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Pion-nucleus size resonances Pai, David Mieng 1980

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PIOH-NDCLEOS SIZE BESONANCES bf DAVID HIENS ^ PAJE B.Sc. , U n i v e r s i t y of B r i t i s h Columbia, 1973 I f . S c , U n i v e r s i t y of B r i t i s h C o l u a b i a , 1975 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES Department of P h y s i c s We accept t h i s t h e s i s as conforming t o the r e q u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA A p r i l , 1980 (S) David Mieng ^Pai In presenting th is thesis in par t ia l fu l f i lment of the requirements for an advanced degree at the Univers i ty of B r i t i s h Columbia, I agree that the Library shal l make i t f ree ly ava i lab le for reference and study. I further agree that permission for extensive copying of th i s thesis for scholar ly purposes may be granted by the Head of my Department or by his representat ives. It is understood that copying or publ icat ion of th is thesis for f inanc ia l gain shal l not be allowed without my written permission. Department of The Univers i ty of B r i t i s h Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 ABSTRACT The p o s s i b i l i t y and properties of pion-nucleus o p t i c a l p o t e n t i a l resonances (or size resonances) are investigated. There are two parts to the th e s i s . The f i r s t i s mainly preparatory work.. Resoaance phenomena i n reactions are examined i n general, with em^nasis on relating the nature of the int e r a c t i o n to the existence of various types of resonances- ranging from compound nucleus resonances to size resonances. It i s argued that of these only size resonances may exist for the pion-nucleus system due to strong pion annihilation. _The pion-nucleus i n t e r a c t i o n in the form of an o p t i c a l potential i s reviewed from basics. Many approximations are examined. In p a r t i c u l a r , the o p t i c a l theorem which governs two-body scattering i n a many-Dody environment and i t s usage for examining the impulse approximation are pointed out. The K i s s l i n g e r p o t e n t i a l i s examined from the continuity equation point of view and. two pathologies are uncovered..First, the p o t e n t i a l f a i l s to be absorptive everywhere; and second, i t becomes singular when the potential exceeds&certain strength*.The second pathology, after being remedied, s t i l l gives amusing r e s u l t s ; for example, an i n f i n i t e number of bound states, a kink i a the wave function, negative e l a s t i c width,...etc. Many of these anomalous results can be understood i n terms of the pion having a negative e f f e c t i v e mass a r i s i n g from the r e a l part i i i of the o p t i c a l p o t ential. The second part deals with size resonances s p e c i f i c a l l y . The $-matrix formalism i s presented v i t h a l l necessary modifications for the Kisslinger p o t e n t i a l . The aje-old controversy on the choice of boundary radius i s readdressed to sort out the resonance part of potential scattering properly. This formalism series two complementary purposes. It enables one to calculate size resonance parameters from a pote n t i a l , as well as to judge whether a calculated resaaance i s narrow enough for observation. T t ' 2 0 8 P b results show resonances of various p a r t i a l wave and pion charge can jccur in the region of 10 to 40 HeV. The corresponding e l a s t i c widths and absorption widths are in the order of 30 to 40 HeV. Lighter nuclei ( 5 8Ni) show even larger e l a s t i c wxdth. Without further analysis these widths are too lar^e to produce any apparent e f f e c t even i n the resonant part ar the scattering amplitude. However, in a new method.of phase s h i f t analyses* which consists e s s e n t i a l l y of extracting the #-function, the broad resonances are shown to be d i s t i n c t and t h e i r parameters can be extracted. This methoa has widespread possible application for analysing resonances i n general. i v TABLE OF CONTENTS Abstract . .. - • .. i i Table of Contents i v L i s t of Tables . v i i L i s t of Figures v i i i Acknowledgement ix 1. Introduction 1 2. Nucleon-Nucleus Interactions and Resonances ....... *.... 11 I. Compound Nucleus Resonances ..11 II..Single P a r t i c l e Propagation and the Optical Model ...13 III. Doorway Resonances ,. . 19 IV. Size Resonances ...22 V. Unified Description of Resonances . . . . . . . i . . . . . . . . . . . 25 3. Pion-Nucleus Interactions 29 I. Single P a r t i c l e Mode 29 II. Doorway Mode ............................30 III . Compound Nucleus Mode .............................. 32 IV. Pion Annihilation ...33 4. Pion-Nucleus Optical Potential ..........40 I. .Multiple Scattering Series ......................... 40 V II . F i r s t Order Optical Potential ....................... 45 III* Higher Order Corrections - 55 IV. Absorption Properties of U|......................... 57 V. Impulse Approximation .............................. 64 VI. Nucleon-Nucleus Optical Potential ................... 69 5. The Kiss l i a g e r Potential ..............................71 I. Derivation of the K i s s l i n g e r Potential .............71 II. Corrections to the Kisslinger Potential ............ 77 III* Local Absorption Anomalies .........................31 IV. E f f e c t i v e Mass Singularity .........................91 6. Size Resonance Analyses 100 I. The $-Function of the Kisslinger Potential ........102 II. .From the (^-Function to Cross Sections .......111 I I I . Eigenstates and Resonant States ...............116 IV. Ambiguities i n the Boundary Radius 120 V. Calculation of Resonance Parameters ............... 125 VI. Extraction of Resonance Parameters ................. 128 7: Results - 139 I. Resonant Energy and E l a s t i c Width 139 II. Absorption Width 143 I I I . Resonance Extraction from Phase S h i f t s ............. 145 IV. . Anomalous Resonances ...............149 v i References ...............................175 Appendix I. Transforming the Ki s s l i n g e r Potential from Momentum to Coordinate Representation ...... 178 Appendix II. Derivation of Sink Functions for the Kisslinger Potential .182 Appendix III Modification of the Two-Potential Formula ...18^ Appendix IV ; Orthogonality Relation when the Ef f e c t i v e Mass Changes Sign 181 v i i LIST OF TABLES Table 1. Kisslinger Potential Parameters and Depths ..... 153 Table 2. Kisslinger Potential Parameters and Depths ..... 154 Table 3. Kisslinger Potential Parameters and Depths ..... 154 V 111 LIST OF FIGURES 1a ...20 Fig. ,1b ...22 2a ................. 24 2b .................^ 9. 25 3a, 3b .............. 35 4a .36 4b ...... 37 4c ..................... 38 4d ................. 39 5 ...................... .48 6 ... ........... 49 7 ..... 50 8a .................. 65 8b 66 9 .................. 94 10 ................... 155 11 a, 11b .......... .156 11c ................... 157 12a, 12b ..... 158, 159 13 ................ i, 160 14a, 14b ........... 161 14c .^................ 162 15a, 15b .......... 163 16 ...."........<,. 164 17a, 17b .......... .165 17c, 17d ............. 166 18a 167 18b, 1Bc ............, 168 19 169 20a, 20b .............. 170 21a, 21b 171 22 17 2 23 ................ 173 24 ................... .174 ACKNOWLEDGE!* ENT I wish to thank my Research Supervisor* Professor .ricich Vogt, for suggesting and directing t h i s thesis work. His guidance and encouragement, as well as his immense patience and personal kindness, are deeply appreciated. I also l i k e to thank Dr..T..Fujita for discussions «raich helped to i n i t i a t e t h i s work; and to Professor Maicolm McMillan and Dr..Tony Thomas for helpful discussions. Special thanks are due to T.T.,For pointing out Ref.[45] and [47j. . 1 CHAPTER 1 INTRODUCTION The atomic nucleus i s commonly perceived as a c o l l e c t i o n of protons and neutrons. Nucleir physics t r a d i t i o n a l l y has been concerned with understanding the nucleus from t h i s ^oint of view. However, i t has also long been thought that t a i s i s necessarily a s i m p l i f i e d picture since the force between •constituent p a r t i c l e s ' must arise from exchanged of particles..The discovery of Yukawa's meson, the pion, i n 1947 e f f e c t i v e l y confirmed t h i s belief..So why has i t not been necessary to bring pions- (and other exchange p a r t i c l e s j , as well as related p a r t i c l e s such as nucleon isobars, e x p l i c i t l y into the description of nuclear processes? Or to pat i t d i f f e r e n t l y , under what situations can a nuclear system no longer be adequately described i n terms of nucleons alone? If so, what other degrees of freedom are needed? And how are they to be introduced? These fundamental questions have recently been reviewed in 'Mesons i n Nuclei' [1].. There have been several approaches to go beyond nucleonic degrees of freedom. Some have looked into electromagnetic and weak nuclear processes for fi n e r efxacts due to exchange of v i r t u a l particles* Others have looked into 'new' nuclear environments of high density, either created experimentally, for example from heavy ion c o l l i s i o n s , or formed naturally such as in neitron stars, for new degrees of freedom whicli they might put into the description; And then there are the pion physics people who seek to introduce a pion d i r e c t l y into the nuclear wave function. Examination of the pion-nucleus i n t e r a c t i o n whea one brings a free pion into the nucleus seems to be the most direc t way to investigate what roles the exchange pions play in the description of a nucleus. What distinguishes a r e a l pion from a v i r t u a l pion i s whether or not i t i s on the mass-shell. A t r u l y general treatment should be able to encompass both i n a u n i f i e d mannerHowever, because ox the lack of basic understandings f o r strong interactions, t h i s has not been possible; In most treatments so far the pion-nucleus interaction has been synthesized only from experimental pion-nucleon scattering information. Nevertheless, t h i s c e r t a i n l y represents a f i r s t step towards understanding the pion-nucleus i n t e r a c t i o n . In addition to probing pionic degrees of freedom e x p l i c i t l y , the pion-nucleus interaction holds the promise of revealing novel aspects of nuclear structure. This i s only too evident when one considers what the photon has done for electromagnetic systems where the v i r t u a l photon plays the role of the exchange p a r t i c l e . Of course, i n t h i s case the basic i n t e r a c t i o n i s well known and photonic degrees of freedom can be properly taken care of, thus making the paoton a tremendously useful probe for structure. This i s the stage of understanding one wants to achieve for pion-nucleus interaction eventually. . F a l l i n g short of basis understanding, pion-nacleus in t e r a c t i o n has been constructed i n terms of an o p t i c a l potential from pion-nucleon scattering data. Much e f f o r t has been devoted to establishing this p o t e n t i a l from the widths and s h i f t s of pionic atom energy l e v e l s . However, for phonic atoms, the Coulomb potential is mainly responsible for the binding, and the o p t i c a l potential i s only of secondary importance. In this regard, i t i s cer t a i n l y desirable to f i n d whether there ex i s t s pion-nucleus systems 'bound' by the o p t i c a l potential i t s e l f . . By 'bound* we mean not only t r u l y bound states but also the temporary binding which occurs i n a pion-nucleus potential resonance; More importantly, i f such bound systems e x i s t at a l l , they would c e r t a i n l y contribute to understanding the pion-nucleus i n general. In t h i s tnesis, we consider the possible existence and properties of o p t i c a l potential resonances for the pion-nucleus system. This type of resonance i s commonly known as s i z e resonance i n nuclear physics and i s , of course, well known, for the nucleon-nucleus system.. The thesis s t a r t s with a general description of resonance phenomena occurring in the nucleon-nucleus system. . Against a variety of resonances i n t h i s well studied system, size resonance can be understood from an o v e r a l l perspective. By examining the existence of various types of resonances i n r e l a t i o n to the nature of the nucleon-nucleus int e r a c t i o n , we i l l u s t r a t e that size resonance measures the extent or the s i n g l e - p a r t i c l e degree of freedom for the incident p a r t i c l e . This i s c e r t a i n l y an important aspect of any pr o j e c t i l e - t a r g e t system. Following that, i n Chapter 3, we carry out si m i l a r discussions on resonance phenomena for the pion-nucleus system. From a q u a l i t a t i v e discussion on the nature of the pion-nucleus i n t e r a c t i o n we conclude that pion annihilation i s so great that the only kind of resonance which can occur i s size resonance. 1 In Chapter 4, extending the q u a l i t a t i v e concepts, we develop a more quantitative discussion f o r the pion-nucleus in t e r a c t i o n . The general theory of constructing the f i r s t order o p t i c a l potential based on the multiple scattering formalism i s reviewed c r i t i c a l l y . In p a r t i c u l a r , we point out the o p t i c a l theorem which governs two-body scattering events in a many-body environment. This theorem puts r e s t r i c t i o n s on the matrix elements of the f i r s t order o p t i c a l potential and i s p a r t i c u l a r l y suitable for examining the impulse approximation and associated corrections. The impulse approximation replaces two-body scattering i n a many-body environment by free two-body scattering. It can not possibly be valid for pion-nucleus i n t e r a c t i o n since under suca an approximation no pion a n n i h i l a t i o n can occur at a l l . By applying the impulse approximation to the f i r s t order o p t i c a l potential and then assuming a p a r t i c u l a r form f o r the pion-nucleon free t-matrix, Kisslinger derived one of the e a r l i e s t pion-nucleus o p t i c a l potentials.. Over the years, assorted corrections to the impulse approximation have beea added to the potential. We review these i n the f i r s t half of Chapter 5. In the second half we examine the Kisslinger potential (including corrections) from the continuity equation point of view..Re are led to t h i s approach by the need of defining the sink function (which measures the absorption at a point) in order to study absorption broadening in d e t a i l . From t h i s examination, two pathologies in the Kisslinger potential are r e v e a l e d i . F i r s t , based on the continuity equation with the •obvious' d e f i n i t i o n of probability density and current, we f i n d the corresponding sink function i s not always positive everywhere. This particular point has not been raised against the Ki s s l i n g e r potential probably because the p o t e n t i a l i s always o v e r a l l ( i f not everywhere) absorptive (as guaranteed by i t s fori* and sign of parameters) aad thus in ordinary usage where only the o v e r a l l absorption- rather then pointwise absorption- matters no flagrant v i o l a t i o n of u n i t a r i t y ever appears. However, since the f i r s t order o p t i c a l potential does not describe any rescattering back into the e l a s t i c channel ( i t accounts only f o r amplitudes for which incident p a r t i c l e s either remain or get scattered away from the e l a s t i c channel), i t i s supposed to be absorptive everywhere for the incident p a r t i c l s . He also try to establish t h i s requirement for the f i r s t order o p t i c a l p o t e n t i a l from the o p t i c a l theorem which governs two body scattering i n a many-body environment, but witn only limited success. At any rate, no reasonable f i r s t order o p t i c a l potential should be creating pions anywhere inside the nucleus. The Ki s s l i n g e r potential does not s a t i s f y t h i s requirement. The second pathology i s more serious..It must be removed before any c a l c u l a t i o n can be carried out. The Kisslinger potential e s s e n t i a l l y contains two parts, one due to pion-nucleon s-wave scattering and the other p-wave scattering._Depending on how the absorptive part of the potent i a l i s removed i n i t i a l l y for calculating resonant energy and e l a s t i c width, the resultant equation can become singular i f the p-wave part exceedsacertain strength._This c r i t i c a l value i s i n fact close to the value of the p-wave potential determined from pionic atom data. A convenient way of removing t h i s s i n g u l a r i t y i s by using the absorptive part of the potential to provide a cut-off to the si n g u l a r i t y . However, the resultant equation i s no longer s e l f - a d j o i n t . ' 1 In f a c t , the si n g u l a r i t y can not be removed without making the equation non s e l f - a d j o i n t . This requires a new d e f i n i t i o n of pr o b a b i l i t y density and current and serious consequences follow. In particu l a r , in order to keep the current continuous, we fi n d that the derivative of the wave function must change sign at the (previously) singular point, i * e . a kink i n the wave function. 2 Further, had the wave function been icept continuous arid smooth as usual, the e l a s t i c width would be negative. There are other amusing r e s u l t s associated witn the si n g u l a r i t y such as an i n f i n i t e number of bound states, resonance narrrowing by 'absorption', ..etc. In f a c t , much of these anomalous r e s u l t s can be understood more e a s i l y in terms of the e f f e c t i v e mass point of view. I t turn$out that the p-wave poten t i a l can be absorbed into giving the pion an ef f e c t i v e mass. When the potential exceeds certain strength, the e f f e c t i v e mass becomes negative and, not su r p r i s i n g l y , anomalous res u l t s follow. With such 'bugs' i a the Ki s s l i n g e r potential under control, we f i n a l l y proceed to size resonance analyses in Chapter 6.. At the beginning, we make i t clear that there are two objectives i n our analyses. The f i r s t i s to be able to calculate size resonance parameters, namely resonant energy, 2 In fact, even when no s i n g u l a r i t y occurs, for the 'obvious' d e f i n i t i o n of probability density and current associated with the Kisslinger potential the waire function must have a kink when the potential i t s e l f i s not smooth (e.g. a square well). 8 e l a s t i c width, and absorption width, from the Kisslinger type of potential. The second objective i s to e s t a b l i s h whetner a calculated resonance i s 'narrow enough' to be 'observable'. This i s p a r t i c u l a r l y important since pion a n n i h i l a t i o n i s strong and i s expected to contribute a large absorption width. A 'narrow' resonance produces rapid energy variation in cross sections and i t s experimental signature i s clear cut. .However, a 'wide' resonance may be completely obscured in cross sections by non-resonant backgrounds. Its extraction depends c r u c i a l l y on how and what type of experimental data one analyses. Towards achieving these two objectives, we present the ^ - m a t r i x formalism of Eisenbud and tfigner, including appropriate modifications demanded by the unusual d e f i n i t i o n of p r o b a b i l i t y density and current associated with the Kisslinger potential..The age-old controversy associated with the ^-matrix formalism on the choice of the boundary radius defining eigenfunctions i s readdressed. Some arguments are given for why the radius must be chosen on the potential surface and i t i s shown that the precise choice i s not c r u c i a l ; Following that, based on the ^-matrix formalism, we show how s i z e resonance parameters are calculated for a given potential. . At the end of the chapter, we go on to the more important task of achieving the second objective..Defining a calculated resonance to be observable i f i t s parameters can be extracted from experimental tfpe of quantities, we discuss how to extract resonance parameters from experimental data. F i r s t dismissing cross sections to be inappropriace for analyzing wide resonances, then pointing out that the conventional quantity based on phase s h i f t s for resonance analyses, the resonant amplitude, has several shortcomings which are p a r t i c u l a r l y serious for wide resonances, we a r r i v e at the conclusion that the most appropriate experimental quantity for exhibiting and yielding resonance parameters i s the $-function..A detailed prescription i s given f o r how the ^ - f u n c t i o n can be calculated from phase s h i f t s . He discuss the many merits the ^ - f u n c t i o n has over the resonant amplitude i t s e l f for resonance analysis* In p a r t i c u l a r , since the ^ - f u n c t i o n contains only the spreading width and not the e l a s t i c width i t i s p a r t i c u l a r l y s u i t a b l e for analyzing resonances of large e l a s t i c width. In the f i n a l chapter we present c a l c u l a t i o n results f o r pion size resonances. F i r s t , with, absorption removed from the potential, we calculate resonant energies and e l a s t i c widths as functions of potential 'depth' for pions incident on 2 0 8 P b . It i s shown that corresponding to the pionic atom potential parameters, resonances of various p a r t i a l wave and pion charge can occur i n the region of 10 to 40 Meif with corresponding e l a s t i c width of the order of 30 to 40 deV. Results on l i g h t e r n u c l e i , ssui s p e c i f i c a l l y , show larger 10 e l a s t i c widths. Next, we demonstrate absorption e f f e c t s by turning on the absorptive part of the potential gradually. The resonant amplitude i s shown v i v i d l y losing i t s resonance p r o f i l e as the absorption increases. Even before reaching the f u l l absorption strength, for which the absorption width i s of the order of 30 MeV, the resonant amplitude (magnitude squared) becomes monotonically varying. However, also snown, the ^ - f u n c t i o n retains i t s resonance p r o f i l e against absorption ranarkablly well. Furthermore, the absorption width extracted from the ^ - f u n c t i o n by assuming a constant background even at f u l l absorption agrees well witu the calculated value. Therefore i t can bs concluded that the Kisslinger potential, with parameters determined from pionic atom data, i s a t t r a c t i v e enough to support pion-nucleus s i z e resonances.„ However, both . the e l a s t i c width and the absorption width are so large that no clear resonance effects can show up in cross sections.. Nevertheless, by using our method of resonance analyses, s p e c i f i c a l l y finding the ^ - f u n c t i o n , one might be able to extract resonance parameters. Most importantly, t h i s method of resonance analyses can be applied to any type resonance. I t has widespread applications i n general. At the end of the chapter, we present some anomalous results f o r the case when the p-wave potential exceeds that strength for which the pion e f f e c t i v e mass becomes negative. 11 CHAPTER 2 NOCLEON-NUCLEUS INTERACTIONS AND RESONANCES In order to understand si z e resonance from an o v e r a l l perspective, i t i s useful to give a general discussion of resonance phenomena i n reactions. In t h i s respect, the nucleon-nucleus system i s ideal since there ex i s t s a variety of structures i n cross sections associated with d i f f e r e n t type of resonances. We therefore discuss the nucleon-nucleus system in t h i s chapter and extend s i m i l a r discussions to the pion-nucleus system i n following chapters. From the start of nuclear physics f i f t y years ago, resonance structures in cross sections of nucleon-nucleus c o l l i s i o n s have been of central importance in understanding nuclear structures and reaction mechanisms. Resonances are invariably associated with comparatively long-lived intermediate states of the. p r o j e c t i l e - t a r g e t system. According to the type of intermediate state, nucleon-nucleus resonances can i n general be categorized i n the following way: compound nucleus resonances, doorway resonances and si z e resonances. We discuss these resonances below i n r e l a t i o n to the nature.of nucleon-nucleus int e r a c t i o n s . . 2 i l - . Compound Nucleus Resonances The e a r l i e s t nuclear experiments showed that there exist many sharp resonances (width of the order of few eV) i n low energy (< a few keV) neutron e l a s t i c and radiative capture" (the only reaction channel open at such low energy) cross 12 sections. Bethe [2] f i r s t t r i e d to explain the large cross sections by a simple model- that the resonances correspond to quasi-stationary states of the incident neutron in a potential well of the s i z e of a nucleus. This model corresponds to assuming the nucleus i s r i g i d and undisturbed by the incident neutron. It f a i l s , however, to reproduce the small spacing between resonances since single p a r t i c l e stationary states are separated by about 10 to 20 MeV. This l e v e l separation i s determined by the size of the nucleus.. Recognizing that stationary states of a many-particle system are clo s e l y spaced, Bohr [ 3 ] suggested the •compound nucleus formation' reaction mechanism to account for the clos e l y spaced resonances..The main concept i s that, since the nucleon-nucleon interaction i s strong, a many-body system (the compound nucleus) i s formed through nucleon-nucleon c o l l i s i o n s a f t e r the incident neutron enters into the nucleus. The incident energy i s distributed uniformly to the whole system. The small width of the e l a s t i c scattering resonances, or the long l i f e time of the compound nucleus, indicates that the probability of f i n d i n g the compound nucleus having the energy concentrated on a single neutron for i t to escape i s very small. This resonance mechanism also explains why the heavier the nucleus the smaller the resonance spacing. The extreme compound nucleus model of Bohr i s the antithesis of the single p a r t i c l e model i t replaces. 13 In t h i s view the single p a r t i c l e picture of the incident p a r t i c l e i s a l l l o s t . The i n d i v i d u a l compound nucleus resonances do not per s i s t to i n d e f i n i t e l y high energy for the following reason. As the incident energy increases, compound nucleus quasi-stationary states become more c l o s e l y spaced and have larger width because more reaction channels are opened up. Eventually the:width becomes larger than the spacing and many states are excited at a single incident energy. As a r e s u l t , l e v e l s overlap and (angle-integrated) cross sections become smooth. 3 However, sometimes a group of nearby states may overlap i n such a manner that together they give a aroad resonance which i s actually associated with degrees of freedom other than (and much lass complex than) the compound nucleus mechanism. We discuss these other mechanisms below. The same phenomenon occurs at lower energy i f one averages the cross sections over i n d i v i d u a l compound resonances. 2 s.11 j. - S in ql e P a r t i c l e Propagation and the Optical Model Long after the discovery of compound nucleus resonances, nucleon-nucleus cross sections beyond the compound resonance region (see Fig.2a) were found to be in apparent disagreement 3 However, angular cross sections s t i l l show variations (of the order of 1 MeV) with energy- a phenomenon usually c a l l e d Ericson f l u c t u a t i o n - but not as a resu l t of any resonance e f f e c t . 14 with the compound nucleus mechanism. In the case of a complete compound nucleus farmation, i . e . traces of the incident p a r t i c l e are lo s t completely, the probability for returning to the incident channel should be neg l i g i b l e when many other channels are opened up. In other words, the nucleus should gradually become.black to the incident neutron as the incident energy increases to beyond the region of in d i v i d u a l resonances. The t o t a l cross section for a alack nucleus i s roughly a constant (twice the geometric cross section area of the target) over energy. This i s d e f i n i t e l y not the behavior observed in experiments which show aroad •resonance' lLlce variations. In view of t h i s experimental fact, Feshbach, Porter and Weisskopf [4], (see also Friedman and Weisskopf [5]) modified Bohr's picture by adding that the compound nucleus formation i s not necessarily complete during the process of i n t e r a c t i o n - that some 'memory' of the incident p a r t i c l e ' s orbit i s retained inside the nucleus. To them the interaction i s a combination ef f e c t of single p a r t i c l e propagation plus compound nucleus formation ana i t s subsequent decays..Because of the single p a r t i c l e component, the incident neutron can always go i n and out of the nucleus no matter how many reaction channels are open. E f f e c t i v e l y , the nucleus i s not black, but rather cloudy, to the incident neutron. Based on t h i s picture they developed the o p t i c a l model approach to nuclear reactions. 15 The o p t i c a l model describes the e f f e c t of the nucleus on the incident p a r t i c l e by a complex p o t e n t i a l . The rea l part of the potential accounts f o r c o l l i s i o n s which do not excite the nucleus. The imaginary part accounts for the loss or the incident flux due to compound nucleus formations through c o l l i s i o n s . This description, however, does not account f o r re-emission of the incident p a r t i c l e back in t o the entrance channel by the compound nucleus. This re-emission i s usually minimal (thus the o p t i c a l potential i s ' r e a l i s t i c ' ) wheu the incident energy i s high enoagh that many kinds of reaction channel are open. However, at low energy where only the rad i a t i v e capture reaction i s open, t h i s re-emission into the incident channel i s a l l important- and indeed responsible f o r the observation of narrow e l a s t i c and radiative capture resonances.,In the low energy region, then, there might appear to be conceptual d i f f i c u l t i e s i n j u s t i f y i n g the op t i c a l potential description*. However, one can take i n t o account the average of re-emission e f f e c t s and esta b l i s h the relevance of the op t i c a l potential i n the compound resonance region i n the following manner.. It can be argued from the uncertainty p r i n c i p l e (or established e x p l i c i t l y from the time-dependent scattering theory [ 6 ] ) that the time-interval for which a system i s subjected to investigation depends inversely on the resolution of the incident energy. I f the energy resolution 16 i s poor, then phenomena associated with the long l i f e time features of the intermediate state can not be probed.. No wonder that in order to s o l i c i t resonance response of a system the incident energy must be well defined i n comparison with the resonance width. A poor resolution experiment i s only capable of probing the short l i f e time features ox the intermediate combined system. Since the o p t i c a l potential i s supposed to describe the part of single p a r t i c l e propagation, i . e . the early response of the system, i t s predictions saould be connected with re s u l t s of poor resolution experiments. Therefore, the opti c a l potential can be defined so that i t reproduces the average scattering amplitude* The i n t e r v a l of average should be large enough such that the probing time i n t e r v a l i s comparable to the single p a r t i c l e t r a n s i t time. It i s i n t h i s averaging sense that the relevance of the o p t i c a l model was established*. In the i n d i v i d u a l compound resonance region, then, although the o p t i c a l potential, can not possibly reproduce the rapid energy variation i t does reproduce the average cross sections* Beyond the i n d i v i d u a l resonance region the o p t i c a l potential reproduces actual cross sections since, due to the absence of re-emissiou (or resonant response of the system), the scattering amplitude i s smoothly varying anyway and the averaging procedure i i not necessary... The . s t a t i s t i c a l theory of nuclear reactions (see Vogt [ 4 3 ] ) concerns with nuclear cross sections averaged over 17 many resonances and the o p t i c a l model forms i t s basis for calcula t i o n s . The o p t i c a l potential was o r i g i n a l l y introduced to account for single p a r t i c l e propagation..But the concept was phenomenological i n the sense that no f u l l j u s t i f i c a t i o n was given for why the potential exists or how i t can be calculated. However, l a t e r , Fashbacli [7] generalized the o p t i c a l potential concept. He formulated a one-body (the incident particle) operator which when used i n the one-body Schroedinger equation reproduces the e l a s t i c scattering amplitude exactly..This i s the generalized o p t i c a l potential operator. We must emphasize that formally t h i s operator (although one-body) i s no longer r e s t r i c t e d to describing single p a r t i c l e propagation. In p r i n c i p l e , for example, i t i s even capable of producing compound e l a s t i c amplitudes..In practice, however, the formal expression for the o p t i c a l potential operator can not be evaluated exactly. In fact, most often approximations must be made such that the f i n a l evaluated operator contains only the single p a r t i c l e propagation aspect and becomes good, then, only for describing gross structures of cross sections.. Such a potential operator i s only the f i r s t order o p t i c a l p o t e n t i a l . However, commonly, i t i s perceived as what an o p t i c a l potential means. More detailed discussions are given i n 4.II where we develop the f i r s t order pion-nucleus o p t i c a l 18 potential from multiple scattering formalisms. So far we have discussed two degrees of freedom for nucleon-nucleus interactions, namely single p a r t i c l e propagation and compound nucleus formation. These two modes are extremes to each other. One involves only the incident p a r t i c l e and the other involves every p a r t i c l e in the system. However, t h i s description i s incomplete since i t asserts that a l l reactions necessarily proceed via the compound nucleus, whereas experiments have shown that many reaction products retain a large portion of tlie incident energy, a result of direct knock-out. Further, i t i s unclear how single p a r t i c l e and compound nucleus can both be ef f e c t i v e modes together as the o p t i c a l model asserts..The single p a r t i c l e picture i s v a l i d only i f the transfer of energy through c o l l i s i o n s i s minimal (say the nucleus i s nearly r i g i d ) as Bethe o r i g i n a l l y suggested..On the other hand, the compound nucleus formation comes about because the incident energy i s thoroughly mixed among a l l p a r t i c l e s , presumably because the nucleon-nucleon inte r a c t i o n i s strong and gives r i s e to e f f i c i e n t energy transfer as Bohr o r i g i n a l l y suggested..It may seem that these two modes should be exclusive to each other. It i s now understood that, due to Pauli's p r i n c i p l e which forbids the c o l l i d i n g nucleons having f i n a l momenta already occupied by other nucleons, the nucleon-nucleon interaction inside the nucleus i s e f f e c t i v e l y weak {or the nucleus i s almost rigid) 19 and lends v a l i d i t y to the single p a r t i c l e picture (aaa the s h e l l model of the nucleus). The compound nucleus must necessarily come from many c o l l i s i o n s . During these c o l l i s i o n s p a r t i c l e s may be knocked out i f there i s s u f f i c i e n t energy. This picture of reaction mechanism gives r i s e to direct reactions and also furnishes a basis for understanding how single p a c t i c l e and compound nucleus can both be e f f e c t i v e i n nucleon-nucleus i n t e r a c t i o n s . Weisskopf [8] introduced a graphical i l l u s t r a t i o n , as shown i n Fig. la, of the series of events following the entry of a p a r t i c l e into a nucleus. In that figure, 'no c o l l i s i o n ' refers to that part of the inte r a c t i o n (through c o l l i s i o n s of course) which does not excite the nucleus.. The ' c o l l i s i o n ' refers to nuclear excitations due to c o l l i s i o n s . Previously we stated s p e c i f i c a l l y that the imaginary part of an o p t i c a l potential acounts f o r compound nucleus formations; Now, with the introduction of dire c t reactions, the imaginary part must be generalized to include a l l i n e l a s t i c processes.* 2iIII-. Doorway_ Resonances The series of c o l l i s i o n s preceding the formation of a 4 This i s , however, s t r i c t l y correct only for the exact o p t i c a l potential.. For the f i r s t order o p t i c a l potential, for example, the imaginary part also accounts f o r a l l non-single p a r t i c l e e l a s t i c scatterings. 20 Shape elastic Direct inelastic scattering scattering Compound elastic scattering No collision Same particle leaving s i. ^ One collision Two collisignsl Compound nucleus formation n collisions' n s ^ . Other particles leaving . v Direct reaction Compound nucleus reaction F i g ^ l j i A graphic i l l u s t r a t i o n of the course of nucleon-nucleus reaction. As long as 'no c o l l i s i o n ' takes place, only s i n g l e - p a r t i c l e e l a s t i c scattering xs possible..After a f i r s t c o l l i s i o n direct reactions take place; l a t e r on, af t e r many c o l l i s i o n s , compound nucleus formation occurs. Reproduced from [8]. compound nucleus gives r i s e to the p o s s i b i l i t y of new nodes of propagation with complexity intermediate between the single p a r t i c l e and the compound nucleus picture. For example, the incident p a r t i c l e may transfer i t s energy p r e f e r e n t i a l l y to a group of nucleons after a few c o l l i s i o n s . This group of nucleons, i f conditions permit, may propagate as an entity for a while and then decay via a number of possible routes. It may re-emit a p a r t i c l e back into the incident channel, or decay into reaction channels, or transfer the energy to more nucleons and lead eventually to compound nucleus formation. The p i c t o r i a l name 'doorway' state has been given to these kinds of intermediate states since they are supposedly on the way to compound nucleus 21 formations. This idea of stopping-over i s p a r t i c u l a r l y plausible when one takes nuclear structure into consideration. For example, due to the s h e l l structure o f ' the nucleus, the incident nucleon may i n i t i a l l y transfer i t s energy to the valence nucleons and leave the core alone. . S p e c i f i c a l l y , the incident energy may be shared through simple excitations into r o t a t i o n a l motion foe deformed nuclei, or v i b r a t i o n a l motion for spherical n u c l e i . Weisskopf's flowchart for reactions i n Fig.1a should be modified to include the p o s s i b i l i t y of quasi-stationary doorway states. In Fig. 1b we show one such revised (and more abstract) representation.. Each box stands for a p a r t i c u l a r mode of propagation and they are connected by l i n e s which represent c o l l i s i o n s . . Since the re-emission i n t o the incident channel from a doorway state constitute a 'semi-late' e l a s t i c process and, similar to the compound e l a s t i c process, i t gives r i s e to resonance structures i n cross sections. iaese structures are intermediate between fi n e structure of the compound nucleus mode and gross structure of the single p a r t i c l e mode. Intermediate structure, as reviewed by Mekjian [9 j , i s quite widespread i n neutron-nucleus cross sections. However, i t s interpretation i n terms of various types of doorway states i s not conclusive.. On the other hand, i n proton 22 s i n g l e doorway compound p a r t i c l e e l a s t i c nucleus e l a s t i c a e l a s t i c > doorway mode I 1 , s i n g l e p a r t i c l e mode compound nucleus mode 1 * \ d i r e c t doorway compound reactions reactions nucleus reactions Fiq,._1b A revised version of Fig. 1a, adding i n tae doorway mechanism.' Each box represents a particular mode. Lines connecting boxes represent c o l l i s i o n which connects various modes. induced reactions, many structures (width ranges from a few keV to hundreds of keV) have been established to be due to isobaric analog states acting as doorway states. Isospin conservation (approximate, broken by Coulomb interaction) prevents these states from decaying quickly and give r i s e to the observed structures. 2 i I V i Size Resonances We have seen how resonances arise due to the existence of quasi-stationary states associated with the compound nucleus and the doorway mode of propagation. Similarly, resonances occur at quasi-stationary energies of the sxngle p a r t i c l e mode.. Since the single p a r t i c l e mode can be described by the o p t i c a l model, the single p a r t i c l e resonant states are just standing waves (precise d e f i n i t i o n given i n 6.Il l ) i n the f i r s t order o p t i c a l p o t e n t i a l . The standing wave phenomenon i s closely associated with the size of the system and thus gives r i s e to the name 's i z e resonance', or potential resonance. Sometimes i t i s more prec i s e l y c a l l e d the s i n g l e - p a r t i c l e resonance* A single p a r t i c l e resonant state can decay via a number of ways. It may emit a p a r t i c l e back into the entrance channel (single p a r t i c l e e l a s t i c process), or decay into reaction channels, or turn into more complicated configurations l i k e doorway state or compound nucleus state. Therefore, the t o t a l width can be conveniently divided into two parts: the e l a s t i c width due to the single p a r t i c l e e l a s t i c process and the absorption width due to a l l the rest. 5 Size resonances have been observed in nucleon-nucleus cross sections. Shown i n Fig.2a are poor resolution neutron t o t a l cross sections as a function of energy f o r various nuc l e i . The variations with energy are quite broad, t y p i c a l l y occuring i n a few MeV, thus r e f l e c t i n g the single- par t i d e nature. Also they vary systematically with atomic weight, or the size of tne nucleus, as expected*. However, i t must be s For pion-nucleus interactions the absorption width nas an additional contribution from pion annihilation which, i n f a c t , dominates over a l l other absorption processes. 24 239 236 Fig.2a Poor resolution neutron t o t a l cross section as a function of energy f o r various nuclei. Reproduced from [9]. pointed out that only those peaks near zero energy are simple s i z e resonances. As explained in 6.VI, the broad pea&s for medium and heavy nuclei near 10 to 15 MeV are actually 'resonance echoes', although they were mistaken for si z e resonances for years. The problem about id e n t i f y i n g and analyzing wide resonances from cross sections (or other types of data) i s indeed a major concern to our pion-nucleus s i z e resonance study. In 6.VI we explain i n d e t a i l how to best approach the problem. Neutron size resonances near zero energy have been studied in great d e t a i l * Shown i n Fig.2b i s the neutron s-wave strength function at zero energy f o r various au c l e i (or as a function of potential s i z e ) . The strength function 10 8 ST 1 6 g 4 OF "2 UNITS 1.0 0.8 r o 0.6 0.4 0.2 0.1 • EXPERIMENTAL POINTS O THEORETICAL POINTS_| —SPHERICAL MODEL 25 0 20 4 0 60 80 100 I 20 140 160 180 200 220 240 TARGET ATOMIC WEIGHT Zi.is.2b_ Neutron s-wave strength function at zero energy as a function of mass number. The s o l i d l i n e i s o p t i c a l model ca l c u l a t i o n s . The broken horizontal l i n e correponds to the black nucleus model. Reproduced from [9]. describes the spreading of the single p a r t i c l e state into compound le v e l s of the same spin and p a r i t y . 6 The o p t i c a l model best f i t i s obtained for a potential depth of -65-3i MeV of the Saxon-Wood type. As shown, zero energy s-wave size resonances occur at mass number 50 and 150 approximately. Some departures from the op t i c a l model prediction have been attributed to doorway type of resonances, as explained i n de t a i l in [9]. 2. Unified Description of Resonances We have given above a descriptive picture for how 6 The strength function i s explained in d e t a i l i n ^9] and L101. nuclear reactions proceed i n general. The main concept i s that there are contributions from various modes to the l i n a l observed channel. The mode can vary from being as complex as involving a l l p a r t i c l e s in the system to as simple as involving only the incident p a r t i c l e . The various modes give r i s e to structures i n cross sections from as fine as a few eV to as wide as many HeV.. I t i s desirable to be able to regard a l l modes on a common basis. The concept of spreading width introduced by Lane, Thomas, and Wigner [10] provides such a view. The idea i s that a pa r t i c u l a r model state can always be represented as a li n e a r combination of true nuclear states (i.e* compound nucleus s t a t e s ) . The extent of spreading among the true states i s inversely proportional to the l i f e t i m e of the model state and i s a measure of how poorly the model hamiltoaian approximates the true hamiltonian; For example, i f Bohr's view of strong interaction among p a r t i c l e s were v a l i d , the single p a r t i c l e model would be u n r e a l i s t i c and the corresponding model states would be spread into a large energy range. No size resonance would then be evident. 3n the other hand, i f Bethe's view of single p a r t i c l e propagation were e n t i r e l y correct, there would be no spreading and the t o t a l width would consist only of the natural width of the 27 model state i t s e l f . 7 The real picture i s i n between the two extremes and one sees giant single p a r t i c l e resonances. .Since the natural width of the single p a r t i c l e model i s the e l a s t i c width, the absorption width i s often refered to as the spreading width of the single p a r t i c l e state. Similar to the example given above, i t i s possible to view any structure in cross sections as ar i s i n g from some model hamiltonian resembling. to some extent the true hamiltonian. The t o t a l width of a modal state then consists of i t s natural width and i t s spreading width.. ' This unified view of structures in cross sections also provides a framework for modeling i n t e r a c t i o n mechanisms.. l o explain a pa r t i c u l a r structure, one can construct one's fa v o r i t e model and proceed to calculate the associated tfidth from the model and the true hamiltonian..In t h i s way, LTW [10] estimated the spreading width of the single p a r t i c l e model in nucleon-nucleus i n t e r a c t i o n s . . Their r e s u l t s were l a t e r improved by Vogt and lascoux [11] and found to be in agreement with the imaginary part of the o p t i c a l potential as f i t t e d to experimental data* This approach has been applied to a variety of problems, for example, i n calculating the spreading width of the alpha-particle doorway states i n iieavy 7 The natural width of a model state i s the width given by the model hamiltonian alone. 28 ion c o l l i s i o n s {121- In p a r t i c u l a r , t h i s author [13J has cooperated with others i n applying the spreading width method to the positron-atom system. 29 CHAPTER 3 PION-NUCLEUS INTERACTIONS In the l a s t chapter we; discussed the nucleon-nucleus system.. In pa r t i c u l a r , we i l l u s t r a t e d the intimate re l a t i o n s h i p between the nature of interaction and the existence of various types of resonances. We have seen that, due to the weak nature of the nucleon-nucleus interaction inside a nucleus, various nodes with d i f f e r e n t degrees of complexity can coexist and give r i s e to various resonance structures i n cross sections. We now extend similar discussions to the pion-nucleus system. In the next chapter, based on the concepts introduced here, we develop the pion-nucleus o p t i c a l potential. 3.1*. Single P a r t i c l e Mode The c r u c i a l question to consider i s i n what ways an incident pion transfers i t s energy to the nucleus. The most obvious mechanism i s multiple c o l l i s i o n s among p a r t i c l e s , just l i k e in the nucleon-nucleus system. This mechanism, however, transfers energy poorly i n the pion-nucleus case.. The pion mass i s about 1/7 of the nucleon mass. C l a s s i c a l l y , a f t e r c o l l i d i n g head-on with a pion, a nucleon r e c o i l s with 1/8 of the available kinetic energy. However, Pauli's principle asserts that the r e c o i l nucleon must have enough energy to get out of the Fermi sea, or else retains i t s o r i g i n a l energy..Effectively, a low energy pion can then transfer energy only to those nucleons on the Fermi surrace. 30 Nucleons much below the Fermi surface do not r e c o i l and act as i f t h e i r mass i s i n f i n i t e . Therefore, to a low energy pion, the nucleus seems rather r i g i d and can hardly be excited by c o l l i s i o n s . .. This i s the reason for small i n e l a s t i c (not including annihilation) cross sections. If the pion-nucleus interaction consists only of multiple simple e l a s t i c c o l l i s i o n s , then the single p a r t i c l e picture should be a good approximation. Size resonances should then consist mainly only of the e l a s t i c width and stand a good chance of being found. However, the pion-nucleus inte r a c t i o n i s much richer than multiple e l a s t i c pion-nucleon c o l l i s i o n s . F i r s t , there i s the p o s s i b i l i t y of pion-nucleon resonance* More importantly, the pion can be annihilated and deposit i t s entire energy (including mass) to the nucleus.. This i s a genuinely new mechanism not encountered in the nucleon-nucleus system. In f a c t , as discussed in Chapter 7, pion annihilation i s so strong that i t introduces a large absorption width to size resonances and very nearly succeeds in washing out resonant effects in cross sections. Doorway Mode At about 200 HeV incident energy, the pion resonates strongly with the nucleon i n the J=3/2 1=3/2 channel (wnere J stands for the t o t a l angular momentum and I the isospin quantum number) forming the well known A . For pion-nucleus interactions, t h i s means that an incident pion, when i t s 31 energy i s correct, tends to couple strongly with a single nucleon to form a A which then propagates and decays inside the nucleus. The A may decay back, to a pion and a nucleon, or other more complicated configurations. This i s just the doorway mechanism mentioned in the l a s t chapter. 3 The A-doorway, also often c a l l e d isobaric doorway, has been considered by many to explain observed cross sections [14J. I t i s useful to compare the -^-resonance in the pion-nucleus i n t e r a c t i o n with the photon-atom resonance in the light-matter interaction. .It i s well known that, i f the photon resonates with i n d i v i d u a l atoms, then the index of re f r a c t i o n varies rapidly with l i g h t frequency. However, i n the pion-nucleus system near the A-resonance region, there i s no such spectacular resonance-like phenomenon. This i s disappointing;. The explanation l i e s in how a _\ propagates and decays inside a nucleus. In free space, a /_ always decays back into the e l a s t i c channel (with a width 123 deV). Inside a nucleus, however, a A i n t e r a c t s with other nucleons and can decay via a number of ways. In p a r t i c u l a r , the pion annihilation channel dominates;.In fact, t h i s decay mode i s so strong that (except i n the l i g h t e s t nuclei) the nucleus i s e s s e n t i a l l y black to pions which enter i n t o the ,4-doorway 8 Doorways i n nucleon-nucleus interactions often lead to compound nucleus formations; . But the _J-doorway leads mostly to pion a n n i h i l a t i o n . 32 state; Indeed, experiments show that d i f f e r e n t i a l cross sections near the A-resonance region resemble those from a black, disk. A black nucleus simply can not produce any resonance structure at a l l . In contrast, photon resonances exis t despite the p o s s i b i l t y of photon anni h i l a t i o n . This i s because photons are often reemitted from the excited atom (indeed t h i s i s the resonance mechanism) due to lack of opened reaction channels;.On the other hand, pions can always be annihilated leading to reactions (often with many nucleons ejected) because of the large mass. Other doorway modes have been contemplated for the pion-nucleus system. For example, Rowe and Vogt £15] considered the p o s s i b i l i t y of heavy fragment emissions ,(e. g. alphas) a r i s i n g from pion annihilation occuring in a pion-many-nucleons doorway state. However, any such subsystem probably suffers from having an absorption width as l a r ^ e as the A does. 3 . I l l ; Compound Nucleus Mode In nucleon-nucleus interactions, at low energy due to the lack of opened reaction channels, compound nucleus resonances show up..However, for pion-nucleus, no matter how low the energy pion annihilation i s always possible. . Considering the strong annihilation and the small l e v e l spacing c h a r a c t e r i s t i c of compound systems, one may dismiss compound nucleus resonance? for pion-nucleus r i g h t away.. 33 However, i t i s i n t e r e s t i n g to contemplate what A i f pion resonant miqkf occur annihilation had been absent. Dae to the small pion mass, as explained in 3.1, energy transfer through multiple c o l l i s i o n s i s i n e f f i c i e n t . Therefore, i t requires many c o l l i s i o n s to form a compound nucleus and subsguently emit a pion back to the e l a s t i c channel. Hence, the e l a s t i c width would be much smaller than in the nucleon-nucleus case. It i s inte r e s t i n g to note that for the electron-atom system, where the electron-electron interaction i s quite weak, compound atom resonances have been observed with width of the order of 0. 1 eV [16 ]. 3. IV,. Pion Annihilation I t i s unfortunate that pion annihilation k i l l s many pote n t i a l l y possible resonances..However, an n i h i l a t i o n xs a novel feature not encountered i n more conventional probes and has captured much, i n t e r e s t . . So far, despite great erxort, only some general understanding has been achieved* Pion annihilation inside nuclei i s l i k e exploding a bomb since i t deposits at least 140 MeV which corresponds to the binding energy of f i v e to ten nucleons. It i s a genuine many-body pheaomenon. In free space, pion a n n i h i l a t i o n on a single nucleon i s not allowed by energy and momentum conservation. On a bound nucleon the same i s true to a xarge extent, as follows.. The independent p a r t i c l e model for the nucleus gives each bound nucleon a binding energy of 34 approximately 10 MeV and a momentum d i s t r i b u t i o n up to i?ermi momentum about 250 MeV/c. When a pion annihilates on a bound nucleon* the nucleon acquires the whole energy and must leave the nucleus with k i n e t i c energy — 'zlHO MeV. This requires the nucleon having had a momantum p * 500 MeV/c prior to an n i h i l a t i o n . That i s about twice the Fermi momentum. Therefore, l i t t l e pion annihilation can take place based on t h i s mechanism. Of course, one may argue that tha independent p a r t i c l e model i s only approximate and there may exist some momentum components of the order of 500 MaV/c due to nucleon-nucleon c o r r e l a t i o n s . Indeed, some hope has been placed i n measuring these high momentum components based on pion an n i h i l a t i o n on a single nucleon. Contrary to the single nucleon mechanism, pion an n i h i l a t i o n on two nucleons can e a s i l y s a t i s f y the conservation requirement (e.g.. two nucleons, back to iaack, each with half of the pion mass) with no extraordinary demand on i n i t i a l nucleon momenta. Two-nucleon emission has been observed to be an important or dominant channel for pion an n i h i l a t i o n . Multi^-nucleon emission may also be explained based on the two-nucleon mechanism with further f i n a l state interactions between the two outgoing nucleons and other residual nucleons. . The two-nucleon annihilation 'black box', as shown i n Fig.3a, has been modeled by many people. The t y p i c a l approach 35 i s unknown. Fig-4 shows one popular guess. Fig.3b Pion-nucleon scattering. i s summarized in Fig.4a. As shown, one assumes the t r a d i t i o n a l Yukawa vertex via which the pion f i r s t annihilates on a nucleon, which afterwards interacts witn the other nucleon to give two f i n a l nucleons compatible with energy and momentum conservation. The nucleon following the annihilation vertex (i.e. the N; in the figure) should be regarded as a guasi-nucleon since i t must decay (bacjc to a pion and a nucleon) due to non-conservation of energy at the annihilation vertex. . Within i t s l i f e t i m e , however, _.f the guasi-nucleon interacts with another nucleon then two f i n a l energetic nucleons s a t i s f y i n g energy and momentum conservation may be produced. 9 In t h i s sense, pion annihilation can only take place when two nucleons are 9 Fig.4a describes just as well the A problem i n 2.II.. One merely needs to replace the guasi-nucleon by the A. In xact, i t i s probably because & has a longer l i f e t i m e than, say a quasi-nucleon formed with a low energy pion, that the nucleus i s more black at the A energy than at low energy. 36 N M ILias-iS One model for pion annihilation on two-nucleoas. Pion f i r s t annihilates on one nucleon which interacts afterwards with the other one to share the excess energy. This i s the single scattering type since pa.cn involves e x p l i c i t l y with only one nucleon. nearby. In Fig. 4a, the guasi-nncleon l i f e t i m e depends on the extent to which energy conservation i s violated at the annihilation vertex.. As described e a r l i e r for pion an n i h i l a t i o n on a single nucleon, the higher the nucleon momentum prior to annihilation, the smaller the energy non-conservation at the annihilation vertex. Further, the longer the guasi-nucleon l i f e t i m e , the higher chance tnat i t i n t e r a c t s with other nucleons to complete the annihilation process. Therefore, to be f a i r to Fig.4a, one snould incorporate nucleon-nucleon correlations i n front of the annihilation vertex in order to include long l i f e t i m e quasi-nucleons..Initial correlations are included, of course, i f 'good' i n i t i a l state wave functions are used. To be e x p l i c i t , however, we draw in Fig.4b two nucleons {say, described by the independent p a r t i c l e model) intera c t i n g through r e s i d u a l interactions, thus getting to be correlated. Note that in Fig.4b i f the vertex describing the f i n a l 37 IT c e r r c l a t i D n Zi3_.i*b A modification of F i g . 4a. Putting in nucleon-nucleon correlation prior to annihilation prolongs the guasi-nucleon l i f e t i m e , so annihilation i s enhanced. state interaction i s taken out, then the diagram describes the single nucleon mechanism f o r pion a n n i h i l a t i o n . . i t i s interesting that both the si n g l e nucleon and the double nucleon mechanism depend on nucleon-nucleon co r r e l a t i o n . However, for the single nucleon mechanism, i t i s the correlation before annihilation; and whereas for the double nucleon mechanism the correlation afterwards (or f i n a l state i n t e r a c t i o n ) , that i s important. Pion an n i h i l a t i o n on the deuteron has been calculated extensively. It i s found (see [17], [18], and [20]) that in addition to the mechanism described i n Fig.4b, a double scattering type of term, as represented in Fig.4c, contributes s i g n i f i c a n t l y to annihilation. In fact, i t turns out to be the more important mechanism. From the multiple scattering equation (Eq.(9) i n 4.1) one can see that the T-matrix for the pion-deuteron system has two single 38 Fig.4c Pion annihilation on two-nucleons, the double scattering type.. Pion f i r s t scatters o f f one nucleon and then gets annihilated on the other. scattering terms and two double scattering terms. 1 0 Fx> 4b, in which the pion i n t e r a c t s only with one nucleon, belongs to the single scattering type. Fig.4c, i n which the pion scatters off one nucleon f i r s t and then gets absorbed by the other, of course belongs to the double scattering type; and i s often referred to as the •rescattering' contribution to annih i l a t i o n . In f a c t , i t i s t h i s scattering off the f i r s t nucleon that gives r i s e to the observed peak i n Tf+(J-»pp cross section near the A-resonance energy. However, associated with the double scattering mechanism for pion annihilation there i s the notorious 'double counting' problem. In Fig. 4d we draw the lowest order diagram of Fig.4c. As shown, depending on how one separates the diagram, Fig;.4d may be regarded either as single scattering or as double scattering. This problem stems from the xo Admittedly, the multiple scattering equation should be presented prior to drawing annihilation diagrams. 39 double scattering stnqle jotfcennj Zi3iMi Lowest order diagram of Fig.4c. However, depending on how one looks at i t , t h i s diagram c o u i l either be the single scattering type in Fig.4b, or rne double scattering type i n Fig.4c. t r i l i n e a r nature of Yukawa interaction. However, as reviewed in [21 ], t h i s problem seems to have been resolved. Nevertheless, i t i s possible to bypass th i s problem by using a phenomenological approach to pion annihilation. . For example, McMillan and Hsieh [19] uses two 'black boxes', as shown i n Fig.3a and 3b, for describing respectively pion-two-nucleon annihilation and pion-nucleon scattering. The t r i l i n e a r type of pion-nucleon i n t e r a c t i o n i s avoided s p e c i f i c a l l y . By f i i d i n g the amplitudes of these two _>asic diagrams, 1 1 one can calculate pion-nucleus processes (both scattering and annihilation) without running into double counting problems. i i However, in extracting "irNN-**iM amplitudes from, say, ir+d-^pp data, there w i l l be complications due to multiple scatterings between the two diagrams in Fig.3a and Fig.3b. See Footnote 13. 40 CHAPTER 4 PION-NUCLEO" S OPTICAL POTENTIAL We have seen i n the l a s t chapter that, aside from pion-annihilation and pion-nucleon resonances, the pion-nucleus interaction can be described mainly by single p a r t i c l e propagation;.We discuss i n t h i s chapter, for single p a r t i c l e propagation, how an o p t i c a l potential can be constructed from elementary pion-nucleon scatterings by use of the multiple scattering theory; In the next chapter we then use the developed theory to derive the popular K i s s l i n g e r p o t e n t i a l , which we use eventually for s i z e resonance calculations; . Extensive references on pion-nucleus work can be found i n two notable reviews, one by Hufner [20] and the otner by Landau and Thomas [21]. 4.. I-. Multiple Scattering Series The problem of scattering of a p r o j e c t i l e oy a c o l l e c t i o n of scatterers occurs e s s e n t i a l l y i n every f i e l d of physics. In t h i s section we present Watson's multiple scattering theory [22]. This theory i s applicable to a f u l l quantum mechanical p r o j e c t i l e - t a r g e t system. Of course, in special situations other s i m p l i f i e d formalisms might be used. For example, when the incident wavelength i s much shorter than the scatterer s i z e , the p a r t i c l e viewpoint i s appropriate. The treatment i s then usually based on Boltzman's i n t e g r o - d i f f e r e n t i a l equation for transport 41 processes, or sometimes on the simpler d i f f u s i o n equation. . Let the free Hamiltonian of the systen be denoted by Ho 5 5 Kir* HT (1) where ICq i s the free Hamiltonian of the p r o j e c t i l e and H-f the target Hamiltonian.^ Further, l e t the p r o j e c t i l e - t a r g e t interaction operator be written as A V * I ^ i (2) 1*1 where "0; i s the interaction operator between the p r o j e c t i l e and the i - t h (i=1,..,A) target p a r t i c l e . 1 2 The T-iaatrix s a t i s f i e s the Lippmann-Schwingec equation T m « V * VQH)T(i) i3) where Grfi) i s the free Green's operator. 1 2 Usually i n the multiple scattering approach "Uf i s r e s t r i c t e d to be a poten t i a l type of operator. However, t h i s r e s t r i c t i o n i s not necessary for derivinq the multiple scattering series solution. Therefore, we leave t); to be e n t i r e l y general. However, i t must be pointed out that i f Vi i s taken to a pion number non-conserving type of operator there s h a l l be renormalization problems i n using the multiple scattering series. 42 Physical amplitudes are of course calculated at'? = E;n where Efn i s the incident energy and Fin stands for linH>E|n+te . From (2) we can write (3) as with T; given by j where means summing over j with j*=i. This equation can be solved i t e r a t i v e l y as follows T; a )r Bi • V; Gr 0; + V; Or * £ £Tj' + "U; ft £Tj j j + U; £ T/r 6, TK & IT; + -Uj C V; & I'TJ + y(- £ $ V which leads to Eight away, one recognizes that Vi(Gr^ «0 resembles the inter a c t i o n operator expansion of a two-body t-operator, c a l l i n g i t fj , which s a t i s f i e s the equation 43 However, we must emphasize that T»U) i s actually a guasi-two-body operator since i s a many-body operator. More s p e c i f i c a l l y , T^Cl) describes the scattering between two p a r t i c l e s i n the nuclear medium. Now we can write •n-Ts-ivGrTj <8> J and obtain the f u l l T-matrix This equation can then be solved i t e r a t i v e l y to obtain the multiple scattering series, where the f i r s t term i s a sum of single scattering events from each nucleon, the second term double scattering events / 44 from two d i f f e r e n t n u c l e o n s , , e t c - * 3 Therefore, the projectile-many-scattarer problem i s solved i n terms of a series of guasi-two-body scattering events. Notice that there i s a f i n i t e number of terms i n the multiple scattering series. The above derivation seems to consist of a series of formal manipulations with l i t t l e appeal to physical i n t u i t i o n . The same results can be derived more transparently (but more tediously), as done in [23], by using wave functions instead of operators. . The multiple scattering solution. Eg. (9), i s an exact solution of the T-matrix.. In p r i n c i p l e , i t i s capaole of producing exact r e s u l t s . In practice, however, one must introduce approximations for doing the summation and taxing care of the many-body characters of 'X; and £ . Nevertheless, extremely simple situations a r i s e for some systems, notably 1 3 We might remark that the pa r t i c u l a r way of separating the interaction in (2), which says the p r o j e c t i l e i n t e r a c t s with each target nucleon separately, i s not a necessary r e s t r i c t i o n to deriving the multiple scattering series. _ For example, for McMillans's method of doing pion annihilation (see 3. IV) for which the pion in t e r a c t s not only with i n d i v i d u a l nucleon but also two-nucleons, the multiple scattering series i s just as applicable. One only needs to interprete each f; as corresponding to the i - t h potential, not necessarily the i - t h nucleon. In t h i s sense, for example, even i f there i s only one target p a r t i c l e one can divide the int e r a c t i o n i n t o two parts and derive a 'multiple scattering' series (up to double scattering only* of course) . This i s an i n t e r e s t i n g alternative to the usual two-potential formula. f o r example, neutron scattering from c r y s t a l s and l i g h t propagation through matter..In these cases, since i n d i v i d u a l scattering i s weak, the single scattering term i s usually s u f f i c i e n t . * * Furthermore, since binding energy i s small compared to the incident energy, '7/j resembles clo s e l y the free-two-body t - o p e r a t o r ; 1 5 These two conditions enable a simple solution. However, for pion-nucleus scattering none of these si m p l i f i n g conditions holds..In 4.II, we show how one manages to sum the multiple scattering series for the special case of e l a s t i c scattering; and i n 4.V, we show how to approximate /K from the free-two-body t-operator; i i i l l i F i r s t Order Optical Potential One pa r t i c u l a r application of the multiple scattering series i s that i t can express Feshbach's formal o p t i c a l potential operator, which as we described i n 2.II produces e l a s t i c scattering amplitude exactly, i n terms of two-body scatterings inside the nucleus.. This provides a means for cal c u l a t i n g the o p t i c a l potential. The formal expression i s derived in [23].. However, not s u r p r i s i n g l y , the formal l * In fact, the single scattering approximation i s always used in scattering experiment at some stage.. For example, f o r pions scattering o f f a f i n i t e size target, one never worries about double scattering over two different nuclei; is This i s usually refered to as the impulse approximation. . He discuss i t i n 4.V. . 46 expression can not be evaluated exactly i n general. Instead of introducing approximations to the formal o p t i c a l p o t e n t i a l , we f i n d i t more transparent to derive the x i r s t order o p t i c a l potential by making approximations direct..y to the multiple scattering series._ This o p t i c a l potential i s good for describing the single p a r t i c l e mode. The e l a s t i c scattering amplitude i s given by <ifeout, 0(-_fc«t)|T lO(-bcn), fetn> where j fee„ti = I feft»|. The i n i t i a l state 10(rbii0 fein> consists of the nucleus i n the ground state Jo> with centre of mass momentum -bioand an incident pion with momentum _!Jn i n the If-nucleus centre of mass frame. A similar interpretation applies to the f i n a l bra-state Itjbnf 0("i?Qtir)j« The pion operator defined by (,6(*_Jo«»r)lTlO(feiVil(call i t Too ) then produces e l a s t i c amplitudes exactly when i t stands between the incoming and the outgoing pion states. In terms ox the multiple scattering series, t h i s operator i s then given by Too ( kin, Uout) * <0(-fee*r) 11 Ti 10 (-fe,n)> + <0(-keur) i Z T;C<7j \ 0(-fe;n)> +'"( 10) Before continuing with the main discussion, we make a l i t t l e digression. Often (if not always) one sees i n the l i t e r a t u r e that Too appears simply as ^ o l T I O ) , i . e . the nuclear centre of mass motion i s e n t i r e l y ignored..This i s not s t r i c t l y correct.. Consider, for example, one term of the single scattering sum in Eg. (10) , i t can be written as <o(-feoiit)if* |o ( - Iz ;* )> ' !<o t - fcuD|n(~fe ;„ ) I?,-1 o(-fe,*n)> < 1 1 ' where jn('fefft)> constitute a complete set of nuclear states. . Since £o(-fewr)Jn(-k,*fi/)does not necessarily vanish for n$0, i t i s not e n t i r e l y correct to replace (11) by ^O lT i ' lO) Therefore* i n calcul a t i n g "f_0 , some correction factor snould be introduced when the centre of mass motion of the target i s ignored. This factor could be important when the target has substantial centre of mass motion (say in heavy ion c o l l i s i o n s ) . . But i t can be safely ignored for the pion-nucleus case due to the small pion mass. Now, without further h e s i t a t i o n , we replace Eg. (10) by Too - <ol I T r lo> + <ol X f; 6 f,- lo> + (12) There are simple physical interpretations to each term of the above series. The single scattering term b^out",017/ (0, fe<n) stands for the amplitude f o r which, after a single scattering event with the i- t h nucleon, the pion gets i n t o fcouf and tne nucleus remains in the ground state. This process i s represented diagramatically i n Fig.5. Double scattering terms are a b i t more involved. Since & i s diagonal with regard to 48 I k,'n> [ \ | kour? I i 3 i 5 Single scattering e l a s t i c amplitude products of nuclear states and pion s t a t e s , 1 6 lTr>|n)<'r)|<lT) where Hf In)* c\,ln> ^1TT>= Cff ltT> a double scattering term can be written as This may be interpreted as the amplitude for which the pion-nucleus system i s excited i n t o |nfTf> by the f i r s t scattering and de-excited into the f i n a l state by the second scattering. This amplitude i s represented in Fig.6. Terms of higher number of scatterings in Eg. (12) can be interpreted 16 We write i n the form of discrete nuclear states and continuous pion states;. 49 Fig._.6 Double scattering e l a s t i c amplitude s i m i l a r l y . Notice that so far we have not specified what forms the p r o j e c t i l e Hamiltonian, l^if , or the i n t e r a c t i o n operator, D; , take. In p a r t i c u l a r , one needs not r e s t r i c t Kif to a singl e pion operator; or Vi to a pion number conserving operator (see Footnote 12). In other words, intermediate pion states Jn,ip may contain any number of pions, and thus pion annihilation can be included in the multiple scattering framework. The vertices in Fig. 5 and Fig.6 conserve momentum but not energy. Therefore Ifi.Tf) may be of any energy of He. However, as i s evident i n (13), those intermediate states too f a r off the energy s h e l l (i.e.. £fo -tftn 'large') do not c o n t r i b u t e . 1 7 Notice that states with more than one pion present, as well as states with no pion present but with low nuclear excitation are far o f f - s h e l l . The intermediate states 17 That i s one can eliminate^ intermediate states of short l i f e t i m e . Notice that-65--$,)' corresponds to the lifetime of the intermediate state. 50 which do contribute are of three types. We use the double scattering diagram i n Fig. 6 to i l l u s t r a t e . As shown i n Fi.g.7, there are three types of 'nearly on-shell' processes for double scattering. The f i r s t two diagrams r e f e r to processes ; Fig.7 Three 'nearly on-shell' double scattering e l a s t i c amplitudes. (a) Ground intermediate nuclear states (b) Excited intermediate nuclear states (c) Pion annihilated and (thus) highly excited intermediate nuclear states in which the intermediate state contains a single pion , with the nucleus respectively i n the ground state , |o> , and the excited state, 1n> . The t h i r d diagram refers to intermediate states with no pion present. It stands for the amplitude f o r which the pion i s annihilated by the f i r s t scattering and re-emitted back to the e l a s t i c channel by the second 51 scattering. Recall that had pion annihilation taken place on a single nucleon, the struck nucleon can retain only i x t t l e portion of the pion energy due to momentum conservation..This type of annihilation leads to a lowly excited nucleus and far o f f - s h e l l intermediate states..Therefore i n Fig.7c we use the subscript 'h' to designate highly excited intermediate nuclear states formed from pion a n n i h i l a t i o n on at l e a s t two nucleons. In order to sum the multiple s c a t t e r i n g series, one often makes the coherent approximation. It amounts to keeping contributions to the e l a s t i c s c attering amplitude only from those terms i n which the nucleus remains i n the ground state throughout scatterings.. For example, for the double scattering amplitude i n (13), one keeps only the following terms, This i s e s s e n t i a l l y equivalent to keeping only the r i r s t diagram of F i g . 7 . 1 8 The above can be simply written as 1 8 Actually lit) s t i l l has the freedom of having any number of pions. But, as pointed already, the intermediate state |0,TI> having other than one pion i s far o f f - s h e l l and does not contribute. 52 where <j(£) i s the pion propagator (with the nucleus i n the ground state) given by Now the coherent part of the multiple scattering sum can be written as This i s the pion operator which, when standing between the incoming and outgoing pion states, produces that part of the e l a s t i c amplitude for which the nucleus remains i n the ground state throughout a l l the intermediate scatterings. Assuming that the constituent nucleons are a l l i d e n t i c a l fermxons, i . e . 10) i s properly anti-symmetrized, we can write <0{/T;'0) = <T> to be indepedent of i . Now the above series can be summed as follows. To* - A<T> + A<T> 5(A-0<T> -f --• u ~Too may or may not be a good approximation to Too (see 4.III). However, i t i s an important quantity by i t s e l f . 53 Recall our description in Chapter 2 about various types of modes i n nucleon-nucleus i n t e r a c t i o n s . . I t was mentioned that, in general, e l a s t i c scattering has contributions from single p a r t i c l e , doorway, and compound nucleus modes. The single p a r t i c l e mode refers s p e c i f i c a l l y to processes in which the nucleus remains i n the ground state throughout a l l of the scatterings. Contributions from more complex modes are associated with excited intermediate states; 1 9 Therefore, too i s simply the single p a r t i c l e part of Too- 2 0 Furthermore, as described i n 2.II, the single p a r t i c l e e l a s t i c amplitude describes the slow energy dependence of the f u l l e l a s t i c amplitude and produces gross structure of the cross section. Therefore io6 i s capable of experimental prediction. . This point seems to be missing in the l i t e r a t u r e . Much e f f o r t has been directed at improving upon loo to get at loo. The single p a r t i c l e s ignificance of \oo i s often overlooked. In f a c t , for the purpose of studying s i z e resonances which are associated with the single p a r t i c l e mode, |oo i s precisely the quantity one needs. Of course, when one deals with a more complicated mode s p e c i f i c a l l y , loo i s then not s u f f i c i e n t . 19 Weisskopfs diagram i n F i g . l a , which depicts p i c t o r i a l i y the relationship between various modes, can be naturally related to the multiple scattering series by i d e n t i f y i n g <h|/T;io) with • c o l l i s i o n ' and iofulo) with 'no c o l l i s i o n ' . 20 This i s the reason for the supercript 's'. 54 The formal o p t i c a l p o t e n t i a l operator i s defined to s a t i s f y T o o - U ^ U ^ T o o (15) This operator allows the e l a s t i c scattering amplitude (TooIb.'n*)to be calculated exactly from solving the one-body equation where ^ depends only on the p r o j e c t i l e coordinate. Altnough possible, as done i n [23], we did not derive the f u l l o p t i c a l potential U i n terms of two-body scatterings inside the target by solving for Too from the multiple scattering formalism. Instead, we obtained Too as shown in Eg. .(14). Nevertheless, (14) can be written i n a form resembling Eq. (15) as follows where U , - l A - 0 < f > (18) This r e s u l t means that the s i n g l e - p a r t i c l e e l a s t i c amplitude We c a l l 0\ the f i r s t order o p t i c a l p o t e n t i a l . Various people use t h i s name d i f f e r e n t l y . For example, in [24], i t refers to the potential obtained after making the impulse approximation (see Eq. .(31)).. We prefer the name i n the present context since the lowest order potential gets associated v i t a the lowest degree of freedom of a projectile*-target system. Higher order corrections to |©o simply correspond to modes more complicated than the single p a r t i c l e mode.. In the following section we discuss higher order corrections for the pion-nucleus system. 4-III.. Higher Order Corrections As described in Chapter 3, pion-nucleus e l a s t i c scatterings are mainly from the single p a r t i c l e mode. Now i n terms of the multiple scattering framework we can maice the argument more d e f i n i t e by establishing that higher orders corrections to "Too a r e small* F i r s t consider two ideal i z e d s i t u a t i o n s i n which "|oo=Too« Obviously, i f terms l i k e <n(/Tj|0) vanish for n¥0, then there i s no contribution from excited intermediate states and "[^ equals "feo • This i s equivalent to the target aaiag <kohHToo ife.o>/ multiplied by hi A ' can be found by solving (19) 56 completely r i g i d so that i t can not be excited at all..Under such r e s t r i c t i v e conditions, of course, no reaction i s possible either. This kind of s i t u a t i o n might be r e a l i z e d , say, when the target p a r t i c l e s are so heavy that they do not r e c o i l under c o l l i s i o n s . . Another situaion i n which "foo=1oo/ but l e s s r e s t r i c t i v e than the above one, i s to allow excitation of the target but no de-excitation back to the ground state. This can happen, for example* when the target p a r t i c l e s are completely independent of each other. In t h i s case, products l i k e <0[ T/i |n> ^" lTj lo> must vanish since the state |D> excited by the f i r s t c o l l i s i o n with the i - t h p a r t i c l e can not be de-excited by a second c o l l i s i o n with the j-th when i and j are completely independent of eacli other. Onder t h i s s i t u a t i o n , of course, reactions are s t i l l possible via terms, say, f o r double scattering l i k e <•>»( I'll f\ >*n(Tj* 10). Consider now the pion-nucleus system. Recall that diagrams such as shown in Fig.7b and 7c are not included i n the f i r s t order o p t i c a l potential. For Fig.7b, as described i n 3.1, due to the Pauli p r i n c i p l e and the small mass r a t i o between the pion and the nucleon there should be l i t t l e nucleon r e c o i l upon c o l l i s i o n s . As a r e s u l t , the amplitude described by Fig. 7b, <o(f» |n>Crt i T j 10> , i s doubly small ieven when ignoring the difference between i and j) since <.nlT'o>'$ are i n d i v i d u a l l y small.. This i s analogous to the above example of i n f i n i t e l y heavy target p a r t i c l e s . However, in 57 contrast, the amplitude ^r\lTj|0> i n Fig.7c due to ann i h i l a t i o n can be substantial. .Nevertheless, r e c a l l that !flh> l i k e l y contains two high energy nucleons. Therefore, unless subseguent f ' s involve these two nucleons again (which xs not l i k e l y ) , there i s no chance of emitting a pion and getting back to the ground state. Hence, s i m i l a r to the second example given above for not returning to the ground state after e x c i t a t i o n . Fig.7c should give small contributions to e l a s t i c scattering*. In summary, there would be l i t t l e nuclear excitation were i t not for pion annihilation.. But excitation due to ann i h i l a t i o n hardly returns to the ground state. .  Consequently, there i s only the single p a r t i c l e mode l e f t f o r giving e l a s t i c scattering and the f i r s t order o p t i c a l potential should be a good approximation to the f u l l o p t i c a l potential. 4.IV. Absorption Properties of Ui The expression f o r the f i r s t order o p t i c a l potential, Eg. (18), can not be evaluated i n general since T i s an unknown quasi-two-body operator.. The impulse approximation (see the following section) i s then almost always made i n the evaluation. In t h i s section, we set down a few rules which must be observed in making approximations for*T . These rules follow from the o p t i c a l theorem associated with T - They can be connected to the absorption content of the f i r s t order 58 o p t i c a l potential and therefore are p a r t i c u l a r l y relevant for examining the impulse approximation which neglects pion annihilations e n t i r e l y . In the following we derive the o p t i c a l theorem fo r T • An o p t i c a l potential i s i n general non-Hermitian and produces non-elastic scattering. This i s c l e a r l y shown by the o p t i c a l theorem associated with Eg. (19) ^Bnf« (e«o)»Jlf(«i»>l*^ + ^r«n<M / lU,|ip> ( 2 0 ) where i s the e l a s t i c scattering amplitude and tne two terms on the right are the e l a s t i c and the t o t a l absorption cross section r e s p e c t i v e l y . 2 1 This o p t i c a l theorem was f i r s t generalized by Lax [25] from the Hermitian potential case. The quantity of the nucleus as a whole and .This condition, as shown nelow. measures the absorption therefore must be positive. 2 1 The two cross sections should be interpreted a l i t t l e d i f f e r e n t l y when the o p t i c a l potential i s not exact.. For example, for the f i r s t order o p t i c a l potential, the f i r s t term should be the single p a r t i c l e e l a s t i c cross section and the second term a l l the rest which.includes a l l absorption cross sections as well as non-single p a r t i c l e e l a s t i c cross sections. This i s explained i n 2.II and 4.II. 59 i s formally s a t i s f i e d by the f i r s t order o p t i c a l potential. From (18), hence Eg. (21) can be written as ! f > = ( f t - 0 < : u / o | (22) Further, since (£,•„)=? f; (fin') » t n e above becomes J o^> (22. 1) As i t stands, Eg. (7) i s not suitable for establishing the sign of (22) since f; appears on the RHS of (7) as well..The (7) and (23) are exactly analogous to the usual two ways of writing the (free) two-body t-matrix, one in terms of the free and the other the in t e r a c t i n g Green's operator. f; of course i s not exactly a two-body t-matrix since G» contains a l l nucleons and their mutual interactions. However, one can s t i l l derive (23) from (7) i n the same way as for the (free) two-body t-matrix. alternative expression to use for "ft i s 2 2 Trie)* Vf-r 1/,- d(U)U, ' (23) where Ul i s a ' p a r t i a l ' Green's operator, 60 0 , 1 U ) * ? - ( H 0 * W ) Let (£) denote the set of eigenstates of He+^i - Inserting t h i s complete set on both sides of C{ , ws can write (22.1) as lyo\^a*'n)'™h he'ie ^o|T)MeS<gV^ te''w)'^?l|e><e|v/io>|;> 2.t J It Since Ci i s diagonal i n l€>, the above reduces to which i s always positive. Therefore, as required, (22j i s always positive and the f i r s t order o p t i c a l potential i s always absorptive. Although the f i r s t order o p t i c a l potential i s formally always absorptive, when *\\ i s treated approximately as always in phenomenological o p t i c a l potentials, t o t a l absorption i s not necessarily guaranteed. In f a c t , the above 'proof can be carried further to obtain an o p t i c a l theorem for %.. Since l<pr> ^k (£r»-ft)» Eg. (24) becomes - tm CO'DI t; (*J ) 10 + )* tt | ^  I «f I > l ' (-44. 1) where = Further, let |(ffft^) be eigenstates o f He o f energy 61 t\t\, i i e . H,l(TTn)p^.n!(irn)p ( 2 5 ) Then (ftVfy)and (ep, which s a t i f i e s are related by Vt\tp T;(^)!07n)p U5.2) Therefore {24. 1) becomes Our derivation so far i s completely independent of Suppose we replace i t by|k;„), then Z Tm < km o l Ti (H&> 1° fe;o > - l\ <(tJty | f? (Et«) 1 0 k-"> I 127) The LHS i s just the on-shell forward amplitude of /W ; and the RHS i s the sum of the magnitude squared of on-shell amplitudes. This i s just the usual o p t i c a l theorem of a T-matrix. 62 This re s u l t may seem surprising when considering that /f;, as in our discussion so f a r , i s merely an a u x i l i a r y operator within the multiple scattering framework, whereas the usual o p t i c a l theorem refers s p e c i f i c a l l y to the f u l l T-matrix of a system. However, *Xi takes on s p e c i a l s i g n i f i c a n c e i n the following hypothetical (but perfectly physical) system. Suppose the p r o j e c t i l e only interacts with the i - t h p a r t i c l e , i n t h i s system then, Ho+Vi i s the f u l l hamiltonian and ft i s the f u l l T-matrix. Once r e a l i z i n g t h i s , one can write down Eg. (27) right away. With hindsight, we have merely gone through the steps proving the usual o p t i c a l theorem of the T-matrix. . We must explain the o p t i c a l theorem for TV i a more d e t a i l . . In (27), J (Tfn)p are a l l possible f i n a l states compatible with energy conservation as indicated in (25) . . That means |(jfn)p contains, f i r s t , lj?'o> (where |b'i=lbin{t i . e . e l a s t i c a l l y scattered f i n a l states) ; and isecond, Iti'n) (where n%0, |b'l* Ifetnl, i . e . inela s t i c a l l y scattered f i n a l states); and t h i r d , pion annihilated f i n a l states, Inh>. Of course, had one r e s t r i c t e d IK to a potential operator, then (Op could not be present i n {(TTO)f>. ^ Since U, = (A->KoiT 10")the o p t i c a l theorem for f leads naturaly to conditions for the matrix elements of Ui . However, (27) can not be used exactly since i t s RHS also refers to excited nuclear states. Nevertheless, the weaker 63 c o n d i t i o n 2 -~rm<fejnlU,| _ttn> > I K k i l mkm>| (28) (where |te||=lfe;n|) must c e r t a i n l y be s a t i s f i e d . The difference between,the. inequality, of course, refers to the i n e l a s t i c 23 and the anni h i l a t i o n part of T; ( i . e. |h'n) and lfty) of |llfri^»' Certainly (28) should be v e r i f i e d when T; i s treated phenomenoloqically. . However, t h i s has not been done i n most cases. The above shows how the o p t i c a l theorem of TJ r e s t r i c t s the momentum space matrix elements of the f i r s t order o p t i c a l potential.. S i m i l a r l y , there exist conditions for the coordinate space matrix elements. The most obvious i s perhaps that the 'forward* amplitude must have a negative imaginary -part, i . e . 2 3 In (28) > may be re p l a c e d by » i f f r has a l a r g e amplitude l e a d i n g to |flh> from lofjtn). Indeed, a n n i h i l a t i o n i s a dominant channel i n pion s c a t t e r i n g o f f n u c l e i . However, i t has been e s t a b l i s h e d (see 3.IV) t h a t the ' r e s c a t t e r i n g ' mechanism ( i . e . n o n - s i n g l e s c a t t e r i n g ) c o n t r i b u t e s the most of a n n i h i l a t i o n . So i t i s not c l e a r how important a n n i h i l a t i o n i s i n f l i t s e l f . . 64 which follows from replacing i n Eq.(26) b y l p . 2 4 This condition at every; point i s d e f i n i t e l y a more s t r i c t t e s t than the t o t a l absorption condition, (21). The form of the LHS suggests i t to be related to the absorption of the pot e n t i a l at a point. This i s discussed more f u l l y i n 5.III.. So, i n conclusion, there are two o p t i c a l theorems for a multi-particle target..One i s the o p t i c a l theorem for the f u l l T-matrix, Eg. (20), and the other f o r the T-matrix, Eq. (27).. The one for T relates to the absorption of the p o t e n t i a l as a whole.. The one for T , as we have seen, relates to the absorption of the f i r s t order o p t i c a l potential at a point.,. It has been brought to our attention that Tandy e t . . a l [47] have derived a s i m i l a r o p t i c a l theorem for the T i n the nucleon-nucleus system. . r 4_.V__ Impulse Approximation Since T i s not known in general, the usual f i r s t approximation in evaluating (18) i s to assume the pion-nucleon c o l l i s i o n takes place within such a short •impulse 1 that the struck nucleon has no time to f e e l the influence of other nucleons. E f f e c t i v e l y , one assumes both the pion and the nucleon are free from other nucleons during the c o l l i s i o n and replaces • *\\ by the free two-body t-operator, 2 4 Remember that (26) can be derived independent cf f . t i il) 1>; * Ui (30) where i s the i - t h nucleon free hamiltonian. Within t h i s approximation, T; i s represented diagramatically by Fig. 8a. Immediately, one recognizes the unsatisfactory aspect or the impulse approximation. I t i s that the free t-operator completely f a i l s to account for that part of TV which describes pion annihilation by two or more nucleons, as depicted in F i g * 8 b . 2 5 In fact, i t i s already clear that, within the impulse approximation, the two sides of (28) are almost equal since the on-shell matrix elements, {(imp)t |obJn>yare mostly e l a s t i c due to the small pion mass* We explain l a t e r Of course, the free t-operator can s t i l l account for pion a n n i h i l a t i o n on a single nucleon, i . e . !b"> i n Fig. 8a could be replaced by a no pion state. However, t h i s kind of an n i h i l a t i o n can only lead to a lowly excited nucleus.. The corresponding intermediate state i s far o f f - s h e l l and does not contribute, as pointed out several times already. EiSiSa The above, ^  ' pi ">< fe',p) | f c > V 10), represents tae impulse approximation to the a mplituda, |o . /y i s replaced by the free two-body t-matrix'i -^ . * 2 5 how t h i s d e f e c t i s u s u a l l y remedied 66 \\i > 10) = F i g . 8b Pion a n n i h i l a t i o n amplitude<£flhl/TI0l£>. T h i s amplitude i s ignored e n t i r e l y i n the impulse approximation shown i n Fig.8a s i n c e ^Df, 11 JO It'? <i o Perhaps the impulse approximation i s most e a s i l y ciiecked by examining s i n g l e s c a t t e r i n g events, thus o f f - s h e l l amplitudes of f i do not m a t t e r . 2 6 One may then compare ikotf n! t{ 10 bin) ( c a l c u l a t i o n ) with f^cour n lfj 1 o ktVo (experiment) d i r e c t l y . Notice t h a t n^l does not n e c e s s a r i l y e q u a l t o (o) s i n c e s i n g l e s c a t t e r i n g can cause n u c l e a r e x c i t a t i o n . 2 7 However, i t i s l i k e l y t h a t only near forward g u a s i - e i a s t i c s c a t t e r i n g s can be s a f e l y i d e n t i f i e d as s i n g l e s c a t t e r i n g events. Based on the impulse approximation as d e p i c t e d i n Fig*8a, the momentum r e p r e s e n t a t i o n of the f i r s t order 2 6 O f f - s h e l l c o n t r i b u t i o n s of Ti come i n the form of i n t e r m e d i a t e s t a t e s between s c a t t e r i n g s . . 2 7 T h i s would not be t r u e were i t not f o r pion a n n i h i l a t i o n . . S e e t h e d i s c u s s i o n f o l l o w i n g (27) . 67 o p t i c a l potential i s then given by f (31) Or (A-0 ^b"o I i I o k'> « (AH)J„y <ol(5,,><ffe,;e'|tlp,1fe,?^ 'lO> where p^'lO> i s the probability amplitude for finding the ground state nucleus having one nucleon of momentum pj . O n l y integration over f_' i s needed since the recoiled nucleon momentum p i s fixed by momentum conservation, R+P = tz*P .. The pion momenta, and \z' , equal to the i n and out momenta, bin and bout, only for single scatterings. Beyond that, they need to range over a l l values. In evaluating (31), one usually further assumes that ^fc? |_* ItIp'.tS depends weakly on the nucleon momenta and takes i t out of the integral..Hence, *fe"iu, i fe'> ' ( A - o *b", i t [*t,)| Po, J?> F, (32) where Po i s some appropriate nucleon momentum, G, the pion (or nucleon) momentum transfer, Q = b''" k' = p'-p" r and F,(0 the nuclear e l a s t i c form factor More f a m i l i a r l y , i n coordinate space, the form factor i s F,(4)-[d 3 i rp.^)e , - ' r ( 3 3 ) 68 where p, (C) i s the one nucleon probability density d i s t r i b u t i o n . 2 8 Many calculations have gone beyond the s t r i c t impulse approximation given here. Notably, treating pion-nucleon c o l l i s i o n s i n the presence of a core (which supposedly represents a l l other nucleons) as a three-body problem, Thomas and Landau [21] modified!" to include Pauli's p r i n c i p l e and nucleon binding e f f e c t s . . Also, others have treated nuclear Fermi motion more exactly by using Eg.(31) d i r e c t l y . However, the most important correction to the impulse approximation i s c e r t a i n l y due to the neglect of the pion annihilation piece in T* . . So far no one has managed to include annihilation i n i0\^\0) from f i r s t p r i n c i p l e s . . That i s one of the most challenging problems i n pion-nucleus o p t i c a l potential work. The d i f f i c u l t y stems from those an n i h i l a t i o n problems outlined i n 3.IV. The usual approach i s to add an ann i h i l a t i o n piece phenomenologically to the free two-body t-operator, i . e . A-» - ' (34) * m i t i o b s . . . a c t " , ! ? 1 ) Notice that flL should be added with condition (28) i n mind.. Without t h i s term, (28) i s almost an equality since((^lt lOkfcis 2 8 This i s the reason for the subscript '1' i n the form factor. mostly e l a s t i c due to the small pion mass. S t r i c t inequality must be obtained a f t e r introducing ft. (see Footnote 23). Also, as for coordinate space considerations, Oi must be added such that (29) i s s a t i s f i e d everywhere. These requirements follow from the o p t i c a l theorem for T ..Most op t i c a l potentials have been constructed with no reference to these requirements at a l l . 4.Vii Nucleon-Nucleus Optical Potential In t h i s chapter we presented the multiple scattering approach f o r constructing pion-nucleus o p t i c a l p o t e ntials..It i s i n s t r u c t i v e to consider the same for the nucleon-nucleus system*.This problem has been discussed extensively i n the well known paper by Kerman, McManus, and Thaler [26]. 2» It was shown that f i r s t order o p t i c a l potentials based on the impulse approximation seem to be adequate for incident energy above 100 MeV. At low energy the discrepancy caa be attributed to the break down of the impulse approximation. 3 0 2 9 In t h e i r formulation the d e f i n i t i o n of T d i f f e r s somewhat from that given here;.It contains an additional projection operator so that only intermediate states of proper symmetry between scatterings are allowed to contribute. 3 0 Remember that, even though higher order corrections migat be important for the nucleon-nucleus system at low energy (as evidenced by the appearance of non-single-particle resonances), the f i r s t order o p t i c a l potential can always describe gross structures of cross sections. The f a i l u r e i s due to the impulse approximation. . 70 At very low energy the scattering strength i s characterized in terms of the scattering length. Since the nucleon-nucleon scattering length i s 5.4 fm for the spin-1 state and -24 fm for the spin-0 state; whereas for pion-nucleon the scattering length i s 0; 25 fm for isospin-1/2 state and -0.15 fm for isospin-3/2 state, multiple scattering i s much more important in the nucleon-nucleus case. Hence i t i s more important to have 'correct' o f f - s h e l l amplitudes of T . This i s the reason for the o r i g i n a l hope of being able to construct low energy pion-nucleus (in c o n t r a d i s t i n c t i o n to nucleon-nucleus) opti c a l potentials using knowledge of elementary s c a t t e r i n g s . 3 1 However, as explained already, the impulse approximation for the pion-nucleus system i s simply not adequate due to pion a n n i h i l a t i o n . Let alone any multiple scattering attempt to construct o p t i c a l potentials, one does not even have good enough knowledge of the on-shell matrix elements of T for doing single scattering calculations. 3 1 Free two-body scatterings only provide on-shell amplitudes of t. CHAPTER 5 THE KISSLINGER POTENTIAL 71 In the l a s t chapter we presented the multiple scattering approach for constructing the f i r s t order pion-nucleus o p t i c a l potential. The f i n a l expression. Eg. (34) , i s i n terms of the free two-body t-matrix and some correction terms. Making use of (34) , Kisslinger constructed one of the e a r l i e s t pion-nucleus o p t i c a l potentials by postulating a pa r t i c u l a r form for the pion-nucleon free t-matrix. x,ater others added i n correction terms. These are described i n 5.1 and 5.II respectively. The K i s s l i n g e r type of pot e n t i a l has enjoyed and continues to enjoy considerable popularity. However, we have encountered two pathologies during resonance analyses while examining the K i s s l i n g e r potential from the continuity o equation point of view;„ It i s found that, f i r s t , the potential f a i l s to be absorptive everywhere; and second, i t a becomes singular when the potential exceeds Acertain strength. The second pathology, after being remedied, s t i l l gives amusing r e s u l t s : for example, an i n f i n i t e number of hound states, a kink in the wave function, a negative e l a s t i c width,... etc;. Many of these r e s u l t s can be understood i n terms of the pion having a negative e f f e c t i v e mass due to the o p t i c a l potential. We discuss a l l of these i n 5.III and 5.IV. 5 i l D e r i v a t i o n of the Kisslinger Potential The general two-body scattering amplitude (ignoring spin 72 and isospin dependences) can be written as a sum of p a r t i a l wave amplitudes where 1^ i s the Legendre polynomial. .The i n and out momenta, and K*' / refer to the two-body center of mass frame (Ik \=-\\£ | of course).. This amplitude i s related to the on-shell t-matrix by -f = - (2IT)*-^5: ton 135) Therefore, keeping only s and p waves at low energy, one can write ton(Y, %') = U(k') + tp(u') (36) where "ts and tp are energy dependent parameters wnich supposedly can be extracted from phase s h i f t s . Mimicking the form of the above on-shell .t-»matrix, Kisslinger [27] simply assumed the t-matrix i n Eg. (34) to be <£, Pc ** 11 (f J ) ! p., b' > -1$ ( M * tp ( M ku-k' (37) There are two d r a s t i c s i m p l i f i c a t i o n s i m p l i c i t i n assuming the above. F i r s t , there i s no reason for the o f f - s h e l l t-matrix elements taking on the same form as the on-shell ones. 3 2 Second, even assuming so, one should not write Eg. (37) since |/ and u' in Eg. (36) refer to the centre of mass of the two c o l l i d i n g p a r t i c l e s , whereas fe' and fe" in Eg. (34) refer to the pion-nucleus centre of mass frame. This complication, which i s generally referred to as 'angle transformation', has been reviewed in d e t a i l i n L21J. Therefore, (37) should be regarded as phenomenologica1 at best. In f a c t , (37) i s already pathological by i c s e l f , independent of a l l those assumptions behind i t . I t diverges wxth and K . This i s the behavior of a zero range interaction* In some work •£$ and tp are taken to be dependent on and b'' so that no divergence appears. Based on (37), the f i r s t order o p t i c a l potential i n momentum representation becomes (38) where t$ and tp are taken to be independent of b' and fe" and are to be extracted from pion-nucleon scattering amplitudes . i ,.i are Recall that I* and g are o f f - s h e l l i n general. They Aegual to the in and out momenta, b;0 and fe^t, only for single scattering calculations. 74 at the incident energy.. We also l e t f = (A-1)f, t Ap, oe the form factor which refe r s to the t o t a l nucleon density.. (38) i s known as the Kisslinger potential. Since pion-nucleon scattering a c t u a l l y depends on the charge state of the p a r t i c l e s , the above should be generalized to include isospin dependences of tj and tp . Taking the pion-nucleon interaction to be r o t a t i o n a l l y invariant i n isospin space, one can write the isospin structures of t$ and tp as t « * t $ o * t « , H'W { 3 9 ) tp * tpo + tp( Lit ' t*4 where and in are pion and nucleon i s o s p i n operators. . For example, for the case of TT+ scattering, one has then, <_"| U^ iir*lb ,>*[(t.o + t*i)+ (tpo + tp,) b". b' J Fp<b"-fe') +[ (tt. -t«i) + (ip0 - tp/) fe1'- b' J F n <^fe 1) where f-p and Fn are proton and neutron form factors, respectively. (p = Fp+Fn' o f course.) For any pion charge the 75 above i s simply generalized to <k"| U * I « ( t $ o + tpo b"- If')F - err (fc< + tpi b"-b') (F 0 - F P) C O Of course, one may use the above potential aad the i n t e g r a l form of Eg.(19) to calculate pion-nucleus e l a s t i c scattering i n momentum space. _However, the usual practice i s to transform the Kisslinger p o t e n t i a l to coordinate space and use the d i f f e r e n t i a l form of Eg-(19). In fact, for size resonance studies* we must carry out coordinate space calculations since the ^-matrix formalism, which we use f o r resonance analyses, refers s p e c i f i c a l l y to coordinate space quantities..We show i n Appendix I that the coordinate space representation for the Kisslinger potential i s < r i u K i r ' > = (2Trf j [ t * p - e» t«, (ft- f io] Hz-?) or uK*(2n)3(B:$«p-c»t«.(Pn-Pp)]-[tpo?./)2- e*tPlv.pi]] <42> This i s a non-local potential. More f a m i l i a r l y , the Kisslinger potential i s expressed in terms of parameters which characterize the scattering amplitude. The s and p-nave part of the pion-nucleon scattering amplitude, including 76 isospin dependence, i s usually written T\,p ( ^ " , K.')=(bt4 b, LTt..M) + (C 0 + C, i * . ^ where b 0 r bi , Co , and C ( are parameters determined from pion-nucleon data. . Hence, by Eqs. (3 6) and (39), then (42) becomes 3&U,c--QGr)+ V-oc(D V i 4 3 ) where Q _ - 4 T ( f b o p - ^ b ^ P n - p p ) ] (44) In place of the non-local Kisslinger potential, one may assume, instead of (37) , the following, ^ \ 11 ) I P., b' > - " M M + tp(b.V) fe.v, - tp(b:^(fe M.fe 1) 1^ (<*5) r and arrive at a l o c a l potential..It can be ea s i l y checked that, s i m i l a r to (37)j, the on-shell value of the above agrees with the known on-shell free t-matrix, (36). Both (45j and (37) are ' v a l i d ' assumptions for the free t-matrix. There i s no reason to prefer one over the other. However, since (45) depends only on fc> and fe through the momentum transfer, b^ '-le' t t n e resultant potential i n coordinate space has the usual l o c a l form ( i * e; . only 6 -functions).. There are no derivatives of the wave function i t s e l f . The dependence on momentum transfer r e s u l t s i n a term i n the potential. This gives r i s e to the name 'Laplacian' p o t e n t i a l . . Other than assuming a pa r t i c u l a r form for the pion-nucleon t-matrix elements, there have been attempts to calculate these matrix elements by assuming a 'separable p o t e n t i a l ' for the pion-nucleon i n t e r a c t i o n , as reviewed i n [21]. 5.IIa Corrections to the Kisslinger P o t e n t i a l Surely the Ki s s l i n g e r potential does not account for any pion a n n i h i l a t i o n . To repair such shortcomings, Ericson and Ericson [28] modified the Kisslinger p o t e n t i a l in the s p i r i t of (34), i . e . add an extra term to the free t-matrix..They proposed GU-lHrftip-fiiy M f t - f p ) + ftp*] A T ( 4 6 ) <*-- M [ C0f - 6 5 C, Co p J where $0 and Co are complex quantities and are supposed to account for pion annihilation e f f e c t s i n If . Tne pZ dependence i s supposed to arise from the pion-two-nucleon 78 an n i h i l a t i o n mechanism, but has not been j u s t i f i e d f u l l y . Recall from 4.V that a general rule must be observed i n adding the annihilation correction term to the impulse approximation. I t i s that the resultant potential must s a t i s f y (28), a resu l t of the o p t i c a l theorem f o r T .The momentum representation of (46) i s 3 3 l U l feS* [ ( b o < u bu. fe' )F(4> + (6.• C. b u V ) F % ] where f ( f : i t p(r)e The above should be checked against (28). Furthermore, the absorption condition i n coordinate space, (29), should be s a t i s f i e d as well. . According to (41), since £(o)>0, L(o)=Q, and i fo)<o (loosely speaking), one requires vxr<0 and tfj>0. 3 + Fortunately these are the signs used in practice.. As explained i n 4.V, the single most important correction to the impulse approximation i s to add i n the anni h i l a t i o n piece. As far as pure scattering (for which the pion survives c o l l i s i o n ) i s concerned, deviations from the 3 3 Ignore b( and C\ , the isovector terms. . 3 4 We s h a l l use subscripts R and I to indicate, respectively, the real and the imaginary part of a quantity from now on.. impulse approximation perhaps are not as important. Nevertheless, i n addition to the above ann i h i l a t i o n terms, the Ericsons [28] introduced v corrections to the o p t i c a l potential by considering the e f f e c t of discrete structure of the nucleus upon the multiple scattering process.. They emphasized the importance of regarding the nucleus as a c o l l e c t i o n of correlated scatterers, rather than as merely a continuous homogeneous piece of matter (or randomly distributed s c a t t e r e r s ) . In p a r t i c u l a r , by examining the e f f e c t s on the multiple scattering process due to the absence of matter around any particular scatterer, they established that the e f f e c t i v e f i e l d incident on a scatterer d i f f e r s from the 'average' f i e l d . 3 5 . This has the e f f e c t of modifying the potential to where $(0 refers to the d i n (46). Notice x that the s-wave part, & i retains the o r i g i n a l form but with values of parameters modified. The value of X depends on the range of 3 5 Understandably, the Ericsons b u i l t t h e i r multiple scattering theory i n coordinate space since s p a t i a l correlations are most ea s i l y accommodated i n such. However, i t i s not clear whether t h i s correlation can be included as well i n the present multiple scattering framework by putting i n Pauli's e f f e c t i n ^olflO). -80 interaction.. I t has the value of 1 for point interaction and i s reduced to smaller values for f i n i t e range. It i s i n t e r e s t i n g to note that the wave eguation, <(1 9) , becomes when only p-wave interactions are taken in t o account ,(with ^=1 and C, r Co ignored) . This i s i d e n t i c a l to the eguation describing l i g h t propagation in a dense medium of polari-zable atoms. The V-tyliCspV term describes multiple dipole scatterings with atoms. The correction factor to t h i s term i s due to atom-atom cor r e l a t i o n s as a result of i n t e r - p o l a r i z a t i o n among neighboring atoms;. Such correlations give r i s e to nonlinear dependences of the r e f r a c t i v e index on the density, the so c a l l e d Lorentz-Lorenz e f f e c t . This p a r a l l e l between nucleon-nucleon and atom-atom correlations upon the effects of p-wave scatterings has been explained i n d e t a i l by Scherk [29 ].. Incidentally, in view of the crude assumptions behind the Kisslinger type of potential (only the on-shell t-\(, 'correct') , i t i s not clear why (49) works so well for p-wave l i g h t scatterings i n atomic media. Perhaps the photou-atom scattering strength i s weak and single scattering dominates. This way, only the on-shell t Amatters. I f t h i s i s true, the Laplacian type of potential should work just as well. 81 Interesting as the Lorentz-Lorenz effect may ae i n pion-nucleus interactions, experimentally i t s existence has not been established firmly. The Ericsons [28] and others, as reviewed by Backenstoss [30], have pursued f i t s or the potent i a l i n (48) (or with some modifications) to a large amount of pionic atom d a t a * 3 6 F i t t e d potential parameters (i>o i b< r C« , and c» ) generally are i n agreement with pion-nucleon data, but with no consensus on the value of Other than pionic atoms, the pion-nucleus o p t i c a l p o t e n t i a l has been tested to pion-nucleus scattering up to several hundred MeV.. However, only the low energy data are expected to be sensitive to th e o r e t i c a l inputs mainly because of the strong absorption through the A-doorway ( i . e. alack disc scattering) above *>100 MeV. . Recently, Strieker, Mcilanus, and Carr [31] have applied the Kis s l i n g e r potential <(with assorted corrections) to low energy data. The potential parameters e s s e n t i a l l y agree with those from pionic atoms. 5.S.III*. Local Absorption Anomalies As explained in 2.IV the width of a size resonance has 3 6 i . e . s h i f t s and widths, due to the strong int e r a c t i o n (in the o p t i c a l model) , of the -p- Coulomb bound levels around nu c l e i . 3 7 One usually compares to zero energy pion-nucleon parameters for atomically bound pions. .. 82 two contributions: an e l a s t i c escape width and an absorption width. . These widths, i f possible, should be kept to a minimum in order for the resonance to be d i s t i n c t . The e l a s t i c width can be controlled somewhat by imposing Coulomb and angular momemtum barriers.. The absorption width, however, depends es s e n t i a l l y on the nature of the interaction* Nevertheless, i t i s ce r t a i n l y desirable i f one can locate where (on the nuclear surface or inside) the absorption mainly occurs so that perhaps, by making use of angular momentum and/or Coulomb barri e r s one can keep pions away from those regions. . To define the quantity which measures the absorption at a point, namely l o c a l absorption, 3 8 one requires the continuity equation. I t i s from constructing t h i s equation for the Kisslinger potential that we encountered two pathologies. They are discussed i n t h i s and the following s e c t i o n . 3 9 For the Kisslinger p o t e n t i a l , because of i t s momentum dependence, the conventional d e f i n i t i o n of probability density and current i s no longer appropriate. This affects 3 8 In general. 4£or non-local potentials, i t i s not clear i f one can always- 4ei:nc a guantity which measures the absorption at a point* 3 9 One may define and discuss l o c a l absorption. However, i t i s not clear how s i g n i f i c a n t the K i s s l i n g e r potential i s to such a quantity. The poten t i a l i a s mainly been tested against experimental data (e. g* pionic atoms) which are only s e n s i t i v e to the t o t a l absorption. 83 d i r e c t l y the d e f i n i t i o n of l o c a l absorption. More importantly, there are also profound effects on the wave function i t s e l f , as we s h a l l show; I t i s useful to discuss f i r s t the simplest type of l o c a l potential, In t h i s case, the t o t a l absorption cross section as given by (20) i s The above being pos i t i v e does not necessarily f i x the sign of r e l a t e W to l o c a l absorption and, consequently, to a sign f i x for W . Consider Eg;(19) in coordinate space to be*o (50) W. However, the l o c a l nature of the p o t e n t i a l allows one to Multiplying the above by and i t s complex conjugate by lp , subtracting the two, one obtains 4 0 Eq. (19) i s time independent. We make i t time dependent by attaching the usual o s c i l l a t o r y time dependence to 84 7-J + ( 5 2 ) where p and J are, of course, the usual p r o b a b i l i t y density and current ; . JL.t\»'i\f. \\> IT) Because of the continuity nature of Eg* (52) , i t i s natural to i n t e r p r e t —W|l^| a s the number of p a r t i c l e s absorbed per unit volume per unit time by the target. Let us c a l l i t the sink function. = ~ w \ (54) In general, the continuity eguation i s then i n the form -17-J ~<i(r) r &~ (=0 for a stationary state) (55) As expected, the integrated sink function divided by the incident flux* -~- , gives simply the t o t a l absorption cross section in (51). Since incident p a r t i c l e s are supposed to diasppear everywhere, the sink function should be positive (or W negative) everywhere*. This sign has always been adhered to i n practice. Notice that t h i s sign reguirement can also be arrived at by considering (29) -, the absorption condition imposed on the f i r s t order o p t i c a l potential by the optic a l theorem associated with /Y •.However, this connection between the sink function and (29) can not always be made.*1 For the Kisslinger p o t e n t i a l , for example, (29) i s s a t i s f i e d as we i l l u s t r a t e d already*. However* as shown below, i t s sink function i s not necessarily always positive. The n o n - r e l a t i v i s t i c equation for the Ki s s l i n g e r p o t e n t i a l i s Y . < H * ) * H ' + - ^ ( Q + Q c ) M ^ ( 5 6 ) where Qc i s the Coulomb potential which we simply add to G l . 4 2 As shown i n Appendix I I , f o r the usual probability and current i n (53, „one can derive the continuity equation, ,(55) , for the Kis s l i n g e r potential with the sink function given by 4 1 (29) can always be applied to any f i r s t order o p t i c a l p o t e n t i a l . However, one may not even be able to define the sink function for, say, a non-local p o t e n t i a l . 4 2 This equation i s very si m i l a r to the 'linearized* Klein-Gordon equation which i s usually used for pions i n o p t i c a l potentials. . See [32]. In fact, the only difference i s a pion energy dependent correction term to Q . Since in our work pion energy i s below 40 MeV, (56) i s not a bad approximation. Also the subscript ' i n ' to the incident energy used i n the l a s t chapter i s dropped from now on. 86 However, also shown La Appendix I I , i t i s possible to arr i v e at the continuity equation based on a di f f e r e n t sat of proba b i l i t y and current, The corresponding sink function i s (58) 2 ^ . A A ill/11 159) Although (57) d i f f e r s from (59) at each point, their i n t e g r a l over the whole nucleus ( i . e . t o t a l absorption,) may be the same. This i s easily seen. Because of (55), the volume i n t e g r a l of 5 equals the (-ve) surface i n t e g r a l of j Provided t-j* • s are the same outside the nucleus, the J's i n (53) and (58) are equal outside the nucleus since o<g vanishes t h e r e . 4 3 Therefore, contrary to l o c a l absorption, the t o t a l 4 3 Or a l t e r n a t i v e l y , i f one l i k e s , since the volume in t e g r a l of the above vanishes. 37 absorption cross section seems to be independent of which d e f i n i t i o n one uses.. One might conclude that only non-observable quantities, such.as l o c a l absorption, depend on the d e f i n i t i o n of density and current. However, i n f a c t , the wave function i t s e l f can not be calculated independent of the d e f i n i t i o n of density and current. 4* In solving the Schroedinger equation, one requires continuous current at each point. Otherwise, according to (55), there would De an i n f i n i t e source or sink at the discontinuity. Therefore, for d e f i n i t i o n (53), the wave function must be kept continuous and smooth at each point*.This i s just the usual requirement, of course. On the other hand, fo r d e f i n i t i o n (58) , one must instead keep ( l - O ^ ) ^ E ^ - ^ t f l f ) continuous at each point. This gives nothing new when ot% i s continuous. But when i t i s not, as i n a square well for example, there are unusual effects at the i n t e r f a c e - one must put a 'kink' into the wave function at the interface i n order to keep the current continuous there* Therefore, for (58) , i n some cases the usual reguirement of having smooth wave function might have to be abandoned. The wave function, thus any observable quantity. 4 4 The importance of t h i s d e f i n i t i o n can not be over emphasized. I t has general i n t e r e s t . Consider, for example, a pion described by the K i s s l i n g e r potential interacts witn the electromagnetic radiation f i e l d . The pion current one uses f o r c alculation should come from such considerations. As we have seen, the current may be modified considerablly from the usual form. . 88 can depend on which d e f i n i t i o n one uses for ca l c u l a t i o n s . Therefore, l e t alone our present need of defining l o c a l absorption, i t i s fundamentally important to have a 'correct' d e f i n i t i o n of probability density and current. In most usage of o p t i c a l potentials, t h i s has been ignored;.Usually one simply solves for a continuous and smooth wave function..That may not be proper, as i n the present case when the potential i s momentum dependent and has sharp edges. In general, the potential may even be non-local. However, i t i s not clear what considerations one must make regarding density and current i n that case. But how to choose the 'correct' d e f i n i t i o n of density and current from among a l l the p o s s i b i l i t i e s ? * 5 We have seen that for each d e f i n i t i o n there exists a corresponding sink function. The reasonable thing to do, then, i s to turn o f f a l l absorption mechanisms and from the resultant potential {real now) derive the continuity equation (unique now) without sink. The obtained current can then be used to derive the sink function from (55) with absorption f u l l y restored i n the potential. For the Kisslinger p o t e n t i a l , i f turning o f f absorption mechanisms simply means keeping the r e a l part of potential parameters ( o( and CL ) , then (58) i s the proper 4 5 Different d e f i n i t i o n s are possible since i n (55) one can separate the sink function and the divergence of the current a r b i t r a r i l y . . 39 d e f i n i t i o n of density and current and (59) the sink function. In comparison, for (53) and (57), turning off absorption means not only getting r i d off the imaginary part of o( and Q , but also the r e a l part of C( , i . e . no p-wave scattering l e f t . That i s ce r t a i n l y not a reasonable thing to do.. He emphasize that which d e f i n i t i o n results i s a natter ox how one arrives at the r e a l p o t e n tial. That should r e a l l y be carried out by turniag off absorption mechanisms, instead of getting r i d of the imaginary part of the potential i n some ar b i t r a r y fashion*. Recall that the f i r s t order o p t i c a l potential does not describe any re-scattering back into the e l a s t i c channel. This means incident p a r t i c l e s are supposed to disappear everywhere, or the sink function of a f i r s t order o p t i c a l potential must be po s i t i v e everywhere..Clearly (59) does not s a t i s f y t h i s requirement i n general. Despite this pathology, however, the t o t a l absorption i s always p o s i t i v e , as shown l a t e r . In most usages of the p o t e n t i a l only the t o t a l absorption matters and t h i s pathology never becomes evident. Contrary to the above f a i l u r e on being l o c a l l y absorptive, the K i s s l i n g e r potential nevertheless s a t i s f i e s the absorption condition imposed by (29), as we have shown already. From i t s l o c a l appearance the quantity, - TW* I U I K. 1 ' •"-n I 2 9) seems to suggest that i t i s related to the sink function. Unfortunately, unlike for the simple l o c a l p o t e n t i a l i n (50), t h i s connection can not be made for the Ki s s l i n g e r p o t e n t i a l . It would cer t a i n l y be appealing had i t been otherwise.. although the sink function i s not positive everywaere, the Kisslinger potential i s always t o t a l l y absorptive. The i n t e g r a l of (59) can be written as [fiiCC)* ( - r f r l W 1 - * M ^ ) ( 6 0 ) which i s always positive f o r Qi<0 and crf-f>0.46 However, the integrands on the two sides c e r t a i n l y are not equal. Further, the integrand on the RHS can not be written as the divergence of a current. It i s probably because the Kisslinger potential i s always absorptive that i t has eluded close examination of i t s absorption properties. We r e i t e r a t e that l o c a l absorption conditions, either in the form of (29) or i n the form of the sink function being positive, must be satisfied... Mo reasonable o p t i c a l potential should be creating pions iu any 4 6 The derivation follows from where the volume i n t e g r a l of the f i r s t term on the RHS vanishes. Notice that (60) also equals to the i n t e g r a l of (57) (xf ao kink occurs ia the wave function) . 91 part of the nucleus.. 5. IV._ E f f e c t i v e Mass Singularity Part of the momentum dependence of the Kisslinger potential can be conveniently absorbed into the pion mass. This r e s u l t s i n the pion having an e f f e c t i v e mass i n a simple l o c a l potential such as (50)..Writing (55) out e x p l i c i t l y , 2m *rn zm then ignoring the o(' term (which i s e f f e c t i v e only on the nuclear surface), we obtain i £ v V + l V s ^ E l p (61, 41*1 where rn = - j ^ and Ui^s i s the s-wave part of the Kisslinger potential*. Therefore, insi d e the nucleus, pions can be es s e n t i a l l y regarded as t r a v e l l i n g i n a simple l o c a l potential (the Ufc,*) with a complex e f f e c t i v e mass. As shown l a t e r , t h i s interpretation of the p-wave term i s useful for understanding many re s u l t s of the p o t e n t i a l . 4 7 In analyzing s i z e resonance, we want to turn on 4 7 However, the Laplacian p o t e n t i a l , does not allow such an inte r p r e t a t i o n . So i t i s not clear how s i g n i f i c a n t the e f f e c t i v e mass picture, as given by the K i s s l i n g e r . i s f o r the pion-nucleus i n t e r a c t i o n . absorption broadening calculating gradually and shift) . from and see According i t s e f f e c t s (resonance to (58), th i s me,ans 0 - 0(0 *V 4 (-tfp.') 4>l4 ( [ ? l - f i r f i t ) f : 0 (62) for non-dissipative properties of the size resonance (resonant energy and e l a s t i c width) . However, i t turns out that i n some sets of parameters determined from f i t t i n g pionic atom data the value of c^ g exceeds 1. This i s p a r t i c u l a r l y true for heavy nuclei where the neutron excess enhances the importance of the C\ term i n (46). In those cases, somewhere inside the nucleus, o(g equals 1. . For Eq. (62) , that i s a sinqular point, the ef f e c t i v e mass s i n g u l a r i t y , where prj' becomes i n f i n i t e . * 8 The si n g u l a r i t y i s more c l e a r l y e x i h i b i t e d by rewriting (62) as ^ + + -T—lf-aa-bt)^ ~~0 (63) 4 8 The s i n g u l a r i t y may not be obvious from the appearance of (62).. But consider the following.. In solving (62) numerically, one obtains g^f f rom If and if'and then proceeds to evaluate and if'at the next point. When <tfg=1» then olows up. An alternative way of looking at the s i n g u l a r i t y i s tnrough (58), according to which J i s i d e n t i c a l l y zero at c*g=1. This i s true no matter what one does to Vlp at that point. This i s cl e a r l y not acceptable. Bethe [33] f i r s t discussed t h i s type of s i n g u l a r i t y and pointed out that i t can not possibly occur due to the u n i t a r i t y requirement associated with two-body c o l l i s i o n s inside a medium. This s i n g u l a r i t y , instead of having any significance of i t s own, merely r e f l e c t s the crudeness of (37, which the K i s s l i n g e r potential i s based on. In practice, t h i s s i n g u l a r i t y does not present any problem because the imaginary part of 0^  i s always kept.. Further, i n ract, possible e f f e c t s due to t h i s s i n g u l a r i t y (e.g. a sudden change in results as the strength of increases beyond 1, are masked by the s i g n i f i c a n t size of (/£ . However, t h i s s i n g u l a r i t y does present a problem to the present study. Eg. (63, can not be solved at a l l at where equals to 1. &n obvious method to remove t h i s s i n g u l a r i t y i s to replace — ! — by R where <| cuts o f f the i n f i n i t y , as shown schematically in Fig. 9. Comparing the above with - j —, one recognizes tnat £ may be i d e n t i f i e d with o<j: when oi i s used f u l l y . _ This indicates to us that, in order to turn off absorption without incurring a s i n g u l a r i t y , one should retain the r e a l part of 1 rather than — — . . Defining 94 (65) we write the f u l l equation as V > + 1-(&C,)4'I+ fi(\?-<X~ ~0 (66) One could, of course, keep only the r e a l part of y^,erf', and Q, i n the above for turning off absorption. However, since our treatment of removing the e f f e c t i v e mass s i n g u l a r i t y i s juite arbitrary, one might as well get r i d of the imaginary part of 95 the equation in the most convenient fashion. One method i s to separate the r e a l and the imaginary parts of the products of parameters i n {66, , i . e . as (67) where fl. =-^ #1 The purpose of the factor ^ i s to turn on any f r a c t i o n of the 'absorption 1 as desired. This way tne so ca l l e d absorption only introduces imaginary terms to the equation. This i s convenient for s i z e resonance analyses since the absorption s h i f t i s minimized and there i s only broadening effect l e f t . Admittedly, t h i s p a r t i c u l a r method of a r r i v i n g at the r e a l potential does not necessarily correpond to the most physical way of turning off absorption mechanisms. However, any other approach for the Kissixnger potential i s probably just as obscure or runs i n t o a s i n g u l a r i t y i n the resultant equation. Conceivably, the d e f i n i t i o n of density and current i n (58, i s no longer compatible with the manner of turning o f f absorption indicated in {67,. In fa c t , due to the removal of the s i n g u l a r i t y , the rea l part of (67, M>'* tfin < fe1"- A« - Ac A * ] l|> *0 (68) i s not even s e l f - a d j o i n t i n contradistinction to ( 6 2 ) . 4 9 One consequence i s that l^ty can no longer serve as p r o b a b i l i t y density. As before, multiplying (68) by If* and i t s complex conjugate by ip , subtracting the two* we f i n d that one can write down the continuity equation (55) (with no sink yet, of course) for the following d e f i n i t i o n s P a A / » * l ^ (69) where /^(rUe/p^-j^t'ftgi*'')) (thus A = A ^ S ) 1 S t n e m u l t i p l i c a t i o n factor which makes (68) s e l f - a d j o i n t . However, we must be careful when the e f f e c t i v e mass, y5e » changes sign. i i n c e p r o b a b i l i t y density, by d e f i n i t i o n , i s required to be positive d e f i n i t e . We must modify the above d e f i n i t i o n s i n the followinq way, which i s our f i n a l r e s u l t , P-- A l^ inf ! 8 , ( 7 0 ) - lien fa 4 9 (68) i s said to be s e l f - a d j o i n t i f i t can be written i n the This requires (j* ) -A . As long as we use $g as the r e a l e f f e c t i v e m a s s R t o avoid s i n g u l a r i t y (no matter how the absorption i n (66) i s turned off) , t h i s condition can not be s a t i s f i e d since fig =o (see Fig.9) at the previously singular point. Notice that - . < the r e a l equation from the previous method of turninq off absorption, (62) , fs self-ad j o i n t . As before, when /•>£ i s discontinuous (like i n a square well) , there must be a kink i n the wave function to ensure the current continuous. . Now, i n addition, there i s a similar effect even when potential parameters are continuous. It arises from the factor in the current d e f i n i t i o n . . This factor i s discontinuous whenever changes sign.. In that case, V lp must change sign as well i n order to keep the current continuous. .The fact that fa plays the role of an ef f e c t i v e mass i t i s l i t t l e wonder that the wave function must change d r a s t i c a l l y whenever changes sign. There are other important e f f e c t s which follow from the new d e f i n i t i o n of density and current. In p a r t i c u l a r , since the wave function must be normalized with a weighting fa c t o r , any calculation involving pion probability density ought to be affected. For example, in Appendix I I I , we show how the two-potential formula must be modified. This has serious implications for the distorted wave type of calc u l a t i o n s involving the K i s s l i n g e r potential. The sink function associated with (70) can be simply derived by using (55).. The result i s i r ' C M - ^ v + M f e ' - e . - f l c ) - / ? ^ ] ! ^ , 7 1 , Again, asymtotically, the above i s equivalent to (57) and (59) i f the wave function in a l l three cases are equal (i.e. 98 provided no kink occurs in any one, . The anomalous aspect of negative e f f e c t i v e mass ia the Kisslinger potential has been exploited by Ericson and iiyhrer [34] to speculate on pion bound states i n nuclei. so They concluded that when 0(g> 1 there exist an i n f i n i t e number of bound states.. This i s easiLy understood i n terms of the negative e f f e c t i v e mass concept. According to (67), when [IffOt the wave function o s c i l l a t e s more rapidly (thus more nodes) the higher the binding energy. In other words, the potential can support an i n f i n i t e number of bound states irrespective of the sign of fig.?1 In t h e i r discussion, however, Ericson and Hyhrer did not make use of the e f f e c t i v e mass concept given here..Further, they f a i l e d to consider at a l l the e f f e c t on the wave function ( r e c a l l the kink, when the e f f e c t i v e mass changes sign. Expectedly, as shown in 6.1 and 7.IV, t h i s negative e f f e c t i v e mass anomaly also produces unusual results for s i z e resonances. If one keeps the wave function continuous and smooth (i.e. (69) i n effect) , the e l a s t i c width turns out to be negative. Further, when absorption i s turned on, the 5 0 This i s very s i m i l a r to our size resonance problem. s 1 However, they further f i n d that by making pion-nucleon interaction range f i n i t e ( r e c a l l that (37) i s associated with zero-range interaction) the number of bound states becomes f i n i t e . Calculations by de Takacsy [35] show similar r e s u l t s . 99 resonance gets narrowed. Both of these odd e f f e c t s are removed when the required kink due to (70) i s put into the wave function. I t must be pointed out that, however exotic the anomalous e f f e c t s may be, they are not by any means genuine r e a l physical phenomena one expects to see i n pion-nucleus inter a c t i o n s . The anomalies are rather consequences of crude assumptions behind the K i s s l i n g e r p o t e n t i a l . Because of t h i s fact, our size resonance r e s u l t s may be robbed^some of their s i g n i f i c a n c e . . The most unsatisfactory aspect i s that one can not separate out absorption cleanly from the pote n t i a l . . f h i s makes the separation of the e l a s t i c width from the absorption width rather a r b i t r a r y . However, i f the po t e n t i a l i s assumed to be phenomenologically correct, the r e s u l t s for the t o t a l width may be more trustworthy.. 100 CHAPTER 6 SIZE RESONANCE ANALYSES A complete size resonance analysis consists of two parts. The f i r s t , obviously, i s to calculate resonance parameters (i.e. resonant energy, e l a s t i c and absorption width) from the given po t e n t i a l . . However, that i s not s u f f i c i e n t . A t h e o r e t i c a l l y calculated resonance i s of any use, and indeed can be j u s t i f i a b l y c a l l e d a resonance only i f i t produces experimentally observable e f f e c t s . Of coarse, a narrow resonance produces rapid variations of cross sections with energy and i t s experimental signature i s clear cut. However, a wide resonance may be completely obscured in cross sections by non-resonant backgrounds. Its observation depends c r u c i a l l y on how and what type of experimental data one analyses. For pion-nucleus size resonances t h i s i s an especially important problem since the \ absorption widtn i s expected to be large due to pion a n n i h i l a t i o n . Therefore, the second part i s to establish whether a calculated resonance i s narrow enough to be observable..This requires finding how and what type of data one should extract for wide resonances. 5 2 Once those are known, calculated resonances can then be tested (with calculated data of course) to determine whether 5 2 Of course, extraction of resonances from data i s of general i n t e r t e s t . But the problem i s not as acute for narrow resonances for which the one-level approximation i s v a l i d and the extraction i s comparatively easy, as shown l a t e r . 101 or not they can be extracted. Towards achieving the two complementary objectives of calc u l a t i n g and extracting resonances, i n the f i r s t three sections of t h i s chapter, we analyse the K i s s l i n g e r potential in the $-matrix formalism. This formalism was o r i g i n a l l y developed by Eisenbud and Wigner [36] for analyzing compound nucleus resonances. I t i s a general formalism and, in pa r t i c u l a r , Vogt [37] has used i t for size resonance studies. The formalism needs to be reproduced here since, f i r s t , the unusual d e f i n i t i o n of probability density (which i s closely t i e d to orthogonality relations) associated witn the Ki s s l i n g e r potential demands considerable modification; and second, the need of pushing to the l i m i t of extracting very broad resonances requires d e t a i l s of the formalism. Associated with the (^-matrix formalism, we are for our aroad resonances also forced to address the age-old controversy on the choice of the boundary radius defining eigenfunctions..In 6.IV some arguments are given for why the radius must be chosen on the potential surface and i t i s shown that the precise choice i s not crucial..Following that, i n 6.V, aased on the $-matrix formalism we show how size resonance parameters are calculated for a given potential* F i n a l l y , i n 6.VI, we go on to the more important discussion on achieving the second objective- how to extract resonance parameters from experimental data*.First dismissing cross sections to be 102 inappropriate for analyzing wide resonances, then pointing out that the conventional quantity based on phase s h i f t s for resonance analyses, the resonant amplitude, has several shortcominqs which are p a r t i c u l a r l y serious for wide resonances, we a r r i v e at the conclusion that the most appropriate experimental quantity for yielding resonance parameters i s the ^ - f u n c t i o n . . A detailed prescription i s given f o r how the ^ - f u n c t i o n can be calculated from phase s h i f t s . We discuss the many merits the tf?-function has over the resonant amplitude. In p a r t i c u l a r , the fact that the (R-function contains only the absorption width and not the e l a s t i c width makes i t p a r t i c u l a r l y suitable for analyzing wide resonancesi. The results of t h i s chapter for analyzing broad resonances are f u l l y demonstrated in the next chapter.. the S( -function of the Kisslinger Potential In the (^-matrix framework, one characterizes the system in terms of a complete set of eigenstates. Each state i t s e l f i s characterized by i t s energy and a set of p a r t i a l widths. From these parameters, one can construct a matrix, the ^-matrix, which r e l a t e s to experimental quantities. The simplest application of the (R-matrix theory i s found i n the system described by a potential.. In t h i s case, the one channel nature reduces the matrix to a function, the ^ - f u n c t i o n . Further, those parameters which characterize the (^-function (for a complex o p t i c a l p o t e n t i a l , there are three 103 parameters per eigenstate) can be calculated e x p l i c i t l y . , V o g t [37] has applied the <#-function to study size resonances of a simple l o c a l p o t e n t i a l . Now we extend the application to the K i s s l i n g e r potential; I t may already be evident from (69) that extensive modifications ace required since normalization (thus orthogonality between eigenstates) must be carried out with a weighting f a c t o r . For a p a r t i c u l a r p a r t i a l wave, after writing lF= ^ ^ , J t y P ) ' w e obtain from (67) It i s best to i l l u s t r a t e with the absorption part turned off f i r s t ( i .e. set =0). At the o r i g i n , ^ s a t i s f i e s the boundary condition, Iim <p. o as y^+l . Since t h i s equation i s r*0 * second order l i n e a r , we can define a complete set of orthogonal functions between two points by using two-point boundary conditions._ F i r s t , we put (72) i n i t s s e l f - a i j o i n t form ( A O ' - A [ ^ f e ^ Q ^ f l c ) + j f e Q * - ^ . . ^ ] f y * 0 (73) Now l e t s a t i s f y the same equation of eigenvalues ^x' s 104 with the two-point boundary condition where R i s a radius and Hp i s a r e a l number. Note that i n order to define a complete set of eigenstates witn r e a l eigenvalues 9p i s required to be r e a l . 5 3 Both d and ^ are completely a r b i t r a r y so f a r . Their natural prescriptions are given l a t e r . . Multiplying (73) by T/fcx and the complex conjugate of (74) b y ^ , subtracting and integrating from the o r i g i n to € , we f i n d A ( « n £ w If/fc)- Qjl*) 3 - (fe^ - ) J A M M H ^ W J K (75) 0 The orthogonality r e l a t i o n between the ^ » s i s imbedded i n the above equation. Taking ^ to be one of the eigenfunctions, say Y^y ( A $ X ), we see that the following orthogonality r e l a t i o n holds (provided @p i s a real number) , i A ^ y ^ Y £ V *r*0 (76) 5 3 Kapur and Peierls [38] developed a s i m i l a r theory using complex boundary condition (thus complex eigenvalues) . . i t i s not clear whether the associated eigenstates form a complete set. . See p. 130 of [ 6 ]. . 105 But i n the case \ -\ , we can not normalize "Y^ in the usual fashion. Instead, we have '6 M * i y i v il * e t l <77> where the sign on the EES depends on the sign of i n the region 0<r< R . In case the e f f e c t i v e n a s s , ^ , i s mostly negative, the negative sign i n the above equation then applies. This has serious consequences, as follows. Expanding f ^ t f o r 0<r<£) i n terms of ^ » s, <P,« I C^X^ (78) where C^are the c o e f f i c i e n t s , we find According to (75) , the above can be written as « V tty'w- t n i a ) ( 8 0 ) kh - £ At r= R, we then obtain (81) 106 Therefore* the logarithmic derivative a t ^ c d a ^ e written as (82, where i s the so ca l l e d $ - f u n c t i o n . According to the above* i t i s given by where The quantity fyyis c a l l e d reduced width. 5 4 M u l t i p l i e d by the penetration factor (see (115,) i t gives e s s e n t i a l l y the e l a s t i c width. Since the penetration factor i s always po s i t i v e , the above shows that the e l a s t i c width i s negative when the e f f e c t i v e mass i s negative inside This i s a puzzling r e s u l t . Of course, when the e f f e c t i v e mass i s positive, there i s no such problem. 5 4 ft*/ Vogt [37] defined the reduced width d i f f e r e n t l y , rather it, in order to*obtain for i t the unit of energy. The penetration factor i s then multiplied by the radius to compensate for t h i s modification.. This problem can be naturally removed by considering how the d e f i n i t i o n of probability density must be modified when the e f f e c t i v e mass changes sign. After a l l , orthogonality relations are t i e d closely to the d e f i n i t i o n of probability density. Recall from 5.IV, i n p a r t i c u l a r (69) and (70) , that i n solving the Schroedinger the eguation, one must put a kink into the wave function where Jl^ changes sign. . In our derivation above, t h i s fact has been ignored. S p e c i f i c a l l y , since the orthogonality r e l a t i o n , (76), applies only to a smooth solution, i t must be modified when the kink occurs. As shown in Appendix IV, and not s u r p r i s i n g l y i n accordance with the d e f i n i t i o n of probability, the replacement i s r « i / j * i M x . * = » < > 0 Now, instead of (77), Y^xcan be normalized by j A 1(1*1 l ) M " » 1 0 Following the same steps as from (78) to (81) , one can derive (82) and (83) again, but with (84) modified to (87) which i s always positive. Notice that " l ^ does not merely (85) (86) 108 change by a sign since is necessarily d i f f e r e n t because of the kink. I t i s g r a t i f y i n g to see that our consideration of the d e f i n i t i o n of probability density and current removes the embarassing r e s u l t of a negative e l a s t i c width. I t i s i n t e r e s t i n g to note that, although the Kisslinger potential d i f f e r s much from the simple l o c a l potential i n (50, , the expression f o r i s quite s i m i l a r . Eq. (87, can be written as the pr o b a b i l i t y density divided by the e f f e c t i v e mass at £ , as opposed to simply the pr o b a b i l i t y density divided by the mass in the case of (50,. Our consideration so f a r f o r a r e a l potential shows that the ifi-function i s a sum of terms, each of whicn i s characterized by two numbers, the eigenenergy and the reduced width. For a complex o p t i c a l potential, the r e s u l t s are i d e n t i c a l except each term i s characterized by one additional number which measures the absorption in the potential. The approach for deriving the f u l l $-function i s e s s e n t i a l l y the same as above. One constructs a complete set of eigenstates and expand the wave function in terms of i t . We s t i l l use the set Xj^' s since 6^ has to be real to s a t i s f y completeness. As before, writing (72, i n i t s s e l f - a d j o i n t form, , (88, 1/ * and multiplying the above by JC^ and the complex conjugate of 109 (74) by ^ JL , subtracting and integrating from 0 to Q. , we f i n d where Again, in the above, the sign change in ^ £ i s taken care of by (jj^jand the associated kink i n the wava function. Following the same steps as from (78) to (81) , one finds (82, again with the $ -function modified to Eg. (90, needs some comment;.. The absolute sign f o r jl ^ i n that expression follows from requiring a kink i n the wave function where fl^ changes sign according to (70) . . This factor i s not necessary i f one uses (69, as the current d e f i n i t i o n which requires a smooth wave function regardless of the sign change i n jig. However, this woald give a negative e l a s t i c width when /^ R i s mostly negative inside the nucleus, as we have shown already..Now l e t us investigate for t h i s case the sign of W/jjj with and without the required kink i n the wave function; Without the absolute sign around jf^ in (90) , (^pis 1 10 negative. This i s e a s i l y seen. .For weak absorption the denominator in (90) i s approximately -1. The numerator i s positive since ^<0 , &j<0, .^>0 and fig- fic-<0.ss With the absolute sign around (corrasponding to having the kiaic i n the wave function) , i s s t i l l negative since both the numerator and the denominator change sign. . Therefore, according to (117), the absorption width i s positive regardless of the kink.. However, without the kink, the e l a s t i c width i s negative. This i s opposite i n sign to the absorption width and gives r i s e to resonance narrowing by absorption- another odd result associated with negative . ef f e c t i v e mass when the required kink i n the wave function i s missing. The $ -function so far i s merely a quantity made of parameters which characterize the potential. I t has no experimental s i g n i f i c a n c e yet*. However, from i t s r e l a t i o n with the logarithmic derivative, (82) , i t can be related to cross sections. In f a c t , since (91) and (82) are i d e n t i c a l to the r e s u l t s for a simple l o c a l potential, the resonant cross ss We ignore the &£ term i n (90) . Also when $g i s positive everywhere, ^U> i s s t i l l negative. The denominator now i s approximately +1. The numerator, however, becomes negative since /fc >0, <S>r<0, fo>0, and kl-Qe-Qc>0. Notice that j^-fiR-fit changes sign with /}r since fe*"— ftp.- fic) must be positive in order to produce o s c i l l a t i n g wave functions for resonances. 111 section for the K i s slinger potential has the familiar Breit-Wigner shape (sometimes c a l l e d the Lorentzian snape). A l l the complications i n the Kisslinger potential have been absorbed into ^ and Uj^. 6.II. From the $-function to Cross Sections In t h i s section we show how to r e l a t e the ^ - f u n c t i o n to experimental cross sections, and consequently see (in the following section) the s i g n i f i c a n c e of those eigenstate parameters.. Since (67) i s second order l i n e a r , i t s solutions can be written as a linear combination of two independent solutions. However, since ^f. has to s a t i s f y the boundary condition at the o r i g i n , Uro ty. f"^{ r there i s only one a r b i t r a r y constant;.We therefore:write (up to a normalization mu l t i p l i c a t i v e factor) ^ ( 0 - L OtLr) (92) where and Oji denote any two independent solutions. The c o e f f i c i e n t C, i s to be determined such that the boundary condition at the o r i g i n i s s a t i s f i e d . I t i s straight forward to derive Ji BL. 0. In parti c u l a r , at the radius £ , we can r e l a t e .. ^  . to the 112 H -function via (82). Therefore In general, ^ and 0^ are not known. On the other hand, their asymptotic behaviors at large distance are well known. In p a r t i c u l a r , i f we choose and 0, such that they have f~ respectively the asymptotic behaviors of incoming and outgoing waves, i . e . IT lim I, -7 e where (f and if are the usual Coulomb phase s h i f t and parameter respectively, then the c o e f f i c i e n t C i s c a l l e d the c o l l i s i o n function and i s related to the usual nuclear phase s h i f t bysfi 5 6 Note that C (or IJL) can be evaluated v i a (93) at any radius,, although only at large distances where and Op s a t i s f y (95) that \jL takes on the s p e c i a l meaning of phase s h i f t . Notice in particular that Cg, and Op are f u l l solutions, not free solutions, of (72). They of course behave l i k e free solutions outside the potential range (in p a r t i c u l a r l i k e (72) asymptotically). I t i s commonly stated that the phase s h i f t i s obtained by matching the wave function with free solutions asymtotically. This matching i n f a c t can be done anywhere with f u l l solutions, and fy. ._This point i s important for the discussion i n 6.IV on the choice of £ for defining eigenfunctions. 113 t - £ * (36) We therefore obtain j j i M t l ^ L ^ M . J i L B ) (97) The above expression can be s i m p l i f i e d further by using another set of independent solutions, defined as follows, (98) These are ,of course, the regular and the ir r e g u l a r solutions. We then have j where The numerator i n the above expression f o r i s the •Wronskian' of (73) .. Therefore, i t multiplied by ^(f)equals to a constant. We evaluate t h i s constant at large r by using the asymptotic behaviors of , ^  , and AM: g , - * Cos ( fer- +(T. - f JUafcr) Or--)cO ^ a. «* This constant i s found to be fe. . Therefore ^ i s simply p - 1 2 ^am ( 1 0 2 ) Now the c o l l i s i o n function i n (97) can be s i m p l i f i e d to where (•+£<?,- ^ c r 2 ) ] i R A - r p ^ ) ^ corresponds to the in t e r n a l part (which contains resonance behavior) of the c o l l i s i o n function and e*'*f/>_ V g)-'7>(ft) (105, corresponds to the external part (which e s s e n t i a l l y equals to the hard-sphere c o l l i s i o n function, see 6.IV). The relationship between the c o l l i s i o n function and cross sections i s usually quite straight forward. However, the Coulomb scattering introduces much complication due to i t s i n f i n i t e range. This complication, however, i s not 115 central to seeing the s i g n i f i c a n c e of the ^ -function and for si m p l i c i t y we use usual expressions. 5 7 The d i f f e r e n t i a l cross section i s given in terms of the c o l l i s i o n function as aft where ~ Z(2f+0 ± - A(W) (106) The p a r t i a l wave e l a s t i c cross section i s given by which we simplify i n terms of two amplitudes (-07) The f i r s t term in J| i s the external amplitude and the second term, we s h a l l see, i s the resonant amplitude. From (104), 5 7 Notice that o^ , in (96) i s the nuclear phase s h i f t . Asymptotic solutions i n (95) contain Coulomb d i s t o r t i o n s . Complications due to Coulomb scattering in rela t i n g the c o l l i s i o n function to cross sections are explained in, for example, p.2o9 of [39]. These complications are important in extracting (nuclear) resonance parameters from experimental data. See 6. VI. 116 the resonant amplitude relates to the (^-function by 1108) The expression for p a r t i a l wave reaction cross sections rs This means, contrary to {107,, one does not have to contend with the external amplitude i n dealing with reaction cross sections. This proves to be advantageous i n extracting resonance from reaction rather than e l a s t i c scattering data. For l a t e r use, we write the p a r t i a l wave reaction cross section i n terms of the resonant amplitude which. functions at £ ( i f £ i s outside the potential range,, i s reduced to (109, (110, 6.. III-. Eigenstates and Resonant States The s i g n i f i c a n c e of the -function in (108,, the 117 resonant amplitude, becomes apparent when the energy i s near a pa r t i c u l a r eigenlevel, say j ^ . Again, we i l l u s t r a t e with no absorption f i r s t . In the one-level approximation, 6^(Q,)^ ^? f (108) becomes . t fh.u _ zP^mfLo .mi) This expression shows that the e l a s t i c cross section has a resonance p r o f i l e at +1 fyW]^5 . However, we must s t i l l make an appropriate choice of the boundary condition number, Djt , since the one-level approximation may not be valid at the resonant energy (£^+[^-^00]^) i f [fy- ^ 0 i s not small compared to the l e v e l separation. . In other words, the boundary value i s not a r b i t r a r y i f one wishes to use the one-level approximation to describe a resonance.. The natural choice of $f, which makes the one-ievel approximation v a l i d for describig the resonance at Bj^0 i s fyi* ^ C Q - | 0 i R > 4112) Physically, t h i s boundary condition corresponds to matched impedance for waves incident on a cavity, as explained by Vogt [37], Since i t s inception the (f?-matrix theory was plagued by the ar b i t r a r i n e s s in the boundary values used to define resonant states u n t i l i t was removed by the above i 118 choice. Notice that, out of the set of eigenstates, only one (the one s a t i s f y i n g (112)) corresponds to a resonant state. It follows from t h i s special choice of the boundary value that near the eigenenergy, * we can write [@^- b^O*)]^ * fy0l£M*$) where if HO (113) Eg. .(111) i s then s i m p l i f i e d to the Breit-Wigner one-level amplitude, e A » C&! , 4 l 1u> where n X ft tf) <VQ This means cross sections near have a resonance p r o f i l e of f u l l width f j ^ . T h i s width i s es s e n t i a l l y a product of two factors. The penetration factor, ^  , as i t s name implies, determines the ease with which the p a r t i c l e penetrates through b a r r i e r s from £ to the outside. I t depends on the incident energy as well as the height and the range.of bar r i e r s . The other factor, (the reduced width), as explained already, i s proportional to the probability of 119 finding the p a r t i c l e at£.. Substituting (114), which refers to no absorption, into (110) one obtains zero reaction cross section, of course. I t i s straight forward to derive the equivalent of (114, with absorption^ restored in the potential. Again, taking a single l e v e l from (91, and using the special boundary condition, (112,, we f i n d (114,, the one-level resonant amplitude, i s modified to Pi:h>Y.\ Coo JL ^ _ _ . _ . (116, ( where ^HO (117, This result means that absorption introduces a l e v e l s n i f t , and an additional width, I go, into the resonance p r o f i l e . . From (107, , the e l a s t i c scattering cross section in. the one-level approximation i s then I T & ^ ° | i « ) B.e.«,.w«l..rJ'.vJ " 1 8> The resonance p r o f i l e i s distorted by the external amplitude. In comparison, from (110,, the reaction cross section i n the one-level approximation becomes There i s ao d i s t o r t i o n at a l l from the external amplitude., One difference between applying the one-xevel approximation to (83) and (91) must be emphasized. The one l e v e l approximation i s always val i d f o r (83) so long as £g0-C" i s small compared to l e v e l separations. However, fo r i 9 i ) , the approximation may be i n v a l i d even at ^ = £ when Vi^ i t s e l f i s comparable to l e v e l separations. This means, for a r e a l potential, there i s always an energy region around Er^ at which the cross section i s resonant l i k e . For a complex potential, on the other hand, the resonance may be completely washed out by absorption. The c r i t i c a l question as to wnen a resonance i s no longer recoqnizable i s discussed i n 6.VI.. 6. IV. .ambiguities i n the Boundary-Radius In the o r i g i n a l $-<-matrix framework, there were assumed to be two regions i n nuclear reactions- an i n t e r a c t i o n region and a free region- separated by an abrupt boundary. In the •natural 1 choice of t h i s separation radius as the boundary radius, R , defining eigenfunctions, the external phase s h i f t then simply equals to the hard-sphere phase s h i f t and represents a sudden hard-sphere r e f l e c t i o n of incident waves 121 at the interaction boundary. 5 8 This hard-sphere r e f l e c t i o n associated with reactions has long been taken to x>e an unsatisfactory aspect of the -matrix framework..Vogt et a l [37,40] have demonstrated s p e c i f i c a l l y the inadequacy of t h i s (supposed) feature of the framework f o r the Saxon^tfood potential (which has an extended edge). They constructed an •equivalent square well' and showed how i t s external quantities d i f f e r from that of the Saxon-Wood well. However, the r e s t r i c t i o n to an abrupt boundary c e r t a i n l y i s not necessary for the i^-matrix framework. In f a c t , our presentation in the previous three sections has shown t h i s point e x p l i c i t l y - the external phase s h i f t (or any other external quantities, e. g. . or Pp ) i s given i n terms of Lp and 0^  (or Xp and %p) , which are f u l l (in contrast to free) solutions of (72). In p a r t i c u l a r , when £ i s on the inside of a diffuse surface edge, r e f l e c t i o n and penetration at that point no longer resemble that of a hard-sphere because £p and 0, d i f f e r from free solutions there. It i s not clear why the For an abrupt surface, the values of Lp and Op (or xp and $(U) on the surface egual to free solution values, thus the external phase s h i f t equals to the hard-sphere phase s n i f t . Other external quantities, such as bp and fy, » then of course also have hard-sphere c h a r a c t e r i s t i c s . Free solutions of (72) are meant to be solutions without the nuclear potential i n the equation. Therefore, s t r i c t l y , for charqed pions the hard-sphere phase s h i f t actually means the phase s h i f t of a r e a l hard sphere dressed with a Coulomb • t a i l " . 122 abrupt boundary has so often been assumed to be part of the i$ -matrix framework. . Perhaps the misconception that the phase s h i f t , as defined by (93, and (96,, means matching the wave function with free solutions has caused i t . . (See also Footnote 56 for t h i s point., Certainly since Cp and Op in (93, are actually f u l l solutions there i s no need at a i l to keep R at radius so large that ^ and Op correspond to free solutions, thus eliminating the need of an abrupt boundary.. The above c l a r i f i e s one old (but s t i l l common, misconception of the $-matrix formalism. However, there s t i l l remains one major problem. I t appears that, within the formalism i t s e l f , the radius ft can be chosen anywhere- t h i s i s true even in the case of a square well. S p e c i f i c a l l y , as shown i n the previous three sections, t h i s means that any choice of £ can define a complete set of eigenstates and one of them (the one s a t i s f y i n g (112,, i s supposed to represent a resonant state.. This feature of the formalism i s c l e a r l y unacceptable. The way to c l a r i f y t h i s ambiguity i s through physical arguments... The c l a r i f i c a t i o n i s based on the idea that the d i v i s i o n of the t o t a l phase s h i f t i n t o two parts- the resonant and the external part- should make physical sense. It can be established that the external phase s h i f t corresponds to 123 describing r e f l e c t i o n . 5 9 Since size resonance phenomenon refers to the penetration and the confinement of the incident p a r t i c l e inside a p o t e n t i a l , i t i s natural to set R on the potential surface: so that r e f l e c t i o n -occurs mere. Subtracting the external phase s h i f t then corresponds to eliminating surface r e f l e c t i o n , which i s unrelated to the resonance phenomenon, from the t o t a l scattering of the incident wave.. Not choosing £ on the surface amounts to an incorrect separation between resonant scattering and surface r e f l e c t i o n . . It i s unambiguous where surface r e f l e c t i o n occurs for a square well..However, r e a l i s t i c nuclear potential surfaces have diffuse edges and the choice of i s not as well defined. However, the following i l l u s t r a t i o n shows that the exact choice of £ i s not c r i t i c a l when the surface i s not too extended. A change in R by £V would displace the resonant energy by "(j^^^V4*^, . comparing t h i s witn the resonance width given i n (121), we find that —~- = / -&bf.tb N( cX \ m T O evaluate the above, over the surface region l e t us approximate and 0^ , by free solutions, r. e. approximate ^^by the hard-sphere phase s h i f t . As an example, for the "if°-Pb jl=0 system f o r which i ^ - ^ f e r and^?C«sfe^ , we obtain - . _ A t y p i c a l surface thickness i s 2 fm. This Via 2 5 9 This i s easily seen for a hard sphere. 124 jo means, say for a 15 Mev "if , the resonant energy wouid be displaced by 30% of the e l a s t i c width when the boundary radius i s changed by as much as the surface thickness. So we conclude, over the surface thickness, the change in the resonant energy divided by the e l a s t i c width (this r a t i o i s given by half of the change in the external phase snift) remains well below unity. 6 <> The most uncertainty the dif f u s e surface can give to the resonant energy i s well below the e l a s t i c width i t s e l f . I t i s not c r u c i a l to make an 'exact' choice for 9- as long as i t i s on the surface. We emphasize again that £ i s chosen on the surface for the purpose of describing resonant scattering. In general, there i s no a p r i o r i reason within the ^-matrix formalism i t s e l f f or choosing R there. For example, even i n the case of a square well, one may choose £ anywhere and obtain two parts of the phase s h i f t . . O f course, when £ i s not on the surface, then the external part does not correspond to surface r e f l e c t i o n and the i n t e r n a l part does not correspond to resonant scattering either. Nevertheless, the (^-function may s t i l l be. described by a set of eigenstates.. In par t i c u l a r , at the eigenenargy s a t i s f y i n g the no-shift condition, (112), the one-level approximation i s s t i l l v a l i d . 6 0 Of course, when the surface i s s u f f i c i e n t l y extended that the r a t i o i s no longer small, then the (2-matrix formalism i s ambiguous for describing resonant scattering. 125 Certainly, however, t h i s energy can not be recognized as a resonant energy since the separation of the to t a l phase s h i f t i s not physical, as far as for describing resonant scattering i s concerned, when R i s not on the surface. Incidentally, the approximation that -f^ and §p are free solutions gradually breaks down as R approaches inward. However, i t can be established that the r a t i o ^./^ remains close to the free solution value because the departures of and from t h e i r free values are similar. Since the external phase s h i f t i s a function of t h i s r a t i o , i . e . ^ ta«' fatyf) * t n e hard-sphere phase s h i f t ( i . e . wita free and 9/1 ) i s a good approximation to the external phase s h i f t even for R considerably i n s i d e the p o t e n t i a l . 6 1 On the other hand, other external quantities such as and ft may change appreciably from free solution values. At any rate, i n our c a l c u l a t i o n (see 6.7), a l l external quantities ( included) are evaluated using exact values of A, and0., instead of free solution values.. 6..V. Calculation of Resonance Parameters Now we are ready to state how to calculate resonance parameters of a p o t e n t i a l . . F i r s t , one turns off absorption 6 1 Often, the external phase s h i f t i s simply c a l l e d the hard-sphere phase s h i f t . However, i t should be distinguished from the true hard-sphere phase s h i f t . . 126 mechanisms and obtain the r e a l p o t e n t i a l . 6 2 Resonance i s defined by (112). That means For the above, of coarse, R is chosen to be on the potential surface. The LHS can be calculated by integrating (73) from the o r i g i n with ^(0)=0 and arbitrary ^ ( 0 ) ; whereas the RHS can be calculated by integrating the same equation backward st a r t i n g with the asymtotic values of §p and Hp (and the i r d e r i v a t i v e s ) , as given by (101), from a large r to Of course, i f the potential i s not a t t r a c t i v e enough, (120) may never be s a t i s f i e d . Next, one calculates the e l a s t i c width from (115).. A l l the required quantities in that expression are already evaluated in solving (120). F i n a l l y , with absorption restored, one calculates the absorption width and s h i f t from (117) where l/U^pis evaluated from (90). Instead of using the wave function i t s e l f , one may use phase s h i f t s which represent asymptotic behaviors of the wave function. I t follows from (114), which i s equivalent to (112), that resonances occur at L ^=(n + 1/2)TT . The in t e r n a l phase s h i f t , , i s calculated by subtracting the external phase s h i f t , ^ b r from the t o t a l phase s h i f t . After locating a 6 2 As explained i n 5.IV, i t i s ambiguous how t h i s should be done fo r the Kisslinger p o t e n t i a l . 127 resonance, one can calculate the e l a s t i c width from P.* " T T ^ - T - H21) which follows d i r e c t l y from (114). The above procedures are carried out, of course, with absorption turned off. As f o r calc u l a t i n g absorption parameters from phase s h i f t s , however, there i s no general way. But under the special condition that the one-level approximation, i. e . . (116), i s v a l i d , one may solve for I £o and treating as known. The result i s Remember that t h i s equation i s applicable only x% the one-level approximation i s v a l i d . The extent to whicn the values calculated from t h i s equation agree with those from (117) indicates the v a l i d i t y of the one-level approximation, or : how well the resonance survives absorption.. This par t i c u l a r question i s dealt more f u l l y i n 6.VI. Often, i n resonance studies, one simply uses the t o t a l phase s h i f t i n place of the in t e r n a l phase shift..However, their d i s t i n c t i o n i s important since i t separates out surface r e f l e c t i o n from resonance e f f e c t . In practice, one may use 128 the t o t a l phase s h i f t f o r convenience provided the e x t e r n a l phase s h i f t i s s m a l l (compared to<^fe) and v a r i e s s l o w l y with energy (compared to f T ^ k I • .. T h e r e f o r e , i n d e a l i n g with broad resonances (such as s i z e r esonances), i t i s u s u a l l y necessary to d i s t i n g u i s h between the i n t e r n a l phase s h i f t and the t o t a l phase s h i f t . 6 . V I . E x t r a c t i o n of Resonance Parameters As mentioned al r e a d y , when the a b s o r p t i o n width i s l a r g e , the one l e v e l approximation breaks down and the o r i g i n a l ' a b s o r p t i o n - f r e e 1 resonance may be completely washed out. For the pion-nucleus system, because of strong pion a n n i h i l a t i o n , i t i s e s s e n t i a l t h a t one examines t h i s aspect. S p e c i f i c a l l y , we must check whether our c a l c u l a t e d resonance, with f u l l a b s o r p t i o n on, can be e x t r a c t e d from experimental data. In t h i s s e c t i o n we i n v e s t i g a t e how and what type of experimental data one should analyse f o r wide r e s o n a n c e s . I n p a r t i c u l a r , we s h a l l present a new method of resonance e x t r a c t i o n . . T h i s method uses the ^ - f u n c t i o n d i r e c t l y and i s s u p e r i o r i n many ways t o the c o n v e n t i o n a l method of a n a l y z i n g the resonant amplitude.. It i s p a r t i c u l a r l y powerful f o r e x t r a c t i n g wide resonances. Experimental data can be o b t a i n e d i n v a r i o u s degree? of s o p h i s t i c a t i o n . The most d i r e c t are c r o s s s e c t i o n s . However, as d i s c u s s e d below, i t i s d i f f i c u l t t o e x t r a c t resonance parameters of broad resonances d i r e c t l y from c r o s s s e c t i o n s . . 129 Under the most favorable conditions, i n which, say, one l e v e l dominates the resonant amplitude (i.e. (116) i s vaxid), and further the resonant amplitude dominates over other amplitudes (the external amplitude and other p a r t i a l wave amplitudes) , then the scattering amplitude, (106) , becomes i t v > - - ( P i , * r?.>/2 Only then, do cross sections ( e l a s t i c , (106) and (107), or reaction, (110)) have simple dependences on resonance parameters and the extraction of these parameters i s straight forward. However, we do not expect t h i s to be true for the pion-nucleus system because of large absorption. The resonant amplitude, f i r s t , i s not given by a single l e v e l ; and second, can not dominate over other amplitudes because of i t s weak energy dependence.. As a r e s u l t , cross sections have complicated dependences on resonance parameters- they do not exhibit simple peaks- which, i n turn, can not be extracted e a s i l y . However, among the cross sections, the reaction cross section i s the most suitable f o r yielding resonance parameters.. In co n t r a d i s t i n c t i o n to the e l a s t i c cross sections, (106) or (107), the reaction cross section in 4,110) depends only on the resonant amplitude. This has the advantage of avoiding undue dependences on the external 13 0 amplitude, and i n the case of charged pions, further complications due to Coulomb scattering (see 6. IV) while extracting resonance parameters. On the other hand, the d i f f e r e n t i a l cross section i s the most unsuitable since each resonant p a r t i a l wave in t e r f e r e s with the other p a r t i a l waves. A good example to i l l u s t r a t e the inadequacy of cross sections for analyzinq resonances can be found in the neutron-nucleus system..As shown i n Fiq.2a, neutron tot a l cross sections show broad 'resonance* l i k e energy variations at about 10 to 15 MeV for medium and heavy nuclei. rhese broad peaks have been attributed t o neutron-nucleus size resonance e f f e c t s for many years. However, Peterson [41J and McVoy [42] noticed that had that been the case the peaks should move toward lower energy for heavier nuclei because of the larger s i z e . Closer examination revealed that the peaks actually correspond to the t o t a l s-wave phase s h i f t passing through an odd multiple of T/2. 6 3 Further, the variation of peaks over nuclei suggests that the t o t a l phase s h i f t r s i n f a c t decreasing with energy at the peak positions,.This More c o r r e c t l y , d passes through the positive imaginary axis since there i s always absorption. This type of peak i s known as Ramsauer maximum, which occurs frequently i n atomic c o l l i s i o n s . I t i s not clear i f reaction cross sections show Ramsauer maxima. I f not, then reaction cross sections are l e s s misleading for resonance studies.. 6 3 131 behavior does not at a l l correspond to Aresonance e f f e c t , but arises as a result of Levinson's Theorem. 6 4 This theorem states that the phase s h i f t at i n f i n i t e energy must return to zero (or -nj| where n i s the number of bound states). Therefore,' i f the phase s h i f t has increased through TT/2 a few times as a r e s u l t of resonances, i t must decrease througn Tf/2 just as many times (or more i f there are bound states). These •returns', appropriately named 'resonance echoes', can give deceiving resonance l i k e peaks. In conclusion, cross sections can either hide r e a l or produce f a l s e resonance peaks. Therefore, i t i s desirable to use more sophisticated quantities such as phase s h i f t s , for finding wide resonances. tde The phase s h i f t has the d i s t i n c t advantage over A c r o s s i section' in that i t no longer has complications due to non-resonant p a r t i a l waves, external s c a t t e r i n g (the external part can be simply subtracted out), and Coulomb scattering.. For example, had phase s h i f t s been extracted from neutron-nucleus cross sections, those 'echo peaks' i n Fig.2a would be i d e n t i f i e d as anything but resonances. In fact, underneath those echo peaks l i e s i z e resonances of higher p a r t i a l waves according to calculations from commonly accepted o p t i c a l potentials [43].. Understandablly, these resonances- hive never been extracted from ';'$ee.t\pn 6 * See p. 195 of [39] for Levinson's theorem. 132 , date!'. because of overwhelming background^ However, i t should become cle a r at the end of t h i s section how these resonances might be extracted from phase s h i f t s . However, the phase s h i f t i s not a d i r e c t experimental guantity. I t must be extracted from cross s e c t i o n s . 6 5 Alternatively, one may regard the reduction to phase s n i f t s as one e f f e c t i v e way of eliminating some of the non-resonant backgrounds i n cross sections. In the following we discuss how to extract resonance parameters from phase s h i f t s . .In fact, t h i s we have already done in 6.IV f o r the s p e c i a l case of weak or no absorption. In that case, the one-level approximation r e s u l t , (116,j, i s v a l i d and resonance parameters can be simply solved i n terms of phase s h i f t s from (121) and (122). However, again for large absorption width, i t i s much more complicated. One must use the f u l l expression, (108), i n which the ^ - f u n c t i o n no longer can be approximated by a single l e v e l . The most obvious, and in fact the conventional, approach in solving (108) i s based on the one-level r e s u l t , (116). One generalizes (116) to 6 5 One extracts phase s h i f t s by decomposing the scattering amplitude obtained from d i f f e r e n t i a l cross sections i n t o p a r t i a l wave amplitudes according to (106). 133 thus e x p l i c i t l y taking higher l e v e l s into account. However, despite i t s seemingly reasonable form, t h i s eguation does not follow from (108). One can, on a phenomenological basis, improve the one-level approximation by . - 5 yJtl * Ao (125) where A^*represents the contribution of higher lev e l s . . The hope i s that, in the v i n c i n i t y of l ^ , , A^ oO varies slowly with energy compared to the f i r s t term and one may solve for resonance parameters from t h i s eguation. An alternative approach, in the same s p i r i t but superior, i s to improve the one-level approximation where i t is f i r s t made, namely the (J^-function. As given in (91) , the {^-function i s a sum of a l l the l e v e l s , One may again single out the interested l e v e l and approximate/ the rest by a slowly varying quantity. That i s , we analyze the ^ -function instead of the resonant amplitude. Of course, one may carry t h i s out provided can be calculated from phase s h i f t s . Recall that 6^ relates to the logarithmic derivative at R via (82), which may be rewritten as 4V H - 6, The quantity s h i f t s , i . e . SL 134 i127) cer t a i n l y can be evaluated from phase ft 1128) where Lp and Op (and their derivatives) can be assumed by free solution values. However, i n order to evaluate the boundary condition, Sp (the logarithmic derivative of eigenfunctions at # ), which i s given by (112) and (100), 6 , b ( p P ) = i ^ ± * l ^ (129) one needs to make an i n i t i a l guess of the resonant energy 9g 0. 6 6 This i s not too serious a shortcoming since f o r any method of phase s h i f t analysis one must choose a suspected energy region f i r s t and applies subsequent analysis during which one probably always require? guessing the resonant energy at some s t a g e . 6 7 Combining the previous four 6 6 The usual assumption that resonance occurs where K cross the positive re a l imaginary axis only applies to weak or no absorption. 6 7 For example, i n solving (125) , one needs to guess £.ft f i r s t . .. 135 equations, one obtains where tj?.^ represents higher l e v e l s and, again, hopefully varies slowly with energy compared to the f i r s t l e v e l near the v i n c i n i t y of E^,. The LHS, of course, can be simply evaluated once phase s h i f t s ara known and the above equation may then ba solved for resonance parameters, ftg and IA|/10. There are several advantages i n using (130) instead of the conventional (125). F i r s t , since (125) follows from (124) which i s not even exact, the quantity A^^may depend on the f i r s t l e v e l and the assumption that i t varies slowly with energy compared to the f i r s t term on the RHS of (125) may not be s t r i c t l y correct.. Second, the parametrs f]^ and f ^ i n (125), i n contrast to tf^o and W^ o i n (130), are energy t dependent through external quantities (see (115) and (117),.. r ^ 0 i s e s pecially energy dependent since i t i s proportional to the penetration factor which increases with energy.. This dependence, as shown i n 7 . I l l , needs to be accounted for and makes (125, more d i f f i c u l t to solve than (130,. Remember that, although and fgo are the e f f e c t i v e widths appearing in cross sections, i t i s TS^ 0 and which characterize the potential and are the quantities involved in a comparison between theory and experiment. Third, and most importantly, 136 the l e v e l width i n (130) equals l/Jj 0 # whereas i n (1<i;>) i t equals to fjio • . . Consequently, i n a plot versus energy what i s resonance l i k e i n j?^ may be simply monotonically varying in • r — • In t h i s respect, a large e l a s t i c width resonance I ( l i k e those of TT ) which o r d i n a r i l y can not be extracted from . H (125) (since /\jx>no longer weakly energy dependent compared to the f i r s t level) may have a chance^from (130). Therefore, the t# -function i s better than the conventional resonant amplitude • : . for displaying and i analyzing resonances, p a r t i c u l a r l y for those of large e l a s t i c width. This res u l t i s natural in the sense now that both t h e o r e t i c a l c a l c u l a t i o n s and experimental extractions refer to the same quantity, the $-function. Perhaps the reason for using the resonant amplitude t r a d i t i o n a l l y i s i t s close r e l a t i o n s h i p with resonance p r o f i l e s i n cross sections. However, there i s no reason not to break away to more fundamental quantities once phase s h i f t s are known. In conclusion, cross sections are straight experimental quantities but are not suitable for yielding resonance parameters. As a r e s u l t , phase s h i f t s should be extracted. However, the conventional quantity associated with phase s h i f t s , the resonant amplitude, i s not the best for resonance analyses. Instead the o?-function, which possesses several advantages over the resonant amplitude, should be used. However, we are not aware of any such work. 137 Phase s h i f t analysis i s a major aspect of any resonance s t u d y . 6 8 Our discussion here is not r e s t r i c t e d at a l l to size resonances only. In pa r t i c u l a r , Wigner*s $-function can always be defined and used to describe the c o l l i s i o n function f o r any resonance phenomenon.69 Consequently, our method of analyzing the (^-function for resonance parameters possesses much poten t i a l f o r general applications. One case come to mind immediately i s the A-resonance i n pion-nucleus scattering,. As reviewed by Hlifner [ u u ] , peaks i n cross sections corresponding to t h i s suspected resonance disappear quickly as the atomic weight increases.7° Certainly t h i s method of phase s h i f t analysis should help to bring out the resonance p r o f i l e s by eliminating the e l a s t i c width whicn by 6 8 For example, i t ranks as one d i s t i n c t topic i n the upcoming (July, 1980) 17 International Conference on Baryon Resonances in Toronto. In general, of course, the l e v e l parameters (Egx, 6jLv» andl^yj can no longer be calculated simply from a potential. However, they can always be defined i f one regards the target as a black box describable by a set of orthogonal eigenlevels.: 7 0 There i s some controversy as to whether peaks i n cross sections at ^ 200 MeV for l i g h t nuclei (no peaks above 2 7 A l ) are a re s u l t of pion-nucleon 4-resonance. As discussed in 3.II, d i f f e r e n t i a l cross sections in these cases resemble those of a black nucleus. . Nevertheless, the A-resonance must per s i s t to some degree beyond the proton nucleus. 138 no means i s s m a l l . 7 1 7 1 The t o t a l width (=the e l a s t i c width) for the A-resonance i n pion-nucleon scattering i s about 120 MeV. CHAPTER 7 RESULTS 139 In t h i s chapter we present c a l c u l a t i o n results obtained for s i z e resonances of the Kisslinger potential. Calculations are carried out according to the procedures l a i d out in 6.V. In order to judge whether these resonances are observable from experimental data, as discussed i n 6.VI, we also simulate extracting t h e i r parameters from (calculated) phase s h i f t s . In the end of the chapter we show some anomalous re s u l t s discussed in 6. I foe negative e f f e c t i v e mass resonances. 7.1. Resonant Energy and E l a s t i c Width Strieker, McManus, and Carr [31] i n t h e i r pion-nucleus scattering c a l c u l a t i o n s summarized a t y p i c a l set of Kisslinger potential parameters which f i t pionic atom data. These parameters are presented i n Table 1. The corresponding potential 'depths' ( i . e . values of &> and fit ) f o r zoap^ a r e also included. The corresponding pion e f f e c t i v e mass d i s t r i b u t i o n , ]%g= (?e "j'>'Bt'"> a r e shown i n Fig. 10. Recall that the sign of /Jg i s c r u c i a l to the c a l c u l a t i o n . In parti c u l a r , where $g changes sign, the wave function must have a kink; otherwise the e l a s t i c width becomes negative. As shown, since Jlfc i s positive everywhere, we expect no such anomaly. However, the potential parameters need not change greatly to make ^ n e g a t i v e - the case we discuss i n 7.IV. We calculate resonant energy for d i f f e r e n t values or &| , 140 keeping a l l other parameters fixed at those values i a Table 1. I t i s ce r t a i n l y desirable to see how resonance results vary with potential depths which, by no means, are well determined..Presented in Fig.11a are calculated resonant energies for various pion charges and p a r t i a l waves, as functions of 0-\ . Reduced widths, tpy,, and e l a s t i c widths, f\)v, corresponding to these resonances are shown in Fig.11n and 11c, respectively. Incidentally, i t i s g r a t i f y i n g to note that values obtained from (121), which calculates e l a s t i c width from phase s h i f t , agree with those from (115), which calculates the e l a s t i c width e x p l i c i t l y from the wave function. .The :fast increase of f j ^ w i t h E ^ , i n contrast to the almost constant tf)L)v, i s due to the penetration factor in (115). As shown i n Fig. 12a, t h i s factor increases witn the incident energy and varies greatly depending on the pion charge and p a r t i a l wave. These are just expected behaviors of p a r t i c l e penetration through potential b a r r i e r s . As a r e s u l t , the IT resonances have extremely large e l a s t i c width. Coulomb and angular momentum barriers are shown in Fig.12b. The penetration factor makes the e l a s t i c width increase rapidly with the incident energy. As shown i n Fig.11a and 11c, corresponding to the pionic atom values of Qi , at resonances occur at energies where the e l a s t i c width i s ^ l e a s t 30 MeV. Since the second l e v e l occurs at only 20 MeV higher i n energy, these resonances are not l i k e l y to show up i n 141 cross sections or even the resonant amplitude, as shown l a t e r . This i s discouraging. However, as mention^alreaay in 6. VI, the <$-function i s oblivious of e l a s t i c widtn i n cont r a d i s t i n c t i o n to cross sections or the resonant amplitude..Its width consists only U^. This means, as shown in d e t a i l i n 7 . I l l , a l l the. resonances shown i n Fig.11, despite t h e i r large difference in e l a s t i c width, have s i m i l a r chance of being found from the function ( i f t h e i r are similar, . This i s most remarkable. Nevertheless, i t i s s t i l l desirable to f i n d a • r e a l ' •the resonance which shows up d i r e c t l y m^cross section. In order to reduce the e l a s t i c width, one needs a more a t t r a c t i v e p o t e n t i a l to lower the resonant energy. According to Fig. 11a and 11c, the e l a s t i c width decreases by .8 MeV for every 1 MeV of additional attraction i n Q.\ . It turns out that i n a paper on possible isospin symmetry breaking in strong interactions, Weinberg [45] estimated s-wave *u*°-n and IT-p scattering lengths (not equal!, to be (131, Putting the maximum of these values into (46,, we find the s-wave potential strength becomes Q,=-.1 fm , This i s anout 142 30 fleV more at t r a c t i v e than pionic atom v a l u e s . 7 2 I t i s clear from Fig. 11 that t h i s value can produce resonances of 10 MeV e l a s t i c width. Similar to the s-wave pote n t i a l , one may also investigate the e f f e c t of the p-wave potential strength on resonance parameters.. Notice that the p-wave potential i s more a t t r a c t i v e i f the corresponding e f f e c t i v e mass i s larger (since ^ wave function o s c i l l a t e s faster) ..Shown i n Fig. 13 i s the pion e f f e c t i v e mass as a function of (Xg . I t i s clear that the optimum value for binding i s p^"'.9. Summarized i n Tanle 2 i s a set of potential parameters Tauscher [46] used f o r f i t t i n g pionic atom data. Compared to Table 1, t h i s set has larger cLg and smaller Ql\ . The corresponding pion e f f e c t i v e mass for sospb i s plotted i n Fig. 10. .Note that for "IT i s even large enough to turn fig negative. Anomalous results associated with negative jl^ are discussed i n 7. IV. Results obtained for t h i s set of potential parameters (for "ft4 andff 0) are presented i n Fig.14. Resonant energies, compared to Fig.11a, are lower for the same value of Ql\ , 7 3 However, for the pionic atom value of &| , the least e l a s t i c width i s 26 MeV- s t i l l too large ' for direct observation i n cross 7 2 In f a c t , i t i s opposite in sign to the pionic atom value. Could the sign i n (131) be wrong? 7 3 This i s due to both smaller Q| and larger ftg . 143 sections. We considered Pb so far..One might wonder i f resonant energy (thus e l a s t i c width) might be reduced for l i g h t e r nuclei, i n spite of their smaller p o t e n t i a l s i z e , since they 4 have the advantage of lower Coulomb repulsion (for IT J . In addition, binding may be enhanced since the isovector term Cj makes larger (for 17*, opposite f o r "If" ) , thus larger, due to smaller neutron excess. Fig.10 shows a comparison between the It e f f e c t i v e mass i n 2 0 8 P b and 3 8 N i . These conditions favorable to bind If* on l i g h t e r nuclei, however, do not seem to improve the s i t u a t i o n at a l l . . For example, i t i s shown i n Fig.15a that for the same potential parameters the ji =1 IT*" resonances occur at even lower n -I energies i n Pb than in Ni for fel,< -.2 fm . Even when the resonant energy i s lower in Ni, a larger e l a s t i c widtn i s always obtained anyway because of the reduced Coulomb barrier, as shown i n Fig.15b. Therefore, as f a r as reducing the e l a s t i c width i s concerned, heavy nuclei afford more favorable conditions. 7.II. Absorption Width For the absorption width one f i r s t calculates from (90).. They are i n the order of -15~1i MeV, as presented i n Fig.11b and 14b. The absorption width i t s e l f , I relates to Wj^by (117).. Since 6 i s found to be small (maximum i n the order of .3), I i s i n the order of 30 MeV. This large value, 144 as shown i n 7 . I l l , e f f e c t i v e l y wipes out any resonance profile" i n cross sections; and to a large extent, even i n the $ - function. Shown in r i g . 16 are the imaginary potential terms contributing to the absorption width as given by (90). I t i s clear that {l^ ( fel-Qg-flt ) i s the most important term. Notice that kl-Q.g~Q~6 i s a constant for a p a r t i c u l a r type (characterized by p a r t i a l wave and pion charge) of resonance, as shown by the straight l i n e graphs i n Fig.11a and 14a; and has a higher value (for the same pion charge) the lower the l i n e s are. This i s why i s almost a constant ror a particular type of resonance; and i s larger f o r nigher p a r t i a l wave, as shown i n Fig.14b. Note also that the value of [}i depends on both the re a l and the imaginary part of # . In Fig. 13 we plot J}% as a function of tfg with oit fixed at .1, As shown, a t 9 , fi% i s p a r t i c u l a r l y s e n s i t i v e to In fact, between Table 1 and 2, although #g merely changes from .84 to .95 (for Tf° ) , [If changes by three times and consequently, as shown i n Fig. 11b . ' o -t and 14b, Wgx (for the Tf dO resonance at ft=-.15 fm ) d i f f e r s by about 1.5 times..Therefore, we note that although both sets of potential parameters are supposed to f i t pionic atoms, they produce quite di f f e r e n t r e s u l t s (especially for absorption) for size resonances. Pionic atoms perhaps are not as sensitive to o p t i c a l potentials as si z e resonances are. 145 7.III. Resonance Extraction from Phase S h i f t s The c r u c i a l question for our calculated resonances, i n view of t h e i r large widths, i s whether they can be found at a l l from experimental quantities.. We already discussed i n o 6. VI the method of analyses. Now we pick the i^iO, Q,(=-.1 fm ) resonance of Fig. 14 for i l l u s t r a t i o n . We plot i n Fig.17a the Argand diagram showing the v a r i a t i o n of phase s h i f t over energy for different absorption strength, \ . 7 4 The corresponding resonant amplitude squared, p . — — , i s ' i plotted i n Fig.17b. I t i s discouraging that resonance p r o f i l e s quickly disappear as increases. Of course, then, cross sections have no hope of showing any resonant variation. On the other hand, as shown i n Fig.17c, the quantity,i$p\, p e r s i s t s i n keeping i t s resonance shape with absorption broadening much strongly. This confirms the sai d superiority of the $-function over the resonant amplitude i n displaying and analyzing resonances. However, to be f a i r to the resonant amplitude, part of i t s non-resonant energy dependences can be removed. As shown in (108), the resonant amplitude contains external quantities in addition to the (^-function, in p a r t i c u l a r the penetration Argand diagram usually refers to t; where ty. i s the t o t a l phase s h i f t . S t r i c t l y , however, one should use the resonant part of the phase s h i f t , L^, , to demonstrate resonance. 146 factor, fy# appears in both the numerator and the denominator of the resonant amplitude. Since Pp. i s strongly energy dependent, as .shown by Fig. 12a, i t di s t o r t s " the resonance p r o f i l e associated with the resonant amplitude.. This d i s t o r t i o n should be accounted for when extracting information from the resonant amplitude, i . e . when solving (125). However, better s t i l l , the penetration factor in the numerator of (108) can be removed beforehand. That i s , one analyses . , instead of : , for resonance h i parameters. Shown i n Fig.17d i s fy^fi? f i L _ — ~ | . Indeed, resonance p r o f i l e s ceemerge. . This demonstrates well the e f f e c t of non-resonant type of energy depences in the resonant amplitude.. If possible, these non-desirable dependences should be removed beforehand. However, for (108, , that can only be done for the penetration factor i n the numerator. Notice that the reemergence of resonance p r o f i l e s in Fig.17d does not mean the d i s t o r t i o n due to the penetration factor i n the denominator of the resonant amplitude i s unimportant. . In fact, as shown l a t e r , t h i s d i s t o r t i o n can be just as overwhelming as the penetration factor i n the numerator when the e l a s t i c width i s no longer dominated by the absorption width. In contradistinction to the resonant amplitude, the (^-function i s given e n t i r e l y 'py resonance parameters with no d i s t o r t i o n of any kind,.In t h i s respect, once again, the 147 (^-function i s superior to the the resonant amplitude. However, the r e a l power of the $ -function over the resonant amplitude l i e s i n that the only width or the (^-function i s the spreading width, fc^. As mentioned several times already, t h i s means the $ - f u n c t i o n can handle resonances of large e l a s t i c width. We choose from Fig.11 a the (TTpO, 9^ = 15 MeV) resonance, which has an e l a s t i c width of 45 MeV, and plot i t s ItK^ I in Fig.18a. As expected, but s t i l l i n c r edibly, the energy variation i s quite s i m i l a r to Fig.17d. In f a c t , s i n c e i s larger i n t h i s case, the resonance behavior in |#jf i s even more pronounced. In contrast, as shown in Fig.18b, the resonant amplitude shows l i t t l e resonant v a r i a t i o n even for no absorption. Now we make a l i t t l e digression to return to the discussion on d i s t o r t i o n s caused by the penetration factor in the resonant amplitude. . In contrast to Fig.17b, Fig. 18b shows that the resonant amplitude (magnitude squared) decreases monotonically with energy when the absorption strength increases.. This behavior can be explained as follows. In the case of Fig.17b, since the absorption width i s larger than the e l a s t i c width (25 MeV to 10 MeV), the e l a s t i c width i n the denominator of the resonant amplitude i s r e l a t i v e l y unimportant. As a r e s u l t , with increasing energy, the resonant amplitude (magnitude squared) increases due to the penetration .factor i n the numerator. However, t h i s i s no 148 longer true i n the case of Fig.58b since the e l a s t i c width i s larger than the absorption width (45 MeV to 25 MeV). The penetration factor in the numerator no longer dominates over energy dependences. In fa c t , as shown i n Fig. 18c, when one removes the penetration factor in the numerator the resultant amplitude (magnitude squared) decreases d r a s t i c a l l y with energy as a resu l t of the penetration factor i n the denominator of the resonant amplitude. So again, we have i l l u s t r a t e d that the resonant amplitude i s infested with non-desirable non-resonant type of energy dependences, i t i s r e a l l y not the best quantity for extracting resonance parameters. So far we i l l u s t r a t e d many advantages of the ^ - f u n c t i o n over the resonant amplitude for resonance analyses. Now using the calculated6sj,lfc), we simulate extracting resonance parameters from i t . For the resonance i n Fig.17a, we solve (130) for Wjjo treatingQpao as a constant. The obtained W^ o are then converted to p*0 by (117).. As shown i n Fig . 19, there i s good agreement with actual values throughout. As a check, the extracted ()<^ o^ i s found to be close to the calculated second l e v e l amplitude,-— ;— . Also shown i n Fig.19 are values of J g 0 found from the one-level approximation, (122) . As expected, there i s agreement only for weak absorption, < .1. We conclude that the Kisslinger potential, with parameters determined from pionic atom data, i s deep enough 149 to support pion-nucleus size rssonancss.. However, botn the e l a s t i c width and the absorption width are i n the order of 30 MeV. Those values are too large to produce any resonant effects in cross sections. Nevertheless, i t i s possible by using a non-conventional method of phase s h i f t analyses, s p e c i f i c a l l y finding the $-function, one might be able to extract resonance parameters. 2 i I Y i Anomalous Resonances In t h i s f i n a l section we show anomalies predicted when the e f f e c t i v e mass term, jig, becomes negative. Already shown in F i g . 10 i^that, for the set of potential parameters in Table 2, the IT e f f e c t i v e mass i s negative. However, since most of our r e s u l t s are f or Tf° # for comparisons we s h a l l discuss negative mass effects in terms of TT° also. In fact, for another set of potential paraneters Tauscher [ 46] used for f i t t i n g pionic atoms, as presented i n Table 3, one merely needs to change the Cq term from .49 fm^ to .51 fm^ i n order to make the IT0 e f f e c t i v e mass negative. The corresponding /?g i s plotted in Fig. 10. F i r s t , we do the usual- keeping the wave function continuous and smooth everywhere. Again, we calculate resonant energies for various values of fit . The re s u l t s are shown i n Fig.20a. I t i s peculiar that the curves are i n the 'wrong* order when compared with those i n Fig.11a or Fig.14a. However, t h i s i s just the expected behavior since larger 150 actually correspond to a mora a t t r a c t i v e well when (1% i s negative (since flpQ.\ appears as the e f f e c t i v e potential). Because of t h i s reverse ordering, the number of resonances depends on how positive (or repulsive i n the common sense) (J, i s . As shown, there i s no resonance (for j2=0) unless Q( i s -1 -7. above 0.14 fm , and only one unless Q\ i s above 0.36 fm . This reverse ordering also applies to curves of bound states. In that case, any horizontal l i n e intersects i n f i n i t e number of curves i n the negative energy region, corresponding to an i n f i n i t e number of bound states for any value of Q| . Shown i n Fig. 21a i s the phase s h i f t (the resonant part) versus energy for one of the resonances. Just as expected, the phase s h i f t decreases with energy at resonance, thus giving r i s e to negative e l a s t i c width. We further notice that the magnitude, of the e l a s t i c width, as shown i n Fig. 20n, i s extremely small compared to that of the positive e f f e c t i v e mass case. This i s due to the extremely large y?g at the nuclear surface (see Fig.10) that makes penetration d i f f i c u l t . This fa c t i s born out by the wave function shown in Fig.23 for one of the resonances. It resembles the wave function of a bound state. Anomalous results also show up for absorption effe c t s . As discussed following (91), without the required kink i n the wave function, the e l a s t i c width and the absorption width have opposite sign.. This shows up most impressively i n 151 Fig.21b which shows the resonance p r o f i l e being narrowed by •absorption*. The reason foe using the small absorption f actor, ^ =. 001, i s to keep Vi^ not too large compared to oV^ . Otherwise, the resonance p r o f i l e quickly disappears..Even though the nucleus i s i n i t i a l l y creative at small r according to (60) which states that the nucleus i s absorptive as a whole at^=1, the nucleus must eventually change to be absorptive as ^ i n c r e a s e s . Indeed, as shown i n Fig. 22, t h i s occurs at ^ =0.2. Now l e t us see how some of the anomalies get to be removed by demanding that the slope of the wave function changes sign at where /ig^O, i n accordance with our discussion on keeping the current in (70) continuous at that point. .This i s demonstrated i n Fig. 23 where we show s p e c i f i c a l l y the required kink i n the wave function. Implementing t h i s modification, we show in Fig.24 new r e s u l t s of phase s n i f t s (the resonant part) versus energy. Sure enough, as predicted in 6.1, the phase s h i f t no* increases with energy and the e l a s t i c width becomes posi t i v e . . The wave function corresponding to t h i s resonance i s shown i n Fig.23. Further due to the kink in the wave function, as discussed following (91), the odd behavior of resonance narrowing by 'absorption* demonstrated in Fig.21b i s expected to be eliminated ny the kink i n the wave function. However, the 'reversed' ordering of curves i n Fig.20a should s t i l l be applicable since larger 152 &I s t i l l corresponds to more a t t r a c t i v e well, despite the kink i n the wave function. The anomalies discussed "in t h i s section, however exotic, merely res u l t from the d e f i n i t i o n of probability density and current which, i n turn, depends on how one turns off absorption in the p o t e n t i a l . , Unfortunately, for the K i s s l i n g e r potential, absorption can not be tur.ned off in a fashion without either incurring a s i n g u l a r i t y i n the resultant equation or giving r i s e to these anomalies (wnen oig exceeds 1).. These problems stem from the crude assumptions behind the K i s s l i n g e r type of potential which, despite i t s popularity, ought to be regarded with suspicion. 153 Table X K i s s l i n g e r potential parameters and depths b' = 0 c = 0 : -.042 fm b = -.11 3 .63 fm c'j = .60 fm fm 3 B = -.17 o C = -.53 0 (1-i) fm 4 (1-i) fm 6 5=1 Q^fm" 2) Q 2(fm" 2) a l a (o) + IT .04 .06(l-i) 1.07 - . 1 9 ( l - i ) .68+.11 i 208 Pb o IT .09 1.34 = .84+.10 i TT .14 1.61 .97+.09 i + TT .08 1.29 _ .80+.10 i 58, N TT .09 .10 1.34 1.39 = The parameters on top are Kisslinger potential parameters defined i n (48). They are taken from [31]. We rewrite (48) below i n order to define the new quantities appearing in the Table. a=-4i [ U/> - e n b, (ft- p?) + B, f 1 ] -- Q, FM + q, F m c V fir [C 0p - e, c, ^ n-/>p) 4 C 0 ^ ] - <*, F M + f V ) 4 1°<c Notice that for s i m p l i c i t y the i n (48) i n not written down e x p l i c i t l y . In calc u l a t i o n s we aake the assumption that fi» , P? , and P are a l l of the Ferai shape. For example, P(')= ftC(r) where p(r) =[l + e^n'>/t]",and p,=.l7(W3 . The value of V at the o r i g i n , <*(o) , i s e x p l i c i t l y given and w i l l be used in Fig.13. ISA-Table 2 K i s s l i n g e r potential parameters and depths b = -.042 fm b, = -.11 fm B = .17 i fm* o 1 o c = .64 fm 3 Ci = -62 fm 3 C = .61 i fm^ E=l o *"1 o _ Q^fm" 2) Q 2(fm" 2) a2 a(o) . T T + .04 ' -.06 i 1.10 .22 i .81+.12 i 2 0 8 P b TT° .09 = 1.38 1 = .95+.11 i T T " .14 = 1.66 = 1.08+ .09 i Potential parameters are taken from [46]. See Table 1 for explanations. Table 3 Kisslinger potential parameters and depths b = o -.042 fm b x = -.11 fm B = o = .17 i fm 4 c = b .51 fm3 3 C j = .62 fm C = o = .29 i Qjffm" 2) Q 2 ( f n f 2 ) a l a2 a Co) + TT .04 -.06 i .80 .10 i .8+.10 i 2 0 8 P b O TT .09 = 1.08 = 1.08+.10 i .14 — 1.36 = 1.36+.10 i Potential parameters are again taken from [46], except Co has been changed from .49 ^  to .51 ^ 3 in order to sake the e f f e c t i v e mass negative. 155 8 + -8-1 . 1 . 1 0 2 4 6 6 K) r (fm) F i g . 10 The re a l part of the pion e f f e c t i v e aass, fit = . The number beside each curve in d i c a t e s which Table oc i s taken from. The nucleus i s «08pb unless sp e c i f i e d otherwise. The parameters in the nucleon density function, Rr> (see Table 1 ) , are taken to be tha following; ft =6 .5 fm (Pb) , 156 E f x (MeV) F i g . 1 la The potential depth reguired to support resonances as a function of the resonant energy. In terns of f a m i l i a r u n i t , the v e r t i c a l scale m u l t i p l i e d by */2m ( = 138. 6) i s i n MeV. The values of , of, , and o(2 are fixed at those i n Tabie 1. The nucleus i s *° ePb. The symbol (TT°dO) means TT° , i =2, and /.= 0 (the f i r s t resonance},... etc. Pionic atcm values of Q, are marked on the v e r t i c a l axis. In doing the calcu l a t i o n we take the matching radius, R , to be 7 fm f o r Pb. .2 .\\-ft 'E 0 M -d -.1 -.2 -3 • CTT" po) - 3Z.6 - 1 0 . 0 (Tr°do) ^2X5 -q.8 - 0.4-17.3 MeVfm Re W/Ax=-H.^ MeV TrnWfcK3- &MeV 1 ,' 1 • 1 i 1 10 20 30 40 F i g - l l b Resonance parameters ( tfj*and Wj^ ) for the resonances in Fig.11a. 158 o io 20 3o 40 E (MeV) F i g . 12a Penetration factor as a function of incident energy for various p a r t i a l wave and pion charge. The nucleus i s 2 0 8 P b . Corresponding b a r r i e r s are shown i n Fig.12b. 153 .6 r ( f m ) F i g . 12b Coulomb p o t e n t i a l , Qc , and the 1 = 1 angular momentum potential for the TT+-Pb system. Q<. i s taken to be where we take g c to be 6.5 fm for Pb and 4.1 ta for Mi. In terms of f a m i l i a r unit, the v e r t i c a l scale multiplied by *Y2*,( = 138. 6) i s i n HeV. The penetration factor evaluated at 7 fm (Pb surface) for these b a r r i e r s i s shown in Fig . 12a. > 160 e < — " — • — i — ' — i — i — i i i i i < i i A J 6 . 8 I 1 . 2 1 . 4 » . 6 0<R(0) Fig..J.3 The r e a l acd the inaginary part of the pion e f f e c t i v e a a s ~ a t the o r i g i n , /fro) = -y—^ , as a function cfoc^o) . Black dots and open c i r c l e s correspond to the values of &(0) i n Table 1 and Table 2 respectively. Curves are drawn for o<r(0) f i x f d at .1. 161 .2 E -.2 (7T° dp) 10 20 30 E^(MeV) 40 Fig.14b Resonance parameters (|£and Kj^) for the resonances i n Fig.14a. 162 163 10 to >e F i g . 15a Comparison between s i z e resonances i n medium and heavy n u c l e i . The s o l i d l i n e s are taken from Fig. 11a, i . e . *° BPb. The broken l i n e s are for * B N i . P i c n i c atoa values of fii are marked on the l e f t v e r t i c a l axis for Pb and the right v e r t i c a l axis for Ni. In doing c a l c u l a t i o n s , ne take the matching r a d i u s , R , to be 6 f a f o r Ni. Fig.15b E l a s t i c widths for the resonances i n Fig.15a. 164 Fig • 16 Absorption terms (see Eq. (90) ) for resonance of Q,=-. 1 & l in Fig. 14a. the (T°<10) 165 F i g . 17a Argand plot f o r the (TT dO) resonance of Q|=-.1 -f* i n Fig.14a for various values of absorption strength, h • E (MeV) F i g . 17b Resonant amplitude magnitude squared foe the resonance i n F i g . 17a. 166 E (MeV) Fig.17d Feaoving the penetration factor froa the nuaeratjr of the resonant amplitude plotted i n Fig. 17b. See Eg. (108). 167 Pig . 11a. Notice that the e l a s t i c width i s 45 HeV! i.o 0 o •» *• >° E (MtV) F i a i 1 8 b Resonance amplitude Magnitude sguared foe resonance i n Fig.18a. E (M*v) yjq,18c Beaoving the penetration factor froa the numerator the resonant aaplitode plotted i n Fi g . 18b. 163 30 Fi3«.J9 Absorption width, calculated froa d i f f e r e n t aethod, as a function of f o r the resonance i n F i g . 17a. The s o l i d l i n e , calculated froa (90) and (117), corresponds to actual values. The broken l i n e i s from (122), the one le vel-approxiaation. Black dots are froa solving (130) by treating^j^o as a constant, and then by using (117) to convert 170 (M 'E o parameters are Fig-.20a Similar plot as F i g . 11a. Potential taken from Table 3 for vhich the TT° e f f e c t i v e mass term, PR , i s negative, as shown i n Fig.10. Fig.20b E l a s t i c widths for the resonances i n Fig.20a. the negative sign on the v e r t i c a l axis. Notice 171 M CM I Fig.21a Resonant ft, =.2 fuf z phase s h i f t resonance i n Fig.20a. E (MeV) vs. energy for the Absorption i s c f f . («-°s0) of F i g . 2.1 b Resonant amplitude magnitude sguared for the resonance in Fig. 21a- Notice the resonance narrowing by •absorption'. Fig.22 Total absorption of the nucleus as a function f o r the resonance i n Fig.21a. Notice the change of sign 173 Fig.23 Solid line (l) is the wave function of the resonance in Fig.21a. The "broken line shows the change of slope at where pg (see Fig. 10) changes sign, as required by (?0). Solid line (2) is the wave function for the resonance in Fig.2k, Notice that (2) oscillates slower than (l) inside the nucleus even though the energy is higher. This is due to the negative effective mass. Also notice the logarithmic derivatives of the two at the boundary radius, |? =7 fm, are quite similar, since both correspond to resonances. The vertical scale is set by lim(D 174 0 -2 Fig. 24- Resonant phase shift vs. energy for the (•ffsO) Q, =0.2 fm resonance. Potential parameters are taken from Table 3t for which pi? is negative for i f 0 . Calculations are carried out with the required kink in the wave function, as shown by (2) in Fig.23. Compare this figure with Fig.2ia where the phase shift decreases with energy at resonance. 175 REFERENCES 1.. Mesons i n Nuclei, 3 Volumes (ed. M.Rho and D.Wilkinson), North-Holland, Amsterdam and New York (1979) 2. H.Bethe, Phys. .Rev..47, 747 (1935) 3. N.Bohr, Nature 137, 344 (1936) 4. H.Feshbach, C.Porter and V.Weisskopf, Phys. 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S.Weinberg, Harvard University Preprint (1977), HUTP-7 7/A05 7, Submitted to appear in a f e s t s c h r i f t i n honor of I.i.Habi published by New lork Academy of Sciences 46. L.Tauscher, Proceedings of tae International Seminar on Pion-Nucleus Interactions, Strassbourg, France (1971) 47. P.Tandy, E.Redish and D.Bolle, Phys. Rev. Lett. 35,921 (1975) 118 Appendix I. Transforming the K i s s l i n g e r P o t e n t i a l from Momentum to Coordinate Representation For s i m p l i c i t y , we transform Eq.(38), ^ " l U l b ' > = (U + tp bM.fe') F(k'-k') instead of the f u l l equation, (40). The coordinate representation of a p o t e n t i a l i n terms of momentum matrix elements i s < / I u iY> = j<:x i b , ,>^fe"iuib ,>^'i/>^'i^>dV A ' A (i) (2) Consider f i r s t the simple case, (k I U Ik'> = t j F (k"-(?) then Ct. u n|/> = ~ , J c '-'^'^ts F( ?) e ' ' ' a W ? A ' where q^b"-fe . S i n c e J e 1 ' ' A" = (?TT) J ) we get << I u up> = ( i <«-<') ts Pft) e'-'• i fV, A A ' From (33), F(1)= j € ~p(<)d* where |9 i s the nucleon density, thus j F ( 6 ) e ' 9 - A = (eir/pc*) ( 3 ) I l l Therefore for the simple case of (12). In the above £5 i s a constant and one has the usual l o c a l form i n the f i n a l expression. I t i s c l e a r from above that i f the momentum matrix elements of U only depend on 4 , then th i s simple form s t i l l r e s u l t s . However, when the momentum matrix elements depend on R or H , as the p-wave part does, t h i s can no longer be the case. Consider <fc"IU lk'>* tp b"-b' f(V-!?') then (II) becomes <, iu 1 if> *fa JV - •(-"*',tp h'-q) F($) e"9 V i W b ' A A» (5) As before, integrate through I? f i r s t , (however, see * on p.181) Notice that* ( eib* dk * ?n Jtt) f it/) fry) = ((o) -.0 -°° *See, for example, p.470 of Quantum Mechanics by Messiah. (6) I 8 o Therefore (7) Notice that d i f f e r e n t i a t i o n i n the 5 -function i s with respect to . (15) now becomes { i ( ) ( z - A ) 5 ( X r ' < ^ K H - ^ ) + iqi<JU-/.)]+e+c.} A A ' ( 8 ) where 'etc' r e f e r s to s i m i l a r expressions as (17) had fe'i-b'ifli or - k'-ify been used. Now integrate through i n (18). From (16), we get j e ! ?" W ) J «»• < i '"I. - <',>]&'= | f c'" -' - iftw] Notice the sign change from (16) i n the second expression. (18) now becomes (i l u i >^ = j t P F «) (- v 2* J ? • ?) (e'?' - ifa] A Evaluating through, we get Further, from (13), J iq f(«ne'-V? -(2TT) 181 Therefore Collecting with (14), we get the f i n a l expression a i u > = [ ts p«> - t P Y - (to i] Yc*) Using properties of the &-function i n (16), we may also write the above as *• An alternative proof of the above, provided by Dr. R.Barrie;is consi-derately shorter. For s i m p l i c i t y , take the one dimensional case of (1.5) Integrate through if* f i r s t , then '1 t>* '> f y , f ^ " ^ l ^ \ ) ^ a " M ) e ' a , W ) , t m Appendix I I . Derivation of Sink Functions for the K i s s l i n g e r P o t e n t i a l We rewrite Eq.(56) below, Mu l t i p l y the above with \^ and i t s complex conjugate with V » subtract the two, we obtain Right away we recognize that the f i r s t term can be written as the divergence of the current i n (53), 2im Therefore, according to the continuity equation, (55) , - V - j - s » o the sink function i s simply However, some quantities i n the above sink function may be written as a divergence of a current themselves. S p e c i f i c a l l y , from the second l i n e , Therefore, combining the above with the previous current, we can define a new current, The corresponding sink function i s simply \2H Appendix I I I . M o d i f i c a t i o n of the Two-Potential Formula In t h i s appendix we want to show how the usual two-potential formula must be modified f o r the K i s s l i n g e r p o t e n t i a l as an i l l u s t r a t i o n f o r the many e f f e c t s which must follow from the unconventional d e f i n i t i o n of p r o b a b i l i t y density and current i n (69). Two-potential formula ref e r s to the expression obtained for the t-matrix amplitudes i n terms of two p o t e n t i a l s separately.* It forms the basis for d i s t o r t e d wave type of c a l c u l a t i o n s . The formula usually r e f e r s to the f u l l amplitude.* However, we f i n d i t easier to i l l u s t r a t e with the p a r t i a l wave amplitudes. Suppose we have a system described by the K i s s l i n g e r p o t e n t i a l and another p o t e n t i a l , Vz • The p a r t i a l wave equation (in the s e l f - a d j o i n t form) i s then From the theory of d i f f e r e n t i a l equations,** a p a r t i c u l a r s o l u t i o n can be written as oo 0 by regarding Vi a s the f o r c i n g term. Ct i s Green's function *See f o r example, p.271 of Scattering Theory by Taylor. **See, for example, p.355 of Methods of Math. Physics by Courant and H i l b e r t . / and can be constructed from two independent s o l u t i o n s , U , and U t » of as where C i s a constant and given by C - A«rV[ U, UiVj - U,Vj ( V ' ) ] . However, (J,and Jt.can not be j u s t any two independent s o l u t i o n s . Tbey must be chosen such that s a t i s f i e s the boundary conditions lift. <P ^ C i + where f and L, are j u s t the usual R i c c a t i - B e s s e l and Riccati-Hankel functions. -f^  of course i s the p a r t i a l wave amplitude. Suppose we choose (J, such that (thus liw U , W t " * Jit) =e (j'e+bt»V)where fa i s the K i s s l i n g e r phase s h i f t and the corresponding p a r t i a l wave amplitude); and such that 186 then f-fao where o O ^ s u.ir'j ^Mii'1) 'o Thus we see that the f u l l s o l u t i o n s a t i s f y i n g the required boundary conditions can be written as so that Therefore the f u l l p a r t i a l wave amplitude i s given by This i s j u s t the two-potential formula. In the l i m i t that i s weak compared to the K i s s l i n g e r p o t e n t i a l , then f , u , ( ^ A ^ M ' ) V ^ O US') This i s the d i s t o r t e d wave Born approximation. Notice the extra /\ $g factor due to the density d e f i n i t i o n . 18 7 Appendix IV. Orthogonality Relation when the E f f e c t i v e Mass Changes Sign We want to derive a new orthogonality r e l a t i o n , i n place of (76), f o r eigenfunctions of (73) when changes sign at some place. R e c a l l that eigenfunctions, XjlAs> s a t i s f y with boundary conditions i = 6, R where ^ i s a radius and B)L i s a r e a l number. Also r e c a l l that i f @R changes sign, say at ( f?o ^  R )» then )C X 1 must change ( i . e . a kink i n the wave function) i n order to keep the current i n (70), i . e . continuous. Now we want to f i n d how t h i s kink a f f e c t s the orthogonality r e l a t i o n between ] ( ^ ' s . Let and Yi^' denote two eigenfunctions, i . e . (A Y t x) • A [ f ? R ( ^ - Q R - Q c ) + B,Qt - - ] X*x=o 188 M u l t i p l y i n g the f i r s t with and the complex conjugate of the second with , subtracting the two, and then i n t e g r a t i n g from the o r i g i n to R , we obtain 0 O A (ti * i - Xn, x*>!) = U $K x*>> where we write the i n t e g r a l into two parts. Now l e t us suppose s p e c i f i c a l l y that i s negative in s i d e (?o and p o s i t i v e outside. Then the quantity, XjlX "" Xj\ XjlV ' e v a l u a t e < * a t ^ 0 a r e °f opposite sign f o r the above two equations due to the kink i n the wave function, x.e, where f ^ a n d Ro^ are approaches to from below and from above res p e c t i v e l y . In order to make and )^» orthogonal i n the region from the o r i g i n to £ , we see that the the i n t e g r a l from ^ 0 to ^. must be m u l t i p l i e d by a negative sign before adding i t with the i n t e g r a l from 0 to (Zo »* i . e . Po J0 *This i s equivalent to changing the d e f i n i t i o n of the current. Then the above becomes Because of the boundary condition Xi)v 's s a t i s f y at 0 and R , the above vanishes. This i s the new orthogonality r e l a t i o n between eigen-functions when ^ £ changes sign. 

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