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A modern potential model calculation of proton-proton bremsstrahlung Workman, R. L. 1984

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A MODERN POTENTIAL MODEL CALCULATION OF PROTON-PROTON BREMSSTRAHLUNG by v RONALD LESTER WORKMAN B. Sc., The University of Victoria, 1981 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in THE FACULTY OF GRADUATE STUDIES (Department of Physics) We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA March 1984 © Ronald Lester Workman, 1984 In presenting t h i s thesis i n p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y a v a i l a b l e for reference and study. I further agree that permission for extensive copying of t h i s t h e s i s for s c h o l a r l y purposes may be granted by the head of my department or by h i s or her representatives. I t i s understood that copying or p u b l i c a t i o n of t h i s t h e s i s for f i n a n c i a l gain s h a l l not be allowed without my written permission. Department of Physics  The University of B r i t i s h Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date March 29, 1984 :-6 (3/81) ABSTRACT We describe a modern p o t e n t i a l model c a l c u l a t i o n for proton-proton bremsstrahlung( ppY ). This c a l c u l a t i o n i s important as i t uses the Paris Nucleon-Nucleon p o t e n t i a l , a modern t h e o r e t i c a l l y based p o t e n t i a l which has not been used i n ppY c a l c u l a t i o n s before. Observables for the ppY process are calculated and compared with r e s u l t s obtained by other authors - both t h e o r e t i c a l and experimental. We calculate both cross sections and analyzing powers for ppy. Asymmetry data i s expected from a new TRIUMF ppy experiment, which w i l l provide the f i r s t opportunity f o r comparison of modern asymmetry data with a modern p o t e n t i a l model c a l c u l a t i o n . Several potentials are used i n the present ppy c a l c u l a t i o n ; r e s u l t s f o r the Reid soft-core p o t e n t i a l , the 'Extended' Reid soft-core p o t e n t i a l and, the Paris p o t e n t i a l are produced. The partial-waved Reid soft-core and Paris potentials are described i n d e t a i l . The c a l c u l a t i o n u t i l i z e s f u l l y r e l a t i v i s t i c kinematics, coplanar and non-coplanar geometries and, can be performed i n either the lab or center-of-mass frame. R e l a t i v i s t i c spin corrections have been added and the e f f e c t of Liou's gauge term has been considered. The numerical c a l c u l a t i o n s are checked i n several ways. Comparisons are made with soft-photon approximation c a l c u l a t i o n s near the soft-photon l i m i t . In addition, e l a s t i c observables are produced and checked at intermediate points i n the c a l c u l a t i o n . We f i n d that r e s u l t s from the Extended Reid and Paris p o t e n t i a l s are q u a l i t a t i v e l y the same. We also f i n d that the analyzing powers are sen s i t i v e to approximations made i n the c a l c u l a t i o n - more s e n s i t i v e than are the cross sections. i i i TABLE OF CONTENTS Page ABSTRACT i i LIST OF TABLES v LIST OF FIGURES v i ACKNOWLEDGEMENT v i i CHAPTER 1 Introduction 1.1 Previous Proton-Proton Bremsstrahlung Calculations 1 1.2 Outline of the Present Calculation 5 CHAPTER 2 Derivation of the Invariant Amplitude 2.1 Introduction 9 2.2 The T-matrix Formalism 10 CHAPTER 3 A Numerical Solution for the N-N T-matrix 19 CHAPTER 4 The Partial Waved Paris N-N Potential 4.1 Introduction 28 4.2 The Paris N-N Potential 29 CHAPTER 5 Contruction of the ppy Observables 44 CHAPTER 6 The ppy Code 6.1 Description of the Code 6.2 Numerical Tests of the Program 49 54 ±v Page CHAPTER 7 Results of the Numerical Calculation 7.1 Introduction 64 7.2 The PPy Results 65 7.3 Summary and Possible Improvements 82 BIBLIOGRAPHY 89 APPENDICES Appendix A Evaluation of the Electromagnetic Matrix Elements 92 Appendix B The Reid Potential 97 Appendix C The Elastic Observables 102 Appendix D The Axis of Quantization 107 Appendix E The Legendre Functions 112 LIST OF TABLES Page Table 1 The Tensor Coupling C o e f f i c i e n t s 3 5 Table 2 Table of Useful Integrals 4 2 Table 3 Comparison of the ^-SQ Reid Soft-Core Phase from 5 5 D i f f e r e n t Sources Table 4 A Comparison of the Reid Soft-Core and Paris 5 6 P o t e n t i a l T=l Phases at Low Energy Table 5 A Comparison of the Reid Soft-Core and Paris 5 7 P o t e n t i a l T=l Phases at Medium Energy Table 6 Integrated Cross Sections at 2 0 0 MeV 03=0^=30° 6 2 v i LIST OF FIGURES Page Figure 1 Diagrammatic Expansion of the T-matrix 13 Figure 2 Algorithm for Checking the ppy Program 51 Figure 3 Soft-Photon Limit at Tl = 25 MeV 59 Figure 4 Soft-Photon Limit at Tl = 280 MeV 61 Figure 5 Calculation at Tl =42 MeV ©3=0^=22° 66 Figure 6 Calculation at Tl = 200 MeV ©3=0^=16.4° 67 Figure 7 Relativistic Spin Corrections, Coulomb contributions and the 1-u exchange at 200 MeV ©3=0^=16.4° 68 (cross sections) Figure 8 Relativistic Spin Corrections, Coulomb contributions and the 1-TT exchange at 200 MeV ©3=9^=16.4° 69 (asymmetries) Figure 9 Relativistic Spin Corrections, Coulomb contributions and the 1—IT exchange at 100 MeV ©3=0^=30° 70 (cross sections) Figure 10 Relativistic Spin Corrections, Coulomb contributions and the 1-TT exchange at 100 MeV ©3=0^=30° 71 (asymmetries) Figure 11 Cross sections at Tl = 200 MeV 03=2O° 0^=40° 73 Figure 12 Asymmetries at Tl = 200 MeV 03=2O° 0^=40° 74 Figure 13 Cross sections at Tl = 280 MeV 03=1O° 0l+=3O° 75 Figure 14 Asymmetries at Tl = 280 MeV 03=1O° 0^=30° 76 Figure 15 Cross sections at Tl = 280 MeV ©3=0^=10° 78 Figure 16 Asymmetries at Tl = 280 MeV ©3=0^=10° 79 Figure 17 Cross sections at Tl = 280 MeV 03=0^=30° 80 Figure 18 Asymmetries at Tl = 280 MeV ©3=0^=30° 81 Figure 19 Off-shell Extension Function Tl = 200 MeV, XS0 State 85 Figure 20 Off-shell Extension Function Tl = 200 MeV, 3P, State 86 ACKNOWLEDGEMENT I would l i k e to thank Dr. H.W. Fearing for his patience and guidance over the past two years. I must also thank Drs. D. Beder, M. McMillan and N. Weiss for giving me the background necessary f or t h i s project. Credit should be given to Dr. A. M i l l e r for providing some computer codes for h a l f - o f f - s h e l l T-matrices. F i n a l l y , I would l i k e to thank the National Science and Engineering Research Council and The University of B r i t i s h of Columbia for f i n a n c i a l assistance. 1 CHAPTER 1 Introduction 1.1 Previous Proton-Proton Bremsstrahlung Calculations In the following, we study the proton-proton bremsstrahlung( pp ppy ) process within the framework of a p o t e n t i a l model c a l c u l a t i o n . Before comparing our c a l c u l a t i o n with previous proton-proton bremsstrahlung( ppy ) r e s u l t s , we should r e f l e c t on the importance of understanding the ppy process. The most important reason for doing ppy ca l c u l a t i o n s i s the p o s s i b i l i t y that they w i l l provide information on the o f f - s h e l l aspects of the nucleon-nucleon ( N-N ) i n t e r a c t i o n . I t i s well known that, given one po t e n t i a l which reproduces the e l a s t i c data, an i n f i n i t e class of potentials may be generated 8' 9 v i a unitary transformations. Each p o t e n t i a l within the class has the same on-shell behavior and thus, i s able to reproduce e l a s t i c data equally w e l l . These potentials do not, however, have the same success i n reproducing ppy data. This has been demonstrated by Nyman8. In addition, such potentials give d i f f e r e n t r e s u l t s when used i n nuclear matter c a l c u l a t i o n s 9 . In order to understand these differences one must r e a l i z e that ppy cal c u l a t i o n s use o f f - s h e l l T-matrix elements, which are not required i n e l a s t i c s c a t t e r i n g . Thus, such cal c u l a t i o n s constitute a test of the o f f -s h e l l behavior of potentials which have reproduced e l a s t i c data. ( The re l a t i o n s h i p between T-matrices and the matrix elements of a p o t e n t i a l w i l l become clearer i n Chapter Two. ) By comparing ppy c a l c u l a t i o n s and experiments i t i s hoped that the aforementioned class of potentials can be reduced i n number. 2 These expectations have not yet been r e a l i z e d . There are two possible reasons for t h i s . F i r s t l y , past cal c u l a t i o n s have used old phenomenological potentials and have usually neglected correction terms which are not always small - hence these c a l c u l a t i o n s may not be s u f f i c i e n t l y accurate. Secondly, there has been i n s u f f i c i e n t communication between th e o r i s t s and experimentalists p r i o r to the planning of experiments. As a r e s u l t , experiments have not always used geometries 1 0 which enhance o f f - s h e l l e f f e c t s . In many cases, soft photon approximation SPA ) c a l c u l a t i o n s , which require only on-shell information, have reproduced the e x i s t i n g data as well as or, better than p o t e n t i a l model c a l c u l a t i o n s 2 3 . The troubled h i s t o r y of ppy theory and experiment began i n the mid-s i x t i e s , when a set of bremsstrahlung experiments was proposed at Harvard 1 . An i n t e r e s t i n the theory of nucleon-nucleon bremsstrahlung( NNy ) was also developing at t h i s time. The early papers of Sobel and Cromer 2 appeared i n Physical Review as r e s u l t s were becoming av a i l a b l e from the experiment-a l i s t s . Unfortunately, these c a l c u l a t i o n s contained several errors, as has been indicated by S i g n e l l and Marker 3. I n i t i a l l y , the disagreement with experimental values caused considerable speculation 4 and prompted a surge of t h e o r e t i c a l input. By 1968, however, proton-proton bremsstrahlung c a l c u l a t i o n s 5 * 6 were reproducing the q u a l i t a t i v e behavior exhibited i n ppy cross section data?. Through the seventies, the basic c a l c u l a t i o n was refined and, although experimental and t h e o r e t i c a l values were converging, agreement was s t i l l only q u a l i t a t i v e at medium energies, below the pion production threshold. The discrepancies at medium energies and other more puzzling problems at lower e n e r g i e s 2 3 remain today. For instance, at 42 MeV, r e l a t i v i s t i c corrections should be small and, one would expect good agreement between 3 p o t e n t i a l model c a l c u l a t i o n s and experimental data. However, there are geometries i n which agreement has only been q u a l i t a t i v e . Assuming that the poor agreement between theory and experiment i s due to an unsatisfactory c a l c u l a t i o n of ppy observables, we should consider what progress has been made on the basic p o t e n t i a l model c a l c u l a t i o n . Soon after S i g n e l l and Marker 3 pointed out errors i n Cromer and Sobel's early paper 2, a second paper was published by Sobel and Cromer 1 1. However, there s t i l l remained q u a l i t a t i v e differences between theory and experiment. In p a r t i c u l a r , the integrated cross section were off by large factors at low energies. This low energy problem was resolved when Drechsel and Maximon published two papers 5 1 2 i n the following year. These c a l c u l a t i o n s were performed i n both the lab and center-of-mass( CM ) systems. As i n reference 11, the "double s c a t t e r i n g " terms were omitted. These terms are shown i n chapter 2, f i g . 1. I t was thus found that dropping the double scattering terms i n the lab frame, using the transverse gauge, was not a good approximation. In the CM system, q u a l i t a t i v e agreement with experimental data was reached. Drechsel and Maximon5 showed that the double scattering terms were small i n the CM system, when the transverse gauge was used. However, the use of the transverse gauge i n the Lab frame corresponded to a Lorentz transformation followed by regauging 1 2. This would not have been a problem i f the ppy amplitude, constructed from single s c a t t e r i n g terms, were gauge i n v a r i a n t . The addition of double scattering terms had e a r l i e r been reported by V. Brown 1 3. Maximon and Drechsel found agreement with Brown's resu l t s to the 10% l e v e l i n the CM system. Other c o r r e c t i o n terms have also been investigated. Liou and Cho have found 1 1* that r e l a t i v i s t i c spin corrections are important at 157 Mev. These ad d i t i o n a l terms, which are r e l a t i v i s t i c corrections to the electromagnetic i n t e r a c t i o n , dramatically improve agreement i n the forward and backward photon d i r e c t i o n s . H e l l e r 1 5 and L i o u 1 6 have drawn from the work of Low 1 7 to produce gauge terms for the ppy c a l c u l a t i o n . These terms help to make the lab system c a l c u l a t i o n s more reasonable since they are equivalent to the leading part of the double sc a t t e r i n g contribution but, do not improve the CM r e s u l t s . It i s c l e a r from Liou's form of the gauge terms that these terms do not contribute i n the CM system. L i t t l e else has been done to improve the basic ppy c a l c u l a t i o n to date. New potentials have been produced recently, for example the Paris p o t e n t i a l 1 8 and the Bonn p o t e n t i a l 2 1 * , which have used meson exchange models as a basis, but have not previously been employed i n ppy c a l c u l a t i o n s . A new ppy experiment i s also being planned at TRIUMF. It w i l l be an i n t e r e s t i n g experiment for several reasons. I t i s expected to run at 280 MeV, which i s a higher energy than most previous ppy experiments yet, s t i l l low enough to be below the threshold for pion production. In addition, analyzing powers w i l l be measured for the f i r s t time i n a modern experiment. One should also note that a thorough comparison of experimental and t h e o r e t i c a l asymmetries has never been made. I t w i l l be seen l a t e r that these asymmetries are quite s e n s i t i v e to the methods used i n t h e i r c a l c u l a t i o n . The next step i n the study of ppy should be a comparison of res u l t s derived from these modern potentials and t h e i r predecessors and modern ppy data. 1.2 Outline of the Present C a l c u l a t i o n In the following section, we w i l l describe our approach to the p o t e n t i a l model c a l c u l a t i o n , as well as the ingredients which make our c a l c u l a t i o n unique i n comparison with previous c a l c u l a t i o n s . The paper of Drechsel and Maximon5 i s a good introduction to the formalism which we have employed and i s c h a r a c t e r i s t i c of the best of the older c a l c u l a t i o n s . Like t h e i r c a l c u l a t i o n , ours i s also based on the Lippmann-Schwinger equation and i s , therefore, inherently n o n - r e l a t i v i s t i c In addition, both c a l c u l a t i o n s : - neglect the double s c a t t e r i n g terms ( see f i g . 1 ) - u t i l i z e f u l l y r e l a t i v i s t i c kinematics. - may use either the lab or center-of-mass system( Unless otherwise stated, we do a l l our c a l c u l a t i o n s i n the CM system i n order to minimize the e f f e c t s of neglecting the double scattering terms. ) - may be performed i n non-coplanar geometries( Unfortunately, the aforementioned experiment, planned at TRIUMF, w i l l probably not be exploring the e f f e c t s of non-coplanarity.) There are, however, many differences between the two c a l c u l a t i o n s . For instance, we generate our h a l f - o f f - s h e l l N-N T-matrices by solving an i n t e g r a l equation for the ' h a l f - o f f - s h e l l function'. This method i s described by Watson and N u t t a l l 1 ^ and, i s deta i l e d i n Chapter 2. Maximon and Drechsel have produced t h e i r h a l f - o f f - s h e l l N-N T-matrices by solving d i f f e r e n t i a l equation for the quasi-phase parameters, which they describe. More importantly, we have included coulomb corrections i n our c a l c u l a t i o n 6 of the N-N T-matrices. We have also added r e l a t i v i s t i c spin corrections to the e l e c t r o -magnetic i n t e r a c t i o n and have investigated the e f f e c t of adding Liou's gauge term. Several potentials have been used i n our p o t e n t i a l model c a l c u l a t i o n . We have made comparisons between the Reid Soft-Core 2 1 p o t e n t i a l , the Paris p o t e n t i a l 1 8 and the 'extended' Reid potential-described i n a paper by B.D. Day 2 2. In addition, the 1—IT exchange p o t e n t i a l 2 5 has been added f or p a r t i a l waves, above which the various potentials are eit h e r not calculated or, are not defined. This i s the f i r s t c a l c u l a t i o n i n which a modern and more ' r e a l i s t i c ' p o t e n t i a l has been used. Very l i t t l e progress has been made on the ppy poten t i a l model c a l c u l a t i o n since the early seventies. Because of t h i s , the theory i s now outdated and i s not adequate for a comparison with the new TRIUMF experiment 2^. The inadequacy of previous c a l c u l a t i o n s i s es p e c i a l l y apparent when analyzing powers are considered. As has been mentioned previously, some of our most i n t e r e s t i n g r e s u l t s have come from asymmetry c a l c u l a t i o n s . References 1. B. Gottschalk, W. Shlaer & K. Wang,"A Measurement of Proton-Proton Bremsstrahlung at 160 MeV," Phys. Lett. 16, 294 (1965); Nucl. Phys. 75, 549 (1966); Nucl. Phys. A94, 491 (1967). 2. A.H. Cromer & M. Sobel,"Theory of Proton-Proton Bremsstrahlung," Phys. Rev. 152, 1351 (1966). 3. P. Si g n e l l & D. Marker,"Proton-Proton Bremsstrahlung Calculations, B u l l . Am. Phys. Soc. 12_, 123 (1967). 4. W.A. Pearce, W.A. Gale & I.M. Duck, "Proton Proton Bremsstrahlung, Nucl. Phys. B3, 241-244 (1967). 5. D. Drechsel & L.C. Maximon,"Potential Model Calculation for Coplanar and Non-Coplanar Proton-Proton Bremsstrahlung," Ann. Phys. 49, 403-444 (1968). 6. V. Brown,"Proton-Proton Bremsstrahlung Including Rescattering," Phys. Rev. 177, 1498 (1969). 7. M.L. Halbert, "Review of Experiments on Nucleon-Nucleon Bremsstrahlung," i n : The Two-Body Force i n Nuclei, eds. S.M. Austin & G.M. Crawley(New York: Plenum Press, 1972). 8. E.M. Nyman,"Bremsstrahlung i n Nucleon-Nucleon C o l l i s i o n s , " Phys. Rep. 9_, 184-187 (1974). 9. F. Coster, S. Cohen, B. Day & CM. Vincent,"Variation of Nuclear Matter Binding Energies with Phase-Shift-Equivalent Two-Body Potentials," Phys. Rev. CI, 769 (1970). 10. H.W. Fearing,"Forbidden Asymmetries and Choices of Geometry i n Proton-Proton Bremsstrahlung," Phys. Rev. Lett. 42, 1394-1397 (1979). 11. M. Sobel & A.H. Cromer,"Potential Model Calculation of Proton-Proton Bremsstrahlung," Phys. Rev. 158,1157 (1967). 12. D. Drechsel & L.C. Maximon,"Potential Model Calculation for Coplanar and Non-Coplanar Proton-Proton Bremsstrahlung," Phys. L e t t . 26B, 477 (1968). 13. V. Brown,"Theory of Proton-Proton Bremsstrahlung Consistent with Experiment," Phys. L e t t . 25B, 506 (1967). 14. M.K. Liou & K.S. Cho,"Non-Coplanar Proton-Proton Bremsstrahlung Calculations Including the R e l a t i v i s t i c Spin Corrections," Nucl. Phys. A160, 417-427 (1971). 15. L. Heller,"Soft Photon Theorem for Nucleon-Nucleon Bremsstrahlung, Phys. Rev. 174, 1580 (1968). 8 16. M.K. Liou, "Low Energy Theorem for Nucleon-Nucleon Bremsstrahlung," Phys. Rev. C2, 131 (1970). 17. F.E. Low, "Bremsstrahlung of Very Low-Energy Quanta i n Elementary P a r t i c l e C o l l i s i o n s , " Phys. Rev. 110, 974 (1958). 18. M. Lacombe, B. Loiseau, J.M. Richard & R.V. Mau, "Parameterization of the Paris N-N Potential," Phys. Rev. C21, 861 (1980). 19. K.M. Watson & J . N u t t a l l , Topics i n Several Body Dynamics(San Francisco: Holden-Day, 1967), Chapter 2. 20. T.R. Mongan, "Separable Potential Models of the Nucleon-Nucleon Interaction," Phys. Rev. 178, 1597 (1969). 21. R.V. Reid, "Local Phenomenological Nucleon-Nucleon Potentials," Ann. Phys. 50, 411-448 (1968). 22. B.D. Day, "Three Body Correlations i n Nuclear Matter," Phys. Rev. C24, 1269-1270 (1981). 23. H.W. Fearing, "Comparison of Proton-Proton Bremsstrahlung Data at 42 and 156 MeV with Soft Photon Calculations," Phys. Rev. C22, 1388 (1980). 24. K. Holinde, R. Machleidt, M.R. Anastasio, A. Faessler & H. Muther, "Role of Noniterative TT exchange i n NN-Scattering," Phys. Rev. C19, 948 (1979). 25. P. C z i f f r a , M.H. MacGregor, M.S. Moravcsik & H.P. Stapp, "Modified Analysis of Nucleon-Nucleon Scattering. I. Theory and p-p Scattering at 310 MeV," Phys. Rev. 114, 880-886 (1958). 26. TRIUMF experiment #208; P. Kitching, spokesman. 9 CHAPTER 2 Derivation of the Invariant Amplitude 2.1 Introduction In t h i s and the following chapter, we w i l l derive r e l a t i o n s for the ppy observables which we have c a l c u l a t e d . Along the way, we w i l l also describe some e l a s t i c p-p scattering observables which have been calculated i n order to test intermediate r e s u l t s produced by our ppy code. In t h i s chapter we w i l l o utline the T-matrix formalism which has been used to c a l c u l a t e the invariant amplitude for ppy. Here, we w i l l compare our c a l c u l a t i o n with the e a r l i e r c a l c u l a t i o n of Drechsel and Maximon 7. In p a r t i c u l a r , we w i l l Indicate where changes and improvements have been made. Many of the more de t a i l e d c a l c u l a t i o n s w i l l be relegated to the appendices. In Chapters 3 and 4, the N-N T-matrices and matrix elements of the Paris N-N p o t e n t i a l w i l l be derived. Then i n Chapter 5, having obtained the ingredients required to produce the invariant amplitude, we w i l l construct the ppy cross section and analyzing powers. We w i l l also give r e l a t i o n s for the e l a s t i c cross section and phase s h i f t s and, w i l l i n d i c a t e how other e l a s t i c observables have been ca l c u l a t e d . 2.2 The T-Matrix Formalism Before deriving the T-matrix equation which we have solved, i t may be b e n e f i c i a l to indic a t e what 'goes int o ' the Lippmann-Schwinger( L-S ) equation. B a s i c a l l y , the L-S equation may be derived from the Schroedinger equation - both equations are n o n - r e l a t i v i s t i c . To solve the L-S equation we must evaluate an i n t e g r a l equation rather than a d i f f e r e n t i a l equation. The i n t e r a c t i o n p o t e n t i a l i s the 'input' which determines the N-N T-matrices. In our case, we have both the electromagnetic i n t e r a c t i o n and N-N potentials to deal with. The methods, based on the two-potential formalism, used here have been outlined by many o t h e r s 1 - 6 . We have used the T-matrix of Drechsel and Maximon7 and w i l l now j u s t i f y i t s usage. We w i l l s t a r t from the N-N T-raatrix equation for N-N T-matrix t(E) t ( + ) ( E ) = V N[ 1 + G(E) t ( + ) ( E ) ] ; G(E) = [ E - H Q ± ie ] _ 1 2.1 H Q i s the free Hamiltonian for the two protons and V N i s the N-N potentials The d e r i v a t i o n given by Drechsel and Maximon uses the two-potential formalism, which has been developed by Gell-Mann and Goldberger 6. This technique i s summarized n i c e l y by Goldberger and Watson 8. We st a r t with a d e f i n i t i o n 9 of the reduced T-matrix, T ^ . < f I T I i > = 6( P. - P. ) T £. 1 1 — f — i f i 2.2 In the above, | i > and | f > are resp e c t i v e l y the i n i t i a l and f i n a l states of the scattering system and, < i | T | f > i s the t o t a l T-matrix. In the two-potential method, we write down the f u l l Hamiltonian, H n + V + V„, u em N ' 11 such that the total Hamlltonlan is split into a free part HQ , a strong N-N interaction which is treated 'exactly' and, a weak electromagnetic interaction V g m which w i l l be taken to f i r s t order only. Using this approximation and the Lippmann-Schwinger equation, Goldberger and Watson show t h a t 1 0 < f I T I i > = < I v U t + > > + < (f)1"' I v |x. > 1 1 T f 1 em 1 t f N ' o where $ f : l~ i satisfies 4>f t - > - X f + (E - HQ - i e ) " 1 VN «(,<-' 2.4 and x is a plane wave state. i|> + satisfies = X- + (E - H + i e ) " 1 (V + V ) i K t + ) 2.5 i <- u em N i Now in ppy, <f>£~5 also contains a photon state. Since V^ does not contain creation operators for photons, the second term in equation 2.3 does not contribute. ^ e m» however, does contain creation operators for photons and so the f i r s t term in 2.3 gives a non-zero result. Drechsel and Maximon neglect the V e m term in equation 2.6 and use the following relation, wherein, t ^ + ) i s is a shorthand for t f + * ( E ). VN ^ i + , = t C + > X i 2 ' 6 Equation 2.6 neglects the effect of V on the N-N T-matrices. I n i t i a l l y , em J ' we also neglect this and calculate the N-N T-matrices without coulomb J contributions. The coulomb contribution is added separately, as described the next chapter. Writing <t>£-> e x p l i c i t l y as ^~>> = | > ® | k > E | !,<->, k > 2.7 where IJJ^  J is the scattering state for two f i n a l protons and lc is the photon momentum, we reproduce Drechsel and Maximon's result?, 6(P f - P. ) T f i = < X f , k | T | X < ; , 0 > 2.8 with the following T-matrix expansion depicted in figure 1. T = V + t C + ) ( E £ ) G(E £) V + V G(E^) t C + )(E.,) + em f f e r n em i i t C + : >(E_) G(E^) V G(E • ) t c +^(E:) 2.9 r r em L «• where we have substituted equations 2.4 and 2.5 into 2.3 and have used both equation 2.6 and the following r e l a t i o n 1 3 . [ t < _ > ( E ) ] t - t < + > ( E ) 2.10 In the absence of r e l a t i v i s t i c spin corrections, we have used Ve™ = (-e/m) Z [£ ± ' A(r- t) + u p o. • VL x A(r-)/2] 2.11 where, Figure 1 Diagrammatic Expansion of the T-matrix Single Scattering Terms tr G(E r) Vem Double Scattering Terms t ? G(E ?) Vem G(E-) t. 14 A(£ ±) = ( 2 i r ) - 3 / 2 Z / d 3k ex [ a x ( k ) expCik/r^) + + a + (k) exp(-ik«r ) ] ( 2 T r / k ) 1 / 2 A 1 2.12 a(k) | 0 > = 0 , a'(k) | 0 > = | k > A(t\) i s the vector potential, e i s the photon polarization vector, 1 A m i s the proton mass, = K + 1 i s the proton magnetic moment where K = 1.793 i s the anomalous magnetic moment of the proton. In the above, ( i ) labels the proton which i s radiating. When radiation preceeds the N-N interaction, 1=1,2, else, i=3,4 for radiation following the N-N inte r a c t i o n . This expression for the vector potential d i f f e r s , by a constant, from the form given by Bjorken and D r e l l l l + . The difference i s due to a choice of units. Drechsel and Maximon have i m p l i c i t l y used units such that e 2 = a ( a being the fine structure constant ) while, Bjorken and D r e l l t a k e 1 5 e 2 = 4TTCX . The r e l a t i o n for i n equation 2.12 may be derived by performing a n o n - r e l a t i v i s t i c reduction on the electromagnetic vertex below 1 5. e u(p')[ y F (k2) - i - k v K F 2(k 2)/2m]u(p)A >' ; ( J - m )u(p)=0 2.13 to 0(l/m) with the choice of a transverse gauge, the form factors being p r o t o n Equation 2.11 may be derived by expanding F 1(0)=F 2(0)=1, for a proton. As was mentioned i n the introduction, r e l a t i v i s t i c corrections to V em are important to ppy sc a t t e r i n g , e s p e c i a l l y at higher energies. These come from keeping terms through 0(m - 2) i n the expansion of equation 2.13. L i o u and S o b e l 1 7 have produced equivalent r e l a t i v i s t i c spin correction( RSC ) terms for V g m by considering a Foldy-Wouthuysen 1 8 transformation applied to the Dirac equation describing the i n t e r a c t i o n of a proton with an external electromagnetic f i e l d . They have taken the expansion to 0(m - 2) and have dropped n e g l i g i b l e terms. Below, we w i l l f i r s t evaluate < k | Vem | £ > without RSC terms to find < k | Vem | 0 > = (-e/2Trmk111) \ -i n iu a . • k x E, 12 1 exp(ik»r.) 2.14 p — t — —A 1 ^ L. This i s also the form of V used by Drechsel and Maximon. Note the em J error i n Drechsel and Maximon's equation 4.20 ( 1/2 •»• i/2 ). Liou and Sobel f i n d the following r e s u l t for < k I V I 0 > — 1 em 1 — Including RSC terms. < k I Vem I 0 > = (-e/2-rfmk1/2) \ [p. • e. + iy a . • e. x {k - (l-[2y J -^k^/m}^ ] e x p d k - ^ ) 2.15 Note that i s equation 2.15, Liou and Sobel have dropped several terms. Both equations 2.14 and 2.15 have been used i n our c a l c u l a t i o n for the purpose of comparison. In order to take the matrix elements < £3 >k | T | jpj ,22 ^> ^t i s 16 convenient to i n s e r t a complete set of states for the two protons and, then convert to r e l a t i v e and t o t a l momentum variables using £l» £2 * L = £l + £2 » £ = (£1 " .E2)/2 2 ' 1 6 The i n t e g r a t i o n i n v o l v i n g the t o t a l momenta gives a de l t a function 6 3(P. - P. ) which enforces the requirement — i —r £ l + £ 2 = £ 3 + £ + + i i 2 ' 1 7 The in t e g r a t i o n i n v o l v i n g the r e l a t i v e momenta i s i m p l i c i t i n the N-N T-matrices. Using equations 2.8, 2.9, 2.16 and 2.17, we have Tf i = < ^ Z ^ - B t ) I c f I ^ U P l > V<£l> t E f " E ( £ l _ i ) " E(£2)] _ 1 + < 1 ; 2(£,-£,) I t f I 1 / 2 ( £ 1 - [ £ 2 - k J ) > V ( & ) [E f - E ( £ l ) - E ^ - k ) ] " 1 + VCpj) [E. - E(p_3+k) - ECp^)]" 1 < 1/z{[23+l.}-£My I \ I ^ C P - l " ^ > + V(p^) [ E ± - £ (£3) - ECp^+k)]"! < V^P^-tj^+k]) I t ± I 1/ 2(J2 1- J2 2) > + Double Scattering Terms 2.18 wherein, E ( p ) 2 = jo 2 + m2 and, E ^ f Is the i n i t i a l / f i n a l energy of the two-nucleon system and, V(p.) = (-e/2irmk 1 / 2) [p. • e - ip a • • e x B-/2 ] 2.19 with B . = k ( No RSC ) , B . - {k - (l-[2u l " 1 ) ! ^ /m} (Including RSC) 2.20 The double s c a t t e r i n g term i n equation 2.9 i s considerably more d i f f i c u l t to handle because, there i s a free momentum( q_ ) over which we V. Brown finds that the contribution of t h i s term increases with energy for coplanar symetric geometries and, contributes to the integrated cross section at the 15% l e v e l for a lab energy of 300 MeV 2 0. We have not yet added t h i s term to the c a l c u l a t i o n . The c a l c u l a t i o n of the N-N t-matrices i s discussed i n the next chapter and i n Appendix C. Note that the T-matrix i s s t i l l i m p l i c i t l y between states of t o t a l spin | S,M > of the two protons. Suppressing the momentum states and e x p l i c i t l y displaying the spin states, we have < S'M' | T f ± | S M > = ES..M,. [ E f-E( £-k)-£(££) ]"1< S'M' | t f | S"M"> • • < S"M" | V(£ x) | S M > + other 3 terms 2.21 The evaluation of the spin matrix elements, < S'M' | V(£ ) | S M >, i s relegated to Appendix A and, the n o n - r e l a t i v i s t i c propagators are, of course, spin independant. 18 References 1. J.R. Taylor, Scattering Theory(New York: J. Wiley, 1972) 2. M.L. Goldberger & K.M. Watson, Collision Theory(New York: J. Wiley, 1964) 3. M. McMillan, lectures in 'Theoretical Nuclear Physics,' University of British Columbia, 1982. (Unpublished) 4. D. Drechsel & L.C. Maximon, "Potential Model Calculation for Coplanar and Non-Coplanar Proton-Proton Bremsstrahlung," Ann. Phys. 49, 403-444 (1968). 5. B.A. Lippmann , "High-Energy Semi-Classical Scattering Processes," Ann. Phys. 1_, 113 (1957). 6. M. Gell-Mann & M.L. Goldberger, "The Formal Theory of Scattering," Phys. Rev. 91, 398 (1953). 7. Drechsel & Maximon, equation 4.1.4 8. Goldberger & Watson, chapter-5.4 9. Goldberger & Watson, p.79 10. Goldberger & Watson, pp.202-207 11. Taylor, pp.270-271 12. Taylor, p.169 13. Taylor, pp.133-135 14. J.D. Bjorken & S.D. Drell, Relativistic Quantum Fields(New York: McGraw-Hill, 1965), p.75. 15. Bjorken & Drell, p.369. 16. J.D. Bjorken & S.D. Drell, Relativistic Quantum Mechanics(New York: McGraw-Hill, 1964), pp.243-245. 17. M.K. Liou & M.I. Sobel, "p-p Bremsstrahlung Calculations and Relativistic Spin Corrections," Ann. Phys. 72, 323-352 (1972). 18. L. Foldy & S. Wouthuysen, "On the Theory of Spin 1/2 Particles and i t s Non-Relativistic Limit," Phys. Rev. 78, 29 (1950). 19. Liou & Sobel, p.330, equation 12'. 20. V. Brown, "Proton-Proton Bremsstrahlung Including Rescattering," Phys. Rev. 177, 1511 (1969). 19 CHAPTER 3 A Numerical Solution for the N-N T-matrices In order to evaluate the total T-matrix of Chapter 2, we require half-off-shell N-N T-matrices. Given a N-N potential, we can calculate the N-N T-matrices by solving a Lippmann-Schwinger (L-S) equation. In order to solve the partial waved L-S equation, we must partial wave the chosen N-N potential in momentum space. The Paris potential is partial waved in the next chapter while, a treatment of the Extended Reid potential is given in Appendix B. The following L-S calculation neglects the effects due to the electromagnetic interaction. Coulomb contributions are added later, as described in Appendix C. Therefore, V, in this chapter is a pure strong Interaction N-N potential. We use the notation t(E,q',q) = < q* | t(E) | q > where, E = E(q) = q2/2mr (Non-rel.) G(E,q') = [ E - E 1 + ie ] - 1 ; E* = E(q') 3.1 v ( q ' ,q) = <q' | v | q > ; | q > Is a plane wave state In the above, m^  is the reduced mass of the 2-nucleon system. Now, in addition to < q' | t(E) | q >, we also require < q | t(E) | q' >. This is easily obtained using the symmetry relation 1 < S'M', p* | t(E') |SM, p > = < S M , p | t(E') | S'M', p' > 3.2 which follows from parity and time-reversal invariance. An added complication is due to the fact that in each N-N T-matrix, spins are quantized^ along the d i r e c t i o n of the i n i t i a l r e l a t i v e momentum, while we want the N-N T-matrices quantized along a common axis, the beam ax i s . We therefore rotate each N-N T-matrix as described i n Appendix D. In order to use Stapp's 3 M-amplitudes and phases, we must evaluate the L-S equation between states of coupled spin and o r b i t a l angular momentum ( | LSJ > ). Throughout this chapter, the | LSJ > matrix elements w i l l be i m p l i c i t with the eigenvalues l a b e l l e d by super/sub-scripts. By p a r t i a l waving the L-S equation, we can write the matrix elements i n terms of a series of eigenvalues ( ' p a r t i a l waves') for each allowed L,S and J combination. For uncoupled waves, we l a b e l only the o r b i t a l angular momentum state - leaving S and J i m p l i c i t . This w i l l not be possible for coupled waves. We, therefore, begin the analysis with a treatment of uncoupled waves. The equations for uncoupled waves are e a s i l y generalized to the coupled case and are much less cumbersome and easier to follow. Below we ca l c u l a t e the N-N T-matrices from the L-S equation t(E,q',q) = V(q',q) + / d 3q" V(q',q") G(E,q") t(E,q",q) 3.3 This equation may be s i m p l i f i e d i f we expand V(q',q), t(E,q',q) and t(E,q",q) i n terms of p a r t i a l waves, using the general r e l a t i o n V(q',q) = £ V ( q \ q ) Y (q) Y L ^ ( q ' ) 3.4 with an analogous expansion for t(E,q',q). Relation 3.3 for uncoupled waves then becomes t L(E,q',q) = V L(q',q) + / dq" q" 2 V^q'.q") G(E,q") t L(E,q",q) 3.5 The solution to equation 3.5 has been given i n s e v e r a l 4 ' 5 compilations of scattering theory. A problem arises i n r e l a t i o n 3 .5, for q" such that E" = E. Here G(E,q") - 1 •»• 0, which could cause problems i n a numerical s o l u t i o n . Kowalski 6 has shown a simple way to avoid t h i s s i n g u l a r i t y . By adding l i n e a r combinations of equation 3.5 for t L(E,q',q) and t L(E,q,q) one o b t a i n s 5 t j E . q ' . q ) = V L(q',q)/V L(q,q) + / dq" q" 2 G(E,q") { V^q'.q) - V^q'.q)-•Vq.q^/Vjq.q) } t L(E,q",q) 3.6 It has been shown6 that t L(E,q',q) can be written as t L(E,q',q) - f j q ' . q ) t j E . q . q ) 3.7 where f L(q',q) i s a r e a l function. Substitution of equation 3.7 into 3.6 y i e l d s for the ' h a l f - o f f - s h e l l ' function f ^ C q ' » q ) the equation f u ( q ' , q ) = V u(q',q)/V L(q,q) + / dq" q" 2 G(E,q") { V j q ' . q " ) - V J q ' . q ) . •V u(q,q")/V u(q,q) } f L(q",q) 3.8 Now i n order to solve for f L(q',q) numerically, the i n t e g r a l i n equation 3.8 i s approximated by a sum. The methods of Gaussian i n t e g r a t i o n are used to pick i n t e g r a t i o n points and weighting f a c t o r s . We can rewrite equation 3.8 i n the form AX=B, where A i s an NxN matrix and, X and B are N-dimensional vectors where, i and j refer to g r i d points i n q. We have X.= f L ( q . ,q) and B.= V L ( q L ,q)/V L(q,q) 3.9 A..= 6.. W j q 2 G(E,q. ) ( V J q L ,q. ) - V^(q ^ ,q) V J q ^ )/V L(q,q)) 3.10 where, w^  i s the weight factor for the Gaussian i n t e g r a t i o n . This i s the form used i n our c a l c u l a t i o n 9 . The system of equations i s solved using the UBC MATRIX subroutine DSLIMP. Having solved for f u ( q ' , q ) , we may obtain the on-shell N-N T-matrix, t ^ C q ' j q ) , i n order to check our c a l c u l a t i o n by s u b s t i t u t i n g equation 3.7 into 3.5 and using the condition, f L(q,q) = 1, to obtain t L(E,q,q) = V L ( q , q ) { l - J dq' q' 2 G(E,q') V j q . q ' ) f j q ' . q ) } " 1 3.11 Note that the right hand side of 3.11 contains only known functions. Since equation 3.8 was solved on a g r i d of points, we have f (q',q) evaluated at points appropriate for the numerical i n t e g r a t i o n on equation 3.11. This i n t e g r a l , however, has a s i n g u l a r i t y at E = E'. If we do the contour i n t e g r a l i n 3.11 by adding and subtracting q 2 V u(q,q) / dq' (E - E' + i e ) " 1 = [i r q m ^ (q,q) ] i 3.12 taking E = q 2/2m r > the numerical r e l a t i o n i s t L(E,q,q) = V u(q,q)/D L(E,q,q) 3.13 wherein D L(E,q,q) = V L ( q , q ) ( l + iTrqmrVL(q,q)) - i wL {q 2 V L(q,q ) f j q . ,q) -q 2 V L(q,q)} G(E,q L) 3.14 This procedure could have problems i f we were so unfortunate as to choose one of the integration points, q L , such that q. = q. We would then be near the indeterminate form, 0/0, which could insert a term with large round-off errors into the summation. Having evaluated equations 3.8 and 3.11, we have the on-shell N-N T-matrix, t L(E,q,q), and the half-off-shell function evaluated at N values of q^. In order to obtain the half-off-shell T-matrix, t L(E,q,q'), where q' is some arbitrary value of momentum, we can use relation 3.7. The appropriate value of f L(q,q') is found using Lagrange interpolation on the grid of f L(q,q^) values. Such an interpolation procedure assumes that f L(q,q') is a f a i r l y smooth function( no spikes ). This has been demonstrated by Mongan10. Now, we w i l l make the theory more general by including coupled waves made necessary by the tensor part of the N-N interaction. When we wrote V L(q 1,q 2) previously, we were implicitly stating that the i n i t i a l and f i n a l angular momenta were identical. In effect, we could have written V^Cq^qg) 6,i L instead. More generally, however, the i n i t i a l and f i n a l angular momenta are different, so that Vt,L(q^,q2) must be treated as a matrix. The coupled L-S equation is tJu(E,q',q) = V^q'.q) +.r,/dq" q" 2 (q' ,q") G(E,q") t? u(E,q",q) 3.15 For coupled waves, the scalar multiplication in equation 3.7 is replaced by matrix multiplication and we have T J + L J T t J L ( E , q ' , q ) - Z ^ f ^ C q ' . q ) t J u ( E , q , q ) 3.16 In order to obtain V^Cq',q) let us f i r s t explicitly produce Vu(q',q) from equations 3.1 and 3.4. We expand 1 2 the plane wave states < q | of < q' | V | q > in terms of spherical harmonics, , and spherical Bessel functions, j , and then carry out the angular integration in order to rewrite the le f t hand side of equation 3.4 as From this, we identify Vu(q',q) = (2/TT) / dr r 2 j ^ q ' r ) V(r) J L(qr) 3.18 which is generalized for L * L' to give VL,L(q*,q) = (2/TT) / dr r 2 ^ ( q ' r ) V(r) ^ ( q r ) 3.19 for states of coupled momentum. The superscript J has also be added to VL.L(q',q) to denote the total angular momentum of coupled waves. Note that for coupled waves the i n i t i a l and fi n a l spin is fixed, S = 1. Here then, the spin index may be left implicit since our N-N amplitudes are diagonal in spin. For the rest of this chapter, as in equation 3.16, summations on angular momenta w i l l also be left implicit and w i l l run from L=J-1 to L=J+1, but not including L=J. For coupled waves we must replace equation(s) 3.9 with the following relations. Respectively, X L f L(j) = f J L ( q j , q ) and B ^ C i ) = v/ L „ (q-,q) Vj,' (q,q) 3.20 In equation 3.20, the angular momentum variable L" is summed over and, V - i denotes the inverse of the matrix V. In the following equation, the angular momenta L and L' are summed. Equation 3.10, for coupled waves, is • V _ 1 L U " <1»^ ^.(q.qj) ] } 3.21 Explicitly, our equations are A? f ^ L ( j ) = B* t(i) 3.22 wherein, the angular momentum L" is summed on. This set of equations may be solved exactly as in the uncoupled case for each choice of i n i t i a l and fi n a l angular momenta. In order to obtain the on-shell N-N T-matrix, t^ u(E,q,q), we evaluate tj^E.q.q) = D-^ L(E,q,q) \*(q,q) 3.23 wherein denotes matrix inversion, with D.J.(E,q,q) = 6 + i ™ q V J (q,q) - Z w, { q 2 V J <q.,q) f J L (q. ,q) - q 2 V J (q,q) } 2m r[q 2 - q 2 ]~1 3.24 In both equations 3.23 and 3.24, L is summed. With the above relations, we can calculate the N-N T-matrices up to some maximum number of partial waves, determined by the N-N potential. In order to compensate for higher partial waves in an approximate way, we have b u i l t i n the option to add 1—rr exchange amplitudes above those calculated with our N-N p o t e n t i a l . These have been conveniently calculated i n reference 11. Note that we are using an on-shell approximation to the true 1—FT exchange amplitudes. Now we have the o n / o f f - s h e l l p a r t i a l wave N-N T-matrices. These are summed to give the f u l l h a l f - o f f - s h e l l N-N T-matrices, i n the | SM > basis, using the summation r e l a t i o n s given by Stapp 3 and equation C.4 of Appendix C. At th i s point, the coulomb corrections are added, as described i n reference 2. B a s i c a l l y , the coulomb scattering amplitudes are added to Stapp's M-amplitudes with a phase added to the d e f i n i t i o n of the p a r t i a l wave N-N scattering amplitudes, i n order to produce interference between the two i n t e r a c t i o n s . The on-shell N-N T-matrices may be used to calculate e l a s t i c observables as i l l u s t r a t e d i n Appendix C. This checks much of the ppy computer program. The o f f - s h e l l N-N T-matrices w i l l be used i n contruction the t o t a l T-matrix leading to the construction of the ppy observables, i n Chapter 5. References 1. D. Drechsel & L. Maximon, "Potential Model Calculation for Coplanar and Non-Coplanar Proton-Proton Bremsstrahlung," Ann. Phys. 49, 424 (1968). 2. H. Stapp, T. Ypsilantis & N. Metropolis, "Phase-Shift Analysis of 310-MeV Proton-Proton Scattering Experiments," Phys. Rev. 105, 304 (1957). 3. Stapp et a l , p.304, equations 3.8 & 3.9. 4. G. Brown and A.D. Jackson, The Nucleon-Nucleon Interaction(New York: Elsevier, 1976). 5. K.M. Watson & J. Nuttall, Topics in Several Body Dynamics (San Francisco: Holden-Day, 1967), Chapter 2 and references therein. 6. K.L. Kowalski, "Off-Shell Equations for Two-Particle Scattering," Phys. Rev. Lett. 15, 798 (1965). 7. Brown and Jackson explain this procedure very clearly. 8. M.I Sobel, "Parametric Expression for p-p Off-Energy-Shell Matrix Elements and p-p Bremsstrahlung," Phys. Rev. B138, 1517 (1965). 9. Credit must be given to Dr. A. Miller for supplying us with his subroutines. 10. T.R. Mongan,"0ff-Energy-Shell Behavior of Partial-Wave Scattering Amplitudes," Phys. Rev. 180, 1514-1521 (1969). 11. P. Cziffra, M. MacGregor, M. Moravcsik & H. Stapp, "Modified Analysis of Nucleon-Nucleon Scattering. I. Theory and p-p Scattering at 310-Mev," Phys. Rev. 114, 880-886 (1959). 12. See A. Messiah, Quantum Mechanics(New York: J. Wiley, 1964), Vol. I, Appendix B. CHAPTER 4 The P a r t i a l Waved Paris N-N P o t e n t i a l 4.1 Introduction As was mentioned i n the previous chapter, we require a p a r t i a l waved N-N p o t e n t i a l i n order to evaluate the L-S equation for the N-N T-matrices. Here we p a r t i a l wave the Paris N-N p o t e n t i a l 1 . Treatment of the Extended Reid p o t e n t i a l i s relegated to Appendix B. Use of the Paris N-N p o t e n t i a l i s a major new feature of this ppy c a l c u l a t i o n . We have chosen t h i s p o t e n t i a l as i t i s one of the most modern and, has been parameterized i n a convenient form. The quadratic spin o r b i t p o t e n t i a l introduces some mathematical problems not seen i n the Reid p o t e n t i a l analysis - these are studied i n d e t a i l . For the uncoupled L-S equation, we require V L(q',q) while, V L i u(q',q) i s necessary i n coupled L-S equation. These matrix elements have been described i n Chapter 3. R e c a l l that they are also to be evaluated between states of coupled L and S, | LSJ >. Once again, V L(q',q) for the uncoupled p o t e n t i a l i s equivalent to V^Cq'.q) <5L«L. The spin index i s l e f t i m p l i c i t , as i s the t o t a l angular momentum J , however, the contributions from d i f f e r e n t spin states w i l l be c l e a r . The notation which we are about to use i s admittedly cumbersome but, i s consistent with that used by Lacombe et a l 1 . 29 4.2 The Paris N-N Potential The Paris N-N potential, in i t s parameterized form 1, i s a result of work by Cottingham et a l 2 on the long and medium range N-N interaction. The long and medium range part of this potential is constructed from the contributions of (TT + 2TT + w) exchanges. The short range part (r £ 0.8fm) has been f i t with a phenomenological model. M. Lacombe et a l 1 claim that the Paris and Hamada-Johnston3 potentials agree very well down to r = 0.8 fm. This is encouraging since the Paris potential has fewer free parameters. For convenience, the Paris P-P potential is presented below. The T=l potential parameters (m; and g. ) are given in Table I of ref. 1. V(r,p_2) = V 0 ( r , p 2 ) f i Q + V 1 ( r , p 2 ) Q 1 + V U J ( r ) n u j + V T ( r ) n T + v s o x ( r ) n S M L 4.1 In the above, «„ = (1 - a 1*a 2)/4 4.2 flj - (3 + a ^ o ^ M ftLS = L«S_ ^soi = l 2 J Z ' — + £ 2 *— —l *—^ ^ 2 In r e l a t i o n 4.1, the following shorthand V Q ^ r . p 2 ) = v \ r ) + (p2/m)V b(r) + V b(r)(p 2/m) 4.3 i s used, with 2 = - -ft2 C 1 d 2 r - L 2 I ; m = 938.2592 MeV 4.4 ( l £ r - L 2 j I r d r 4 r* ) The operators fiQ and , project out the S = 0 and S = 1 parts of the central p o t e n t i a l r e s p e c t i v e l y . and Q T are the usual spin-o r b i t and tensor operators, ftSoz i s a quadratic spin-orbit operator, not present i n the Reid p o t e n t i a l but s i m i l a r to L 1 2 used i n the Hamada-Johnston f i t . Each component V(r) i s parameterized through a sum of Yukawa terms V(r) = E, g. F(m. r)exp(-m.r)/m. r 4.5 j j » i J wherein F(m-r) = 1 for VQ^ 4.6 FCm^r) = l/Cmjr) + l / ( m - r ) 2 f o r V u £ F(m..r) = 1 + 3/(m.r) + 3/(m-r) 2 for V T FCm^r) = ( l / ( m i r ) 2 ) ( l + 3/(m>r) + 3/Cmjr) 2) for V $ 0 1 The p o t e n t i a l i s regularized at r = 0 by constraints set on the g. and m-J J values. B. Loiseau has performed 4 a Fourier transform on the p o t e n t i a l . The r e s u l t i s 2 ^ V<£L »£f > = V£; »£f )fio + vi<£; .2* + v,.,(AOn i n which ftQ 1 i s equivalent to ftQ j , A = £,- £^ and, ftLS = (1 / 2 ) ( O j + £2)'£. x £ ^ ^ T = A2°j. '£2 ~ 3£i *L I2' — ^sox = £ i - ( £ C x £ * ) 2 2 * ( £ t ; x ^ The v e l o c i t y dependance of the cen t r a l components (V Q simpler form v ( £ ; » £ r ) " V*(A 2) + (* 2/m)(p 2 + p 2) V b ( A 2 ) with V(A 2) - [2TT 2]" 1 (gj/m-) (m2 + A 2 ) " 1 f(m^) and f(m-) = 1 for V 0 ) 1 f(m<) = 1/m2 f o r V,. and V r f(m. ) = 1/m? for V t I. The Central Potential Using equations 4.7, 4.9 and 4.10, the central part of the Paris potential may be written as follows, iX [2*2]"! ,E( [(m.2 + A2)m^ ] - 1 ( { g j° d + g° b * 2 ( p 2 + + {gf + g l b * 2 ( p 2 + p2)/m}jJ1 ) 4.12 where, as before, m = 938.2592 MeV. Now, the central potential applies to diagonal matrix elements only. Rewriting equation 3.4 as an expression f or V (p ,p.), we have V (p, ,p. ) = 2TT Jd6 sine P (cos6) V(p ,p. ) 4.13 o where, V(pf ^  ) = < p^ | V | p. > and 0 = ( £ • • p_^ , )/ | p_ . • p_ |. If we write the term [UK 2 + A 2 ] - 1 in the following way [m.2 + A 2 ] - 1 = [A. + B cos6] - 1 ; A• = m-2 + p. 2 + p 2 and B = -2p. p 4.14 then, by changing variables, x = cos6, and using the integral /dx P u(x)[z-x]" 1 = 2Q L(z) 4.15 we may write the partial-waved central potential as f [TTp.p^m.]-! Q L( Z j){[g° a + g f * 2 ( p 2 + P2)/m]H0 + 4.16 [g] a + g j b <h2(p2 + p^/m]?^} 33 wherein Q is a Legendre function of the second kind and L Zj = [P? + P 2 + mf ]/[2p.pf ]. 4.17 Now consider the spin matrix elements of £2Q and ft^ , the operators described in equations 4.2. We can evaluated these easily, using Si'Ss. I LSJ > = 2[ S(S+1) - 3/2 ] | LSJ >. 4.18 Here, as in Chapter 3, | LSJ > denotes a state with L and S coupled to give J. Using the above relation i t is easy to see that ftQ and ft1 project out the singlet and tr i p l e t states respectively, as mentioned previously. II. The Spin-Orbit Potential In this case i t is more convenient to start with the coordinate space representation of the spin-orbit part of the Paris potential( V L J ). Combining equations 4.1, 4.2 and 4.6 we may write V l 5 ( r ) 8 L S = E g ^ ([m.r]" 1 + [m-r ]~2) (expC-m^ r)/(m. r) S 4.19 The spin-orbit operator may be evaluated using L«S = [ J 2 - L 2 - S 2]/2 4.20 and L«S_ | LSJ > = 1/ 2[J(J+1) - L(L+1) - S(S+1)] | LSJ > 4.21 This part of the Paris potential is then partial-waved using equation 3.19 for the partial waved potential. Using the integral 5 (2/TT)/ J L ( p f r ) {([m.r]" 1 + [m^r]"2) exp(-m. r)/(m.r)} j^p.^) r 2 d r = 4.22 [irmj3(2L + D ] " 1 ^ . , (* J > " <Wzj>} the partial waved spin orbit potential is ? g j u s [Trm_;3(2L+ l ) ] " 1 {Q ^  ( z • ) - QL + 1 (z- )}L-S_ 4.23 III. The Tensor Potential A coordinate space treatment is again used for the tensor potential. We write the tensor potential as V T ( r ) f i T = I g7 ( l + Stm^r]"1 + 3[m-r]-2)(exp(-m-r)/(mjr)) 4.24 with ftT defined in equations 4.2. The matrix elements of Q- are obtained via the Wigner-Eckart theorem6. We find the following result fiT | L=J±1,S=1,J > = N J t [ J M | L=J±1,S=1,J > + N J t v 3_ ( | L=J+1,S=1,J > 4.25 with the values of N , given below for different values of L and L*. 35 Table 1 The Tensor Coupling Coefficients L' J - l J - l J+l J - l 2 J + I X J * I kUCJ^Oj 7* o -202.) 2. J + | Then using the following integrals 5 (2/TT)/ j l . ( p f r ) { (1 + S ^ r ] " 1 + 3 [ m j r ] - 2 ) expC-nij r)/(m-r)) j j p - r ) r*dr f [mu^l-MlPf /P L 1 Q^ p + tP-t/Pf 1 Qu<zj> " 2Vzj>l ; L*L' 4 , 2 6 [ i r a j P . p f ] - l Q u ( Z j ) + 3[irm.3(2L + l ) ] " 1 ^ , (z. ) - Q ^ ( Z j ) } ; L-L* we can c o n s t r u c t the p a r t i a l waved tensor p o t e n t i a l . IV. The Quadratic Spin-Orbit Potential The quadratic spin-orbit part of the Paris potential is unlike any term in the Reid potential. In coordinate space, this term is W r ) a s o z 4.27 wherein ft S o^ is given by equation 4.2. Since we are dealing with states of total spin S , we rewrite nso^ , as ftSoZ = 2 [ L » S J 2 - L 2 4.28 using, _a«A _a«A = A 2. This leads to ftSo;L | LSJ > = { 1/ 2[J(J+1) - L(L+1) - S(S+1)]2 - L(L+1)} | LSJ > 4.29 This term w i l l be partial-waved in coordinate space, where 4.27 becomes Z g5ox [m3r]-l{l + 3[m 3 r]-l + S ^ r ] " 2 } ( e x p [ - m . r ] / [ m j r ] )SlSoX 4.30 As with the tensor and spin-orbit parts of the potential, we must evaluate an integral, of the form I = 2 T T 1 / r 2 d r ( j L . ( p f r ) [ m j r ] - 2 { l + S t n ^ r ] " 1 + 3 [ m j r ] - 2 } . 4.31 (expf-mjrj/tmjr]) j ^ p ^ r ) ) The evaluation of equation 4.31 may be carried out i f we f i r s t use the recursion relation for spherical Bessel functions. J L ( p r ) = pr[2L+l]-M j ^ C p r ) + J L + 1 ( p r ) } 4.32 Applying equation 4.32 to 4.31 produces four integrals of the form 2p. P p[ 1im j 2(2L+l)]- 1/ r 2 d r J ^ C p , r ) {1 + S ^ r ] " 1 + 3 [ m j r ] ~ 2 } J L t | ( P L r ) 4.33 3 7 These may be evaluated individually 1 5 • We note that matrix elements of the S^OX operator give a 6UU> constraint. Setting L=L', we have the following result for L > 2 For L=0, the quadratic spin-orbit potential does not contribute, therefore, we need only consider the L=l case. Unfortunately, the L=l case of this potential is a source of d i f f i c u l t y . For L=L'=1, the integral in equation 4.31 is divergent and, we are only saved by the constraints given in reference 1, equation 8, which make the f u l l potential f i n i t e at the origin. These are listed below. The divergence problem arises In the following integral from equation I 0 = 2 T T - 1 / r 2 d r j Q ( P f ^ { l+Stm . r l - l+atm . r l ^JCexp t - m j r l / t m j r D J Q C p . r ) I = [ 7 r m i 3 ( 2 L + l ) 2 ] - 1 { (1 + [p 2 + p 2 I m . " 2 ) ^ , (z). + Q u. v(z.)) " 4p.p m.-^QCz) + 3p p. [m. 2(2L+3)]- 1(Q L(z) - Q.^.U)) -3 P p [m 2(2L-1)]~ 1(Q (z) - Q (z)) } 4.34 4.35 4.33. = [Trm^p. p^  r^oCz^) + 6 T T - 1 / r 2 d r { j Q ( p f r ) ( [m. r ] ~ l + [m^r]" 2 )-(expt-m^rl/tm^rDjgCp.r)} 4.36 Part of the remaining integral can be evaluated, using / r 2dr j Q( P < ;r)(exp[-m jr]/[m jr] 2)J 0(p.r) 4.37 = [ m j 3 a a ' d x exp(-x) x - 2 sin(ax)sin(a'x) = [m^aa'] - 1! 0.5 a tan - 1([2a'][l+a 2-a* 2] - 1) + 0.5 a' tan - 1 ( [ 2 a ] [ l + a , 2 - a 2 ] - 1 ) + 0.25 ln([l+(a-a') 2][l+(a+a') 2] - 1) } where x=m-r, a=p. m"!1 and a'=p m.71 • (See reference 7 as a check on the J t o P J above integral.) This leaves an integral of the form II = ! r 2dr J 0(Ppr) [m-r] - 3 expf-m^r] j 0(p. r) 4.38 = [m. 3aa'] - 1/ dx x - 3 exp(-x) sin(ax)sin(a'x) With x, a and a' defined as before. Now the integral in 4.38 is divergent, however, we can separate out the divergent part and show that the constraints in 4.35 make this term vanish. We f i r s t write, I, = / lim { dx x exp(-x) sin(ax)sin(a*x)[x 2+g 2] - 2 } 4.39 and note that by interchanging the integral and limit we can obtain a fin i t e integral for g>0. Note that this is a bit cavalier but, the interchange of limit and integration is justifiable here. If we le t , I 9 = lim / dx x exp(-x) [x 2 + 3 2 ] - 2 sin(ax)sin(a*x) 4.40 * p-+o then, we may write, Ir, = - lim / dx x [x 2 + g 2] 2 { exp(u,x)+exp(u1x)- 4.41 (J-*o A 1 exp(y3x)-exp(yltx) }/4 wherein, \il = 1 - i(a'+a) , u 2 = 1 + i(a*+a) , y 3 = 1 - i(a'-a), y^ = 1 + i(a'-a). 4.42 We must now evaluate I 2 using 8 I 3 = / dx x [x 2 + g 2 ] - 2 exp(-yx) = 20 - 2{l -py[ ci(3y)sin(gy) - 4.43 si(0y)cos(f3y) ]} where, x si(x) = the sine integral = —rr/2 + / sin(t) t - 1 dt 4.44 o c i (x) = the cosine integral = C + ln(x) + / (cos ( t ) - l ) t - 1 dt 4.45 with C being Euler's constant. By expanding c i , s i , sin and cos in equation 4.43 for small 3, we may write I 3 as I 3 = [2g] - 1 + y 2 ( C - 1 + ln(3y) )/2 - T r y[43] _ 1 + Order(g) 4.46 Having obtained I 3 , we can now evaluate I 2 , by summing the contributions from y 1 , y 2 , y 3 and y ^ . After some algebra, we have I, = lim {(1 - C - ln(f3)) + [8aa' ] - 1 ( [l-(a'+a) 2] ln[l+(a+a') 2 ] 4.47 (3-»0 - [l-(a'-a) 2] ln[l+(a'-a) 2] - 4(a'+a)tan-1(a'+a) - 4(a'-a)tan- 1(a'-a) ) }/8 From I 2 , we have 3^  since II = [aa'm 3]-l{ ^ } 4.48 Note that by using the relations in 4.35, the coefficient of the constant term ( which is divergent as 3 >0 ) vanishes when we sum up a l l pieces of the potential while, the other terms do not, since a and a' are functions of the parameter m^ . We have thus demonstrated that constraints in 4.35 effectively remove the divergent part of 4.40. Different methods of regularizing equation 4.39 were found to give different constant terms. This is of l i t t l e consequence, since these terms do not contribute to the potential. We can summarize the result as follows. For L>2 the partial waved quadratic spin orbit potential is E g s ° l [7rm.3(2L+l)2]-l{ (1 + [p? + p2 ]m."2)[ Q f z ) + 4.49 Qu_, <*> 1 - 4P. P f m.2QL(z.) + 3p.pf ^ 2(21+3)]" 1[ Qjz.) -Q L n U ) ] " 3p.Pf [ m ^ ^ L - l ) ] - ! [ Q L(z) - Q ^ C z ) ] } n$ox while for the L=l case, we must replace the last term in 4.49 with 3 a tan- 1([2a'][l+a 2-a , 2]-l) + 3 a' tan - 1([2a][1+a 1 2- a 2]"1) + 0.75 {[l-(a'+a)2] ln[l+(a'+a)2] - [l-(a'-a) 2] ln[l+(a'-a) 2] -4(a'+a)tan-1(a'+a) + 4(a'-a)tan"1(a'-a)} 4.50 In summary, we have now obtained the partial waved Paris N-N potential. The f u l l partial waved potential is the sum of the individual pieces which we have evaluated. Specifically, the f u l l partial waved potential may be obtained by adding the results from equations 4.15, 4.18, 4.21, 4.23, 4.25, 4.26, 4.29, 4.49 and 4.50. This is the quantity required in the Lippmann-Schwinger equation for the N-N T-matrices, described in Chapter 3. Having obtained a l l the tools required to produce the invariant amplitude for proton-proton bremsstrahlung, we w i l l construct the ppy observables in the next chapter. 4 2 Table 2 Table of Useful Integrals J J L ( P ' r ) (exp[-mr]/[mr]) J L(pr) i^dr = [2pp'm]-1 QL(z) 1.1 / J L(p' r) (exp[-mr]/[mr]){ [mr]"1 + [mr]"2 } J L(pr) r 2dr = [2m3(2L+l)]-1{ Q^U) ~ Q L + 1 ( Z ) } 1-2 / J L(pr) (exp[-mr]/[mr]) J L,(p*r) r 2dr = -[2pp,m]-1 QJz ) + [2L+3H2L+1]-1 { [2 P' 2m]-l Q L + 1 ( z ) + P'-1P/ J L _ l ( p r ) (exp[-mr]/[mr]) ^ . ^ ( p ' r ) r 2dr } 1.3 / J L(pr) (exp[-mr]/[mr]){ 1 + 3[mr]~l +3[mr]-2 } J L,(p'r) r 2 d r = m~3{ P'^p]- 1 QL(z) + p[2p']"l Q L , (z) - QjCz) } 1.4 / j Q(pr) (exp[-mr]/[mr]) J 2(p'r) r 2dr = [4p'p3 m]-l { 6p'p + (3m2 + p' 2 - 3p 2) ln( [ z+l ] / [ z-l]) + 6mp( tan"1[m(p'+ p)" 1] - tan~1[m(p - p') _ 1] ) } 1.5 Note; In the above, z = (p' 2 + p 2 + m2)/(2p'p) and, 1.4 is used in cases where J=(L+L')/2 , L=J-1 & L'=J+1. A l l integrals are evaluated from 0 to + °°. References 1. M. Lacombe, B. Loiseau, J.M. Richard & R.V. Mau, "Parameterization of the Paris N-N Potential," Phys. Rev. C21, 861 (1980). 2. W.N. Cottingham, M. Lacombe, B. Loiseau, J.M. Richard & R.V. Mau, "Nucleon-Nucleon Interaction from Pion-Nucleon Phase-Shift Analysis," Phys. Rev. D8, 800 (1973). 3. T. Hamada & J.D. Johnston, "A Potential Model Representation of Two-Nucleon Data Below 315 MeV," Nucl. Phys. 34, 382 (1962). 4. B. Loiseau, Ph.d. Thesis, Universite Pierre et Marie Curie, Paris, 1974. 5. See Table 1. 6. A. Messiah, Quantum Mechanics(New York: J. Wiley, 1964), p.573. 7. l.S. Gradshteyn & I.M. Ryzhik, Table of Integrals, Series and  Products(New York: Academic Press, 1980), p.491, 3.947.2. 8. Gradshteyn & Ryzhik, p.312, 3.355.2. CHAPTER 5 Construction of the PPy Observables Thus f a r , we have indicated the form of our ppy amplitude and, have calculated the i n d i v i d u a l pieces required i n i t s construction. We w i l l now use the amplitude i n order to produce some ppy observables, s p e c i f i c a l l y the asymmetries and cross s e c t i o n . We begin with the cross section which may be written i n the manifestly invariant form 1 da = (2TT) 1 +{[£ |-£ 2- E 1E 2]2-m'*}l/2 . J d3p 3d3p 1 +d3k[E 3E l tk]-l6 , + (P 1+P 2-P3-p 1 +-k) (fl u x ) (phase space) • | | ( E 3 E 4 k ) l ' 2 T ( E , E 2 ) l / 2 | 2 5 > 1 Spin (amplitude) (The momentum subscripts r e f e r to the reaction: p 1 4- p 2 •*• p 3 + p^ + k ) Both the f l u x and phase space factors are i n d i v i d u a l l y i n v a r i a n t . This leads to the Lorentz invariance of the amplitude term 2. This invariance feature i s useful when a change of coordinate system i s desired. C l e a r l y , a l l the i n t e r e s t i n g 'physics' i s contained i n the amplitude term of equation 5.1. Here we should i n d i c a t e how we have been able to c a l c u l a t e the cross section i n both the lab and CM systems. The fl u x and phase space factors present no r e a l problem as they can e a s i l y be evaluated i n any system. For the ppy amplitude, however, i t i s easier to calculate the N-N T-matrices i n the CM system. We, therefore, repeat the arguement of reference 2 f or the e l a s t i c p-p amplitude. By multiplying and d i v i d i n g by ( E 1 E 2 E 3 E 1 + ) 1 / 2 w e c a n 45 construct ( E 3 E 1 + ) 1 / 2 < f | t | i > ( E ^ ) 1 ' 2 5.2 which i s i t s e l f i n v a r i a n t . This part of the ppy amplitude i s then always evaluated i n the CM system while the remainder of the ppy amplitude may be evaluated i n e i t h e r the lab or CM system. Thus i n the lab there w i l l be a factor { [ E 1 E 2 E 3 E i + ] L a b / [ E 1 E 2 E 3 E i t ] C M } 1 / 2 i n equation 5.1, with the amplitude of equation 5.2 evaluated i n the CM system and the f l u x and phase space evaluated i n the Lab system. Now, for the ppy amplitude J I T n I2 = lU S E [ < §M | TH I SM > f< SM | T n | SM > Spin X SM 5.3 J the sum on spins i s e a s i l y performed while, the sum on the photon p o l a r i z a t i o n i s evaluated as follows. We f i r s t define 5.4 For e n * 0, we would have had T, 5.5 By u t i l i z i n g the gauge invariance r e l a t i o n V 1 * = 0 5.6 we find T° = k • T ; k = k / | k | 5.7 Combining relations 5.3, 5.4 and 5.7, we have I I TfL I = T«T - k«T k«T 5.8 Having constructed the ppy cross section, we w i l l next produce the asymmetries which have been calculated. F i r s t , we must change the spin basis for the total T-matrices using | m: m2 > = | S M > < S M | m1 m2 > 5.9 so that < S'M* | T f L | S M > < m3 m4 | T f : | ml m2 > 5.10 wherein, m = ± 1/2 is the spin projection of the j proton. Next, for simplicity, let us write < m3 m^  I T f i, | m1 m2 > as simply T in the following discussion. We use the same asymmetry definition as McGuire and Pearce 3 - modulo a sign. As was mentioned previously, McGuire and Pearce are the only ones to have published ppy asymmetries calculated in a potential model. Our asymmetries have a l l been calculated in coplanar geometries, in which the asymmetry is equivalent to the polarization 3> 4. McGuire and Pearce have used A = Tr { o • n M Mf } / Tr. { M M1" } 5.11 wherein, A is their asymmetry, n is a unit vector perpendicular to the scattering plane and, M is their coplanar ppy scattering amplitude. 'A', in this convention, is equivalent to our A^ with a ± sign depending A on the choice of orientation for n.. Our convention i s A ± = Tr{ a_ • rL T T* }/ Tr{ T T f } 5.12 where we continue to use 'T' to denote the ppy amplitude and (i) labels the axis along which 1^  is directed. Below are explicit formulae for our asymmetries: A% , A^ and A^ . A x = Re(T 1 2 2+T 2 1 2)/Re(T u 2+T 2 2 2) 5.13 A y = R e { i ( T 1 2 2 - T 2 1 2 ) } / R e ( T n 2 + T 2 2 2 ) A = R e ( T 1 1 2 - T 2 2 2 ) / R e ( T u 2 + T 2 2 2 ) where the subscripts 1 and 2 mean m equal to +1/2 and -1/2, Re(x) denotes the real part of x and T a b 2 = I ^ < m 3 * \ I T \ I * m 2 > < m 3 I T V J b ^ ^ V l " ^ - h * X 1 1 1 X i 5.14 We may now calculate cross sections and asymmetries for ppy using the methods of this chapter or, as a check, we may also calculate elastic p-p observables using the formulae of Appendix C. In the following chapter we w i l l describe the FORTRAN code written to do the numerical calculation required to generate these observables. 48 References 1. M. Goldberger & K. Watson, C o l l i s i o n Theory(New York: J . Wi ley , 1968), p .91 , equation 141. 2. Goldberger & Watson, p .85 . 3. J . McGuire & W. Pearce, " C a l c u l a t i o n of ppy Cross Sections and Asymmetries," N u c l . Phys. A162, 562 (1971). 4. Goldberger & Watson, Chapter 7 .5 . CHAPTER 6 The PPy Code 6.1 Description of the Code If a l i s t i n g of a l l programs and subroutines used in the preparation of this thesis were included here, 200 pages could be added to the text. Annexing such a bulk, of FORTRAN code serves no real purpose - the author has learned that the untutored use of another researcher's code can be very hazardous. The following is an outline of the numerical calculation performed by the abovementioned code. The potential model program has been incorporated into a large 'MAIN' routine which was written to perform soft-photon approximation (SPA) calculations. This MAIN program is due to Dr. H.W. Fearing. This meshing of routines has saved much duplication of labour in the calculation of cross sections and analyzing powers. As was Illustrated in Chapter 2, the cross section calculation can be spl i t into 3 invariant parts. The relations giving the invariant flux and phase space are identical for the SPA and potential model calculation. The kinematics are obviously also the same. The subroutine 'AMPL' is called by the MAIN program in order to obtain: (1) the invariant radiative amplitude for ppy, (2) the square of the elastic amplitude for P-P scattering and, (3) the polarization matrix, described in Chapter 5. The radiative amplitude is used to calculate the cross section for ppy while, other elastic observables are produced along with the P-P elastic cross section in order to check the calculation in the on-shell l i m i t . The polarization matrix is used to produce the asymmetries Ax, Ay, Az for the ppy process. Let us next concentrate on the AMPL routine i t s e l f . Common blocks pass a l l the required kinematic variables from the MAIN routine to AMPL, along with control variables which choose options within the subroutine. One such option is to perform the calculation in the CM system, rather than the lab system. If this is desired, a subroutine ''VECTOR' does a simple Lorentz transformation on the kinematic variables generated in the MAIN routine. AMPL then begins the task of evaluating the total T-matrix expansion, given in Chapter 2. A separate routine 'EM' is called by AMPL in order to evaluate the electromagnetic matrix elements, as described in Appendix A. EM uses the functions 'WIGN3J' and 'WIGN6J' in order to evaluate the various 3 and 6-J symbols required here. These functions are available in the library f i l e TRMF:CERNLIB( on the UBC MTS system ). Also called from within AMPL is the subroutine 'NUSC' - a modified and double precision version of Dr. A. Miller's 'NUSCAT' subroutine. NUSC calculates N-N T-matrices and elastic observables. Several options are available in this subroutine. See f i g . 2 for a schematic description of the program flow. On-shell N-N T-matrices can be calculated either from a potential or from Arndt's (1977) set of P-P elastic phases. The on-shell N-N T-matrices are then modified by the 'off-shell function', defined in Chapter 3. The option to calculate elastic P-P observables also exists here. Since the on-shell N-N T-matrices and elastic phases are produced in intermediate calculations, i t is easy to calculate elastic cross sections and asymmetries. Now in order to produce the off-shell function and the on-shell N-N T-matrix from a potential calculation another routine FLH0FF is called. Figure 2 Algorithm for Checking the PPy Program TLON = on-shell N-N T-matrix , TLOFF = o f f - s h e l l N-N T-matrix , FL = o f f - s h e l l function Calculate TLON ( FL=1 ) from Exper. phases Calculate TLON & FL from a P o t e n t i a l Calculate E l a s t i c Observables Check with Experimental Observables Calculate TLOFF=FL*TLON Calculate PPy Observables Compare with past PPy Calculations Calculate E l a s t i c Observables Check with Quoted Results Check with Experimental Observables Compare with Experimental Observables An argument of t h i s subroutine c a l l i s the p a r t i c u l a r p o t e n t i a l to be used. B r i e f l y , each p o t e n t i a l i s written as a Fortran function. The i n i t i a l and f i n a l r e l a t i v e momenta of the two protons are passed to t h i s routine, as are parameters which specify the required p a r t i a l wave. As has previously been mentioned, such functions have been produced for the Reid, extended Reid and Paris p o t e n t i a l s . FLHOFF evaluates the o f f - s h e l l function at a g r i d of energies. As discussed i n Chapter 3 , t h i s routine c a l l s UBC MATRIX subroutine DSLIMP to solve the r e s u l t i n g matrix equation and, uses the methods of Gaussian integration to choose i n t e g r a t i o n points and weights e f f i c i e n t l y . A FLHOFF entry point 'TMATON' i s c a l l e d from within NUSC i n order to construct the on-shell N-N T-matrices. O f f - s h e l l N-N T-matrices are then produced i n NUSC and passed through a common block back into AMPL. The e f f i c i e n c y of t h i s method i s related to the number of times that the Legendre functions must be generated i n the p o t e n t i a l function. This requires a judicious choice of i n t e g r a t i o n points for each p a r t i a l wave. A routine 'ANGLE' produces the Euler angles <x,f3,Y required to rotate the N-N T-matrices to a common axis of quantization. The subroutine 'ROTATE' performs t h i s transformation, as described i n Appendix D. Propagators are calculated within AMPL and are combined with the N-N T-matrices and matrix elements of V to produce the t o t a l T-matrix. em The value of Tr{ | T p } i s then c a l c u l a t e d . Factors are added i n order to match units with Dr. Fearing's program which used the d e f i n i t i o n of a cross section given by Bjorken and D r e l l 1 . As i s indicated i n f i g u r e 2 , there are several points at which the p o t e n t i a l model c a l c u l a t i o n may be checked. Since Arndt's experimental phases are b u i l t into the NUSC subroutine, e l a s t i c observables can be calculated at any energy where the po t e n t i a l model c a l c u l a t i o n i s v a l i d . These can then be compared with e l a s t i c observables derived from the po t e n t i a l c a l c u l a t i o n s . This test i s very useful since the on-shell N-N T-matrices are produced v i a an i n t e g r a l involving the o f f - s h e l l function. By constructing e l a s t i c observables we are then i n d i r e c t l y t e s t i n g the ca l c u l a t i o n of the o f f - s h e l l function. The condition that the o f f - s h e l l function should approach unity on-shell i s also tested. The o f f - s h e l l c a l c u l a t i o n may, to a lesser extent, also be checked. Previous p o t e n t i a l model c a l c u l a t i o n s can be approached by using the old Reid p o t e n t i a l and by dropping various correction terms from our c a l c u l a t i o n . By reproducing the ca l c u l a t i o n s of other authors, we can check that our c a l c u l a t i o n i s consistent with those done previously. Comparisons may also be made between our c a l c u l a t i o n and the e x i s t i n g experimental data. PPy cross sections and integrated cross sections have been calculated by many others - analyzing powers have not. Asymmetries have also only r a r e l y been measured. Unfortunately, analyzing powers seem to be more s e n s i t i v e 2 to the d e t a i l s of the c a l c u l a t i o n than are the cross sections, so i t i s more d i f f i c u l t to check t h i s part of the program by comparison with other c a l c u l a t i o n s - which d i f f e r i n d e t a i l from ours. Comparison with Dr. Fearing's SPA c a l c u l a t i o n has also been very useful throughout the construction of this FORTRAN code since we have a requirement that r e s u l t s from p o t e n t i a l model c a l c u l a t i o n s must reduce to those derived from SPA cal c u l a t i o n s i n the soft-photon l i m i t . This check was e s p e c i a l l y useful i n the 'debugging' stage. Some tests of the ppy c a l c u l a t i o n are presented i n the next section. Further r e s u l t s are given i n the next chapter. 6.2 Numerical Tests of the Program In this section, we w i l l show the results of some tests made on the ppy code. Since i t is d i f f i c u l t to differentiate between 'interesting results' and 'tests of the program', some of this section could have been included in the next chapter. We have, however, attempted to leave the more interesting calculations for Chapter 7. Although the ppy calculation requires off-shell N-N T-matrices, on shell checks are equally important. This point was briefly discussed in the previous section. Here we merely emphasize that our calculation of elastic quantities u t i l i z e s every part of the program required to produce the off-shell N-N T-matrices. Following is a table il l u s t r a t i n g a check made on the Reid soft-core ( RSC ) potential. In table 3 we compare the values of the 1S Q phase shift obtained from several sources. Here, we compare the results of Reid 3 and Halftel and Tabakin 4 with our calculation. The variable phase 1 0 calculation was included as an independant check. The 'Matrix Inversion' column contains the results of our on-shell calculation, using N=24 Gaussian integration points. In comparing these numbers one should note that Reid's phases3 include coulomb contributions while the others do not. In tables 4 and 5, we compare the Paris and Reid soft core phases at lab energies of 24 and 208 MeV, respectively. The Reid phases are from reference 3. The corresponding Paris phases7 were obtained from SAID, an interactive program provided by Dr. R. Arndt which produces scattering observables and phases for the N-N system. It is interesting to note the difference in the on-shell character of these N-N potentials - especially at 208 MeV. The numbers appearing in brackets are the results of our Table 3 Comparison of the Reid Soft Core Phase from Different Sources (in radians) E(Lab) Reid Paper3 Halftel/Tabakin 4 Variable Phase5 Matrix Inversion 6 24 MeV 0.862 0.86055 0.860566 0.861 48 MeV 0.696 0.68459 0.684554 0.685 96 MeV 0.454 0.44019 0.440170 0.441 144 MeV 0.277 0.26303 0.263029 0.263 208 MeV 0.093 0.08033 0.080337 0.081 304 MeV -0.188 - -0.129618 -0.129 352 MeV -0.205 -0.21638 -0.216363 -0.216 Table 4 A Comparison of the Reid Soft-Core and Paris P o t e n t i a l T=l Phases at Low Energy ( i n degrees) E(Lab) 24 MeV Reid Soft-Core 3 P a r i s 7 b 0 49.4 ( 49.3) 49.02 ( [ 48.95) % 8.08 ( 8.37) 8.64 ( ; 8.88) -4.24 (-4.45) -4.88 ( ; -5.11) 2.18 ( 2.33) 2.30 < ; 2.55) 0.63 ( 0.67) 0.70 ( : 0.74) 3 F 0.11 ( 0.10) 0.10 ( : 0.11) - -0.23 ( [ -0.24) - 0.02 ( : 0.02) - 0.04 ( ; o.o4) - 0.01 ( ; o.oo) - -0.02 < -0.02) e 2 -0.75 (-0.78) -0.80 ( : -0.84) eU - -0.05 ( . -0.05) Table 5 A Comparison of the Reid Soft-Core and Paris P o t e n t i a l T=l Phases at Medium Energy ( i n degrees) E(Lab) 208 MeV Reid Soft--Core 3 P a r i s 7 5.33 ( 4.62) 4.31 ( 3.58) 3 " o -0.69 (-0.94) 0.00 (-0.81) -22.12 (-22.3) -21.19 (-21.58) 15.87 ( 15.99) 16.46 ( 17.13) 7.05 ( 7.11) 7.57 ( 7.68) F 2 1.26 ( 1.28) 1.07 ( 1.04) 3 F - -2.93 (-2.94) - 1.67 ( 1.79) - 1.08 ( 1.13) - 0.37 ( 0.36) - -0.85 (-0.86) E 2 -2.98 (-2.95) -2.72 (-2.65) - -1.22 (-1.23) 5 8 on-shell potential model calculation. The differences are probably due mainly to the treatment of coulomb effects. The Paris paper7 is particularly vague about coulomb effects. In addition, the Paris group has dropped some terms in the momentum space quadratic spin-orbit potential. We have retained a l l non-zero terms. One should also note that the L=0,1 phases were found to be sensitive to the methods of integration. We made certain that our results were stable. In order to test our program in the soft-photon limit, the potential model calculation has been compared to Dr. Fearing's soft-photon approximation (SPA) calculation. Before presenting our results, we w i l l attempt to make our figure notation clear. The following conventions have been used in labelling the figures. TI = Kinetic energy of the incident proton ( Lab system ). THTA3 = 0 3 , THTA4 = 0 ^ ( Lab angles of the fi n a l protons ) DSIG/D03D04DTHK = da/dSl^^dQ^ THETA GAMMA = 0 ^ ( Lab angle of the photon ) MB = millibarns *E-3 = X 10 - 3 In figure 3, we plot the ppy differential cross section da/dftgdfi^d©^ versus 0 ^ in a coplanar geometry, with a small photon momentum. The reaction is defined by: P^+P^Pg+P^+k where k is the photon 4-momentum. Here we see that the Paris potential does indeed approach the SPA calculation is the limit k*0. Exact agreement should not be expected in this case since, the experimental phases used in the SPA calculation do not have exactly the same on-shell charactor as the Paris p-p phases. Figure 3 Soft Photon Limit TI = 25 MeV T l - 2 5 M E V T H T R 3 - T H T F H - 4 4 . 5 D E G 0.0 20.0 40.0 60.0 80.0 100.0 120.0 140.0 160.0 190.0 THETfl GflMMfl In figure 4, the soft-photon limit is again tested. This time, however, the incident proton energy is much higher and the Extended Reid potential is used in the potential model calculation. The difference is larger here than at 24 MeV. This is mainly due to the fact that the Paris N-N potential gives a better f i t to the phases used in the SPA calculation than does the Extended Reid potential. Table 6 compares integrated cross sections calculated with an incident lab energy of 200 MeV and 03=0^=30°. In this table, we have attempted to reproduce the integrated cross sections of Drechsel and Maximon8. The calculation has been done in both the lab and CM systems. No r e l a t i v i s t i c spin corrections have been added and, we have used the old Reid soft-core potential which only extends up to J=2. Unlike Drechsel and Maximon, we have included the coulomb interaction in our N-N T-matrices. Coulomb effects, however, make only a small contribution to the integrated cross section at this energy and geometry9. The agreement is reasonable considering the remaining differences between the calculations. Note that the lab results are not re a l i s t i c since the double scattering terms are not included. They can be neglected only in the CM system. Having gained some confidence in our calculation, we w i l l proceed to explore the nature of ppy cross sections and asymmetries in the next chapter. Figure 4 Soft Photon Limit Tl = 280 MeV T 1 - - 2 8 0 M E V T H T f l 3 - T H T f l 4 - 4 3 . Q D E G a "1 0.0 20.0 . 40.0 60.0 60.0 100.0 120.0 140.0 160.0 1B0.0 THETfl GflrlMR TABLE 6 Integrated Cross Sections at 200 MeV 03=0^=30° (yb/sr 2) Drechsel & Maximon Our Calculation Energy CM Lab CM Lab 100 MeV « 5.7 » 10.7 5.2 11.2 200 MeV - 13 -13 12.1 15.5 Note; Only the CM calculations should be compared with experimental data. See the text. References 1. J.D. Bjorken & S.D. Drell, Relativistic Quantum Mechanics(Toronto: McGraw-Hill, 1964), Appendix B. 2. H.W. Fearing & R.L. Workman, proceedings of the 10th International  Conference on Few Body Problems in Physics, Karlsruhe, 1983. 3. R.V. Reid, "Local Phenomenological Nucleon-Nucleon Potentials," Ann. Phys. 50, 424 (1968). 4. M.I. Haftel & F. Tabakin, "Nuclear Saturation and the Smoothness of Nucleon-Nucleon Potentials," Nucl. Phys. A158, 8 (1970). 5. Results of a Variable Phase calculation using the Reid potential with no coulomb potential added. 6. Results given by our on-shell potential model calculation - no coulomb. 7. M. Lacombe, B. Loiseau, J.M. Richard & R.V. Mau, "Parameterization of the Paris N-N potential," Phys. Rev. C21, 868 (1980), Table III. 8. D. Drechsel & L.C. Maximon, "Potential Model Calculation for Coplanar and Non-Coplanar Proton-Proton Bremsstrahlung," Ann. Phys. 49, 439 (1968), Figure 7. 9. L. Heller & M. Rich, "Proton-Proton Bremsstrahlung: Coulomb effect," Phys. Rev. 10_, 479 (1974). 10. F. Calogero, Variable Phase Approach to Potential Scattering(New York: Academic Press, 1967) CHAPTER 7 Results of the Numerical C a l c u l a t i o n 7.1 Introduction Having made a check of the ppy program, both i n the soft photon l i m i t and against the c a l c u l a t i o n of Drechsel and Maximon 1°, we w i l l now compare our c a l c u l a t i o n with experimental data and with other p o t e n t i a l model c a l c u l a t i o n s . F i r s t , we compare our p o t e n t i a l model c a l c u l a t i o n with the r e s u l t s of two previous ppy experiments 1' 2 at 42 and 200 MeV. At 200 MeV, previous p o t e n t i a l model c a l c u l a t i o n s 3 have given a very poor f i t to the experimental data. At 42 MeV, the p o t e n t i a l model c a l c u l a t i o n s have had better success. However, i n some geometries, the data has been f i t 4 by neither p o t e n t i a l model nor SPA (Soft Photon Approximation) c a l c u l a t i o n s . At 100 and 200 MeV, we have calculated both cross sections and asymmetries, with and without c o r r e c t i o n terms. The f u l l p o t e n t i a l model c a l c u l a t i o n i s compared with c a l c u l a t i o n s which drop the 1—rr exchange amplitudes, coulomb contributions to the N-N T-matrices and, the r e l a t i v i s t i c spin c o r r e c t i o n s . Next we check our c a l c u l a t i o n against an old c a l c u l a t i o n from McGuire and Pearce 5 who has made a p o t e n t i a l model c a l c u l a t i o n of asymmetries f o r ppy. Such c a l c u l a t i o n s have been r a r e 5 . The only published r e s u l t s we have found, are due to these authors. F i n a l l y , some SPA and p o t e n t i a l model ca l c u l a t i o n s w i l l be presented at a lab energy of 280 MeV - the energy of the TRIUMF experiment 1 1. 65 7.2 The PPy Results In this section, the bulk of our ppy results w i l l be displayed. The implications of our calculations/comparisons w i l l be discussed in section 7.3. We begin, in figure 5, by comparing the SPA and Paris potential model at 42 MeV 63=0^=22° - the energy and geometry of some experimental data from reference 2. (The labelling conventions used in our figures have been listed in section 6.2.) Here, we see that the f i t to this experimental data is not appreciably improved through use of the Paris potential. The Hamada-Johnston calculation of reference 2 is very close to our Paris potential model calculation for this choice of kinematics. In figure 6, the SPA calculation is displayed along with the Paris and Extended Reid potential model calculations. The 200 MeV 03=0^=16.4° data has been taken from reference 1 and, the corrections for a f i n i t e detector solid angle have been included. Here, our Paris potential model calculation gives a f i t to the data which is no better than the older Hamada-Johnston calculation of Bohannon7 and which is only marginally better than the Extended Reid calculation. The effects of several refinements to the basic ppy potential model calculation have been explored in figures 7 to 10. Figures 7 and 8 indicate the importance of various correction terms for kinematics corresponding to the 200 MeV experimental data while figures 9 and 10 repeat the comparison at 100 Mev 03=0^=30°. Each figure indicates the importance of coulomb and 1—rr exchange contributions to the N-N T-matrices and, r e l a t i v i s t i c spin corrections to the electromagnetic interaction. Several points are immediately obvious. It is clear that at these energies and geometries, r e l a t i v i s t i c spin corrections are very important. These Figure 6 Calculation at T l = 200 MeV © 3 = GH = 16.4 T l - 2 0 0 M E V T H T R 3 - T H T R 4 - 1 6 . 4 D E G X - Extended Reid P o t e n t i a l to i UJ tn x •. o crz or x on to x Ql (T) V. OO a X (-» a a a to S"! \ a C3 • — i CO cn A - Paris P o t e n t i a l 4- - SPA Calculation o I 1 r -60.0 ••—I—•— 120,0 ••—I—•"-140,0 ••—I— 160.0 —I 1B0 0.0 20.0 40.0 BO.O 100.0 THETfl GRMMR Figure 7 R e l a t i v i s t i c Spin Corrections, Coulomb contributions and the 1-TT exchange at 200 MeV 63 = © 4 = 16.4° T l - 2 0 0 M E V T H T R 3 - T H T R 4 - 1 6 . 4 D E G / ty + - F u l l Calculation y / tf • - No R e l a t i v i s t i c Spin Corrections X - No 1-TT exchange A - No Coulomb 20.0 40.0 60.0 80.0 100.0 THETfl GflMMFl 120.0 140.0 160.0 180.0 ON OO Figure 8 R e l a t i v i s t i c Spin Corrections, Coulomb contributions and the 1-TT exchange at 200 MeV ©3 = &H = 16.4° T l - 2 0 0 M E V T H T R 3 - T H T R 4 - 1 6 . 4 D E G -+• - F u l l Calculation a - No R e l a t i v i s t i c Spin Corrections X - No 1-TT exchange & - No Coulomb 0.0 20.0 40.0 60.0 80.0 rtXT.O , S L20.0 THETfl Gflflflfl" 140.0 160.0 1B0.0 as Figure 9 R e l a t i v i s t i c Spin Corrections, Coulomb contributions and the 1-TT exchange at 100 MeV 63 = 6 ^ = 30° o Figure 10 R e l a t i v i s t i c Spin Corrections, Coulomb contributions and the 1-TT exchange at 100 MeV 63 = 6^= 30° T l - 1 0 0 M E V T H T R 3 - T H T R 4 - 3 0 . 0 D E G - F u l l C a l c u l a t i o n No R e l a t i v i s t i c Spin Corrections No 1 -TT exchange No Coulomb 0.0 20.0 40.0 60.0 80.0 100.0 THFJTR GflMfIA 120.0 140.0 160.0 180.0 a d d i t i o n a l terms i n V greatly improve the f i t to the 200 MeV cross em section near 0° and 180°. It also appears that the asymmetries are more sen s i t i v e to c o r r e c t i o n terms than are the cross sections, p a r t i c u l a r l y at higher energies. Once again, the deletion of r e l a t i v i s t i c spin c o r r e c t i o n terms produces the largest v a r i a t i o n i n the asymmetries. However, figure 8 indicates that every contribution to the asymmetries i s important at 200 MeV 83=0^=16.4°. F i n a l l y , figures 7 and 9 show that dropping the coulomb contribution has very l i t t l e e f f e c t on the cross sections. This i s consistent with the r e s u l t s of L. H e l l e r and M. R i c h 8 . Contributions from the 1-ir exchange amplitudes, used for the higher p a r t i a l waves, appear to be small here as w e l l . In figures 11 and 12, we compare our cross section and asymmetry c a l c u l a t i o n with a c a l c u l a t i o n made by McGuire and Pearce 5. These authors have done a p o t e n t i a l model c a l c u l a t i o n i n which th e i r N-N T-matrices where taken o f f - s h e l l using a pure 1—rr exchange p o t e n t i a l . Most importantly, r e l a t i v i s t i c spin corrections were not included i n t h e i r c a l c u l a t i o n . Here, the Extended Reid p o t e n t i a l model c a l c u l a t i o n of the cross sections and asymmetries i s close to the SPA r e s u l t but, quite d i f f e r e n t from the c a l c u l a t i o n of McGuire and Pearce. The largest q u a l i t a t i v e difference i s seen i n figure 12, which again demonstrates how s e n s i t i v e the ppy asymmetries are to the methods and approximations used i n t h e i r c a l c u l a t i o n . Considering how close our p o t e n t i a l model c a l c u l a t i o n i s to the SPA c a l c u l a t i o n i n this geometry, this old c a l c u l a t i o n from McGuire and Pearce may not s a t i s f y the soft photon l i m i t . Figures 13 and 14 show another SPA versus p o t e n t i a l model c a l c u l a t i o n with 0 3 * 0 ^ at the energy of the new TRIUMF experiment. The cross sections and asymmetries are q u a l i t a t i v e l y s i m i l a r with the largest difference i n Figure 11 Cross sections at Tl = 200 MeV 0, = 20° = 40 T l - 2 0 0 M E V T H T R 3 - 2 0 . 0 T H T R 4 - 4 0 . 0 D E G a X - Extended Reid Potential Figure 12 Asymmetries at TI = 200 MeV © 3 = 20° 9H = 40° T l - 2 0 0 M E V T H T R 3 - 2 Q . 0 T H T A 4 - 4 0 . 0 D E G r - i — • — • — • — • — i — • — • — • — • — i — • — • — • — • — i — • — • — • — • — i — • — • — • — • — i — • — • — • — • — i 0.0 60.0 120.0 180.0 240.0 3QD. 0 3€0.Q THE7R GAMMA Figure 13 Cross sections at TI = 280 MeV ©3 = 10° © M = 30 asymmetry unfortunately coinciding with the lowest cross section, thus making an experimental test more d i f f i c u l t here. Figures 15 and 16 show more cross sections at 280 MeV. The SPA calculation is compared with both the Paris and Extended Reid potential model calculations. A comparison of figures 4, 15 and 17 illustrates the variation of ppy cross section with increasing photon momentum. The photon carries away the most momentum in figure 15, corresponding to 10°-10° , where the potential model calculation shows the greatest departure from the SPA curve. The least departure occurs at proton angles of 43°-43°. In figures 15 and 17, cross sections are calculated at a lab energy of 280 MeV with symmetric proton angles of 10°-10° and 30°-30° respectively. As was seen in figure 6 at a lab energy of 200 MeV ©2=0^=16.4°, figures 15 and 17 also show that the Paris potential model calculation consistently f a l l s below the Extended Reid calculation near 0 equal to 0° and 180°. Another feature common to symmetric angle geometries is the gap between SPA and potential model calculations. The SPA calculation always f a l l s below our potential model calculations. In figures 16 and 18, we display two more asymmetry calculations at a lab energy of 280 MeV with symmetric proton angles of 10°-10° and 30°-30° respectively. While there is s t i l l a qualitative similarity between the SPA and potential model calculations at 30°-30°, these disappear at 10°-10°. In the next section we w i l l summarize the results of our potential model calculations and comparisons with experimental data. Figure 15 Cross sections at T l = 280 MeV &j = &H= 10 T l - 2 8 0 M E V T H T H 3 - T H T R 4 - 1 0 . 0 D E G A - Extended Reid Poten t i a l / / 20.0 40.0 100. 0 THE7A GflMfin 120.0 140.0 1 6 0 . 0 180.0 00 Figure 16 Asymmetries at T l = 280 MeV © 3 =9^= 10 T l - 2 8 0 MEV THTR3-THTFH-10 .0 DEG 1HE7H GflMMfi Figure 17 Cross sections at TI = 280 MeV ^ = = 30° T l - 2 8 0 MEV THTFI3-THTFH-30.0 DEG OO o Figure 18 Asymmetries at T l = 280 MeV 63 = 0H= 30° 7.3 Summary and Possible Improvements In the previous section we have given the results of a new potential model calculation for ppy. The Paris potential, a modern, theoretically based potential, and the Extended Reid potential have been used, both for the f i r s t time in a ppy calculation. We have included a number of corrections and improvements such as r e l a t i v i s t i c spin corrections, coulomb effects, the 1—IT exchange potential for higher partial waves, gauge terms, r e l a t i v i s t i c transformations of the amplitudes which have been considered before, but never combined into one calculation. We have also emphasized the asymmetries, which have only rarely been calculated. One of the most interesting results is the fact that our new calculations are qualitatively similar to the older potential model calculations and, thus, are s t i l l quite different from the SPA calculations and, at 200 MeV, the data. Clearly the improvements have not solved the problem of this disagreement. We have, however, learned more about the ppy calculation and the Paris N-N potential. As was mentioned in the previous section, the asymmetries appear to be more sensitive to the method of calculation than are the cross sections. This should perhaps be expected since, the asymmetries involve cancellations between pieces of the ppy amplitude. This is an important observation, though, since i t provides some justification for the measurement of asymmetries in the new TRIUMF experiment. The disagreement between potential model and SPA calculations and the data at 42 MeV is s t i l l puzzling. Relativistic corrections should be very small here and according to V. Brown9, the double scattering terms should contribute only about 0.2% to the integrated cross section at this low energy. This suggests that there is something fundamentally different or missing in the potential model approach. At 200 MeV, the large contribution due to the r e l a t i v i s t i c spin corrections suggests that r e l a t i v i s t i c corrections in general should be given more study. It is interesting to note in figure 4, however, that at 280 MeV 03=0^=43°, near the soft photon limit, the Paris potential model calculation approaches the SPA calculation. If important r e l a t i v i s t i c corrections have been neglected at 200 MeV, their importance at 280 MeV should be even greater. Of course, this argument neglects the fact that the photon momentum approaches zero In figure 4 - masking any r e l a t i v i s t i c corrections which might depend on the photon momentum. As has been mentioned in Chapter 6, when making comparisons with the SPA calculation, one should recall that the SPA code uses experimental p-p phases as an on-shell input while, the Paris and Extended Reid potential model calculations do not have exactly the same on-shell character. The comparisons between SPA and potential model calculations would be more consistent i f the N-N potential was used to generate elastic phases for the SPA calculation. There are several other areas in which our calculation could be improved. The most obvious addition would be the double scattering terms which are expected to contribute on the order of 15% to the integrated cross sections at the energy of the new TRIUMF experiment, based on the results of V. Brown9. In light of the large contributions from r e l a t i v i s t i c spin correction terms at 200 MeV, we should investigate the effect of adding additional terms. Another point of improvement centers on the 1—rr exchange amplitudes which were calculated in an on-shell approximation. We should instead use the true off-shell 1-TT exchange amplitudes. So far, however, the major contribution to ppy appears to come from the lower partial waves, so this is not expected to be a large effect. Consider the N-N potentials which were used in the calculation. The Extended Reid potential produces results which are not greatly different from those obtained using the much more complicated Paris potential. In order to better understand this behaviour, we have compared the off-shell behavior of these two potentials via the half-off-shell extension function. We find that the off-shell extension functions are very similar in the neighbourhood of the on-shell point, f = 1. In figures 19 and 20, L i we have compared the half-off-shell functions for the Paris and Extended Reid potentials at a lab energy of 200 MeV, in the and 3 s t a t e s . In figure 19, for example, the two functions vary considerably for an off-shell momentum ( POFF ) of * 6 fm - 1. At this energy, however, the N-N T-matrices are calculated for a range of off-shell momenta between approximately one half and twice the on-shell value. Perhaps, since the two potentials have similar on/off-shell behavior, we should not expect very different ppy results. It would be interesting to see i f the Bonn potential has a similar off-shell behavior as the Bonn potential is calculated from meson exchange diagrams and has not been parameterized in terms of Yukawa-like potentials. It is possible -that any potential, constructed from Yukawa terms, w i l l have a similar off-shell character in the neighbourhood of the on-shell point. Aside from trying other potentials, the effect of phase-equivalent transformations could also be considered. This approach has the advantage of producing potentials which have exactly the same on-shell behaviour. Then in comparing ppy calculations, which use phase-equivalent potentials, Figure 19 O f f - S h e l l Extension Function T l = 200 MeV, 'S0 State. T l - 2 0 0 MEV HflLF-OEF-SHELL FUNC. + - Extended Reid P o t e n t i a l X - Paris P o t e n t i a l o - On-Shell Point 12.0 POPF t f m ] oo Figure 20 O f f - S h e l l Extension Function TI = 200 MeV, 3P, State. T l - 2 0 0 MEV HALF-OFF-SHELL FUNC. no differences are masked by differences i n purely o f f - s h e l l e f f e c t s . A r e c o n c i l i a t i o n of the fundamental differences between SPA and p o t e n t i a l model c a l c u l a t i o n s may eventually help to explain the d i f f i c u l t i e s we have encountered in our attempts to f i t ppy data. Unlike the p o t e n t i a l model c a l c u l a t i o n , the SPA approach i s f u l l y r e l a t i v i s t i c and therefore requires r e l a t i v i s t i c propagators. In addition, the N-N amplitudes are evaluated on-shell at an average energy for the SPA c a l c u l a t i o n . Gauge invariance i s also b u i l t into the SPA approach. It i s d i f f i c u l t to i s o l a t e the most important differences between the two methods because of t e c h n i c a l differences between r e l a t i v i s t i c and non-r e l a t i v i s t i c theory. From another viewpoint, i t might be i n t e r e s t i n g to see what would r e s u l t from the a p p l i c a t i o n of Dirac phenomenology to the ppy problem. Such a c a l c u l a t i o n would reveal more of the s e n s i t i v i t y of PPY to r e l a t i v i s t i c e f f e c t s . Although this work has not solved the ppY problem, i t has raised many questions regarding the methods used i n ppY c a l c u l a t i o n s . However, since every thesis has an end, any speculation w i l l have to await further i n v e s t i g a t i o n . References 1. J.G. Rogers, J.L. Beveridge, D.P. Gurd, H.W. Fearing, A.N. Anderson, J.M. Cameron, L.G. Greeniaus, CA. Goulding, C.A. Smith, A.W. Stetz, J.R. Richardson & R. Frascaria, "Proton-proton bremsstrahlung at 200 MeV," Phys. Rev. C22, 2512-2522 (1980). 2. H.W. Fearing, "Comparison of proton-proton bremsstrahlung at 42 and 156 MeV," Phys. Rev. C22, 1388-1393 (1980). 3. Rogers et a l , p.2517. 4. Fearing, P.1390. 5. J.H. McGuire & W.A. Pearce, "Calculation of ppy Cross Sections and Asymmetries," Nucl. Phys. A162, 561-572 (1971). 6. G. Bohannon has made asymmetry calculations, but these have remained unpublished. (Dr. H.W. Fearing, private communication.) 7. Rogers et a l , P.2516-7, Table I and f i g . 6. 8. L. Heller & M. Rich, "Proton-Proton Bremsstrahlung: Coulomb Effect," Phys. Rev. C10, 479 (1974). 9. V. Brown, "Proton-Proton Bremsstrahlung Including Rescattering," Phys. Rev. 177, 1498 (1969). 10. D. Drechsel & L.C. Maximon, "Potential Model Calculation for Coplanar and Noncoplanar Proton-Proton Bremsstrahlung," Ann. Phys. 49 (1968). 11. TRIUMF experiment #208; P. Kitching, spokesman. BIBLIOGRAPHY Bateman, H. The Bateman Manuscript Project. New York: McGraw-Hill, 1953, Vol. I. Bjorken, J.D. & S.D. D r e l l . R e l a t i v i s t i c Quantum Mechanics. New York: McGraw-Hill, 1964. Brown, G. & A.D. Jackson. The Nucleon-Nucleon Interaction. New York: Elsevi e r , 1976. Brown, V. "Proton-Proton Bremsstrahlung Including Rescattering," Physical  Review, 177, 1498 (1969). Calogero, F. Variable Phase Approach to Potential Scattering(New York: Academic Press, 1967). Coster, F., S. Cohen, B. Day & CM. Vincent. "Variation of Nuclear Matter Binding Energies with Phase-Shift-Equivalent Two-Body Potentials," Physical Review C, 1_, 769 (1970). Cottingham, W.N., M. Lacombe, B. Loiseau, J.M. Richard & R.V. Mau. "Nucleon- Nucleon Interaction from Pion-Nucleon Phase-Shift Analysis," Physical Review D, 8_, 800 (1973). Cromer, A.H. & M. Sobel. "Theory of Proton-Proton Bremsstrahlung," Physical Review, 152, 1351 (1966). C z i f f r a , P., M.H. MacGregor, M.S. Moravcsik & H.P. Stapp. "Modified Analysis of Nucleon-Nucleon Scattering. I. Theory and p-p Scattering at 310 Mev," Physical Review, 114, 880-886 (1958). Day, B.D. "Three Body Correlations i n Nuclear Matter," Physical Review C, 24_, 1213 (1981). Drechsel, D. & L.C. Maximon. "Potential Model Calculation for Coplanar and Noncoplanar Proton-Proton Bremsstrahlung," Annals of Physics, 49, 403, (1968). Drechsel, D. & L.C. Maximon. "Potential Model Calculation for Coplanar and Noncoplanar Proton-Proton Bremsstrahlung," Physical Letters B, 26, 477, (1968). Edmonds, A.R. Angular Momentum i n Quantum Mechanics. Princeton: Princeton University Press, 1968. Fearing, H.W. "'Forbidden' Asymmetries and Choices of Geometry i n Proton-Proton Bremsstrahlung," Physical Review Letters, 42, 1394 (1979). Fearing, H.W. "Comparison of Proton-Proton Bremsstrahlung Data at 42 and 156 MeV with Soft Photon Calculations," Physical Review C, 22, 1388 (1980). Fearing, H.W. & R.L. Workman. "A Modern Potential Model Calculation of Proton-Proton Bremsstrahlung" in Proceedings of the 10th International  Conference on Few Body Problems in Physics. Karlsruhe, 1983. Foldy, L. & S. Wouthuysen. "On the Theory of Spin 1/2 Particles at i t s Non-Relativistic Limit," Physical Review, 78, 29 (1950). Gell-Mann, M. & M.L. Goldberger. "The Formal Theory of Scattering," Physical Review, 91_, 398 (1953). Goldberger,M.L. & K.M. Watson. Collision Theory. New York: J.Wiley, 1964. Gottschalk, B., W. Shlaer & K. Wang. "A Measurement of Proton-Proton Bremsstrahlung at 160 MeV," Physical Letters, 16, 294 (1965). Gradshteyn, I. & I. Ryzhik. Table of Integrals, Series, and Products. Toronto: Academic Press, 1980. Haftel, M. & F. Tabakin. "Nuclear Saturation and Smoothness of Nucleon-Nucleon Potentials," Nuclear Physics A, 158, 40-42 (1970). Halbert, M.L. "Review of Experiments of Nucleon-Nucleon Bremsstrahlung," in The Two-Body Force i n Nuclei, eds. S.M. Austin & G.M. Crawley. New York: Plenum Press, 1972. Hamada, T. & J.D. Johnston. "A Potential Model Representation of Two-Nucleon Data Below 315 Mev," Nuclear Physics, 34, 382 (1962). Heller, L. & M. Rich. "Proton-Proton Bremsstrahlung: Coulomb effect," Physical Review, C10, 479 (1974). Heller, L. "Soft Photon Theorem for Nucleon-Nucleon Bremsstrahlung," Physical Review, 174, 1580 (1968). Holinde, K., R. Machleidt, M.R. Anastasio, A. Faessler & H. Muther. "Role of Noniterative TT exchange in NN-Scattering," Physical Review C, 19,, 948 (1979). Kowalski, K.L. "Off Shell Equations for Two-Particle Scattering," Physical  Review Letters, 15, 798 (1965). Lacombe, M., B. Loiseau, J.M. Richard & R.V. Mau. "Parameterization of the Paris N-N Potential," Physical Review C, 21, 861 (1980). Liou, M.K. & M.I. Sobel. "p-p Bremsstrahlung Calculations and Relativistic Spin Corrections," Annals of Physics, 72, 323-352 (1972). Liou, M.K. "Low Energy Theorem for Nucleon-Nucleon Bremsstrahlung," Physical Review C, 2, 131 (1970). Liou, M.K. & K.S. Cho. "Non-Coplanar Proton-Proton Bremsstrahlung Calculations Including the Relativistic Spin Corrections," Nuclear  Physics A, 160, 417-427 (1971). Lippmann, B.A. "High Energy Semi-Classical Scattering Processes," Annals of  Physics, 1, 113 (1957). Loiseau, B. Ph.d. Thesis, Universite Pierre et Marie Curie, Paris, 1974. McGuire, J.H. & W.A. Pearce. "Calculation of ppy Cross Sections and Asymmetries," Nuclear Physics, A162, 561-572 (1971). McMillan, M. Lectures in 'Theoretical Nuclear Physics,' University of British Columbia, 1982 ( Unpublished ). Messiah, A. Quantum Mechanics. New York: J. Wiley, 1967, Vol. I,II. Mongan, T.R. "Separable Potential Models of the Nucleon-Nucleon Interaction," Physical Review, 178, 1597 (1969). Mongan, T.R. "Off-Energy-Shell Behavior of Partial-Wave Scattering Amplitudes," Physical Review, 180, 1514-1521 (1969). Pearce, W.A., W.A. Gale & I.M. Duck. "Proton-Proton Bremsstrahlung," Nuclear Physics B, _3» 241-244 (1967). Reid, R.V. "Local Phenomenological Nucleon-Nucleon Potentials," Annals of  Physics, 50, 411-448 (1968). Rogers, J.G., J.L. Beveridge, D.P. Gurd, H.W. Fearing, A.N. Anderson, J.M. Cameron, L.G. Greeniaus, CA. Goulding, CA. Smith, A.W. Stetz, J.R. Richardson & R. Frascaria. "Proton-proton bremsstrahlung at 200 MeV," Physical Review, C22, 2512 (1980). Signell, P. & D. Marker. "Proton-Proton Bremsstrahlung Calculations," Bulletin of the American Physical Society, 12, 123 (1967). Sobel, M. & A.H. Cromer. "Potential Model Calculation of Proton-Proton Bremsstrahlung," Physical Review, 158, 1157 (1967). Sobel, M. "Parametric Expressions for p-p Off-Energy-Shell Matrix Elements and p-p Bremsstrahlung," Physical Review B, 138, 1517 (1965). Stapp, H.P., T.J. Ypsilantis & N. Metropolis. "Phase-Shift Analysis of 310 Mev Proton-Proton Scattering Experiments," Physical Review, 105, 302 (1957). Taylor, J.R. Scattering Theory. New York: J. Wiley, 1972. Ulehla, I. "Solution of the Reaction Matrix in Nuclear Matter," Nuclear  Physics A, 181, 270-272 (1972). Watson, K.M. & J. Nuttall. Topics in Several Body Dynamics. San Francisco: Holden-Day, 1967. Wolfenstein, L. "Polarization of Fast Nucleons," Annual Review of Nuclear  Science, 6, 43 (1956). APPENDIX A Evaluation of the Electromagnetic Matrix Elements We must evaluate the following r e l a t i o n , the matrix element of the electromagnetic i n t e r a c t i o n given i n equation 2.22 < S'M' | (-e/2Trmk 1 / 2)(£»e^ - i U ^ k x £ _ A / 2 ) | SM > A . l i n which S,M and S',M' are the t o t a l spin and spin p r o j e c t i o n of the i n i t i a l and f i n a l states r e s p e c t i v e l y . The p_»e_^  term i n A . l does not contain spin operators and i s , therefore, easy to evaluate. The r e s u l t i s (-e/2 1rmkl/2) £.i x6 s s,6 M M, . A.2 Next, we c a l c u l a t e the contribution from the second, spin dependant term. In the following, spherical components w i l l be used and, the conventions of Edmonds1 w i l l be followed. Consider the product: a_ • k x £. In spherical components we have 0 • k x £_ = £ q ( - l ) 4 a(q) V(-q) A.3 with V = k x E and q = -1, 0, 1. R e c a l l that a i s represented by 2 o(±l) = +( a ± la )/f? , a(0) = a . A.4 x y z With this notation, we can write the spin dependant term of A . l as (-e/2Trmk 1 / 2)(-iy Z (-l) qV(-q) < S'M' | a ^ 1 ) 2 ) ( q ) I SM » A.5 p q Now the terms < S'M' | o ^ » 2 ^ ( q ) | SM > can be written as 'reduced' matrix elements 3 where, (1,2) denotes the rad i a t i n g proton, and <S'M* | a c l ) ( q ) |SM> = ( - 1 ) S ' " M V S' 1 S \ < S' | | 0 C n (q) ||S> A.6 \ -M' q M j Here, we have introduced the '3-J symbol' which may be defined, i n terms of Clebsch-Gordon c o f f i c i e n t s 4 , as 'Jl h J \ = (- 1> d l J 2 M ( 2 J + l ) 1 / 2 < J 1 m 1 J 2 m 2 | j 1 J2JM> . A.7 , m^  m2 M The factor < S' | | a ^ ^ ( q ) | | S > i n A.6 may be expressed i n terms of 6-J symbol 5 using < S '||a(q) || S > = (-1) S ((2S+1)(2S'+1)) 1 / 2U S' U < i || a(q) | | i > A.8 The 6-J symbol i n A.5 may be expressed i n terms of 'recoupling c o e f f i c i e n t s 6 which, i n turn, are constructed from Clebsch-Gordon c o e f f i c i e n t s . In Edmonds' convention we have {3\ J 2 Ji2 j J j K.J3 J23 = ( ( 2 j 1 2 + l ) ( 2 j 2 3 + l ) ) - l / 2 ( - 1 ) J 1 + J 2 + J 3 + J . •<( 3i j 2 ) h i ' h >J bi(J2 ^ 3^23 » J > A * 9 Now, the reduced matrix element i n A.8 i s e a s i l y evaluated 7. 94 < i l l°- ( 1 )(q) M i > - f e 1 (we set /fi = c = 1) A.10 Now, using the following symetries 8 of the 3-J and 6-J symbols, we can write A.l in the form of Drechsel and Maximon's r e s u l t 9 . ! 1 X 1 S 1 ± ( X 1 S' A . l l A.12 Finally, after changing variables q -»• -q and noting that (_ 1 ) M' +q-M = L A.13 we find < S'M' | ( - e / 2 - r r m k 1 / 2 )(£•£- i u a ( l ) • k x E / 2 ) | SM > = P— (-e/2Trmk1/2)[£4X6ss'V' + i ( y D / 2 ) ^ 6 ( 2 S + 1 ) ( 2 S ' + 1 ) ^ / 2 ( S 1 S ) .(-1) M 1 (kxe x) n q A q A.14 In the above, o^was carried through the matrix element calculation. Had we inserted o^ 2*instead, a factor of (-1) would have resulted 5. The above equation is not, however, convenient for our calculation. A more appropriate form factors out the polarization vector to give, e" • M . 95 This form i s obtained i f we write e • a x k i n sp h e r i c a l components as e, • a x k = - i 6 1 / 2 E (e, ) a k / 1 1 1 — A — — m,m^  tvci2 A m m^  m^  m2 m^  ) A.15 Now, since A.2 also contains only a scalar product with e , 'e may be A A factored out of the matrix element. Then, i n the Invariant amplitude we can use E,e e * = - g„ , A.16 A u v 6MV to eliminate the p o l a r i z a t i o n vector. Replacing r e l a t i o n A.3 with A.15 gives the following r e s u l t for the matrix element. < S'M' | ( - e [ 2 T r m k 1 / 2 ] _ 1 )(£•£, - ip o ( 1 ) ' k x e./2) | S M > = ( - e / 2 T r m k 1 / 2 ) ( £ . i x 6 s s,6 M M,+ 3y f ( 2 S + l ) ( 2 S ' + l ) ) 1 / 2 C S' 1 S (-1) M' £ (e\) k /S' 1 S\ / l 1 1\ ) m^  ,m2 ,m3 —A m^  m2 / > ' > ' M' m^  -My y m2 m3y A. 17 This i s the form which we use while, r e l a t i o n A.14 was used by Drechsel and Maximon. As described i n Chapter 2, the addition of Liou's RSC terms to V , makes the following change i n equation A.17. em k •»• [ k - ( 1 - (2y kp /m ] A.18 m2 L _ ^ p 1 Jm2 References 1. A.R. Edmonds, Angular Momentum In Quantum Mechanics(Princeton: Princeton University Press, 1968), p.72. 2. Edmonds, p.69 3. Edmonds, p.75 4. Edmonds, p.46 5. Edmonds, p . I l l 6. Edmonds, p.91 7. Edmonds, p.76 8. Edmonds, p.47, pp.94-95 9. D. Drechsel & L.C. Maximon, "Potential Model Calculation for Coplanar and Noncoplanar Proton-Proton Bremsstrahlung," Annals of Physics, 49, 419-420 (1968). 10. Edmonds, pp.70-72 (note the sign error in relation 5.1.8) APPENDIX B The Reid P o t e n t i a l In the late s i x t i e s 1 , R.V. Reid presented a phenomenological N-N potential which became popular for p-p bremsstrahlung calculations. Reid produced a potential for each state of d i s t i n c t isotopic spin T , t o t a l spin S and t o t a l angular momentum J . Both hard and soft core potentials were f i t t e d . Here, however, we have considered only the Reid soft core potential, which was f i t t e d independantly i n each p a r t i a l wave channel with J < 2. Below are the Reid soft core T=l potentials, as given by Reid 2. In the following, X = ur where, Reid sets u = 0.7 fm - 1. V^Sg) = -10.463exp(-X)/X - 1650.6exp(-4X)/X - 6484.2exp(-7X)/X B.l V( 1D 2) = -10.463exp(-X)/X - 12.322exp(-2X)/X - 1112.6exp(-4X)/X + 6484.2exp(-7X)/X B.2 V( 3P Q) = -10.463exp[(l + 4/X + 4/X 2)exp(-X)/X - (16/X + 4/X 2)exp(-4X)/x] +27.133exp(-2X)/X - 790.74exp(-4X)/X + 20662exp(-7X)/X B.3 V( 3P X) = 10.463exp[(l + 2/X + 2/X 2)exp(-X)/X - (8/X + 2/X 2)exp(-4X)/x] -135.25exp(-2X)/X + 472.81exp(-3X)/X B.4 V ( 3 P 2 - 3 F 2 ) = V + V f S ^ + V L S L*S B.5 The terms V , V and V T„ are defined below, c ' T LS A A, S 1 2= OJ «r_ £2 _ 2.\'_l2 B.6 V £ = (10.463/3)exp(-X)/X - 933.48exp(-4X)/X + 4152.1exp(-6X)/X B.7 V T = 10.463[(l/3 + 1/X + 1/X2)exp(-X)/X - (4/X + 1/X 2)exp(-4X)/x] -34.925exp(-3X)/X B.8 V = -2074.1exp(-6X)/X B.9 An extended version of the RSC potential has been used by B. Dayd. Following are the T=l potentials which have been given in reference 3. These additional potentials produce partial waves up to J=5. V( 3F 3) = 10.463[(1 + 2/X + 2/X2)exp(-X)/X - (8/X + 2/X2)exp(-4X)/x] - 729.25exp(-4x)/X + 219.8exp(-6X)/X B.10 - -10.463exp(-X)/X - 39.025exp(-2X)/X + 6484.2exp(-7X)/X B . l l V( 3H 5) = 10.463[(1 + 2/X + 2/X2)exp(-X)/X - (8/X + 2/X2)exp(-4X)/x] B.12 V( 3F 1 +- 3H l t) = V c + V TS 1 2 + V L g L'S B.13 In B.13, V c and are defined by relations B.7 and B.8 respectively. V__ is defined below. V = -1037.05exp(-6X)/X B.14 In order to obtain the partial-waved potential, we must evaluate the following integral 4 for each potential in equations B.l to B.14. <pfL'SJ |V( 2S+l L j ) | P ±LSJ> = i L _ L ' 2TT-1 / j ( 1 ( P f r){6 L L,(r) Vc (r) + <L'SJ | L-S | LSJ> 6 T T, V T C (r) + <L*SJ I S,« | LSJ> V f r ) } j (p, r) r*dr B.15 wherein, < T I L S J > = ^ < L m S M | LSJJ 2 > Y L m ( r) b # 1 6 < r | pLSJ > = i L ( 2 / n ) 1 / 2 j T(pr) < r | LSJ > Some integrals, useful for the evaluation of equation B.15, have been collected in table 1. The explicit evaluation of <L'SJ | S 1 2 | LSJ> and <L'SJ | _L-_S | LSJ> is standard and, to avoid duplication, is shown only in Chapter 4. The expression <p L'SJ | V ( 2 S + 1 L ) | p LSJ> w i l l be abbreviated as V ( 2 S + 1 L ) I X J below. In addition, the following conventions w i l l be used, z =[p2 + p 2 +(nu) 2]/[2p ip f] , u = 0.7 fin - 1 , h = 10.463 MeV , B = iry 3 We find the following results 5 V(lSQ) = A[-h Q Q C Z ^ - 1650.6 Q 0(z l f) + 6484.2 Q 0 ( z ? ) ] V( 3P Q) = A[-h Q J C Z J ) + 27.133 Q 1 ( z 2 ) - 790.74 Q 1(z l +) + 20662 Q 1 ( z 7 ) ] " 4h{ Q Q ( Z L ) _ Q 2 ( Z L ) JI3B]- 1 + 4h{ Q Q ( Z L + ) - Q ^ ) } [3B]"1 V( 3P X) = A[h Ql(zl) -135.25 Q 1 ( z 2 ) + 472.81 Q 1 ( z 3 ) ] + 2h{ Q 0 ( Z ] L ) - Q 2 ( Z L ) } [3B] _ 1 - 2h{ Q 0 ( Z L + ) - Q 2 ( 2 L T ) } [3B]"1 V( 1D 2) = A[-h q2(z1) - 12.322 Q 2 ( z 2 ) - 1112.6 Q 2(z l f) + 6484.2 Q 2 ( z ? ) ] V( 3P 2- 3F 2) = { (h/3) F L , L ( D - 937.48 + 4152.1 F L, L(6) } + { (h/3) G L, L(1) -(h/3) G L, L(4) + (16h/3) F i l L(4) - 34.925 F L u(3)}. •<L',S=1,J=2 | S 1 2 | L,S=1,J=2> - { 2074.1 F L, L(6) } < L',S=1,J=2 | L*S_ |L,S=1,J=2 > B.22 where, A = [iryp^p^] - 1 and in B.22 we have used F L, L(n) = 2T T — 1 / J L,(p fr) [ exp(-nyr)/(yr) ] J L(p Lr) r 2dr B.23 which may be evaluated for the case L'=J-1 and L=J+1 using integrals 1.3 and 1.5 from Table 1. Here, of course, J = 2. As we l l , we have used G L , L ( n ) = 2 T r - 1 / j L , ( P f r)exp(-nur)(ur)- 1{n 2+3n(ur)- 1+3(yr)- 2} J L ( p L r ) r 2 d r B.24 which i s i n t e g r a l 1.4 of Table 1. The i n t e g r a l s required to extend the Reid p o t e n t i a l to J=5 are e a s i l y performed. The cal c u l a t i o n s of V ( 3 F 3 ) and V^G^) are e s s e n t i a l l y the same as those required for V C 3 ? ^ and V ^ S Q ) r e s p e c t i v e l y . S i m i l a r l y , V C 3 ^ ) and V C ^ - 3 ^ ) are calculated as were V ^ P j ) and V ( 3 P 2 ~ 3 F 2 ) . These r e s u l t s are l i s t e d below. V ( 3 F 3 ) = Ah Q 3 ( z 1 ) + 2h{ Q 2 ( Z l ) - Q 1 + (z 1 ) J ^ B ] " 1 B.25 - 2h{ Q 2 ( Z l + ) - %(zk) U7B]"1 - A[ 729.25 Q ^ ) - 219.8 Q 3 ( z 6 ) ] = A[ -h Q^Cz^ - 39.025 Q 1 + (z 2 ) + 6484.2 Q^Czg) ] B.26 V( 3H 5) = Ah Q 5 ( Z l ) + 2h{ (^(zj) - Q 6 ( Z l ) }[11B]~ 1 - 2h{ Q ^ ) - Q 6 ( Z l t ) }[11B]~1 B.27 V ( 3 F ^ _ 3 H i t ) i s g i v e n by B.22 with the replacement, 2074.1 -»- 1037.05. References 1. R.V. Reid, "Local Phenomenological Nucleon-Nucleon Potentials," Ann. Phys. 50, 411-448 (1968). 2. R.V.Reid, Ann. Phys. 50, 422-423 (1968). 3. B.D. Day, "Three Body Correlations in Nuclear Matter," Phys. Rev. C24 1269-1270 (1981). 4. B.D. Day, Phys. Rev. C24, 1213-1218 (1981). 5. In this case, "we" should include Dr. A. Miller, as he has also done some of the preliminary calculations for the Reid potential. 6. G. Brown & A.D. Jackson, The Nucleon-Nucleon Interaction(New York: Elsevier, 1976), P.228 (Note: there are errors in this compilation); Dr. A. Miller, Private Communication; I. Ulehla, "A Solution of the Reaction Matrix in Nuclear Matter," Nucl. Phys. A181, 270-272 (1972); M. Haftel & F. Tabakin, "Nuclear Saturation and the Smoothness of Nucleon-Nucleon Potentials," Nucl. Phys. A158, 40-42 (1970); H. Bateman, The  Bateman Manuscript Project(New York: McGraw-Hill, 1953), Vol. I. APPENDIX C The E l a s t i c Observables Having constructed the N-N T-matrices i n Chapter 3 , we may c a l c u l a t e e l a s t i c as well as ppy observables. As has been mentioned previously, the c a l c u l a t i o n of e l a s t i c observables e f f e c t i v e l y checks both the on-shell and the o f f - s h e l l N-N T-matrices. In th i s appendix, we have indicated how e l a s t i c observables are calculated and how they are rela t e d to the N-N scattering phase s h i f t s . The standard phase-shift analysis for e l a s t i c scattering of protons i s well known. The works of Goldberger and Watson1 and Wolfenstein 2 are useful references, as i s the often quoted paper of Stapp, Y p s i l a n t i s and M e t r o p o l i s 3 . These sources w i l l be followed throughout this s e c t i o n . We w i l l f i r s t define the scattering amplitude through the asymptotic relation 1* lim ¥pC+> (r) = <K£- l)U S M + [ 2 n ] - 3 / 2 {exp(ip. r ) / r } . r-*oo 1 •E < SM | f(£,£; ) | SM > u S M C . l iy<+) Was defined i n Chapter 2. u g M i s the eigenfunction of t o t a l spin S and projection M, <Kp.-) i s a plane wave state and, < SM | f ( r , p . ) | SM > i s the scattering amplitude - a matrix i n spin space. The sca t t e r i n g amplitude can be written i n a more revealing form, as f ( r , £ l ) = f ( r . £ i , P_L) C.2 103 and for f i n a l asymptotic states, _ •*• , so that f<£'iL» P;> * f < E / £ L » P i ) C' 3 We can obtain the N-N T-matrix from the scattering amplitude using 6 < S'M' | f(£,£ L) | S M > = - ( 2 T T ) 2 p [ d E / d p t ] _ 1 < S'M' | t(p.) | S M > C.4 In equation 3.4, E i s the energy of the 2-nucleon system and, we have p^[dE(p^)/dp^] - 1 = ( mr ( n o n - r e l a t i v i s t i c ) C.5 I E(p ) = [p_ L 2 + m p 2] 2 ( r e l a t i v i s t i c ) We have used the r e l a t i v i s t i c expression i n r e l a t i o n C.5. Drechsel and Maximon6 have used the n o n - r e l a t i v i s t i c expression wherein, m^  i s reduced mass of the two proton system. We may construct our scattering amplitudes from e i t h e r the basic phase s h i f t s or the N-N T-matrices. This allows us to compare both the observables and the phase s h i f t s with experimental values. Phase s h i f t s can be obtained d i r e c t l y from the N-N T-matrices using the following r e l a t i o n s ^ . (uncoupled waves) S J P 2 ) = ll2 t a n " 1 ! -2Trmrp Re( t * ( p 2 ) ) [1 + 2Trmfp Im( t * ( p 2 ) ) ] _ 1 } C.6 (coupled waves) SL ( P 2 ) = l/2 tan"M " 2 ™ r p Re( t j t | , J M ( P 2 ) )• .[1 + 2Trmrp Im( ^ ( p 2 ) } C.7 (mixing parameter) E j = l / 2 sin"l{ - 2 ™ r p Re( t j . ( , J +, (p 2) )[cos( 6*, + 6 ^ )]"1 } C.8 Here e is the mixing parameter in Stapp's8 S-matrix S = )] isln(2e)exp[i(6^ + 6*+1 )] cos(2e)exp(2ifij + 1 ) C.9 The phase-shifts listed in equations C.6 and C.7 correspond to Stapp's nuclear 'bar' phases and, do not include coulomb contributions. We use Stapp's3 relations for the construction of scattering amplitudes from phase-shifts. Stapp writes his scattering amplitudes as 'M-amplitudes' with the following correspondence. wherein 0 and <|> are the CM scattering angles. Relations for the various M-amplitudes, in terms of our phase shifts, are given in Table III of Stapp et a l . As was discribed in Chapter 3, we include the coulomb interaction via Stapp's method. The coulomb scattering amplitude is added to the N-N scattering amplitude with an additional phase being added to the partial wave N-N scattering amplitudes to account for interference. Using equation C.4 we produce the on-shell N-N T-matrices. These can then be taken off-shell by the 'half-off-shell' function described in < S=0,M=0 | f | S=0,M=0 > = M5S (0,<|>) < S=1,M | f | S-l.M > = M_ (9,4>) C I O 105 Chapter 3. In order to test our calculation, we have taken the half-off-shell N-N T-matrices calculated from a potential and evaluated them in the on-shell limit, using the result to produce the elastic cross section. For spinless particles, the elastic cross section is simply 9 I 0 = do/dft = | f(r,£ ) p C U By adding spin to equation 3.11, we obtain h = % Vn 1 fP C*12 - l U { |Mgs P + |M0Q p } + l / 2 { |M1Q p + |MQ1 P + l M n P + l Mi-i F 1 c-13 using the symmetry relations I Mil 1= I I » IM1"1 I" |M-ll I. I M01 1= I M 0 ~ l I > I Mio I = I M-io I • . c-14 The polarization, P, and triple scattering correlation coefficients A, R, D, C , C, are also constructed by Stapp and, are discussed by nn kp Wolfenstein 2. These coefficients are useful as, they test relations between the individual M-amplitudes. Explicit relations for these coefficients are given in Table II of reference 3. References 1. M.L. Goldberber & K.M. Watson, Collision Theory(New York: J. Wiley, 1964). 2. L. Wolfenstein, "Polarization of Fast Nucleons," Ann. Rev. Nucl. Sci. £, 43 (1956). 3. H.P. Stapp, T.J. Ypsilantis & N. Metropolis, "Phase-shift Analysis of 310-MeV Proton-Proton Scattering Experiments," Phys. Rev. 105, 302 (1957). 4. Goldberger & Watson, p.353. 5. Goldberger & Watson, p.236. 6. D. Drechsel & L.C. Maximon, "Potential Model Calculation for Coplanar and Non-coplanar Proton-Proton Bremsstrahlung," Ann. Phys. 49, 403-444 (1968). 7. T.R. Mongan, "Separable Potential Models of the Nucleon-Nucleon Interaction," Phys. Rev. 178, 1597 (1969). 8. Stapp et a l , p.309. 9. Goldberger & Watson, p.237. 10. Stapp et a l , p.305, equation 3.11. APPENDIX D The Axis of Quantization We have chosen an axis of quantization along the beam d i r e c t i o n as t h i s choice s i m p l i f i e s the c a l c u l a t i o n of p o l a r i z a t i o n q u a n t i t i e s . This d i f f e r s from the choice made by Drechsel and Maximon 1. The photon momentum was used to define the axis of quantization i n t h e i r c a l c u l a t i o n . Their choice s i m p l i f i e s 2 the c a l c u l a t i o n of the electromagnetic matrix elements but complicates the construction of p o l a r i z a t i o n q u a n t i t i e s . Before giving the r e l a t i o n for rotated N-N T-matrices, we w i l l f i r s t review the theory which i s b a s i c a l l y given by Rose 3. We consider the ef f e c t of ro t a t i n g a coordinate system by Euler angles a,B,y. A 3-vector V_ i n the unrotated system S transforms to V' i n the new system S' , v i a V = M • V , M: a 3x3 matrix. D.l where M = cosacosBcosy-sinasiny sinacosBcosy+cosasiny -sinBcosy D.2 -cosacosBsiny-sinacosy -sinacosBsiny+cosacosy sinBsiny cosasinB sinasinB cosB Now change to a spher i c a l basis with a unitary transformation. TJ.V = U«M'U _ 1'(U«V) D.3 The spherical equivalents of V, V' and M are given by V . = U • V , V' = U • V* and M„ . = U • M • U - 1 _ sph — ' — spW — sph D.4 where u = -1 - i 0 D.5 2-1/2 0 0 2 1 - i 0 and the r o t a t i o n matrix, i s just the transpose o f M sph D = e i ( a + Y ) [ l + c o s 3 ] 2" 1/2 e - i Y s i n 3 2-1/2 e i ( a - Y ) [ l - c o s g ] 2-1/2 -e" i a [ s i n 3 ] 2-1/2 cosg e i a s i n 3 2-1/2 e" i(a-y)r-i oi ''[1-cosgJ 2~ 1/2 - e a s i n g 2-1/2 e i ( a + Y ) [ l + c o s B ] 2~l/2 V . . L . has components: V • spK '1 V0 D.6 D.7 In the following, we use the subscripts S' and S to l a b e l the associated Cartesian coordinate system. Dirac bras and kets are used to represent spin states and i n Rose's notation, ., denotes the mm' r o t a t i o n matrix. The superscript (1) indicates that the matrix rotates spin objects. We have I 1 m > = E ,D^n , | l m > D.8 Inverting equation D.8 gives 1 m > = E [ D t l 5 ] J . | 1 m'> D.9 m' m m ' S Using the above r e l a t i o n , we can write the N-N T-matrices, represented i n a rotated basis, as follows. < 1 m I t I 1 m > = E D a ) , < 1 m' I t I 1 m"> [ D c l ) J1" .. D.10 ' ' YA'W\" mm' 1 1 1 J mm This i s the form given by Maximon and Drechsel 1*. Now choose the coordinate systems S and S'. Let S be the f i x e d lab coordinate system with a z-axis along the incident beam d i r e c t i o n and, with basis vectors {x, y, z}. Let S' be the coordinate system along which the i n d i v i d u a l N-N T-matrices ( <q^ | t | q^ > ) have been quantized 5. 2 = » y = q ± x q f » x = y x z ; where a = aj | a I In order to obtain oc,B,Y> we use equations D.l and D.2. We f i n d the following r e l a t i o n s . D.H cosg = z • z* D.12 sincx = y • z'/sin3 cosa = x • z'/sing siny = z • y'/sinB cosy = -z • x /sing Note that the above relation for cosy differs, by a sign, from that given by Drechsel and Maximon's equation 5.15. Obviously, the above relations f a i l i f 3 = 0. In this special case, only one Euler angle i s necessary to determine the rotation. For 3 = 0, we have 3=y=0 , x' • x = cosa D.13 These transformations allow us to calculate the N-N T-matrices in the CM system, quantized along the i n i t i a l CM momentum, (which differs for the various N-N T-matrices required for the ppy amplitude) and then express a l l such N-N T-matrices in terms of a fixed quantization axis. I l l References 1. D. Drechsel & L.C. Maximon, "Potential Model Calculation for Coplanar and Noncoplanar Proton-Proton Bremsstrahlung," Ann. Phys. 49, 403 (1968). 2. Drechsel & Maximon, see p.420. 3. M.E. Rose, Elementary Theory of Angular Momentum(New York: J. Wiley, 1957), Chapter 4. 4. Drechsel & Maximon, p.425. 5. See Drechsel & Maximon, p.424; H.P. Stapp, T.J. Ypsilantis & N. Metropolis, "Phase-Shift Analysis of 310-MeV Proton-Proton Scattering Experiments," Phys. Rev. 105, 302-310 (1957). In < q ± | T | q >, q^ and q^ represent the i n i t i a l and fin a l relative momenta of the two nucleon system. APPENDIX E The Legendre Functions Legendre functions of the second kind( Q (z) ) are generated in both the Paris and Reid function subprograms. Since there are numerical problems associated with these functions, this short appendix has been written. Here we discuss some properties of the Legendre functions and, their treatment in this calculation. Fi r s t recall the series expansion for (^(x) 1. (^(x) = [ r ( v + i ) r ( i / 2 ) ] [ r ( v + 3/2)'2v+1]~1» (i/x) v+i • • F((v+2)/2, (V+1)/2; V + 3/2; 1/x2) E.l where F(a,b;c;l/x 2) is a hypergeometric function defined by2 F(a,b;c;l/x 2) = 1 + [<ab)/(c»1)](1/x2) + [a(a+l)b(b+l)/(c(c+l).l»2)](l/xl+)+... E.2 Now consider the ratio, Q L+ 1(x)/Q L(x) for large x and L. We have Q L + 1(x)/Q L(x) = [L+1][2L+3]"1 x"l [F([L+3]12, [L+2]/2; L + 5/2; x" 2)/ F([L+2]/2, [L+l]/2; L + 3/2; x~ 2)] E.3 For large x, we can approximate the ratio of hypergeometric functions using (1 + e)(l + 6 ) " 1 « 1 + e - 5 ; e,6 « 1 E.4 113 Thus we f i n d , for large x and L, Q L(x)/Q L_ 1(x) * LI2L+1]- 1 x-1 { 1 + [ 4 x 2 ] " l } E.5 A better approximation to this r a t i o , which retains the next expansion term, i s given for a l l L by Q T(x)/Q T_ 1(x) = M2L+1]- 1 x" 1 { 1 + [L+l] 2[(2L+l)(2L+3)]" 1 x" 2 + [L+1] 2[(2L+1)(2L+3)(2L+5)]~ 1 x"1* } E.6 Now, i n order to produce Q (x), we can proceed i n one of two ways. We L i could use the recurrence r e l a t i o n for the Q (x) functions 3 Lt [2L+1] x Q L ( x ) = [L+l] Q L + 1 ( x ) + Q ^ C x ) Q Q ( x ) = 0.5 ln{[l+x]/[l-x]} E.7 Q ^ x ) = 0.5 x ln{ [l+x]/[l-x]} - 1 i n order to recur up from Q Q and , to any desired L-value. For large values of x, however, i t i s more accurate to use equation E.6 and, to recur down from some large L-value. The method involves choosing a value for Q L0 and using equation E.6 i n order to produce Q . Equation E.7 i s then L0 A V used to recur down from L Q to 0. The values are then normalized by the r a t i o Q Q / Q Q* This normalization procedure i s v a l i d as, the recursion r e l a t i o n i n E.7 i s l i n e a r i n Q . Lt A numerical problem i s associated with these methods. For example, i f the i n i t i a l Q T i s not chosen c a r e f u l l y , the recursion process w i l l produce L0 unnormalized Q values which exceed the r e a l number l i m i t i n FORTRAN. This JL 1 1 4 problem occurs for very large values of x because the recursion r e l a t i o n E.7 i s l i n e a r i n x so that Q i s of order x»Q . I t can be handled by Li 1 ii a judicious choice of Q or by m u l t i p l i c a t i o n of the Q values by some LQ L factor R < 1 at intermediate steps i n the recursion procedure. Currently, we multiply the i n i t i a l values of Q and Q by a very small number i n L 0 L 0 order to stop the unnormalized Legendre functions from overflowing. References 1. I. Gradshteyn & I. Ryzhik, Table of Integrals, Series, and Products (Toronto: Academic Press, 1980), p.999 2. Gradshteyn & Ryzhik, p.1039 3. Gradshteyn & Ryzhik, p.1018-1019 

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