 3 ) fm (52) F o r b o t h n u c l e o n s and heavy i o n s the d i f f e r e n c e i n the r a d i i i s u s u a l l y between one and two f e r m i s . I t i s our c o n c l u s i o n t h a t the l a r g e r ' r a d i i were a r e s u l t o f the square w e l l t r e a t m e n t . B e f o r e embarking on an e x p l a n a t i o n o f the d i f f e r e n c e s between o l d and new r a d i i we make some remarks on the c u r r e n t f a s h i o n s . The charge r a d i i o f n u c l e i , as measured by e l e c t r o n s c a t t e r i n g and 1/3 me s i c atoms are 1.09 x A f e r m i s . I f t h i s i s t a k e n t o r e f l e c t the n u c l e a r d e n s i t y as w e l l as the charge then a n u c l e o n s h o u l d ' f e e l the same r a d i u s - the same v a l u e s h o u l d a p p l y t o the o p t i c a l p o t e n t i a l . l / 3 ] / ~ S i m i l a r l y 1.09 (A-j + A.-, ) s h o u l d then be the r a d i u s f o r heavy i o n r e a c t i o n s . Now t h e r e a r e some a d d i t i o n a l e f f e c t s w h i c h t e n d t o i n c r e a s e the o p t i c a l model r a d i u s s l i g h t l y and t o make i t s l i g h t l y dependent on the s h e l l s t r u c t u r e . The f i r s t i s cor e • p o l a r i z a t i o n - an i n c o m i n g n u c l e o n p u l l s the t a r g e t n u c l e o n s t o w a r d i t . The second i s the n e u t r o n e x c e s s i n the s u r f a c e o f a n u c l e u s •- an ex c e s s w h ich l e a d s t o the i s o t o p l c s p i n term i n the o p t i c a l p o t e n t i a l . . Such a term means t h a t the d e n s i t y extends beyond the ch a r g e . Both o f t h e s e e f f e c t s can be e s t i m a t e d o n l y r o u g h l y and depend on the p a r t i c u l a r n u c l e u s i n v o l v e d . T h e i r magnitude r o u g h l y . j u s t i f i e s the s m a l l amount by w h i c h c u r r e n t r a d i i exceed the charge r a d i i . There i s some e x p e r i m e n t a l u n c e r t a i n t y i n the r a d i i as w e l l . A l t h o u g h the wave resonance 2 e f f e c t s d e t e r m i n e the p r o d u c t V q R v e r y a c c u r a t e l y - perhaps to 1% - we know o f no e x p e r i m e n t w h i c h unambiguously d e t e r m i n e s the r a d i u s o f the r e a l p a r t o f the o p t i c a l p o t e n t i a l t o a n y t h i n g l i k e , ] /3 t h i s a c c u r a c y . T h e r e f o r e the c h o i c e t o r n u c l e o n s of 1.25 A i s n o t o n l y a r e a s o n a b l e c h o i c e b u t a l s o has about a 10% u n c e r t a i n t y . By the same t o k e n , a.n a l p h a p a r t i c l e r a d i u s o f 1.6 i s a l s o r e a s o n a b l e From our a n a l y s i s o f wave p r o p e r t i e s we see t h a t t h e r e a r e s e v e r a l ways i n w h i c h the change from an o p t i c a l p o t e n t i a l to a square w e l l m o d i f i e s the r a d i u s . F i r s t o f . a l l , t h e r e i s the d i f f e r e n c e i n r e f l e c t i o n between the w e l l s w h i c h can be compensated f o r by a d i f f e r e n c e i n r a d i i ; s e c o n d l y , t h e r e i s the wave o s c i l l a t i o n i n the t a i l o f the r e a l p a r i ; o f the o p t i c a l p o t e n t i a l w h i c h l e a d s ( F i g . 5 ) t o a d i f f e r e n c e i n r a d i u s between, the o p t i c a l p o t e n t i a l and. i t s e q u i v a l e n t s q uare w e l l ; t h i r d l y , there i s a b s o r p t i o n i n the t a i l o f the i m a g i n a r y p a r t o f the p o t e n t i a l . A r e a s s e s s m e n t o f the o l d a n a l y s e s needs t o examine only the heavy i o n a b s o r p t i o n c r o s s s e c t i o n s f o r which the j dependence on the n u c l e a r r a d i u s i s unambiguous. F o r n u c l e o n s i t has t u r n e d out t h a t t h e r e are s i z e - r e s o n a n c e e f f e c t s w h i c h Vary through, the p e r i o d i c t a b l e (see F i g . 2) . F o r a l p h a p a r t i c l e s and o t h e r heavy i o n s a l l the s i z e - r e s o n a n c e e f f e c t s a r e washed 2 out - p a r t i c l e s r e a c h i n g the n u c l e u s a r e a bsorbed. E a r l y e v i d e n c e " i 27 f o r 1 the l a r g e r a d i u s came from charged p a r t i c l e r e a c t i o n s as w e l l as n e u t r o n r e a c t i o n s . I n the b l a c k - n u c l e u s model f o r a l p h a p a r t i c l e a b s o r p t i o n the s t r e n g t h f u n c t i o n was shown ( i n Sec. 2) t o be t h a t o f a square w e l l . I t has no g i a n t r e sonance s t r u c t u r e b u t n e i t h e r does the d i f f u s e - e d g e o p t i c a l model p o t e n t i a l s u i t a b l e f o r a l p h a p a r t i c l e r e a c t i o n s . We might t h e r e f o r e be tempted t o say t h a t the b l a c k -n u c l e u s r a d i u s s h o u l d be chosen t° be t h a t o f the e q u i v a l e n t square w e l l o f the a p p r o p r i a t e o p t i c a l p o t e n t i a l . I n such a c h o i c e the A R o f F i g . 9 has a v a l u e o f about 0.5 f e r m i s . But such a c h o i c e i g n o r e s wave r e f l e c t i o n . From F i g . 9 the r e f l e c t i o n f a c t o r f o r a l p h a p a r t i c l e s has a. v a l u e between 3 and 5. The t r a n s m i s s i o n f u n c t i o n s f a r below the b a r r i e r are d i r e c t l y p r o p o r t i o n a l t o the r e f l e c t i o n f a c t o r . To enhance the b l a c k - n u c l e u s t r a n s m i s s i o n f u n c t i o n s by a f a c t o r o f 3 t o 5 we can i n c r e a s e the n u c l e a r r a d i u s , t h u s i n c r e a s i n g the p e n e t r a t i o n f a c t o r . The r e q u i r e d i n c r e a s e i n - 52 -the r a d i u s i s about 0.5 f e r m i s * . l' The p e n e t r a t i o n f a c t o r a t an energy, E, f a r below the Coulomb b a r r i e r , B, depends on the n u c l e a r r a d i u s r o u g h l y as P k 11 G ~ 2 ( k R) o c k R " 2 k R where F o r a l p l i a ' p a r t i c l e s k t y p i c a l l y has v a l u e s between 1.0 and 2.0. Thus the wave o s c i l l a t i o n . («/\R i ; 0.5) and the wave r e f l e c t i o n t o g e t h e r a c c o u nt f o r the 1.0 fm d i f f e r e n c e between the b l a c k - n u c l e u s r a d i u s and t h a t o f modern o p t i c a l p o t e n t i a l s . The e a r l y a n a l y s i s o f a l p h a decay r a t e s i n heavy n u c l e i r e q u i r e d s i m i l a r anomalous l a r g e r a d i i w h i c h have been b r o u g h t i n t o agreement w i t h modem " 1 v a l u e s by our-wave a n a l y s i s . " I n our view t h e r e i s no e v i d e n c e a t a l l t h a t any n u c l e a r r e a c t i o n r a t e s r e q u i r e a nomalously l a r g e r a d i i o r t h a t they throw i n t o q u e s t i o n the i n d i v i d u a l p a r t i c l e p i c t u r e o f n u c l e a r s t r u c t u r e . The sum-rule l i m i t s a s s o c i a t e d w i t h r e d u c e d w i d t h s a r e square w e l l v a l u e s w h i c h need t o be m o d i f i e d t o take i n t o a c c o u nt the r e f l e c t i o n f a c t o r o f a d i f f u s e - e d g e w e l l . A c c o r d i n g t o (20) above, f o r any r e a c t i o n c h a n n e l we can w r i t e a s i n g l e p a r t i c l e w i d t h as r V = 2P f(.v4V (53) - S3 -Where F, i s the-: c o n v e n t i o n a l p e n e t r a t i o n f a c t o r , ( 6 ) , and ( y ) " I the s i n g l e - p a r t i c l e r e d u c e d w i d t h of the w e l l . E q u a l l y , we can w r i t e ( y V = f ( V X p ) 2 = f f i 2 M 2 (SM) 1\ 2 where f i s the r e f l e c t i o n f a c t o r d i s c u s s e d above and ( y ' ) i s I SW the s i n g l e - p a r t i c l e r e d u c e d w i d t h o f the e q u i v a l e n t square w e l l (ESW) h 2 I t i s fy ) which, has the c o n v e n t i o n a l s u m - r u l e • v a l u e o i 2 SW di /mR ^ where R i s the r a d i u s o f the ESW. T h e r e f o r e a t r u e s i n g l e ' o o p a r t i c l e l e v e l s h o u ld'have a r educed width, e x c e e d i n g the c o n v e n t i o n a l sum-rule l i m i t by the r e f l e c t i o n f a c t o r f which t y p i c a l l y has a v a l u e between two and f i v e . Our work shows t h a t t h i s f a c t o r a p p l i e s to a l l c h arged p a r t i c l e r e a c t i o n s as w e l l as t o - n e u t r o n s and. g i v e s q u a n t i t a t i v e e s t i m a t e s o f f . F a r below the b a r r i e r the a b s o r p t i o n i n the t a i l o f the o p t i c a l p o t e n t i a l can dominate the whole a b s o r p t i o n p r o c e s s . We have warned o f the dangers o f u s i n g c o n v e n t i o n a l o p t i c a l models to c a l c u l a t e r e a c t i o n r a t e s w h ich ar e dominated by t h e i r t a i l s . We can, however, t u r n the argument around and s u g g e s t t h a t the measurement o f a b s o r p t i o n c r o s s s e c t i o n s f a r below the b a r r i e r be u sed t o d e t e r m i n e n u c l e a r p r o p e r t i e s a t l a r g e r a d i i . T h i s i s a p r o b l e m o f c o n s i d e r a b l e i m p o r t a n c e i n modern n u c l e a r p h y s i c s 28 29 because o f f i s s i o n i s o m e r i s m , mesonie atoms " and the g e n e r a l q u e s t i o n o f n u c l e a r c l u s t e r s i n r e g i o n s o f low d e n s i t y . From our s t u d y i t i s c l e a r t h a t any n u c l e a r d e n s i t y (whether n u c l e o n s or - 5'-l -o t h e r c l u s t e r s ) e x t e n d i n g t o l a r g e r a d i i , can dominate the a b s o r p t i o n p r o c e s s . 11 wou l d be v e r y i n t e r e s t i n g and v a l u a b l e t o p e r f o r m n u c l e a r r e a c t i o n s w i t h i n t e n s e beams a t low e n e r g i e s t o examine, s a y , the p r o b a b i l i t y o f a l p h a p a r t i c l e a b s o r p t i o n i n the bombardment o f many l i g h t n u c l e i . ' S i m i l a r l y , e x p e r i m e n t s w i t h • p r o t o n . o r heavy i o n beams might a l s o y i e l d a n o m a l o u s l y l a r g e 3 6 2 0 2 M c r o s s s e c t i o n s . I s " 0 much more t i g h t l y bound t h a t Ne o r Mg? Does the d i s t a n t t a i l o f the n u c l e a r d e n s i t y i n the l a t t e r p r e s e n t n u c l e o n s o r a l p h a p a r t i c l e c l u s t e r s ? I t i s l i k e l y t h a t the k i n d 12 12 22 o f s u r p r i s e s f o u n d i n the C + C r e a c t i o n s w i l l abound i n many o t h e r c a s e s . The r e s u l t s may a l s o change many o f our n o t i o n s c o n c e r n i n g s t e l l a r r e a c t i o n r a t e s . R e f e r e n c e s i I G. Bre.il: and E.P. Wigner, Phys. Rev. 49", 519, 642 (1936). H. A. B e t h e , Rev. Mod. Phys. 9 , 69 (1937). V.W. W e i s s k o p f and D.H. Ewing, Phys. Rev. 57 , 472 (1940) and J . B l a t t and V.W. W e i s s k o p f , T h e o r e t i c a l N u c l e a r P h y s i c s (John W i l e y & Sons, I n c . , New Y o r k , 1952) . H.H. B a r s c h a l l , Phys. Rev. 86, 431 (1952). s e e , f o r example, E. A l m q v i s t , D.A. Bromley, J . Kuchner and E. V o g t , i n P r o c e e d i n g s o f the I n t e r n a t i o n a l C o n f e r e n c e on N u c l e a r S t r u e t u r e , K i n g s t o n , e d i t e d by D.A. Bromley and E. Vogt ( U n i v e r s i t y o f T o r o n t o P r e s s , T o r o n t o , Canada, 1960), p. 736. K.W. McVoy, Phys. L e t t e r s 17, 42 (1965), A n n a l s o f P h y s i c s 43, 91 (196 7) . H. Feshbach, C.E. P o r t e r and V.W. W e i s s k o p f , Phys. R e v / 96, 448 (19 54) . G.R.. S a t c h l e r , i n I n t e r n a t i o n a l N u c l e a r P h y s i c s C o n f e r e n c e , G a t l i n b u r g , e d i t e d by R.I,. B e c k e r e t a l . (Academic P r e s s , New York,. 196 7) , p. 1. L. E i s e n b u d and E.P. Wigner, P r o c . N a t l . Acad. S c i . U.S. _27, 281 (1941); E.P. Wigner, Phys. Rev.' 70 > 15 (1946), Phy Rev. 70, 606 (1946); E.P. Wigner and L. E i s e n b u d , Phys Rev. 72, 29 (1947); T. Teichmann and E.P. Wigner,'Phys • R e v - 1Z» 1 2 3 0° r , ;-) • A.M. Lane and'D. Robson, Phys. Rev. 151, 774 (1966). 10. E.W. V o g t , i n Advances i n N u c l e a r P h y s i c s , V o l . 1, e d i t e d by M. Bar a n g e r and E.W. V o g t , (Plenum P r e s s , New Y o r k , 1968) , p. 261. 11. A.M. Lane, R.G. Thomas and E.P. Wigner, Phys. Rev. 98, 693 , (1955) . 12. E.W. V o g t , Rev. Mod. Phys. 34, .723 (1962). 13. E.W. V o g t , G.J. Michaud and H. Reeves, P h y s . L e t t e r s 19, 570 (1965) . 14. L. S.cherk and E.W. V o g t , Can. J . Phys. 46, 1119 (1958) . 15. N. A u s t e r n , A. 'Prakash and R.M. D r i s k o , Ann. Phys. (N.Y.) _39, 253 (1966) . 16. D.C. P e a s l e e , N u c l . Phys. _3, 255 (1957) . 17. R.G. Thomas, Phys. Rev. 8JL, 148 (1951). 18. G. B r e i t , Handbuch der Phys., V o l . X L I , ( S p r i n g e r V e r l a g , B e r l i n , 1.9 59) , p. 1. 19. . M.A. P r e s t o n , P h y s i c s -of the N u c l e u s ( A d d i s o n Wesley P u b l . Co. , New Y o r k , 1962) . 20. R.V. Wagoner, W.A. F o w l e r and F. H o y l e , Ap. J . _148, 3 (1967) . • 21. J.W. T r u r a n , A.G.W. Cameron and A. G i l b e r t , Can. J . Phys. _44, 563 (1966); D. Bodansky, D.D. C l a y t o n and W.A. F o w l e r , Phys. Rev. L e t t e r s 2h, 161 (1968) and W.D. A r n e t t , ' E x p l o d i n g S t a r Models and Supernovae, K e l l o g P r e p r i n t , 1968. 22. J.R. P a t t e r s o n , H. W i n k l e r and C.S. Z a i d i n s , t o be p u b l i s h e d (1969) . 23. J.R. P a t t e r s o n , H. W i n k l e r and H. S p i n k a , P r i v a t e communication (1968). 211 - J.W. T r u r a n , C.J. Hansen, A.G.W. Cameron and A. G i l b e r t , Can. J . Phys. 151 (1966) . 25. P.M. Endt and C. Van der Lenn, N u c l . Phys. A105, 1, 1967. 26. W.A. F o w l e r and F. H o y l e , A s t r o p h y s . J . S u p p l . 91, 201 (196'l) .27. s e e , f o r example, J.P. B l a s e r , F. Boehm, P. Marmier and D.C. P e a s l e e , Phys. A c t a 24, 3 (1951). A complete discussion of e a r l y n u c l e a r r e a c t i o n s i s found i n B l a t t and W e i s s k o p f s book, r e f . 3. 28. V.M. S t r u t i n s k y , N u c l . Phys. A_95, l!20 (1967); E. Mlgneco and J.P. T h e o b a l d , N u c l . Phys. A112, 603, 1968. 29. S. Devons and I . ' D u e r d o t h , Advances i n N u c l e a r P h y s i c s , V o l . 2, e d i t e d by M. Baranger and E.W. V o g t , (Plenum P u b l . Co., New Y o r k , 1969), p. 29 5. F i g i i r e C ap 11 ons F i g u r e 1. C o m p a r i s o n o f t h e low l y i n g r e s o n a n t s t a t e s o f t h e s q u a r e w e l l and o f t h e c o r r e s p o n d i n g d i f f u s e edge w e l l f o r s-wave n e u t r o n s . Once t h e d e p t h and r a d i u s o f t h e s q u a r e w e l l a r e a d j u s t e d so t h a t b o t h w e l l s h a v e a r e s o n a n c e ' w i t h t h e same r e d u c e d w i d t h a t E = - O.M-3 MeV, a l l the o t h e r low l y i n g r e s o n a n c e s i n t h e two w e l l s o c c u r a t v e r y c l o s e l y t h e same e n e r g y and w i t h v e r y n e a r l y the same r e d u c e d w i d t h ( e x c e p t f o r t h e r e d u c e d w i d t h o f t h e r e s o n a n c e a t E -- 53 MeV) I t i s t h e n p o s s i b l e f o r a s q u a r e w e l l t o r e p r o d u c e t h e i n t e r i o r p r o p e r t i e s o f r e s o n a n c e o f a d i f f u s e w e l l . The d i f f e r e n c e b e t w e e n t h e two w e l l s i s t h e n c o n t a i n e d o n l y i n t h e p e n e t r a b i l i t y and s h i f t f u n c t i o n s . F i g u r e 2.,. C o m p a r i s o n o f o f p r o t o n s f o r d i f f e r e n t v a l u e s o f t h e s u r f a c e t h i c k n e s s a . B e c a u s e t h e t r a n s m i s s i o n f u n c t i o n s a r e c l o s e .to 1. , i t i s n e c e s s a r y t o compare t o o b t a i n t h e r e f l e c t i o n f a c t o r s . The 'T ? a r e o b t a i n e d f r o m the t r a n s m i s s i o n f u n c t i o n s by u s i n g T« = / ( I l - t ; / 1 ! ) . I t i s s e e n t h a t by m u l t i p l y i n g ^2 ( a " 0) by a f a c t o r f (a.) , i n d e p e n d e n t o f e n e r g y , one g e t s an e x c e l l e n t a p p r o x i m a t i o n . t o T ^ (a ¥ 0) • The I n t e r i o r o f t h e two w e l l s must t h e n have v e r y s i m i l a r r e s o n a n c e p r o p e r t i e s . O n l y t h e p e n e t r a b i l i t i e s j i I v a r y . I t then, seems p o s s i b l e , as i t was f o r n e u t r o n s , t o r e p l a c e a d i f f u s e w e l l , by a square w e l l . ! F i g u r e 3. The shape o f the r e a l p o t e n t i a l f o r s-wave p r o t o n s 32 ( • S + p) . The Coulomb p o t e n t i a l (V ) does add a r a d i u s dependent term. However i t s r a d i u s dependence i s much weaker t h a n the r a d i u s dependence o f the n u c l e a r p o t e n t i a l ( V ^ ) • I n f i r s t a p p r o x i m a t i o n the Coulomb f i e l d adds o n l y a c o n s t a n t term w h i c h does n o t change the r e f l e c t i o n p r o p e r t i e s o f the two w e l l s . P r o t o n s s h o u l d behave l i k e n e g a t i v e energy n e u t r o n s . As i s the case f o r n e u t r o n s , i t s h o u l d then be p o s s i b l e t o f i n d an e q u i v a l e n t square w e l l 0^ .^,) f o r c h a r g e d p a r t i c l e s . The d i f f e r e n c e between the two w e l l s s h o u l d a g a i n r e s i d e o n l y in. the r e f l e c t i o n f a c t o r . T h i s i s v i n d i c a t e d by F i g . 2. I n t h i s example, the n u c l e a r p o t e n t i a l i s chosen t o have a Saxon-Woods shape w i t h d e p t h SO MeV, r a d i u s 3.96fm, and. s u r f a c e t h i c k n e s s O.Sfm. • ' F i g u r e 4. R a t i o o f t r a n s m i s s i o n f u n c t i o n s o f an o p t i c a l p o t e n t i a l 4 32 s u i t a b l e f o r He -I- S f o r two v a l u e s o f the s u r f a c e t h i c k n e s s . ( a = -^"V'^J^Q (a = 0 . 0 )) . The two w e l l s have the same V and the same R . I t seems o o e i t h e r t h a t t he i n t e r i o r o f the two w e l l s show d i f f e r e n t 1! r e s o n a n c e , b e h a v i o u r o r t h a t d i f f e r e n t phenomena' o c c u r u n d e r t h e Coulomb b a r r i e r . I t w i l l be s e e n i n t h e n e x t s e c t i o n s t h a t most o f t h e e n e r g y v a r i a t i o n s o f t h e r a t i o comes f r o m c h o o s i n g two w e l l s w h i c h d o n ' t have t h e same I n t e r i o r p r o p e r t i e s . A b s o r p t i o n . I n t h e b a r r i e r w i l l be s e e n t o become d o m i n a n t o n l y a t v e r y l o w e n e r g y (E < 1 MeV i n t h i s case) . F i g u r e 5 . A d i f f u s e (SOLID) and a s q u a r e (DASHED) p o t e n t i a l w e l l and t h e c o r r e s p o n d i n g s-wave z e r o - e n e r g y wave f u n c t i o n s . The d e p t h and. r a d i u s o f t h e s q u a r e w e l l were c h o s e n so t h a t i t h a d a r e s o n a n c e a t E := 0 , and t h a t t h e r e d u c e d w i d t h o f t h a t r e s o n a n c e be t h e same as t h a t o f t h e d i f f u s e w e l l a t t h e s q u a r e w e l l r a d i u s . The i n t e r i o r p r o p e r t i e s o f t h e two w e l l s w i l l t h e n be t h e same as can be s e e n f r o m F i g . 1 . The wave f u n c t i o n , o f t h e d i f f u s e w e l l c o n t i n u e s t o r i s e o u t s i d e o f t h e s q u a r e w e l l r a d i u s . As d i s c u s s e d i n t h e t e x t t h i s can be r e l a t e d t o t h e p e n e t r a b i l i t y o f . t h e d i f f u s e w e l l and the d i f f e r e n c e s b e t w e e n t h e s q u a r e and d i f f u s e p o t e n t i a l c a n a l l be r e l a t e d t o t h e r a t i o t h e t r a n s m i s s i o n f u n c t i o n s o f complex p o t e n t i a l s ( S e c . 5) . The d i f f u s e p o t e n t i a l h a s been c h o s e n t o h a v e a Saxon-Woorls • shape w i t h i n t e r i o r wave number K , r a d i u s R , and s u r f a c e t h i c k n e s s a . w h e t h e r we be s t u d y i n g n a r r o w r e s o n a n c e s f a r b e l o w the Coulomb b a r r i e r ( S e c . ll) o r o I I i F i g u r e 6. The v a l u e o f AR/R and of the r e f l e c t i o n f a c t o r f , as a f u n c t i o n o f a/R and K R /IT ( - 2m V R 2 / 1 I . 2 T T 2J 7) . o. o - o o 1 T h i s i s the most f r e q u e n t l y used s e c t i o n o f F i g . 1 o f R e f . 1.3. F o r a w i d e r range o f v a l u e s , the r e a d e r - j i s r e f e r r e d t o Ref. 13. /\ R and f a l l o w one to det e r m i n e the c r o s s s e c t i o n s and the resonance w i d t h o f d i f f u s e w e l l s , once those o f square w e l l s a r e known. F i g u r e 7. The energy v a r i a t i o n o f the r e f l e c t i o n f a c t o r was c a l c u l a t e d from the r a t i o o f the t r a n s m i t t e d waves f o r a g i v e n i n c o m i n g wave. We have made o t h e r c a l c u l a t i o n s v a r y i n g the Coulomb p o t e n t i a l , b u t k e e p i n g the r e d u c e d mass and the n u c l e a r p o t e n t i a l the same. The r e f l e c t i o n f a c t o r was then found t o depend on ( B - E ) o n l y , where B i s the h e i g h t o f the b a r r i e r i n the d i f f u s e w e l l - The r e f l e c t i o n f a c t o r t a k e s i t s maximum v a l u e a t the energy E = B ^ ^ , where i s the maximum v a l u e o f the d i f f u s e b a r r i e r . L e t t i n g the r e f l e c t i o n f a c t o r v a r y with, energy w i l l , i n some c a s e s , improve the a c c u r a c y o f the e q u i v a l e n t square w e l l model (not f o r the c a l c u l a t i o n o f t r a n s m i s s i o n f u n c t i o n s however) . The energy v a r i a t i o n o f the r e f l e c t i o n f a c t o r depends m a i n l y on the r e d u c e d mass and the f i g u r e may be used t o e s t i m a t e i t , by i n t e r p o l a t i o n , f o r cases 12 32 n o t c o v e r e d by our c a l c u l a t i o n s . Note- t h a t f o r C + " S the energy s c a l e , i s g i v e n by I' = E + 1 0 . 0 MeV. i d C -I- J^S F i g u r e 8. The r a t i o o f d i f f u s e - e d g e o p t i c a l mode], t r a n s m i s s i o n f u n c t i o n s t o ' e q u i v a l e n t s q u a r e - w e l l t r a n s m i s s i o n ' • ; • 32 f u n c t i o n s as a f u n c t i o n o f the energy f o r 'S + ^ . The d i f f u s e p o t e n t i a l i s t h a t o f F i g . 'I and the I - e q u i v a l e n t square w e l l has been c a l c u l a t e d from F i g . 6 (/^  R = 0.S8 fm) . ' The r e f l e c t i o n - f a c t o r ( f = '1.62) i s the r a t i o o b t a i n e d a t z e r o energy i n the absence o f Coulomb and a n g u l a r momentum b a r r i e r s . I t i s seen t h a t i f one chooses a c o r r e c t v a l u e o f /\E f o r the e q u i v a l e n t square w e l l the o b s e r v e d r e f l e c t i o n i s i n s e n s i t i v e t o charge and c e n t r i p e t a l b a r r i e r s over a wide range o f e n e r g i e s and i s i n c l o s e agreement w i t h the r e f l e c t i o n f a c t o r ; i n c o n t r a s t , i t was seen i n F i g . 'I f o r a d i f f e r e n t square, w e l l ( A R = 0) t h a t the o b s e r v e d r e f l e c t i o n depended s t r o n g l y on a l l t h e s e f a c t o r s . (The d o t t e d l i n e s g i v e the c o r r e s p o n d i n g r a t i o s f o r . - *> 7f *\ J_j / c o n n e c t e d t o the t r a n s m i s s i o n f u n c t i o n s by the 1 2 a p p r o x i m a t e r e l a t i o n T^ ~ 1^/(1 + '/^ ) . •I ] 2 F i g u r e 9 . ,aR and i: f o r p, n, He and C i n c i d e n t on t a r g e t n u c l e i * o f d i f f e r e n t masses. The d i f f u s e p o t e n t i a l s u sed had R = 1.2 5 h]'/3, 1.6 + 1.25 A,)/ 3, and o T T 1.25 ( 1 2 1 / 3 + A^/ 3) and V q = 50 MeV, 75 MeV and 100 MeV, ] 2 •for n u c l e o n s , a l p h a p a r t i c l e s and. C r e s p e c t i v e l y and a = 0. 5fm i n a l l eases. A R and f were obtained. from F i g . 6. Wo have a l s o p l o t t e d w i t h a dashed l i n e (- - ~) the a p p r o x i m a t i o n to f o b t a i n e d by P e a s l e e f o r s-wave n e u t r o n s (16) . The agreement i s seen t o be q u i t e c l o s e f o r n e u t r o n s and p r o t o n s . F i g u r e 10. The r e s o n a n t s-wave f u n c t i o n s (upper p a r t o f F i g . 10a) and 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 ( F i g . 10b) f o r a d i f f u s e p o t e n t i a l (SOLID) a p p r o p r i a t e 32 to the r e a c t i o n "S + and i t s e q u i v a l e n t square w e l l (DASHED) ( l o w e r F i g . 10a) . The r a t i o (3.2'l) o f the c r o s s s e c t i o n s i s s i m p l y r e l a t e d t o the wave r e f l e c t i o n f a c t o r (5.2) c a l c u l a t e d i n Sec. 3 thr o u g h the energy dependence o f the l e v e l s h i f t . I t i s seen from the r e s o n a n t wave f u n c t i o n s t h a t t h e • r e f l e c t i o n o f the d i f f u s e w e l l i s now d i s t r i b u t e d between the i n t e r i o r ( r e s o n a n t ) and e x t e r i o r ( p e n e t r a t i o n ) p r o p e r t i e s o f the w e l l ; n e v e r t h e l e s s , the sum o f the e f f e c t s i s what we would e x p e c t from the z e r o - e n e r g y b a r r i e r c a l c u l a t i o n s . The d i f f u s e w e l l has a Saxon-Woods shape with, depth - 6M.9 MeV, r a d i u s 5.5685 fm., and s u r f a c e t h i c k n e s s 0.5 fm.; the e q u i v a l e n t square w e l l i s f o u n d t o have depth - 56.3 MeV and r a d i u s 6.25 fm.. F i g u r e 11. F r a c t i o n o f the a b s o r p t i o n as a f u n c t i o n o f r a d i u s f o r d i f f u s e p o t e n t i a l s . w i t h parameters a p p r o p r i a t e t o the r e a c t i o n s 2 0 8 ] ? b -f- c < (SOLID) and. 3 2 S + (DASHED) . The r a d i i used f o r the Saxon-Woods w e l l s a r e i n d i c a t e d by v e r t i c a l l i n e s . The r e l a t i v e amount o f a b s o r p t i o n i n the b a r r i e r i s seen t o be g r e a t l y i n c r e a s e d - b y the extreme b a r r i e r s a s s o c i a t e d w i t h heavy n u c l e i . I n f a c t , a t low e n e r g i e s ( ~ 7 MeV) a l l . a b s o r p t i o n o c c u r s i n the b a r r i e r i f we use a c o n v e n t i o n a l Saxon-Woods shape f o r the . i m a g i n a r y p o t e n t i a l ( w i t h a r a d i u s ^ R , o f 5.568 fm f o r 3 2 S + and o f 8.8 fm f o r 2 0 8 P b + ) . Hence, '. ' the n a t u r e o f the i m a g i n a r y p o t e n t i a l i n the b a r r i e r r e g i o n i s c r i t i c a l i n d e s c r i b i n g a b s o r p t i o n c r o s s s e c t i o n s . . I f the a b s o r p t i o n d e s c r i b e d by the f i g u r e i s , i n f a c t , p h y s i c a l , the e q u i v a l e n t square w e l l model w o u l d f a i l b a d l y s i n c e most p a r t i c l e s would be absorbed, b e f o r e r e a c h i n g the n u c l e a r s u r f a c e . F i g u r e 12. R a t i o s o f t r a n s m i s s i o n f u n c t i o n s f o r a v a r i e t y o f r e a c t i o n s . Even i n the p r e s e n c e o f moderate b a r r i e r s ( F i g . a) the b a r r i e r a b s o r p t i o n i s seen t o depend s t r o n g l y on the s u r f a c e t h i c k n e s s o f the i m a g i n a r y p o t e n t i a l . I n f a c t , even a s l i g h t i n c r e a s e i n the b a r r i e r ( F i g . b) i s seen t o c o n s i d e r a b l y enhance the b a r r i e r a b s o r p t i o n ; on the o t h e r hand, i t i s r a t h e r i n s e n s i t i v e t o the r e d u c e d mass ( F i g . c) . I n a l l .these c a s e s the e q u i v a l e n t square w e l l mode], i s seen t o have c o n s i d e r a b l e v a l i d i t y p r o v i d e d we choose a c o n v e n t i o n a l s u r f a c e t h i c k n e s s f o r the i m a g i n a r y I . • • I I i . i p o t e n t i a l ( a ^ = = 0.5 fm). I n the p r e s e n c e o f the extreme b a r r i e r s a s s o c i a t e d w i t h heavy i o n r e a c t i o n s and a l p h a p a r t i c l e s c a t t e r i n g from heavy n u c l e i ( F i g . d) i t i s seen t h a t the e q u i v a l e n t ! "square w e l l model f a i l s b a d l y even f o r c o n v e n t i o n a l s u r f a c e t h i c k n e s s e s . Hence, i t i s o f c r u c i a l i n t e r e s t t o u n d e r s t a n d ' the d e t a i l s o f the i m a g i n a r y p o t e n t i a l i n the b a r r i e r i f one w i s h e s t o e x t r a p o l a t e h i g h energy d a t a t o the e n e r g i e s o f i n t e r e s t to a s t r o p h y s i c s . F i g u r e 13. F r a c t i o n o f the a b s o r p t i o n as a f u n c t i o n o f the r a d i u s f o r a d i f f u s e p o t e n t i a l whose r e a l and i m a g i n a r y p a r t s have Saxon-Woods shapes w i t h p arameters a p p r o p r i a t e t o 1 2 1 2 the r e a c t i o n C + "C . The comparison o f the a b s o r p t i o n f o r s-waves (SOLID) and g-waves (DASHED) shows t h a t the r e l a t i v e amount o f a b s o r p t i o n w h ich o c c u r s i n the b a r r i e r i s ' i n c r e a s e d , when the b a r r i e r i s • h e i g h t e n e d . I t i s a l s o seen t h a t i n c r e a s i n g the s u r f a c e t h i c k n e s s a V I, o f the i m a g i n a r y p o t e n t i a l (DOTTED) c o n s i d e r a b l y enhances the r e l a t i v e amount o f b a r r i e r a b s o r p t i o n . Hence, low energy a b s o r p t i o n c r o s s s e c t i o n s may y i e l d d e t a i l e d i n f o r m a t i o n about the shape o f the a b s o r p t i v e p o t e n t i a l i n the n u c l e a r s u r f a c e . (The energy i s '! MeV.) A.E.C.L. Rcf. it A-2792-K N O R M A L I Z E D R E S O N A N C E S T A T E S E N E R G Y L E V E L S (MeV) R E D U C E D WIDTH 10 8 6 2 h 0.01 I J I I I I |. 20 - . 24 28. 32 36 40 . co CO st CVJ O P O T E N T I A L , MeV W A V E F U N C T I O N • I I I • i r , fm. CD 

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.831.1-0084757/manifest

Comment

Related Items