‘ lllllllllljlllllllfllllllljll' 3 1293 1.13 RAR fl MiG" "3n Sty-m / Marrow...“ *- ff mgr... This is to certify that the thesis entitled A Study of the Pion-Nucleus Optical Potential presented by Karen Sue Stricker has been accepted towards fulfillment of the requirements for Ph.D. degree in Physics 794% MflW Major professor Date October 23, 1979 0-7639 RETURNING MATERIALS: IV‘ESI_] Place in 500E drop tof remove this checkout rom w your record. FINES wiIl be charged if book is returned after the date stamped below. A STUDY OF THE PION-NUCLEUS OPTICAL POTENTIAL By Karen Sue Stricker A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Physics 1979 ABSTRACT A STUDY OF THE PION-NUCLEUS OPTICAL POTENTIAL By Karen Sue Stricker The optical potential model is a convenient means of charac- terizing the interaction of the pion with the nucleus. Its simplicity makes it practical for the calculation of elastic scattering and pion distorted waves for more complicated processes. Its success in reproducing the early pion data and the existence of new, higher quality data motivate the present investigation. An Optical potential for pion-nucleus interactions in the energy range 0-250 MeV pion kinetic energies is constructed with the Watson multiple scattering series and the nN transition ampli- tude as starting point. The pion-nucleon to pion-nucleus center of mass transformation is calculated to first order in the ratio of total pion energy to nucleon mass. Multiple scattering correc- tions in low energy approximation are included to second order in the s—wave and to all orders in the p-wave (the Lorentz-Lorenz or Ericson-Ericson effect). True pion absorption terms, proportional to the square of the nuclear density, are included in both 5 and p-wave parts of the potential. Pauli blocking is approximated, and an energy shift due to the Coulomb interaction is incorporated. Karen Sue Stricker The potential parameters are taken from the experimental nN phase shifts and theoretical calculations. The potential, of Kisslinger type, is incorporated in coordinate space computer codes which cal- culate pionic atom level shifts and widths, elastic scattering dif- ferential cross sections, and total and partial cross sections. These calculations are compared to the current experimental data. It is found that at low energies (0-50 MeV) the potential produces elastic cross sections which fit the data provided the s-wave repul- sion is increased. The pionic atom level data require more absorp- tive strength than that given by current calculations, as well as increased repulsion consistent with the scattering results. The general features of the resonance region elastic scattering and total cross sections are well reproduced. ACKNOWLEDGMENTS My thesis advisor, Professor Hugh McManus, deserves much credit and many thanks for his constant guidance and encouragement through- out this work. I also wish to thank James A. Carr, whose research on a related thesis t0pic afforded numerous opportunities for pro- ductive collaboration, and Professor Dan 0. Riska, for many helpful discussions. The various experimental groups whose data are included in this work were most generous in providing their results prior to publication. I am also indebted to several other theorists in the field of pion physics who freely shared their ideas and calculations. Most of the computations for this thesis were performed on the MSU Cyclotron Laboratory computer, the staff of which gave much valuable assistance. The MSU Physics Department provided time on the main University computer for some of these calculations. I also wish to acknowledge the dedicated, courageous, and expert typing of the rough draft by Mrs. Shari Conroy and the final copy by Mrs. Carol Cole. Finally, I am pleased to acknowledge my parents, whose support and encouragement were invaluable through my academic life, and my friends at MSU, whose concern and cheerful camaraderie smoothed many rough spots. To Daniel Bauer especially I owe more than I can say for a friendship which has become something much greater. ii TABLE OF CONTENTS Page LIST OF TABLES ........................ v LIST OF FIGURES ........................ vi Chapter I. INTRODUCTION ...................... 1 II. THE FIRST ORDER OPTICAL POTENTIAL ............ 9 1. The Pion-Nucleon Interaction ............ 9 2. The Pion Wave Equation ............... 15 3. The Multiple Scattering Series ........... 19 4. The Optical Potential-~Simplest Assumptions ..... 26 III. THE FULL OPTICAL POTENTIAL ............... 31 1. Kinematics ..................... 32 2. Multiple Scattering Corrections ........... 45 3. True Pion Absorption Terms ............. 61 4. Pauli and Coulomb Corrections ............ 77 5. The Optical Potential ................ 83 IV. PIONIC ATOM LEVELS ................... 87 1. General Features .................. 88 2. Details of the Calculations ............. 91 3. Calculated Shifts and Widths ............ 93 V. ELASTIC SCATTERING CROSS SECTIONS ............ 102 1. General Features--Low Energy Scattering ....... 103 2. General Features--Resonance Region Scattering . . . . 109 3. Details of the Calculations ............. 112 4. Calculations--Low Energy Region ........... 114 5. Calculations--Resonance Region ........... 133 iv Chapter VI. VII. TOTAL AND PARTIAL CROSS SECTIONS ............ 1. Extraction of Total Cross Sections and Scattering Amplitudes ..................... 2. Total Cross Section and Scattering Amplitude Calculations .................... 3. Theoretical Expressions for the Partial Cross Sections ...................... 4. Partial Cross Section Calculations ......... CONCLUSIONS ....................... Appendix A. B. C. D. THE PION-NUCLEON SCATTERING AMPLITUDE .......... DETAILS OF THE DERIVATION OF THE MULTIPLE SCATTERING SERIES ......................... RELATIVISTIC POTENTIAL THEORY .............. LIST OF REFERENCES ...................... 173 178 200 LIST OF TABLES Table Page 1. Symbols used throughout this work and their interpretation 8 2. The Coulomb energy shift E , evaluated at the nuclear surface, for various nuclei ............... 82 3. The forms and parameters of the matter and charge density distributions for various nuclei ............. 85 4. Experimental pionic atom energy level shifts and widths in keV from Refs. 59 and 60 ................. 89 5. Parameters used in the pionic atom calculations ..... 99 6. Parameters used in the low energy elastic scattering calculations ....................... 115 Figure 10. 11. LIST OF FIGURES The quantities "i and "ij as a function of pion lab kinetic energy ............ . ........ The optical potential parameters b0, b1, c0, and c1 as a function of pion lab kinetic energy .......... Comparison of calculations with the Kisslinger (solid curve) and Laplacian (dashed curve) optical potentials . Comparison of calculations with no kinematic corrections (dashed curve), full kinematics (solid curve), and alternative choices for the kinematic corrections (dotted and dot—dashed curves) . . ........... The integral I as a function of pion lab kinetic energy, and the approximation Im(I) = kO ............ The first order s-wave optical potential parameter b0, and the same parameter but with the second order correc- tion included, b0, as a function of pion lab kinetic energy ......................... The effect of the second order s-wave term (solid curve) compared to calculations with the first order term only (dashed curve) and with the second order term in zero energy approximation (dotted curve) . . ........ The parameter c' compared to the first order p-wave parameter c0 as a function of pion lab kinetic energy The effect of the Ericson-Ericson correction to the p-wave, A = 1.6 (solid curve) and A = 1.0 (dotted curve), compared to calculations with no correction (dashed curve) and with the correction included to second order only, A = 1 (dash-dotted curve) ...... The effect of the c' term (solid curve) compared to cal- culations without this term (dashed curve) ....... Diagrams of the s and p-wave processes included in the calculation (50) of the absorption parameters ..... vi Page 12 13 35 46 51 52 54 59 60 62 71 Figure 12. 13. 14. 15. 16. 17a. 17b. 17c. 17d. 18. 19. 20. 21. vii Page The absorptive parameters 80 and C0 as a function of pion lab kinetic energy ................ 74 The effect of the absorption terms (solid curve) com- pared to calculations without these terms (dashed curve) and with the absorption terms included to first order only (dotted curve) .................. 76 The Pauli factor Q as a function of pion lab kinetic energy ......................... 79 Comparison of calculations with (solid curve) and with- out (dashed curve) Pauli corrections .......... 80 Comparison of calculations of n and n scattering on 16O and 208Pb at 162 MeV with (solid curve) and without (dashed curve) the Coulomb shift. The references for the n data are the same as for the corresponding fl+ data .......................... 84 Calculations of the s-wave shift as a function of Z compared to the experimental data, with the parameters of set 1 of Table 5 with A = 1 (dashed line), A = 1.6 (dotted line), and with the parameters of set 2 (solid line). Data are from Ref. 59 ............. 94 Calculations of the s-wave width. Data are from Ref. 59 95 Calculations of the p-wave shift. Data are from Ref. 60 96 Calculations of the p-wave width. Data are from Ref. 60 97 Amplitudes and corresponding cross sections for the first order real optical potential in Born approxima- tion. The vertical scales are arbitrary ........ 106 Elastic scattering cross sections for 50 MeV n on 120 with (a) the full Born amplitude and (b) the full opti- cal model calculation, for Re(bo) = 0., -O. 04 fm, -O. 08 fm, and -0. 12 fm ................. 107 Elastic scattering cross sections for 50 MeV n" on 120 with curves same as in previous figure ......... 108 Comparison of black disk model calculation (dashed curve) and full optical potential calculation 2Ssolid curve) for NT and n scattering from150 and2 8Pb at 162 MeV ........................ 111 Figure 22a. 22b. 22c. 22d. 22e. 22f. 23a. 23b. 23c. 23d. 23e. 23f. 24. 25a. viii Page Elastic scattering of n+ from 120 at 30, 40, and 50 MeV with the optical potential parameters of set 1 (dashed curves) and set 2 (solid curves). Data are from Refs. 32 (diamonds), 34, 69, 7O (triangles), and 68 (circles) ....................... 117 Elastic scattering of n+ from 160. Data from Refs. 32 (diamonds), and 33, 34, and 69 (triangles) . . . . . . . 118 Elastic scattering of n+ from 40Ca. Data from Refs. 34, 69 . . . . . ................. . . . . . 119 Elastic scattering of n+ from 90Zr. Data from Refs. 34, 69 ................... . ....... 120 Elastic scattering of n+ from 208Pb. Data from Refs. 34, 69 ...................... 121 . . - 12 208 Elast1c scatter1ng of n from C and Pb at 30 MeV. Data from Refs. 71 and 72 ............... 122 Elastic scattering of “T from 12C at 30, 40, and 50 MeV, with the optical potential parameters from set 2 of Table 5 (solid curves), set 3 of Table 6 (dotted curves), and set 3 but with Im(80) and Im(C0) from set 2 of Table 5 (dashed curves) ............... 124 Elastic scattering of NT from 16o ........... 125 Elastic scattering of fl+ from 4OCa ........... 126 Elastic scattering of “T from 90Zr ........... 127 Elastic scattering of "T from 208Pb .......... 128 208 Elastic scattering of n' from 12C and Pb at 30 MeV . 129 . . . . + Compar1son of elast1c scatter1ng calculations for n on 12C with two values of Re(B ) (solid and dashed curves) and a series of values for e(b0) ........... 132 Elastic scattering of n+ and n' from 160. Optical potential parameters are taken from the RSL phase shifts and Riska calculations with A = 1.6 (solid curves) and A = 1.0 (dashed curves). Also shown are first order Kisslinger potential calculations (dotted curves). Data are from Refs. 35, 74 ................. 135 Figure 25b. 25c. 26. 27a. 27b. 27c. 28. 29a. 29b. 30a. 30b. 31a. 31b. 32. 33. ix Page Elastic scattering of n+ and n' from 400a. Data are from Ref. 35 ...................... 136 Elastic scattering of n+ and n' from 208Pb. Data are from Refs. 36 (162MeV) and 75 (all other energies) . . . 137 Imaginary absorption parameters extrapolated from pionic atom values (dashed curves) compared to those of Ref. 50 (solid curves) ..................... 141 Elastic scattering of 1+ and n' from 160. Absorption parameters are taken from Ref. 50 (solid curves) and extrapolated from pionic atom values (dashed curves) with other parameters from the RSL phase shifts and = 1.6 ........................ 143 Elastic scattering from 400a .............. 144 Elastic scattering from 208Pb ............. 145 Total cross section calculations using the absorption parameters of Ref. 50 (solid curves) and extrapolated pionic atom parameters (dashed curves) compared to the data of Carroll et al. (78) .............. 155 The amplitude %§-Re[fN(0)] for n+ (circles) and n' (x's) n 27Al, asa function of energy. Curves are same as in Figure 28. Data are from Jeppeson et al. (79) ..... 157 The amplitude% TrIm[fN (0)] for n+ and n on 27Al . . . . 158 The amplitude %:-Re[f: (0)] for n and n on 208Pb. Curves are same as 1n Figure 28. Data are from Ref. 79 159 The amplitudei — "Nrmfi (0)] for r and 1' on 208% . . . 160 The amplitude {T Re[fN (0)] for n+ and n as a function ofA at 165 MeV. Curves are same as in Figure 28. Data are from Ref. 79 .................... 161 The amplitude% 1TIm[fN (0)] as a function of A ..... 162 Partial cross section calculations for n on 120 as a function of energy compared to the data of Navon et al. (81). Curves are same as in Figure 28 ......... 169 Partial cross section calculations as a function of A for n+ at 165 MeV, compared to the data of Ref. 81. Curves are same as in Figure 28 ............ 171 CHAPTER I INTRODUCTION In the past few years the field of pion—nucleus interactions has advanced rapidly. A large amount of excellent quality data has come out of the intermediate energy laboratories; LAMPF at Los Alamos, New Mexico, USA; TRIUMF in Vancouver, Canada; and SIN in Switzerland. This data includes not only elastic and inelastic differential cross sections, but also total and partial cross sections and single and double charge exchange measurements as well as more complicated reac- tions. Much progress has also been made in the theoretical descrip- tion of the pion-nucleus interaction, with characterizations which vary from the phenomenological to the fundamental and microscopic. The present work will focus on the optical potential model, which takes a middle ground between these two approaches and has had a fair amount of success in the description of the early pion-nucleus data. The concept of an optical potential, that is, a complex poten- tial describing the interaction between the projectile and the nucleus as a whole, and in which the imaginary part accounts for flux lost to other channels from the elastic channel, is due to Bethe (1). An optical model for scattering of high energy particles by nuclei was first introduced by Fernbach, Serber, and Taylor (2) to describe the scattering of 90 MeV neutrons. They proposed a constant complex potential inside the nucleus, the imaginary part of which can be related to the mean free path A of the nucleon in nuclear matter (3), ko Im(u )=— Opt 2 9 (1-1) >4]... where k0 is the particle momentum and M its mass. The mean free path can, in turn, be expressed in terms of the total collision cross section and the nuclear density, §= 0T0 . (1-2) The optical model was first applied to low energy scattering by Feshbach, Porter, and Weisskopf (4) in the analysis of resonances in 0-3 MeV neutron total cross sections. A theoretical basis for the optical model was provided by Natson (5), who derived the optical potential from a multiple scattering theory. A simplification of the theory, due to the antisymmetry of the target states, was given by Kerman, McManus, and Thaler (6). The study of the pion-nucleus interaction began with the dis- covery of the pion in 1947, since early pion experiments usually involved nuclear targets in cloud chambers and emulsions. An optical model for pion-nucleus elastic scattering which included both 5- and p-wave terms was first given by Kisslinger (7), and used in the analysis of differential cross section data (8) for 62 MeV1FIand n- on 12C. The analysis of the measured energy shifts and widths of pionic atoms made clear the necessity of including higher order terms in the optical potential. The inclusion of true pion absorp- tion terms, first suggested by Brueckner (9), and the calculation of the Lorentz-Lorenz effect were made by Ericson and Ericson (10) and greatly improved the agreement between calculated and measured levels. The emphasis shifted to the resonance region with the appear- ance of n'-12C elastic scattering cross sections at 120, 200, and 280 MeV from CERN (11), followed by data on other nuclei. Although the Kisslinger potential was originally derived for low energy scat- tering, it was found to give reasonable results in the resonance region also (12). A local optical potential form, the Laplacian model, gave similar results for scattering near resonance (13). Glauber theory (14) was also successfully applied to scattering data in this region (15). Very little was known of the low energy (0-50 MeV) pion scat- tering cross sections, and few calculations beyond first order existed for these energies (16) until about 1975, when more accurate data of 50 MeV n+ elastic scattering from 120 appeared (17). Thies (18) showed that the inclusion of kinematic effects, higher order multiple scattering terms, and s-wave absorption greatly improved the agreement between the calculated and measured cross sections. Since 1975 the theoretical activity in pion-nucleus interactions has been intense. The most successful microscopic calculations have been the isobar-hole calculations (19) which treat the dominant channel, with A33 intermediate states, by means of a spreading potential, the parameters of which are fitted to the data. The phenomenological input is small, and the results are encouraging; 16O are impractical. however, calculations for nuclei larger than The phenomenological optical model has also received a great deal of attention. It has been shown (20) that a first order Kisslinger potential with four free parameters can be fitted to the elastic scattering data for pion kinetic energies around 50 MeV. Four parameters are also sufficient to describe the pionic atom data for a wide range of nuclei (21). The elastic scattering cross sections in the resonance region can also be fit by optical model calculations, requiring, however, a somewhat more sophisticated potential with more than four parameters (22). The phenomenological approach, although successful in describ— ing various classes of data, has almost no predictive power and is most unsatisfying to a theorist. The microscopic theories have a strong theoretical base and a minimum of approximations but are extremely complicated, tedious calculations and have been made for only a few light nuclei at a few energies. Thus the need at present for a simpler approach based on theoretical considerations but with a simple optical potential form. Such a model, if carefully con- structed, should be valid over a fairly wide energy range, say 0-250 MeV, and for all nuclei large enough to justify the optical model assumptions, certainly carbon and all heavier nuclei. The theoretical basis gives the model predictive power; its simplicity makes it a useful tool in calculations of more complicated processes. The important physical content of the theory appears in the optical potential in a straightforward way, not buried in vast computer calculations, giving a feel for the important features of the prob- lem. It is to be hoped that the microscopic calculations will even- tually become sufficiently tractable and accurate to be applicable to most nuclei and energies. However, an optical potential type model, taking input from the more sophisticated theories with suita- ble approximations, will almost certainly be the basis of most practical calculations. The early work on theoretically based optical potentials by the Ericsons (10) and Thies (18) has already been mentioned. Pieces of the problem have been much discussed by various authors. A review of all such research will not be attempted in this brief introduc- tion; the interested reader is referred to the proceedings of the several recent pion conferences (23). The purpose of this disserta- tion is to bring together all aspects of the problem in a coherent framework, to construct an optical potential with a broad range of validity. The form chosen for the potential is a coordinate space form of Kisslinger type, local in the sense of depending on only one pion coordinate. Previous investigations suggest that the essential physics survives the approximations necessary to obtain such a form, which is chosen for its simplicity. The more important test of the validity of the theory is, of course, the accuracy with which it predicts the experimental results, hence the inclusion in this work of calculations of pionic atom level shifts and widths, differential elastic scattering cross sections, and total and partial cross sections for a variety of nuclei and energies. The reason usually given for the study of the pion-nucleus interaction is the hope that the pion can be used as a probe of nuclear structure once the pion-nucleus dynamics are understood. The nature of the pion, with three isospin states and no spin, and the fact that pions can be absorbed on nucleons, make the pion a unique tool in, for example, the determination of neutron and proton distributions and perhaps the study of correlations between nucleons. There is a growing interest, however, in the pion-nucleus problem itself. The field of intermediate energy physics, of which pion physics is an important part, has become a meeting ground for the nonrelativistic many-body theories of low energy nuclear physics and the relativistic field theories developed in elementary particle physics. Although the present study does not delve deeply into these questions, the development of the potential indicates where these elements enter and provides a base for more detailed calcula- tions. Any improvements to the optical potential model discussed here will almost certainly involve a more careful synthesis of these two aspects of the problem. The dissertation is divided into six main parts. In the first of these, Chapter II, the information is presented which forms the basis of the optical potential theory: the pion-nucleon inter- action, the form of the pion wave equation, and the multiple scat- tering formalism. From these the first order Kisslinger optical potential is constructed. In Chapter III, the optical potential is refined with the addition of kinematic effects, multiple scat- tering corrections, true absorption terms, and Pauli and Coulomb effects. This completes the construction of the optical potential; the comparisons to data are discussed in the following three chap~ ters. The first of these, Chapter IV, is a discussion of the optical potential applied to the analysis of pionic atom shifts and widths. Chapter V presents the calculations of elastic differential cross sections, compared to a selection of the available data. In Chap- ter VI a discussion is given of the calculation of total and partial cross sections, with the results compared to the data. In Chapters V and VI, two different approaches are taken to the choice of param- eters for the optical potential. The first is to adopt the param- eters as calculated theoretically in the earlier chapters. The second is to extrapolate by simple means the information gained from the pionic atom analysis discussed in Chapter IV to non-zero energies, extending the work of reference 24. The major conclusions of this work are discussed in the final chapter. The symbols used for some common quantities are given in Table 1. These will be used throughout, unless otherwise noted. Table 1. Symbols used throughout this work and their interpretation. Symbol Meaning (k,w) Momentum and energy of incoming pion (k',w') Momentum and energy of outgoing pion (p,E) Momentum and energy of incoming nucleon (p',E') Momentum and energy of outgoing nucleon (P,EA) Momentum and energy of incoming nucleus m Mass of pion M Mass of nucleon MA Mass of nucleus t Isospin operator for pion I Isospin operator for nucleon Spin operator for nucleon 0 cm subscript 2cm subscript no subscript Quantities in pion-nucleon center of mass Quantities in pion-two nucleon center of mass Quantities in pion-nucleus center of mass (Subscripts are often dropped when only one frame is being considered.) CHAPTER II THE FIRST ORDER OPTICAL POTENTIAL The optical model provides a method by which the pion-nucleus many body problem can be reduced to a one particle equation for the pion, interacting with an optical potential which describes the nucleus. The optical potential can be derived from knowledge of the measured pion-nucleon scattering amplitude and a multiple scattering formalism which relates the pion-nucleon amplitude to the pion-nucleus interaction. In Section 1 of this chapter the pion-nucleon interaction is described. The pion wave equation is discussed in Section 2. The development of the multiple scattering series for the optical potential is given in Section 3, and the first order coordinate Space optical potential is given irISection‘4. 1. The Pion-Nucleon Interaction The most prominent feature of pion-nucleon scattering in the energy region 0-300 MeV pion lab kinetic energy is the effect of the nN resonance, denoted A33, at 1236 MeV total center of mass energy or about 180 MeV pion kinetic energy. The subscripts 33 refer to the isospin and total angular momentum of the resonance, both of which have the value 3/2. The orbital angular momentum of the state is L = 1. This channel dominates the nN interaction 10 at these energies, giving rise to a large p-wave term in the scat- tering amplitude. The most general scattering amplitude for this problem can be expanded in orbital angular momentum, isospin, and total angular momentum. The s- and p-wave terms of this expansion are f N = (b0 + bIE°I) + (CO + c 1E I)Ecmk ~ cm + (50 + S E ~)~ (Ecmx E'cm) ' (11.1) The relationships between the coefficients bi’ c. 1, and 5i and the measured pion-nucleon phase shifts are derived in Appendix A, and are given by cm c =-—l—-l (4w + 2w + 2w + w ) 0 k3 3 33 31 13 11 9“ (II-2) c = ~l—-l-(2w + w - 2w - w ) 1 k3 3 33 31 13 11 cm s = —l——l-(2w - 2w + w - w ) O k3 3 33 31 13 11 cm s = 1 1 (w w w + w ) 1 k3 3' 33 31 13 11 11 The wi and wij are related to the a21,2J of Appendix A by _ o wi - kcmail (II-3) _ 1 wij ‘ kcmaij where L exp(2i8%I’ZJ) - 1 O‘21,2.1 ‘ 21 kcm ' (11‘4) Here I, L, and J are respectively the isospin, orbital angular momentum, and total angular momentum of the system. The first two terms of equation II-l, referred to as the s- and p-wave terms, are the most important terms for the calculation of pion-nucleus scattering. The third term, also a p-wave term, is usually negligible for pion-nucleus calculations since the nucleon spin is summed over, and will not be discussed further. The d-wave and higher partial waves do not contribute appreciably until ener- gies well above resonance. The "i and wij and the s- and p-wave parameters are shown in Figures 1 and 2, as a function of pion lab kinetic energy Tn. They are computed from the parametrization of the n-nucleon phase shifts given by Rowe, Salomon, and Landau (25), in which an analytic function of energy is fitted to the quantity kc (2L+1) tan 621 20' over the energy range 0-400 MeV. The advantage of such a parametri- zation is that the scattering parameters derived from it vary 12 —— b0 — CO -_... b1 -__ Cl H r E ’ E 0.50- x -0005» it: y I? L 5' O > 0 CC -0.10~ (r 0.0 y i 1.00 H 0.30b r E > E 3; > .2. b” ’ 6' 0.05. 0.50 E e H H 0.00 : ,1. 447' .2 : g * “““ 0.00 a 100 200 l Tallobl [MeV] Figure 1. The quantities "i and wij as a function of pion lab kinetic energy. 13 '—-'“33 — “3 --- ~31 -.... H1 ------ N13 "" N11 03-1»- i L ____________ i ’x” 0.0 ¢ 4 : i : fit 0 CE -0.H> p 0.8} id > E H 0.9» Tallobl [MeV] Figure 2. The optical potential parameters b , b1, c0, and c1 as a function of pion lab kinetic energy. 14 smoothly with energy, even at low energies, whereas the scattering parameters calculated directly from phase shifts, even those that have been smoothed, such as the CERN Theoretical set (26), are quite noisy below about 80 MeV. Several things may be noted. Although the phase shifts 521,20 are purely real below the threshold for pion production, ”cm + Ecm = 2m + M, the scattering parameters are not; the imaginary parts are zero only at zero pion kinetic energy. The real part of the isoscalar s-wave parameter b0 is negative in this energy region, corresponding to a repulsive s-wave interaction, and is nearly zero at low energies due to a near cancellation of the two terms w1 and 2w3. The p-wave parameters c0 and c1 are dominated by the 6§3 phase shift and display characteristic resonant behavior, the real part crossing zero at the resonance energy and the imaginary part reach- ing a maximum at this point. The parameter Re(c0) 1'5 positive 13910" resonance, hence an attractive p-wave interaction in this region. The quantity reguired for the pion-nucleus calculations is the pion-nucleon transition amplitude tWN. This is related to the pion-nucleon potential v by a Lippmann-Schwinger type equation. nN _- - O nN _ t - v + v2wcmg t , (II 5) 0 where g is the propagattn‘ for a free pion; in momentum space 90 = 2 12 , (II-6) -k + k0 + is 15 The factor ZEEm’ where 5' is the relativistic equivalent of the cm reduced mass =_fl_c_m__ ’ (11-7) is a result of the use of the Klein—Gordon equation rather than the Schrddinger equation for the pion. This point will be considered in more detail in the discussion of the pion-nucleus scattering equation. Matrix elements of tTTN between momentum states of the pion and nucleon can be written = 3 _ 1_ 1 ‘TTN I lk’9> (2") 6(5 + E E B )t (Ecm’E cm) ‘ (II-8) The transition amplitude and scattering amplitude can then be shown (27) to be related in the nN center of mass by TIN . _ 4'” I - t (Ecm’gcm) - ' 25cm an(Ecm’Ecm (II 9) Thus, the required pion-nucleon T-matrix is related in a simple way to the experimental phase shifts. 2. The Pion Nave Equation Before discussing the multiple scattering series expansion for the optical potential, it is necessary to consider what form the wave equation for the pion will take. The nucleus is quite 16 massive and can be treated nonrelativistically. However, the rest mass of the pion is not large compared to its momentum at the ener- gies considered here and must be treated relativistically. Thus, the Hamiltonian for the system must include rest masses and can be written p2 A+mx+v (Ham H=uz+¥fi+n where P and MA are the momentum and mass of the nucleus, and V characterizes the interaction between the nucleons in the target and the pion. The part of the Hamiltonian which describes the internal dynamics of the nucleus has been neglected, assuming that the excitation energies of the various nuclear states do not play an important role. The Schrddinger equation for the system is then 2 [(k2 + m2)1 + M + J:—-+ VJv = E (II-11) A 2MA T‘i’ Goldberger and Watson (27) have shown that for |V|< I 00 00 where G , (II-34) 00 with similar definitions for 100 and 080. Note that GO is diagonal in the nuclear states, 0 = Enamn . (II-35) 23 Equation II-33 is now a one particle equation for the pion and can be written as a Schrodinger-like equation with U00 as potential. Thus the problem can be solved "exactly" (by computer), if U00 is known. The second rearrangement is motivated by the fact that the nN T-matrix, not the potential, is the quantity closely related to the experimental data. The terms in the expansion for O which involve only the potential of the ith nucleon can be grouped together to define a quantity similar to the free tTTN of equation II-5. Writing = 2 O1. (II-36) and noting that = 2 Q. , (II-37) . 1 1 equation II-31 can be rewritten using the second result of Appendix B, =Z¥i+2 Z iméoi-Ph. i' i jfi (II-38) + 2 .2. 2. 1100(1 - P0)TJ-GO(1 - P0)Tk + 1 321 kfa 24 where r. is defined by 0(1 - PO)¥. . . = v. + . T v1G 1 1 1 (II-39) In order to relate 11 to the free pion-nucleon T-matrix tflN, define __1_" d -.L" Ti - 25 T1 an vi - 25 vi . (II-40) Then Ti = vi + vi[GO(1 - P0)2mii . (II-41) The first result of Appendix 8 applied to equations II-41 and II-5 gives the relationship of 11 and the free nN T-matrix for the ith nucleon th, Ti = t?” + t?“ [00(1 - P0)23 - gOZEEm] Ti . (II-42) As was pointed out by Kerman, McManus, and Thaler (6), the antisymmetrization of the intermediate states can be exploited in order to simplify equation II-38 for O. Let A be a projection operator projecting onto completely antisymmetrized target states. Note that A commutes with V, P0, and(¥)since these are totally symmetric in the nucleon coordinates. Thus, assuming T and U will always be taken between properly antisymmetrized states, equa- tions II-29 and II-31 can be written 25 i = {i + 960.? (11-43) 0 = v + 960(1 - PO)AO . (II-44) Equations II-38, II-39, II-41, and II-42 can be rederived with the operatoruaincluded, yielding equations of the same form but with 50(1 - P0) replaced by 00(1 - P0)A. Note that the $1 thus defined are somewhat different than those defined in equation II-39. With this change the matrix elements of the T1 in the equivalent of equation II-38 are the same for all i, since with the antisymmetriza- tion all nucleons are equivalent. Equation II-38 becomes A A- AAO A U - Ari + A(A - 1)TiG (1 - PO)ATj (II-45) 22 0 A )Aer (1 - P0)ATk + ... “O + A(A - 1) TiG (1--P0 where i f j, j # k, and so on. The equation giving Ti in terms of t?" is now 1. = t?" + tIN [G O 1 1 1 ( 1 - P0)126'- gozagm] Ti . (II-46) The difference between Ti and t?” will be neglected; this is known as the impulse approximation (30). The final result, equation II-45, is the multiple scattering series for O. The first term describes the scattering to all orders from one nucleon, summed over all nucleons. The second term 26 describes scattering to all orders by one nucleon, propagation, and then scattering to all orders by a second nucleon. The third term describes three such scatterings, and so on. The optical poten- tial is given by A ZZMOPt = U00 5 <0|U|O> . (II-47) Writing equation II-33 in the form of a Schrodinger equation and including the Coulomb potential gives a wave equation much like equation II-17 but involving pion coordinates only, 2 2 (v +kO - 2; (u + Vc) + g vE) ¢(r) = o (II-48) opt where o is the pion wavefunction. It is, of course, impossible to calculate all terms of the series for U, equation II-45. However, the first two terms and a partial summation of the rest can be calculated if some approxi- mations are made. This is the subject of the next chapter. 4. The Optical Potential-~Simplest Assumptions To see the general features of the pion-nucleus optical poten- tial, it is useful to construct the first order potential, arising from the first term in equation II-45. The impulse approximation is made, and kinematic corrections due to the transformation of nN t to the pion-nucleus center of mass, and to the difference between w and '5, are ignored. Then 27 nN 210(1) = A (II-49) which, by equations II-8 and II-9, is 1 | _ 3 I I I ZwUépg(§.f ) - A<0|(2n) 5(k + p1 - k - Bl)('4")an(E’E )|0> (II-50) * Here |0> represents 00(p1,p2...pA) and 1 = coiya1 leadsix>Re(c0)>-.47 fm3, a condition satisfied by c0 computed from phase shifts in the entire low energy region. Higher order correc- tions to be discussed in the next chapter, in particular the Ericson-Ericson effect, reduce the strength of the p-wave term. It is not clear, however, whether this reduction is sufficient to avoid difficulties. It is to be noted that although the pion wave- function is singular in the interior, its exterior behavior is not anomalous, and the calculated scattering cross sections are perfectly reasonable. The anomalous behavior Of the Kisslinger potential is due, Of course, to the approximations made, and in particular to the 3O Off-shell extrapolation chosen. The introduction of form factors, which eliminate the high momentum components, is one possible remedy. CHAPTER III THE FULL OPTICAL POTENTIAL As noted in the introduction, the first order treatment of the Optical potential described in the previous chapter was found to be inadequate for the description Of pion-nucleus processes, in particular the pionic atom level shifts and widths. This led to studies Of kinematic and second order effects in the Optical potential. In this chapter the various corrections tO the first order Optical potential which are incorporated into the calculations are derived. The first section Of this chapter deals with the kinematic transformation which was ignored in the simple Optical potential of the last chapter, that is, the transformation of the pion-nucleon T-matrix from the pion-nucleon to the pion-nucleus center Of mass. In the second section, the higher order multiple scattering terms are considered, in particular the second order s-wave term and a partial summation Of the p-wave terms known as the Ericson-Ericson effect. Terms which arise from true absorption are discussed in Section 3. Other corrections, due tO the Pauli exclusion principle and Coulomb distortion, are described in Section 4. Finally, the full Optical potential is stated in Section 5. 31 32 In order to give some indication of the effect Of the various kinematic and higher order corrections in pion-nucleus calculations, a representative set of differential elastic scattering calculations is shown where appropriate. Calculations for the nuclei 160 and 208Pb at 50 and 162 MeV are given to illustrate the nucleon number and energy dependence of the effects. Only n+ scattering is shown in most cases, since the h' scattering shows similar changes. The low energy data shown is that of Ref. 32 (diamonds), and Refs. 33 and 34 (triangles). The data at 162 MeV is from Ref. 35 (160) and Ref. 35 (208Pb). 1. Kinematics The pion-nucleon transition matrix T. = tIN which is required 1 for the Optical potential is simply related to the experimentally measured scattering amplitude an’ defined in the nN center Of mass, where |k| = |k'|. However, Ti must be known for k + p f O as well. If one ignores the Fermi motion of the nucleons within the nucleus Ti must be calculated in the pion-nucleus center Of mass. This is sometimes referred to as the angle transformation, since it involves the transformation of the angle between k and k' in the p—wave term, among other things. When Fermi motion is included the transformation depends on p as well as k. It is a straight- N forward matter tO relate tn for k + p = Q to the nN center of mass ~ amplitude for which 5 + p = 0. If, however, [kl f Ik'l in the frame in which E + p = 0, some assumption must be made about the off energy shell behavior Of tn”. An infinite number Of such assumptions exist. 33 A complete theory of the pion-nucleon system would provide a unique Off-shell extrapolation; however, such a theory does not yet exist. N The Kisslinger potential assumes that t1T is proportional to b0 + c k-k' for all k and k'. A different off shell amplitude can be Obtained if the scattering amplitude fflN = b0 ,, Cok'k' (III—1) is rewritten using 5-5' = %(k2 - qz) . (III-2) which gives - 1 1 2 - an - b0 + c0 q (III 3) 1.. 12 1-1 =1_ ° where b0 - bO +-§ k c0, c0 - - 2 c0, and q k k. Th1s leads to an Optical potential, with the simplifying assumptions of the previous chapter, 2duopt(r) = -4n{béo(r) + c6[Vzo(r)]} . (III-4) which is generally called the Laplacian model. Note that the V2 acts only on p(r), making this a local potential. The Equation III-2 34 is only true on shell, therefore the off shell behavior Of these two potentials is somewhat different. Figure 3 shows a comparison of differential cross sections calculated with these two potentials. The differences are pronounced, especially at 50 MeV, where the two curves are Of quite different character. At 162 MeV the curves have different magnitude but more or less the same shape. The partial cross sections also show large differences at 50 MeV. For 16 0 the reaction cross section calculated with the Kisslinger potential is three times that for the Laplacian potential, with a corresponding inequality in the total cross sections; the elastic cross sections are about equal. For 208 Pb the Kisslinger reaction cross section is also greater than the Laplacian; however, the total elastic scattering cross section for the Kisslinger poten- tial is only about half that of the Laplacian, leading to a smaller total cross section. At 162 MeV the cross sections are much more similar; those of the Laplacian potential are slightly larger. The Kisslinger and Laplacian potentials have been the most popular models for pion-nucleus scattering and are easily transformed to coordinate space. Another type of potential is known as the separable potential because the k and k' dependence of the pion- nucleon potential v is assumed to be of the form ig(k)g(k') in each channel, leading to a t"N Of the form :g(k)g(k')D(E) in each channel. The g's, known as form factors, reflect the finite range of the nN interaction. This assumed form is quite useful in that the form factors and D can be related to the nN phase shifts (37) and give 35 3 loTrtthTIIVVIT1[Ir: lostfrvt111IrvIty'1 3 ; 162 MeV I IO"I 11+ 100_ 1 103 E I 100, v- T 180 10, 1 10 E E If\\ : 2 1 : ‘. a-Is " “.1 I; f. ‘ /‘\ ° 1- _ 04 fl \ :i E 9" .D L‘ 1 '~ E r .1 (gr, \\ _. . I 10* 01 - a “O \ 100.- : 10" b = m ‘o L .1 b . 103 208Pb . ,1 100 , 10»— o E -. -’\ . 10 ._ . :5; , 1- .. O \‘_\ ~ 1 I \_\ 1r I; 3 ~ "‘~_. : 0.1 I 1- 10'2 0.1 . . 10-3 0 30 60 90 120 150 0 3O 60 90 120 150 ec.m. [deg] Figure 3. Comparison of calculations with the Kisslinger (solid curve) and Laplacian (dashed curve) Optical potentials. 36 a more realistic Off shell behavior for tn"; however, the coordinate space potential derived from such a theory is of an awkward non- local form. The Kisslinger form will be adopted for the Optical potential in this work. Because it is explicitly separated into s- and p-wave parts it is best for pionic atom analysis and low energy scattering. The form is convenient, also, for the calculation of higher order terms. In the low energy region the Kisslinger potential parameters vary slowly with energy. This is not true in the resonance region, however. It has been shown that when higher order multiple scatter- ing terms are included, taking account Of the correlations between nucleons, Kisslinger and Laplacian potentials give similar results (30). This is true since the interaction is Of short range and the cor- relations insure that nucleons are not close together, so that the potentials due to different nucleons are almost non-overlapping. Thus Beg's theorem (38) is applicable, which states that for non- overlapping potentials the scattering depends only on the on-shell part of the potential. Once the Off-shell behavior Of the pion-nucleon T-matrix is chosen, all necessary T-matrix elements can be calculated. One method Of doing this relativistically is given by relativistic poten- tial theory (39). The process is more complicated than a Lorentz transformation, since the T-matrix does not have well-defined trans- formation properties (see Appendix D). The relativistic potential theory provides a prescription (40) for relating . 37 where ET is the total energy of the pion-nucleon system, to t(w,q',q), where w is an energy parameter related to ET' The momentum q is related to k and p, and q' is related to k' and p', by equation D-2 of Appendix D, which is equivalent to a Lorentz transformation to the two-particle center of mass frame. Thus 3 = 5cm and q' = EOm' It is to be noted that the B for the transformation (k,p)-+ (3,-3) is not in general the same as the B for (k',p') +-(q',-q'). The quantity t(w,q,g') with Iql = Ig'l is just the on-shell T-matrix and with [3] # Iq'l is its Off-shell extrapolation. The exact expression for in terms of t(w,q,q') is given by equations D-5, D-6, and D-IO, along with an expansion of the result in powers Of 02, where Q = k + p = k' + 2', given by equation D—11. Keeping only the first term in the expansion, (Ei’Ellt(ET)lE’E> = (2W)3O(E| + E. - E - E)Nt(W,E'cmsEcm) (III-5) where - EwE'w' (Ecm + wcm)(EOm + mlcm) '1 N ‘ E w E' w' (E + w)(E' + w') (III-6) cm cm cm cm and 38 Note that s-02+¥fi gm=mg+mhi (III-8) _ 2 21 _ 2 21 w - (k + m ) wcm - (kcm + m ) are not the same as the on-shell values usually represented by these symbols. The off-shell forms discussed previously were given for an' Since tTIN is related to fTTN by TIN . _ __ I . t (W’Ecm’Ecm) ‘ ' zwcm an(w’Ecm’Ecm) (III 9) where w _ wcmEcm cm wcm + Ecm an Off-shell form is needed for the reduced energy 5cm' This is chosen tO be symmetric in incoming and outgoing particle energies, d E d' E' a = [ mfg .Cmf'g. ]* . (111-10) wcm cm wcm cm Taking the impulse approximation T. = ti" (III-11) 39 and recalling T. = 251- , (III-12) it is seen that Off-shell matrix elements of Ti also require an Off- shell form for 5, taken as __ w 'E' A w = [wwEA + EA 0). + 511] (111-13) with EA and EA the initial and final energies of the nucleus. Putting all these factors together gives (5"EilTi(ET)IE’Ei> = '4i(2">3 (III-14) E. x 6(5' + P% ‘ E ' pi)==“ N f hN(w’ Ecm’ 5cm) In order to simplify this result, the nucleons and nucleus are assumed nonrelativistic, so that II M E = E' = Eém = M and EA = EA = MA = AM . (III-15) cm This is not sufficient to make equation III-14 usable in a coordi- nate space calculation, as w and w' still depend on k and 5'. There- fore, w and w' are set equal to their on-shell values w = mi = “o 2 (kg + m2)*, (III-16) 40 With these approximations equation III-14 becomes = ’4“(2“)3 (III-17) 1 + e x 5(E' + Bi ' E ' Bi) 1 + e/A an(w’Ecm’Ecm) where “o e = TW' (111-18) “0 w0cm 2 and the difference between WT-and -7W—3 of order a , has been neglected. At 50 MeV e = .20; at 200 MeV e = .36. The arguments of f-nN’ Ecm and kc'm, must be expressed in terms of k, p, k'. and 2'. For this equation D-2 or equivalently the Lorentz transformation can be used. The latter gives ~(:m ~ ~ y+1~o~ (III-19) Eém = ’5' + é'Y'Wryfié' '5' ' m" where 5*? E'+F g = E w and § = E + w' (III-20) To make these expressions tractable, only terms of first order in g will be kept. (This is nearly equivalent to expanding to first order in 6.) At T“ = 200 MeV 3 .25 and the error due to the 41 dropped terms is about 6%. With the additional approximations given hiIII-ISand III-16, equation III-19 becomes Mk - mop k - ED 5cm = M + mo = 1 + E (III-21) Mk' - wop' k' - cp' k' = ~ ~ = :;____;;. ~cm M + “0 1 + e ’ and the p-wave term in fTTN is . . 1 2 =-—-— [5°E'-€+€ p-p'] . (III-22) ~Cm ~Cm (1 + €)2 ~ ~ The last term is of order 62 and should be dropped. However, it is an induced s-wave term and, as noted by Brown, Jennings, and Ros- tokin (41), is important since the first order s-wave term is unusually small. A more careful calculation, too tedious to be given here, indicates that this is the only important 52 term. Because the nucleon is part of a moving nucleus, the nucleon momentum should be separated into a part due to the momentum of the nucleus as a whole and a part due to the momentum of the nucleon relative to the nucleus, _ 1 E"KE+EO ””4” However, this separation complicates greatly the process of trans- forming from momentum space to coordinate space and only contributes 42 terms of order e/A which are negligibly small for all but the lightest nuclei. Therefore p and EU will be considered equivalent. ~ Let 2+9' 3*" E: 2 E: 2 (III-24) ?=E-B 9=F-E Then equation III-22 can be written .I .. 1 .|_2 o 5cm 5 cm - (1 + 8)2 E E 1 + e E E (III-25) The T-matrix is now expressed in terms of pion-nucleus center of mass quantities, and the first order term of the optical potential, -(1)- - zouopt - A (III 26) can be calculated. Omitting the spin term this is 26033.3; = -4wA<0!<2w>36<21+s-e;- 5'>{<1+e> 2 1 g 2 I + (Co + “13'3"“ e 5"}. ' 23'“ " % q + 1'1. 2'2 “10> ~ (III-27) 43 where it is assumed that p3 = pj for j f i. The integrations over ground state nucleon momenta of the various terms are given in Appendix C. (This process is called Fermi averaging.) The result is azuggfi (~.g') = ~4n{pl[boo(Q) + eflb1-pon1 Opt ~ 1 O n 1 p n - pily-[coo)1y +l<1-'h¥uoo)+ecm(o-ocoo(r) + - k COK(r)} where isovector terms have been suppressed for simplicity. It should be noted that any choice of kinematics must treat 5 and k' sym- metrically; otherwise the potential is not Hermitian and the results may violate unitarity. The energy parameter w, given by equation III-7, is the energy at which the scattering parameters should be evaluated and should also be expressed in terms of Bi and p; before the integrals over nucleon momenta are performed. This is not practicable, however, and w2 is evaluated with ET and Q the total energy and momentum of the nucleon and pion before collision, w2 = (E + w)2 - (E + 5)2 . (III-31) This is just the total energy in the pion-nucleon center of mass. 45 It should be noted that the same results can be obtained from much simpler assumptions (see for example Ref. 24). It is instruc- tive, however, to begin with a theory which claims to be relativis- tically correct and consistent. The approximations made are all explicit and the calculations required to improve the model are obvious, if not simple. Figure 4 illustrates the effect on elastic scattering calcula- tions of several choices for the kinematic transformation. The calculation with a first order optical potential with no kinematic terms, equation II-53 (dashed line), is compared to that with kine- matics as in equation III-29 (solid line), equation III-30 (dash- dotted line), and equation III-29 but without the K(r) term (dotted line). It is clear that the choice of kinematics has a non-negligible effect on the scattering from both light and heavy nuclei, not only at low energies, but in the resonance region as well. 2. Multiple Scattering Corrections Thus far only the first term of the multiple scattering series has been used in the construction of the optical potential. In this section the second and higher order terms of the series are con- sidered. These modify both the s- and p-wave parts of the optical potential. As the s-wave parameter bO is nearly zero, the second order s—wave correction, first derived by Ericson and Ericson(10), is quite important. The p-wave parts of the multiple scattering series can be summed to all orders in the low energy limit, giving rise to what is termed the Ericson-Ericson or LLEE effect (10) first 103 46 100 10 E 1 \ D s E. ca ‘0 \\ 100 b 'O 10 1 o. 1o 30 Figure 4. I'VTrYVIVI'VIIWY: losfiIfiv11r1rvy11't 50 MeV a ; 182 MeV f1+ “10"” «+- 1 103 I 100 160 . 160 1 10 . ZOBPb .“ . lv/‘uS .§. ‘ I § ‘ i. ' , ..... :‘ . '0'..... . ~. .: § 4 .‘f‘\ . /// ‘ a .0 \. I 9' :1 \l '1' h .' I 80 90 120 150 9mm. [deg] Comparison of calculations with no kinematic corrections (dashed curve), full kinematics (solid curve), and alterna- tive choices for the kinematic corrections (dotted and dot-dashed curves). 47 discussed by M. Ericson (43), analogous to the Lorentz-Lorenz effect in electromagnetism (44). The second term in the multiple scattering series for U, equa- tion 11-45, is A(A-1)¥iEO(1 - P0)A¥j . (III-32) Making the impulse approximation as before, the second order optical potential can be written 2$U(2)(k,k') z = f <0lA(A - 1)(2w)36(gi+5'-91-5") opt .. ... x (-4n)?(k',k") 2 12 [1 ‘ |0><0l] ~ ~ “k" + k0 + 15 3 ' n _ " d3kll x (2w) 6(22 +5 -22 - w—W M» (2 ,3 TI (III-33) where p% = pi is assumed for i f 1, 2, and f'is the pion nucleon scattering amplitude with the kinematics derived in the previous section included, We?) = p1(bo + has) * p17 (.0 + elm-'5' (III-34) - %(1 - p11)(c0 + C1E'I)q2 + p1(1 - p'l'l)2 c0 K(Q) - 48 . “0 Here p1 = 1 + a with e = 1T“ As the last two terms are already small, their contribution to the second and higher order optical potential is neglected. The second order s-wave term is generated from the s-wave parts of f(k',k") and f(k",k). The two terms of equation III-33, from the 1 and |0>2 1Lir-l- PE “3 + 2b§>c -i(k'-k")°r -i(k"-k)-r' x o(r)p(r')e ~ ~ ~ e " ~ * (111-35) 3 3 1 d3k" r d r' -k"2 + k + is (2n)3 xd 2 0 where 2 3j (k Ir - r'l) Note that the first term on the right-hand side of equation C-27 exactly cancels the |0> - 4an A (no + 2b1) [pm -i(k'-k)-r -ik-x eikolfl 3 3 x e ” ” ” [(-) C(x)p(r')e ” ”-——T—T—- d x]d r X 2 2 '1(k"k).r 3 =4np1——A——(b0+2b1) fp(r)e ” " ~Idr (III-37) The integral I, sometimes denoted <%>corr’ can be performed assuming: (1) an on-shell approximation, 5 = ko; (2) a specific form for the correlation function; and (3) p(r') approximately constant over the region in which C(x) is large. With the Fermi gas model value for C(x), equation C-28, and p(r') z 35%;-, the integral I can be done analytically for k0 = 0 or numeri3cflally for any given k0. For k0 = 0 the result is 3kF 10 =71?- . (III-38) With I approximated by a constant for a given k0, equation III-37 is a function of q only and can be Fourier transformed, giving ~ 25U§§)(5) = 4np§ fl—g—l (b3 + 2b§)1p(r) . (III-39) Ericson and Ericson (10) made somewhat different assumptions in their derivation of the s-wave effect, leading to a term in p2(r). 50 The form given here is taken from Krell and Ericson (45), and requires slightly less radical assumptions. The value of I, obtained by numerical integration, is shown in Figure 5 as a function of pion lab kinetic energy (solid line). In order to obtain a value of I for low energy scattering Thies (18) ikox ikO-r expanded e and e ” to first order in k0, resulting in 3k ____5 IT - 2n .+ ik0 (III-40) As can be seen in the figure, this is not a good approximation above about 10 MeV, as the real part of I falls quickly from its zero energy value égg” and the imaginary part does not follow k0 (dashed line). Both real and imaginary parts of I go to zero at high ener- gies, the imaginary part falling off more slowly than the real part. Because the second order s-wave term is proportional to p(r), it can be combined with the first order s-wave term, giving zauéglcg) = -4np1[30p(r) + eflblopm - pnnm (III-41) where -- _ A -1 2 2 __ b0 - b0 - p1 A (bO + 2b1)I . (III 42) Figure 6 shows b0 and Eb as a function of pion kinetic energy. As expected, the difference is greatest at low energies and also of greatest importance, as b0 is small there. Um") I Figure 5. (L8 09+ (L2 Oil The integral I as a function of pion lab kinetic energy, 51 L 1 Irn II FRe I 100 lTnUob) 1 200 [h4eVfl and the approximation Im(I) = k0. 52 (foo) [49 b 0.10 (fml law b Tallob) [MeV] Figure 6. The first order s-wave optical potential parameter b0, and the same parameter but with the second order correc- tion included, b0, as a function of pion lab kinetic energy. 53 In Figure 7 differential cross sections calculated with the optical potential including the second order s-wave term (solid curve) are compared with those calculated with the first order optical potential equation III-29 (dashed curve). Also shown at 50 MeV is the effect of treating I in equation III-42 in the zero energy approximation, I = :;E-(dotted curve). At 50 MeV the second order s-wave term makes an appreciable difference, especially at backward angles. It has no effect at all, however, in the resonance region. The second term of equation III-33 to be considered is the s-p interference term. This arises from ilE',E")?(k".5)==(bo i b1E'I)(Co * c 5'3)(5'°5" + EHOE) ~ (III-43) and is zero by symmetry. The second order term due to the p-wave parts of f(5'.5") and f(k",k) in equation III-33 is -m) 2 5(Wkuu"p 211Upp (E, E )= ( 4'”) 2(A A ' 1)p1 f<0l(2fl) _2ku + k3 + i8 XMfi+E'-h-kWU-IW¢HM%+ +k"- W'k) ~ d3k" (hfi x (Co + C13'11)(C0C13 32’1”) (III-44) 54 3 lo TIVYIYfTTTTYUIIV: losrT'UUTfthitTilr so MeV : ; 182 MeV 11* : 1°“ «4’ '1 : 100 1 103 I 100 1 180 10 1 10_ I 1 \'\ a II t .. 0.1 ‘ ~‘ (I 10'? 100 10“ dcr/dQ [mb/erl 103 100 10 i 10 " l l '1 g on I -4 10-2 0.1 IO-BJLILIILJILLIAL 0 30 80 90 120 150 0 30 60 30 120 150 9mm. [deg] Figure 7. The effect of the second order s-wave term (solid curve) compared to calculations with the first order term only (dashed curve) and with the second order term in zero energy approximation (dotted curve). 55 The operator (k'-k")(k"'k) can be written as the sum of a second rank tensor and a scalar in k", I. II n. = I II II _ l 112 l 112 . l _ (15 '5)“: 5) 123 kikjEkikj 3 513k ]+ 3 k 55 (11145) The tensor part is negligible at low energies and will be ignored at all energies considered here. As in the s-wave term, the inte- grals over nucleon momenta give the Fourier transforms of functions of r and r'. The remaining terms in k" can be integrated over as before, . u. . ik x ..2 1 ‘5 (5’5) d3k" _ 2 1 e 0 k 2 2 e --j§ - 'Vx[' zf"‘jf" ] -k" + k0 + 15 (2n n (III-46) where x = r - r' and x = |x|. This gives two terms, L v2 elkox = -5(x) - 539—159: (III-47) 4n X X 4n x ° In the zero energy limit only the first term contributes, giving r = r'. As was noted in Appendix C, the first term in brackets in equation III-44 gives zero contribution for r = r', assuming hard core repulsion between nucleons. Equation C-9 gives for the second term 1:15.15.» =1 >2——- *1 m .(k k.” (III-48) + t3c1(pp(r) - pn(r))]2 e1 ~ ” k k' 'd3r in momentum space or Zwéghr) = -(4 4102 éLA—l V-[cooM + t3c1(op(r)- on(r))]2y (III-49) in coordinate space. The p-wave terms can be calculated to all orders 'hi the zero energy limit, assuming tensor terms in the intermediate momenta do not contribute. The delta functions which appear,6(r-r')6(r'-rf0 and so on, insure that only the P0 pieces of the (1 - P0) operators are nonzero. Thus the Nth order p-wave term is proportional to pN(r). To all orders the p-wave terms are ziupph) = 3°4npilcoo(r) m§0 [- fl31 LA—l‘ copilo(r)]"‘y -1 (111-50) 411pl cop(r) V 1 1 - [- 531 A 5,1—1 pilcooh‘fl ” =V° where the isovector terms have been suppressed for simplicity. This is the Ericson-Eriscon or LLEE effect, with A = 1. The Ericson-Ericson effect can be calculated more carefully, giving a term of the same form but with A different from one. The 57 best value for A has been a matter of dispute. Recent calculations by Brown, Jennings, and Rostokin (41), including n, p, and w meson intermediate states and taking into account the finite range of the pion-nucleon interaction, yield a value for A greater than one. They note that although the finite range of the interaction causes a reduction in the Ericson-Ericson effect, as noted by Eisenberg, HUfner, and Moniz (46), other terms strengthen it, the net effect being a value of A which is about 1.6 or higher (47). As their calculations were done in the low energy limit, their conclusions apply to the low energy region only. Oset and Weise (48) have also made an estimate of A for low energies, based on calculations in the isobar-hole model. They give a value for 1 in the range A = 1.2 - 1.6. For this work the value A = 1.6 is ad0pted as reasonable. In the case of nonzero pion kinetic energy each term in the sum, equation III-50, should be modified by the effect of the second term in equation III-47. However, the second term is much more difficult to manage than the first, therefore its effect is calcu- lated here to second order only, giving a rough idea of its impor— tance. The second order term is quite similar to the s-wave term, and can be evaluated in a similar manner, giving “1'! 3 2 % kgk-k'kg + 2%): fp(r)e d r , - 41T A—A"l p1 (III-51) 58 where I is defined in equation 111-37. The p-wave optical potential in coordinate space is, with the inclusion of this term, %gufl—fi—l 1 p11 [cop(r)i-t3cl(pp(r)"On(r))] ~ -1 U (r) = 4nV°{ ”1 [cop(r’ * t3¢1(op - on(r))] pp 1 + + A A 1 PIIC'D(P)} y (III-52) where + 2c§)1 . (111-53) Note that the isovector part of the c' term has been neglected. Figure 8 gives the energy dependence of c', as compared with that of co. As c' goes as k3 it is small at low energies, and dies away at high energies due to the falling off of both I and the p-wave scattering parameters. It is, however, a large effect in the resonance region. In Figure 9 the effect of the Ericson-Ericson correction is illustrated. The differential cross sections calculated with only the first order p-wave term (dashed curve) are compared with those calculated with the full Ericson-Ericson effect with A = 1.6 (solid curve), with A = 1.0 (dotted curve), and with the Ericson-Ericson effect to second order only, A = 1.0 (dash-dotted curve). At 50 MeV the differences are quite large; at 162 MeV there is a difference at backward angles between calculations with and without the 59 0.50 lfm'l C 0.001 Re 0.50 > (fm'l C o-oc : ‘A : 4'” 4‘ In1 100 200 TnIlob] (MeV) Oviviv Figure 8. The parameter c' compared to the first order p-wave parameter cO as a function of pion lab kinetic energy. 60 103 5 T 1 I I I 1' I I V I I fat I T 7 lo 50 MeV 162 MeV fl+ 10“: «+ 100 \ 103. #111W11t1fvyivjw 100 _ O O I 1 1 11141 180 10 10 O i J 1 LL111I 0.1 ~ 10‘? 100 10“ dG/dQ [mb/srl 103 100 10 10 g-o .... 0.1 10'2 1 11$4110-311111IJ1L11 l 1 l 1 L4! 1 80 90 120 150 0 30 60 90 120 150 Own. [deg] Figure 9. The effect of the Ericson-Ericson correction to the p-wave, A = 1.6 (solid curve) and A = 1.0 (dotted curve), compared to calculations with no correction (dashed curve) and with the correction included to second order only, A = 1 (dash-dotted curve). 61 Ericson-Ericson effect, but almost none between the full effect and the second order approximation to it. Figure 10 shows calculations with (solid curve) and without (dashed curve) the c' term. Because of the crudeness of the calcula- tion it is likely that the effect is greatly overestimated, espe- cially at higher energies. Other terms such as the tensor terms may also begin to contribute in the resonance region. Therefore this term is dr0pped in future calculations. One other comment should be made about the Ericson-Ericson effect. The pion-nucleus T-matrix given in equation II-30 can be expressed in terms of :1, using the same reasoning that leads to II-38. The result is (29) £=~r. +22 $.80; +2 2 Z $160$jéoT i i jfi j i jfi kfj (III-54) This series, the Watson multiple scattering series for T, is just the Born series with an effective potential :1. Using the same arguments which lead to the Ericson-Ericson effect, it is seen that the second term of equation III-54 is zero. Thus the first Born approximation should be a reasonable approximation to the full cal- culation. This argument breaks down, of course, for 1 f 1.0. 3. True Pion Absorption Terms The conventional multiple scattering series includes only intermediate states consisting of one pion plus the nucleus, since 62 3 5 lo TIV'Tf'TTrterVT10_1111T[vt[#rltuyu 182 MeV q 10 : “4» 100 160 10 ... 100 dcr/dQ [rnb/arl 103 - 209% 10 0.1 10'3 o 30 so so 120 150 o 30 so so 120 150 9cm. [deg] Figure 10. The effect of the c' term (solid curve) compared to calculations without this term (dashed curve). 63 the T1 in the multiple scattering series correspond to pion nucleon scattering. Because the pion is a meson, however, it can be absorbed on one or more nucleons, resulting in intermediate states with no pion. This process gives a non-negligible contribution to both the real and imaginary parts of the optical potential, and is especially important at low energies. The parametrization of this process as terms in the Optical potential is discussed in this section. Analogous to the scattering T-matrix T, define an absorption operator 1 and emission operator 1+ for the pion, describing respec- tively the processes Nn + N and N +an. Although there is some contribution from pion absorption on one nucleon at very low energies, the dominant absorption mechanism is two nucleon absorption, in which the pion scatters from one nucleon and is absorbed by the second, with the two nucleons sharing the kinetic energy due to the disappearance of the pion mass. The lowest order optical poten- tial term due to such a process can be written <01A(A - 1)TGO(1 - P0)MTGN(1 - P0)ATGO(1 - P0)ATIO>, (III-55) where GN is the propagator for the nucleus. Although this is, in a sense, a fourth order term, it has a large imaginary part, describing flux lost from the elastic channel due to true absorption of the pion. At low energies the other imaginary terms in the 64 optical potential, due to quasielastic processes, are nearly zero, and the absorption terms make the dominant contribution. In their discussion of pionic atoms Ericson and Ericson (10) introduce absorption by defining a pion-two nucleon amplitude describing pion absorption and reemission, W = B0 + C 'k' (111-56) 2) 0E2cm ~2cm where the subscript 2cm refers to the pion-two nucleon center of mass frame in which f(2) is defined. The B0 and C0 can be deter- mined in principle from the various amplitudes of the angular momen- tum and isospin states of the pion—two nucleon system. A fair amount of data does exist for the reaction n + d +rN + N, but it is difficult to eliminate effects of deuteron structure from the amplitudes. The amplitude f(2) can be related to expression III-55 by defining ¥(2) a $80(1 - P )AT+G (1 - P )118°(1 - P )1? (111-57) 0 N o o and T(2) 1 1 ;(2) (III-58) “(2) where'Ekz) is the reduced energy of the pion-two nucleon system in that system's center of mass. Then the impulse approximation gives 65 ;(2) 1 _4flf(2) (111-59) Equation III-57 may include more than s- and p-wave terms; however, the first two partial waves are assumed to dominate. As the amplitude f(2) is defined in the pion-two nucleon center of mass, a kinematic transformation to the piononucleus center of mass is required for 1(2). The results derived in Section 1 for T can be immediately applied to 1(2) with the substitution M +42M in the kinematic factors. Thus the lowest order absorption terms in the optical potential are zauggis) (5,5') 2 <0|A(A - 1)r(2)10> = x A(A - 11113230 + pglcog-g' - %(1- pglxoqzilw where the term analogous to K(r) has been neglected. Note that the details of the two nucleon absorption process are buried in the parameters B0 and CO. By expression C-31 of Appendix C the Fourier transform of equation III-6O is -— (abs) _ -1 . 2onpt (r) - -4n[p28062(r,r) - p2 COY 02(r,r)Y (III-61) + %(1 - PEI)COV202(r.r) 66 The two-body density p2(r,r) is replaced with 92(r), assuming all correlation effects are included in B0 and Co. This rather awkward situation is due to the simplified form of equation III-60. A more careful derivation is given by Rockmore, Kanter, and Goode (49). They write the absorption term of the optical potential schematically as — (abS) I 2 * I I + I I I zwuopt (§ ’§) ” A .lfw (EI’CZ’E3"°EA)U (EI’CZ’ 5 ) (III-62) x G (r' r' r r )U(r r ° x)w(r r r r )d3r'd3r21‘ Id3r N ~l’12’~1’~2 11’12’ ~ ~1’~Z’~3°"~A 1 'i where U and U+ are the two nucleon absorption and emission operators and x and x' are the pion coordinates. The two nucleon propagator GN is diagonal in momentum space, 1 G~ = 6(21 - Ei)5(92 - 21> p§ pg w-m-ffi-ie (III-63) neglecting interactions between the two nucleons. Let r + r r' + r' _ _ ~1 ~2 . _ ~1 ~2 E'ENEz B 2 3""2' P - P 2=3%fi [=Q‘E1 :'=Q-ri (III-64) 67 Then the Fourier transform of equation III-63 is H "(28' RH“ ' E1) G~ = f6 .1 , . "(23* E) (E2 ‘ '32) 1 d3P d3p 2 2 3 3 w _ %fi__ %T__ is (2n) (2n) (III-65) Xe The dinucleon momentum E is, by momentum conservation, roughly the same size as k, the incoming pion momentum. The relative momentum of the two nucleons, p, is large, however, because of the kinetic energy gained when the pion is absorbed. Therefore the 2; term in GM is dropped. This gives ip-(r - r') 3 G~(‘3"E"E’E)=4(B-E') fe‘ “' “ d” 1 2 3 = 6(R - R')G (r - r') 1 ~ pO 1 1 (III-66) where p0 =‘VBM and eipx Gp(x) ~ x (III-67) Putting this in equation 111-62 and making the coordinate trans- formations III-64 gives 68 —-(abs) . 2 E. E. zwopt (5 .5) ~ A [1*(3-7,3+-§—, 5 5A) x U+(R,r',x')6(R - R')Gp (r - r')U(R, ,x) .... .. .. .. O- ., 11.. r , 3+ %, r 5A)d3Rd3R'd3 mlz-s . 3 x¢(8- rdrigzdri. (III-68) The wavefunction product is expanded about 5, giving (III-69) where the functions f insure the proper correlations between the two nucleons, f(0) = O and f(r) +-1 for r large. With zero range interactions equation III-68 becomes a local potential and the U+ and U depend only on r' and r. Thus the absorption term of the 2 with the effect Of correlations included optical potential goes as p in the integrals over 5 and 5' which give rise to the absorption parameters. The lowest order absorptive terms in the Optical poten- tial are then 26U§§25’(r) = -4w[p280o2(r) - pglcoy-p2(r)y (III-70) + 111 - p511c0v2o2(r)] 69 The only higher order multiple scattering terms due to absorp- tion to be calculated here are those which contribute to the Ericson- Ericson effect. The other terms involve long range correlation functions of three or more particles, which are not well known. It is to be hoped that they make only small contributions to the Optical potential and can be ignored. The absorptive terms can be included in a simple way in the Ericson-Ericson effect by replacing 2) given by equation III-59. T in the derivation by T + 1(2), with T( As 1(2) is a two nucleon Operator, the delta functions Of position, 6(r - r'), 6(r‘ - r"), and so on, become delta functions of the coordinates Of one particle and the center of mass of a pair Of particles, or of the centers of mass of two pairs. The two body density p2(r,r') is replaced by three and four particle densities. It must then be argued that the nucleons in the pair are close enough so that the three particle density is zero when the coordinate of one particle corresponds to the center of mass of a pair, and simi- larly with the four particle density and two pairs. With these assumptions the Ericson-Ericson term is Pilcoob‘) + PEICODZU‘) 25U(EE)(r) = -4nV° 4" V opt ~ A-l -1 -1 2 ~ 1 + 3 A—'_A [p1 COD(Y')+P2 CDC (7‘)] (III-71) for an N = Z nucleus. With the form of the absorption terms determined, it is neces- sary to choose the parameters B0 and C0. Rather than attempt to construct these directly from n + d + N + N amplitudes, the results 70 of calculations Of B0 and CO for nuclear matter by Riska, Bertsch, Chai, and Ko (50) will be adopted. Only a brief summary of their calculations will be given, as they are fully described in the ref- erences cited, and are fairly long. Riska and collaborators write the lowest order two particle absorptive term of the Optical potential U(abs)_ 1 t. E? 2:T f{ E. + - E op ffi Ti w f - in6(Ei + w - Ef)}Tfi (III-72) where P stands for principle value. The Tfi are the two nucleon absorption and emission operators, with i and f labelling the nuclear states. Note that this expression is simply related toequation III-57 for 1(2), with the expression in brackets above equal to GN' The Tfi are Obtained from the evaluation of the diagrams shown in Figure 11. For the s-wave, the rescattering vertex is described by a phenomenological Hamiltonian (51), H = 4.”. Tn—w ¢.¢w +411- __g.+-[ WT 9 11¢ , (III-73) where 4+ and w are nucleon field Operators and ¢ and n are the pion field Operator and the momentum operator conjugate to it. The coup- ling constants A1 and 12 are determined by the n-N phase shifts. The pion absorptiOn vertex is described by H = - %N(Zo§7’)g.gp , (111-74) 71 S-WOVG ff . 1---... ,. ,,...::.... N N N N p-wove A ’f/IA i N N N N Figure 11. Diagrams of the s and p-wave processes included in the calculation (50) of the absorption parameters. 72 f2 where 4“ =0 081. The p-wave rescattering is assumed to be dominated by the A33 resonance, as shown in the p-wave diagrams Of Figure 11. Both n and p intermediate states are included. The Lagrangians for the various vertices are LnNN ‘%“’+3'V(? I” gO LONN=-2-fi(1+l()1p(oxv):6 (III-75) f + + LnNA=FATZTVWThHC 90131., LpNA = T13 (‘7 x B” T h°c° where w is the nucleon operator, Q the delta Operator, which is a vector-spinor in spin and isospin spaces, 9 the pion operator, and E the rho Operator, which is a vector in spin and isospin spaces. The symbols "+" and "~" to denote vectors are both used in order to distinguish vectors in differint spaces.2 The values of the coupling constants adopted are 4g = O. 32.-2% = 0.55, K = 6.6, and 90A = 9%: gp(l + K). The vertex factors are combined with the propagator for the intermediate particle to give the two nucleon absorption Operator T. The initial and final nuclear states for which T is evaluated are two nucleon states in nuclear matter, i.e. plane wave states, 73 which are symmetric or antisymmetric in the two nucleon coordinates, chosen so that the total wavefunction including space, spin, and isospin parts is antisymmetric. The isospin and spin sums over the two particle initial and final states are performed first, assum- ing equal numbers of protons and neutrons. Multipole expansions are made of the initial and final wavefunctions and T, and the nucleon coordinates are expressed in terms of the center of mass R and relative coordinate r, with the integral over R giving conserva- tion Of momentum. The sums over initial and final spatial states are converted to integrals over momenta, and the final integrals are performed numerically. The s- and p-wave optical potential parameters thus Obtained are shown as a function of energy in Figure 12. The imaginary part of 80 increases with energy; the real part is approximately zero at zero energy and becomes more negative with increasing energy. The imaginary p-wave parameter shows a peak near resonance, as does the real part. It should be noted that the real parts of the absorptive terms of the Optical potential are, unlike the imaginary parts, quite sensitive to Pauli blocking effects, and somewhat sensitive to the radius assumed for the hard core repulsion, and the form factors assumed for the pion and rho meson. The inclusion of Pauli blocking increases the values Of the real parts; thus the values given here can be considered lower bounds on these numbers. 74 5“ E I: T f I 1 1 r T l 1 3.0-” 2.0— (I? E 2: 1.0- _ Re Co 1 0.0 4 L 1 l 1 L 1 l l 0 $0 100 150 200 Figure 12. The absorptive parameters BO and Co as a function of pion lab kinetic energy. 75 There are several problems with calculations of this type. Because it is a nuclear matter calculation, the form of the absorp- tion terms for a finite nucleus must be derived separately; hence the inclusion of the argument of Rockmore et al. A more serious problem is the possibility of double counting in the p-wave, as part of the amplitude equation III-57 looks like a third order mul- tiple scattering term with far Off shell components, and is there- fore already included in the Ericson-Ericson effect derived in the previous section. This problem is perhaps best resolved by a dif- ferent approach, in which absorption and scattering terms are con- sidered together at the outset. Other calculations of these parameters have been performed. The s-wave absorption parameters have been calculated at threshold by Hachenberg and Pirner (52), who Obtain the value B0 = (0.375 + i 0.144)fm4. This has a somewhat larger imaginary part and much larger real part than the Riska value B0 = (0.0067 + i O.080)fm4. G.A. Miller (53) has calculated the p-wave parameter; his values are 00 = (-0.05 + i 0.57)fm6 at 50 MeV and c0 = (-0.03 + i 0.72)fm6 at 150 MeV. Oset, Weise, and Brockmann (54) have also calculated C0, obtaining (0.96 + i 0.64)fm6 at threshold, (1.20 + i O.88)fm6 at 50 MeV. These numbers can be compared with those calculated by Riska et al., 00 = (0.287 + i 0.343)fm6 at threshold, (0.373 + i 0.622)fm6 at 50 MeV, and (1.20 + i 2.55)fm6 at 150 MeV. Figure 13 shows the effect of the absorptive terms on the elastic scattering cross sections. Shown are calculations without 76 dU/dQ (mb/sr) Figure 13. 30 5 lo ‘jir'trr'jrrfiII 162 MeV T1| 10 _ “4» 103i 100’ 160 10 04_ urz, 10“ 103 . ‘ ZOBPb 100 / 10 Aq 1 Q J\ 0.1 10"2 -3 60 90 120 150 10 0 30 60 90 120 150 own. [deg] The effect of the absorption terms (solid curve) compared to calculations without these terms (dashed curve) and with the absorption terms included to first order only (dotted curve). 77 absorption (dashed curve), with absorption as discussed (solid curve), and with absorption included to first order only, i.e. with the C0 term not included in the Ericson-Ericson effect but added separately (dotted curve). Clearly absorptive terms Of the size calculated by Riska and collaborators are not negligible even in the resonance region. Although the absorption terms have both real and imaginary parts, most of the effect on the scattering cross sections is due to the imaginary parts both at low energies and in the resonance region. Near resonance the effect of the real parts is not seen at all except at very backward angles. The reaction and total cross sections are not greatly changed by the inclusion of absorptive terms. The reaction cross section for 160 at 50 MeV increases by 30%; the total by 14%. For 208Pb at 50 MeV the increase in the reaction cross section is only 7%; the total cross sections are nearly equal. The differences are much less pronounced at 162 MeV, with almost no difference at all 208 for Pb. 4. Pauli and Coulomb Corrections Two corrections to the optical potential remain to be made. The first is due to the Pauli exclusion principle, limiting the intermediate states accessible to the struck nucleon. This correc- tion has already been made for the isoscalar s-wave part by explicitly calculating the second order term which includes the Pauli correla- tions. The calculation of the corresponding p-wave term could only be made in second order, but the Ericson-Ericson effect includes 78 terms Of all orders. Therefore this method is abandoned and the effect of the Pauli principle is approximated by reducing the imaginary parts Of the parameters c0 and c1 by a factor O which corresponds to the fraction of phase space available to the nucleon. This is also done for b1, as the second order isovector term was not calculated. The Pauli factor Q is taken from the Goldberger (55) classical calculation as given for pions by Landau and McMillan (56). The particles in the nucleus can be thought of as occupying a sphere in momentum space of radius kF = 1.36 fm'l. When a pion of given momentum strikes a nucleon the nucleon may or may not gain enough momentum to displace it into the allowed momentum region outside the Fermi sphere. Q is the probability that for a given pion momen- tum the nucleon will scatter into the allowed region, averaged over nucleon initial momentum and scattering angle. Figure 14 shows 0 as a function of pion kinetic energy. As expected, 0 goes to zero at zero energy, as no states are accessible, and approaches one at high energies. The absorption parameters also have imaginary parts; however, as the nucleons gain a large amount of momentum when the pion is absorbed, they are likely to be in unoccupied states. Therefore, no correction is made. Figure 15 shows the effect of the Pauli corrections just dis- cussed (solid curve) compared to calculations with no Pauli factor, i.e. Q = 1 (dashed curve). At 50 MeV 0 =0.31; at 162 MeV Q = 0.75. 79 0.0 1 l L 0 100 200 Figure 14. The Pauli factor Q as a function of pion lab kinetic energy. 80 103 . 105 Y j r Y j I V I h I T I I T r 182 MeV I. 10 ; 11+ 100 160 10 H I dcr/dQ [mb/srl 103 .’ zoepb 10 I,. ‘ , , 4 100 ‘ / \ 10 I 1 01 T; 10'2, 0.1 10-3 0 30 60 90 120 150 0 30 60 90 120 150 6mm. (deg) Figure 15. Comparison of calculations with (solid curve) and without (dashed curve) Pauli corrections. 81 Although the Pauli effect is larger at 50 MeV, the changes resulting from its inclusion are more pronounced at higher energies, where the scattering is more sensitive to the magnitude of the imaginary p-wave terms. As is evident in both Figures 13 and 15, the effect of increasing the imaginary part of the Optical potential in the resonance region is to raise the differential cross section curve and make the minima shallower. The second correction to be discussed is due to the Coulomb interaction. This correction is necessary because the electromag- netic part of the interaction was not treated consistently in the derivation Of the Optical potential; the Coulomb potential was ignored until the end of the calculation and then put back in. The needed correction can be considered in the following way: When a negative pion approaches the nucleus it is accelerated by the Coulomb field; a positive pion is decelerated. Thus positive and negative pions strike the nucleus with different effective energies. TO account for this the scattering parameters are calculated at an energy different from the incoming pion energy by EC’ the magnitude of the Coulomb field at the nuclear surface, assuming the inter- actionis.surface peaked. This Coulomb shift is an important effect for large nuclei in the resonance region. It is not so important for light nuclei and at low energies, where the parameters vary only slowly with energy. Values Of Ec for various nuclei are given in Table 2. 82 Table 2. The Coulomb energy shift EC’ evaluated at the nuclear surface, for various nuclei. Nucleus Ec (MeV) 4He 1.6 7Li 2.1 93a 2.5 120 3.4 160 4.2 24Mg 5.4 27Al 5.7 28Si 6.0 400a 7.5 56Fe .9 58Ni 9.5 gozr 11.7 12OSn 13.3 208 Pb 18.1 83 TO illustrate the effect of the Coulomb shift, the elastic scattering of 0+ and n- from 16O and 208Pb at 162 MeV is shown in Figure 16 with (solid curve) and without (dashed curve) the Coulomb shift in the energy of the parameters. The main effect of this is to deepen the minima for n" and make them more shallow for n+. This is important, as the data for n+ and n' in the resonance region are quite similar, but the calculations give pronounced differences when the Coulomb shift is not included. 5. The Optical Potential The various pieces can now be put together to give the full Optical potential, zwuopt - on(r))] + pglcopz(r) } ~ 4— 1.;lcc..(r>+e.c.-4401454044} ~ + %(1 - pil)V2[cOp(r) + eflc1(pp(r) - on(r))] (III-76) + p1<1 - p1112c0K1 where all i-terms have been dropped. The parameters b0, c0, b1, and c1 will be denoted single nucleon parameters, to distinguish them from the absorption parameters B0 and C0. The parameters are 84 0.1 10‘? dcr/dQ [nib/er] Figure 16. 30 vv[vt[1fi]vv'v lofitltvlfijt111itv 160 ZOBPb 152 MeV 1°" 162 MeV 103 100 . 11' 11' 10 0.1 ”A U l I -—--- .. o'" I \ \ I \ I 10'? 10" 100 \\ 10 1 A \ “I ----- 0.1 104 -3 60 90 120 150 10 0 30 60 90 120 150 own. [deg] . + - . Egmparis% of calculations of n and n scattering on 0 and8 88Pb at 162 MeV with (solid curve) and without (dashed curve) the Coulomb shift. The references f r she n data are the same as for the corresponding n ata. 85 to be calculated at the energy w - eflEc. The imaginary parts of b1, c0, and c1 are multiplied by the Pauli factor Q. For all calculations the difference in radii between the proton and neutron distributions is ignored; thus pp(r) - pn(r) can be replaced by Zifl-Mr). The parameters of the charge distribution are taken from the available tables (57,58). For the light nuclei the matter distributions are obtained from the charge distributions by adjusting the size parameter such that = Rc - 0.64 fm , (III-77) where R is the radius of the equivalent uniform distribution, R2 = §-. The density forms and parameters for various nuclei are given in Table 3. This completes the construction of the optical potential. The next three chapters will deal with the calculation of various experimental quantities using the potential equation III-76. Table 3. 86 The forms and parameters of the matter and charge density distributions for various nuclei. 2 2 2 _ 2 2 2 -r la o<==(1+ot-r—2)e'Hal por(1+a-"—2)e c a C a C a(fm) ac(fm) a 7L1 1.67 1.77 .327 12C 1.57 1.66 1.33 16o 1.75 1.83 1.54 oaoc“[1 + 6(r'R)/Z]'1 R(fm) z(fm) 27A1 3.07 .519 40Ca 3.51 .563 55Fe 3.97 .594 63Cu 4.21 .586 9°2r 4.83 .496 1205n 5.32 .576 208 Pb 6.46 .549 CHAPTER IV PIONIC ATOM LEVELS Measured pionic atom level shifts and widths provide an impor- tant test of the optical potential at essentially zero pion kinetic energy, as the data furnish information about the overall strengths of the s and p-wave parts of the real and imaginary potential. As several approximations were made in the low energy limit, the poten- tial form should be most reliable at zero energy. The parameters derived from "N phase shifts are not well known at low energies, however; the values used in this analysis are taken from the Rowe, Salamon, and Landau (RSL) fit (25), extrapolated to zero energy, and are known only within rather large error bars. Pionic atom calculations are not new; however, a new analysis is worthwhile and necessary because the data has improved greatly in quality in the last few years, because the form of the optical potential used in the present analysis is slightly different from that used in earlier analyses, and because detailed calculations of X and the absorption parameters are available for the first time. It will be found that the optical potential derived in the previous chapter, with parameters taken from the sources indicated there, does not reproduce the pionic atom data. However, the form can be used with fitted parameters to give an excellent description of the data. 87 88 The first section of this chapter describes the general features of pionic atom levels. The details of the calculations of level shifts and widths are discussed briefly in Section 2. The results of calculations with the optical potential as derived, and the same optical potential but with fitted parameters, are discussed in Section 3. 1. General Features A pionic atom is an atom in which a negative pion is bound in the Coulomb potential of the nucleus in the place of an electron. Because of its large mass, the pion orbitals are much closer to the nucleus; the pion Bohr radius is £§-=-§%§ times the electron Bohr radius. Thus, in the lower orbitals the pion is close enough to interact strongly as well as electromagnetically with the nucleus. This causes a shift in and broadening of the energy levels relative to the positions and widths to be expected from the electromagnetic interaction alone. The energy shifts and widths are obtained from measurements of the pion transition x-rays. As the strong interaction effects are largest for the lowest 2 states, in which the pion is closest to the nucleus, most of the difference between measured and electro- magnetic transition energies and widths is. due to the lower state. The general features of the data are evident in Table 4, which gives the calculated electromagnetic energy of the transition, experimental s-wave shifts and widths from Ref. 59, and experimental p-wave shifts and widths from Ref. 60. Some d and f wave levels have been measured 89 Table 4. Experimental pionic atom energy level shifts and widths in keV from Refs. 59 and 60. S'Wave Nucleus (Ep - ES)EM AES(6XP) Ps(exp) 108 68.714 - 2.977 1 0.085 1.59 1 0.11 120 99.066 - 5.874 1 0.092 3.14 1 0.12 14M 134.740 - 9.915 1 0.144 4.34 1 0.24 160 175.413 -15 03 1 0.24 7.64 1 0.49 19F 220.952 -24.46 1 0.35 9.4 1 1.5 2ONe 270.952 -33.34 1 0.50 14.5 1 3.0 23Na 327.131 -49.93 1 0.71 10.3 1 4.0 Q'Wave Nucleus (Ed - Ep)EM AEp(€XP) Pp(€XP) 27A1 87.270 0.201 1 0.009 0.120 1 0.007 2851 101.283 0.308 1 0.010 0.192 1 0.009 325 132.510 0.635 1 0.016 0.422 1 0.018 400a 207.674 1.929 1 0.014 1.590 1 0.023 55Fe 352.356 4.368 1 0.088 6.87 1 0.21 63Cu 439.016 6.67 1 0.24 11.4 1 0.8 64Zn 469.995 6.44 1 0.33 12.4 1 1.4 90 also (61), but will not be discussed here. As might be expected, the level shifts and widths increase with increasing Z. The s-wave shifts are negative, indicating a repulsive s-wave interaction; the p-wave shifts are positive, indicating an attractive p-wave potential. In the calculation of pionic atom shifts and widths the various parts of the optical potential are correlated with the measured quantities: the values of the real s- and p-wave optical potential parameters determine the s- and p-wave level shifts, and the imaginary parameters determine the corresponding widths, to a good approxima- tion. As there are only four independent quantities to be measured, only four optical potential parameters can be determined correspond- ing to the overall real and imaginary s- and p-wave strengths. Thus Re(b0) and Re(BO) cannot both be determined, nor can Re(c0) and Re(C0) from data on one nucleus. Although it is possible in prin- ciple that certain combinations of these related parameters reproduce the A dependence of the data better than others, in practice all reasonable combinations give similar results. That is, the combina- tion Re(b0)p(r) + Re(BO)pz(r) acts like [Re(b0) + Re(BO)pav]p(r) with pav independent of A. The same is true for the p-wave param- eters. Although there are correlations between particular parameters or combinations of parameters and particular measured quantities, there is a fair amount of mixing. Some of this is due to the kine- matics, which introduce s-wave terms with p-wave coefficients, some to the Ericson-Ericson effect which mixes real and imaginary terms 91 in the p—wave. Even without these, however, the s-wave optical poten- tial strength affects the p-wave quantities, and to a lesser extent, vice versa. It was evident from early measurements of pionic atoms that the first order optical potential with parameters determined from phase shifts could not describe the data. The calculated s-wave strength was too small, the p-wave strength was too large, and the widths could not be explained at all, as the imaginary parts of the single nucleon parameters are zero. This was the original moti- vation for the calculation of the second order s-wave term, the Ericson-Ericson effect, and the s- and p-wave true pion absorption terms. 2. Details of the Calculations The pionic atom level shifts and widths were calculated using the program MATOM written by R. Seki (62). This is a position space code which sets up and solves the eigenvalue equation for the pion wavefunction, obtaining the complex eigenvalue corresponding to the energy and width of a given state. The wave equation is first reduced to an equation in r only, noting that the Optical potential is independent of angle for a spherical nucleus, u; + f(r)ui + [g(r) - £1£—%—ll]u£ = 0 (IV-1) r where 92 f(r) CC; r_ 1 (IV-2) and 9M (1- c(r))'1{c—'-§.fl + 1.12- m2 - 25% + v3 - b(r)} (IV-3) with b(r) and c(r) the s- and p-wave terms of the optical potential, Zonpt(r) = b(r) + Y-c(r)Y . (IV-4) The first derivative term in equation IV-l is eliminated by a change of variables, m1 = (1 - c6016) (Iv-5) and the equation becomes u 1 C' 2 1 C" _ y + {3(j—j—E) + §'T-:-E'- f}y - 0 (IV'G) where f = 1.12 - m2 - szC + vE - b(r) - 81%;) (IV-7) r Equation IV-6 is set up in the nuclear interior and exterior, with a matching radius chosen to be well outside the range of the strong interaction. A guess is made of the eigenenergy w and the equation is integrated in from infinity and out from zero using Milne's predictor-corrector method (63), with the two wavefunctions compared 93 at the match point. A new energy guess is made from the size of the mismatch and the process repeated until convergence is reached. The number of nodes is checked against the number proper to the state being calculated at the end of each iteration and a correction to the energy guess is made accordingly. The program includes electro- magnetic effects due to the finite charge distribution of the nucleus, vacuum polarization, and electron screening. The program has been modified to include the full optical potential, equation III-76. One defect has not yet been remedied, however; only one form for the nuclear density is available, the woods-Saxon form. It was therefore necessary for small nuclei to derive the size parameters from the experimental rms radius and skin thickness. 3. Calculated Shifts and Widths Calculations of shifts and widths for the nuclei listed in Table 4 have been performed using the optical potentialequation III-76 with single nucleon parameters derived from the RSL phase shifts and the absorptive parameters of Riska and collaborators. Two dif- ferent values of the parameter A were used, the Ericson-Ericson value (10), 1=1,and a value in the range suggested by Weise (48) and Brown (47), A = 1.6. The results are shown in Figure 17, where the lines are simply drawn from one calculated point to the next. The dashed line corresponds to A = 1, the dotted to 1 = 1.6. It is not possible to say with certainty what parameter changes are necessary to bring the curves to the data, but it appears that the 100 10 [keV] 94 III] I fi I 111] l 5. ~— (I. J l 1 l Figure 17a. 8 8 10 Z Calculations of the s-wave shift as a function of 2 compared to the experimental data, with the parameters of set 1 of Table 5 with A = 1 (dashed line), A = 1.6 (dotted line), and with the parameters of set 2 (solid line). Data are from Ref. 59. 95 10 IIII' [keV] *— l 14 1J l Figure 17b. Ref. 59. 8 Z Calculations of the s-wave width. Data are from 96 loo T T r [keV] 0.1 1 1 1 l #1 1%. L1 1L1 1 Lil 111141 I I 11111 I l Figure 17c. Calculations of the p—wave shift. Data are from Ref. 60. 97 1 l 1 1 1 r l j I T 1 I 1 1 I I 1 r _ P . 10— _ _ I I- d I—\ I- a- > <1) 4‘ U 1_ —I I I 1- .. I— .1 b d |- ‘1 001— ~— - -1 l— .. 1 1 1 1 1 1 1 J 1 1 #1 1 1 1 J L 4 16 20 2‘1 28 Figure 17d.. Calculations of the p-wave width. Data are from Ref. 60. 98 s-wave is not repulsive enough, while the p-wave is too attractive, especially for the value A = 1. The s- and p-wave absorption strengths appear to be too weak. A fit was then made to the data, varying the parameters Re(b0), Re(c0), Im(B0), and Im(C0). No attempt was made to minimize the X23 the fit was done only in order to give some idea what parameter changes were necessary to give reasonable results. The parameters thus obtained are given as set 2 in Table 5 along with the parameters of the first calculation, set 1. The results with the fitted param- eters are shown as the solid curve in Figure 17. The calculated quantities follow quite well the A dependence of the data, except for the p-wave shift for the larger nuclei. This deviation is not surprising, as the isovector parameters were not fitted. Comparison of the numbers in Table 5 indicates that the parameters Re(b0), Im(Bo), and Im(C0) must be increased greatly in magnitude in order to reproduce the data. The parameter Re(c0) must be increased slightly, although the required value is well within the error bars of the value given by RSL for co at zero energy. Alternatively the RSL value for c0 can be used if A is reduced to X = 1.5, with a corresponding small decrease in Im(C0). Although the discrepancies between the required single nucleon parameters and their phase shift values can be resolved by assuming different values for Re(B0) and Re(C0), no such remedy exists for the imaginary parameters, except to the extent that Im(C0) is sensitive to the value of A. The one nucleon absorption mechanism, not included in the optical potential, 99 Table 5. Parameters used in the pionic atom calculations. Real Imaginary Set 1 b0(fm) -0.029 0. b1(fm) -0.13 0. 80(fm4) 0.007 0.08 c0(fm3) 0.65 0. c1(fm3) 0.43 0. Co(fm6) 0.29 0.34 1 1.0 or 1.6 Set 2 b0(fm) -0.050 0. b1(fm) -0.13 0. 80(fm4) 0 007 0.19 c0(fm3) 0.66 0. c1(fm3) 0.43 0. C0(fm6) 0.29 0.90 A 1.6 100 is not large enough to make up the difference; it is estimated at not more than 30 percent of the two nucleon strength (64).' Thus there is a real discrepancy at zero energy of about a factor of two between the absorption parameters calculated by Riska and col- laborators and those required to fit the pionic atom widths. Other calculations of these quantities are somewhat nearer the fitted values: Hachenberg and Pirner (52) obtain Im(BO) = 0.144 fm4; Oset, Weise, and Brockmann (54) calculate Im(C0) = 0.64 fm6. This is clearly a subject which requires further investigation. The fitting procedure was also carried out for A = 1 so that comparisons could be made with previous work. Comparison is still 20, V202, and K(r) terms in the optical difficult because of the V potential used here, which have not been included in other analyses. The effect of these terms is to decrease the s-wave repulsion, thus requiring a more negative Re(b0) to compensate. It is clear that the value of the p-wave absorption parameter thus obtained, 6, is smaller than that obtained by, for instance, 6 Im(CO) = 0.77 fm Krell and Ericson (45) in their fits, Im(C0) = 1.12 fm , and used in the calculations of Ref. 24. Krell and Ericson give an alternate value, which gives better results for certain nuclei in their sample, Im(C0) = 0.56 fm6. Fortunately the recent p-wave data is much better than that available to Krell and Ericson, making a more definitive result possible. From the calculations it can be seen that the objective has not yet been reached; the optical potential with parameters from 101 theoretical predictions does not reproduce the data. The deficiency in the s-wave repulsion is an unsolved problem, and will appear alsoirlthe low energy elastic scattering calculations. The inadequacy of the absorption parameters is a less serious matter, requiring further refinements in the calculations and a more careful treatment of the LLEE effect including absorption and multiple scattering on an equal footing. It is encouraging that the A dependence of the data is reproduced by the optical potential with fitted param- eters. Because the theoretical parameters vary slowly with energy in the range zero to 50 MeV, the information gained from the pionic atom analysis is also relevant for the low energy scattering calcula- tions to be discussed in the next chapter. CHAPTER V ELASTIC SCATTERING CROSS SECTIONS In this chapter the pion-nucleus elastic differential cross sections calculated using the optical potential equation III-76 are discussed and comparisons are made to the existing data. The energy region considered in this study, 0-250 MeV, will be divided into two regions: the low energy region, defined roughly as 0-50 MeV, and the resonance region, around 180 MeV. The elastic scattering cross sections from these two energy ranges have quite different characteristics. The low energy scattering shows evidence of inter- ference between the s-wave, p-wave, and Coulomb amplitudes; the real parts of the optical potential are most important. In the resonance region the scattering has a diffractive character, due to the large imaginary part of the p-wave optical potential. The scattering cross sections for energies between these two regions have some characteristics of both. The data in the low energy region at present consists of cross sections for n+ on various targets at 30, 40, and 50 MeV. Unfor- tunately there is as yet very little n' data, only on 12C and 208Pb at 30 MeV. There is no data for pion energies below 30 MeV. Between 208% at 50 and 100 MeV there are measurements of n+ on 160 and 80 MeV. Above 100 MeV there is an abundance of 0+ and n' scattering data from various nuclei, clustered about the energies 115, 162, 102 103 180, and 240 MeV. A characteristic sample of these data will be compared with the theoretical calculations. The first section of this chapter describes in a simple model the general features of the low energy elastic scattering cross sections. Section 2 includes a model for diffraction scattering, relevant to scattering in the resonance region. In Section 3 the computer program used for the calculations is briefly described. Finally, the theoretical and experimental differential cross sections are compared in Sections 4 and 5. 1. General Features--Low Energy Scattering The low energy elastic scattering cross sections are charac- terized chiefly by the interference between the s- and p-wave ampli- tudes, the strengths of which are determined by the real parts of the optical potential parameters, and the Coulomb amplitude. This can be seen most easily in Born approximation. As was noted in Section 2 of Chapter III, the second order p-wave term in the Born series is suppressed due to short range correlations. Therefore the Born approximation with a potential which includes only the first order p-wave terms should give a good approximation to the scattering from the full p-wave part of the optical potential includ- ing the Ericson-Ericson effect. For an N = Z nucleus the amplitude in Born approximation for scattering from the simple first order optical potential equation II-53 is 219. f: = 150 + cok2 cos 61601) 1 2 6cm) (H) q 104 where the + or - refers to the pion charge and a is the fine struc- ture constant. This can be rewritten f = C[1-——l;——.+ 4y2 sin2«9 - 2y2(1 - x)] (v-2) 1 . 2 8 2 s1n -— 2 b c k4 with C = - 299, x = - ——9—, and y2 = —9——-. Thus, y measures the 4k2 kzc 9“ relative strength of the pgwave and Coulomb potentials and x measures the strength of the s-wave repulsion relative to the p-wave attrac- tion. The nuclear and charge form factors p(q) and pc(q) have been ignored in equation V-Z, as they decrease only slowly over the range 2 of q relevant for a light nucleus and low pion energy. For 50 MeV pions, the RSL phase shift values for the real parameters are Re(Eb)= 3 -0.042 fm and Re(c0) = 0.75 fm , giving C = -0.0083 fm, x = 0.13, and y = 4.4. The parameters derived from the fit to pionic atoms give the values x = 0.18, y = 4.1 at 50 MeV, assuming the energy dependence of the parameters can be ignored. The behavior of f is considered separately for 0+ and n'. 2 9-_1_. §-- Zy’ thus the position of the minimum is determined only by the p-wave strength. For positive pions f has a minimum at sin 39°. The zeros of f are III For the RSL value of c0 this gives 0 given by 51112; {(1 - x) 1 [(1 - x)2 i?) (M) bio—- For y < 2 there are no zeros. For y > 2 two zeros exist for x < 1 - g, none for x > 1 - §* This behavior is illustrated in 105 Figure 18 which shows f+ (upper left) as a function of x. Here the parameters were arbitrarily chosen to be y = 5 and x = 0.4, 0.6, and 0.8. The square of this amplitude (bottom left) indicates how changes in the s-p interference parameter x produce one minimum, which, with decreasing x, broadens, then becomes two minima separated by a hump. For negative pions f is monotonically increasing with 6 and has a zero at sin2 g-= %{(1 - x) + [(1 - x)2 +-£§]§} . (V-4) Y For the RSL values of the parameters this gives 0 = 86°. The ampli- tude f_ is also shown as a function of x in Figure 18 (upper right) along with If_|2 (lower right). Note that the position of the zero of f_ becomes the position of the minimum in the differential cross section, and thus depends on both the s- and p-wave strengths. The full Born amplitude, neglecting the K(r) term, is given by F = [ 5' + '1c k~k' - p1 - 1 c 2] (q) B p10 “10.... 2pl 0qp - 1 , -1 . ' pz 2 2 - + [p280 + P2 C05 5 ' 2P2 Coq JD ((1) (V 5) 2w€ - p-1 n“ oc(q) . 1 02 The Ericson-Ericson effect is not included, as noted above. Fig- 12 ures 19 and 20 show calculations for 0+ and n' scattering from C. 106 114] 1111 [1 40° 40° 80° 11 Figure 18. Amplitudes and corresponding cross sections for the first order real optical potential in Born approxima- tion. The vertical scales are arbitrary. 102 I I l I I I I I T T I l j I T I I 10 10 Figure 19. dcr/dQ '[mb/sr) 107 1 1 L1111 I I I W 1-, I l I TIIITT] I T: I “ I I llIlll CT 0 H O 1 1 141111 111 I / _l \\ [b] I I IIIIII 1 1 111111 I l 1 1 l 1 1 l 1 1 1 g l 0 30 80 90 ec.n'1. [deg] _L l l L I L 120 150 Elastic scattering cross sections for 50 MeV 0+ on 12C with (a) the full Born amplitude and (b) the full optical model calculation, for Re(Eb) = 0., -0.04 fm, -0.08 fm, and -0.12 fm. 10 H D Figure 20. dU/dQ [nob/6r] 108 r’ I I IIIII 1 1 11111 1 I I [IIII' 1 1 111111, “ [o] L “I ‘ 1 1 111111 .1 1 I r[ IIIII I l [b] I I IIIIII 1 1 111111 I l 1 1 1 1 1 1 1 1 1 0 30 80 SO ec.r1n. [deg] l 1J1114 120 150 Elastic scattering cross sections for 50 MeV n' on with curves same as in previous figure. 109 The curves labeled (a) are Born approximation calculations with reasonable values of the optical potential parameters and Re(Bb) = 0., -0.04 fm, -0.08 fm, and -0.12fm. The curves labeled (b) are the full calculations, with the optical potential equation III-76 minus the K(r) term, and the same parameters as were used in calculations (a). It can be seen that the characteristic interference effects persist when the kinematic and absorption terms are added to the Born amplitude, and in the full calculation. Although the imaginary part of the Optical potential has some effect, the differential cross sections are most sensitive to the overall real s-wave and p-wave strengths. 2. General Features--Resonance Region Scattering Because the imaginary part of the p-wave optical potential is large in the resonance region, the nucleus appears nearly "black" to the pion. This gives rise to differential cross sections with a distinctive shape, known as diffractive or shadow scattering. A semiclassical description can be given for this type of elastic scattering (65). Assume that the nucleus is completely absorbing, so that all pions with impact parameters smaller than R, the radius of the nucleus, are absorbed; those with impact param- eter greater than R are transmitted. The scattering amplitude can be written ”1 - 1 f(6) = Z (21 + 1) 711—— P£(cos 8) (M) 1 110 where complete absorption is characterized by n = 0, complete trans- mission by n = 1. The impact parameter b can be expressed in terms of the angular momentum of the incoming pion, so that n2 = 0 for 2‘: kR, n2 = 1 for 1 > kR. The differential cross section is just the square of f, kR do _ 1 1 2 _ d0 ' k2 192:0 (2 T 2)Pz(c°5 6” ' (v 7) Assuming kR large and the angular range small, the discrete variable 1 becomes continuous, 1 +-%-+ kb, and P£(cos 9) is approximated by Jo(kb sin 6), where do is the zeroth order Bessel function. With these replacements equation V-7 becomes 01(kR sin 0) 2 sin e R do_1 2 . 2:2 d0 - k2 [.12 k bJO(kb Sln 0)db| R [ (V-8) which corresponds to the classical formula for the diffraction scat- tering from a black sphere. In Figure 21 this approximate form is compared to optical 208Pb at 163 MeV. The radius used model calculations for 160 and is the effective nuclear radius, R = Ru + 1, where Ru is the radius of the equivalent uniform distribution and A = %u The momentum k is calculated for the incoming pion energy minus the Coulomb poten— tial at the surface, k = [(w - ENEC)2 - m2]§, where EC is discussed in Section 4 of Chapter III. The simple model does reasonably well 111 ‘05 TIIIjrfrfiTerIIT IOSIIrfiIIVIfi'1fi'fi1 160 ZOBPb 11 ‘1 ‘0 152 MeV . 1° \ 152 MeV 10 ' \ 10 \ _ 100 100 } f - ' \ .. f-“- ‘1 \ f’. ‘1 10 l/ 10 = y \I/ 1 1 1 1 1 A 1 ' 1 1 u‘ A O 3; . I I 01 I OJ \\ .o ' 1 e ' ’ W 10-2 : 10-2 8 \\ 10‘ 10“ £3 - 10 103 ‘\ ,,+ 100 100 10 10 1 1 01. 01 10-2 10-2 10°3 10'3 . - J . . 0 o 0 '0 120 1 0 0 30 so 90 120 150 e1:.11‘1. [deg] Figure 21. Comparison of black disk model calculation (dashed curve) agd full_optical potential calculation (solid curve) for n and n scattering from 160 and 208Pb at 162 MeV. 112 in describing the magnitude of the curves and the position of the 208Pb. As it is derived in small angle minima, especially for approximation, the model is not expected to do well near 90°. The inclusion of the Coulomb shift in k reproduces fairly accurately the differences in the position of the minima for 0+ and n'. It is evident from these calculations that the imaginary part of the optical potential is of principle importance in the resonance region. 3. Details of the Calculations The differential cross sections were calculated using a modi- fied version of the program PIRK, written by R.A. Eisenstein and G.A. Miller (66). PIRK is a position space code which solves the wave equation II-17 and determines the phase shift between that wavefunction and the exterior Coulomb wavefunction. The wave equa- tion is reduced to a set of equations in r only, noting that the optical potential is independent of angles, 113+ f(r)u,; + [g(r) - 553—11111, = o (M) where f(r) = ccr' r_ 1 (HO) and 9(1) = (1- c(r))'119—'—'§Il + 13 - 25vC - b(r)} (v-11) 113 with b(r) and c(r) the s- and p-wave parts of the optical potential. This is, of course, the same equation as was discussed in Chapter IV, except that the VE term has been dropped, for reasons to be explained below. Two complex coupled first order equations are formed from the second order equation V-9, by defining v(r) = u'(r). These are solved numerically by a fourth order Runge-Kutta method (63). The differential cross section is obtained from do HI? = Ifc(0) + fN(0)|2 (V-12) where the Coulomb amplitude is fC(e) = ";_;?§2fg exp {2i[o0 - ncln(sin §)]} (V-13) with he = Zeflo‘i— (v-14) and the nuclear amplitude is 1 21% em" - 1 fN(e) = W E: (21 + 1)e [ 2 JP£(cos 9) (V-15) where 02 is the Coulomb phase shift and 62 is the phase shift between the Coulomb wavefunction and the solution to equation V-9. The Coulomb wavefunctions and Coulomb phase shifts are deter- mined for a nonrelativistic particle, i.e. they are solutions to 114 the Schr5dinger equation with a Coulomb potential; no VE term is included. Because the exterior wavefunction is calculated in this way, it is deemed necessary to drop the V3 term from the calculation of the interior wavefunction as well. It is found that the inclusion of the V2 term in the equation for the interior causes a small amount of instability of the results with matching radius. In any case, the inclusion or exclusion of the V5 term in equation V-9 makes only a small change in the differential cross sections for 208Pb at low energies, where it is expected to be important, and none at all for the light nuclei. Cooper, Jeppeson, and Johnson (67) look at the effect of the VS term not only in the interior equation but in the Coulomb wavefunctions and phase shifts as well. For 208Pb at 100 MeV they find discrepancies larger than the experimental errors. It is not clear that the discrepancies are larger than the uncertainties in the theoretical calculations. however. There is a need for further investigation on this point. The program PIRK has been modified to include all the terms of the optical potential, equation III-76. A routine to calculate total and partial cross sections has also been added, as discussed in Chapter VI. 4. Calculations-~Low Energy Region Several elastic scattering calculations using different param- eter sets are discussed in this section. The first of these calcula- tions is that with the theoretical potential derived in Chapter III and used in the pionic atom calculations of Chapter IV. The 115 theoretical Optical potential parameters at 30, 40, and 50 MeV are given as set 1 in Table 6. As before, the single nucleon parameters are taken from the RSL phase shift fit (25), the absorption param- eters from the calculation of Riska and collaborators (50), and 1 = 1.6. Note that the imaginary parts of b1, c0, and c1 listed in the table are the RSL values multiplied by the Pauli factor Q. The elastic scattering cross sections calculated with set 1 are shown as the dashed curves in Figures 22. Clearly, the curves do not bear much resemblance to the data. The simple analysis of Section 2 can be used to give an indication Of what is amiss in the Optical potential. Comparison of the shapes of the 12C and 160 curves with those shown in Figure 19 suggests that the s-wave repulsion in the Optical potential is too weak. The same conclusion 12C calculation in Figure 22f can be drawn by comparison of the n"- and the curves Of Figure 20. The solid curves in Figures 22 are the result of calculations with more negative values of Re(5b), listed as set 2 in Table 6. These were chosen to give reasonable eyeball fits to the 12C and 160 data, and to have the same slope as a function of energy as the original values. The other parameters of the Optical potential were left unchanged. Although the fits are not perfect, the energy and A dependence is well reproduced and, on the basis of the two cases available, the n' data is also well described, including the 208 diffractive appearance of the 0' data for Pb at 30 MeV. 116 Table 6. Parameters used in the low energy elastic scattering calculations. 30 MeV 40 MeV 50 MeV Set 1 ‘Bb(fm) -0.035 + i0.003 -0.038 + i0.004 -0.042 + i0.006 b1(fm) -0.132 - i0.001 -0.131 - i0.001 -0.131 - i0.002 80(fm4) -0.005 + 10.115 -0.010 + iO.130 -0.020 + iO.14O c0(fm3) 0.70 + 10.007 0.72 + i0.015 0.75 + i0 029 c1(fm3) 0.44 + i0.004 0.45 1 i0 007 0.45 + i0.014 C0(fm6) 0.32 + i0.46 0.34 + 10.52 0.37 + i0.62 A 1.6 1.6 1.6 Set 2 'Bb(fm) -0.070 + i0.003 -0.073 + i0.004 -0.077 + i0.006 Other parameters as in Set 1. Set 3 ‘Bb(fm) -0.057 + i0.003 -0.06O + i0.004 -0.064 + i0.006 b1(fm) -0.132 — i0.001 -0.131 - i0.001 -0.131 - i0.002 80(fm4) -0.005 + i0.22 -0.010 + i0.24 -0.020 + i0.25 c0(fm3) 0.70 + i0.007 0.72 + i0.015 0.75 + i0.029 c1(fm3) 0.44 + i0.004 0.45 + i0.007 0.45 + i0.014 C0(fm6) 0.32 + i1.02 0.34 + i1.08 0.37 + i1.18 A 1.6 1.6 1.6 0 0.19 0.24 0.31 100 dU/dQ [mb/sr] 100 Figure 22a. 10 100 10 10 ec.ri'1. [deg] 12 Elastic scattering of n+ from C at 30, 40, and 50 MeV with the optical potential parameters of set 1 (dashed curves) and set 2 (solid curves). Data are from Refs. 32 (diamonds), 34, 69, 7O (triangles), and 68 (circles). 10 dU/dQ [mb/srl IT'V I 100 I I I'IIII' .— d d .1 .1 q - ‘ G 10 I I IIIIIII 1 1111111 O e<:.rin. [deg] Figure 22b. Elastic scattering of 1+ from 160. Data from Refs. 32 (diamonds), and 33, 34, and 69 (triangles). 3 10 I I l I I l I T l I f] I rj r I E HOCO E 100 'E 30 MeV 5 10 e a A 1 T Q (0 100 2 ..E. 95‘ \ 10 b D 100 g 10 1 \ ’ 3 \ // - 1 1 1 1 L 1 1 1 1 l 1 1 \-1/ 1 l 1 1 O 30 60 80 120 150 e1;.n'i. [deg] 40 Figure 22c. Elastic scattering of n+ from Ca. Data from Refs. 34, 69. I IIII 1 1111111 100 I I IIIIII] 1 11111111 7‘"l L111l11111 100 10 dU/dQ [mb/sr) IIIV 100 I I IIIIII] 10 I IIIIIII] 1 1 1 1 l 1 120 150 O to O O) C) LO 0 9mm. [deg] Figure 22d. Elastic scattering of 0+ from 90Zr. Data from Refs. 34. 69. 100 10 /_1_1_LL1.1.111__1_L_1_L1.111L___1_1_1_L1111 100 dU/dQ [mb/sr] I I rif/ 100 I I I [IIIII 1 1 1 111111/ 10 1 1 1 111111 1 I I rIIIn] —’ l 1 1 1 1 l 1 O 30 60 80 120 150 9m. Ides] 208 Figure 22e. Elastic scattering of n+ from Pb. Data from Refs. 34, 69. 3 10 I I I I I I I T I I I I I I I = :- ,,- a F 30 MeV ‘ 100:- _. 10:. - I I '2: F ‘ < L:— \/ ‘1 .0 \K : _§, 1‘ : 8 ~ . \ \ 103E \ b . 13 I 100? I 10:- C 1. 1 1 1 O 208 Figure 22f. Elastic scattering of n' from C and Pb at 30 MeV. Data from Refs. 71 and 72. 123 A X2 fit to some of the 50 MeV data using the same optical potential as used here but without the K(r) term gives a minimum in x2 3 for A = 1.6 and Re(c0) = 0.75 fm , with Re(bb) = -0.06'fin(73). The value for Re(c0) is almost the phase shift value at 50 MeV, 3 Re(c0) = 0.74 fm . The value for Re(Bb) is not as negative as that required in the analysis presented here, due to the absence of the K(r) term. Note that all the induced s-wave terms, the V20, V202, and K(r) terms, are attractive, requiring more repulsion in the Ebb term. Another approach can be made to the question of choosing opti- cal parameters for the low energy region. The analysis of pionic atom shifts and widths gives the overall strength of the s- and p-wave parts of the optical potential. This information can be extrapolated from zero energy to the required energies assuming some reasonable prescription. As a first approximation, the optical potential parameters are assumed approximately energy independent in the energy range zero to 50 MeV. This is the approach taken in Ref. 24. The param- eters for the calculations are taken from the fits to the pionic data, set 2 of Table 5, with the exception of the imaginary parts of the single nucleon parameters which are taken from set 1 of Table 6. The results of these calculations are shown as the solid curves of Figures 23. These are in fact rather close to the data for the light nuclei and nearly reproduce the s-p interference. They become steadily worse, however, with increasing A. 3 10 T I T T r l l I I r I l T I I I E 12C E 100 __ 30 MeV : 10 __ F: .. (D N 100 ‘\ E E g: L+0 MeV \ 10 b \> ‘-‘ ’04" 99;..2 _ U I. ; ‘~‘ J}? : :r' E 5 E 50 MeV j 105. __ a -------- ’9 5 l- -T- l L L l 1 1 LL 1 L L L L Figure 23a. L l 1 l 1 0 30 60 90 120 150 9mm. [deg] Elastic scattering of n+ from 120 at 30, 40, and 50 MeV, with the optical potential parameters from set 2 of Table 5 (solid curves), set 3 of Table 6 (dotted curves), and set 3 but with Im(BO) and Im(Co) from set 2 of Table 5 (dashed curves). 100 10 100 10 dU/dQ [mb/er] 100 10 l L 1 0 30 80 90 120 ec.m. [deg] 16 Figure 23b. Elastic scattering of n+ from 0. 100 10 E 100 2 .5. g \ 10 8 \_> F- 100: :5 105' '5 1 1 L l 1 1 l 1 1 1 L C) (A) O 0) O (0 O ...... N C) .....- Ul- O ec.rI'1. [deg] . . + Figure 23c. Elast1c scattering of n from Ca. 1 1111111 100 30 MeV a A ‘ 1 10 5‘.-. . “...-iv“;- 2‘ d (D N \ 100 1: .5. g \ 10 3 l? 1005- l: 10? A A _= 1 1 1 I 1 L l 1 0 30 60 90 120 150 ec.m. [deg] 90 Figure 23d. Elastic scattering of n+ from Zr. 100 10 100 10 dcr/dQ [mb/sr] j IIIV 100 I I IIIIIII 10 I I IIIIIII O ec.m. [deg] 208 Figure 23e. Elastic scattering of n+ from Pb. 100 10 1 11111 I III/ [1 1111111] dU/dQ [mb/sr] T I I IIIIII 100 T l I III". 10 I I IIIIIII L 1111L111Ll11 J 1 1 l 1 60 80 120 150 C) (D O ec.m. [deg] 12 208 Figure 23f. Elastic scattering of n' from C and Pb at 30 MeV. 130 The optical parameters are not hifact energy independent, even in this energy range. One should, therefore, make some estimate of the energy dependence of these parameters. For the p-wave param- eter c0 this is a straightforward task; as the pionic atom value of Re(c0) is close to the zero energy RSL phase shift value, these values of c0 are adopted at all energies. The pionic atom value for Eb is not near the value calculated from the RSL phase shifts; therefore, it is extrapolated assuming the same slope as a function of energy as the calculated value, i.e. the difference between the fitted value of 56 and the RSL value at zero energy is added to the RSL value at all energies. The real parts of the absorption parameters were fixed in the pionic atom analysis at the values calculated by Riska et al., therefore the calculated values are used at all energies. In the first set of calculations, the imagi- nary parts of the absorption parameters were kept at their zero energy fitted values. These are the dashed curves of Figures 23. In the second set of calculations, the dotted curves of Figures 23, the absorption parameters Im(Bo) and Im(C0) were extrapolated also. This can be done in two ways. The first is simply to assume the slope as a function of energy of Im(BO) and Im(C0) is that of the Riska calculations. A more general method is discussed in the next section. The two give nearly identical results for 30-50 MeV. The complete set of extrapolated parameters is given in Table 6 under the heading set 3. A comparison of the dashed and dotted curves indicates that the smaller absorption strength gives quitereasonable 131 results, especially for the light nuclei, while the calculations with the larger absorption strength are somewhat worse. The cal- culations are rather insensitive to the imaginary absorptive param- eters, and small adjustments of the real 5 and p-wave strengths can at least partially compensate for differences due to the absorp- tive strength. Note that both calculations with extrapolated param- eters do better than those with the zero energy values. From these calculations one can conclude that although the smaller imaginary absorption parameters calculated by Riska and collaborators seem to give Somewhat better fits to the data, the absorption parameters deduced from pionic atom analysis are not inconsistent with the data. In fact, the set of optical potential parameters deduced from pionic atom shifts and widths fits the scattering data rather well, considering the approximate treatment of the energy dependence. As yet nothing has been said about the relative strengths of the single nucleon and absorption pieces of the real part of the potential. In fact, the scattering is quite insensitive to this; the overall real 5 and p-wave strengths are the important quantities, just as was the case for the pionic atom shifts. This is illustrated for the s-wave in Figure 24, where the elastic scattering cross 12C at 50 MeV are calculated for two different sections for n+ from values of Re(BO) (solid and dashed curves) and several values of Re(Bb), with the highest curves corresponding to the most negative values of Re(Eb). The curves are equally spaced except at the most backward angles, indicating the insensitivity of the calculation 132 :10 t ‘0 fl :3 . "-‘ - é // \\ x/ \ l 03 " \ .. 'o F \ // \\ \ \ \ / \ . \ \ .. b \ / \ \ 'C / /’~\ \ \ ./ ‘\\‘ “ r- \ / \ - / \\ \/ \‘ 1 IIIII 11111 1 1 l 1 1 J 4 1 l 4 1 l 1 60 90 120 ec.m. [deg] l l 1 L1 IESO C) (1) C) Figure 24. Comparison of elastic scattering calculations for n+ on C with two values of Re(B ) (solid and dashed curves) and a series of values for Re(9b). 133 to the origin of the s-wave strength. A similar situation exists for the p-wave. Thus, the low energy elastic scattering data yield only a limited amount of information about the parameters of the optical potential, and even less about the form of the potential. The overall real 5 and p-wave strengths are roughly indicated, and some wide limits are placed on the imaginary strength of the Optical potential. 5. Calculations--Resonance Region In this section several sets of calculations of elastic scat- tering for energies above 50 MeV are compared with a sample of the data in this energy region. The first set is again that with the theoretical optical potential. It is not clear what value of the LLEE parameter A should be used at these energies as both the simple model(10), giving A = 1, and the calculations of Weise (48) and Brown, Jennings, and Rostoken (41), giving A s 1.6, are made in the low energy limit. There is as yet 1K) theoretical value for A above about 50 MeV. It is convenient to choose the value A = 1.6, as this was used in the low energy calculations. Calculations using both values, A = 1.6 (solid curve).and A = 1 (dashed curve), are shown in Figures 25. The differences between these two curves are in each case not large, so that the choice is not a crucial one. Both curves reproduce well the general features of the data, but not the details. The best fits occur when the nucleus is blackest, around 163 MeV,and for the larger nuclei. It is difficult to analyze the optical potential in the intermediate energy range 134 .eu .mm .mmmm Eocm mgw mama .Amm>gzu umppouv mcowpmpsupmu meucmpoa mecwpmmwx gwugo umgwm mew czosm omp< .Amm>c=u umnmmuv o.H n K use Amm>c=u uwpomv o.H u x ;p_2 meowpwpaupao mxmwm new mpcwem omega 4mm an“ soc» :mxma mcm mcmumsmcma meucmuoa quwuao .0 ma sect .: can +: we mcwgwupmum ovummpm .mmm mcamwm 135 mmm mgamwu oc— no. .9. 2: an. ac— Inuulli_ >01 arm om. GN— - u q q a 3.3 .Edo em .q-Tq (“AU-I) tsp/0p 136 .mm .h—bm EOLL. 0L0 mama .60 >0: :— DUO? I II 3.3 .Edo am am a 2 an: eN— 8 am am a ..r.u..«1..fiq - .... A a . _ d a d a . MIG— we. we. >oz ...N ..o ..o .J _ _ o >0: mm— A a. . p :9. m, 9. no m mg. m. q / C 2 wt 8. i\.... MO— 8— _ 2 fa. ow soc» I: new += we mcwcmuumum ovumMFN .nmm wgzuwu 137 .Ammwmcmcm gmguo Fpmv mu Dcm A>m2 NQHV mm .mmmm SOL"... mLm mumo .nmmom E0}... 1:. “EM +2. ....0 OCwLmHHwUm Umeme .UmN 92.5?- 303 .560 3c... .56: :2 av. am cm on .- >02 mm— >oz m: (as/aw) tsp/no 2: co. no. no. ...: ...: €8~--.. aooew-... hrbbr-bthppbhp-binn: ithprbprppphbppimG— 138 80-150 MeV, as the dependence of the scattering on the parameters is not given by a simple model. It is not clear, therefore, what changes in parameters would be required in order to fit the data. Detailed fitting of this data, with a first order optical potential which includes form factors, has been done by Stephenson et al.(76), who vary the radius and skin thickness of p and the overall normali- zation as well as the optical potential and form factor parameters. Thus, it is difficult to correlate their information with the poten- tial used in this investigation. As the early calculations of resonance region scattering were made with the Kisslinger potential with no modifications, it is interesting to see whether the addition of the higher order correc- tions, which are extremely important at low energies, improves the fits at these energies. The results of Kisslinger potential calcula- tions, with parameters from n-N phase shifts, are shown as the dotted curves in Figures 25. Although these curves are closer to the data in places, the inclusion of higher order corrections causes an over- all improvement in fit for all nuclei at all energies shown here. The second method discussed in the previous section for choosing the potential parameters, using the zero energy parameters derived from pionic atom analysis, can be applied at these energies also. However, a more sophisticated method of extrapolation is required. As noted before, the real p-wave parameter obtained from pionic atom fits is quite close to the RSL phase shift value, when the value 1.6 is adopted for A. Although the real s-wave parameter is 139 not near the phase shift value, the resonance region scattering is not very sensitive to the s-wave strength, and the RSL value for Eb will be adopted. As the real parts of the absorption param- eters were fixed at the values given by Riska and collaborators, the only parameters for which an extrapolation procedure is required are Im(BO) and Im(CO). In order to derive a simple procedure, the assumption is made that in the matrix element of 1(2),equationIIII-57, the dominant contribution to the energy dependence is from the scat- tering operators T. Thus, the absorption parameters are assumed proportional to the square of the single nucleon parameters. For the s-wave it is necessary to include the isovector contribution, as the isoscalar term b0 is small. The imaginary parts of the s-wave parameters can be neglected, however, in this simple estimate. The s-wave absorption parameter is therefore taken as (52) Im(BO) = K1{[Re(b0)]2 + [Re(b0) + 2Re(b1)]2} (V-16) where K1 is a constant of proportionality. For the p-wave, the isovector terms are proportional to the isoscalar terms, due to the dominance of the A33 channel, and need not be considered. The imaginary part of the p-wave single nucleon parameter must be included, however, as it dominates in the resonance region. The p-wave absorption parameter is taken to be 2 Im(CO) = Kzlcol . (V-17) 140 The constants K1 and K2 can be determined by evaluating equationsV>16 and v—17 at zero energy. With Im(Bo) = 0.19 fm4 and Im(CO)= 0.90 fm6 from Table 5, and the zero energy RSL values for b0, b1, and co, the constants K1 and K2 are determined to be K1 = 2.6 fin2 and K2==2.1. The absorption parameters obtained in this way are shown as a func- tion of pion energy in Figure 26 (dashed curves), compared to the Riska values (solid curves). Both absorption parameters derived from pionic atom fits are higher than the calculated values at low energies, but the simple extrapolation gives a flatter energy depen- dence so that the calculated values overtake the extrapolated ones in the resonance region. The peak in the extrapolated value of Im(C0) is also shifted in energy compared to the calculated values. Although the imaginary parts of the absorption parameters can be estimated in this simple way, the real parts cannot, due to the more complicated form of the real part of the nuclear propagator GN in equation 111-57. (The imaginary part of GN is just a delta function giving energy conservation.) The calculations of Riska and collaborators must be used to give the energy dependence of the real absorptive terms. Calculations using this second set of parameters are shown as the dashed curves in Figures 27, and are compared to the previous calculations with A = 1.6, shown as the solid curves. Again, there are not large differences between the two sets of calculations, although the extrapolated parameters give slightly better fits on the whole. 141 57‘ E 2". I I I r I I I l I 1'? E 3:. 0.0 l L 1 l 1 __I_ 1 l 1 0 50 100 150 200 T«(lobl [MeV] Figure 26. Imaginary absorption parameters extrapolated from pionic atom values (dashed curves) compared to those of Ref. 50 (solid curves). 142 .o.H u < new muewgm mmmgn 4mm msu so»; mcmumsmgma gmsuo saw: Amm>g=u uwsmmuv mmzpm> scum uwcowa Eocm cmpmpoamguxm vcm Amm>csu twpomv om .wmm soc» :mxmp mew mcmumEmcma cowuacomn< .o2 Eocm .: vcm +: $0 mc_cmupmom uwpmmpu .mnm mcamwu 143 mum mgzmwu 9 am e 9 DW— ON—q . flu u Om GM 9 IO— 4 4°? «OuNu . cm. a q . . _Mq q mieu 1111 Nia— Ill - II w \ Nio— >oz arm. . .c .a: ; >0: mm. >9: mm. :— 2: no. .c. .cc. >.z y: a. 8— pupppppppbrtrpr no. me. (u/aw) tsp/on 144 A >02 =.N / >0: mm— .~ >02 7: 2 oo— L . .m 08 30... .....oo .omr..ow_ om om cm 9 . 1.44-1-‘uuMID— >0: =.N >0: mm. cc— >02 3: Eocw mcwcmpumUm uwpmmpm 140/qu tsp/on .nkm acamwu .nawom seem mcwcwppmum uwpmmpm .uum mczmwu 3.3 .246 3.3 .660 cm: cN— am am cm a , OWL cN— I¢m am am a dq-qq-dufiqqddeWIOm quudqqd dd-«4.¢q~:°— ,_ ._ ~ .... .... >0: mm: _ >02 mm. _ O- o— 2: 2: MO— an: 9 5 >02 w: )3. w: fl 4 1|. ’6— ..Q— m I w. 8. 8. fl . Ml MA: MO— 20— I ..6— 9: cc— MO— ND— ..2 ...: prL-pnpb 146 It is not clear whether the parameters or the optical potential itself should be blamed for the lack of detailed agreement between theory and data. The form of the Optical potential is open to ques- tion, as several terms were calculated in the low energy limit. The concept of an optical potential in the resonance region has been questioned by Weise (77), who deduces from isobar-hole calcula- tions that the potential is too nonlocal to be treated in this simple way at these energies. Unfortunately, detailed calculations, such as the isobar-hole calculations, are quite complex and must be done on a case by case basis. The optical model is the best simple approach, allowing calculations over a wide range of energies and nuclei. CHAPTER VI TOTAL AND PARTIAL CROSS SECTIONS Before discussing the importance of total and partial cross section calculations as a test of the optical potential it is useful to define these quantities. The total cross section OT’ the cross section for any kind of interaction to occur, can be divided into two pieces due to elastic scattering and to all other processes, CT = OE] + CR , (VI-1) where o is the differential cross section integrated over angles. El The reaction cross section OR can again be divided into two parts, due to true pion absorption, OA’ in which there is no pion in the final state, and to quasielastic scattering, COE’ which includes all processes with a pion in the final state. Note that in this discussion the charge exchange cross section, OCX’ is included in OQE' Thus CT = OEl + GA + OQE (VI-2) gives the various partial cross sections to be discussed in this chapter. 147 148 The total cross section and the partial cross sections 0A and OQE are important because they are sensitive to parts of the optical potential that the differential elastic cross section may not be. In particular, 0A and OQE are sensitive to the imaginary parts of the absorption and single nucleon parameters respectively and can provide information about the relative strength of these two parts of the potential which cannot be obtained from elastic scattering. Unfortunately the total and partial cross sections are difficult to measure, and the data are sparse. It is also not clear how to divide the calculated reaction cross section into its absorptive and quasielastic pieces in the model considered here. Thus the analysis can yield only tentative conclusions at present. The first section of this chapter gives a discussion of the methods used in extracting total cross sections and forward nuclear amplitudes from the data and of the theoretical quantities to which these numbers correspond. In Section 2 the calculations of the total cross sections and nuclear amplitudes are compared to the data. The third section includes possible techniques for calculating the partial cross sections. Calculations of GA and oQE are compared to the experimental data in Section 4. 1. Extraction of Total Cross Sections and Scattering Amplitudes Before calculations can be made the term total cross section must be redefined for any problem which involves the Coulomb inter- action, because the total Coulomb cross section is infinite. A 149 quantity must be defined which has this Coulomb cross section sub- tracted, and which can be extracted from some set of experimental data. Two experimental groups have defined and measured such a quantity, Carroll et al. (78) and Jeppesen et al. (79). Their choices will be described in the discussion given below. Both begin with data obtained in a transmission experiment, in which the beam flux is measured before striking the target, and the flux of particles within a solid angle 9 centered on the beam axis is measured after the target. The difference can be expressed in terms of a cross section 0(9). Several cross sections can be defined relative to this. (This discussion follows that of Cooper and Johnson (80).) One cross section which can be extracted from transmission data is the reaction cross section. Define don CRUZ) = C(Q) - £9 (19 71-5.— (VI-3) That is, the flux lost due to reactions other than elastic scattering is the total flux lost minus the flux that went into elastic pro- cesses with angle greater than 9. The reaction cross section is the limit of oR(O) as 9 goes to zero. The calculation of the reac- tion cross section for c(Q) requires either a model for the elastic scattering or measured differential elastic cross sections at all angles. 150 The cross section usually referred to as the total cross sec- tion, also called the removal cross section, is defined as the limit of 1 2 l *- on) = 0(9) - £9 an Ifc| - 2Re [£9 an fch] (v1-4) as Q + O. This is the quantity extracted by Carroll et al. (78). Here the scattering amplitude has been separated into a Coulomb and a nuclear part, f(9) = fN(e) + fC(e) (VI-5) where f (e) = - ————:EL——-exp {2i[ 0 - n ln(sin 9)]} (VI-6) C . 2 e O C 2 2k Sln -7 with nc = Zena? (VI-7) and . 2i6 210 R __l_ 2 e - 1] _ we) - 1k § (21 + 1)e [ 2 P£(cos a) (v1 3) Note that the nuclear amplitude cannot be completely separated from the Coulomb; fN still contains the Coulomb phase shifts 01' The removal cross section measures flux lost to reaction processes and the |le2 part of the elastic scattering. The extraction of 0T '2 is difficult because one needs not only IfC which is calculable, 151 * but also Re(foN), for which a model of fN is required. It can be shown (80) that the cross section OT defined in equation VI-4 can be related to the scattering phase shifts by 0T = -k—- Im[fN(O)] (VI-9) where 215 ~ 2 - me) = 712% (29. + 1)[§——2—l]P£(cos e) (v1-10) Note that fN differs from fN in the absence of the Coulomb phase shift factor. The quantity extracted by Jeppesen et al. (79) is oN(Q), defined and discussed by Cooper and Johnson (80) 2 0N(Q) 0(Q) .I:Q Q I Cl ( ) The advantage of this definition is that only the known function fc is required; no model is necessary for the full pion nucleus interaction. Cooper and Johnson show that the limit of oN(Q) as Q + O is given by UN = 4T"- Im[fN(O)] (VI-12) The quantity UN has no direct physical significance, and in fact is not always positive. If equation VI-ll is rearranged and a 152 polynomial expansion in Q is made for the various terms, the follow- ing form results, on> = (0N + § Anahcos w + if ReEfN(0)]+ § Bnan]sin w + §cnsz" (VI-13) where N = y log (Q/4n) - 200 (VI-14) with Y = if; (VI-15) for positive pions. Here v is the pion velocity. The quantity fN is given in equation VI-8. Thus if sufficient data exists at small enough D so that the sums have only a few terms, the param- eters 0N, Re[fN(O)], An, Bn’ and Cn can be determined. For large Z the cos N and sin w terms can be distinguished and both Re[fN(O)] and Im[fN(O)] (from 0“) can be derived. For small nuclei only Im[fN(O)] is determined. A model for elastic scattering can be used as an aid to extracting fN(O); the number of fitted parameters can thereby be reduced. The role of the model in this analysis is much less important than in the extraction of OT, however. The chief difficulty in the approach of Cooper and Johnson is the problem LL.____ .. _ 153 of interpreting the results, as fN includes the Coulomb phase factors eZIOl. This completes the description of the two experimental quan- tities related to the total cross section, OT and fN(O), and their relation to the calculated pion-nuclear phase shifts. Theoretical calculations of these quantities are compared to these data in the next section. 2. Total Cross Section and Scattering Amplitude Calculations In this section two sets of calculations are presented for CT and fN(O), corresponding to the two sets of parameters discussed in Section 5 of the previous chapter. These are the theoretical set including the absorption parameters of Riska, shown in the figures as solid curves, and the set in which Im(B0) and Im(C0) are extrap- olated from pionic atom fits, shown as dashed curves. The first set of data to be considered is that of Carroll et al. (78), who have extracted 0T, as defined in the previous sec- tion, from data on natural Li, C, Al, Fe, Sn, and Pb at energies from about 65 to 250 MeV. For the quantity fN required in their analysis they used an optical model calculation with a first order Laplacian potential and parameters from n-N phase shifts. They adjusted the energy and width of the (3,3) resonance contribution, however, to fit the position and width of the peak in the total cross section as a function of energy. 154 A comparison of the experimental and calculated total cross sections is given in Figure 28. Although the calculations reproduce the data for the light nuclei, there are significant discrepancies for the larger nuclei, especially for n+. The calculations show a much greater difference between n+ and n" total cross sections at the lower energies than is seen in the data. It is to be noted that the calculated cross sections are insensitive to differences in the absorption parameters of the size to be found in the two parameter sets. It is difficult to draw conclusions from this data, as the effect on the experimental cross sections of the model used in their extraction is unknown. The dashed curves of Figure 3 give the result of calculations with the first order Laplacian model with parameters from n-N phase shifts. As can be seen, the fits are somewhat random, 208Pb. The effect of the varia- being reasonable for 160 but not for tions made in the parameters in the fitting of the total cross sec- tion peak is not clear. Thus no definite conclusions can be drawn until the accuracy of the data has been assessed. The second set of data to be discussed is that of Jeppesen et al. (79). The data are from targets of Al, 40 Ca, Cu, Sn, Ho, and Pb and pion energies from 63 to 215 MeV. Although their analysis is less model dependent, the interpretation of the trend in energy or A of the quantity fN(O) that they extract is complicated by the Coulomb phase, as noted in the previous section. 155 S... 33 c... Ll— C .9 c I; 8 3_ (.0 8 L .. L) '6 23 2‘ t— 1.. l. 0 1 I 1 l 1 l 1 l 0 100 200 0 100 200 TflUOb) (MeV) Figure 28. Total cross section calculations using the absorption parameters of Ref. 50 (solid curves) and extrapolated pionic atom parameters (dashed curves) compared to the data of Carroll et al. (78). 156 The experimental and calculated values of fN(O), defined in equation VI-8, are compared in Figures 29-31. Figures 29 and 30 27Al and 207Pb respec- illustrate the energy dependence of fN(O) for tively; Figures 31 show the A dependence at 165 MeV. In each case the data for n+ are indicated by circles, for n' by X's. Although the general features of the data are well described, there are again problems with details. The poor quality of the fits to the data for 207Pb may be due in part to the neglect of the V3 term in the potential, and its absence in the calculation of OR and the Coulomb wavefunctions. The A dependence of the data at 165 MeV is quite well reproduced, suggesting that the large discrepancies between theory and the Carroll data seen for n+ on the large nuclei may be exaggerated. A more detailed comparison of the two data sets is difficult, due to the fundamental differences between the quanti- ties OT and fN(O). As in the case of the resonance region elastic scattering, the poor quality of the fits and the insensitivity to moderate parameter changes demand a closer scrutiny of the optical potential itself. 3. Theoretical Expressions for the Partial Cross Sections Measurements have recently been made of the components of the reaction cross section: the quasielastic, charge exchange, and absorption cross sections, both as a function of energy for one nucleus and as a function of A at a particular energy (81). Although the reaction cross section can be calculated from the simple expression (82) (mbl [Llfl/k] R8 fN[0] -1000 1 0 Figure 29a. 800- 600- 400* 200- 200 l 300 l l l 800 800 % _ l 1 1 100 200 TnUob] U48V] The amplitude %§~Re[fN(O)] for n+ (circles) and n" (x's) on 27Al, as a function of energy. Curves are iame as in Figure 28. Data are from Jeppeson et al. 79 . 158 27M 1200_ _ // ’-\\\ 3 // . \\\ .é . / \ / . - c3 * / \ ‘2 ll \\ “' 1000- , \ E / \ l—l __ \ ,— \ :1 \ + \\ L \ O t \ j— \ _. . \ 800- % F L 1 l 1 1 0 100 200 TflUOb] “WGVJ . . 4n + - U Figure 29b. The amplitude T Im[fN(O)] for n and TI on Al. T I I T 208pb 1 I ‘ 2000- 4 - E ff ii 1 - g - 1 Z r o i 1 1 1 5? i \ L J t I. ‘2000- J b J ‘HOOO A l 1 l 0 100 200 TflUOb) [MeV] 208 Figure 30a. The ampiitude-%F Re[fN(O)] for n+ and n' on Pb. Curves are same as in Figure 28. Data are from Ref. 160 ZUBPb -1000 I {mbl l -2000 [Lin/k] Im {NW} 1 -3000 -4000 1 I J 1 0 100 200 TnUOb) [MeV] Figure 30b. The amplitude gig-r- ImLfN(0)1 for "Jr and Tl- 0“ 208Pb° 161 T1 I T I r T rfi I I 185 MeV l 3000 2000 l I 1000 [mb] -1000- “111/11] R9 fN[0] l '2000 -3000 l -L+000 1 1 L] 1 1 L1] 1 10 100 A. Figure 31a. The amplitude‘51 Re[fN(O)] for n+ and n" as a function of A at 165 MeV. Curves are same as in Figure 28. Data are from Ref. 79. 162 T T I ITTWTI I 165 MeV 3000- 2000- [mb] I 1000 -1000- [Lin/k) Im le01 '2000 l I-O-i l -3000 'HOOO 1 1 1 I 1L1 1 1 l L 10 100 Figure 31b. ' The amplitude 5'} Im[fN(O)] as a function of A. 163 0R = - % f 1*(rixmtzauoptJ¢(:>d3r . (VI-16) where ¢(r) is the distorted pion wavefunction, there is no well defined prescription for calculating the various components of 0R within the framework of the optical model. Thus some approximate means of calculating 00E and 0A must be devised. As the optical potential can be divided into single nucleon and absorption terms, a first guess at the form of the partial cross sections is 005 = - % f¢*([)Im[2®zgt([)1¢(f)d3r (vi-17) and 0A = - % [f(r) )1m[2;u"pt(r)]¢( (3r)d r (v1-13) This technique is perhaps reasonable in the low energy region, where the absorption terms dominate. It is not a good prescription in the resonance region, where the imaginary single nucleon and absorp- tion parameters are both large. The reason is that the two processes are not equivalent; a pion which scatters quasielastically can still be absorbed, but an absorbed pion cannot later scatter. This prob- lem was considered in Glauber theory (14), which is a good approxi- mation in the limit of high energy, projectile wavelength short compared to the nuclear size, and strong forward scattering. The Glauber result can be recast in the form of equations VI-17 and 164 VI-18. The result is that only the imaginary absorptive terms should be used in calculating the distorted wavefunctions in the expression for CA, * _ o, = --,% f¢A(r>1mtzuu§pt(r>J¢A A O V II O o 166 where x = O is the edge of the nucleus at which the pion enters, with x increasing across the nucleus. The solutions to equa- tions VI-21 with these boundary conditions are _ -x/A P0 - e P = e (1 - e ) (VI'23) QE -x/A _ A PA -1- e where A is defined by 1-_1_ .1. - “A — A + A (VI 24) A S The cross sections for these processes are obtained by integrating the probabilities at the end of the pion path through the nucleus over all impact parameters and angles, -L b /A -L b /A =2nfbdbe ()A(1-e H S) -L b /A 2nfbdb(1-e()A) OQE (VI-25) 0A where L(b) is the length of the path through the nucleus at impact parameter b, L(b) = ZVR2 - b2. Here R is the radius of the nucleus, taken to have a uniform matter distribution. The reaction cross section is just the sum of these two expressions, 167 OR = 2n f b db(1 - e'm’m) (VI-26) with A given by equation VI-24. Comparison of the expressions for 0A and OR indicates the Glauber result, that the absorption cross section is calculated in the same way as the reaction cross section, but with only absorptive processes taken into account. The transport calculation just described can itself be used to give the quasielastic and absorption cross sections, with the mean free paths estimated from the optical potential. The results, however, give the wrong ratio of quasielastic and absorption cross sections and the wrong A dependence of the quasielastic cross sec- tions. Various improvements to the transport theory are possible (83); however, these will not be discussed here. The transport theory result is used only as an approximate justification for equations VI-19 and VI-ZO. Partial Cross Section Calculations In this section the calculations of oQE and 0A are compared to the data of Navon et al. (81). The two approximations discussed in the last section are adopted in their respective energy domains: equations VI—17 and VI-18 are employed for energies through 50 MeV, equations VI-19 and VI-ZO in the resonance region, 160-220 MeV. No reliable method is known for extracting OQE and GA between 50 and 160 MeV. The limitations of the model should be kept in mind when comparisons of calculations and data are made. 168 In Figure 32 calculations of partial cross sections, using the two parameter sets described in Section 2, are compared with the data for n+ on 12C as a function of pion energy. Note that smooth curves have been drawn to connect the low energy and resonance regions. The reaction cross sections are well described by the In calculations. The differences between the results of the two param- eter sets for OR at low energies are due almost entirely to dif- ferences between CA for the two sets; the quasielastic cross sections are about the same. The calculated absorption cross sections mirror to some extent the differences between the p-wave absorption param- eters of the two sets, with the extrapolated pionic atom parameters giving much higher absorption cross sections at low energies, peak- ing earlier, and falling faster than the calculations with the Riska parameters. It is clear that both sets of calculations overestimate the absorptive and underestimate the quasielastic pieces of the reac- tion cross section. The absorption cross section has the correct shape, peaking near the resonance and falling off at higher energies. The data are not yet precise enough to determine the energy at which 0A peaks, thus little can be said of the relative merits of the two parameter sets. The calculated quasielastic cross section appears to rise too slowly between 50 and 150 MeV; however, as the curves are pure interpolation in this region no conclusions can be drawn from this. Insufficient evidence exists at present to determine whether the overestimated absorption cross sections are due to the strength of the absorption parameters themselves or to 3 10 r I i 41 “-0- _12C .. - J ' l :3 E _ i (D — .. Z % C) H I— L) LIJ ’ - (f) (D (D O [I L) _J 100- < l— H - 1— _ .. CC < - - O. .. O'OE .- + 0'A .. X O'R _ 1 J 1 L 0 100 200 TflUOb) [MeV] 12 Figure 32. Partial cross section calculations for n+ on C as a function of energy compared to the data of Navon et al. (81). Curves are same as in Figure 28. 170 the method of calculating the cross sections. It should be noted that calculations of 0A using equation VI-17, that is, with the fully distorted wavefunctions, produce much smaller absorption cross sections. The true value may lie between these two calculations. Figure 33 shows comparisons of calculated and experimental partial cross sections as a function of A. The data were taken at 165 MeV; however, the calculations are done at the resonance energy, 180 MeV, where the conditions required for the transport theory to be a good approximation are best met. The differences between cross sections at 165 and 180 MeV are presumed small. The solid and dashed curves are the optical potential calculations with the two parameter sets previously described. Note that the data are preliminary and have not been assigned errors as yet. The reac- tion and total cross sections are well reproduced by both sets of calculations. Although the calculated absorption cross section has about the right A dependence, it is too large in comparison with the data. The calculated quasielastic cross section is much too small and exhibits a flattening or saturation effect at large A which is not seen in the data. Part of this effect may be due to noise in the calculation, as OQE is the difference of two large terms. Again, the uncertainties in the method of determining OQE and 0A preclude any definite conclusions about the optical potential at this stage. It is possible that the problem of separating the reaction cross section into absorptive and quasielastic pieces could be solved 171 LI IOFTITTTI I I IIIIIII I I . 180 MEV j :3 E .- W Fl 2 O H ’— U LIJ “7 103_ m .— (f) 1- C3 _ C L) . _J _ < H I. ’— m < - 0. 100111111 1 LL111111 1 1 10 100 A. Figure 33. Partial cross section calculations as a function of A for n+ at 165 MeV, compared to the data of Ref. 81. Curves are same as in Figure 28. 172 by a different approach. The quasielastic cross section can perhaps be calculated directly, as was done for nucleon scattering by Bertsch and Tsai (84). Studies of charge exchange reactions indicate that the quasielastic scattering, of which charge exchange is assumed representative, looks like quasi-free scattering in energy and angle dependence, but with an effective number of nucleons Neff which is less than A (85). Thus the quasielastic scattering is a simple process, and could be calculated by summing the distorted wave impulse approximation results to all final states, using the full optical potential to distort the incoming wave and the optical poten- tial with only absorptive imaginary parts for the outgoing wave, as suggested by Koltun (83). A great deal of work remains to be done on this subject. CHAPTER VII CONCLUSIONS The focus of this work has been the construction and testing of a pion-nucleus optical potential which is simple in form but which includes the important physics. In this chapter the main features of the model are reviewed and its accuracy in reproducing the relevant experimental data is summarized. Unsolved problems and areas for further study are also discussed. The optical potential was constructed from the experimentally determined pion-nucleon transition amplitude using the Watson mul- tiple scattering series. The off-shell form of the T matrix was taken to be of Kisslinger type, without form factors. This form is convenient because it leads to a coordinate space potential which is local (although velocity dependent), making calculations much simpler. The kinematic transformation of the T matrix was treated inéuiapproximate way, expanding in w/M and keeping only zero and first order terms. The second order s-wave term of the multiple scattering series as calculated in low energy approximation was included, as was a sum of the p-wave series in the same approxima- tion. True pion absorption was represented by terms quadratic in the nuclear density, for which an approximate theoretical justifi- cation exists. Pauli blocking was roughly included, and an energy 173 174 shift due to the Coulomb potential was incorporated. The parameters for the various terms in the optical potential were taken from the experimental IN phase shifts and from various theoretical calcula- tions (47.48.50). An alternative set of parameters was derived from fits to pionic atom level shifts and widths with suitable energy extrapolation. This method of constructing the optical potential has several weaknesses. The first is that the starting point contains insuf- ficient information; the IN data can only give the on-shell behavior of the T matrix (unless a separable form is assumed (37)). A funda- mental theory of the nN interaction is necessary to give the correct off-shell dependence. Such a theory, if fully relativistic, would provide the proper kinematics for the problem as well as the neces- sary framework for treating the higher order multiple scattering and absorption terms consistently. No completely satisfactory nN theory exists, although some form of Chew-Low description (86) is probably adequate. To carry out this program of optical model con- struction is an impossibly complicated task; a pr0per first order calculation in a finite nucleus would be very difficult. The com- plicated dependence of the full T matrix on the nucleon momenta requires the use of realistic nucleon wavefunctions in the integral over the ground state, limiting the resulting potential to one nucleus as well as one energy. It must be hoped that such a complete calculation can be used to justify approximations which lead back to an optical potential of sufficient simplicity to be used in 175 practical calculations. The success of the simple potential pre- sented here suggests that this hope is not groundless. The second weakness in the derivation of the potential given here is the number and severity of approximations required. Some of these are necessary to keep the potential form simple; others are required to make calculation of a given term feasible. Most of the approximations are easily justified for low pion energies but are not obviously valid in the resonance region. Fortunately they are also less important there, as the scattering calculations are not as sensitive to the terms in question. It should be noted that knowledge of the experimental phase shifts is vital for studies of pion-nucleus processes, whether incor- porated directly, as in the present study, or used as a test of the fundamental NN theory which generates the pion-nucleus inter- action. The resonance region phase shifts are fairly well deter- mined; however, the low energy phase shifts are difficult to measure and not well known at present. This is a serious problem, as the low energy scattering calculations are quite sensitive to these numbers. The validity of a model such as the one presented here rests ultimately on its ability to reproduce the experimental results. The initial comparisons at low energies were somewhat disappointing. The pionic atom level shifts and widths indicated too little 5 and p-wave absorptive strength in the potential at zero energy and too little s-wave repulsion. The problem of the absorption parameters 176 is not serious, as the calculation of these parameters is still subject to uncertainty. The missing real s-wave strength is more disturbing, since it is not clear what mechanism could provide the required repulsion. This problem appears also in the low energy scattering, for which the potential gives a good description pro— vided the s-wave repulsion is increased. A reasonable fit is also given by the potential with parameters extrapolated from the pionic atom fit. These two potentials have quite different absorptive strengths; the elastic scattering data at low energies are not very sensitive to the imaginary part of the optical potential. Unfor- tunately, there is as yet no low energy absorption cross section data, which would provide more information about the absorption parameters. Thus the form of the potential gives an excellent . framework for studying the systematics of the low energy data; how- ever, the theoretically derived parameters do not satisfactorily reproduce the experimental quantities. The optical potential does reasonably well in the resonance region. The general features of the elastic scattering and total cross sections are reproduced over a wide range of energies and nuclei. No attempt has been made to improve the fit by parameter searches, as the Coulomb energy shifts incorporated in the poten- tial would necessitate a separate search for each nucleus at each energy. The results of such a study would be very difficult to correlate. The evidence from the absorption cross section measure- ments cannot be interpreted unambiguously because ofiflueuncertainties 177 inherent in the calculation of the absorption cross sections within the context of the present theory. Thus much work needs to be done, both experimentally and theoretically. Studies of more complicated pion-nucleus processes offer new tests of the simple optical potential concept, while micro- scopic calculations can give new insight into the appropriate form and approximations for the problem. APPENDIX A THE PION-NUCLEON SCATTERING AMPLITUDE 178 "L APPENDIX A THE PION-NUCLEON SCATTERING AMPLITUDE In this Appendix a review is given of the origin of the form of the scattering amplitude used to describe pion-nucleon scattering. The expansion of the scattering amplitude in terms Of the phase shifts is given, and the terms Of this expansion which are important for the pion-nucleon interaction are expressed in simple form with the parameters related to the relevant phase shifts. The scattering amplitude can be expanded (3) in terms Of iso- spin 1, orbital angular momentum L, and total angular momentum J, . _ L _ “5,5 ) ... 1213.1 QI PLJ (21 +1)u21’2J PL (cos a) (A 1) where . L OL = exp(21621’2J) - l (A-2) 21,20 21 kcm is the scattering amplitude for the (I,L.J) partial wave,and QI and PL J are projection Operators projecting onto states Of given I or L and J, 179 180 -1 01/2 ' 3(1 ‘ E I) Q = 1-(2 + t°T) 3/2 3 .. (ll-3) L-Sii PLJ=L--1—= 2L” . 2 L + 1 + O'g PL.J=L+%- 2L” Here g is the relative angular momentum Operator, which acts on the Legendre polynomials in equation A-1, EP£(cos e) = r x (-iV)P (cos 6) ~ ~ 2 _ - “ 3L - 1r x 6 39 P£(cos e) . Note that the plane defined by r and 6 is the same as that defined by Ecm and Ecm’ A A k x k. A ~cm ~cm _ -r X e = | : n . (A-S) IEcm x ECWJ_ Thus gP2(cos O) = inPi(cos O) . (A-5) The important terms in the pion-nucleon amplitude for the energies considered here are the L = O and L = 1 terms of equa- tion A-l. The L = 0 terms are —_ --_H._ _~.— ...... A . .....—--. -— 181 o o Q1/2Po,1/2O‘11 Po(COS 9) + Q3/2PO,1/20‘31 P0(C°S 9) (A-7) ‘ 511 ‘ E'I)091 + 112 + E'I)“g1 = bO + blt'T where - 1_ o 0 b0 ‘ 3(011 I 20‘31) _1_o 0 b1 - 3( all + e31) . (A-8) The p-wave terms are 1 1 3[P1,1/2ml/2O‘11 + Q3/2831) + 1 1 + P1,3/2(Ql/2a13 + Q3/2a33)]P1(COS 6) - 1 1 1 1 ' [(01/2811 + Q3/2031) * 2(01/2813 T Q3/20‘33)]C°S e 1 1 1 1 . A . + ['(01/2811 + Q3/2831) + (01/2913 T Q3/20‘33H‘9'n 5‘" 9 ° (A-9) Noting that lkcml = [kéml 5 k0 for elastic scattering, the functions of a can be written k 'k' cos 0 = 353L€?Zfl (A-lO) k 0 and A n sin 0 = 182 _1_ k k x ' . ~cm 2cm 2 0 Therefore the p-wave terms are where (to + cmk .1. + ($0 + s tmils... x Eém) , ..cm ..cm ... ~ ... )+2(a +2011 )1 1 13 33 jL_1_ _ 1 1 _ 1 1 k2 3[( 0‘11 + “31) + 2‘ 0‘13 + 033)] o ;L_1 _ 1 1 1 1 k2 §{ (“11 l 20‘31) + (“13 + 20‘33)] o _1__1 _ _ 1 1 _ 1 1 k2 3[ ( 0‘11 + “31)*'( 0‘13 + 0‘33)] ° 0 (A-11) (A-12) (A-13) APPENDIX 8 DETAILS OF THE DERIVATION OF THE MULTIPLE SCATTERING SERIES 183 APPENDIX 8 DETAILS OF THE DERIVATION OF THE MULTIPLE SCATTERING SERIES In this Appendix two results are derived which are required in the development Of the Optical potential formalism given in Chap- ter III. These are: (1) Let A, B, C, D, and F be Operators in an arbitrary space. If A = B + BCA (B-1) and D = B + BFD (8-2) then D = A + A(F - C)D (B-3) (2) Let A, B be many particle Operators and ai’ Bi one particle Operators. If ' ‘A = 2 a]. + 2 oiBA (3-4) i i and A = 2 A. (B-S) 184 185 then A=2 31+: 81.32 A]. (B-6) 1 i in where 8i = 0i + aiBBi (3-7) TO prove the first result, rewrite equation B-l, B = A(1 + CA)"1 (B-8) and substitute for B in equation B-2 0 = A(1 + CA)'1(1 + F0) . (3-9) By writing 1 = 1 + CA - CA this becomes 0 = A(1 + CA)(1 + CA)'1(1 + F0) + ACA(1 + CA)‘1(1 + F0) = A(1 + F0) - ACB(1 + F0) (B-lO) where equation B-8 was used in the last step. The second term can be simplified using equation B-2, giving C II A + AFD - ACD (B-ll) 01“ C II A + A(F - C)D . (B-12) This proves the first result. 186 For the second result, equations B-4 and B-5 imply Ai=ai+aiBA=ai+aiBJzAj° Grouping the A1 terms gives (1- 111-BM]. = 011- + 011.8 2 A 01“ Equation B-7 can be written _ -1 B. - (1 - GiB) on]. and substituted in equation B-15, yielding Ai=Bi(1+B Z A. ). J'fi J Summing both sides over i gives the second result, A=Zei+ZBiB Z A.. l 'I #1 J (B-13) (B-14) (B-15) (B-16) (B-17) (B-IB) APPENDIX C INTEGRATION OVER NUCLEON MOMENTA 187 IL.__ APPENDIX C INTEGRATION OVER NUCLEON MOMENTA The derivation of the Optical potential requires the integra— tion of several one and two particle Operators over the coordinates (or momenta) Of the nucleons in the ground state nucleus. In this Appendix the required integrals are performed. The Fourier trans- forms Of the two common momentum space forms for the Optical poten- tial are also given. As mentioned in the text, the momentum Of the nucleus as a whole is ignored in the evaluation of the ground state terms. The one particle operators to be evaluated are 1:1) = A , (C-l) II?) = A , (c-z) and 113) ‘ A ' (C-3) Let A('T-i)‘ao + a1E'Ii 0(”(Ei’Ei) = 1 188 189 N V A ('U do U (U do - v I II ('0 —l - 11 (C-4) 11 d pj X (p(Plsgz... PA 3 (21103 (C'S) where q = k' - k. The i = 1 term has been chosen and the spin and isospin variables suppressed to simplify notation. Transforming to coordinate space gives 11")<:) = A fwri’rz rAmrvrz EMMA) . . 3 1(p -q)-r' -1p -r . . d p 3 xdn)(p1 _ q,p1)e ~1 ~ ~1 e ~1 ~1 e13: 13 (143 7 ~ ~ (2w) (2") x d3ri II d3l". j .1 (C-6) The operator 0(n)(p1 - 3,21) can be replaced by OWN}??? - 3Y1) acting on the exponentials, allowing the momentum integrals to be performed 190 11”’<:> = A )1 w*<:i’:2 --- EA)W(31’C2 --- CA) ((2-7) HORN-1:71, -%-:(1>=Af1“Oh-~2090+hrnN012~-m> n: 3 * j>1 d "i (M) The aO term is just the nuclear density p(r). As there is no change in the isospin projection for the nucleus, the only non-zero contri- bution from t-: is from t313, where 13 is +1 for protons, —1 for neutrons. Expression C-B is therefore 1(1’15) = aoo(r) + alt3[op(r) - on(r)] . (c-9) where p is normalized tO A, pp to Z, and on to N. Expression C-l is the Fourier transform of this, 1 If )(g) = 600(4) + alt31op(q) - on(q)] (c-10) V'-V For the second expression, CA2) =-11Efi—ll , and integration by parts gives (C-11) This is, aside from the factor A(T), the current density and is zero for a spherically symmetric nucleus. Assuming the neutron and proton distributions are spherically symmetric, expression C-2 is zero. For n = 3, CA3) = Vl-Yi and integration by parts twice gives 113)(E) ‘ aoA [1714952 EA)°‘Z‘1’(I’52 EA) {>11 ‘3'} a0K(r) (C-12) where K(r) is 2M times the kinetic energy density Of the nucleons, and the isovector part Of A(1) has been dropped. The quantity K(r) is evaluated in the Thomas-Fermi approximation (41), (<0) = %(—3- 1,2)2/ 305/ 3 043) This completes the evaluation Of the one-body Operators. The first two-particle Operator to be evaluated is 1%” = MA - 1>66<21+r - 21 - .1">«i<21+:s" - p2 - 1) x (a0 + alt-IlflaO + a1t°:2)lO> . (C-14) 192 which can be written Iél’us' w = MA- 1)] w*<21- “5' - 5"),92- 0.2"- Wm) d3p. x A(T1)A(T2)‘l’(‘21"32433 PA) I} 3 (2n) (C-15) This is only a function of the variables (k' - k") and (k" - k) and can be Fourier transformed resulting in a function of r and r', 1(1)(r r') = A(A - 1) ¢*(r r' r r ) (r r' r r ) 2 ~,~ ~,~ ,~3 ... ~A u; ~,~ ,~3 ... ~A x A(11)A(12) 3.132 d3!“j . (C-15) Without the isospin factors this is just the two particle density p2(r,r'), which can be written in terms of the two body correlation function C(r,r'), 02([,[') = [1 + C([.[')]p(f)o([') - (C-17) In order to simplify expression C-16, assume that u can be expressed as an antisymmetrized product of single particle wavefunctions, 1 w = 2%? det {¢i(rj)} (C-18) 193 Then expression C-16 becomes Iélhmu) =1}; ¢;(:)¢3(:')A(T)A(r')[¢i([)¢j(§')- ¢i([')¢j(:)l (C-19) The first term is just [aoo(§) + t3a1(op([) - on([))][aoo(§')+t3a1(op(§') - on(['))] (C-ZO) The spin and isospin dependence of the second term can be made explicit and factored out, replacing ¢k(r) by ¢k(r)xm(o)ns(1) where X and n are Pauli spinors in spin and isospin space. For this term the spin and isospin projection of the nucleus will be approximated as zero. Then the sum over states can be separated into sums over space, spin, and isospin states. The sum over spin dependent factors gives x;(o>x; m,n=1,2 The sum over isospin factors gives 3:1 2 n:(T)n:(T')(ao + alg-g)nsj(g)=-o(g)p(:.)li6[ :FIE': 51" ] (C-26) Combining equations C-19, C-20, C-21, C-22, and C-26 gives for expression C-14 195 I§1)w<2pr2£3 IA) X e ( ) (C-30) -ip ° r - r' -iq°r' X e ~2 ~2 ”2 e ~ “'2 d3r' d3r' II d3r. 1 2 i 1 3 . 3 3 d pl d p1 d p2 X (2n)3 (2n)3 (2n)3 ' and the remaining momentum integrals lead to delta functions. The result is the Fourier transform of the two particle density 02(r,r), 197 2) 'lS'E 3 3 I( (q) = f w*(r,r,r r )w(r,r,r r )e d r IId r. (C-31) This completes the discussion of two-body operators. The simple Kisslinger and Laplacian models for the optical potential include the momentum space terms 5°5'o(Q) (C-32) and 020(q) (C-aa) respectively. The Fourier transform of C-32 is ik'r -ik'-r' 3 3 . k'k'p(k' - k)e * ” e * ” d k3 d “:3 (c-34) ” ~ ” ~ (2n) (2“) which Can be written iK’X iq R 3 3 jmwmmm~~e~~ dg dg «an ~ ” (2") (2“) where 5 = 2 ’ 3 ll ’3 I (7" o I” II N e (X ll (‘5 I ('1 The integrals over 5 and q give ~ 198 y-y'tao; - [')o(” 2 ~ )1 (c-ss) Note that this is an operator of the form . The gradients which act on the delta and density functions can be turned around to act on whatever functions of r and r' are to the right and left of 0. This gives r + r' .+ PM: - [')o(“ 2 ~ )Jv