✍ Scribed by Nicolas Michel, Marek Płoszajczak
No coin nor oath required. For personal study only.
This book provides the first graduate-level, self-contained introduction to recent developments that lead to the formulation of the configuration-interaction approach for open quantum systems, the Gamow shell model, which provides a unitary description of quantum many-body system in different regimes of binding, and enables the unification in the description of nuclear structure and reactions. The Gamow shell model extends and generalizes the phenomenologically successful nuclear shell model to the domain of weakly-bound near-threshold states and resonances, offering a systematic tool to understand and categorize data on nuclear spectra, moments, collective excitations, particle and electromagnetic decays, clustering, elastic and inelastic scattering cross sections, and radiative capture cross sections of interest to astrophysics.
The approach is of interest beyond nuclear physics and based on general properties of quasi-stationary solutions of the Schrödinger equation – so-called Gamow states. For the benefit of graduate students and newcomers to the field, the quantum-mechanical fundamentals are introduced in some detail. The text also provides a historical overview of how the field has evolved from the early days of the nuclear shell model to recent experimental developments, in both nuclear physics and related fields, supporting the unified description. The text contains many worked examples and several numerical codes are introduced to allow the reader to test different aspects of the continuum shell model discussed in the book.
Preface
Contents
About the Authors
1 Introduction: From Bound States to the Continuum
References
2 The Discrete Spectrum and the Continuum
2.1 Definition of One-Body States
2.2 The Harmonic Oscillator Potential
2.3 Coulomb Potential and Coulomb Wave Functions
2.3.1 Continued Fractions
2.3.2 Analytic Continuation of Coulomb Wave Functions
2.3.3 Asymptotic Forms of Coulomb Wave Functions for Small and Large Arguments
2.4 Pöschl-Teller-Ginocchio Potential
2.4.1 Calculation of the Function y(r)
2.4.2 Different Terms of the Pöschl-Teller-GinocchioPotential
2.4.3 Wave Functions of the Pöschl-Teller-GinocchioPotential
2.4.3.1 Energies of Bound, Antibound, and Resonance States
2.4.4 Modified Pöschl-Teller-Ginocchio Potential
2.5 Basic Properties of Bound States
2.5.1 Number of Bound States of a One-Body Hamiltonian
2.5.2 Virial Theorem
2.6 Analytical Properties of the Wave Functions
2.6.1 Decomposition in Incoming and Outgoing Wave Functions
2.6.2 Asymptotic Behavior of Complex-Momentum Wave Functions
2.6.3 Analyticity of Complex Momentum Wave Functions
2.6.4 Jost Functions
2.6.5 Wave Functions in a Screened Potential
2.6.6 The Scattering Matrix
2.6.7 Radial Equation and Shooting Method
2.6.8 Calculation of the Width of a Metastable State
Solutions to Exercises
References
3 Berggren Basis and Completeness Relations
3.1 Normalization of Gamow Functions
3.2 One-Body Completeness Relation
3.2.1 One-Body Completeness Relation for = 0 Neutrons
3.2.2 Completeness Relation of Coulomb Wave Functions
3.2.3 Completeness Relation for the General Case
3.3 Normalization and Orthogonality of Gamow States and One-Body Matrix Elements
3.4 Matrix Elements Involving Scattering States
3.5 Completeness Relation Involving Single-Particle Gamow States
3.5.1 Domain of Applicability of the Berggren Completeness Relation
3.5.2 Berggren Completeness Relation for Complex Potentials
3.5.3 Overcompleteness Relations Involving Single-Particle Gamow States
3.6 Newton and Berggren Completeness Relations Generated by Nonlocal Potentials
3.7 Numerical Implementation of the Berggren Completeness Relation
3.7.1 Completeness Relations Involving Proton States
3.7.2 Completeness Relations Involving Antibound s1/2States
3.8 Complex-Symmetric Operators and Matrices
3.8.1 Diagonalizability of Complex-Symmetric Matrices
3.8.2 The Two-Dimensional Complex Symmetric Matrix
3.8.3 Numerical Studies of Complex-Symmetric Matrices
Solutions to Exercises
References
4 Two-Particle Systems in the Berggren Basis
4.1 Exact Formulation of the Two-Particle Problem Using Relative Coordinates
4.2 Study of Two-Nucleon Systems with Realistic Interactions
4.2.1 Numerical Studies of Dineutron, Diproton, and Deuteron in Berggren Basis
4.3 The Particle-Rotor Model in the Berggren Basis
4.3.1 Mathematical Formalism of the Particle-Rotor Model
4.3.1.1 Use of Pseudo-Potentials in Weakly Bound Anions
4.3.2 Dipolar Anions
4.3.3 Quadrupolar Anions
4.3.4 Weakly Bound and Unbound Atomic Nuclei
4.3.4.1 Deformed Proton Emitters: The Example of 141Ho
4.3.4.2 Rotational Bands in 11Be
4.3.4.3 Effects of Deformation and Configuration Mixing on E1 Electromagnetic Transition in 11Be
Solutions to Exercises
References
5 Formulation and Implementation of the Gamow Shell Model
5.1 Mathematical Foundation of the Gamow Shell Model
5.1.1 Rigged Hilbert Space Setting of the Gamow Shell Model
5.1.2 Complex Observables and Their Interpretation
5.1.3 Many-Body Berggren Completeness Relation
5.1.4 Complex Observables: The Generalized Variational Principle
5.2 Translationally Invariant Shell Model Scheme: The Cluster Orbital Shell Model
5.3 Truncation of the Many-Body Berggren Basis in the Gamow Shell Model
5.3.1 Natural Orbitals
5.4 Determination of Eigenvalues in Gamow Shell Model: The Overlap Method
5.5 Optimization of the Gamow Shell Model One-Body Basis
5.6 Calculation of Two-Body Matrix Elements in the Gamow Shell Model
5.7 The Jacobi-Davidson Method in the Gamow Shell Model
5.8 Memory Management and Two-Dimensional Partitioning
5.8.1 Slater Determinants and Partitions in Shell Model Codes
5.8.2 Memory Management of the Hamiltonian
5.8.3 2D Partitioning Parallelization Scheme of the Hamiltonian
5.8.4 Treatment of the Angular Momentum Projection
5.9 Diagonalization of Very Large Gamow Shell Model Matrices with the Density Matrix Renormalization Group Approach
5.10 Statistical Evaluation of Modeling Errors and Quality of Predictions in the Gamow Shell Model
5.10.1 Penalty Function and Linear Regression Approximation
5.10.2 Bayesian Inference of Parameters
Solutions to Exercises
References
6 Physical Applications of the Gamow Shell Model
6.1 Effective Interaction in the Vicinity of the Particle-Emission Threshold
6.1.1 Monitoring Branch Points and Avoided Crossings in the Complex Plane
6.1.2 Effect of Continuum Coupling on the Spin–Orbit Splitting
6.2 T=0,1 Nuclear Matrix Elements in the Berggren Basis
6.2.1 Dependence of Nuclear Matrix Elements on Angular Momentum and Continuum Coupling
6.2.2 Influence of Continuum Coupling on Effective Nuclear Matrix Elements
6.3 Effective Nucleon–Nucleon Interactions for Gamow Shell Model Calculations in Light Nuclei
6.3.1 Study of the Helium, Lithium, and Beryllium Isotope Chains
6.3.1.1 Energy Spectra
6.3.1.2 Mirror Symmetry Breaking
6.3.1.3 Pairing Correlations and Correlation Density in 6He and 6Li
6.4 Three-Body Model in Berggren Basis
6.4.1 Berggren Basis Expansion in the Three-Body Model
6.4.2 Convergence Properties of Eigenvalues Calculated in Jacobi and Cluster Orbital Shell Model Coordinates
6.4.3 Correlation Densities in a Three-Body Gamow Model
6.5 Inclusion of Continuum Couplings in the Pairing Model
6.5.1 Numerical Solution of the Generalized Richardson Equations
6.5.1.1 Physical Applications of the Generalized Richardson Equations
6.6 Comparison of Gamow Shell Model and Hamiltonian Complex Scaling Formalisms
6.6.1 Complex-Scaled Hamiltonians and Their Eigenstates
6.6.2 Basis Dependence in Truncated Model Spaces
6.6.2.1 Basis Optimization
6.6.2.2 Dependence of Observables on the Rotation Angle in Practical Calculations
6.6.3 Numerical Comparison of Eigenenergies of Gamow Shell Model and Hamiltonian Complex Scaling Method
Solutions to Exercises
References
7 Wave Functions in the Vicinity of Particle-Emission Threshold
7.1 Configuration Mixing in the Vicinity of the Dissociation Threshold
7.2 Halo Nuclei
7.2.1 Halo States in a Single-Particle Potential
7.2.1.1 Neutron Halo States in a Spherical Potential
7.2.1.2 Proton Halo States in a Spherical Potential
7.2.1.3 Neutron Halo States in Deformed Nuclei
7.2.2 Effects of Nucleon–Nucleon Correlations on Halo States
7.3 Asymptotic Normalization Coefficient in Mirror Nuclei
7.4 Near-Threshold Behavior of Wave Functions
7.4.1 Two-Particle Reaction Cross Sections and Wigner Cusps
7.4.2 Spectroscopic Factors and Radial Overlap Integrals
7.4.2.1 Spectroscopic Factors and Pairing Correlations in 11Li
7.5 Mirror Nuclei and Isospin Symmetry Breaking
Solutions to Exercises
References
8 No-Core Gamow Shell Model
8.1 Formulation of the No-Core Gamow Shell Model
8.2 Benchmarking of No-Core Gamow Shell Model Against Other Methods
8.2.1 Binding Energies and Center-of-Mass Excitations
8.2.2 5He Unbound Nucleus
8.2.3 Radial Overlap Integral and Asymptotic Normalization Coefficient
8.3 Multi-Neutron Resonances
8.3.1 Tetraneutron Energy from the Extrapolation of Overbinding Interactions
8.3.2 Trineutron and Tetraneutron in the Auxiliary Potential Method
8.4 Renormalization of Realistic Interactions in a Finite Model Space Using -Box Method
8.4.1 Application of the -Box Method in Oxygen and Fluorine Chains of Isotopes
References
9 The Unification of Structure and Reaction Frameworks
9.1 Nucleon–Nucleus Reactions
9.1.1 Coupled-Channel Hamiltonian for Structure and Reaction Observables
9.1.1.1 Orthogonality of the Composite States with Respect to the Core States
9.1.1.2 Coupled-Channel Equations
9.1.2 Electromagnetic Transitions in a Gamow Coupled-Channel Formalism
9.1.3 Cross Sections
9.1.3.1 Cross Sections of Direct Reactions
9.1.3.2 Projectile Energies in the Different Reference Frames
9.1.3.3 Cross Sections of Radiative Capture Reactions
9.2 Generalization to Many-Nucleon Projectiles
9.2.1 Clusters in the Berggren Basis Formalism
9.2.2 Numerical Treatment of Well-Bound Projectiles
9.2.3 Inclusion of Break-up Reactions
9.2.4 Berggren Basis Expansion of Center-of-Mass States
9.2.5 Orthogonalization of Composites with Respectto the Core
9.2.6 Asymptotic Cancellation of Correlations at High Projectile Energy
9.2.6.1 Cancellation of High Energy Matrix Elements with Plane Waves
9.2.6.2 Cancellation of High Energy Matrix Elements with Real-Energy Neutron States
9.2.6.3 Cancellation of High Energy Matrix Elements with Real-Energy Proton States
9.2.6.4 Suppression of Antisymmetry at High Projectile Energy
9.2.6.5 Generalization to Complex-Energy States
9.2.7 Coupled-Channel Equations and Wave Functions for Structure and Reactions with Multi-Nucleon Projectiles
9.2.7.1 Many-Body Wave Functions
9.2.7.2 Hamiltonian Coupled-Channel Equations
9.2.8 Treatment of Faddeev Equations with the Berggren Basis
9.2.9 Elastic and Inelastic Scattering Cross Sections
9.2.10 Radiative Capture Cross Sections
9.2.10.1 Electromagnetic Hamiltonian with Many-Nucleon Clusters
9.2.10.2 Radiative Capture Cross Section
9.3 Computational Methods for Solving the Coupled-Channel Gamow Shell Model
9.3.1 Orthogonalization of Target–Projectile Channels
9.3.2 Solution of Coupled Equations in Coordinate Space
9.3.2.1 Coupled-Channel Equations Using Local Coupling Potentials
9.3.2.2 Use of Nonlocal Coupling Potentials
9.3.3 Representation of Coupled-Channel Equations in the Berggren Basis
9.3.3.1 Bound and Resonance Solutions of Coupled-Channel Equations
9.3.3.2 Scattering Solutions of Coupled-Channel Equations
9.4 Calculation of Spectra and Reaction Cross Sections in the Coupled-Channel Gamow Shell Model
9.4.1 Spectra of Unbound Nuclei and Excitation Functions of Proton–Proton Elastic Scattering
9.4.1.1 18Ne(p,p) Elastic Scattering Cross Section and 19Na Spectrum
9.4.1.2 14O(p,p) Elastic Scattering Cross Section and 15F Spectrum
9.4.2 Radiative Capture Cross Sections in the Mass Region A=6–8
9.4.2.1 7Be (p,γ) and 7Li (n,γ) Mirror Radiative Capture Reactions
9.4.3 4He(d,d) Cross Section and Deuteron EmissionWidth
9.4.4 Inclusion of Nonresonant Channels: The Resonant Spectrum of 42Sc
Solutions to Exercises
References
Index
📜 SIMILAR VOLUMES
Unified theory of nuclear models
<p>Modern nuclear physics is a well developed branch of physical science, with wide-ranging applications of its results in engineering and industry. At the same time, the development of a consistent theory of nuclei and nuclear processes presents certain problems. It is well known that the most impo
<p><p></p><p>This book highlights a major advance in low-energy scattering theory: the Multi-Channel Algebraic Scattering (MCAS) theory, which represents an attempt to unify structure and reaction theory. It solves the Lippmann–Schwinger equations for low-energy nucleon-nucleus and alpha-nucleus sca
The authors decided to write this book when they could find no other texts for a course on non-relativistic nuclear reactions. It combines a thorough theoretical approach with applications to recent experimental results. The main formalisms used to describe nuclear reactions are explained clearly an