NATURAL BOUNDARY CONDITION METHODS FOR NUCLEAR REACTIONS by Syed Saeed Ahmad A thesis submitted for the degree of Doctor of Philosophy at the Australian National University, Canberra, December 1976 R.L.L. (i) STATEMENT The author has collaborated with his supervisor, Dr B.A. Robson, in developing the IRM (Iterative i?-matrix) method which is being reported with the BD (Barrett and Delsanto) method in Chapter 3 of this thesis. The two 12 12 methods were applied to the C(n, n) C reaction (Chapter 4) and the coupled square well problem (Chapter 5) in close collaboration with Dr R.F. ) 1 2 Barrett and Dr B.A. Robson and have been reported in two publications 5 12 12 The author has independently applied the IRM method to the C(p, p) C reaction (Chapter 6) and some of the preliminary results were presented at 3) the last AINSE conference No part of this thesis has been presented for a degree at any other university. 1) S.S. Ahmad, R.F. Barrett and B.A. Robson, "Natural boundary condition methods for nuclear reactions", Pud. Phys. A257 (1976) 378. 2) S.S. Ahmad, R.F. Barrett and B.A. Robson, "Natural boundary condition methods for nuclear reactions II", Pud. Phys. A270 (1976) 1. 3) S.S. Ahmad and B.A. Robson, "Application of an I?-matrix method with 12 12 natural boundary conditions to C(p, p’) C ", unpublished, presented at Sixth AINSE Conference, University of Melbourne, 1976. (ii) ACKNOWLEDGEMENTS It is a pleasure to thank iny supervisor, Dr Brian A. Robson, for his continuous encouragement, patient guidance and valuable assistance in computation throughout the development of the present work. I am highly indebted to him for a critical reading of the manuscript and several useful suggestions and discussions which have greatly influenced the overall presentation of this thesis. I would like to express my thanks to Dr Ross F. Barrett who has clarified several concepts related to the eigenchannel and BD methods. Many thought provoking discussions with him were the means of visualizing the actual physical content in connection with several problems. Dr Fred C. Barker has always been very kind to explain everything:. either any concept in nuclear structure and reaction theories or any problem related to the computation. I am grateful to him for numerous fruitful discussions, several right answers at the right time and a critical reading of the manuscript. I would like to take this opportunity to express my gratitude to Professor K.J. Le Couteur for his keen interest in my studies leading to the degree of Ph.D. I would also like to thank him for some useful discussions which have highlighted the proper distinction between the structural and dynamical theories of nuclear reactions together with the main difficulties in extending them to the atomic- and molecular-reaction problems. Several discussions with the staff members of the department and visitors are acknowledged to have influenced different aspects of the formulation of the present work. In particular, I would like to thank Drs R.J. Baxter, Kailash Kumar, K.S.J. Nordholm, P.C. Tandy, L.J. Tassie, T. Terasawa, P.B. Treacy, W.S. Woolcock and Professor D.C. Peaslee for many helpful discussions. (iii) It is a pleasure to thank Professors J.O. Newton and Sir E.W. Titterton for the hospitality at the department of Nuclear Physics (R.S. Phys.Sci.), where I have enjoyed free discussions with many speakers on several interesting problems in nuclear physics; either during the seminars or at the "Nuclear Tea". Especially, I benefited from the series of lectures by Dr F.C. Barker, Dr A.M. Lane, Professor J. Cerny and Professor S.S. Hanna. I am pleased to thank Mrs L. Nicholson for her kind assistance and advice which have been very useful throughout the preparation of the thesis. The necessary computation for the present work was mainly carried out at the PDP-10 computer (R.S.Phys.Sei.). Thanks are due to Professor K.J. Le Couteur, for allowing me to lavishly use the precious computing time through the terminals in the department of Theoretical Physics, and to Mr Roger Brown and Miss Sussan Murray for several useful comments and necessary technical assistance. A part of the calculations was also done at the UNIVAC 1100/42 (A.N.U. Computer Centre) where I have enjoyed discussions with Mr Ian Simpson and Mr Leslie Landau. It is a pleasure to thank Mrs B.M. Geary for typing the manuscript and using the best of her skills in appropriately arranging the complicated formulae, lengthy footnotes and diagrams. Last but not the least are the sincere efforts of my parents for inducing in me the desire for learning and I am still unable to find suitable words to thank them appropriately. (iv) ABSTRACT The BD (Barrett and Delsanto) and IRM (Iterative R-matrix) methods for calculating cross sections are discussed. Both methods are characterized by their use of natural boundary conditions at the surfaces separating internal and external regions of configuration space and the employment of energy-dependent basis states. An energy correction which greatly improves the rate of convergence of the BD method is also given. The methods are compared with both standard and generalized A-matrix calculations with energy-independent basis states for the reaction 12 12 C(n, n) C at incident energies below the inelastic threshold using a weak vibrational model. The convergence of the natural boundary condition methods was found to be substantially better than for the other cases. Moreover, the methods are used to calculate both the elastic and inelastic scattering cross sections for an exactly soluble model comprising two square well potentials coupled by a square well interaction. The methods are investigated for weak, intermediate and strong coupling interactions and the results are compared where possible with those of other related methods. It is concluded that for the practical calculation of reaction cross sections from a basic physical model, the natural boundary condition methods offer the most tractable approach particularly for problems involving strong channel coupling. 12 12 The IRM method is also applied to the C(p, p) C reaction below 8 MeV using a collective rotational model for the target nucleus. The predictions of the method are again substantially better than the standard Z?-matrix method and are in good agreement with the equivalent coupled- channels calculations. (v) CONTENTS STATEMENT......................................................... (i) ACKNOWLEDGEMENTS ................................................. (ii) ABSTRACT................................................. .. .. (iv) CONTENTS......................................................... (v) CHAPTER 1 INTRODUCTION ......................................... 1 CHAPTER 2 REVIEW OF i?-MATRIX TYPE METHODS ..................... 12 2.1 Introduction..................................... 12 2.2 Ä-matrix theory ................................. 15 2.3 Standard i?-matrix method......................... 24 2.4 Generalized R-matrix method ..................... 31 2.5 Corrections to the SRM method..................... 40 2.6 Summary and discussion ......................... 48 CHAPTER 3 NATURAL BOUNDARY CONDITION METHODS ...................... 51 3.1 Introduction..................................... 51 3.2 The natural boundary conditions ................. 52 3.3 Barrett and Delsanto method ..................... 55 3.4 Iterative i?-matrix method ..................... 59 3.4.1 One open channel ......................... 59 3.4.2 Several open channels..................... 61 3.5 Energy correction to the BD method ............. 64 3.6 Discussion ..................................... 68 CHAPTER 4 APPLICATION TO 12C(n, n)12C REACTION .................... 71 4.1 Introduction..................................... 71 4.2 The nuclear model................................. 73 4.3 Calculations and results......................... 75 4.4 Discussion and conclusion......................... 83 CHAPTER 5 APPLICATION TO AN ANALYTICALLY SOLUBLE MODEL ......... 85 5.1 Introduction..................................... 85 5.2 Analytically soluble model ..................... 85 5.3 Calculations and results ......................... 87 5.3.1 Intermediate coupling ..................... 87 5.3.2 Strong coupling ......................... 89 5.3.3 Weak coupling............................. 94 5.4 Discussion and conclusion......................... 96 CHAPTER 6 APPLICATION TO 12C(p, p)12C REACTION ................. 99 6.1 Introduction..................................... 99 6.2 The IRM method formulation ..................... 10 5 6.3 The nuclear model................................. 109 6.4 Calculations and results......................... 115 6.5 Discussion and conclusion......................... 127 CHAPTER 7 CONCLUSION ............................................ 131 APPENDICES ..................................................... 138 Appendix (2.2A1) 138 Appendix (2.2A2) 139 Appendix (2.2A3) 140 Appendix (2.5A) 142 Appendix (3.3A) 144 Appendix (3.4A) 147 Appendix (4.2A) 148 Appendix (5.2A) 150 Appendix (5.3A) 157 Appendix (6.2A) 160 Appendix (6.3A1) 163 Appendix (6.3A2) 165 REFERENCES 167 1 CHAPTER 1 INTRODUCTION The various stages in the development of the theory of nuclear reactions incorporating resonance phenomena seem to resemble very much the following pattern which appears to be followed by most physical theories that have taken a considerable period to develop: (i) Evolution of a "hypothesis" from the available experimental and related theoretical evidence. (ii) Translation of the hypothesis into a rigorous "mathematical # language". (iii) Identification of the "transient" and "inherent" character­ istics of the physical formulation. (iv) Reorganization of the inherent characteristics in the form of a "theory" which enables one to describe the physical phenomenon in a logically coherent and mathematically consistent way. (v) Deduction of the basic physical content of the theory from very general principles by other alternative "methods". (vi) Search for the "best" amongst all the available methods so that the physical phenomenon can be explained in the framework of the most rational theoretical approach. The occurrence of sharp resonances in the total scattering cross section was pointed out about forty years ago in connection with the systematic eozperimental studies of a variety of low-energy nuclear 1-9) reactions . Since then a tremendous amount of accurate experimental data on low- and intermediate-energy resonances has been accumulated. Throughout this period the theorists’ interests were to understand the underlying resonance mechanism and to incorporate it in the framework of a sound and 2 coherently formulated reaction theory. In analogy with the "compound molecule" approach employed for the interpretation of a few chemical reactions’^ \ the first successful attempt on these grounds was the "compound, nucleus" hypothesis of Bohr, Breit and Wigner^ . Accordingly, a reaction process may be considered as being the succession of two causal events. The first is the union of the projectile and target particles to form a relatively long-lived compound nucleus having well defined virtual energy levels. The second event is the disintegration of this compound system into final product. Thus the resonances observed in the total reaction cross sections can be described in terms of a rapid sharing of the projectile’s (kinetic) energy among all the target constituents. The importance of this hypothesis lies in the fact that it is still regarded as a basic postulate of many reaction theories. The compound nucleus hypothesis was simultaneously translated into a 13) reasonable mathematical language i.e. the Breit-Wigner formula for calculating the resonance cross sections proceeding via an intermediate state. Based upon a development analogous to the time-dependent perturbation theory employed for the emission and absorption of optical radiations, the formula gave satisfactory fits to the isolated resonances observed in IG IV) several nuclear reactions 5 . However, apart from its success, the under­ lying assumptions were open to criticism and have been widely discussed in P0—24) the literature . It is perhaps worth emphasizing that by taking into account both the failures and the successes of the Breit-Wigner formalism, 25-29) theorists were able to identify the inherent as well as the general characteristics necessary for the foundation of a reasonable reaction theory. The reorganization of these initial efforts in understanding the mechanism of nuclear resonance reactions led eventually to the development of two well defined theories of nuclear reactions: the Kapur-Peierls