Table Of ContentClustering Phenomena in Nuclei
V.I. Kukulin · V.G. Neudatchin
I.T. Obukhovski · Yu.F. Smirnov
Clusters as
Subsystems in
Light Nuclei
Direct Cluster Reactions– Progress
Towards a Unified Theory
V. I. Kukulin, V. G. Neudatchin,
I. T. Obukhovski, Yu. F. Smirnow
Clusters as Subsystems
in Light Nuclei
D. F. Jackson
Direct Cluster Reactions
Progress Towards a Unified
Theory
Clustering Phenomena in Nuclei
Edited by
K. Wildermuth
P. Kramer
Volume 3
Volume 1
Wildermuth/Tang
A Unified Theory of the Nucleus
Volume 2
Kramer/John/Schenzle
Group Theory and the Interaction
of Composite Nucleon Systems
Volume 3
Kukulin/Neudatchin/Obukhovski/Smirnov
Clusters as Subsystems in Light Nuclei
Jackson
Direct Cluster Reactions - Progress Towards a Unified Theory
V.I. Kukulin!V. G. Neudatchin/1. T. Obukhovski/
Yu. F. Smirnov
Clusters as Subsystems in
Light Nuclei
D. F. Jackson
Direct Cluster Reactions -
Progress Towards a Unified
Theory
With 52 Figures
Springer Fachmedien Wiesbaden GmbH
CIP-Kurztitelaufnahme der Deutschen Bibliothek
Clusters as subsystems in light nuclei/V. I.
Kukulin ... Direct cluster reactions - progress
towards a unified theory/D. F. Jackson.
Braunschweig; Wiesbaden: Vieweg, 1983.
(Ciutering phenomena in nuclei; VoL 3)
ISBN 3-528-08493-6
NE: Kukulin, V.I. (Mitverf.];Jackson, Daphne F.:
Direct cluster reactions - progress towards a
unified theory; GT
All rights reserved
© Springer Fachmedien Wiesbaden 1983
Originally published by Friedr. Vieweg & Sohn Verlagsgesellschaft mbH, Braunschweig 1983
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ISBN 978-3-528-08493-6 ISBN 978-3-663-14197-6 (eBook)
DOI 10.1007/978-3-663-14197-6
Preface
Volume 3 of the series deals with nuclear reactions on the basis of subsystem or cluster
dynamics.
In the first contribution, the authors formulate this dynamics within the rigorous frame
work of many-body scattering theory due to Faddeev. The implications of the Pauli
principle on cluster dynamics, the role of virtual cluster excitations, and the synthesis of
this approach with the shell model are studied. New Approximation schemes are developed,
compared to known computational methods, and applied to various types of nuclear
reactions.
The theory of subsystem dynamics is then transfered to the subnuclear level by con
sidering nucleons as being made out of quarks. The algebraic tools for systems of quarks
are developed. Basic properties of the two-nucleon system are derived from an analysis of
the system of six quarks.
The second contribution of this volume presents a review of direct cluster reactions. The
specific features of these reactions are exposed and the corresponding reaction theory is
discussed. The relation of these reactions to nuclear structure theory and corresponding
models is analyzed, with special attention devoted to the spectroscopic factors. Applications
include alpha-particle knock-out, many-body trasnfer, and heavy-ion peripheral reactions.
Karl Wildermuth
Peter Kramer
Ti.ibingen, November 1982
Contents
Clusters as Subsystems in Light Nuclei
V. I. Kukulin, V. G. Neudatchin, I. T. Obukhovski, Yu. F. Smirnow
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2 Quantum Theory of Scattering in the System of Few Composite Particles . . . . . 8
3 Two Particles in the Field of a Core (The Model of Direct Nuclear Reaction
with Deuterons) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4 Interaction between the Lightest Clusters . . . . . . . . . . . . . . . . . . . . . . . . . . SO
5 Ghost States as Generalization of the Concept of Forbidden States in the
Resonating Group Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
6 The Systems of Three Composite Particles . . . . . . . . . . . . . . . . . . . . . . . . . . 72
7 The NN-System in the Quark Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
Direct Cluster Reactions- Progress Towards a Unified Theory
Daphne F. Jackson
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
2 Reaction Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
3 Nuclear Structure Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
4 Application to Selected Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216
5 Outstanding Problems ...... ·. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264
V.I. Kukulin, V. G. Neudatchin, I. T. Obukhovski, Yu. F. Smirnov
Nuclear Research Institute, Moscow State University
Clusters as Subsystems in Light Nuclei
2 Contents
Contents
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2 Quantum Theory of Scattering in the System of Few Composite Particles 8
2.1 On-and Off-Shell Scattering; Equation for Three-Body Scattering ......... . 8
2.2 The Method of Orthogonal Projecting in Two-Body Scattering; Construction of the
Off-Shell t-Matrix and Green Function in Orthogonal Subspaces ......... . 13
2.3 Faddeev Equation (FE) in the Orthogonality Condition Model; Construction of
Many-Body Projecting Operators ............... . 18
2.3.1 Derivation of Orthogonalized FE .......................... . 18
2.3.2 Construction of Many-Body Projectors on the Basis of FE ............ . 20
2.4 Faddeev Equations in Three-Body Orthogonal Subspace; Rearrangement and
Improvement of the Born Series Convergence in the Scattering Theory .. 22
2.4.1 General Considerations ............... . 22
2.4.2 Two-Body Problem; Potential Scattering .... . 24
2.4.3 Three-Body Scattering ............... . 26
3 Two Particles in the Field of a Core (The Model of Direct Nuclear Reactions
with Deuterons) ................................... . 29
3.1 Preliminary Discussion ................................. . 29
3.2 The Faddeev Reduction for the Two-Body System in the Field of a Heavy Core;
Account of the Filled Orbitals of the Core ........... . 32
3.3 Inclusion of the Pauli Principle by the Schrodinger Approach;
Deuteron Break-up in the Field of Nucleus ........... . 36
3.4 Inclusion of the Core Excitations and the Effect of Nucleon Reduced Widths 39
3.5 Conclusions ..................................... . 47
4 Interaction Between the Lightest Clusters 50
4.1 The Orthogonality Condition Model (OCM); Optical Potential With Forbidden States . 50
4.2 Generalization of the Levinson Theorem for the Case of Composite Particle Scattering .. 59
5 Ghost States as Generalization of the Concept of Forbidden States in the
Resonating Group Method ......................... . 62
5.1 Ghosts in the Theory of Low-Energy n-d and d-d Scatterings . . . . . . . . . . 62
5.2 Substantiation of the Generalized Orthogonality Condition Model (GOCM) ...... . 67
6 The System of Three Composite Particles 72
6.1 The 12C Nucleus as a 3a System With Forbidden States ................ . 72
6.1.1 The Oscillator Basis for a Three-Boson System with Eliminated Forbidden States ... . 72
6.1.2 Levels of the 12C Nucleus in the 3a Model with Forbidden States ....... . 75
6.2 Projecting Through the Orthogonalizing Pseudopotentials; the Structure of the
6He-6 Li-6 Be Nuclei in the a + 2 N Model (the Variational Analysis) ..... . 78
Contents 3
7 The NN-System in the Quark Model . 85
7.1 Introductory Comments ..................... . 85
7.2 Spectroscopy of the Six-Quark System in the MIT Bag Model 91
7.2.1 Connection with the Problem of a Core in the NN Forces .... 91
7.2.2 The MIT Bag. The Equations of Motion and the Conservation Laws .... 93
7.2.3 The Quark Eigenmodes. The Quantization ........... . 95
7.2.4. Effective Hamiltonian in the Second Order in the Constant g ... . 99
Ns Np
7.2.5 The Mass Formula for the s112 p112 Configuration .. 104
7.2.6 Level Succession in the Six-Quark Systems ..... . 110
7.3 Construction of the Fractional Parentage Expansions in the Multiquark Systems 113
7.3.1 The Complementarity of the Unitary Urn and Permutation SN Groups ..... . 114
7.3.2 The Standard Young-Yamanouchi Basis in Terms of Irreducible Representations
of the Unitary Group .................................. . 115
7.3.3 The Kaplan Transformation Matrices for the Transition to Nonstandard Basis .. 117
7.3.4 Relation Between the Kaplan Transformation Matrices and the 3nf-Symbols
for the Unitary Groups ................................. . 118
7.3.5 Factorization of the Fractional Parentage Coefficients as the Product of TM and CGC
of the Complementary Groups SN and Urn ..................... . 120
7.3.6 Total Fractional Parentage Coefficient ........................ . 121
7.3.7 Calculations of CGC for Urn using the Formalism of the Permutation Group SN. 123
7.3.8 Orbital States ........................ . 130
7.4 Results and Discussion; Some Physical Applications ................ . 144
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
