Group Theory and the Interaction of Composite Nucleon Systems

233 Pages·1981·3.192 MB·German

by  P. Kramer, G. John, D. Schenzle (auth.)

P. Kramer
G. John
D. Schenzle
Group Theory and
the I nteraction of
Composite Nucleon
Systems
Clustering Phenomena in Nuclei
Edited by
K.Wildermuth
P. Kramer
Volume 2
Volume 2
Kramer/John/Schenzle
Group Theory and the Interaction
of Composite Nucleon Systems
P. Kramer
G. John
D. Schenzle
Group Theory and the
Interaction of
Composite Nucleon Systems
With 40 Figures
Vieweg
CIP-Kurztitelaufnahme der Deutschen Bibliothek
Kramer, Peter:
Group theory and the interaction of cc.mposite
nucleon systems/Po Kramer; G. John; D. Schenzle.
- Braunschweig, Wiesbaden: Vieweg, 1980.
(Clustering phenomena in nuclei; Vol. 2)
NE: John, Gero; Schenzle, Dieter
1981
All rights reserved
© Friedr. Vieweg & Sohn Verlagsgesellschaft mbH, Braunschweig, 1981
Softcover reprint of the hardcover 1s t edition 1981
No part of this publication may be reproduced, stored in a retrieval
system or transmitted, mechanical, photocopying, recording or
otherwise, without prior permission of the copyright holder.
Set by Vieweg, Braunschweig
Cover design: Peter Morys, Salzhemmendorf
ISBN-13: 978-3-528-08449-3 e-ISBN-13: 978-3-322-85663-0
DOl: 10.1007/978-3-322-85663-0
v
Preface
The study which fonns the second volume of this series deals with the interplay of
groups and composite particle theory in nuclei. Three main branches of ideas are de
veloped and linked with composite particle theory: the pennutational structure of the
nuclear fermion system, the classification scheme based on the orbital partition and the
associated supennuitiplets, and the representation in state space of geometric trans
fonnations in classical phase space.
One of the authors (p. K.) had the opportunity to present some of the ideas under
lying this work at the 15th Solvay Conference on Symmetry Properties of Nuclei in 1970.
Since this time, the authors continued their joint effort to decipher the conceptual struc
ture of composite particle theory in tenns of groups and their representations. The
pattern of connections is fully developed in the present study. The applications are
carried to the points where the impact of group theory may be recognized. The range of
applications in our opinion goes far beyond these points.
We appreciate the criticism, suggestions and contributions with respect to this work
offered by the members of our institute. In particular, we would like to mention here
K. Wildennuth, M. Brunet and W. Siinkel. Our work was shaped in many respects through
presentations and discussions at other institutions. We are indebted in particular to
M. Moshinsky and to A. Grossmann for their criticism. It is a pleasure to acknowledge
Mrs. S. EI Sheikh, Mrs. I. Roser and Mrs. R. Adler for their assistance in preparing the
manuscript. Finally we thank the Verlag Vieweg for its excellent cooperation and for
the get-up of this book.
Peter Kramer
GeroJohn
Dieter Schenzle
Tiibingen, Germany
August 1980
VI
Contents
1 Introduction ......................................... .
2 Permutational Structure of Nuclear States 5
2.1 Concepts and Motivation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 The Symmetric Group Sen) .............................. 8
2.3 Irreducible Representations of the Symmetric Group Sen) . . . . . . . . .. 19
2.4 Construction of States of Orbital Symmetry, Young Operators. . . . . .. 25
2.5 Computation of Irreducible Representations of the Symmetric Group .. 28
2.6 Spin, Isospin and the Supermultiplet Scheme .................. 36
2.7 Matrix Elements in the Supermultiplet Scheme ................. 38
2.8 Supermultiplet Expansion for States of Light Nuclei ............. 40
2.9 Notes and References .................................. 46
3 Unitary Structure of Orbital States ......................... 48
3.1 Concepts and Motivation ............................... . 48
3.2 The General Linear and the Unitary Group and Their Finite-Dimensional
Representations ..................................... . 49
3.3 Wigner Coefficients of the Group GL G, 0:) ................... . 54
3.4 Computation of Irreducible Representations of GLG, 0:) from Double
Gelfand Polynomials ................................. . 55
3.5 Computation of Irreducible Representations of GL G, 0:) from Repre-
sentations of the Symmetric Group S (n) .................... . 60
3.6 Conjugation Relations of Irreducible Representations of GL G, 0:) .... . 60
3.7 Fractional Parentage Coefficients and Their Computation ......... . 62
3.8 Bordered Decomposition of Irreducible Representations for the Group
GLG, 0:) ......................................... . 65
3.9 Orbital Configurations of n Particles ....................... . 69
3.10 Decomposition of Orbital Matrix Elements ................... . 70
3.11 Orbital Matrix Elements for the Configuration f = [4j] ..•......•.. 73
3.12 Notes and References ................................. . 74
4 Geometric Transformations in Classical Phase Space and their
Representation in Quantum Mechanics ..................... 75
4.1 Concepts and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 75
4.2 Symplectic Geometry of Classical Phase Space ............. ,... 76
4.3 Basic Structure of Bargmann Space ......................... 79
Contents VII
4.4 Representation of Translations in Phase Space by Weyl Operators 85
4.5 Representation of Linear Canonical Transformations ............. 90
4.6 Oscillator States of a Single Particle with Angular Momentum and
Matrix Elements of Some Operators ........................ 95
4.7 "Notes and References .................................. 99
5 Linear Canonical Transformations and Interacting n-particle
Systems .............................................. 100
5.1 Orthogonal Point Transformations in n-particle Systems and their
Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 100
5.2 General Linear Canonical Transformations for n Particles and State
Dilatation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 105
5.3 Interactions in nobody Systems and Complex Extension of Linear
Canonical Transformations .............................. 108
5.4 Density Operators .................................... 111
5.5 Notes and References .................................. 115
6 Composite Nucleon Systems and their Interaction 116
6.1 Concepts and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 116
6.2 Configurations of Composite Nucleon Systems ................. 116
6.3 Projection Equations and Interaction of Composite Nucleon Systems .. 118
6.4 Phase Space Transformations for Configurations of Oscillator Shells and
for Composite Nucleon Systems ........................... 121
6.5 Interpretation of Composite Particle Interaction in Terms of Single-
Particle Configurations ................................. 125
6.6 Notes and References .............................." .... 127
7 Configurations of Simple Composite Nucleon Systems .......... 128
7.1 Concepts and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 128
7.2 Normalization Kernels ................................. 128
7.3 Interaction Kernels ................................... 132
7.4 Configurations of Three Simple Composite Nucleon Systems ........ 135
7.5 Notes and References .................................. 138
VIII Contents
8 Interaction of Composite Nucleon Systems with Internal Shell
Structure ............................................. 139
8.1 Concepts and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 139
8.2 Single-Particle Bases and their Overlap Matrix .................. 139
8.3 The Nonnalization Operator for Two-Center Configurations with a
Closed Shell and a Simple Composite Particle Configuration ........ 144
8.4 The Interaction Kernel for Two-Center Configurations with a Closed
Shell and a Simple Composite Particle Configuration ............. 148
8.5 Two Composite Particles with Closed-Shell Configurations ......... 154
8.6 Two-Center Configurations with an Open Shell and a Simple Composite
Particle Configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 157
8.7 Notes and References .................................. 159
9 Internal Radius and Dilatation. . . . . . . . . . . . . . . . . . . . . . . . . . . .. 160
9.1 Oscillator States of Different Frequencies . . . . . . . . . . . . . . . . . . . .. 160
9.2 Dilatations in Different Coordinate Systems ................... 160
9.3 Dilatations of Simple Composite Nucleon Systems .. . . . . . . . . . . . .. 165
9 A Notes and References .................................. 168
10 Configurations of Three Simple Composite Particles and the
Structure of Nuclei with Mass Numbers A = 4-10 ............. 169
10.1 Concepts and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 169
10.2 The Model Space ..................................... 170
10.3 The Interaction ...................................... 172
lOA Convergence Properties of the Model Space ................... 175
10.5 Comparison with Shell Model Results ....................... 179
10.6 Absolute Energies .................................... 180
10.7 The Oscillator Parameter b .............................. 183
10.8 Results on Nuclei with A = 4-10 .......................... 184
10.9 Notes and References .................................. 214
References for Sections 1-9 ................................. 215
References for Section 10 ................................... 218
Subject Index ............................................ 221
1
1 Introduction
In composite particle theories of nuclear structure and reactions one tries to under
stand those aspects of the nuclear many-body system which originate from the internal
structure and the interaction of subsystems of the full system. These subsystems or clusters
will be called composite particles. The full system with a specified subdivision will be called
a composite nucleon system. If the internal states of the subsystems are assumed to be
given, it is possible to study the interaction of the composite particles which arises from
the two-body interaction of their constitutents.
The introduction of such subsystems is indespensable for the theory of nuclear reac
tions since the specification of a reaction channel necessarily presupposes a splitting of the
full system into at least two separate subsystems. In the analysis of nuclear structure, the
composite particle theories compete with other theories as for example the nuclear shell
model. In the nuclear shell theory, the nucleons occupy states of a single nucleon. For the
lowest excitations in the harmonic oscillator model it is well-known that the shell and
composite particle theories yield the same states. The difference of the theories appears
with increasing excitation and in the reaction theory. If the excitations are chosen from
the composite particle configurations, they involve selective and non-spurious states which
are not easily recognized in the shell theory. With respect to the transition from the reac
tion region to the reaction channels, the composite particle theories provide a smooth and
analytic connection up to the full separation of the system. A description of the separated
states in a single-center shell theory would be of no physical interest.
The development of the composite particle theory into a unified theory of the nu
cleus is treated in the first volume of this series by Wildermuth and Tang [WI 77]. For the
interplay between theory and experiment we refer to the proceedings of the three con
ferences on Clustering Phenomena held at Bochum in 1969, at Maryland in 1975 and at
Winnipeg in 1977 [IA 69, GO 75, OE 78].
The subdivision of the system of A nucleons implied by composite particle theories
may be conceived in a geometric setting. The 3 A coordinates of the system fix a point in
configuration space, and the (orbital) state of the system may be taken as a complex-valued
function defmed on this configuration space. The introduction of geometric transformation
groups in configuration space and their representations by operators in state space is the
starting point of the present investigation. The analysis to be presented follows three
branches whose common origin is found in the composite particle theory.
The first branch of ideas arises in relation to permutational structure. When we spoke
of the subdivision of the system of A nucleons, we did not mention the correlations in the
system implied by the fermion nature of the nucleons. The permutations act as geometric