A Unified Theory of the Nucleus

ISBN-13: 9783528083731 / Niemiecki / Miękka / 1977 / 389 str.

Karl Wildermuth

Opis

The purpose of this monograph is to describe a microscopic nuclear theory which can be used to consider all low-energy nuclear phenomena from a unified viewpoint. In this theory, the Pauli principle is completely taken into account and translation ally invariant wave functions are always employed. Also, this theory is quite flexible; it can be utilized to study reactions initiated not only by nucleons but also by arbitrary composite particles. Throughout this monograph, we have endeavoured to keep the underlying physical ideas as easily comprehensible as possible. Consequently, it becomes frequently necessary to sacrifice mathematical rigour in favour of clarity in presenting these ideas. In this way, it is our hope that this monograph could be useful to many research physicists in the nuclear field, experimentalists and theorists alike. In chapters 1 through 4, the formulation of this theory is presented. Numerical examples concerning bound-state, scattering, and reaction calculations are mainly described in chapters 5 through 7. In chapters 8 through IS we discuss, within the framework of this theory, general properties of nuclear systems. Finally, in chapters 16 and 17, we show in specific cases how one can achieve, without carrying out explicit calculations, a qualitative or even semi-quantitative understanding of these cases by applying the general physical concepts contained inherently in this theory.

Spis treści

1. Introduction.- 1.1. General Remarks.- 1.2. Difficulties of Some Reaction Theories.- 2. Reformulation of the Schrödinger Equation.- 3. Discussion of the Basis Wave Functions for Nuclear Systems.- 3.1. General Remarks.- 3.2. Qualitative Discussion of Cluster Correlations.- 3.3. Construction of Oscillator Cluster Wave Functions.- 3.4. Discussion of 8Be as an Illustrative Example.- 3.5. Effects of Antisymmetrization.- 3.5a. Fermions without Mutual Interaction in a Square-Well Potential.- 3.5b. The Lowest 4+ ?-Cluster State of 8Be.- 3.5c. Mathematical Equivalence of the Lowest 6Li States Described in the t + 3He and the d + ? Oscillator Cluster Representations.- 3.5d. Summary.- 3.6. Applications of Oscillator Cluster Representations to a Qualitative Description of Low-Lying Levels in Light Nuclei.- 3.6a. 7Liand7Be.- 3.6b. 6He, 6Li, and 6Be.- 3.6c. ?-Cluster States of 16O.- 3.6d. Brief Remarks.- 3.7. Construction of Generalized Cluster Wave Functions.- 3.7a. Introduction of Jacobi Coordinates.- 3.7b. Introduction of Parameter Coordinates.- 3.7c. Introduction of Jastrow Factors.- 3.7d. Summary.- 4. Formulation of a Unified Microscopic Nuclear Structure and Reaction Theory.- 4.1. General Remarks.- 4.2. Specific Examples.- 4.2a. n + ? Scattering.- 4.2b. d + ? Scattering.- 4.2c. Discussion.- 4.3. Extension to General Systems.- 5. Bound-State Calculations.- 5.1. General Remarks.- 5.2. Calculation of Matrix Elements.- 5.2a. Evaluation of Matrix Elements by the Cluster-Coordinate Technique — Example of 8Be.- 5.2b. Evaluation of Matrix Elements by the Generator-Coordinate Technique — Example of 8Be.- 5.3. Ground and Low Excited States of 6Li.- 5.3a. Introduction.- 5.3b. Calculation With a Nucleon-Nucleon Potential Containing a Hard Core — Accurate Treatment of the Jastrow Factor.- 5.3c. Calculation with a Nucleon-Nucleon Potential Containing a Soft Core — Approximate Treatment of the Jastrow Factor.- 5.3d. Calculation with a Nucleon-Nucleon Potential Containing no Repulsive Core.- 5.3e. Summary.- 5.4. Low-Energy T = 0 States of 12C.- 5.5. Low-Lying Levels of 7Be.- 5.6. Concluding Remarks.- 6. Further Comments About the Pauli Principle.- 6.1. General Remarks.- 6.2. Cluster Overlapping and Pauli Principle.- 6.3. Energetical Favouring of a Cluster Inside a Large Nucleus.- 7. Scattering and Reaction Calculations.- 7.1. General Remarks.- 7.2. Derivation of Coupled Equations.- 7.2a. Single-Channel Problem.- 7.2b. Coupled-Channel Problem.- 7.2c. Reaction Calculations Using Hulthén-Kohn-Type Variational Functions.- 7.3. Quantitative Results.- 7.3a. 3He + ? Elastic Scattering.- 7.3b. l = 0 Phase-Shift in ? + ? scattering.- 7.3c. Specific Distortion Effects in d + ? Scattering.- 7.3d. Effect of Reaction Channels on 3He + 3He Scattering Cross Sections.- 7.3e. ? + 16O Scattering — Utilization of the Generator-Coordinate Technique.- 7.3f. p + ? Scattering Around 3/2+ Resonance Level in 5Li.- 7.3g. ? + ? Scattering with Specific Distortion Effect and a Nucleon-Nucleon Potential Containing a Repulsive Core.- 7.3h. p + 3He and n + t Scattering Calculations.- 7.3i. Coupled-Channel Study of t(p, n) 3He Reaction.- 7.4. Concluding Remarks.- 8. Introductory Considerations About the Derivation of General Nuclear Properties.- 8.1. General Remarks.- 8.2. Introduction of Effective Hamiltonians.- 8.3. Elimination of Linear Dependencies.- 8.4. Concluding Remarks.- 9. Breit-Wigner Resonance Formulae.- 9.1. General Remarks.- 9.2. Single-Level Resonance Formula for Pure Elastic-Scattering.- 9.2a. Derivation of the Resonance Formula.- 9.2b. Discussion of the Resonance Formula.- 9.2c. Existence of Sharp Resonances.- 9.2d. Discussion of a Simple Resonance Model.- 9.3. Many-Level Resonance Formula for Pure Elastic-Scattering.- 9.4. Single-Level Resonance Formula Including Inelastic and Rearrangement Processes.- 9.4a. Derivation of the Resonance Formula.- 9.4b. Application of the Resonance Formula to a Specific Example Involving Two Open Channels.- 9.5. Mutual Influence of Resonance Levels in Inelastic and Rearrangement Processes.- 9.5a. Derivation of a Two-Level Breit-Wigner Formula.- 9.5b. A Specific Example.- 9.6. Behaviour of the Partial Level Width Near a Threshold and Energy-Dependent Width Approximation.- 10. Resonance Reactions and Isobaric-Spin Mixing.- 10.1. General Remarks.- 10.2. Isobaric-Spin Mixing in the Compound Region.- 10.2a. Derivation of a Two-Level Resonance Formula.- 10.2b. The 16.62 and 16.92 MeV States in 8Be as a Specific Example.- 10.3. Isobaric-Spin Mixing in the Incoming Channel.- 10.3a. Qualitative Description.- 10.3b. Quantitative Formulation in the Case of a Single Open Channel.- 10.3c. Brief Discussion in the Case of Many Open Channels.- 11. Optical-Model Potentials for Composite Particles.- 11.1. General Remarks.- 11.2. Optical-Model Description of Elastic-Scattering Processes.- 11.2a. Preliminary Remarks About the Optical-Model Potential.- 11.2b. Optical-Model Potential for Pure Elastic Scattering.- 11.2c. Optical-Model Potential in the Presence of Reaction Channels.- 11.2d. Mean Free Path of a Cluster in a Target Nucleus.- 11.3. Specific Examples.- 11.3a. 3He + ? Scattering.- 11.3b. p + 16O Scattering.- 11.3c. ? +16O Scattering.- 11.4. Features of Effective Local Potentials between Nuclei.- 11.4a. Wave-Function Equivalent Local Potentials.- 11.4b. Phase-Equivalent Local Potentials.- 12. Direct Reactions.- 12.1. General Remarks.- 12.2. Derivation of the General Formulae.- 12.3. Specific Examples.- 12.3a. 3He(d, p) ? Reaction.- 12.3b. 6Li(p, 3He) ? Reaction.- 12.4. Influence of the Pauli Principle on Direct-Reactions.- 12.4a. Study of Direct-Reaction Mechanisms in the Plane-Wave Born Approximation.- 12.4b. Study of Direct-Reaction Mechanisms with the Coupled-Channel Formulation.- 12.5. Concluding Remarks.- 13. Some Considerations About Heavy-Ion Transfer Reactions.- 13.1. General Remarks.- 13.2. Specific Examples to Study the Influence of Antisymmetrization.- 13.2a. ? + 6Li Elastic Scattering at Low Energies.- 13.2b. 6Li (p, 3He) ? Reaction in States of Large Orbital Angular Momentum.- 13.3. Further Discussion of the Odd-Even Feature in the Effective Potential between Nuclei.- 13.4. Concluding Remarks.- 14. Collective States.- 14.1. General Remarks.- 14.2. Rotational States of Even-Even Nuclei with K = 0.- 14.3. Generalization of Rotational Wave Functions.- 14.4. Energetical Preference of Rotational Configurations.- 14.5. Electromagnetic Transitions between Rotational Levels.- 14.6. Relationship with other Descriptions of Nuclear Rotational States.- 14.7. Construction of Intrinsic Wave Functions for Quantitative Studies of Collective States in Medium-Heavy and Heavy Nuclei.- 14.8. Specific Examples.- 14.8a. ? + 16O Cluster States in 20Ne.- 14.8b. Rotational States in 22Ne.- 14.8c. Backbending.- 14.9. Concluding Remarks.- 15. Brief Discussion of Time-Dependent Problems.- 15.1. General Remarks.- 15.2. Connection between the Lifetime of a Compound State and Its Level Width.- 15.2a. Relationship between Phase Shift and Time Delay.- 15.2b. Quantitative Relationship between the Level Width and the Lifetime of a Compound Nuclear State.- 15.2c. Calculation of the Level Width — 6Li as an Example for a Decaying System.- 15.3. Time-Dependent Projection Equation with Time-Dependent Interaction.- 16. Qualitative Considerations of Some Nuclear Problems.- 16.1. General Remarks.- 16.2. Coulomb-Energy Effects in Mirror Levels.- 16.3. Reduced Widths and ?-Transition Probabilities.- 16.3a. Reduced Widths of Nuclear Levels.- 16.3b. ? -Transition Probabilities.- 16.4. Level Spectra of Neighbouring Nuclei.- 16.5. Optical Resonances in Nuclear Reactions.- 16.5a. Optical Resonances in the Incoming Channel.- 16.5b. Optical Resonances in Reaction Channels.- 17. Nuclear Fission.- 17.1. General Remarks.- 17.2. Substructure Effects in Fission Processes.- 17.3. Mass Distribution of Fission Fragments.- 17.4. Deformation Energy of Fissioning Nucleus.- 17.4a. Dynamical Consideration of the Fission Process.- 17.4b. Calculation of the Deformation Energy — Strutinsky Prescription.- 17.4c. Calculation of the Deformation Energy-Cluster Prescription.- 17.4d. Discussion.- 18. Conclusion.- Appendix A — Cluster Hamiltonians and Jacobi Coordinates.- Appendix B — Designation of Oscillator States.- Appendix C — Demonstration of the Projection Technique.- Appendix D — Connection with Conventional Direct-Reaction Theory.- References.