computational and group-theoretical methods in nuclear physics

Computational

Methods in

Group-Theoretical and

I .

Nuclear Physics

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editors

Jutta Escher Nuclear Theory &Modeling Group,

Lawrence Livermore National Laboratory, USA

Octavio Castaiios Instituto de Ciencias Nucleares,

Universidad NacionalAutdnoma de Mixico, Mexico

Jorge G. Hirsch Instituto de Ciencias Nucleares,

Universidad Nacional Autdnoma de Mixico. Mexico

Stuart Pittel Bartol Research Institute,

University of Delaware, UJ]A

Gergana Stoitcheva Physics Division,

Oak Ridge National Laboratory, USA

Playa del Carmen, Mexico

Nuclear Physics Proceedings of the Symposium in Honor of

Jerry P Draayerk 60th Birthday

v World Scientific N E W J E R S E Y L O N D O N * S I N G A P O R E - S H A N G H A I H O N G K O N G T A I P E I - B A N G A L O R E

Computational andGroup-Theoretical

Methods in

18-21 Febbruary 2003

Published by

World Scientific Publishing Co. Re. Ltd. 5 Toh Tuck Link, Singapore 596224 USA once: Suite 202,1060 Main Street, River Edge, NJ 07661 UK once: 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

COMPUTATIONAL AND GROUP-THEORETICAL METHODS IN NUCLEAR PHYSICS Proceedngs of the Symposium in Honor of Jerry P Draayer’s 60th Birthday

Copyright 0 2004 by World Scientific Publishing Co. Re. Ltd. All rights reserved. This book, or parts thereoj may not be reproduced in any form or by any means, electronic or mechanical, includingphotocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

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ISBN 981-238-596-7

Printed in Singapore by World Scientific Printers (S) Pte Ltd

Playa del Carmen and the Mayan Rivera

Jacek Dobaczewski Joe Ginocchio K.T. (Ted) Hecht Calvin Johnson Marcos Moshinsky Witek Nazarewicz George Rosensteel Ed Zganjar

Conference Organization

International Advisory Committee

Jorge G. Hirsch

Stuart Pittel

Octavio Castaiios Jutta Escher Gergana Stoitcheva Dirk Troltenier

(Warsaw University) (Los Alamos National Laboratory) (University of Michigan) (San Diego State University) (Universidad Nacional Autdnoma de Mtxico) (University of Tennessee) (Tulane University) (Lousiana State University)

Organizing Committee

(Co-Chair, Universidad Nacional Autdnoma de

(Co-Chair, Bart01 Research Institute,

(Universidad Nacional Autdnoma de MCxico) (Lawrence Livermore National Laboratory) (Oak Ridge National Laboratory) (SAP, Chicago)

Mtxico)

University of Delaware)

Sponsors

Conacyt (Consejo Nacional de Ciencia y Tecnologia, MCxico) Instituto de Ciencias Nucleares, Universidad Nacional Autdnoma de Mtxico

Louisiana State University (USA) NSF (National Science Foundation, USA) Sociedad Mexicana de Fisica (Mtxico)

(Mtxico)

Conference Photos and Book Cover

The photos appearing throughout this proceedings volume were taken by Victoria Cerdn, Lois Draayer, Jutta Escher, Vesselin Gueorguiev, and Stuart Pittel. The book cover includes an illustration designed by Gergana Stoicheva.

X i

Jerry and Lois Draayer

Preface

The purpose of the Symposium on Computational and Group- Theoretical Methods in Nuclear Physics was to honor Professor Jerry P. Draayer on his 60th birthyear, a most appropriate honor since Jerry Draayer is one of the world's masters in combining powerful modern computational technology with esoteric group theoretical methods to elucidate the structure of nuclei. The conference was held February 18-21, 2003 in Playa del Carmen, Mexico, the choice of location being dictated partly by the many collaborators from Mexico, particularly in the past 15 years, who have aided Jerry Draayer in these endeavors. The beauty of the Mexican Riviera led to a relaxed atmosphere and interactions that were spontaneous, lively, and most fruitful.

Marcos Moshinsky in the first contribution to this conference will explain why the groups SU(3) and Sp(6,R) and their pseudo-SU(3) and symplectic analogs have been of particular applicability in Jerry's life work.

Jerry's interest in SU(3) goes back to his Ph.D. thesis at Iowa State University (1968) with S. A. Williams. This was followed by an important contribution on the SU(4) Wigner supermultiplet symmetry done a t the Niels Bohr Institute in Copenhagen during the first year of a two-year National Science Foundation postdoctoral fellowship. This was followed by a second year at the University of Michigan and a subsequent further three-year stay at Michigan as postdoctoral research associate and instructor. Here, his computational skills led to a technological breakthrough with a computer code for SU(3) Wigner and Racah coefficients in collaboration with Y. Akiyama which contains an essentially canonical solution of the vexing outer multiplicity problem. This code was followed in later years with a further computational breakthrough with a computer code for the needed one and two-part icle operators which have made possible many sophisticated shell model calculations in nuclei with rotational spectra. Also at Michigan, Jerry with Ratna-Raju became one of the founders of the pseudo-SU(3) symmetry model with equally important implications for heavy deformed nuclei.

The Michigan years were followed by a two-year post-doctoral appointment at Rochester University with 3. B. French which led to two seminal papers on statistical spectroscopy ("Strength Distributions and Statistical Spectroscopy: I. General Theory. 11. Shell Model Comparisons" in Annals of Physics, 1977) and work on possible alpha-

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cluster structure in l ight nuclei. Perhaps one of the l itt le known aspects of Jerry's career, but typical of his general attributes, is that Jerry took regular turns with the experimentalists on the night shifts of the Rochester accelerator in their studies of (6Li, d) reactions. He is the only theory postdoctoral fellow whom I have known to have contributed in this fashion.

Since 1975 Jerry Draayer has been on the staff of Louisiana State University. Again, Marcos Moshinsky's contribution to this conference lists in detail Jerry's many administrative activities not only for the Department of Physics and Astronomy but also for the university, and his services at the national and international level. Despite this huge administrative load, particularly a fruitful ten-year stint as chairman of the Department of Physics and Astronomy and his recent role as chair of SURA (Southeastern Universities Research Association), Jerry has kept very active in both teaching and research. Several of Jerry's former Ph.D. students have told me that they were first motivated to choose him as a thesis advisor because they were very much impressed by the clarity and insightful character of his lectures in graduate courses. Despite a heavy administrative load in the past 18 years his research efforts have continued to lead to a veritable stream of important publications as illustrated by his collaboration with three physicists from the University of Mexico as highlighted by Marcos Moshinsky's contribution to the conference.

Jerry Draayer's use of his so-called horizontal couplings, based on judicious choices of SU(3) representations, and vertical couplings, based on symplectic excitations, have led to some of the most detailed and successful shell model calculations of rotational spectra in both medium-weight and heavy deformed nuclei. Jerry's mastery of group theory and computational technology have served him as tools to answer important nuclear structure questions. To cite only a couple of examples: What are meaningful measures of deformation in nuclei with intrinsic non-spherical mass and charge distributions? What are the origins of the seemingly accidental pseudo-spin and pseudo-SU(3) symmetry? Here Jerry has been one of the pioneers in the use of relativistic mean field theory in an attempt to answer this question.

The days at Playa del Carmen have strongly confirmed the following opinion: All of Jerry's former students, Ph.D. candidates, postdoctoral fellows, and research collaborators have greatly enjoyed working with Jerry and have found him most generous of spirit and of his time. We wish him the best for the next 60 years.

Ted Hecht

Introduction

This volume contains the Proceedings of the International Conference on “Computational and Group-Theoretical Methods in Nuclear Physics” held in Playa del Carmen, Mexico, from 18-21 February 2003. The conference, which highlighted recent developments in these two areas of contemporary importance in nuclear physics, honored the 60th birthday of Jerry Draayer, who has been such a major contributor to the field throughout his career.

The program of the conference was divided into five themes: (1) SU(3) and Symplectic Models and their Applications; (2) Pseudo-spin in Nuclear Physics; (3) Collective Phenomena; (4) Computational Physics and Large-Scale Nuclear Models; (5) Mathematical Physics. Each of the themes was represented by an overview talk followed by a series of invited presentations. I n addition, there was a special session in which Marcos Moshinsky summarized Jerry’s myriad of contributions, both to physics research and to science administration. There was a session on Physics Outside Academia, in which a former student that was trained by Jerry in nuclear physics described his subsequent professional experiences outside the field. There were also special lectures presented by Peter Hess and Walter Greiner on forefront scientific topics outside the main themes of the conference. Lastly, there was a very stimulating Poster Session, in which other participants at the conference were able to present their recent results.

A l l speaker and poster contributions are contained in these Proceedings, with page guidelines defined by the type of presentation. Also included is a Preface written by Ted Hecht, one of Jerry’s most valued collaborators and mentors. Throughout the volume can be found photos taken during the conference, which we feel reflect both the very serious scientific nature of the presentations and discussion and the spirit and joy of the participants during their week on the Mayan Riviera.

We would like to express our appreciation to all of the contributors to these Proceedings for providing the uniformly excellent manuscripts contained herein.

Jutta Escher Jorge Hirsch Stuart Pittel

Octavio Castaiios Gergana Stoicheva

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Contents

Conference Photograph

Conference Organization

Preface K. T. Hecht

Introduction J. Eschel; J.H. Hirsch, S. Pittel, 0. Castafios, and G. Stoicheva

I Opening Session

The Work of Jerry P. Draayer M. Moshinsky

I1 SU(3) and Symplectic Models and Their Applications

Computational and Group Theoretical Methods in Nuclear Physics J.19 Draayer

Pseudo + Quasi SU(3): Towards a Shell-Model Description of Heavy Deformed Nuclei

J.G. Hirsch, C.E. Vargas, G. Popa, and J.P. Draayer

Partial Dynamical Symmetry in Nuclear Systems J. Escher

I11 Random Hamiltonians

Systematic Correlations and Chaos in Mass Formulae K Velcizquez, A. Frank, and J.G. Hirsch

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3

17

19

31

40

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Proceedings contributions with multiple authors were presented by the person whose name is underlined in the Contents.

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Shape Phase Transitions and Random Interactions R. Bijker

N Pseudo-Spin in Nuclear Physics

Pseudospin as a Relativistic Symmetry in Nuclei J.N. Ginocchio

Pseudo-Spin Symmetry in Nuclei f? Van hacker

V Collective Phenomena

Shape Evolution in Nuclei R.E Casten

New Exactly Solvable Models of Interacting Bosons and Fermions J. Dukelsky C. Esebbag, and S. Pittel

Exact Solutions of the Isovector Pairing Interaction R Pan and J.P Draayer

Relativistic RPA and Applications to New Collective Modes in Nuclei P Ring N. Paal; T Nikvji2, and D. Vretenar

Superallowed Beta Decay of 74Rb and Shape Coexistence in 74Kr: A Test of the Standard Model

E.E Zaaniar and A. Piechaczek

VI Computational Physics and Large-Scale Nuclear Models

Collectivity, Chaos, and Computers C. W Johnson

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108

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126

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137

Large-Scale Computations Leading to a First-Principles Approach to Nuclear structure

W.E. O m n d and F! Navrdtil

Computational Challenges of Quantum Many-Body Problems in Nuclear Structure: Coupled-Cluster Theory

D. J. Dean

VII Mathematical Physics

Embedded Representations and Quasi-Dynamical Symmetry D.J. Rowe

Shape-Invariance and Exactly Solvable Problems in Quantum Mechanics A.B. Balantekin

Nonlinear Resonant States and Scattering in a One-Dimensional BEC-model

A. Ludu

VIII Special Topics

Vacuum, Matter, Antimatter and the Problem of Cold Compression Greiner and T Buervenich

A Toy Model for QCD at Low and High Temperatures S. Lerma H., S. Jesgarz, RO. Hess. 0. Civitarese, and M. Reboiro

M Poster Session

Analysis of the 196Pt(&)’95Pt Transfer Reaction in the Framework of the IBA and IBFA Models

J. Barea, C. E. Alonso, and J. M. Arias

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X X l V

Nuclear Wave Functions for Spin and Pseudospin Partners RJ. Bopcki, J. Ginocchio, W Nazarewicz, and M. Stoitsov

Finite Well Solution for the E(5) Hamiltonian M A . Capri0

Neutrinoless Double Electron Capture with Photon Emission KE, Cerdn and J. G. Hirsch

Symplectic Mean Field Theory J.L. Graber and G. Rosensteel

Structure of 159Gd C. Grania and D. Nosek

Oblique-Basis Calculations for 44Ti KG. Gueorauiev, J.P Draayel; WE. Omand, and C.W Johnson

Application of Ground-State Factorization to Nuclear Structure Problems i7 Pauenbrock and D.J. Dean

Microscopic Interpretation of the K' = 0; and Kp = 2; Bands of Deformed Nuclei within the Framework of the Pseudo-SU(3) Shell Model

G. Pova, A. Georgieva, and J.R Draayer

Sp(4) Dynamical Symmetry for Pairing Correlations and Higher-Order Interactions in Atomic Nuclei

K.D. Sviratcheva, C. Bahri, A.1. Georgieva, and J.P. Draayer

Excited Bands in Odd-Mass Rare-Earth Nuclei C.E. Varaas, J.G. Hirsch, and J.R Draayer

The Geometry of the Pb Isotopes in a Configuration Mixing IBM C.E. Varaas, A. Frank, and R Van hacker

Photos from the Banquet

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257 List of Participants

I. Opening Session

THE WORK OF JERRY P. DRAAYER

M. MOSHINSKY* Instituto de Fisica, U N A M Apartado Postal 20-364, 01000 Mdxico, D. F.

MEXICO Email: moshi @j?sica. unam.mx

The present meeting deals with group theory and nuclear physics but its main objective is to celebrate the 60th birthday of Jerry Draayer. One could ask why this meeting takes place in Playa del Carmen rather than at Louisiana State University or some other place in the US, due to the many scientific and administrative positions that Jerry holds in the US.

A good reason for the selection of a venue here in Mexico is the fact that of the 158 refereed journal articles that Jerry has participated through 2002, 27 were carried out in collaboration with physicists from Mexico, a number that is considerable higher than with collaborators from any other country, including the US. Another reason is of course that from many standpoints Playa del Carmen is a more pleasant spot.

I will restrict myself to giving some statistics on the many papers and administrative positions Jerry has held, give a very brief survey of some of the essential ideas of his referred papers, with a more extensive analysis of the papers he has written with collaborators in Mexico from information presented by Castaiios, Hirsch and Hess.

Jerry was born in Hollandale MN, USA on August 18, 1942 so we are a bit late in celebrating his 60th birthday but of course this conference concerns his work and not a particular date. While his primary and high school education is not available in his curriculum vitae, we know that he got his Bachelor of Science (1964) and his Ph. D. (1968) at Iowa State University.

Later, in chronological order, he had appointments at Niels Bohr Insti- tute (NSF Fellow 68-69), the University of Michigan (Research Associate and Instructor 70-73) where he further developed his interest in Group Theory by his contact with Prof. K. T. Hecht, University of Rochester

*Member of El Colegio Nacional and Sistema Nacional de Investigadores.

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(Research Associate 73-75), and from 75 to the present he has been a Pro- fessor at Louisiana State University, rising from Assistant to Full Professor in Physics and Computer Science.

He has held other Special Academic Appointments as a Fellow of the American Physical Society, member of the Academia Mexicana de Ciencias and guest professor in the Universities of Tubingen, (Germany) and Louis Pasteur in Strasburg (France).

In Louisiana he had 8 long term (> 6 months) and 8 short term (< 6 months) visitors and 10 postdoctoral associates and directed 15 Ph. D. theses with students from all over the world and received numerous awards.

Coming now to his scientific activities, which include 4 books and book chapters in addition to his 158 refereed journal articles, let me speak about the latter.

In almost all of Jerry’s papers in which he deals with the structure and deformation of nuclei, he employs group theoretical methods. The first question that comes to one’s mind is why group theory plays such an important role in many body problems in quantum mechanics? The reason lies in the fact that, while, in classical mechanics, we deal with a set of particles as points in a phase space and describe their motion by Hamilton- Jacobi theory, in quantum mechanics we start by associating with each particle a set of states which we shall enumerate by the index p. Thus we need first an operator that creates a particle in the state which we denote by up+ and another one that annihilates a particle in that state which we denote by up. While these problems are non relativistic it is useful to employ the covariant (lower index) and contra variant picture (upper index).

The first thing that we note is that the above representation applies to bosons if the commutation relations

[ U P , u;] = upu+ P‘ - u;up = 6 P p , ; [ U P , U P ’ ] = [ U t , up”] = 0

{ U P , u, i } = .”up” + u;up = 6 P p , , { U P , U P ’ } = {up’,u,i} = 0.

hold, while for fermions we must have the anticommutation relation

Independent of whether the relation (1) or (2) holds the operator

CP’ = P P P’

obeys the commutation rules

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c;,’, ”” (4)

which correspond to the Lie algebra of the unitary or linear group of dimension N if the latter is the number of states we consider. Thus, from the very beginning, group theory is involved in quantum mechanical many body problems.

To illustrate how these group and their subgroups characterize our states, let assume that they are associated with the levels of the harmonic oscillator in Fig. 1 and besides the states of the particle have also spin and isospin 4. If we take all states of our unitary or linear group it will be 00 dimensional and thus useless for calculations. We can however restrict our oscillator states to a definite level of the harmonic oscillator like the 2s - Id level indicated in Fig. 1. This level has 1 + 5 values for the orbital states and four value for spin-isospin so the total number of states is 24.

N

L

Figure 1. dashed contour) of the states associated with the SU(3) representation (Xp) = (20).

The degeneracy of states in the harmonic oscillator and example, (under the

The chain of group, is then

U“‘(4) 3 S U ( 2 ) @ S U ( 2 )

where the index becomes p = p ~ r , with p = nlm with n fixed and 1 the orbital angular momentum of the state and m its projection while 0, T are

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the projection of the spin and isospin. The states could also be characterized by total angular momentum combining the spin and orbital part.

In this example, first fully analyzed by Elliott, the chain of groups determine the wave function in terms of polynomials in a,’ acting on the ground state and Hamiltonians, transition operators, etc., can be expressed in terms of C t generators so their expectation values with respect to the wave functions mentioned above can be evaluated through a group theory analysis. This gives the procedure for calculating all process of interest, while physics enters mainly through our judicious choice of the initial set of states.

Let me now go back to Jerry who initiates his research work in 1968 at a time when the appearance of Elliott’s work on SU(3) structure in 2s - I d shell nuclei was the rage in nuclear physics. I probably heard of the work of Jerry about that time because I was interested in similar problems but with a different technique than that used by Elliott which I developed with Valentin Bargman. One thing that impressed me most of Jerry work was not only his understanding of the SU(3) group, but also his computational ability which led to programs for the Wigner and Racach coefficients of this group.

In its original form the SU(3) symmetry was applied to light nuclei where the spin-orbit interactions is weak and thus the energy states can be considered as degenerate as shown in Fig. 1. For heavier nuclei the spin orbit coupling breaks this degeneracy and the levels of the harmonic oscillator become doublets with total angular momentum j = 1 + 3 in the lower one and j = 1 - in the upper one for a given orbital angular momentum 1, as shown in Fig. 2. In each level of the harmonic oscillator then there is an intruder state coming from the last state of the next shell and the set of all the levels surrounded by a closed curve can also be described through SU(3) to which the word pseudo is attached. The importance of this group (or Lie algebra) is that it allows the extension of the SU(3) analysis to heavier nuclei. Many people contributed to the development of pseudo SU(3) but as I learned of this technique through Jerry’s work I consider him as one of the main contributors in this field and many of his papers deal with application of this scheme both to the structure and deformation of nuclei.

Continuing with Jerry’s interest in group theory we can consider in the oscillator the vertical set of levels of fixed 1 rather than as before those of fixed energy. The group Sp(6 , R) (known also as Sp(3, R) by other authors) is related with our fundamental group C;, of Eq. 4, if we add to it the generators

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Figure 2. the dashed lines) of those associated with an irrep of pseudo SU(3) .

The effect of spin-orbit coupling on oscillator states and an example (under

for all the values of indices p,p’ that we consider. While the use of the Sp(6,R) group has mainly been the work of Rowe and Rosensteel, Jerry has also made important contributions in this field as in his work with Rosensteel on “Centroids and widths in the symplectic collective nuclei” and “Symplectic shell model calculations of 2o N e with horizontal configuration” in which in the latter levels of Sp(6 ,R) and SU(3) are also considered in Figs. 2 and 3.

There is no space in this note for many other collaborations of Jerry on these subjects.

The Sp(6, R) group plays with respect to collective variables a similar role that of C ( N ) for many body systems as, in its subgroups it has on the one hand the symmetry of the Bohr-Mottelson problem (ie. that of the five dimensional oscillator) and on the other the SU(3) subgroup. This allowed Jerry to discuss the triaxial deformations associated to the rotor model and, in particular, establish the following relations between the collective parameters (P, y) and irreducible representation (Xp) of SU(3) at least for large (A7P)

P2 = (47r/5)(AF2)[A2 + Ap + p2 + 3(X + p) + 31

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2 F 1H N

.1G I

I

1F

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Figure 3. inside the dashed contour.

Truncated basis of states for an irrep of S p ( 6 , R) 3 S p ( 2 , R) by the levels

My somewhat superficial emphasis on the work of Jerry in group theory in nuclear physics is based on the fact that this is the field that I am more familiar with.

Let me now pass to the interaction of Jerry with physicists in Mexico. Due to time limitations I will speak briefly only of his collaboration with Octavio Castaiios, Peter Hess and Jorge Hirsch.

Castaiios, who spent two years with him in 1986-1988 as a postdoctoral fellow, collaborated on several papers with Jerry that are indicated in Table I, and of particular interest to me was the one on “Contracted symplectic model with ds-shell applications” I As we mentioned the Sp(6, R) has emerged as the appropriate dynamical symmetry group for a many- body description of nuclear collective motion. Applications of the theory have encountered difficulties in establishing a complete orthonormal set of states that are an arbitrary but fixed irreps of this group.

Thus Draayer and Castaiios made a contraction of this group in terms of the semi-direct product

where the bosonic generators of Ub(6) are related with the creation and annihilations operators bk, blm, 1 = 0,2, m = 1 . . .-1 of the core excitations.

The U(6) group admits SU(3) subgroups different from the one originally introduced, whose irrep will be denoted ( X b , p b ) while the ones cor-

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responding to the non-core excited states are denoted by ( A S , p s ) . We can denote the state of total irrep of (Xp) using the tabulated Wigner coefficients of SU(3) to get our states.

INb(Xb, pb) Ns(&, / ls) ; p(X, p)a >=

[ Nb(Abpb) x Ns ( A s , Ps) 1”’ (9)

where p differentiates the multiplicities of the (Xp) and the a index indicate the component connected with the chain of groups given by

With respect to the states (9) one can calculate the spectra of nuclei as well as the electromagnetic transitions between their levels, where, for the former two forms of the Hamiltonians H - I and H + I are proposed, and in the case of 20Ne the level picture is given as well as the electromagnetic transition probabilities. For the papers of Castaiios and Draayer we give in Table I the list with their names underlined.

Table I. References of the collaboration of 0. Castafios with J.P. Draayer

0. Castaiios, J.P. Draayer and Y. Leschber. Collective 1+ States in Rare Earth and Actinide Nuclei. Nucl. Phys. A473 (1987) 494. 0. Castaiios, J.P. Draayer and Y. Leschber. Towards a Shell-Model Description of the Low-Energy Structure of Deformed Nuclei. 11. Electromagnetic Properties of Collective M1 Bands. Ann. of Phys. 180 (1987) 290. 0. Castaiios, J.P. Draayer and Y. Leschber. Shape Variables and the Shell Model. 2. Phys. A329 (1988) 33. 0. Castaiios, J.P. Draayer and Y. Leschber. Quantum Rotor and its SU(3) Realization. Comp. Phys. Commun. 52 (1988) 71. 0. Castaiios and J.P. Draayer. Contracted Symplectic Model with ds-Shell Applications. Nucl. Phys. A491 (1989) 349-372; D. J. Rowe and G . Rosensteel, Phys. Rev C25 (1982) 3236. J.P. Draayer, S. C. Park, and 0. Castaiios. Shell-Model Interpreta- tion of the Collective-Model Potential-Energy Surface. Phys. Rev. Lett. 62 (1989) 20.

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(7) C. Bahri, J.P. Draayer, 0. Castarios, and G. Rosensteel. Resonant Modes in Light Nuclei. Phys. Lett. B 234 (1990) 430.

(8) 0. Castaiios, P.O. Hess, J.P. Draayer and P. Rochford. Pseudo- Symplectic Model for Strongly Deformed Heavy Nuclei. Nucl. Phys. A524 (1991) 469.

The collaboration of Jerry with Peter Hess started in the late eighties on the pseudo symplectic model of the nucleus with considerable success in reproducing experimental data. They looked also at the connections of the symplectic model to the geometrical model of Gneuss and Greiner and where able to obtain collective potentials as function of the deformation.

Many other contributions came from their collaboration as can be seen from the references on their joint work, to be shown in Table 11, in which we also give Hess collaboration with Greiner on similar subjects.

Table 11. References of the collaboration of P.O. Hess with J.P. Draayer and with W. Greiner

(1) R. Lopez, P.O. Hess, P. Rochford, J.P. Draayer. Young diagrams as product of symmetric and antisymmetric components. Jour. Phys. A23 (1990) L229.

(2) 0 Castaiios, P.O. Hess, P. Rochford, J.P. Draayer. Pseudo symplectic model for strongly deformed nuclei. Nucl. Phys. A524 (1991) 469.

(3) 0. Castaiios, P.O. Hess, J.P. Draayer, P. Rochford. Microscopic interpretation of potential energy surfaces. Phys. Lett. B277 (1992) 27.

(4) D. Troltenier, J.A. Maruhn, P.O. Hess, W.Greiner. A general numerical solution of collective quadrupole motion applied to microscopically calculated potential energy surfaces. Zeit. f. Phys. A343 (1991) 25.

(5) D. Troltenier, J.P. Draayer, P.O. Hess, 0. Castaiios. Investigation of rotational nuclei via the pseudo symplectic model. Nucl. Phys. A576 (1994) 351.

(6) H. van Geel, P.O. Hess, J.A. Maruhn, W. Greiner, D. Troltenier. Microscopic derived potential energy surfaces for the chain of Sm- isotopes. Nucl. Phys. A577 (1994) 605.

(7) C. Vargas, J.G. Hirsch, P.O. Hess, J.P. Draayer. Description of the spin-orbit interaction. Jour. Phys. G25 (1999) 881.

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In the collaboration of Jorge Hirsch with Draayer the emphasis was on heavy nuclei where the SU(3) truncation is useless because of the strong spin-orbit interaction, while the pseudo spin emerges as a good symmetry whose origin has been traced back to the relativistic mean field.

For example they give a Hamiltonian

that contains the standard quadrupole interaction and total orbital angular momentum plus a residual interaction that allows the fine tuning of the Hamiltonian to low-lying spectral features of the r;-band splitting.

A technical breakthrough was achieved with a computer code able to calculate reduced matrix elements of physical operators between different irreps of SU (3).

The pseudo SU(3) scheme allows for a very elegant generalization of the geometrical picture as part of a two rotor model. This model considers, for example, the case of the molecular like structure of two I2C nuclei, as rotors with a scissors like relative motion parametrized by an angle 8. Due to this developments, a fully microscopic description of low-energy bands in even-even and odd-A heavy deformed nuclei is now possible.

In the next Table 111 we show some of the references in which Hirsch and Draayer have collaborated with their names underlined.

Table 111. References of the collaboration of J.G. Hirsch with J.P. Draayer

(1) J.G. Hirsch, C. Bahri, J.P. Draayer, 0. Castaiios, P.O. Hess. Re- duced Matrix Elements for the Leading Spin Zero States in the SU(3) Scheme. Rev. Mex. Fis. 41 (1995) 181.

(2) D. Troltenier, J.P. Draayer and J.G. Hirsch. Correlations between the quadrupole deformation, B(E2;01 -+ 21) value, and total GT+ strength Nucl. Phys. A 601 (1996) 89.

(3) T. Beuschel, J.P. Draayer, D. Rompf, J.G. Hirsch Microscopic description of the scissors mode and its fragmentation. Phys. Rev. C 57 (1998) 1233.

(4) D. Rompf, T. Beuschel, J.P. Draayer, W. Scheid, J.G. Hirsch. To- wards understanding the Scissors mode in the Pseudo SU(3) Model - Part I: Phenomenology. Phys. Rev. C 57 (1998) 1703.

( 5 ) J.G. Hirsch, 0. Castaiios, P.O. Hess, 0. Civitarese, D. Troltenier, T. Beuschel, J.P. Draayer, D. Rompf and Y. Sun. Shell model calcu-

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lations for heavy deformed nuclei. Czech. Journ. Phys. 48 (1998) 183.

(6) J.P. Draayer, T. Beuschel, D. Rompf, J.G. Hirsch. The Shell Model - Dead or Alive? Rev. Mex. Fis. 44 Supl. 2 (1998) 70.

(7) C. Vargas, J.G. Hirsch, P.O. Hess and J.P. Draayer. Interplay between the quadrupole-quadrupole and spin-orbit interactions in nuclei. Phys. Rev. C 58 (1998) 1488.

(8) J.P. Draayer, T. Beuschel, D. Rompf, and J. Hirsch. Fragmentation of the Scissors Mode in Deformed Nuclei. Yadernaya Fizika 61 (1998) 1749-1756; Phys. At. Nuclei 61 (1998) 1631.

(9) J.P. Draayer, T. Beuschel and J.G. Hirsch. M1 strengths in deformed nuclei. Jour. Phys. G - Nucl. Part. Phys. 25 (1999) 605.

(10) C. Vargas, J.G. Hirsch, P.O. Hess and J.P. Draayer. SU(3) description of the spin-orbit interaction. Jour. Phys. G - Nucl. Part. Phys. 25 (1999) 881.

(11) J.G. Hirsch, P.O. Hess, C. Vargas, L. Hernhdez, T. Beuschel and J.P. Draayer. The Elliot SU(3) model in the fp-shell. Rev. Mex. Fis. 45 Supl. 2 (1999) 86.

(12) C. Vargas, J.G. Hirsch, T. Beuschel, J.P. Draayer. Shell model description of normal parity bands in odd-mass heavy deformed nuclei. Phys. Rev. C 61 (2000) 31301-1/4.

(13) T. Beuschel, J.G. Hirsch, and J.P. Draayer. Scissors mode and the pseudo SU(3) model. Phys. Rev. C 61 (2000) 54307-1/10.

(14) C.E. Vargas, J.G. Hirsch and J.P. Draayer. Pseudo SU(3) shell model: normal parity bands in odd-mass nuclei. Nucl. Phys. A 673 (2000) 219.

(15) J.G. Hirsch, C.E. Vargas, and J.P. Draayer. Low energy spectra of A=159 and 161 nuclei. Rev. Mex. Fis. 46 Supl. 1 (2000) 54.

(16) J.P. Draayer, G. Popa, J.G. Hirsch and C.E. Vargas. M l strengths (scissors and twist modes) in heavy deformed nuclei (15s-1s0Gd). Rev. Mex. Fis. 46 Supl. 1 (2000) 71.

(17) G. Popa, J.G. Hirsch and J.P. Draayer. Shell model description of normal parity bands in even-even heavy deformed nuclei. Phys. Rev. C 62 (2000) 064313-1/6.

(18) J.P. Draayer, G. Popa, and J.G. Hirsch, E2 and M1 strengths in heavy deformed nuclei. Acta Physica Polonica B 32 (2001) 2697.

(19) C.E. Vargas, J.G. Hirsch, and J.P. Draayer. Quasi-SU(3) truncation scheme for even-even sd-shell nuclei. Nucl. Phys. A 690

13

(2001) 409. C.E. Vargas, J.G. Hirsch, and J.P. Draayer. Interband B(E2) transition strengths in odd-mass heavy deformed nuclei. Phys. Rev. C 64 (2001) 034306-1/4. C.E. Vargas, J.G. Hirsch, and J.P. Draayer. Quasi-SU(3) truncation scheme for odd-even and odd-odd sd-shell nuclei. Nucl. Phys. A 697 (2002) 655. Jorge G. Hirsch, Gabriela Popa, Carlos E. Vargas, and Jerry P. Draayer. Microscopic description of odd- and even-mass Er isotopes. Heavy Ion Physics 16 (2002) 291. C.E. Vargas, J.G. Hirsch, and J.P. Draayer. Excited bands in odd- mass rare-earth nuclei. Phys. Rev. C 66 (2002) 064309-1/7. C.E. Vargas, J.G. Hirsch, and J.P. Draayer. Microscopic description of the scissors mode in odd-mass heavy deformed nuclei. Phys. Lett. B 551 (2003) 98.

I have not mentioned Jerry’s extensive work on application of the shell model to a wide class of nuclei and his work on nuclear reactions or on the scissors model and pseudo SU(3) and many other aspects he touches on his 158 refereed journal publications. Besides these referred papers Jerry has more than 115 publications, 19 with colleagues from Mexico, based on invited talks.

I strongly recommend to the participants of the conference to at least read Jerry curriculum vitae so they get a good idea of what research in nuclear physics is all about. Incidentally I have all the papers of Jerry xeroxed and will have them bound and available in the Library of the Physics Institute for all those interested.

No description of Jerry’s career would be complete without some mention of the enormous contributions that he has made outside of research, teaching and student development. So as not to take up too much space, I will limit myself to a list of the myriad of positions that he has held, both within LSU and outside.

Administrative Positions

Leadership Roles

Internal Associate Commissioner for R&D, LA Board of Regents (98/00); Vice Pres-

14

ident, Faculty Senate (96/98); Director, Office of Federal Programs, LA Board of Regents (95/98); Chair, Council of Chairs, College of Basic Sci- ences, LSU (87/93); Chair, Department of Physics and Astronomy, LSU (85/94); Chair, Ad Hoc Committee on Computing Charges, LSU (89/90); Chair, Chancellor’s Committee on Supercomputing, LSU (85/89); College Policy Committee, LSU (83/84); Chair, University Task Force on Com- puting, LSU (82/85); Chair, System Network Computer Center Advisory Council, LSU (80/82).

External Southeastern Universities Research Association (SURA), President and CEO, (1999-present); Chair, SURA Board of Trustees (98/00); Chair-elect, SURA Board of Trustees (95/97); Chair, SURA New Projects Committee (88/95).

Committee Service

Internal CAMD Search Committee, Chair, LSU (98/99); Vice Chair Faculty Sen- ate, LSU (95/98); University Planning Committee, LSU (91/95); Work- ing Group, CAMD, LSU (88/91); Site Selection Committee, CAMD, LSU (88/89); Student Code of Conduct Committee, LSU (83/87); Faculty Sen- ate, LSU (83/85); College Policy Committee, LSU (81/84).

External Contract Negotiation Team-DOE/CEBAF Facility, SURA (91/93); Pro- gram Advisory Committee-Holifield Accelerator (89/92); Executive Com- mittee, SURA (88/00,); External Reviewer, Punjab University, Lahore, Pakistan (88/92); Louisiana Science and Technology Commission (87/89); Board Member, SURA (85/00).

Professional Service

Nat ional/International American Physical Society-Committee on International Affairs (03/present); American Physical Society-Forum on International Physics (Ol/present); Associate Editor-Physical Review C (91/93); Program Advi- sory Committee-Holifield Accelerator (89/92); External Reviewer-Punjab University, Lahore, Pakistan (88/92).

15

Scientific Program Committees International Conference on Nuclear Structure at the Extremes, Lewes, England (98); Oaxtepec Symposia on Nuclear Physics, Oaxtepec, Morelos, Mkxico(92/present); Workshop/Symposium on Future Directions in Nu- clear Physics Strasbourg, France (91).

Organizing Committees Institute for Nuclear Theory: Nuclear Structure for the 21st Century, Seat- tle, WA (00); National EPSCoR Conference: Strong Sciences & EPSCoR Alliances, New Orleans, LA (97); Institute for Nuclear Theory: Nuclei Un- der Extreme Conditions Seattle, WA (95); Workshop o Nuclear Structure and Models, Oak Ridge, T N (92); Symposium on Group Theory and Special Symmetries in Nuclear Physics, Ann Arbor, MI (91); Seventh Symposiumn on Gamma Ray Spectroscopy Asilomar, CA(90).

State Science and Technology Commission, Member, LSU.

University Vice President, Faculty Senate, LSU (96/97); Faculty Senate Representa- tive, LSU (95/96); Ad Hoc Committee on Computing Charges, Chair, LSU (89/90); Chancellors Committee on Supercomputing, Chair, LSU (85/89); Student Code of Conduct Committee, LSU (83/87); Faculty Senate Repre- sentative, LSU (83/85); University Task Force on Computing, Chair, LSU (82/85); SNCC Advisory Council, Chair, LSU (80/82).

College Council of Chairman, Chair, LSU (87/93); College Policy Committee, Chair, LSU (84/85); College Policy Committee, Member, LSU (83/84).

Department Chairman (85/94), Numerous others: faculty search, general exam, etc.

It is hard to see when he had time for physics. Jerry, congratulations on your career and on your 60th birthday!

11. SU(3) and Symplectic Models and their Applications

COMPUTATIONAL AND GROUP THEORETICAL METHODS IN NUCLEAR PHYSICS *

J. P. DRAAYER Department of Physics and Astronomy, Louisiana State University,

Baton Rouge, L A 70803-4001, USA E-mail: draayerQ2su. edu

Science is not done in a vacuum - occasions like this give one an opportunity to step back and take an inventory of those individuals who have been part of one’s scientific journey: ‘Masters’ from whom one has learned, ‘Students’ of whom one must always be one; ‘Nexgens’ to whom one looks to carry major campaigns forward. Though circuitous my path at times did seem, I would claim to have enjoyed the best of all - Masters who led by example, Students who grew beyond their teacher, and Nexgens who have stayed the course!

1. Introduction

The challenge we as nuclear physicists face might well be described as making ‘order out of chaos’ - in a literal as well as figurative sense! Indeed, understanding from a scientific perspective what one means by ‘order’ or ‘chaos’ takes one down interesting paths. I t is especially important, given the title of this symposium, that the ‘students’ among us - and here I include myself - resist thinking that ‘Computational Methods’ alone will provide answers to fundamental questions or that ‘Algebraic Methods’ by themselves will necessarily lead t o deeper insights into nature’s secrets. Eu- gene Wigner, who I had the privilege of interacting with when for many years he spent six weeks or so each spring at Louisiana State University (LSU) working with various members of our faculty, is famous for his fre- quent comment on this matter: ‘I am happy to learn that the computer knows the answer; now I would like to understand it as well!’ Notwith- standing’ what I find truly exciting about the world in which we live is that one is beginning to see in tangible ways that the blending of numeric and algebraic methods holds promise of yielding gains that one has not been

*This work is supported by the U.S. National Science Foundation (Grant No. 0140300).

19

20

able to achieve from either separately. From my perspective, the title of this symposium - selected by the organizers - is on target and representative of the physics that over the years I have grown to appreciate and love!

A bit of background: My parents, Bert H. Draayer (1912-1999) and Hattie Kuiters (1914-1979), were both born of Dutch immigrants - the Draayer name tracks back to Apeldoorn and Kuiters on my mother’s side to Arnhem and Rotterdam. I grew up in a farming community in South- ern Minnesota, called Hollandale, which was settled in the early 1900s by immigrants mainly drawn from two Dutch communities in Iowa, one (Kuiters) from around Oskaloosa (Central) and the other (Draayer) from around Sioux City (Northwest). I attended a two room elementary school for grades 1-8 that was a mile’s walk from our family farm, and traveled to Albert Lea for high school, grades 9-12. My high school math teacher convinced me to enter engineering studies at Iowa State University (ISU) and after my second year in college I shifted from electrical engineering into physics, graduating in 1964 with a major in physics and minors in mathematics and electrical engineering. My college roommate from Sully, Iowa - another Dutch community - introduced me to Lois Van Wyk in the summer of 1962 and we married in 1964. Lois and I are the parents of three daughters - Leah McDowell, who is with us a t this symposium was born in Ames, Iowa while I was in graduate school at ISU, and Sarah Milligan and Martha Duncan who were both born while I was a postdoc at the Uni- versity of Michigan. Leah and Sarah have given us four grandchildren, two each - James Patrick and Claire Elizabeth McDowell and Benjamin Miles and Anna Katherine Milligan, respectively.

As this family chronology suggests, I continued my education at ISU, supported by a three-year NASA fellowship followed by a one-year NSF fellowship. I complete the PhD degree in nuclear theory in 1968 under the supervision of Stanley Williams. With the aid of another (postdoctoral) fellowship from the US National Science Foundation, I spend a year at the Niels Bohr Institute in Copenhagen, Denmark which was followed by a year that stretched into four at the University of Michigan working with Karl T. (Ted) Hecht. From Michigan I went to the University of Rochester and worked closely with J. B. (Bruce) French for two years. In 1975, I accepted a faculty position at LSU where I served as chair of the Department of Physics and Astronomy from 1985 through 1994 and as the Associate Commissioner of Higher Education for the State of Louisiana from 1995 through 1999 when the Southeastern Universities Research Association (SURA) , which runs the Jefferson Lab - a major nuclear physics research facility located

21

in Newport News, Virginia - for the US Department of Energy, asked me to assume the position of President and Chief Executive Officer of SURA, a position I now hold along with my faculty position at LSU.

As a graduate student at Iowa State University (1964-68), my thesis advisor was Stanley A. Williams who was a student of J. P. Davidsonl and himself fresh off of a postdoc with J. P. (Phil) Elliott'. He let me define my own thesis research which led to a project entitled: 'A Deformed Po- tential Many-Body Theory'. Those were challenging times for me - strug- gling to understand the Princeton Lecture notes of H. Weyl (1934) and G. Racah (1951), M. Hamermesh's book3, as well as papers that were rolling off the press by individuals like Marcos Moshinsky*, Larry Biedenharn5, Ted Hecht', and others! But it did establish a research philosophy that I have lived by and tried to pass on to my own students, letting each define his or her own project while at the same time trying to advance a general campaign that has over the years been focused on obtaining a better understanding of the intertwining of single-particle and collective degrees of freedom in atomic nuclei. What I did not realize while a graduate student, and this may be the downfall of too much independence, is that my thesis research flurted with a concept that has become known as the geometrical symplectic model. Indeed, while I worked with stretched and squeezed harmonic oscillator states and realized that these were generated by a non-compact algebraic structure that took one beyond SU(3), it was not until the definitive work of David Rowe7 and his resourceful student George Rosenstee18 that we were able to gain a full appreciation for what is now called the Symplectie Shell Model. It was in those early years at ISU that I also met Walter Greiner for the first time, another Lmaster' who has had significant influence on our field as well as on my career, especially through his students Peter Hessg and Dirk Troltenier."

Nuclear physics is truly an international discipline, and I am very proud to be part of such a dynamic community. One of my greatest joys has been working with colleagues from abroad (several in attendance at this meeting) - Bulgaria, Canada, China, Czech Republic, Egypt, Germany, India, Indonesia, Korea, Pakistan, Lithuania, Mexico, Romania, Russia, and the Ukraine - and recruiting good students. Front and center among the first set are my Mexican collaborators who have become a source of real joy and pride for me and with whom I have published more than thirty papers. These are students and colleagues of Marcos Moshinsky who I count it a real privilege to have worked with over the years. While the most notable of these are Octavio Castanos, Peter Hess and Jorge Hirsch,

22

my appreciation for members of the Instituto de Fisica and Instituto de Ciencias Nucleares of UNAM with whom I have had numerous discussions has been and continues to be a real inspiration - thank you all!

With regard to students, I have had but one US national, Calvin Coun- tee, who I am proud to note was also the first African-American student to graduate in physics from LSU. The list includes: Calvin Ray Countee (1981), Yorck Leschber (1987), Husney A. Naqvi (1992), Chairul Bahri (1994), Andrey Blokhin (1996), Jutta Escher (1997), Thomas Beuschel (1998), Gabriela Popa (2001), Gergana Stoitcheva (2002), Vesselin Gue- orguiev (2002), with five others ‘in the shoot’: Kristina Sviratcheva (2003), Kalin Drumev, Hovhannes Grigoryan, Hrayr Matevosyan, and Tomas Dytrych. In addition there was one other student, Dirk Rompf (1998) who while officially a student of Werner Scheid of Giessen, Germany did most of his thesis work at LSU under my supervision. Of these Andrey, Jutta, Thomas, Vesselin, Kristina, Kalin and Tomas are all present while Chairul, Gabriela, Hovhannes and Hrayr would have been if it were not for the tragedy of 9/11 that is having a negative impact on everyone’s ability to travel freely, especially internationals. To me the ‘ideal thesis’ is one that has an introduction and a conclusion with three or more middle chapters, each based on a paper that has been published. In what follows I will present some of the results of my most recent graduate, Vesselin Gueorguiev, whose work focused on pushing the integration of computational and group theoretical methods to the limit as we continue to probe our understanding of the coexistence of collective and independent particle degrees of freedom in nuclei.’l

2. The Mixed-Mode Concept

Two dominant and often competing modes characterize the structure of atomic nuclei. One is the single-particle structure that is demonstrated by the validity of the mean-field concept; the other is the many-particle collective behavior manifested through nuclear deformation. The spherical shell model is the theory of choice whenever single-particle behavior dominates. l2 When deformation dominates, the Elliott SU(3) model or its pseudo-SU(3) extension is the natural choice.2 This duality manifests itself in two dominant components in the nuclear Hamiltonian: respectively, the single-particle term, Ho = Ci E ~ T Z ~ , and a collective quadrupole- quadrupole interaction, HQQ = Q . Q. It follows that a simplified Hamilto- nian H = Xi E i n i - xQ . Q has two solvable limits.

23

perturbed spectrum of particle in 1D box

m harmonic oscillator spectrum

-L L -L L particle in harmonic particle in a 1D box subject to

1D box oscillator harmonic oscillator potential

Figure 1. A schematic representation of the potential for a particle in one-dimensional (1D) box at high energies and a harmonic oscillator (HO) restoring force for low energies.

To probe the nature of such a system, one can consider a simpler prob1em:ll the one-dimensional harmonic oscillator in a box of size 2L . As for real nuclei, this system has a finite volume and a restoring force of a harmonic oscillator type, w2x2 /2 . For this model, shown in Figure 1, there is a well-defined energy scale that measures the strength of the potential at the boundary of the box, E, = w2L2/2 . The value of E, determines the nature of the low-energy excitations of the system. Specifically, depending on the value of E, there are three spectral types: 1) For w -+ 0 the spectrum is simply that of a particle in a box; 2 ) at some value of w , the spectrum begins with E, followed by the spectrum of a particle in a box perturbed by the harmonic oscillator potential; and 3) for sufficiently large w the spectrum is that of a harmonic oscillator below E, which is followed by the perturbed spectrum of a particle in a box.

The last scenario is the most interesting since it provides an example of a two-mode system. For this case the use of two sets of basis vectors, one representing each of the two limits, has physical appeal, especially at energies near E,. One basis set consists of the harmonic oscillator states; the other set consists of basis states of a particle in a 1D box. We call this combination a mixed-mode / oblique-basis approach. In general, the oblique-basis vectors form a nonorthogonal and overcomplete set. Even

24

1 .o

0.8

0.6 c 0 .- c .I 0.4

d > p 0.2 a Q rT -

0.0

-0.2

-0.4

-0- Harmonic Oscillator

I

I

f State Number

Figure 2. The relative deviations from the exact energy eigenvalues for the 1D box plus HO potential (Figure 1) with w = 16, L = r / 2 , h/2n = m = 1. The open circles represent deviation of the exact energy eigenvalue from the corresponding harmonic- oscillator eigenvalue (1 - Eh,,/Eezact), the solid diamonds are the corresponding relative deviation from the energy spectrum of a particle in a 1D box, and the solid squares are the first-order perturbation theory estimates using particle in a 1D box wavefunctions.

though a mixed spectrum is expected around E,, our numerical study that includes up to 50 harmonic oscillator states below E,, shows that the first order perturbation theory in energy using particle in a 1D box wave functions as the zero order approximation to the exact functions works quite well after the breakdown of the harmonic oscillator like spectrum. This observation is demonstrated in Figure 2 which shows the relative deviations from the exact energy spectrum for a particle in a 1D box.

Although the spectrum seems to be well described using first order perturbation theory based on a particle in a 1D box wave functions, the exact wave functions near E, have an interesting structure. For example, the zero order approximation to the wave function used to calculate the energy may not be present at all in the structure of the exact wave function as shown in Figure 3. Another feature also seen in Figure 3 is the common shape of the distribution of the non-zero components along the particle in a 1D box basis. The graph in Figure 4 shows this same effect in nuclei, which is usually attributed to coherent mixing.13>14

25

3. Applications of the Theory

An application of the theory15 to 24Mg, using the realistic two-body interaction of Wildenthal" , demonstrates the validity of the mixed-mode concept. In this case the oblique-basis consists of the traditional spherical states, that yields a diagonal representation of the single-particle interaction, together with collective SU( 3) configurations, that yields a diagonal representation of the quadrupole-quadrupole interaction. The results shown in Figures 5 and 6 illustrate typical outcomes. For example, a SM(2)+(8,4)&(9,2) model space (third bar in Figure 6 reproduces the binding energy (within 2% of the full-space result) as well as the low-energy spectrum. For this case the calculated eigenstates have greater than 90% overlap with the full-space results. In contrast, for a pure rn-scheme spherical shell-model calculation one needs about 60% of the full space, SM(4) - the fourth bar in Figure 6, to obtain comparable results.

Studies13 of the lower pf-shell nuclei 44-48Ti and 48C~, using the realistic Kuo-Brown-3 (KB3) interaction17, show strong SU(3) symmetry breaking due mainly to the single-particle spin-orbit splitting. Thus the KB3 Hamiltonian could also be considered a two-mode system. This is further supported by the behavior of the yrast band B(E2) values that seems to be insensitive to fragmentation of the SU(3) symmetry. Specifically, the quadrupole collectivity as measured by the B(E2) strengths remains high even though the SU(3) symmetry is rather badly broken. This has been attributed to a quasi-SU(3) symmetry14 where the observables behave like a pure SU(3) symmetry while the true eigenvectors exhibit a strong coherent structure with respect to each of the two bases. This provides justification for further study of the implications of mixed-mode shell-model studies. Results from oblique basis calculations for 44Ti are reported in Vesselin Gueorguiev's contribution to the proceedings.18

Future research may provide justification for an extension of the theory to multi-mode shell-model calculations. For example, an immediate extension of the current scheme could use eigenvectors of the pairing interactionlg within an Sp(4) algebraic approach to the nuclear structure2' in addition to collective SU(3) configurations and spherical shell model states. Alterna- tively, Hamiltonian driven basis sets could be considered. For example, the method could use eigenstates of near closed shell nuclei obtained from a full shell-model calculation to form Hamiltonian driven 3-pair states for mid- shell nuclei.'l This would mimic the Interacting Boson Model (IBM)" and the so-called broken-pair theory.'l Likewise, the three exact limits of the

26

c 0 3 P

C 0 0

.- CI

.- L CI

0.4 ~

0 .5 I State 27 i

State 25

0.3

0.2

0.1

0

9 1 1 1 3 1 5 1 7 1 9 21 23 25 27 29 31 33 35 37

1D box state

Figure 3. Bar charts that illustrate the similar, coherent structure of the 25th, 27th and 29th exact eigenvectors in the basis of a free particle in a 1D box. Parameters of the Hamiltonian are w = 16, L = 71.12, h/2n = m = 1.

IBMZ3 can be considered to comprise a three-mode system. Nonetheless, the real benefit of the mixed-mode approach is expected when the spaces encountered are too large to allow for exact calculations.

27

0.12

0.1

0.08

0.06

0.04

0.02

0 0.12 1 i

0.1

0.08

0.06

0.04

0.02

0

J=2

"I

yrast state

0.12

0.1

0.08

0.06

0.04

0.02

0 O b b r Q b O r n b O t D r n t D

~ m ~ o b e o r n m ~ o ~ r n r r r r C Y C U m m * )

Values of the second-order Casimir operator of SU(3) Figure 4. Bar charts that illustrate the very similar, coherent structure of the first three yrast states in 48Cr calculated using the realistic Kuo-Brown-3 interaction (KB3). The horizontal axis is C2 of SU(3) while the height of each bar gives the contribution of that configuration to the corresponding yrast state.

4. Discussion

So can one make 'order out of chaos'? Perhaps a mixed-mode analysis of the type suggested can help to sort out dominant structures. Initial results

28

-76 n

3 E -79

p -82

W * Q) c Q)

(ZI

-85 4d

5 -88

m S 3 -91

E -94

-2000 4000 10000 16000 22000 28000

Number of Basis States

Figure 5. The graph shows the calculated ground-state energy for 24Mg as a function of various model spaces. SM(n) denotes a spherical shell model calculation with up to n particles outside of the ds,z sub-shell. Note the dramatic increase in binding (3.3 MeV) in going from SM(2) to SM(2)+(8,4)&(9,2) (a 0.5% increase in the dimensionality of the model space). Enlarging the space from SM(2) to SM(4) (a 54% increase in the dimensionality of the model space) adds 4.2 MeV to the binding energy.

seem promising, but one must resist advancing a single solution. The mixed- mode concept is an example of what we can do today: computationally complex and algebraically intensive, but definitely doable. There is a need to ensure that the Hamiltonian is chosen appropriately. Current effective interaction theories fall short of allowing for anything other than relatively simple P and Q spaces - this is a requirement that must be addressed if one is to continue such analyses. It also is clear that a team effort is required; no individual or group has all the tools needed to advance such an ambitious program: it is the ‘Nexgens’ to whom we must all look - the ‘Masters’ as well as the ‘Students’ among us!

Results of Sections 2 and 3 are from Vesselin Gueorguiev’s thesis. His help with the preparation of this contribution is gratefully acknowledged.

29

100 _I

Q W 2 ) 57.77

B(8 .4) 63.02

mSM(2)+(8,4)&(9,2) 91.58

0 ~ ~ 4 ) 93.25

SM(4)+(8,4)&(9,2) 98.57

2

53.02

63.77

90.95

92.81

98.73

71.49 59.46

87.72 89.06

89.98 92.47

97.92 98.41

Eigenvectors

70.15 54.14

87.35 82.23

91.10 88.33

98.55 96.59

Figure 6. Bar chart that shows representative overlaps of pure SM(n), pure SU(3), and oblique-basis results with the exact full sd-shell eigenstates. A number within a bar denotes the state with the overlap shown by the bar if it is different from the number for the exact full-space calculation shown on the abscissa. For example, for SM(2) the third eigenvector has the largest overlap with the fourth exact eigenstate, not the third, while the fifth SM(2) eigenvector has greatest overlap with the third exact eigenstate.

References 1. J. P. Davidson, Rev. Mod. Phys. 37, 105 (1965) 2. J. P Elliott, Proc. Roy. SOC. A 245, 128 and 562 (1958); A 272, 557 (1962);

A 302, 509 (1968) 3. M. Hamermesh, Group Theory and Its Application to Physical Problems

( Addison-Wesley, Reading, Massachusetts, 1962) 4. M. Moshinsky, Rev. Mod. Phys. 34, 813 (1962) 5. G. E. Baird and L. C. Biedenharn, J. Math. Phys. 5, 1730 (1964) 6. K. T. Hecht, Nucl. Phys. 62, 1 (1965) 7. D. J . Rowe, Prog. Part. Nucl. Phys. 37, 265 (1996) 8. G. Rosensteel, Nucl. Phys. A 341, 397 (1980) 9. 0. Castaiios, P. Hess, P. Rochford, and J. P. Draayer, Nucl. Phys. A 524, 469

10. D. Troltenier, J. P. Draayer, P. 0. Hess and 0. Castafios, Nucl. Phys. A 576, (1991)

351 (1994)

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11. V. G. Gueorguiev, Ph.D. Dissertation, Louisiana State University (2002) 12. R. R. Whitehead, Nucl. Phys. A182, 290 (1972); R. R. Whitehead, A. Watt,

B. J . Cole, and I. Morrision, Advances in Nuclear Physics 9, ed. M. Baranger, and E. Vogt (Plenum Press, New York, 1977)

13. V. G. Gueorguiev, J . P. Draayer, and C. W. Johnson, Phys. Rev. C 63, 14318

14. P. Rochford and D. J. Rowe, Phys. Lett. B210, 5 (1988); A. P. Zuker, J . Retamosa, A. Poves, and E. Caurier, Phys. Rev. C 52, R1741 (1995); G. Martinez-Pinedo, A. P. Zuker, A. Poves, and E. Caurier, Phys. Rev. C 55, 187 (1997); D. J. Rowe, C. Bahri, and W. Wijesundera, Phys. Rev. Lett. 80, 4394 (1998); A. Poves, J. Phys. G 25, 589 (1999); D. J. Rowe, S. Bartlett, and C. Bahri, Phys. Lett. B 472, 227 (2000)

15. V. G. Gueorguiev, W. E. Ormand, C. W. Johnson, and J. P. Draayer, Phys. Rev. C 65, 024314 (2002)

16. B. H. Wildenthal, Prog. Part. Nucl. Phys. 11, 5 (1984) 17. T. Kuo and G. E. Brown, Nucl. Phys. A114, 241 (1968); A. Poves and A.

18. V. G. Gueorguiev, J . P. Draayer, W. E. Ormand, and C. W. Johnson,

19. J. Dukelsky, C. Esebbag, and P. Schuck, Phys. Rev. Lett. 87, 066403 (2001) 20. K. D. Sviratcheva, A. I. Georgieva, V. G. Gueorguiev, J. P. Draayer, and M.

I. Ivanov, J. Phys. A 34, 8365 (2001) 21. K. L. G. Heyde, The Nuclear Shell Model, ed. J. M. Irvine (Springer-Verlag,

Berlin Heidelberg, 1990) 22. F. Iachello, The Interacting Boson Model, (Cambridgeshire University Press,

New York, 1987) 23. M. Moshinsky and Y. F. Smirnov, The Harmonic Oscillator in Modern

Physics, Contemporary Concepts in Physics Volume 9, ed. H. Feshbach (Har- wood Academic Publishing, Amsterdam, 1996)

(2001)

P. Zuker, Phys. Rep. 70, 235 (1981)

Oblique-basis Calculations For 44 T, this volume.

PSEUDO + QUASI SU(3): TOWARDS A SHELL-MODEL DESCRIPTION OF HEAVY DEFORMED NUCLEI*

JORGE G. HIRSCH, CARLOS E. VARGAS Instituto d e Ciencias Nucleares, Universidad Nacional Auto'noma de Mixico,

Apartado Postal 70-543 Mixico 04510 DF, Mixico E-mail: [email protected], [email protected]

GABRIELA POPA Department of Physics, Rochester Institute of Technology,

Rochester, N Y 14623-5612, USA E-mail: [email protected]

JERRY P. DRAAYER Department of Physics and Astronomy, Louisiana State University,

Baton Rouge, Louisiana 70803, U.S.A. E-mail: draayer@Isu. edu

The pseudo-SU(3) model has been extensively used to study normal parity bands in even-even and odd-mass heavy deformed nuclei. The use of a realistic Hamil- tonian that mixes many SU(3) irreps has allowed for a successful description of energy spectra and electromagnetic transition strengths. While this model is powerful, there are situations in which the intruder states must be taken into account explicitly. The quasi-SU(3) symmetry is expected to complement the model, allowing for a description of nucleons occupying normal and intruder parity orbitals using a unified formalism.

1. Introduction

The SU(3) shell modell has been successfully applied to a description of the properties of light nuclei, where a harmonic oscillator mean field and a residual quadrupole-quadrupole interaction can be used to describe dominant features of the nuclear spectra. However, the strong spin-orbit interaction renders the SU(3) truncation scheme useless in heavier nuclei, while at the

*This work was supported in part by CONACyT (Mexico) and the US National Science Foundation.

31

32

same time pseudo-spin emerges as a good symmetry2. The pseudo-SU(3) model2l3 has been used to describe normal parity

bands in heavy deformed nuclei. The scheme takes full advantage of the existence of pseudo-spin symmetry, which refers to the fact that single- particle orbitals with j = 1 - 1 / 2 and j = ( I - 2 ) + 1 / 2 in the 77 shell lie close in energy and can therefore be labeled as pseudo-spin doublets with quantum numbers j = j, fj = 77 - 1 and I = 1 - 1. The origin of this symmetry has been traced back to the relativistic Dirac equation4.

A fully microscopic description of low-energy bands in even-even and odd-A nuclei has been developed using the pseudo-SU(3) model5. The first applications used pseudo-SU(3) as a dynamical symmetry, with a single irreducible representation (irrep) of SU(3) describing the yrast band up to the backbending region5. On the computational side, the development of a computer code to calculate reduced matrix elements of physical operators between different SU( 3) irreps6 represented a breakthrough in the development of the pseudo-SU(3) model. With this code in place it was possible to include symmetry breaking terms in the interaction.

Once a basic understanding of the pseudo-SU(3) model was achieved and computer codes enabling its application developed, a powerful shell-model theory for a description of normal parity states in heavy deformed nuclei emerged. For example, the low-energy spectra and B(E2) and B(M1) electromagnetic transition strengths have been described in the even-even rare

I -

earth isotopes 154Sm, 156,158,160Gd 160,162,164D and 164,166,168~~7,8 ,9

and in the odd-mass 157Gd, 1591161Tb, 159,163Dy, 1 5 9 ~ U , 1619169Tm, and 1 6 5 , 1 6 7 ~ ~ nucleil 1,12,13

In the present contribution we review recent results obtained using a modern version of the pseudo-SU(3) formalism, which employs a realistic Hamiltonian with single-particle energies plus quadrupole-quadrupole and monopole pairing interactions with strengths taken from known systematicsl1>l2. Its eigenstates are linear combinations of the coupled pseudo-SU(3) states. The quasi SU(3) approach for intruder states is also discussed, together with its implications regarding a unfied description of a system with nucleons occupying normal and intruder parity orbitals.

2. The Pseudo SU(3) Basis and Hamiltonian

Many-particle states of n, active nucleons in a given normal parity shell qa, ct = v (neutrons) or T (protons), can be classified by the group chain U(@) 3 U ( f 2 3 2 ) x U ( 2 ) 3 SU(3) x S U ( 2 ) 3 SO(3) x S U ( 2 ) 3 S U J ( 2 ) ,

33

where each group in the chain has associated with it quantum numbers that characterize its irreps.

The most important configurations are those with highest spatial ~ y m m e t r y ~ y ~ ~ . This implies that ST,” = 0 or 1/2, that is, the configurations with pseudo-spin zero for an even number of nucleons or 1/2 for an odd number are dominant. In some cases, particularly for odd-mass nuclei, states with 3, = 1 and 3, = must also be taken into account, allowing for coupled proton-neutron states with total pseudo-spin 3 = $, or 5. Since pseudo-spin symmetry is close to an exact symmetry in the normal parity sector of the space, a strong truncation of the Hilbert space can be invoked. However,the pseudo spin-orbit partners are not exactly degenerate and this introduces a small pseudo-spin mixing in the nuclear wave function.

The Hamiltonian,

includes spherical Nilsson single-particle energies for 7~ and u as well as the pairing and quadrupole-quadrupole interactions, with their strengths taken from s y ~ t e m a t i c s l ~ > ~ ~ . Only the parameters a, b and Asym are used fit to the data. A detailed description of each term in the Hamiltonian (1) can be found in Ref.12.

The electric quadrupole operator is expressed as5

with effective charges e, = 2.3 and e, = 1.35t7. The magnetic dipole operator is

where the ‘quenched’ g factors for 7~ and u are used. To evaluate the M1 transition operator between eigenstates of the Hamiltonian (l), the pseudo SU(3) tensorial expansion of the T1 operator (3)16 was employed.

3. Some Representative Results

The experimental and theoretical ground-, beta- and gamma-bands in 166Er are shown in Fig. 1. Having a close connection with the rotor Hamiltonian,

34

2

1.5

F - z a - A - t? 00.5

0 -

6' 6+ - - - 4+ 4+ - - 8+ 2+ - - 2+ 8+ -

7+- ,7+ o+ - -o+

8+ - - 8' 3+- -3+ 2+- -2+

- Er 166 -

6+- -6+

4+- 5+- -5+ -4+ Kn =O+ - 1-

6' - - 6' - - Kn =2+

4+ - - 4+ 2+ - - 2+ 0 ' - -o+ -

g.s.

Figure 1. Energy spectra of 166Er

Ji + J f

Ogs -+ 2,, 29s -+ 49s 49s -+ 69s 6,, -+ 8,, 2, -+ 4,, 2, + 2,, 0,s -+ 2, 49s -+ 5,

the pseudo SU(3) model is particularly well suited to describe these bands. The term proportional to IT; allows the position of the gamma ( K J = 2) band-head to be fit, a particularly difficult task in many fermionic models.

Experimental and theoretical B(E2) transition strengths in lssEr are shown in Table 1. Effective charges used are 1.25e and 2 . 2 5 ~ Transitions between states in the ground-state band are of the order of e2b2, while those from the 7- to ground-state bands are far smaller. The agreement with the experimental information is remarkable.

B(E2)[e2b2 x Exp. Th.

580 f 27 580 303 & 20 299 273 k 35 265 258 f 35 251

0.363 k 0.027 1.485 4.915 f 0.038 10.310

15 & 1 17 2 0.27 3.22

Table 1. Experimental and theoretical

35

b V 1

1,600

1’200

- 13- band F

163Dy band E ,11- -9-

band D 15- ‘13- -11 .

- ‘13- -7- -5-

-11- -9- -7-

’ band A 3-

band C band B

. (g.s.b.) ( 7 1 -1- 3-

17- (5-l- (9-7 ( 3 - k

Fig. 2 shows the yrast and six excited normal parity bands in 163Dy. The integer numbers denote twice the angular momentum of each state. Experimental17 energies are plotted on the left-hand-side of each column, while their theoretical values are shown in the right-hand-side. These results should be compared with the three bands described in an earlier study13, where the same Hamiltonian and parametrization was employed but the Hilbert space was restricted to ST = 0 and 3” = !j states. The present description reproduces almost all the data reported for normal parity bands in this nucleus.

Table 2 shows the B(E2) intra- and inter-band transition strengths for 167Er, in units of e2b2 x

Experimental data are shown with error bars in the figure and in paren- thesis in the table. As usual, the intra-band transitions are in general two orders of magnitude larger than the inter-band transition strengths. In both cases the agreement with experiment is very good.

Effective charges were 1.3e and 2.3e.

400

0

(I-) ( 7 7 - (15-1 - (9-) -9- - -9- 5- - - (7-1 -7- 03-1 (5;) 3-(5-)- -5- - - 13- 3-- - 5-(37 - 3- 1 1 1 - (1:) -1-

15- 7; -7-

*

11- . 9-- -9- 7-- -7- - 5-- -5-

Figure 2. Energy spectra of 162Dy

36

Table 2. Theoretical B(E2) transition strengths for 167Er, in e2b2 x

Jt: t J; 112; --f 312; 312; --f 512; 512; --f 712; 712; t 9/2; 912; --f 1112; 1112; --f 1312; 112; -+ 512; 312; t 712; 512; t 912; 712; t 1112; 912; + 1312;

B(E2) 275 59 25 15 10 10

415 353 328 308 30 1

J, t J T B(E2) 512; -+ 712; 310 7 / 2; --f 9/23 252

9/23 + 1112; 186 5/22 t 9/23 100 7/22 + 11/24 151

The results shown above faithfully display the usefulness of the pseudo SU(3) model in the description of normal parity bands in heavy deformed nuclei. However, as already mentioned, the role of nucleons in intruder parity orbitals cannot be underestimated. The quasi SU(3) symmetry offers the possibility to describe them in similar terms as those occupying normal parity orbitals.

4. Quasi SU(3) Symmetry

The “quasi SU(3)” symmetry, uncovered in realistic shell-model calculations in the pf-shell, describes the fact that in the case of well-deformed nuclei the quadrupole-quadrupole and spin-orbit interactions play a dominant role and pairing can be included as a perturbation. In terms of a SU(3) basis, it is shown that the ground-state band is built from the S = 0 leading irrep which couples strongly to the leading S = 1 irreps in the proton and neutron subspaces. Furthermore, the quadrupole-quadrupole interaction was found to give dominant weights to the so-called “stretched” coupled representations, which supports a strong SU(3)-dictated truncation of the model space.

The interplay between the quadrupole-quadrupole and the spin-orbit interaction has been studied in extensive shell-model calculations18 as well as in the SU(3) basis14. In the former case1* the authors studied systems with four protons and four neutrons in the pf and sdg shells, and compared the mainly rotational spectra obtained in a full space diagonalization of the realistic KLS interaction with those obtained in a truncated space and a Hamiltonian containing only quadrupole-quadrupole and spin-orbit inter-

37

actions. They found that for realistic values of the parameter strengths the overlap between the states obtained in the two calculations is always better than (0.95)2. They also found that while additional terms in the interaction change the spectrum, the wave functions remain nearly the same, suggesting that the differences can be accounted for in a perturbative way.

Following these ideas, a truncation scheme suitable for calculations in a SU(3) basis was worked out14. In contrast with what was done in19, systems with both protons and neutrons were analyzed, and the interplay of the quadrupole-quadrupole and spin-orbit interactions was emphasized, while the pairing interaction was not included in the considerations for building the Hilbert space.

The SU(3) strong-coupled proton-neutron basis span the complete shell- model space, and represents an alternative way of enumerating it. In order to define a definite truncation scheme that clei, inI4 we investigated the Hamiltonian

H = - - Q . Q X - C 2

is meaningful for deformed nu-

i

where

is the total mass quadrupole operator, which is just the sum of the proton (T) and neutron (v) mass quadrupole terms, restricted to work within one oscillator shell, and I:;, .?i are the orbital angular momentum and spin of the i-th nucleon, respectively. An attractive quadrupole-quadrupole Hamilto- nian classifies these basis states according to their CZ values, the larger the C2 the lower the energy. The spin-orbit operator can be written as l4

(l,l)L=S=l,J=O (6)

1 (7 + 3)! X I : ; . Si = -- 2 [ 2(7- l)! ] [4q,0) , /2~(0 , , ) l /2 ] i

Results for 22Ne are presented in Table 3. Modern shell-model calculations20 exhibit more mixing of SU(3) irreps than previous ones21. The ground-state band, often described as a pure (8,2) state, has important mixing with the spin 1 (9,O) irrep. The J = 1 state with dominant (6,3) L = 1, S = 0 components mixes strongly with (7,l) S = 0 and others, in agreement with the shell-model results.

Extensive calculation of the energy spectra and electromagnetic transitions in many even-even, even-odd and odd-odd nuclei along the sd-she1122

38

Table 3. lated wave functions for J = 0 and 1 states of 22Ne

Comparison of the main components of calcu-

confirm that the quasi S(3) symmetry can be used as a useful truncation scheme even when the spin-orbit splitting is large.

5. Summary and Conclusions

A quantitative microscopic description of normal parity bands and their B(E2) intra- and inter-band strengths in many even-even and odd-mass heavy deformed nuclei has been obtained using a realistic Hamiltonian and a strongly truncated pseudo SU(3) Hilbert space, including in some cases pseudo-spin 1 states.

In light deformed nuclei the interplay between the quadrupole- quadrupole and spin-orbit interactions can be described in a Hilbert space built up with the leading S=O and 1 proton and neutron irreps, in the stretched SU(3) coupling. In heavy deformed nuclei this quasi SU(3) truncation scheme will allow the description of nucleons occupying intruder single-particle orbits.

Using the pseudo + quasi SU(3) approach it should be possible to perform realistic shell-model calculations for deformed nuclei throughout the periodic table.

This work was supported in part by Conacyt and DGAPA-UNAM (IN119002), Mbxico, and the U.S. National Science Foundation, Grants Numbers 9970769 and 0140300.

References

1. J. P. Elliott, Proc. Roy. SOC. London Ser. A 245, 128 (1958); 245, 562 (1958).

39

2. K. T. Hecht and A. Adler, Nucl. Phys. A 137, 129 (1969); A. Arima, et al. Phys. Lett. B 30, 517 (1969).

3. R. D. Ratna Raju, J. P. Draayer, and K. T. Hecht, Nucl. Phys. A 202, 433 (1973).

4. A. L. Blokhin, et. al., Phys. Rev. Lett. 74, 4149 (1995); J . N. Ginocchio, Phys. Rev. Lett. 78, 436 (1997); J. Meng, et. al. Phys. Rev. C 58, R632 (1998).

5. J. P. Draayer, et. al., Nucl. Phys. A 381, 1 (1982). 6. C. Bahri and J. P. Draayer, Comput. Phys. Commun. 83, 59 (1994). 7. T. Beuschel, J. P. Draayer, D. Rompf, and J. G. Hirsch, Phys. Rev. C 57, 1233

(1998); J. P. Draayer, T. Beuschel, D. Rompf, J. G. Hirsch, Rev. Mex. Fis. 44 Supl. 2 , 70 (1998) ; ibid Phys. At. Nuclei 61, 1631 (1998); J. P. Draayer, T. Beuschel, and J. G. Hirsch, Jour. Phys. G - Nucl. Part. Phys. 25, 605 (1999).

8. T. Beuschel, J. G. Hirsch, and J. P. Draayer, Phys. Rev. C 61, 54307 (2000). 9. G. Popa, J. G. Hirsch, and J. P. Draayer, Phys. Rev. C 62, 064313 (2000). 10. J. P. Draayer, G. Popa, and J. G. Hirsch, Acta Phys. Pol. B 32, 2697 (2001);

J. G. Hirsch, G. Popa, C. E. Vargas, and J. P. Draayer, Heavy Ion Physics 16, 291 (2002).

11. C. Vargas, J. G. Hirsch, T. Beuschel, and J. P. Draayer, Phys. Rev. C 61, 31301 (2000); J. G. Hirsch, C. E. Vargas, and J. P. Draayer, Rev. Mex. Fis. 46 Supl. 1, 54 (2000).

12. C. E. Vargas, J. G. Hirsch, and J. P. Draayer, Nucl. Phys. A 673, 219

13. C. Vaxgas, J. G. Hirsch, and J. P. Draayer, Phys. Rev. C 66, 064309 (2002);

14. C. Vargas, J. G. Hirsch, P. 0. Hess, and J. P. Draayer, Phys. Rev. C 58,

15. P. Ring and P. Schuck. The Nuclear Many-Body Problem Springer, Berlin

16. 0. Castaiios, et. ai., Ann. of Phys. 329, 290 (1987). 17. National Nuclear Data Center, http://bnlnd2.dne.bnl.gov 18. A. P. Zuker, J. Retarnosa, A. Poves, and E. Caurier, Phys. Rev. C 52, R1741

19. J. Escher, C. Bahri, D. Troltenier, and J. P. Draayer, Nucl. Phys. A 633,

20. J. Retamosa, J. M. Udias, A. Poves, and E. Moya de Guerra, Nucl. Phys A

21. Y. Akiyama, A. Arima, and T. Sebe, Nucl. Phys. A 138, 273 (1969). 22. C. E. Vargas, J. G. Hirsch, and J. P. Draayer, Nucl. Phys. A 690, 409 (2001);

(2000).

ibid, Phys. Lett. B 551, 98 (2003).

1488 (1998).

(1979); M. Dufour and A. P. Zuker, Phys. Rev. C 54, 1641 (1996).

(1995).

662 (1998).

511, 221 (1990).

ibid. Nucl. Phys. A 697, 655 (2002).

PARTIAL DYNAMICAL SYMMETRY IN NUCLEAR SYSTEMS

JUTTA ESCHER Nuclear Theory and Modeling Group, N Division

Lawrence Livermore National Laboratory, P.O. Box 808, L-414, Livermore, CA 94551, USA

E-mail: escherl @llnl.gov

Partial dynamical symmetry (PDS) extends and complements the concepts of exact and dynamical symmetry. It allows one to remove undesired constraints from an algebraic theory, while preserving some of the useful aspects of a dynamical symmetry, and to study the effects of symmetry breaking in a controlled manner. An example of a PDS in an interacting fermion system is presented. The associated PDS Hamiltonians are closely related with a realistic quadrupole-quadrupole interaction and provide new insights into this important interaction.

1. Introduction

Algebraic, symmetry-based theories provide an elegant and practical a p proach for describing and understanding a variety of physical systems. These theories offer the greatest simplifications when the interaction under consideration is symmetry preserving in the selected state labeling scheme, that is, when the Hamiltonian either commutes with all the generators of a particular group (‘exact symmetry’) or when it is written in terms of and commutes with the Casimir operators of a chain of nested groups (‘dynamical symmetry’). An exact or dynamical symmetry not only facilitates the numerical treatment of the Hamiltonian, but also its interpretation and thus provides considerable insight into the physics of a given system.

Algebraic models can also be of value in situations where it is necessary to introduce symmetry-breaking terms in the Hamiltonian in order to achieve agreement with experimentally observed features. Pragmatically, it is often possible to decompose the offending terms into basic parts (“irreducible tensor operators”) which exhibit specific transformation properties. Provided the appropriate group coupling coefficients and the matrix elements of some elementary tensor operators are available, matrix elements

40

41

of operators that connect inequivalent irreducible representations can be determined and the exact eigenvalues and eigenstates can then be obtained (at least in principle). Furthermore, by studying the effects of symmetry- breaking terms, one gains new insights into the dynamics of the system, the relevance and robustness of the symmetries under consideration, and their limitations. In some cases new symmetries emerge from a broken-symmetry regime. Their identification and interpretation is often simplified in the framework of an algebraic model. Examples of such emerging symmetries include quasi-dynamical symmetry' and pseudo-spin symmetry2.

One can also consider introducing some intermediate structure that allows for symmetry breaking but retains some aspects of the dynamical symmetry. Partial dynamical symmetry (PDS) provides such a structure3. Two types of partial dynamical symmetry have been studied in recent years. Partial dynamical symmetry of the first kind preserves the advantages of a dynamical symmetry for a part of the system. It corresponds to a particular symmetry breaking for which the Hamiltonian is not invariant under the symmetry group and hence various irreducible representions (irreps) are mixed in its eigenstates, yet it possesses a subset of 'special' solvable states which respect the symmetry. PDS of the second kind corresponds to a symmetry breaking for which all eigenstates of the Hamiltonian preserve part of the dynamical symmetry4. In this scenario, the dynamical symmetry associated with an intermediate group G2 in a subchain GI 3 G2 3 G3 is broken for all states of the system, while the remaining (dynamical) symmetries are preserved. The resulting Hamiltonian is in general not analytically solvable, but its eigenstates can still be (partly) classified by quantum labels associated with the groups GI and Gs. Further generalizations of the partial symmetry concept have been considered as well5.

This contribution will discuss an example of a partial dynamical symmetry in an interacting fermion system. More specifically, in the framework of the symplectic shell model (SSM), there exists a family of PDS Hamiltoni- ans which are closely related to the nuclear quadrupole-quadrupole interaction. The Hamiltonians and their eigenstates are discussed and applications to the deformed light nuclei "Ne and 24Mg are presented.

2. Quadrupole-Quadrupole Interaction in the SSM

The quadrupole-quadrupole interaction is an important ingredient in models that aim at reproducing quadrupole collective properties of nuclei. A model which is able to fully accommodate the action of the collec-

42

tive quadrupole operator, Q 2 , = ~ C , T ~ Y Z , ( ~ . , ) , is the symplectic shell model (SSM), an algebraic scheme which respects the Pauli ex- clusion principle6. In the SSM, this operator takes the form Q2m =

generators with good SU(3) [superscript (A, p)] and SO(3) [subscript I,m] tensorial properties. The As:) (B;:)), I = 0 or 2, create (annihilate) 2hw excitations in the system. The C::), I = 1 or 2, generate a SU(3) subgroup and act only within one harmonic oscillator (h.0.) shell (dCi:) = Q,",, the symmetrized quadrupole operator of Elliott, which does not couple different h.0. shells7, and Cj:) = i,, the orbital angular momentum operator). A fermion realization of these generators has been givens.

A basis for the symplectic model is generated by applying symmet- rically coupled products of the 2tiw raising operator A('') with itself to the usual O t i w many-particle shell-model states. Each O h starting configuration is characterized by the distribution of oscillator quanta into the three Cartesian directions, or, equivalently, by its U(l) xSU(3) quantum numbers Nu (A,,,p,,). Here (A,,, p,,) are the Elliott SU(3) labels, and N,, = ~1 + 6 2 + 03 is related to the eigenvalue of the oscillator number operator. 20Ne, for instance, has N,, = 48.5 (after removal of the center- of-mass contribution) and (A,,,p,,) = (8,O). For '*Mg, one finds Nu = 62.5 and (A,,,p,,) = (8,4). The product of N/2 raising operators A('') generates Ntiw excitations for each starting irrep Nu (A,,, p,,). Each such product operator PN('-+n), labeled according to its SU(3) content, ( A n , p n ) , is coupled with IN,, (A,,, p,,)) to good SU(3) symmetry p(A, p ) , with p denoting the multiplicity of the coupling (An, p n ) @ (A,,,pu). To complete the basis state labeling, additional quantum numbers CY = nLM are required, where L denotes the angular momentum with projection M , and 6 is a multiplicity index, which enumerates multiple occurrences of a particular L value in the SU(3) irrep (A,p). The group chain corresponding to this labeling scheme is Sp(6,R) 3 SU(3) 3 SO(3) which defines a dynamical symmetry basis.

The quadrupole-quadrupole interaction connects h.0. states differing in energy by O h , f 2 h , and f 4 h , and may be written as

&($;) + $20) *(02)) , where A(2o) Im , B,, A ( o 2 ) , and ($0 are symplectic 2 m + '2,

Q 2 ' Q 2 = 9Csu3 - 3Csp6 + Hi - 2Ho - 3L2 - 6 A o B o

+{terms coupling different h.0. shells} , (1)

where C.9~3 and Csp6 are the quadratic Casimir invariants of SU(3) and Sp(6,R) with eigenvalues 2(A2+p2+Ap+3A+3p)/3 and 2(A: +p:+A,p,+

43

3A,+3pu]/3+N,2/3-4N,, respectively. These operators, as well as the h.0. I^r, and L2 terms, are diagonal in the dynamical symmetry basis. Unlike the Elliott quadrupole-quadrupole interaction, Q f . Qf = 6 6 ~ ~ 3 - 3L2, the Q2. Q2 interaction of Eq. (1) breaks SU(3) symmetry within each h.0. shell since the term AoBo = Ar’By) = ( { A x B}y) - &{A x B}g2))/& mixes different SU(3) irreps.

A n

3. Partial Dynamical Symmetry in the SSM

In order to study the action of Q2 . Q2 within a h.0. shell, we consider the following family of Hamiltonians:

H(P0, P2) = PoAoBo + P2A2 . B2 (2)

For PO = P2, one recovers the dynamical symmetry, and with the spe- cia1 choice PO = 12, P2 = 18, one obtains Q2 . Q2 = H(Po = 12,Pz = 18) + const(N) - 3L2 + terms coupling different shells, where const(N) is constant for a given h.0. N i b excitation.

It has been showng that H(Po,,&) exhibits partial SU(3) symmetry of the first kind. Specifically, one finds that among the eigenstates of H(Po,P2), there exists a subset of solvable pure-SU(3) states, the SU(3)>SO(3) classification of which depends on both the Elliott labels (A,,, p,) of the starting state and the symplectic excitation N . In general, one observes that all L-states in the starting configuration ( N = 0) are solvable with good SU(3) symmetry (A,, p,). For excited configurations ( N > 0 and even) one distinguishes between two possible cases:

(a) A, > p,: the pure states belong to (A, p ) = (A, - N , pa + N ) and have L = p, + N , p, + N + 1 , . . . ,A, - N + 1 with N = 2,4, . . . subject to 2N 5 (A, - p, + 1).

(b) A, 5 p,: the special states belong to (A ,p ) = (A, + N , p , ) and have L = A, + N , A, + N + 1 , . . . ,A, + N + p, with N = 2,4,. . ..

The special states have well-defined symmetry Sp(6,R) 2 SU(3) 3 SO(3) and are annihilated by SO. This ensures that they are solvable eigenstates of H(P0, P2) with eigenvalues E ( N = 0) = 0, E ( N ) = P2N(N, - A, + p,, - 6+3N/2)/3 for family (a), and E ( N ) = P2N(N,+2X,+pu-3+3N/2)/3 for family (b). All O i b states are unmixed and span the entire (Ac, p,) irrep. In contrast, for the excited levels ( N > 0), the pure states span only part of the corresponding SU(3) irreps. There are other states at each excited

44

level which do not preserve the SU(3) symmetry and therefore contain a mixture of SU(3) irreps. All eigenstates respect the Sp(6,R) and SO(3) symmetries. The partial SU(3) symmetry of H(Po,,&) is converted into partial dynamical SU(3) symmetry by adding to it SO(3) rotation terms which lead to L(L+1)-type splitting but do not affect the wave functions. The solvable states form rotational bands and since their wave functions are known, one can evaluate the E2 rates between themg.

4. Applications

To illustrate that the PDS Hamiltonians discussed here are physically relevant, applications to realistic nuclear systems have been considered. Here the results for 20Ne and 24Mg are summarized. In particular, energy spectra and eigenstates of H p ~ s = h ( N ) + EH(Po = 12, ,& = 18) + y2L2 + y4L4 are compared to those of HQ.Q = f i 0 - xQ2 .Qz + d2L2 +d4Z4, where h ( N ) is a constant for a given Ntiw excitation and contains the h.0. term Ho.

Figure 1. Comparison between experimental values (left), results from a symplectic 8 h calculation (center) and a PDS calculation (right). The angular momenta of the positive parity states in the rotational bands are L=0,2,4,. . . for K=O and L=K,K+l,K+2, . . . otherwise.

Energy spectra for 20Ne.

4.1. The "Ne Example

In Fig. 1, energy spectra of HPDS are compared to those obtained from an 8tiW symplectic calculation (labeled Q 2 . &2), and Fig. 2 shows the decomposition for representative (2+) states of the five lowest rotational bands. The PDS Hamiltonian H p ~ s acts only within one oscillator shell, hence its eigenfunctions do not contain admixtures from different Ntiw configurations. As expected, H p ~ s has families of pure SU(3) eigenstates which can be organized into rotational bands, Fig. 1. The ground band belongs entirely to N = 0, (X,p) = (8,0), and all states of the K=21 band have quantum labels N = 2, (X,p) = (6,2), K, = 2, see Fig. 2. A comparison

45

with the symplectic case shows that the Ntiw level to which a particular PDS band belongs is also dominant in the corresponding symplectic band. In addition, within this dominant excitation, eigenstates of HPDS and HQ.Q have similar SU(3) distributions; in particular, both Hamiltoni- ans favor the same (A, p ) ~ values. Significant differences in the structure of the wave functions appear, however, for the K=02 resonance band. In the symplectic calculation, this band contains almost equal contributions from the O h , 2 h , and 4tiw levels, with additional admixtures of 6fw and 8tiW configurations, while in the PDS calculation, it belongs entirely to the 2tiW level. These structural differences are also evident in the interband transition ratesg and reflect the action of the inter-shell coupling terms in Eq. (1). Increasing the strength x of Qz . Qz in HQ.Q will also spread the other resonance bands over many N h excitations. The K=21 band (which is pure in the PDS scheme) is found to resist this spreading more strongly than the other resonances. For physically relevant values of x, the low-lying bands have the structure shown in Fig. 2.

0 2 4 6 8 0 2 N

Figure 2. Decomposition for calculated 2+ states of 20Ne. Individual contributions from the relevant SU(3) irreps at the O h and 2 h levels are shown for both a symplectic 8fiw calculation (denoted Q2 . Q2) and a PDS calculation. In addition, the total strengths contributed by the N h excitations for N > 2 are given for the symplectic case.

46

4.2. The 2 4 M g Example

For the triaxially deformed nucleus 24Mg additional terms X3 = (ex Q E ) . e and X4 = (e x QE) . (2 x QE) are required in the Haniiltonian in order to reproduce the experimentally observed 'K-band splitting' between the ground and y band of 24Mg. Although these extra terms break the partial symmetry, for realistic interaction parameters the amount of symmetry breaking is very small (- 1%). In Fig. 3, energy spectra of H;,, =

H ~ D s + c ~ X ~ + C ~ X ~ and Hb,Q = H Q . Q + C 3 3 3 + C 4 2 4 are shown. HbD, has families of pure (and nearly pure) SU(3) eigenstates which can be organized into rotational bands; they are indicated in the figure.

60

50

2 40 z

30 ?

20

10

0

Figure 3. Energy spectra for 24Mg. Energies from a PDS calculation (PDS) are compared to symplectic 6tW results ( Q z . Q z ) . Both Oh-dominated bands (K=01,21,41) and some 2 h resonance bands are shown. The K=Oi and K=21 labels indicate the ground band and y band, respectively.

The results are qualitatively similar to those €or "Ne. The PDS Hamil- tonian cannot account for intershell correlations, but it is able to reproduce various features of the quadrupole-quadrupole interaction, as can be seen in Fig. 4, where the structure of selected PDS eigenstates is compared to that of the corresponding Q2 .Q2 eigenstates: PDS eigenfunctions do not contain admixtures from different N f w configurations, but belong entirely to one level of excitation. For reasonable interaction parameters, the N f w level to which a particular PDS band belongs is also dominant in the corresponding band of exact Q2 . Q2 eigenstates. Within this dominant excitation, eigenstates of both Hamiltonians have similar SU(3) distributions. Structural differences, nevertheless, do arise and are reflected in the very sensitive interband transition ratesg. Furthermore, due to the presence of X3 and X4, HbDs is only an approximate PDS Hamiltonian - the K=61 band has small admixtures from irreps other than (A, p) = (6,6). Overall, it may be concluded that PDS eigenstates approximately reproduce the structure of the exact Q2 . Q2 eigenstates, for both ground and resonance bands.

47

Figure 4. Decompositions for calculated Lrr = 6+ states of 24Mg. Eigenstates resulting from a symplectic 6tW ( Q 2 . Q z ) calculation are decomposed into their O h , 2tW, 4tW, 6tW components. At the O t W and 2 b levels, contributions from individual SU(3) irreps are shown, for higher excitations (N > 2) only the summed strengths are given. Eigenstates of H b D S belong entirely to one NtW level of excitation, here O h or 2tW; members of the K=01 and K=21 bands are pure and K=61 states are very nearly (> 99%) pure.

5 . Concluding Remarks

The notion of partial dynamical symmetry extends and complements the familiar concepts of exact and dynamical symmetry. It allows one to remove undesired constraints from an algebraic theory while preserving some of the useful aspects of a dynamical symmetry. As a result, the effects of symmetry breaking can be studied in a controlled manner and new insights into dynamics of the system under consideration are gained.

The work presented here focuses on a family of PDS Hamiltonians which are closely related to the deformation-inducing quadrupole-quadrupole interaction. For a particular parametrization, the PDS Hamiltonians take a form that is intermediate between the full quadrupole-quadrupole interaction, which couples states belonging to different harmonic oscillator shells, and the Elliott quadrupole-quadrupole interaction, which acts only within a shell. The intermediate scheme considered here extends the Elliott picture in that it includes (specific) SU(3) symmetry-breaking contributions.

48

At the same time, it is simpler than the full collective picture since it does not allow for mixing between different oscillator shells.

The PDS scheme sheds light on the in-shell behavior of the quadrupole- quadrupole interaction. For example, the symplectic model predicts the existence of states that are primarily dominated by one N f w level of excitation as well as states that contain strong multi-shell correlationsa. The states that resist the deformation-induced spreading over several N h levels of excitation the strongest are those for which the associated PDS structure exhibits good (or almost good) SU(3) symmetry.

Acknowledgments

It has been a pleasure to present this work on the occasion of Jerry Draayer’s 60th birthday. The research follows a tradition of using symmetry principles to gain insights into physical systems. Throughout his career, Jerry Draayer has contributed much to this tradition and has inspired many to seek the simplicity, symmetry, and beauty hidden within complex physical systems. The work presented here was carried out in collaboration with A. Leviatan (Hebrew University, Jerusalem). It was performed in part under the auspices of the U. S. Department of Energy by the University of California, Lawrence Livermore National Laboratory, under contract No. W-7405-Eng-48.

References 1. C. Bahri, D.J. Rowe, and W. Wijesundera, Phys. Rev. C 58, 1539 (1998);

C. Bahri and D.J. Rowe, Nucl. Phys. A 662,125 (2000); D.J. Rowe, Embedded representations and quasi-dynamical symmetry, contribution in this volume.

2. See J.N. Ginocchio’s and P. Van Isacker’s contributions in this volume and references therein.

3. Y. Alhassid and A. Leviatan, J. Phys. A25, L1265 (1992); A. Leviatan, Phys. Rev. Lett. 77, 818 (1996).

4. P. Van Isacker, Phys. Rev. Lett. 83, 4269 (1999). 5. A. Leviatan and P. Van Isacker, Phys. Rev. Lett. 89, 222501 (2002). 6. G. Rosensteel and D.J. Rowe, Phys. Rev. Lett. 38, 10 (1977); Ann. Phys.

126, 343 (1980); D.J. Rowe, Rep. Prog. Phys. 48, 1419 (1985). 7. J.P. Elliott, Proc. Roy. Soc. A245, 128 (1958); A245, 562 (1958). 8. J . Escher and J.P. Draayer, J. Math. Phys. 39, 5123 (1998). 9. J. Escher and A. Leviatan, Phys. Rev. Lett. 84, 1866 (2000); Phys. Rev. C65,

24305 (2002). 10. C. Bahri et al., Phys. Lett. B234, 430 (1990).

aThis result extends the findings of Bahri et aZ.lO, who studied the structure of the giant quadrupole resonance in 24Mg.

111. Random Hamiltonians

SYSTEMATIC CORRELATIONS AND CHAOS IN MASS FORMULAE*

V~CTOR VELAZQUEZ, ALEJANDRO FRANK, AND JORGE G. HIRSCH Instituto de Ciencias Nucleares, Uniuersidad Nacional Autdnoma de Me'xico,

Apartado Postal 70-543, 04510 Me'xico, D.F., Me'xico E-mail: uic @nuclecu. unam. m x , frank @nuclecu.unam. mx ,

hirs ch @nuclecu. unam. m x

We make a systematic study of correlations in the chart of calculated masses of Moller and Nix and we find that it is possible to reduce the rms by 20%. The correlations can have important consequences in the errors as signaling the presence of chaos, as was recently proposed.

The importance of nuclear masses to understand diverse processes in nuclear physics and astrophysics is very well known '. Moller and Nix 2 ,

Duflo and Zuker among many others have developed mass formulae that calculate and predict the masses (and often other properties) of as many as 8979 nuclides. Recently, the problem of the mass deviations was analyzed from a new angle: in Ref. the errors among experimental and calculated masses in was interpreted in terms of two types of contributions. The first contribution was associated with the the regular part of the error, related to the underlying collective dynamics, and the other arising from some higher order interactions among nucleons 4, that lead to chaotic behavior. The second component of the error would be associated with an impossibility of obtaining an arbitrary precision in the calculation of the masses, independently of the model, setting up limits which cannot be improved upon. It was shown in that the regular part has characteristic fluctuations of 3 MeV without displaying any mass dependence, while the chaotic part has fluctuations of the order of 0.5 MeV and a mass behavior close to AP1l3. In

a systematic study of masses was carried out using the shell model, in an attempt to clarify the nature of the errors. This was achieved by employing realistic Hamiltonian with a small random component.

*This work was supported in part by CONACyT (MBxico).

51

52

The main motivation for the present work arose from Fig. 1, which displays the error in the 1654 isotopes of and the unexpected symmetrical distribution with two peaks around the origin. In this article we carry out a systematic study of the nature of this double peak to show the robustness of its presence and the implication in regards to the average deviation. Fig. 1 shows the distribution of errors obtained by Moller and Nix for different distribution intervals. The intermediate figure (lines and triangles) corresponds to a distribution of 0.1 MeV where the double peak is quite evident. Enlarging the distribution interval to 0.2 MeV softens the curva-

-3 -2 -1 0 1 2 3

Mexp-Mth

Figure 1. Distribution of Mth - Mexp Moller errors for three intervals.

ture but the presence of the double peak is still very clear. In the opposite direction, an interval of 0.05 MeV produces larger fluctuations and some apparent interference, but it is evident that the distribution still displays the peaks. We can thus conclude that the effect is real and not an artifact of a particular distribution interval.

it was pointed-out that the chaotic part of the error can have a dependence of the form A-lI3, reflecting the fact that that the light isotopes have bigger errors. It suggests that light nuclei could contribute in an unexpected way. To investigate their contribution, Fig. 2 shows the full distribution compared with two distributions built with a cut in the group of nuclei indicated. The curve with lines and circles corresponds to the

In

53

n 4 a 8 a

n Y

-0- begin in A=40 120

100

80

60

40

20

0 -3 -2 -1 1 2 3

Figure 2. Distribution of Mth - Mezp for different mass cutoffs.

distribution where all the isotopes with A < 40 have been excluded, and in the curve with lines and triangles we excluded those with A < 60. We conclude again that the double peak survives with only slight variations.

A third test is to normalize the distribution according to the rule AP1l3 4 9 5 . It could be that the elimination of this A-lI3 general trend would leave only the “chaotic” residue and hence the double peak should disappear, arising somehow from this “smooth” behavior. It is clear from Fig. 3 that the double peak remains, implying the possible presence of other regularities in the data

A fourth test was made to check whether experimental errors could influence the distribution. To test this possibility we built error distributions where the experimental mass was obtained as a random variable in a Gaus- sian distribution, with center in the reported experimental mass and width equal to the experimental error reported. In Fig. 4 we show the results for five error distributions where it is appreciated that the presence of the two peaks does not depend on the experimental errors.

When studying the distribution in the errors from the chart of Duflo and Zuker (DZ), we find that no double peak exists. It is also evident that the width of the DZ distribution is smaller, as it should be, since in the case of MN the r.m.s. error is of the order of 0.669 MeV , while for DZ the corresponding error is of 0.375 MeV. In Fig. 5 the mass error distribution in DZ is shown for the three intervals 0.05,O.l and 0.2 MeV. The difference among the form of the two distributions, MN and DZ, begs for a more

54

I004

(M =xP -M,,)A1”

Figure 3. Distribution of (Mhh - Me,,)A; for Moller and Nix masses

120

100

80

60

40

20

0 -2 0 2

M*xp‘Mth

Figure 4. Gaussian distribution centered in Mezp and u = uezp

detailed analysis. Figure 6 contains the mass error distribution (see gray code in the set)

in the N - 2 space, for the semiempirical Bethe-Weiszacker mass formula inspired in the liquid drop model (LDM). We can see large domains with a

Distribution of Mth - Meap for a random experimental values taken from a

55

similar error (each tone is associated to the magnitude of the error). This is related with the strong mass correlation in the LDM: the shell structure is clearly present in the mass deviations, with the lighter regions associated with shell closures.

More refined calculations are depicted in Fig. (7,8). Figure 7 displays the distribution of errors of MN in the N - 2 space. It is remarkable that very defined correlated areas of the same gray tone exist. This clearly

400 -

300 -

200 -

100-

Duf lo-Zu ker -A- int=0.10 -0- int=0.20

Figure 5 . Distribution of Mth - Mezp Duflo errors for three intervals.

manifests some kind of remaining systematics and correlation. Figure 8 is the corresponding graph for the error pattern of DZ. We observe a similar behavior, although the regions of similar magnitude are narrower in comparison to the MN case. As already mentioned, it is apparent that the distributions maintain a long range oscillatory behavior in the magnitude of the error.

In Fig. 9 the mass errors associated with chains of isotopes are plotted as a function of the neutron number N. We find again the oscillations, which are more marked for light isotopes. We can identify two dominant oscillation frequencies, a local one with A N M 20 and a long range one with A N M 80. Since we are speaking of the remaining errors we would not expect these oscillations to be present, but rather show a pretty random behavior. This suggests the implementation of an empirical correction.

A fit with sinusoidal functions can reproduce the average oscillatory

56

4 w o dwo 4wo

.2 wo 0

2Mo

l w o 6wO

I w o

20 40 60 80 100 120 140 N

Figure 6. Distribution of Liquid drop model Mth - Mezp in the N - 2

100-

90 - 80 - 70 - 60 -

z 507 40 - 30 - 20 - 10-

7 I ,

I , 1 ' I - I * 1 ' 1

20 40 60 ao 100 120 140

N Figure 7. Distribution of MN ibfth - Mezp in the N - 2

behavior in many of the chains of isotopes shown in Fig. 9. In these fits we observe two types of frequencies that repeat regularly in an approximate way. In Fig. 10 the correction made to the error using the adjustment is shown, where an interval of 0.1 MeV has been used. Although this corrected distribution has a small peak near 0.5 MeV, it has its maximum at zero.

In Fig. 11 we use the same correction for an interval of 0.2 MeV. It is evident that the double peak has disappeared and that the width of the distribution is smaller than in the original case of MN. With this correction

57

-3 -

-4

90 1

i

70 - 60 -

z 50: 40 - 30 -

-2 5m .ZWD

-1 5m -1 ow

0

ZOW

2 5 m

20 40 60 80 100 120 140 N

Figure 8. Distribution of DZ k i t h - Memp in the N - 2

20 40 60 80 100 120 140 N

3 I I I I I I I

Moller & Nix

Figure 9. Distribution of MN k i t h - MemP vs. N

the width in the error distribution decreases by 20%. In this paper we have analyzed in detail the error distribution for the

mass formulae of Moller and Nix and have found a clear long range regularity that manifest itself as a double peak in the distribution. By assuming a simple sinusoidal correlation, we can empirically correct these correlations and make the average deviation diminish by 20%. For comparison, the fit

58

160

140

120

100

80

60

40

20

0 -3 -2 -1 0 I 2 3

Mexp-Mth

Figure 10. intervak0.1

Distribution of Mhh - Meap for normal and corrected MN masses with

formula of Duflo and Zuker has also been analyzed and shown to possess smaller correlations of this kind. We believe that the physics-motivated formula of MN can be corrected with the corresponding improvement in its predictability. While we have not yet made a detailed analysis, we believe that the remaining regularity has its origin in the Strutinsky shell correction method and particularly in the definition of regular (or smooth) and fluctuating parts, something that can be corrected using a methodology first suggested in '. This could provide a significant improvement to the MN approach. The method of DZ lies somewhere in-between the model-motivated MN methodology and the purely statistical fit of Audi and Wapstra 7. What about the proposal of Bohigas and Leboef 4? We believe that we have convincingly shown that the MN errors still contain long-range correlations which are not consistent with chaotic behavior, but it may still be true that after subtracting these correlations a chaotic signature may remain. A newly suggested test * of the nature of such chaotic signals may prove very useful to test these ideas. We are currently exploring these matters and will report them elsewhere

Acknowledgements: Relevant comments by R. Bijker, 0. Bohigas, J. Flores, J.M. Gomez, P. Leboeuf, P. van hacker, and A. Zuker are gratefully acknowledged. This work was supported in part by Conacyt and DGAPA-UNAM, M&co.

59

Figure 11. Distribution of Mth - Merp for normal and corrected MN masses with interval=0.2

This work is dedicated to Jerry P. Draayer for his friendship and his great interest in establishing strong links with the Mexican Nuclear Physics Com- munity.

References

1. 2.

3.

4. 5.

6 . 7. 8.

9.

S. Aberg, Nature 417, 499 (2002). P. Moller, J.R. Nix, W.D. Myers, W.J. Swiatecki, At. DataNucl. DataTables 59, 185 (1995). J. Duflo, Nucl. Phys. A 576, 29 (1994); J. Duflo and A. P. Zuker, Phys. Rev. C 52, R23 (1995). 0. Bohigas, P. Leboeuf, Phys. Rev. Lett. 88, 92502 (2002). Victor Velhquez, Jorge G. Hirsch, and Alejandro Frank, Rev. Mex. Fis. in press. Gaetan J . H. Laberge and Rizwan U. Haq, Can 3. Phys. 68, 301 (1990). G.Audi and A.H.Wapstra, Nucl. Phys. A595, 409 (1995). A. Relaiio, J. M. G. Gbmez, R. A. Molina, and J . Retamosa, Phys. Rev. Lett. 89, 244102 (2002). Jorge G. Hirsch, Alejandro Frank, and Victor Velkquez, to be published.

SHAPE PHASE TRANSITIONS AND RANDOM INTERACTIONS

ROELOF BIJKER ICN-UNAM, A P 70-543, 04510 Mixico, DF, MCxico

The phenomenom of emerging regular spectral features from random interactions is addressed in the context of the interacting boson model. A mean-field analysis links different regions of the parameter space with definite geometric shapes. The results provide a clear and transparent interpretation of the high degree of order that has been observed before in numerical studies.

1. Introduction

Recent shell model calculations for even-even nuclei in the sd shell and the pf shell showed, despite the random nature of the two-body matrix elements, a remarkable statistical preference for ground states with angular momentum L = 0 l . A similar dominance of L = 0 ground states was found in an analysis of the Interacting Boson Model (IBM) with random interactions ’. In addition, in the IBM there is strong evidence for both vibrational and rotational band structures. According to the conventional ideas in the field, the occurrence of regular spectral features is due to a very specific form of the interactions. The studies with random interactions show that the class of Hamiltonians that lead to these ordered patterns is much larger than is usually thought.

The basic ingredients of the numerical simulations, both for the nuclear shell model and for the IBM, are the structure of the model space, the ensemble of random Hamiltonians, the order of the interactions (one- and two-body), and the global symmetries, i.e. time-reversal, hermiticity and rotation and reflection symmetry. The latter three symmetries of the Hamiltonian cannot be modified, since we are studying many-body systems whose eigenstates have real energies and good angular momentum and parity. It was found that the observed spectral order is a rather robust property which does not depend on the specific choice of the (two-body) ensemble of random interactions 19394,5, the time-reversal symmetry 3 , or the restriction

60

61

of the Hamiltonian to one- and two-body interactions '. This suggests that that an explanation of the origin of the observed regular features has to be sought in the many-body dynamics of the model space and /or the general statistical properties of random interactions.

The purpose of this contribution is to investigate the distribution of ground state angular momenta for the IBM in a Hartree-Bose mean-field analysis '.

2. Phase transitions

The IBM describes low-lying collective excitations in nuclei in terms of a system of N interacting quadrupole ( d t ) and monopole (st) bosons '. The IBM Hamiltonian spans a wide range of collective features which includes vibrational, rotational and y unstable nuclei. The connection with potential energy surfaces, geometric shapes and phase transitions can be studied by means of Hartree-Bose mean-field methods 9,10 in which the trial wave function is written as a coherent state. For one- and two-body interactions the coherent state can be expressed in terms of an axially symmetric condensate

with -7r /2 < a 5 T /2. The angle a is related to the deformation parameters in the intrinsic frame, ,8 and y '. First we investigate the properties of some schematic Hamiltonians that have been used in the study of shape phase transitions.

2.1. The U ( 5 ) - S 0 ( 6 ) case

The transition from vibrational to y unstable nuclei can be described by the Hamiltonian

sin x H = S d t . ( 2 + (stst - dt . dt) (SS - 2- (2) , (2) N 4N(N - 1)

which exhibits a second order phase transition at xc = 7r/4 g. For the present application, we extend the range of the angle x to that of a full period -7r/2 < x 5 37rJ2, so that all possible combinations of attractive and repulsive interactions are included. The potential energy surface is given by the expectation value of H in the coherent state

( 3 ) 1 4 ~ ( a ) = cosx sin2 a + - sinx cos2 2a .

62

The equilibrium configurations are characterized by the value of Q = cy0 for which the energy surface has its minimum. They can be divided into three different classes or shape phases

CYO = 0 - ~ / 2 < x 5 ~ / 4 cos2cro = cotx nf4 5 x 5 3n/4 (4)

a0 = n/2 3n/4 5 x 5 3n/2 which correspond to an s-boson or spherical condensate, a deformed condensate, and a d-boson condensate, respectively. The phase transitions at the critical points xc = n / 4 and 3n/4 are of second order, whereas the one at 3n/2 is of first order.

The angular momentum of the ground state can be obtained from the rotational structure of the equilibrium configuration, in combination with the Thouless-Valatin formula for the corresponding moments of inertia lo.

w For QO = 0 the equilibrium configuration has spherical symmetry, and hence can only have L = 0.

For 0 < (YO < n/2 the condensate is deformed. The ordering of the rotational energy levels L = 0,2, . . . ,2N

is determined by the sign of the moment of inertia 3N(sinx - cosx)

sin x cos x I3 =

For n/4 5 x 5 n/2 the moment of inertia I3 is positive and hence the ground state has angular momentum L = 0, whereas for for n/2 5 x 5 3n/4 it is negative corresponding to a ground state with L = 2N. w For QO = n/2 we find a condensate of N quadrupole or d-bosons, which corresponds to a quadrupole oscillator with N quanta. Its rotational structure is characterized by the labels r , nA and L. The boson seniority r is given by r = 3na + X = N , N - 2 , . . . , 1 or 0 for N odd or even, and the values of the angular momenta are L = A, A+ 1,. . . ,2X - 2,2X 8 . In general, the rotational excitation energies depend on two moments of inertia

1 1 Er,t = -r(r + 3) + -L(L + 1) .

215 213 (7)

For the special case of the Hamiltonian of Eq. ( 2 ) only the first term is present

2N sin x I 5 -

63

L=O

For 37r/4 5 x 5 7r the moment of inertia Is is negative and the ground state has T = N, whereas for IT 5 x 5 31~12 it is positive and the ground state has 7 = 0 ( L = 0 ) for N even, and T = 1 ( L = 2) for N odd.

100

80

* 60 3 s 0 F:

40

6 8 10 12 14 16

N

Figure 1. Percentages of ground states with L = 0 and L = 2 for the schematic IBM Hamiltonian of Eq. (2) with -7r/2 < x 5 3 ~ / 2 calculated exactly (solid lines) and in mean-field approximation (dashed lines).

In Fig. 1 we compare the percentages of ground states with L = 0 and L = 2 as a function of N obtained exactly (solid lines) and in the mean-field analysis (dashed lines). The results were obtained by assuming a constant probability distribution for x on the interval -7r/2 < x 5 3 ~ 1 2 . We have added a small attractive L' . L' interaction to remove the degeneracy of the ground state for the T = N solution. There is a perfect agreement for all values of N . The ground state is most likely to have angular momentum L = 0: in 75% of the cases for N even and in 50% for N odd. In 25% of the cases, the ground state has the maximum value of the angular momentum L = 2N. The only other value that occurs is L = 2 in 25% of the cases for N odd. The oscillation in the L = 0 and L = 2 percentages is due to the contribution of the d-boson condensate. The sum of the L = 0 and L = 2 percentages is constant (75%) and does not depend on N .

64

2.2. The U(5) -SU(3) case

A second transitional region of interest is the one between vibrational and rotational nuclei. In the IBM, it can be described schematically by

+(2st x d t f & ' d t ~ d ~ ) ( ~ ) . ( 2 d x S f J j d x ~ ) ( ~ ) ] . (9)

In the physical region 0 5 x < ~ / 2 , H i exhibits a first order phase transition at xc = arctan1/9 '. As before, here we consider the interval -n/2 < x < 37r/2. The results for the distribution of ground state angular momenta are presented in Fig. 2. For N = 3k the ground state has L = 0 in 75% of the cases and L = 2N in the remaining 25%. For N = 3k + 1 and N = 3k + 2 the ground state angular momentum is either L = 0 (50%), L = 2 (25%) or L = 2N (25%). The variation in the L = 0 and L = 2 percentages is due to the contribution of the d-boson condensate, whereas the sum of the two is constant (75%).

100

80

60 4 4 2 B 0

2 40

20

0

L=O

L=2

6 8 10 12 14 16 N

Figure 2. As Fig. 1, but for the schematic IBM Hamiltonian of Eq. (9).

65

2.3. The S U ( 3 ) - S 0 ( 6 ) case

The transitional region between rotational and y unstable nuclei described by the Hamiltonian

H* = cos x (st . ,t - dt . dt) (5 . 5 - 2. 2)

4(N - 1)

+ ( 2 d X d t f J ; i d t X d t ) ( 2 ) . ( 2 d X 5 f ~ d X ( i i ( 2 ) ] 7 (10)

does not show a phase transition in the physical region 0 x 5 7rf2 g.

Fig. 3 shows that the distribution of the ground state angular momenta is very similar to the previous case.

100

80

a, 60 2 2 c)

Li a 40

20

0 6 8 10 12 14 16

N

Figure 3. As Fig. 1, but for the schematic IBM Hamiltonian of Eq. (10).

3. Random interactions

Finally, we apply the mean-field analysis to the general one- and two-body IBM Hamiltonian

66

in which the nine parameters of this Hamiltonian are taken as independent random numbers on a Gaussian distribution with zero mean and width (T. The distribution of geometric shapes for this ensemble of Hamiltoni- ans is determined by the distribution of equilibrium configurations of the corresponding potential energy surfaces

E(Q) = a4 sin4 LY + a3 sin3 Q cos Q + a2 sin2 Q + a0 . (12) The coefficients ai are linear combinations of the Hamiltonian parameters. The spectral properties of each Hamiltonian of the ensemble of random one- and two-body interactions are analyzed by exact numerical diagonalization

and by mean-field analysis '.

100

80

5 60

E 40

Y

20

0

Figure 4

6 8 10 12 14 16 N

As Fig. 1, but for the random IBM Harniltonian of Eq. (11).

In Fig. 4 we compare the percentages of L = 0 and L = 2 ground states obtained exactly (solid lines) and in the mean-field analysis (dashed lines). There is a dominance of ground states with L = 0 for - 63 - 77% of the cases. For N = 3k we see an enhancement for L = 0 and a corresponding decrease for L = 2. Also in this case, the equilibrium configurations can be divided into three different classes: an s-boson or spherical condensate, a deformed condensate, and a d-boson condensate. For the spherical and deformed solutions the ground state has L = 0 (- 63%) or L = 2N (- 13%). The analysis of the d-boson condensate is a bit more complicated

67

due to the presence of two moments of inertia, Is and 4. There is a constant contribution to L = 2N ground states (- lo%), whereas the L = 0 and L = 2 percentages show oscillations with N 7. Just as for the schematic Hamiltonians, the sum of the L = 0 and L = 2 percentages is constant and independent of N .

4. Summary and conclusions

In this contribution, we have investigated the origin of the regular features that have been observed in numerical studies of the IBM with random interactions, in particular the dominance of ground states with L = 0.

In a mean-field analysis, it was found that different regions of the parameter space can be associated with particular intrinsic vibrational states, which in turn correspond to definite geometric shapes: a spherical shape, a deformed shape or a condensate of quadrupole bosons. An analysis of the angular momentum content of each one of the corresponding condensates combined with the sign of the relevant moments of inertia, provides an explanation for the distribution of ground state angular momenta of both schematic and random forms of the IBM Hamiltonian.

In summary, the present results show that mean-field methods provide a clear and transparent interpretation of the regular features that have been obtained before in numerical studies of the IBM with random interactions. The same conclusions hold for the vibron model For the nuclear shell model the situation is less clear. Despite the large number of studies that have been carried out to explain and/or further explore the properties of random nuclei no definite answer is yet available 12.

Acknowledgments

It is a great pleasure to dedicate this contribution to the 60th birthday of Jerry P. Draayer. Congratulations, Jerry! This work was supported in part by CONACyT under project No. 32416-E.

References

1. C.W. Johnson, G.F. Bertsch and D. J. Dean, Phys. Rev. Lett. 80, 2749 (1998). 2. R. Bijker and A. Frank, Phys. Rev. Lett. 84, (2000), 420. 3. R. Bijker, A. Frank and S. Pittel, Phys. Rev. C 60, 021302 (1999). 4. C.W. Johnson, G.F. Bertsch, D.J. Dean and I. Talmi, Phys. Rev. C 61,

014311 (2000). 5. D. Dean, Nucl. Phys. A 682, 194c (2001).

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6 . R. Bijker and A. Frank, Phys. Rev. C 62, 014303 (2000). 7. R. Bijker and A. Frank, Phys. Rev. C 64, 061303 (2001). 8 . F. Iachello and A. Arima, The interacting boson model (Cambridge University

Press, 1987). 9. A.E.L. Dieperink, 0. Scholten and F. Iachello, Phys. Rev. Lett. 44, 1747

(1980); A.E.L. Dieperink and 0. Scholten, Nucl. Phys. A 346, 125 (1980). 10. J . Dukelsky, G.G. Dussel, R.P.J. Perazzo, S.L. Reich and H.M. Sofia, Nucl.

Phys. A 425, 93 (1984). 11. R. Bijker and A. Frank, Phys. Rev. C 65, 044316 (2002). 12. see e.g. R. Bijker and A. Frank, Nuclear Physics News, Vol. 11, No. 4, 15

(2001); V.G. Zelevinsky, D. Mulhall and A. Volya, Phys. Atom. Nucl. 64, 525 (2001).

IV. Pseudo-spin in Nuclear Physics

PSEUDOSPIN AS A RELATIVISTIC SYMMETRY IN NUCLEI

JOSEPH N. GINOCCHIO MS B283, Theoretical Division, Los Alamos National Laboratory

Los Alamos, NM 87545, U S A E-mail: gino @lanl. gou

The evidence that pseudospin symmetry is a relativistic symmetry is reviewed. Search for pseudospin symmetry beyond the mean field approximation is motivated.

1. Introduction

About thirty years ago, pseudospin doublets were introduced into nuclear physics to accommodate an observed near degeneracy of certain normal- parity shell-model orbitals with non-relativistic quantum numbers (n,, [, j = C+ 1/2) and (n, - 1, C+ 2, j = C+3/2) where n,, C, and j are the single- nucleon radial, orbital, and total angular momentum quantum numbers, respectively 1>2. The doublet structure, is expressed in terms of a LLpseudo” orbital angular momentum e = C + 1 coupled to a “pseudo” spin, S = 1/2. For example, (n,s1/2, (n,-l)&l2) will have 2 = 1, (7L#3/2, (n,-l)fs/z) will have 2 = 2, etc. Since j = i& S the energy of the two states in the doublet are then approximately independent of the orientation of the pseudospin. In the presence of deformation the doublets persist with asymptotic (Nilsson) quantum numbers [N,n3,A,R = A + 1/21 and [N,n3,h + 2 , R = A + 3/21, and can be expressed in terms of pseudo-orbital and total angular momentum projections = A + 1, R = f 1/2 37475. This pseudospin LLsymmetry” has been used to explain features of deformed nuclei, including identical normal and superdeformed rotational bands 617,8.

Jerry Draayer contributed to pseudospin and its generalization to pseudo-SU(3) from the very beginning ’. Jerry has continued to be very much involved in pseudo-SU(3) as is discussed in a number of other contributions to this volume. We shall not discuss pseudo-SU(3) in this paper.

71

72

In addition to demonstrating that pseudospin symmetry occurs in both axially and triaxially deformed nuclei 4,5 , he and his collaborators showed that, in the harmonic oscillator, pseudospin is conserved if the spin-orbit single particle potential and angular momentum single-particle potential are in the combination

Hs, ,=2 j .B+O.5 j . i (1)

and that the shell model, the Nilsson model, and relativistic field theory in nuclear matter approximately satisfy this condition l o . Furthermore he and his collaborators introduced the helicity transformation

5.5 up = - P

to transform from the spin basis to the pseudospin basis and showed that the large spin-orbit interaction in relativistic mean field calculations were transformed into a small pseudospin-orbit interaction l l . This fortitutous approximate cancellation of the pseudospin-orbit single particle interaction has been shown recently to result from an approximate relativistic symmetry that occurs in nuclei because the vector mean field is approximately equal to the magnitude of the scalar mean field, but opposite in sign 12.

This relavistic symmetry is valid if the potentials are spherical, axially deformed or tri-axially deformed.

2. Pseudospin Symmetry and the Dirac Hamiltonian

The Dirac Hamiltonian, H ,

H = ir'. c@+ Vv(F') + pVs(F') + ,B Mc'. (3)

with an external scalar potential, Vs(F'), and an external vector potential with vanishing space components and a non-vanishing time component, Vv(F'), is invariant under an SU(2) algebra if the scalar potential, V s ( 3 , and the vector potential VV(<), are related l 3 7 l 2 :

where Cp, is a constant. In nuclear relativistic mean field models l4 and relativistic optical models 15, Vs(<) -VV(T:), and the space components of the vector fields are zero due to parity conservation. These conditions approximately satisfy (4) and lead to approximate pseudospin symmetry in single-nucleon levels and nucleon-nucleus scattering.

73

The pseudospin generators l6

.+ s = (is). (5)

form an SU(2) algebra - - -

[si, sj] = i f i j k s k , (6)

[si, Hps] = 0, (7)

and commute with the Dirac Hamiltonian Hps satisfying conditions (4)

where = Up s’ Up and Up is the helicity transformation (2) introduced by Jerry and collaborators ‘I, si = 5 and are the usual Pauli matrices. Thus the operators si generate an SU(2) invariant symmetry of H p s . Therefore each eigenstate of the Dirac Hamiltonian has a partner with the same energy,

H p s @Iyfi(q = Ek@Iyfi(q, (8)

3, @&(?) = ji qp).

where Ic are the other quantum numbers and ji = &$ is the eigenvalue of 3 z 7

(9)

The eigenstates in the doublet will be connected by the generators $,

3. Pseudospin Symmetry Conditions on the Dirac Eigenstat es

3.1. General Potentials

We can write the eigenstates @&(q of the Dirac Hamiltonian (3) as a four dimensional vector,

where g:,(?) are the “upper Dirac components” where + indicates spin up and - spin down and f&(F) are the “lower Dirac components” where + indicates spin up and - spin down.

74

Since these eigenstates must also belong to the spinor representation of the pseudospin group, these amplitudes in the doublets with I. = *i must be related as indicated in (9, 10). Clearly from the fact that the lower component of the spin generators (5) is simply L?, (9) implies that

f,+@ = f,7+(r3 = 0,

f&(q = f L - + ( f l = . f k ( q .

( 1 2 )

(13)

while (10) implies that

Thus the pseudospin symmetry implies that the lower components of the pseudospin partners have the same spatial wavefunction.

For the upper components the relationships are more complicated because the operator g intertwines spin and space due to the dependence on the momentum, which produces differential relations:

= -9- k , - i (3 = g d r 3 , (14)

2L (pz + ' p Y ) g ; - $ ( q = (pz - i p Y ) g i , + ( q , p Z g k , f f ( q = (pZ iPy)gk(T3(15)

Thus, the upper components g l + ( T ) , g , - + ( F ) have the same spatial wavefunction but differ by a sign. However, the other upper components g l - + (3, g - (3 can have very different spatial wavefunctions.

k , 5 The Dirac wavefunctions in the doublet then become

Thus, instead of eight amplitudes for the two states in the doublet, there are four non-zero amplitudes, three upper and one lower, and the three upper are related by first order differential equations (15) 17.

3.2. Spherically Symmetric Potentials

If the potentials are spherically symmetric, V ( 3 = V ( T ) , then the Dirac Hamiltonian has an additional S U ( 2 ) symmetry. The Dirac Hamiltonian will be invariant with respect to rotations about all three axes, [zi, Hps] = 0 where

75

+ . . The Dirac eigenstates will then be an eigenfunction of L . L . Also the total angular momentum, J’ = e + 3, will be conserved as well and the Dirac eigenstates will be eigenfunctions of J’. J’ and J,. Rather than using the four row basis for the doublet eigenfunctions, in the spherical symmetry limit it is more convenient to introduce the spin function x p explicitly. The states that are a degenerate doublet are then the states with j = i* and in ref ’’ we have shown that the Dirac egenfunctions (16) have the two row form

. - . *

The spherical symmetry thus reduces the number of non - zero amplitudes in the doublet from four to three. The number of independent first order differential relations (15) between the two upper components is reduced from three to one

For pseudospin the lower component radial amplitudes in the doublet are equal. Therefore f i r is the number of radial nodes of the lower amplitudes, not the upper amplitudes. On the other hand, the upper amplitude with j = 2 - $ in the the doublet has f i r radial nodes while the upper amplitude with j = i-+ will have f i r - 1 radial nodes l8 which agrees with the pseudospin doublets observed in nuclei.

4. Test of Realistic Eigenfunctions

The lower components of the Dirac eigenfunctions for the pseudospin doublets using realistic eigenfunctions determined in relativistic mean field calculations have been shown to be approximately equal in a number of papers 1 4 , 1 9 9 2 0 . These papers also confirm the fact that the lower components are small compared to the upper components which is consistent with the fact that nuclei are primarily non-relativistic quantum systems. However relativistic quantum mechanics is necessary for the understanding of pseudospin symmetry.

Recently the differential relations (19) satisfied by the upper components of the Dirac eigenfunctions in the pseudospin symmetry limit have been tested for the pseudospin doublets in spherical nuclei using realistic eigenfunctions determined in relativistic mean field calculations 7,21,22. In Figure l a the upper components for the 1s; and Qd3I2 eigenfunctions are

76

0.2

4.1

-0.2 4.3 r (fm)

Figure 1. a) The upper component g ( r ) for the 1 s ~ (solid line) and Oda (dashed line) eigenfunctions, b) the differential equation on the right hand side of Equation (19) with = 1 for the I s 1 (solid line) eigenfunction and the differential equation on the left hand

side of Equation (19) with 2 = 1 for the Oda (dashed line) eigenfunction, c) the upper component g ( r ) for the 25-1 (solid line) and I d s (dashed line) eigenfunctions, and d)

the differential equation on the right hand side of Equation (19) with 2 = 1 for the 2sl eigenfunction (solid line) and the differential equation on the left hand side of Equation (19) with 2 = 1 for the l d l (dashed line) eigenfunctions.

2 2

2

2 2

plotted (fi, = 1, i = 1); these eigenfunctions are very different in shape with different numbers of radial nodes. In Figure l b the differential relations for these eigenfunctions are plotted and we see a remarkable similarity between the two differential relations except near the nuclear surface. In Figure l c the upper components for the 1s; and Od3/2 eigenfunctions are plotted (f i, = 2, i = 1); likewise these eigenfunctions are very different in shape. In Figure Id the differential relations for these eigenfunctions are plotted and we see even better agreement between the two differential relations than for fi, = 1. Similar tests are made for higher radial quantum numbers

77

and larger pseudo-orbital angular momentum ”. These results are for neutrons in zosPb but similar conclusions hold for the protons as well. The pseudospin admixing decreases for increasing radial quantum number but decreasing pseudo-orbital angular momentum, the same pattern followed by the binding energies l2 and the lower amplitudes of the eigenfunctions

In the limit of small lower components, the upper components are the non-relativistic approximation to the eigenfunctions. The differential relations (19) have been tested as well for the non-relativistic eigenfunctions of the phemenological Woods-Saxon potential and self-consistent Hartree- Fock mean field 22. The non-relativistic eigenfunctions are shown also to approximately conserve pseudospin symmetry which is consistent with the fact that these models reproduce the single-nucleon spectrum well.

14

5. Magnetic dipole and Gamow-Teller Transitions

In the non-relativistic limit magnetic dipole transitions and Gamow-Teller transitions are forbidden between the two states in the doublet because the states differ by two units of angular momentum. However, relativistically these transitions are allowed. Furthermore pseudospin symmetry relates the magnetic dipole transitions rates to the magnetic moments of the states in the doublet, and relates the Gamow-Teller transitions between states to the transitions between the same states in different nuclei 23. These relationships for magnetic dipole transtions have been tested for several nuclei and found to be approximately valid 24. The Gamow-Teller relationships have not been tested yet.

6. Nucleon-Nucleus Scattering

The condition for pseudospin symmetry given in Eq. (4) is valid even for complex potentials. Relativisitic optical models approximately satisfy this condition 15. Pseudospin symmetry has been found to be approximately conserved for medium energy nucleon-nucleus scattering 15,26 and is expected to improve as the nucleon energy incereases 27,28.

7. QCD Sum Rules

QCD sum rules in nuclear matter have been used to derive

78

using accepted values of the average quark mass in the proton (= 5 MeV) and the value of sigma term, O N (= 45 MeV), measured in pion-nucleon scattering 29. This value is uncannily close to the ratio of the scalar and vector potentials in the interior of the nucleus. This suggest that perhaps pseudospin has a more fundamental foundation in terms of QCD.

8. Summary and the Future

More than thirty years after the discovery of pseudospin symmetry in the energy spectra of single-nucleon states, its relativistic foundations have been established. Realistic Dirac eigenfunctions and non-relativistic eigenfunctions for spherical nuclei approximately satisfy the conditions for pseudospin symmetry. Although some investigations on deformed nuclei 19,20

have been carried out for the lower components, extensive investigations of the derivative relations for the upper components for deformed nuclei are needed. Empirical evidence other than spectra, such as magnetic dipole transitions and nucleon-nucleus scattering, have also shown signs of pseudospin symmetry. QCD sum rules suggest a more fundamental rationale for pseudospin symmetry. This has motivated a search for pseudospin symmetry beyond the mean field by investigating the nucleon-nucleon interaction. Such investigations did not indicate pseudospin conservation 30. However , the pseudospin generators considered to date ignore the contribution of the spatial components of the Lorentz vector potential assuming that it is zero. This assumption is reasonable for a mean field approximation but may not be valid for a two-nucleon interaction. The nucleon-nucleon interaction is being investigated to determine if it conserves the more general form of pseudospin which has non-zero spatial components of the Lorentz vector potential. Finally, the relativistic origin of pseudospin symmetry has vali- dated the helicity transformation Up over other suggested transformations 3

Acknowledgments

This work was supported by the U S . Department of Energy under contract W-7405-ENG-36.

References 1. A. Arima, M. Harvey and K. Shimizu, Phys. Lett. B 30 517 (1969). 2. K.T. Hecht and A. Adler, Nucl. Phys. A 137 129 (1969).

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3. A. Bohr, I. Hamamoto and B.R. Mottelson, Phys. Scr. 26 267 (1982). 4. D. Troltenier, W. Nazarewicz, Z. Szymanski, and J. P. Draayer , Nucl. Phys.

A567, 591 (1994). 5. A. L. Blokhin, T. Beuschel, J . P. Draayer, and C. Bahri, Nucl. Phys. A612,

163 (1997). 6. W. Nazarewicz, P.J. Twin, P. Fallon and J.D. Garrett, Phys. Rev. Lett. 64

1654 (1990). 7. F.S. Stephens et al., Phys. Rev. Lett. 65 301 (1990); F.S. Stephens et al., Phys.

Rev. C 57 R1565 (1998). 8. A.M. Bruce et. al., Phys. Rev. C 56 1438 (1997). 9. R. D. Ratna Raju, J. P. Draayer and K. T. Hecht, Nucl. Phys. A202, 433

(1973). 10. C. Bahri, J. P. Draayer, and S. A. Moszkowski, Phys. Rev. Lett. 68, 2133

(1992). 11. A. L. Blokhin, C. Bahri and J. P. Draayer, Phys. Rev. Lett. 74, 4149 (1995). 12. J.N. Ginocchio, Phys. Rev. Lett. 78 436 (1997). 13. J. S. Bell and H. Ruegg, Nucl. Phys. B98, 151 (1975). 14. J.N. Ginocchio and D. G. Madland, Phys. Rev. C 57 1167 (1998). 15. J. N. Ginocchio, Phys. Rev. Lett. 82, 4599 (1999). 16. J.N. Ginocchio and A. Leviatan, Phys. Lett. B 425 1 (1998). 17. J.N. Ginocchio, Phys. Rev. C 66 064312 (2002). 18. A. Leviatan and J.N. Ginocchio, Phys. Lett. B 518, 214 (2001). 19. G. A. Lalazissis, Y. K. Gambhir, J . P. Maharana, C. S. Warke and P. Ring,

20. J. Meng, K. Sugawara-Tanabe, S. Yamaji, P. Ring and A. Arima, Phys. Rev.

21. J.N. Ginocchio and A. Leviatan, Phys. Rev. Lett. 87 072502 (2001). 22. P.J. Borycki, J. Ginocchio, W. Nazarewicz, and M. Stoitsov to be published

23. J. N. Ginocchio, Phys. Rev. C59, 2487 (1999). 24. P. von Neumann-Cosel and J. N.Ginocchio, Phys. Rev. C62, 014308 (2000). 25. K. Sugawara-Tanabe, S. Yamaji, and A. Arima, Phys. Rev. C 62 054307

26. H. Leeb and S. Wilmsen, Phys. Rev. C 62, 024602 (2000). 27. J. B. Bowlin, A. S. Goldhaber and C. Wilkin, Z. Phys. A331, 83 (1988). 28. H. Leeb and S. A. Sofianos, to be published in Phys. Rev. C (2003); Electronic

29. T. D. Cohen, R. J. Furnstahl , K . Griegel, and X. Jin, Prog. in Part. and

30. J.N. Ginocchio, Phys. Rev. C 65, 054002 (2002).

Phys. Rev. C58, R45 (1998).

C58, R628 (1998).

in Phys. Rev. C (2003); Electronic Archives: nucl-th/0301098.

(2000).

Archives: nucl-th/0304009.

Nucl. Phys. 35, 221 (1995).

PSEUDO-SPIN SYMMETRY IN NUCLEI

P. VAN ISACKER GANIL, B.P. 55027, F-14076 Coen Cedes 5, France

E- ma kl: is a cke r Oga n i1.k

The nuclear shell model allows several analytical solutions which broadly can be divided in two classes: pairing models and rotational models. The latter are based on Elliott’s SU(3) symmetry which presupposes LS coupling. The search for solvable rotational models that can accommodate a departure from LS coupling has been an important theme of nuclear structure in which pseudo-spin symmetry has played a pivotal role. In this contribution the arguments that justify a departure from SU(4) symmetry and a move towards a pseudc+LS or pseudo-SU(4) scheme are reviewed.

1. A symmetry triangle for the shell model

Symmetry considerations have played an important role in the develop ment of nuclear physics. Important landmarks in this development include Wigner’s SU(4) supermultiplet model which assumes invariance in spin and isospin, Racah’s SU(2) pairing model leading to the concept of seniority, Elliott’s SU(3) model which provides an understanding of rotation in the context of the spherical shell model and the U(6) interacting boson model of Arima and Iachello which gives a unified description of collective structures observed in nuclei.

These different models, which were developed over a period of more than half a century, can be understood from a common perspective using the concept of dynamical symmetry or spectrum generating algebra (for a review, see Ref. ’). This approach is formulated in terms of the theory of Lie algebras and can be characterised in words as follows. Given a system of interacting particles (bosons or fermions) a mathematical procedure exists to construct a set of commuting operators which supply the quantum numbers of a classification scheme. Furthermore, to each set of commuting operators corresponds a class of many-body hamiltonians which can be solved analytically by requiring t,hat, they be written in terms of these commuting operators.

80

81

The main characteristics of the shell model of the atomic nucleus are the following: (a) the shell structure generated by the nuclear mean field, (b) the spin-orbit term in the mean field, and (c) the competition between the pairing and the quadrupole residual interactions . These three features can be represented algebraically as is illustrated schematically in Fig. 1 where each vertex corresponds to an analytic solution of the shell model. A

independent - par ticle shell model

SU(2) pairing SU(3) rotation in j j coupling in LS coupling

Figure 1. alytically solvable vertices.

Schematic representation of the shell-model parameter space with three an-

hamiltonian corresponding to the top vertex yields uncorrelated Hartree- Fock type wave functions. This limit is reached if the single-particle energy spacings are large in comparison with a typical matrix element of the residual interaction. The transition between j j and L S coupling is controlled by the strength of the spin-orbit coupling (in relation to that of the residual interaction). And, finally, the transition from SU(2) pairing to SU(3) rotation requires a change of the character of the residual interaction from pairing to quadrupole. The triangle in Fig. 1 summarises schematically the basic symmetries of the shell model; many extensions are possible as discussed comprehensively in the review by Jerry Draayer ‘.

In this meeting in honour of Jerry Draayer I will focus on the rotational vertex of Fig. 1, leaving the discussion of the pairing vertex for the meeting in March 2003 in honour of Franco Iachello. Elliott’s SU(3) model of rotation is an LS coupling scheme and presupposes the existence of SU(4) symmetry, the topic of the next section

2. Wigner’s SU(4) symmetry

Wigner’s SU(4) symmetry represents an embedding of the spin and isospin SU(2) algebras to a larger combined SU(4) algebra ’. A nuclear hamiltonian

82

with SU(4) symmetry satisfies

A A d

where sp (k) and t,, (Ic) are the spin and isospin operator components of nii- cleon k. The 15 operators Ck s , ( k ) , xk t , , ( k ) and X I , s,(k)t,(Ic) generate the Lie algebra SU(4) and, consequently, any hamiltonian satisfying the conditions (1) has SU(4) symmetry.

It is well known that the spin-orbit term in the nuclear mean field is responsible for the breaking of SU(4) symmetry and increasingly so as the nuclear mass increases ’. A simple way of illustrating this breaking requires taking a double difference of nuclear binding energies ’,

SV,, ( N , 2) = -[B( N , 2) - B( N - n, 2) - B( N , 2 - z ) + B( N - n , 2 - z ) ] ,

where n = 2- N mod 2 and z = 2 - 2 mod 2, that is, n = 2 if the number of neutrons N is even and n = 1 if N is odd, and similarly for z . Particularly large values of SV,, are found for N = 2 even-mass and N = 2 f 1 odd- mass nuclei ’.

It was argued that this N - 2 enhancement is a consequence of Wigner’s SU(4) symmetry and the degree of this enhancement provides a test of the goodness of this symmetry lo. An example is shown in Fig. 2 which shows on the left SV,,(N, 2) (where known from Ref. for the sd shell. The SU(4) result of Fig. 2(b) is obtained by assuming a binding

1 n z

(2)

(a) Experiment (c) Broken SU(4)

Figure 2. Barchart representation of double binding energy differences (a) as observed in even-even sd-shell nuclei, (b) as predicted by Wigner’s unbroken SU(4) symmetry, and (c) as obtained by taking a mixture of first- and second-favored SU(4) representations (see text). The I and y coordinates of the center of a cuboid define A’ and 2 and its height z defines &V,,,(N, 2); an empty square indicates that data are lacking.

83

energy of the form n + hg(X, p , v) where o and h are coefficients depending smoothly on mass number l 2 and g(X, p , v) is the eigenvalue of the quadratic Casimir of SU(4) in the favoured SU(4) representation (A, p , v),

g(X, p , v) = 3X(X + 4) + 3v(v + 4) + 4p(p + 4) + 4p(X + v) + 2 h . (3)

As long as the departure from SU(4) symmetry is not too important, its breaking can be investigated by assuming a nuclear ground state that does not correspond entirely to the favoured SU(4) representation but contains an admixture of the next-favoured SU(4) representation. These admixtures will modify the behaviour of SV,, at N - 2. This is illustrated in Fig. 2(c) where the SV,, of even-even nuclei is plotted by taking a varying mixture of first- and second-favoured SU(4) representations [usually (0, T , 0) and (2,T - l , O ) ] . As the mass of the nucleus increases, one notes indeed a decrease of the N = 2 enhancement effect for SV,,, roughly consistent with the experimental observations. An exceptional point occurs for N = 2 = 20 where the calculation is unrealistic since 4"Ca is taken as doubly closed and hence corresponds to a unique SU(4) representation (0, 0,O) with no possible admixtures.

This example illustrates that the breaking of SU(4) symmetry increases with mass A . A similar conclusion is reached on the basis of slimmed Gamow-Teller strength from N = 2 - 2 into N = 2 nuclei which is subject to SU(4) selection rules 13. And herewith the conundrum of the application of SU(3) to the structure of heavy nuclei is made apparent: How to devise such an SU(S)-type model while a t the same allowing for a departure from L S coupling; this is discussed in the next section.

3. Pseudo-SU(4) symmetry

A successful way to extend applications of the SU(3) model to heavy nuclei (and to high angular momentum) is based upon the concept of pseudo-spin symmetry. To understand the nature of this symmetry, consider the unitary transformation

. s . r r

u = 22-. (4) If this transformation is applied to a single-particle hamiltonian that, includes a harmonic-oscillator mean field with a spin-orbit and an orbit-orbit term, one finds (up to a constant,) a transformed hamiltonian with a modified spin-orbit coupling '. In particular, it can be shown that the spin-orbit, term disappears entirely if in the original hamiltonian the strength of the orbit-orbit coupling is four times that, of the spin-orbit coupling.

84

Pseudo-spin symmetry has a long history in nuclear physics. The existence of nearly degenerate pseudo-spin doublets was noted already thirty years ago by Hecht, and Adler '4 and, independently, by Arima et al. 1 5 .

These authors also realised that,, because of the small pseudo-spin-orbit splitting, pseudo-LS (or is) coupling should be a reasonable starting point in medium-mass and heavy nuclei where L S coupling becomes unaccept- able. With L,? coupling as a premise, an SU(3) model can be constructed in the same way as Elliott's SU(3) model can be defined in L S coupling. The ensuing pseudo-SU(3) model was investigated seriously for the first time in Ref. l 6 with many applications following afterwards (for a review, see Ref. '). The formal definition of the pseudo-spin transformation (4) in terms of a helicity operator was given by Bohr et al. l 7 and later gener- alised by Castafios et al. 18, to include transformations that not only act on the spin-angular part of the wave function-as does ($)-but also on its orbital part. Finally, it is only recently that an explanation of pseudo-spin symmetry was suggested based on the relativistic mean-field model of the nucleus. First, the relation between the strength of the spin-orbit and the orbit-orbit coupling (necessary for pseudo-spin symmetry) was found to be approximately valid in numerical calculations l9 and later the pseudo-spin symmetry was proven to be a symmetry of the Dirac equation which occurs if the scalar and vector potentials are equal in size but opposite in sign 20,21.

Just as the SU(2) symmetries of spin and isospin can be combined to yield the larger SU(4) symmetry, one can equally well combine the SU(2) symmetries of pseudo-spin and isospin to give what can be called a pseudo- SU(4) [or s?i(4)] symmetry 2 2 i 2 3 . A hamiltonian with pseudo-SU(4) symmetry satisfies the following commutations relations:

k = l k = l k = l

where S, are the transformed spin operator components, s",, = i-'s,i. The hamiltonian in (5) conserves the total pseudo-orbital angular momentum ?, and the total pseudo-spin 5, which result from the separate coupling of all individual pseudo-orbital angular momenta 1 (k) and pseudo-spins s"(k).

The existence of a pseudo-SU(4) symmetry requires minimally that the valence shell coincides with a pseudo-oscillator shell. A region where this is possibly the case concerns N - 2 nuclei at, the beginning of the 28-50 shell where the dominant, orbits are 2pl12, 2~312 and l f 5 1 2 which can be considered as a pseudo-sd shell. The assumption of ?,,? coupling can be verified by analysing the wave functions of a standard shell-model calcu-

-

85

lation. The procedure for testing the validity of pseudo-SU(4) is similar to the one followed by Vogel and Ormand 24, except that it, is done for pseudo instead of standard SU(4). In Fig. 3 the results of this analysis are shown for :$1129. A shell-model space consisting of the p f 5 / 2 g 9 / 2 orbits

3

2

E (MeV)

1

0

Expt

- 4+

- 5+

1+ -

- 3+

O+

-1+

- T = l

T=O

Shl

4+

-2+

- 5+

- 4+

- 3+

-

- 2+

- 2+

1+ -

- 3+ O+ -

T = l

I + - T=O

1 0.5 0 0.5 1

Figure 3. Left panel: Low-lying experimental levels (Expt) in 58Cu compared with the shell model (SM). Rightpanel: The amplitudes of the two possible pseudo-SU(4) representations, ( K O ) and (200), are indicated with open bam Thefilled bars give the overlap with the leading pseudc+SU(3) representation contained in (010).

86

is considered with a realistic G-matrix interaction with its monopole part phenomenologically adjusted ". Application of this interaction to 58C1~ shows that, although the g9 /2 orbit is important for obtaining the correct energy spectrum, its admixtures in the low-energy levels remain fairly small; the interaction can thus be renormalised to the pf5/2 shell. The energy spectrum obtained with the renormalised interaction agrees reasonably well with the data 26,27 . The right panel of Fig. 3 shows the decomposition of the renormalised shell-model wavefunctions in terms of the two possible pseudo-SU(4) representations, (010) and ( Z O ) , open bars indicating their amplitudes. It. is seen that all lowest eigenstates carry a large component in (6). This component can be further analysed according to its orbital character; the figure illustrates this by indicating with the filled bars the overlap with the leading pseudo-SU(3) representation in (m).

Intuitively, this result is a combined effect of the short-range nature of the residual interaction and the fact that nucleons interact predominantly at the surface of the nucleus. Because of the latter property, matrix elements of the residual interaction are not very sensitive to the radial structure of the wave function at, the interior of the nucleus. As a result, the problem of n nucleons in a pseudo harmonic oscillator shell is approximately equivalent to that of n nucleons in a normal harmonic oscillator shell. In fact, this equivalence is exact for the surface delta interaction 2 8 1 2 9 which is known to be a reasonable approximation to the true effective interaction in nuclei.

Pseudo-SU(4) symmetry leads to predictions concerning Gamow-Teller ,8 decay that differ considerably from the usual SU(4) selection rules. These effects are discussed in Ref. 23.

-

4. Conclusion

The main argument of this contribution has been to show that the characteristic features of a nucleus can be represented algebraically. The limitations of the symmetry approach should not, however, be forgotten: It is clear that i t can only account for gross properties and that, any detailed description of the nucleus requires more involved numerical calculations. In this way symmetry techniques can be used as an appropriate starting point for detailed calculations. A noteworthy example of this approach is the pseudo-SU(3) model which, starting from its initial symmetry ansntz, has grown into an adequate and powerful description of the nucleus in terms of a truncated shell model.

87

Acknowledgements

It, is a pleasure to offer my congratiilations to Jerry Draayer a t the occasion of his sixtieth birt,hday. I believe this contribution illustrates the impact Jerry’s research has mads in the field of nuclear structure through the application of symmetry ideas. The work reported has been done in collaboration with Olivier Juillet and Dave Warner.

References

1. E.P. Wigner, Phys. Rev. 51, 106 (1937). 2. G. Racah, Phys. Rev. 63, 367 (1943). 3. J.P. Elliott, Proc. Roy. SOC. [London) A 245, 128 & 562 (1958). 4. A. Arima and F. Iachello, Phys. Rev. Lett. 35, 1069 (1975). 5. P. Van Isacker, Rep. Progr. Phys. 62, 1661 (1999). 6. J.P. Draayer in Algebraic Approaches to Nuclear Structure. Interacting Boson

and Fernion Models (Harwood, New York, 1993) p 423. 7. A. B o b and B.R. Mottelson, Nuclear Structure I. Single-Particle kfotion

(Benjamin, New York, 1969). 8. J.-Y. Zhang, R.F. Casten, D.S. Brenner, Phys. Lett. B 227, l(1989). 9. D.S. Brenner et at., Phys. Lett. B 243, 1 (1990).

10. P. Van Isacker, D.D. Warner, and D.S. Brenner, Phys. Rev. Lett. 74, 4607 (1995).

11. G. Audi and A.H. Wapstra, Nucl. Phys. A 595, 409 (1995). 12. P. Franzini and L.A. Radicati, Phys. Lett. 6, 322 (1963). 13. P. Halse and B.R. Barrett, Ann. Phys. [ N Y ) 192, 204 (1989). 14. K.T. Hecht and A. Adler, Nucl. Phys. A 137, 129 (1969). 15. A. Arima, hl. Harvey and K. Shimizu, Phys. Lett. 8 30, 517 (1969). 16. R.D. Ratna Raju, J.P. Draayer and K.T. Hecht, Nucl. Phys. A 202, 433

(1973). 17. A. Bohr, I. Hamamoto and B.R. Mottelson, Phys. Scr. 26, 267 (1982). 18. 0. Castaiios, hf. Moshinsky and C. Quesne, Phys. Lett. B 277, 238 (1992). 19. C. Bahri, J.P. Draayer and S.A. hfoszkowski, Phys. Rev. Lett. 68, 2133

(1992). 20. J.N. Ginocchio, Phys. Rev. Lett. 78, 436 (1997). 21. J.N. Ginocchio and A. Leviatan, Phys. Lett. B 425, 1 (1998). 22. D. Strottman, Nucl. Phys. A 188, 488 (1972). 23. P. Van Isacker, 0. Juillet and F. Nowacki, Plays. Rev. Lett. 82, 2060 (1999). 24. P. Vogel and W.E. Ormand, Phys. Rev. C 47, 623 (1993). 25. E. Caurier, F. Nowacki, A. Poves and J. Retamosa, Phys. Rev. Lett. 77, 1954

(1996). 26. R.B. Firestone, S.Y.F. Chu, V.S. Shirley, C.hf. Baglin and J. Zipkin, Table

of Isotopes, 8th Edition (John Wiley & Sons, New York, 1996). 27. D. Rudolph et al., Phys. Rev. Lett. 80, 3018 (1998). 28. I.M. Green and S.A. Moszkowski, Phys. Rev. 139, 790 (1965). 29. R. Arvieu and S.A. Moszkowski, Phys. Rev. 145, 830 (1966).

V. Collective Phenomena

SHAPE EVOLUTION IN NUCLEI

R. F. CASTEN

New Haven, CT 06520-8124, USA E-mail: [email protected]. yale. edu

Wright Nuclear Structure Laboratory, Yale University

The development of collectivity and shape transitions in nuclei is discussed, from the perspective of structural evolution, both with nucleon number and spin.

1. Introduction

Residual interactions amongst valence nucleons lead to correlations and ultimately to the onset of deformation. The nature of nuclear collectivity has long been a major theme in Jerry Draayer’s research and it is an honor to contribute this perspective in tribute to his impressive career.

Traditionally, the onset of deformation is viewed in terms of neutron and proton number. Examples abound in regions near A = 100, 150 and 220. These transition regions have recently become of greatly renewed interest with new ideas of phase transitions and phase with the development8>’ of new, critical point, symmetries that describe nuclei at the phase transition point analytically, and with the discovery of empirical manifestations of these s y r n m e t r i e ~ l ~ - ~ ~ . At the same time, shape evolution as a function of spin has become an important issue.

In this paper, we address both topics. Although second order, or continuous, phase transitions are often of interest in macroscopic systems, we will focus on first order phase transitions which appear to be of greater significance in understanding the low lying levels of nuclei. In the study of shape evolution with spin, we will discuss very recent developments in the use of E-GOS plots” and the study of nuclei that are axially asymmetric.

2. First Order Phase Transitional Behavior in Nuclei

Consider the symmetry triangle in Fig. la, which shows the three dynamical symmetries of the algebraic IBA model, and the two geometric critical point symmetries, E(5) and X(5), that are solutions of the differential Bohr

91

ccoexisstence

92

Hamiltonian for particular potentials V(p, y). E(5) represents a second order phase transition while X(5) is first order. The line connecting them is also a locus of first order phase transitional behavior.

O(6) r-Unstable

W 3 ) K(E=O), x=-1.32

Ut5) &(K=O)

Phase transition

Figure 1. definitions of variables.

a) Symmetry triangle for nuclear structure. b) Spanning the triangle-

As shown in Fig. lb, any point in the triangle can be specified in terms of two parameters, typically chosen to denote a “radial” distance from the U(5) (vibrator) vertex to the point, and an “angular” variable specifying the direction of this radial trajectory. With the IBA Hamiltonian

(1) E

K, H = ~ n d - K,Q . Q = K , [ - - Q . Q]

where Q = (st 2 + dt s ) + ~ ( d t J)(’), (2)

the structural evolution is specified solely by the ratio E / K , and x. Here, K, = 0 gives the vibrator or U(5) symmetry, and E = 0 gives a prolate symmetric rotor for x = 4 1 2 and a y-flat rotor for x = 0.

One obvious but little discussed implication of Fig. la is that it is impos- sible to go from the spherical sector (lower left) to deformed nuclei (upper right) without crossing the line of first order phase transitions [except in the singular case of completely y-soft potentials spanning the x = 0 trajectory from U(5) to 0(6)]. Consider now Fig. 2a which shows R4/2 E

E(4;)/E(2?) for the Sm and Ba isotopes above N = 82. The data for Sm are those which originally inspired the idea of phase transitional behavior in this region. R4/2 is linked to the shape of the nucleus and shows a sharp

93

rise near N = 90, reminiscent of the change in an order parameter near the critical point. The smoother behavior of Ba seems to suggest a gradual onset of collectivity without an actual phase transition. This is the traditional view but is inconsistent with the structure of the symmetry triangle in Fig. 1. Hence, in some way, the Ba isotopes must also be undergoing a first order phase transition. This, in turn, implies that a sudden rise in R4/2 is n o t a proper signature of a phase transition.

Control Parameter N

Figure 2. particle number on the suddenness of phase transitional behavior.

a) Empirical R4/z values for Srn and Ba. b) Illustration of the effects of finite

There are three ways to understand this. The first is that phase transitional behavior in finite systems is muted, as illustrated very schematically in Fig. 2b. In this view the Ba behavior would reflect the fact that Ba (Z = 56) has fewer valence protons and, therefore, fewer valence nucleons for a given neutron number, than Sm (Z = 62). Two other scenarios recognize that while a phase transition must be sudden as a function of the control parameter , it need not be sudden as a function of N. Alternately phrased, the control parameter (e.g., E / K ) need not be a linear function of N, and may depend on Z. For example, E / K may change faster with N for Sm than for Ba. Another possibility is that, while the Sm isotopic trajectory lies along the bottom leg of the triangle, for Ba the trajectory may be along a radial vector at some angle to the bottom axis (ie., x # -fi/2).

We test these three scenarios using the IBA with eqs. 1, 2. The first scenario is easy to test. Parameters are well known16 for Sm, namely x = -f i /2 and various C / K values, as a function of N. For each N, the boson numbers, NB, for Ba are three less than for Sm (e.g., for N = 90, NB (Is2Sm) = 10, while N B ( ' ~ ~ B ~ ) = 7). We compare the calculated R4/2 values for Ba and Sm for x = - 8 / 2 (bottom leg of the triangle) using the same E / K

94

values for a given N. The results are in Fig. 3a. The shape transition is transferred to higher N and in no way resembles the data of Fig. 2a. Hence, while boson number effects are certainly present, they do not account for the difference in how Ba and Sm navigate the phase transition region.

N

x = -1.32 x = -0.35

. . . . . . . .

5 1500

t 1

O ' 86 a8 90

N

Figure 3. a) Predicted R ~ / Z values for Sm and Ba using identical parameters except for different boson numbers. b) E(0;) for Ba compared to calculations for x = - a / 2 and -0.35.

We test the other scenarios by fitting R4/2 by varying for each of several x values. By definition we obtain perfect fits. Fixing K. to reproduce E(2:), we then inspect other data, such as E(Oi), the only other useful observable available for these Ba isotopes. Typically, in a spherical-deformed transition region, the 0; state drops in energy up to the shape transition point, where it minimizes and then it increases into the deformed region. For Sm, an excellent fit with x = - 8 1 2 is obtained. For Ba, as seen in Fig. 3b, the calculations with x = - 8 / 2 do not yield a good fit, while x values roughly around -0.4 are acceptable, at least for N = 88 and 90. Hence we conclude (rather tentatively) that the Ba isotopes follow an internal trajectory in the symmetry triangle something like that illustrated in Fig. lb. Clearly, much more extensive data on non-yrast levels, and B(E2) values involving them, would be highly useful to confirm this.

95

3. Structural Change With Spin

3.1. E-GOS Plots and Nuclear Phenomenology

Recently, the concept of E-GOS plots, that is, plots of E,/I vs. I for band or quasi-band structures was proposed by P. Regan and collaborator^^^. They provide a model-independent way of displaying the data and allow easy distinction between rotational and vibrational regimes. For a harmonic vibrator, E,/I - 1/I, decreasing monotonically to zero. For a symmetric rotor, E,/I - (4 - 2/1), which increases monotonically from 3 to 4. A vibrator to rotor transition as a function of spin therefore gives an E-GOS plot that decreases hyperbolically to the shape transition point and then changes course, increasing slightly with further increase in spin.

Analytic expressions for E-GOS trajectories also exist for an anharmonic vibrator (AHV), where the state with n phonons has

n(n - 1) E(I) = nE(2:) + €4 (3)

where n = I/2, and €4 is the 4: anharmonicity: €4 = E(4:) - 2E(2:). If €4

= (4/3)E(2:), eq. 3 recovers the rotor result R 4 / 2 = 3.33. If €4 = E(2:), then R4/2 = 3.0. Using eq. 3, one obtains

For R4/2 = 3.0, EY(I)/I = E(2:)/2 which is constant. As shown in Fig. 4a, E-GOS plots therefore have 3 regimes: R4l2 < 3 (AHV), 3.0 (the phase transition point), and > 3.0 (rotor). Note that eq. 4 can be rewritten

which is a function only of €4. It was shown in ref.l that a plot of E(4:) against E(2:) for all collective nuclei from Z = 38 - 82 lies on a universal bilinear plot with slopes of exactly (least squares fits) 3.33 and 2.00. The latter regime (with 2.0 5 R412 5 3.15) describes an AHV spectrum with finite but constant €4, even though these nuclei, and their R4/2 values, vary with E(2:). The fact that all these AHV nuclei can be described with the same €4 implies, via eq. 5 , that their modified E-GOS plots ( E7(1’ _ I E (2:1 vs. I) are identical to within the spread in empirical €4 values. Examples of such modified E-GOS plots are shown in Fig. 4b. Although there is a spreading (of - 20 keV) among the curves such plots allow one to identify anomalous nuclei, such as l g 4 0 s . The slopes in Fig. 4b are also interesting. From

96

eq. 5, a modified E-GOS plot should increase with spin. However, clearly, the preponderance of curves decrease with I, suggesting the need for higher order anharmonicities in the AHV (e.g., €6 = E(6:) - 3E(2:) - 3 ~ ) .

a) 1.0

3

\

w* 0.5

0.0

Figure 4.

R.4, < 3.00

\ R,/,= 3.00

/ R,, > 3.0 0

I 0 2 4 6 8 1 0 1 2 1 4

I

a) The three structural regimes of E-GOS plots. b) Modified E G O S plots (see eq. 5) for nuclei with 50 < Z < 82, 82 < N 5 104 and 2.16 5 R4/2 5 3.15. The downsloping trajectory at I = 6 - 8 is for IB40s. Modified E-GOS plots for nuclei with 2.0 < R4/2 < 2.16 show many patterns deviating from the general trend in this figure.

Thus, a simple analysis of E-GOS plots allows a classification of nuclei according to R4/2 values and a modified E-GOS plot allows an analysis of higher order interactions in the AHV model.

3.2. Wobbling Motion

Recently, there has been evidence for wobbling phonon excitations in odd-A at intermediate rotational angular momenta in the range I = 8 -

20. Wobbling motion entails axial asymmetry and existing discussions of this new collective mode have involved rigid triaxial shapes. However, most predictions (energy staggering in the quasi-y-band being the outstanding exceptionlg) of y-soft and y-rigid models are similar as long as < y > is the same. We have calculated2' B(E2) values in a y-flat potential using the IBA in the O(6) limit, for N -+ 03, to simulate the Wilets-Jean21 model in order to test if traditional signatures of wobbling really point to rigid y. The calculations were done for even-even nuclei where the odd spin y- band levels become the 1-phonon wobbling mode with increasing I. Figure 5a shows a comparison of B(E2; I;dd 4 (I + l)g) values for y-rigid22 and y-soft models. Clearly, this key signature of wobbling is nearly identical for y-soft and y-rigid models, the y-soft result also follows nearly the same 1/I dependence, and the B(E2),,t/B(E2)in staggering is the same as is seen in rigid triaxial rotor calculations. These results thus preclude the use of this

nucclei

97

signature in establishing the y-dependence of the potential (for even-even nuclei).

-* 0

+- y-soft t -- 1.2 a)

2- & I N

t 1.0 b)

0.6 - 0.4

0.2

6

'-- &l O.O 3 m

Figure 5 . a) Predictions for y-soft and y-rigid models with < y > = 30°. b) B(E2; I;dd + (I - 1),)/B(E2; I;dd + (I - 2)7).

However, another simple observable does do so. This is shown in Fig. 5b where y-soft and y-rigid models are seen to differ by an order of magnitude for the branching ratio B(E2; I;dd -+ (I - 1),)/B(E2; I;dd -+ (I - 2)y). It would be of interest to see if such a distinction persists robustly for the moderate y values (say, < y > N 10-15") typical of many deformed nuclei.

4. Conclusions

We have discussed the behavior of shape/phase transitions as a function of nucleon number at low spin, showing that, except for the singular spherical- deformed trajectory that preserves 0 ( 5 ) , all such transitions are first order. The behavior of different nuclei (illustrated by Sm and Ba near A = 150) was used to assess different trajectories in the nuclear phase diagram.

Secondly, a discussion of E-GOS plots led to a recognition of three classes of nuclei in E-GOS plots, namely those with R4/2 < 3.0 (downsloping), R4/2 = 3.0 (constant) and those with R4/2 > 3.0 (slightly upsloping). Moreover, it was shown that most nuclei from Z = 38 - 82 lying in the AHV structural region and with the same €4 values have similar modified E-GOS plots. Finally, a new signature distinguishing y-soft and y-rigid nuclei at intermediate spin in highly asymmetric nuclei was identified.

98

Acknowledgments

I a m grateful t o my many collaborators, especially N.V. Zamfir, E.A. Mc- Cutchan, and P. Regan. Work supported by USDOE Grant No. DE-FG02- 91ER-40609.

References

1. R.F. Casten, N.V. Zamfir, and D.S. Brenner, Phys. Rev. Lett. 71, 227 (1993) 2. A. Wolf, R.F. Casten, N.V. Zamfir, and D.S. Brenner, Phys. Rev. C49, 802

(1994) 3. F. Iachello, N.V. Zamfir, and R.F. Casten, Phys. Rev. Lett. 81, 1191 (1998) 4. R.F. Casten, D. Kusnezov, and N.V. Zamfir, Phys. Rev. Lett. 82, 5000 (1999) 5. N.V. Zamfir et al., Phys. Rev. C60, 054319 (1999) 6. J. Jolie, R.F. Casten, P. von Brentano, and V. Werner, Phys. Rev. Lett. 87,

162501 (2001) 7. J . Jolie, R.F. Casten, S. Heinze, A. Linnemann, V. Werner,

Phys. Rev. Lett. 89, 182502 (2002) 8. F. Iachello, Phys. Rev. Lett. 85, 3580 (2000) 9. F. Iachello, Phys. Rev. Lett. 87, 052502 (2001)

10. R.F. Casten and N.V. Zamfir, Phys. Rev. Lett. 85, 3584 (2000) 11. R.F. Casten and N.V. Zamfir, Phys. Rev. Lett. 87, 052503 (2001) 12. R. Kriicken et al., Phys. Rev. Lett. 88, 232501 (2002) 13. P.G. Bizzeti and A.M. Bizzeti-Sona, Phys. Rev. C66, 031301 (2002) 14. N.V. Zamfir et al., Phys. Rev. C 65, 044325 (2002) 15. P.H. Regan et al., Phys. Rev. Lett. (in press) 16. R.F. Casten et al., Phys. Rev. C 57, R1553 (1998); 0. Scholten, F. Iachello,

A. Arima, Ann. Phys. (N.Y.), 115, 325 (1978) 17. S. Bdegkd et al., Phys. Rev. Lett. 86, 5866 (2001) 18. H. Amro et al., Phys. Lett. B 553, 197 (2003) 19. N.V. Zamfir and R.F. Casten, Phys. Lett. B 260, 265 (1991) 20. R.F. Casten, E.A. McCutchan, N.V. Zamfir, C.W. Beausang, and Jing-ye

Zhang, to be published 21. L. Wilets and M. Jean, Phys. Rev. 102, 788 (1956) 22. AS . Davydov and G.F. Filippov, Nucl. Phys. 8, 237 (1958)

NEW EXACTLY SOLVABLE MODELS OF INTERACTING BOSONS AND FERMIONS

J. DUKELSKY* Instituto de Estructura de la Materia, Consejo Superior de Investigaciones

Cientijkas, Serrano 123, 28006 Madrid, Spain E-mail: [email protected]

c. ESEBBAG~ Departamento d e Matemdticas, Universidad d e Alcald, 28871 Alcald de Henares,

Spain Email: Carlos. [email protected]

s. PITTEL~ Bartol Research Institute, University of Delaware, Newark, Delaware 1971 6,

USA E-mail: [email protected]

Richardson’s exact solution of the pairing model can be generalized to three families of exactly solvable models for interacting bosons and fermions. We focus on the rational family and show how to map these models onto classical two-dimensional electrostatic problems. In the case of fermions, we use the electrostatic mapping of the pairing model to provide a new perspective on the superconducting phase transition in finite nuclei. In the case of bosons, we show that this class of models displays a second-order phase transition to a fragmented state in which only the two lowest boson states are macroscopically occupied and suggest that this provides a new mechanism for sd dominance in interacting boson models of nuclei.

1. Introduction

Exactly-solvable models (ESM’s) have played an important role in clari- fying the physics of quantum many-body systems’. In condensed matter

*Work partially supported by grant BFM2000-1320-c02-02 of the Spanish DGI. t Work partially supported by grant BFM2000-1320-c02-02 of the Spanish DGI. t Work partially supported by grants PHY-9970749 and PHY-0140036 of the National Science Foundation.

99

100

physics, well-known examples include the Heisenberg, Hubbard and Yang Models, which are solved using the Bethe Ansatz, the Tomonaga and Lut- tinger models, which are solved using bosonization, and the Calogero and Sutherland models, which are solved by introducing a Jastrow type of wave function. In nuclear physics, important examples include the Elliot SU(3) model2 and the three dynamical symmetry limits of the Interacting Boson Model3, which are solved using group theoretical methods.

The Pairing Model (PM) with non-degenerate single-particle energies can also be solved exactly, as was shown in the 60’s in a series of papers by Richardson4. Considering the importance of pairing in both condensed matter and nuclear physics and the history of important insight from ESM’s in these fields, it is hard to believe that the exact solution of this model passed virtually unnoticed for over three decades, until its recent rediscovery in a study of the physics of ultrasmall superconducting grains5.

In this talk, we first discuss the generalization of Richardson’s solution to three families of exactly-solvable pairing-like models6. We then develop a classical electrostatic analogy for the rational family of models, and use this to address two issues of importance in nuclear physics. One concerns the superconducting phase transition in finite nuclear systems7 and the other concerns sd dominance in interacting boson models of nuclei’.

2. Three families of Exactly Solvable Pairing Models

We begin our discussion of the three families of exactly-solvable pairing-like models by defining the elementary operators of the pair algebra,

which in turn are the generators of SU(2) for fermions, or SU(1,l) for bosons, and close the corresponding commutator algebras,

The operator K/ in (1) creates a pair of particles in time-reversed states with at(.) the particle creation (annihilation) operator and RZ = 1 + l/2. For fermion systems, 01 is the pair degeneracy of level 1. Throughout the paper, the upper sign refers to bosons and the lower sign to fermions.

Assuming that there are L single-particle states and recognizing that each S U ( 2 ) or SU(1,l) group has one degree of freedom, a model is integrable if there are L independent hermitian operators that commute with

101

one another. These operators are the quantum invariants and their eigenvalues, the constants of motion, completely classify their common eigenstates. To find them, we first define the most general hermitian and number- conserving one- and two-body operator in terms of the K generators:

To this point, the matrices X and Y from which the R operators are defined are completely free. Here we fix them by imposing the condition that they must mutually commute to define an integrable model. This condition is fulfilled if the X and Y matrices are antisymmetric and satisfy

Solving (4) leads to three families of solutions6. so-called rational family of models, for which the X and Y satisfy

We focus here on the

The q ' s that enter are an arbitrary set of non-equal real numbers. Any choice of them leads to an integrable rational model and any combination of the corresponding R operators produces an integrabIe rational hamiltonian. The rational family was used in ref.g to demonstrate the integrability of the PM hamiltonian, which can be obtained as a linear combination of its R operators, viz: H ~ M = 2 El qR: plus an appropriate constant.

For the rational family, as for the other two families, the exact eigenstates in the seniority-zero subspace can be expressed as (a similar ansatz can be used for other seniorities)

M

a=l 1

where M is the number of pairs. The function u, which depends on a set of unknown pair energies e , must fulfill the L eigenvalue equations Ri IS) = ri 19). The resulting expressions are

102

For a given set of the L + 1 parameters qi and g of the model, one must solve the A4 coupled nonlinear equations (7) for the pair energies e,. There are as many independent solutions as states in the Hilbert space.

3. An electrostatic analogy for Pairing-like Models

As we have seen, the eigenvalues and eigenfunctions of the three families of pairing-like hamiltonians can be obtained using the Richardson approach, for fermion and boson systems. From this, we can establish an exact electrostatic analogy for such problems by introducing the energy functional

It can be readily shown that when we differentiate U with respect to the pair energies e , and equate to zero we recover precisely the Richardson equations (7). Furthermore, differentiating U with respect to 2771 leads to the eigenvalue rl of the Rl operator (8).

To appreciate the physical meaning of U, we should remember that the Coulomb interaction between two point charges in two dimensions is

v (r17 r2) = -qlq2 In lr1 - r2) , (10)

where qi is the charge and ri the position of particle i .

electrostatic system with the following ingredients: Thus, U is the energy functional for a classical two-dimensional (2D)

0 A set of jixed charges, one for each single-particle level, located at the positions 277i and with charges f4. We call them orbitons. N free charges, one for each collective pair, ocated at the positions e, and with positive unit charge. We call them pairons.

0 A Coulomb interaction between all charges. 0 A uniform electric field in the vertical direction with intensity &&.

103

Orbiton

d 5 / 2 (17 1’7

The key point is that there is an exact analogy between this classical 2D electrostatic problem and the quantum pairing problem. This suggests that we might be able to use the positions that emerge for the pairons in the classical problem to gain insight into the quantum problem.

Some other properties of the electrostatic problem that we will use are:

0 Since the orbiton positions are given by the single-particle energies, which are real, they must lie on the vertical or real axis.

0 For fermion problems, the pair energies that emerge from the Richardson equations are not necessarily real. They can either be real or they can come in complex conjugate pairs. Thus, a pairon must either lie on the vertical axis (real pair energies) or be part of a mirror pair (complex pair energies).

0 For boson problems, the pair energies are always real and, thus, like the orbitons lie on the real axis.

Position Charge 0.0 -1.5 0.44 -2.0

4. A new pictorial representation of nuclear superconductivity

We now apply the electrostatic analogy to the problem of identical nucleon pairing to address how superconductivity arises in such systems. Because of the limited number of active nucleons in a nucleus, it is extremely difficult to see evidence for the transition to superconductivity in such systems.

Table 1. Position and charges of the orbitons appropriate to a pairing treatment of ll*Sn.

-1.0 hi, 2 5.60 -3.0

We will discuss what happens when we apply the electrostatic analogy to the PM hamiltonian

m m

104

0.6 -

0.4 - \:/ 0.2 -

0.0 - =a. (a) : -0.2. g=-0.02.

0.6.

0.4-

(b) .

/ O i Q) Y 0) 0.2.

0.0 1 =.:.. / -0.2 - g=-0.04 -

0.6.

for the semi-magic nucleus l14Sn. The calculations are done as a function of 9 , using single-particle energies ~j from experiment. Table 2 shows the corresponding information on the orbiton positions and charges.

Fig. 1 shows the positions of the pairons in the 2D plane as a function of 9 . Since '14Sn has 14 valence neutrons, there are seven pairons in the classical picture. In the limit of very weak coupling, six neutrons fill the d5/2 orbit and eight fill the g7 /2 . The corresponding electrostatic picture (Fig. l a ) has three pairons close to the d 5 / 2 orbiton and four close to the g7 /2 . In the figure, we draw lines connecting each pairon to the one that is closest to it. These lines make clear that at very weak coupling the pairons organize themselves as artificial atoms around their corresponding orbitons.

0.2

-0.2

-0.6 -0.4 -0.2 0.0 0.2 0.4 0.6

W e ) Figure 1. Two-dimensional representation of the pairon positions in 114Sn for three selected values of g. The orbitons are represented by open circles; only the lowest two, the d5,2 and 9712, are shown at the positions dictated by Table 1.

What happens as we increase the magnitude of g (Figs. lb-c)? [The physical value is M -0.092 MeV.] As g increases, the pairons repel, causing the atoms to expand. For g M -0.04, a transition takes place from two isolated atoms to a cluster, with all pairons connected to one another. We claim that this geometrical transition from atoms to clusters in the classical problem is a reflection of the superconducting transition in the quantum problem.

105

The above remarks relate to the classical 2D analogy to the pairing problem. At the quantum level, the corresponding message is that superconductivity is realized when collective (Cooper) pairs develop which involve the cooperative participation of all active orbits and in which all connection to individual orbits is lost.

5 . A new mechanism for sd dominance in the IBM

The electrostatic analogy can also be applied to boson pairing, with the pairons now confined to the real axis. Fig. 2 shows the pairon positions for a model involving 10 bosons moving in all even-l boson states up to L = 12 and interacting via repulsive boson pairing with strength g. The single-boson energies are assumed to increase linearly with 1.

Orbiton

0.0 0.2 0.4 0.6 0.8 '

g 0

Figure 2. Evolution of pairons for a model involving 10 bosons in all even-l states up to L = 12 subject to a hamiltonian with linear single-boson energies and a repulsive boson pairing interaction

For weak pairing, the pairons sit near the s orbiton, reflecting the fact that the bosons are almost completely in the s state. As pairing increases, a transition takes place to a scenario in which the pairons no longer sit near the s orbiton. However, even after the phase transition all pairons are confined to the region between the lowest two orbitons, the s and d. Thus, after the phase transition the boson pairs that define the corresponding quantum ground state are most likely of primary s and d character.

106

0.03- S 0 = 0.02- a P - p" 0.01-

0.00-

'\,, Model I / ',.. .. /

--. - _ -. .- -. -. ; -_ Model II

0.03- S 0 = 0.02- a P - p" 0.01-

0.00-

1.01

'\,, Model I / ',.. .. /

--. - _ -. .- -. -. ; -_ Model II

3 o 0.4 0

O 0.2-

0.0,'-

- _ _ _ _ _ _ - - - - - -

_- 4 8 12 16 20 24

M Figure 3. Occupation probabilities for a system of bosons and maximum angular momentum L = 12 and pairing strength g = 0.5 as a function of the number of boson pairs. The upper graph shows the sum of occupation probabilities for high-spin bosons ( I > 2) while the lower graph gives the occupation probabilities for s and d bosons.

What is the relevance of this to the IBM? As a reminder, in the IBM the s and d bosons model the lowest pair degrees of freedom for identical nucleons, those with J" = O+ and 2+. The key assumption is that all other bosons, reflecting energetically higher pairs, can be ignored, except for their renormalization effects on the sd space. A second point to note is that in any effort to model composite objects by structureless particles, there invariably arises a repulsive interaction between these particles, to reflect the Pauli exchange between their constituents.

The results in Fig. 2 suggest that in the presence of such a repulsive interaction between bosons only the two lowest boson degrees of freedom can correlate, namely the s and d. This in turn suggests that repulsive pairing between bosons provides a new mechanism for sd dominance in interacting boson models of nuclei.

These points can be made more quantitative by looking directly at the quantum results. In Fig. 3, we show results for the same interacting boson model as above, but with two choices for the single-boson spectrum. In

107

addition to the choice ql = 1 used before (Model I), we also consider ql = 1’ (Model 11). In this way, we can assess whether sd dominance is a generic feature of boson models involving repulsive pairing.

The figure plots the occupation probabilities of the s and d boson states (lower panel) and those for all other boson states (upper panel) as a function of the number of boson pairs M . The calculations are done at a value of g for which the system is well within the mixed phase. Both models show that as M grows, the systems become increasingly more sd dominated. Indeed, in the thermodynamic limit, both systems reach the usual O(6) limit of the IBM, with 50% s and 50% d bosons.

6. Summary

In this presentation, we have shown that (1) Richardson’s exact solution for the pairing model can be extended to three families of pairing-like models; (2) all such models can be mapped onto classical 2D electrostatic problems; (3) the exact solvability of these models coupled with the insight from the associated electrostatic mapping can be used to obtain a new perspective on several issues of importance in nuclear structure. The two examples we discussed concerned the mechanism for realizing superconductivity in finite nuclear systems and the role of the nucleon Pauli principle in producing sd dominance in interacting boson models of nuclei.

References

1. Z.N.C. Ha, in Quantum Many-Body Systems in One Dimension (World Sci- entific, 1996).

2. J.P. Elliot, Proc. Roy. SOC. (London) A245, 128 (1583). 3. F. Iachello and A. Arima, in The interacting Boson Model (Cambridge Uni-

versity Press, Oxford, 1995). 4. R.W. Richardssn, Phys. Lett. 3, 277 (1963); R.W. Richardson and N. Sher-

man, Nucl. Phys. 52, 221 (1964); R.W. Richardson, J. Math. Phys. 9, 1327 (1968).

5. G. Sierra, J. Dukelsky, G.G. Dussel, J. von Delft and F. Braun, Phys. Rev. B 61, R11890 (2000).

6. J. Dukelsky, C. Esebbag, P. Schuck, Phys.Rev.Lett. 87 (2001) 066403. 7. J. Dukelsky, C. Esebbag and S. Pittel, Phys. Rev. Lett. 88 (2002) 062501. 8. J. Dukelsky and S. Pittel, Phys. Rev. Lett. 86 (2001) 4791. 9. M. C. Cambiaggio, A. M. F. Rivas and M. Saraceno, Nucl. Phys. A424 (1997)

157.

EXACT SOLUTIONS OF THE ISOVECTOR PAIRING INTERACTION *

FENG PAN Department of Physics, Liaoning Normal University,

Dalian 116029, P. R. China E-mail: [email protected]

J. P. DRAAYER Department of Physics and Astronomy, Louisiana State University,

Baton Rouge, L A 7080%~001, USA E-mail: [email protected]

Exact solutions for low-lying J = 0 states of 2k nucleons interacting with one another through an isovector chargeindependent pairing interaction are derived by using the Bethe ansatz method. The results show that a set of highly nonlinear equations must be solved for Ic 2 3.

1. Introduction

Pairing has long been considered to be an important interaction in nuclei. The concept was first introduced by Racah within the context of a seniority coupling scheme.['] Various applications to realistic nuclear systems have been carried out[2] following suggestions from Bohr, Mottelson, and Pines.[3] A lot of effort has been dedicated to the pure neutron or pure proton pairing interactions using various techniques. Extensions to neutron-neutron, neutron-proton, and proton-proton pairing interactions have been f~rmula ted . [~-~] It is well-known that the isovector charge independent pairing Hamiltonian can be built by using generators of the quasi-spin group Spj(4), where j labels the orbits considered in the model space, and from this it also follows that the pairing Hamiltonian can be diagonalized within a given irreducible representation (irrep) of the direct

'This work is supported by the U.S. National Science Foundation (Grant Nos. 9970769 and 0140300) and the Natural Science Foundation of China (Grant No. 10175031).

108

109

product group Spl(4) x . . . x Spp(4), where p is the number of orbits. In this case exact solutions - even if only generated numerically - can be given.[*] It is also well-known that approximate numerical solutions can be obtained by using the BCS formali~rn.[~-~'l

A lot of effort has been devoted to finding exact analytic solutions of the nuclear pairing Hami l t~n ian . [ l~ -~~I Extensions to a consideration of generalized and orbit-dependent pairing interactions have been the focus of recent work based on the algebraic Bethe ansatz and infinite dimensional Lie algebraic method^.['^-'^] A method for finding roots of the Bethe ansatz equations for the equal strength pairing model that was solved earlier by Richardson has also been proposed.[20] However, these exact solutions are for proton-proton or neutron-neutron pairing interactions only. In this paper, exact solutions for the mean-field plus isovector charge independent equal strength pairing interaction are revisited using the Bethe ansatz method. We find that the solutions offered by Richardson[21] and Chen and

are only valid when the number of pairs is less than or equal to two.

In this paper we introduce a new formalism for solving the problem. In Section 2, the mean-field plus isovector pairing Hamiltonian and its Sp(4) quasi-spin structure are reviewed. In Section 3, a general procedure for solving the isovector charge independent pairing Hamiltonian is outlined and detailed results for seniority-zero states. The results show that a set of highly nonlinear equations will enter whenever the number of nucleons is greater than or equal to six. Section 4 is reserved for a short discussion regarding implications of our findings.

2. The isovector pairing Hamiltonian and the Sp(4) quasi-spin structure

The mean-field plus isovector charge independent pairing Hamiltonian can be expressed in terms of generators of quasi-spin groups Spj(4), where j labels the total spin of the corresponding orbits. Generators of Spj(4) are the pair creation, Af(p), and the pair annihilation, A j ( p ) , operators with p = +, -, 0; the total nucleon number operator Nj for orbit j ; and the isospin operators Tp( j ) :

110

( 2 . l b )

(2.2b)

m m

where (a jmmt) is the creation (annihilation) operator for a nucleon in the state with angular momentum j , angular momentum projection m, and isospin projection mt with mt = +$, -;. According to the Wigner-Eckart theorem, the pair creation operators Af ( p ) with {A:(+) =

-AT(+), Af(0) = A;(O), AT(-) = A T ( - ) } and the pair annihilation operators Aj(p) with {Aj(+) = A j ( - ) , Aj(0) = -Aj(0), Aj(-) = - A j ( + ) } are T = 1 irreducible tensor operators, that satisfy the following conjugation relation:

(2.4) Aj(p) = (-l)'-' (A;(-p)) t .

The mean-field, with single-particle energies ~j from the spherical shell model, plus isovector charge independent pairing interaction Hamiltonian can be expressed as

H = C E ~ N , - G ~ A ; ( ~ ) A ~ + ) , (2.5) j j j J P

where G > 0 is the overall pairing interaction strength. Since H is invariant under isospin rotation, both the isospin quantum number T and its

111

third component To with eigenvalue MT are good quantum numbers of the system.

3. Exact solutions

In this paper, we only consider seniority-zero states. Hence, the lowest weight state is an isospin scalar. A k-pair excitation eigenstate can be written as

where 10) is the seniority-zero and isospin scalar state satisfying

Aj(,u)lO) = 0 for ,u = +, - , O , (3.2)

and [A] is an irrep of s k . It has been confirmed in exact solutions of the equal strength pairing problem with only neutron-neutron or proton-proton pairing- intera~tion['~-'~] that the building blocks A: (x) can be expressed as elements of the non-linear Gaudin algebra G(SU(2)) with

It suffices to use the non-linear Gaudin algebra G(Sp(4 ) ) to construct the eigenstates, which is generated by

(3.4)

where p is the total number of orbits, gj(,u) are the Spj(4) generators, and ~j is the single-particle energy of the j-th orbit.

It should be noted that the possible irreps [A] occurring in (3.1) should be determined by properties of the AT(,u) operators. Because ,u can only take on three different values, ,u = +, - , O , Young diagrams constructed from those AT(,u) operators can have at most three rows. Furthermore, because the Schur-Weyl duality relation between the permutation group s k

112

and the unitary group U ( N ) , the irrep [A] with exact k boxes of s k can be regarded as the same irrep of U ( N ) . Since the irreps [A] contain at most three rows, in this case they can be considered to be equivalent to the same irreps of U(3). Therefore, the possible isospin quantum number T for a given irrep [A] of s k can be obtained by the reduction[23] of U(3) 3 SO(3). The remaining problem is to find the expansion coefficients Q['](xl, 5 2 , . . . , x k ) and to establish the Bethe ansatz equations based on the corresponding eigenvalue equation. The following elements of the Gaudin algebra G(Sp(4)) will be useful:

, A,(x) = a, T,(x) = l-E. ( j ) (3.5) Aj+W At( . ) = C - 1 - & j X 1 - &jX 3 2 j j j

for p = +, - ,O, and

Nj 1 - E j X

N ( x ) = c -. j

Then, the Hamiltonian (2.6) can be rewritten as

Solving the eigenvalue equation

[A]kM,T To) = Ep1"l<; [A]kM,T TO) (3.8) with the Bethe ansatz wavefunction (3.1), implies that one simultaneously determines the expansion coefficients Q['](x~, 22,. .. , x k ) and the Bethe ansatz equations that the spectral parameters X I , 2 2 , . .. , x k , should satisfy.

Our results confirm that solutions reported in [21-231 are valid for k 5 2 and for any k with T = k . In other cases, however, the solutions are not correct. For k = 3, using the building blocks A+(.), one can construct the Bethe ansatz wavefunction for different irreps of the permutation group S3 by using the induced representation m e t h ~ d . [ ~ ~ - ~ ~ ] For example, wavefunctions for the symmetric k = 3 and T = 1 case should be written as

Ic;[3,0,0], T = ~ , M T = 1) =

113

where gi (i = 1,2) are generators of S3, which are nothing but nearest neighbor permutations defined by gi = (i, i + 1) for i = 1 , 2 , . . . , k - 1, Q[xI(xl, 5 2 , 53) is the expansion coefficient, and

gT=O ( 5 1 , ~ ~ ) = A f ( z i ) A - ( ~ 2 ) + + A?(xi)A:(x~) + A$(xi)A$(xz). (3.10)

It is obvious that x1 and 5 2 in the primitive vector B o ( x l x 2 ) are symmetric with respect to 51 cs x2 permutation. Up to a constant, the coefficients in (1 + g2 + 9192) are taken from the Induction coefficient^[^^-^^] (IDCs) of Sz x S1 1 S3 for the coupling [2] @ [l] [3] . It should be emphasized that, generally, Q[3](2152; 53) # Q[3](x1xg; x2) # Q[3](x2x3; X I ) , where

Q [ 3 1 ( ~ 1 ~ 3 ; ~ 2 ) = g 2 Q [ 3 1 ( ~ 1 ~ ~ ; 53), Q [ 3 1 ( ~ 2 ~ 3 ; ~ 1 ) = S I S ~ Q [ ~ ~ ( ~ ~ X(3.11)

For the [3,0,0] and T = 1 case, we need the following commutation relation:

Using (3.12) with (3.8), one can prove that the eigen-energies are given by

(3.13)

However, in this case, there are nine independent basis vectors in the final expression. Except for the original eigenstate, (3.9), all other coefficients

114

in front of these basis vectors should vanish. Therefore, 5 + R(rc,) should be chosen to satisfy the same condition,

for i = 1,2,3, where

d31 = Q [ 3 1 ( ~ 1 , ~ 2 ; ~ 3 ) , p[31 = Q[31(51 ,~3 ;~2) , yl31 = Q [ 3 ] ( ~ 2 , ~ 3 ; ~ 1 ) (3.15)

are functions of zi (i = 1,2,3) satisfying conditions (3.14), and

metrization, we get

Fi 131 (21, 1c2,z3; d31, p[31,y[31) for i = 1,2,3, is a function of xi. After sym-

x2 p[31 + y[31 - z3 *[31 + y[31 - p[31 F1 - x2 - z1 a[31 + p[31+ 4 3 1 x3 - z1 a[31 + p[31+ 4 3 1 7

131 - +-

x2 a[31 + pi31 - y[31 21 (y[31 + +31 - pi31 [31 -

- 53 ai3i + pi + +I . +- (3.16)

The cancellation of unwanted terms requires that d31, ,d31, and y[31 satisfy the following equations:

F3 - x2 - z3 a[31 + pi31 + 4 3 1

115

Due to relations (3.15), the three sets of equations (3.17a), (3.17b), and (3.17~) can be changed into one another through the permutation g2 and glg2, and therefore, they are not independent. Substituting (3.16) into (3.17), one can get relations among PI3] and y[31, which lead to three sets of complicated polynomial solutions depending on the spectral parameters 2 1 , x2 and x3. Thus, we get three set of solutions of a!] and in terms of yl3l. By substituting them into Eq. (3.16), the final expressions for F’’ will be yI3] independent; and the corresponding Eq. (3.14) provides solutions for the spectral parameters 1c1, z2 and x3 of the problem. Similar results were also derived for [21] irrep.

4. Discussion

A general procedure, based on the Bethe ansatz, for building algebraic solutions for seniority-zero J = 0 states of 2k nucleons interacting through an isovector charge independent pairing interaction has been introduced. We used the procedure to generate explicit results for seniority-zero J = 0 states.

The results derived for k 5 2 as well as for 2k nucleons for symmetric irreps of s k with T = k agree with those given by Richardson[21] and by Chen and R i ~ h a r d s o n . [ ~ ~ - ~ ~ ] However, it is showed that the results given in [21-231 are not valid for six or more nucleons in non-symmetric irreps of the permutation group. The main difference lies in the fact that in the present work the expansion coefficients Q[’l are considered to be functions of the spectral parameter xi and different from one another for non-symmetric irreps of the permutation groups, while the expansion coefficients in the work of Richardson and Chen and Richardson were assumed to be spectral parameter independent. In fact, for 2k nucleon configurations, the present calculation shows that the expansion coefficients Q[’]k can be taken to be the same only for totally symmetric irreps [k] of the permutation groups s k with T = k or totally anti-symmetric irreps [Ik] with k = 1,2,3. But for other cases, general solutions of the type introduced here are required; those offered in [21-231 as solutions for general irreps are not possible.

One of us (PF) would like to express his sincere thanks to Professor Jerry Draayer for his long term support and hospitality during many visits to LSU, and all the best wishes to him on the occasion of his 60th birthday celebration.

116

References

1. G. Racah, Phys. Rev. 62, 438 (1942). 2. S. T. Belyaev, Mat. Fys. Medd. 31, ll(1959). 3. A. Bohr, B. R. Mottelson, D. Pines, Phys. Rev. 110, 936 (1958). 4. K. Helmers, Nucl. Phys. 23, 594 (1961). 5. B. H. Flowers and S. Szpikowski, Proc. Phys. SOC. (London) 84, 193 (1964). 6. A. M. Lane, Nuclear Theory (W. A. Benjamin, Inc., NY, 1964). 7. K. T. Hecht, Nucl. Phys. 63, 214 (1965). 8. K. T. Hecht, Phys. Rev. 139, B794 (1965). 9. M. Baranger, Phys. Rev. 122, 992 (1961). 10. B. Bredmond and J. G. Valatin, Nucl. Phys. 41, 640 (1963). 11. B. H. Flowers and M. Vujicic, Nucl. Phys. 49, 586 (1963). 12. R. W. Richardson, Phys. Lett. 14, 325 (1965). 13. R. W. Richardson, Phys. Rev. 141, 949 (1966). 14. R. W. Richardson and N. Sherman, Nucl. Phys. 52, 221 (1964). 15. R. W. Richardson, J . Muth. Phys. 6, 1034 (1965). 16. Feng Pan, J. P. Draayer, and W. E. Ormand, Phys. Lett. 422,l (1998). 17. Feng Pan and J. P. Draayer, Phys. Lett. 442, 7 (1998). 18. Feng Pan and J. P. Draayer, Ann. Phys. (NY) 270, 120 (1999). 19. Feng Pan, J. P. Draayer, and L. Guo, J. Phys. 33A, 1597 (2000). 20. J. Dukelsky, C. Esebbag, and S. Pittel, Phys. Rev. Lett. 88, 062501 (2002). 21. R. W. Richardson, Phys. Rev. 144, 874 (1966). 22. H.-T. Chen and R. W. Richardson, Phys. Lett. 34B, 271 (1971). 23. H.-T. Chen and R. W. Richardson, Nucl. Phys. A212, 317 (1973). 24. J. Q. Chen, Group Representation Theory for Physicists (World Scientific,

25. Feng Pan and J. Q. Chen, J . Math. Phys. 34, 4305 (1993). Singapore, 1989).

RELATIVISTIC RPA AND APPLICATIONS TO NEW COLLECTIVE MODES IN NUCLEI.*

P. RING AND N. PAAR Physik-Department der Technischen Universitat Munchen, 0-85748 Garching,

Germany

T. NIKSIC AND D. VRETENAR Physics Department, Faculty of Science, University of Zagreb, Croatia, and

Physik-Department der Technischen Universitat Munchen, 0-85748 Garching, Germany

We describe collective vibrations of nuclei within the framework of relativistic quasi-particle random phase approximation with non-linear meson couplings. Pair- ing correlations are described by a finite range effective particle-particle interaction of Gogny type. The quasi-particle random phase equations are solved in the canonical basis and collective strength distributions are discussed for the recently discovered low-lying collective El-modes and for isoscalar dipole excitations.

1. Introduction

The investigation of nuclei far from stability has gained worldwide interest in recent years. These nuclei are characterized by unique structure properties: the weak binding of the outermost nucleons and the effects of the coupling between bound states and the particle continuum. On the neutron rich side, in particular, the modification of the effective nuclear potential leads to the formation of nuclei with very diffuse neutron densities, to the occurrence of the neutron skin and halo structures. These phenomena will also affect collective vibrations of unstable nuclei, in particular the electric dipole and quadrupole excitations, and new modes of excitations might arise in nuclei near the drip line.

A quantitative description of ground-states and properties of excited states in nuclei characterized by the closeness of the Fermi surface to the

'This work has been supported by the Bundesministium fur Forschung und Bildung under the contract No. TM 979 and by the Alexander von Humboldt Foundation.

117

118

particle continuum, necessitates a unified description of mean-field and pairing correlations, as for example in the framework of the Hartree-Fock- Bogoliubov (HFB) theory. In order to describe transitions to low-lying excited states in weakly bound nuclei, in particular, the two-quasi-particle configuration space must include states with both nucleons in the discrete bound levels, states with one nucleon in a bound level and one nucleon in the continuum, and also states with both nucleons in the continuum. This cannot be accomplished in the framework of the BCS approximation, since the BCS scheme does not provide a correct description of the scattering of nucleonic pairs from bound states to the positive energy particle continuum. Collective low-lying excited states in weakly bound nuclei are best described by the quasi-particle random phase approximation (QRPA) based on the HFB framework. The HFB based QRPA has been investigated in a number of recent theoretical studies. In Ref. a fully self-consistent QRPA has been formulated in the HFB canonical single-particle basis. The Hartree-Fock-Bogoliubov formalism in coordinate state representation has also been used as a basis for the continuum linear response theory '. In Ref. the HFB energy functional has been used to derive the continuum QRPA response function in coordinate space. HFB based continuum QRPA calculations have been performed for the low-lying excited states and giant resonances, as well as for the ,D decay rates in neutron rich nuclei.

In this talk we concentrate on the relativistic QRPA in the canonical single-nucleon basis of the relativistic Hartree-Bogoliubov (RHB) model. The RHB model is based on the relativistic mean-field theory and on the Hartree-Fock-Bogoliubov framework. It has been very successfully applied in the description of a variety of nuclear structure phenomena, not only in nuclei along the valley of P-stability, but also in exotic nuclei with extreme isospin values and close to the particle drip lines. Another relativistic model, the relativistic random phase approximation (RRPA) , has been recently employed in quantitative ana.lyses of collective excitations in nuclei. Two points are essential for the successful application of the RRPA in the description of dynamical properties of finite nuclei: (i) the use of effective Lagrangians with nonlinear self-interaction terms, and (ii) the fully consistent treatment of the Dirac sea of negative energy states.

The RRPA with nonlinear meson interaction terms, and with a configuration space that includes the Dirac sea of negative-energy state, has been very successfully employed in studies of nuclear compressional modes 4 1 5 , 6 ,

of multipole giant resonances and of low-lying collective states in spherical nuclei 7, of the evolution of the low-lying isovector dipole response in nuclei with a large neutron excess ',', and of toroidal dipole resonances lo.

119

2. The relativistic quasi-particle random phase approximat ion

The equations of motion of relativistic mean field theory can be derived starting from a density functional ERMF [ j? , 4m], which depends on the density matrix 6 and the meson fields &, = u, w, p and A. From the classical time-dependent variational principle

one derives the equations of motion for the Dirac spinors and for the meson fields. The equation of motion for the density matrix reads

The single particle Hamiltonian is the functional derivative of the energy ERMF with respect to the single particle density matrix j?.

Pairing correlations can be easily included in the framework of density functional theory, by using a generalized Slater determinant I (a) of the Hartree-Bogoliubov type. It is characterized by the single particle density matrix j? and pairing tensor R. The energy functional depends not only on the density matrix j? and the meson fields &, but in addition also on the pairing tensor. It has the form

E[@, R, h] = ERMF[@, 4m] + E p a i Y [ R ] , (2)

The pairing energy Epair[R] is given by

(3) 1 4

V P P is a general two-body pairing interaction. In the framework of Re- laivistic Hartree Bogoliubov Theory (RHB)l1 we use the finite range Gogny interaction in the pairing channel . Finally, the total energy can be written as a functional of the generalized density matrix

Epair[R,l = -Tr [ R * V P P R ] .

.=( - K * 1 - p* ) , (4)

which obeys the equation of motion

ia,. = [‘FI(R), R] ( 5 )

120

The generalized Hamiltonian 'Ft is a functional derivative of the energy with respect to the generalized density R.

By eliminating the meson degrees of freedom the generalized Hamilto- nian 'Ft can be expressed as a functional of the generalized density R only. In the linear approximation the generalized density matrix is expanded

R = Ro + dR(t), (6)

where Ro is the stationary ground-state generalized density. The lin- earized equation of motion (5) reduces to the relativistic quasi-particle RPA- (RQRPA) equations is

the RQRPA matrix elements A and B are given as second derivatives of the energy functional with respect to the generalized densities R. We solve these equations in the canonical basis, the basis where the single particle density matrix p is diagonal.

3. Applications in the 0-region

In order to illustrate the RHB+RQRPA approach and to test the numerical implementation of the RQRPA equations we discuss the isoscalar monopole, isovector dipole and isoscalar quadrupole response of 22 0. Similar calculations for the neutron-rich oxygen isotopes were recently performed by Mat- suo in the framework of the non-relativistic continuum linear response theory based on the Hartree-Fock-Bogoliubov formalism in coordinate state representation. The two theoretical frameworks differ, of course, both in the physical contents, as well as in the numerical implementation. The results can, nevertheless, be compared at least at the qualitative level. In the HFB+QRPA model of Refs. a Woods-Saxon parameterization is adopted for the single-particle potential, and a Skyrme-type density dependent delta force is used for the residual interaction in the ph-channel of the QRPA. Since the calculation of the single-particle potential and ph-interaction is not self-consistent, the interaction strength of the residual interaction is renormalized for each nucleus in such a way that the dipole response has a zero-energy mode corresponding to the spurious center of mass motion. The present RHB+RQRPA calculations are fully self-consistent: the same combination of effective interactions, NL3 in the ph-channel and Gogny D1S in the pp-channel, are used both in the RHB calculation of the ground state and as RQRPA residual interactions. The parameters of the RQRPA

2

s 3 ' 4

z

a

- full pairing no dynamical pairing

0 22

h

. I I

I

121

800 -1

-._.--_ -c\ - .-..- '-

I a I ii 10>l2 I , I , I , I , I , I ,

E [MeV] E [MeV]

Figure 1. The strength function for the neutron number operator (left), and the isoscalar strength function for the monopole operator (right) in 220. The curves correspond to the RMF+RRPA calculation without pairing (dotted), with pairing correlations included in the RHB calculation of the ground state, but not in the RRPA residual interaction (dashed), and to the fully self-consistent RHB+RQRPA calculation (solid).

residual interactions have exactly the same values as those used in the RHB calculation.

In the left panel of Fig. 1 we display the monopole strength function of the neutron number operator in 220. There should be no response to the number operator since it is a conserved quantity, i.e. the Nambu-Goldstone mode associated with the nucleon number conservation should have zero excitation energy. The dashed curve (no dynamical pairing) represents the strength function obtained when the pairing interaction is included only in the RHB calculation of the ground state, but not in the residual interaction of the RQRPA. The solid line (zero response) corresponds to the full RHB+RQRPA calculation, with the pairing interaction included both in the RHB ground state, and in the RQRPA residual interaction. The same result was also obtained in the HFB+QRPA calculation for 240 in Ref. 2: the spurious strength of the number operator appears when the pairing interaction is included only in the stationary solution for the ground

122

1.2 -

1 -

5 0.8 -

F - IE 0.6 - N b) Y

I%

0.4 -

0.2 -

' 1 ' 1 ' 1 ' 1 '

- neutrons protons

' 1 ' 1 ' 1 '

- full pairing - -. no dynamical pairing

E [MeV] r tfml

Figure 2. The isovector strength function of the dipole operator in 220 (left). The fully self-consistent RHBSRQRPA response (solid line) is compared with the RMF+RRPA calculation without pairing (dotted line), and with the RHBSRRPA calculation that includes pairing correlations only in the ground state (dashed line). The proton and neutron transition densities for the peak at E = 8.65 MeV are shown in the right panel.

state, i.e. when the dynamical QRPA pairing correlations are neglected. r: in

220, shown in the right panel of Fig. 1, correspond to three different calculations: a) the RMF+RRPA calculation without pairing, b) pairing correlations are included in the RHB calculation of the ground state, but not in the RQRPA residual interaction (no dynamical pairing), and c) the fully self-consistent RHBfRQRPA calculation. Just as in the case of the number operator, by including pairing correlations only in the RHB ground state a strong spurious response is generated below 10 MeV. The Nambu-Goldstone mode is found at zero excitation energy (in this particular calculation it was located below 0.2 MeV) only when pairing correlations are consistently included also in the residual RQRPA interaction. When the result of the full RHB+RQRPA is compared with the response calculated without pairing, one notices that, as expected, pairing correlations have relatively little in-

The isoscalar strength functions of the monopole operator

123

200

5 *

*E E 2 e?: %

100 d

'0 10 20 30 40 50

- -. no dynamical pairing

'0 5 10 15 20 25 30 35

E [MeV] E [MeV]

Figure 3. The RHB+RQRPA isoscalar and isovector quadrupole strength distributions in 220 (left panel). In the right panel the full RHBSRQRPA isoscalar strength function (solid) is compared to the RMF+RRPA calculation without pairing (dotted), and with the response obtained when the pairing interaction is included only in the RHB ground state (dashed).

fluence on the response in the region of giant resonances above 20 MeV. A more pronounced effect is found at lower energies. The fragmentation of the single peak at M 12.5 MeV reflects the broadening of the Fermi surface by the pairing correlations.

The isovector strength function (J" = 1-) of the dipole operator

N+Z C r p y i m - - QT;' = N

C T n K m (8) z

N + Z

Z N

n=l p= 1

for "0 is displayed in the left panel of Fig. 2. In this example we also compare the results of the RMF+RRPA calculations without pairing, with pairing correlations included only in the RHB ground state (no dynamical pairing), and with the fully self-consistent RHB+RQRPA response. A large configuration space enables the separation of the zero-energy mode that corresponds to the spurious center of mass motion. In the present calculation for 220 this mode is found at E = 0.04 MeV.

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The isovector dipole response in neutron-rich oxygen isotopes has recently attracted considerable interest because these nuclei might be good candidates for a possible identification of the low-lying collective soft mode (pygmy state) , that corresponds to the oscillations of excess neutrons out of phase with the core composed of an equal number of protons and neutrons 12113. The strength functions shown in Fig. 2 illustrate the importance of including pairing correlations in the calculation of the isovector dipole response. Pairing is, of course, particularly important for the low-lying strength below 10 MeV. The inclusion of pairing correlations in the full RHB+RQRPA calculation enhances the low-energy dipole strength near the threshold. For the main peak in the low-energy region (M 8.65 MeV), in the right panel of Fig. 2 we display the proton and neutron transition densities. In contrast to the well known radial dependence of the IVGDR transition densities (proton and neutron densities oscillate with opposite phases, the amplitude of the isovector transition density is much larger than that of the isoscalar component), the proton and neutron transition densities for the main low-energy peak are in phase in the nuclear interior, there is no contribution from the protons in the surface region, the isoscalar transition density dominates over the isovector one in the interior, and the strong neutron transition density displays a long tail in the radial coordinate. In the left panel of Fig. 3 we display the RHB+RQRF'A isoscalar and isovector quadrupole (J" = 2+) strength distributions in 220. The low- lying J" = 2+ state is calculated at E = 2.95 MeV, and this value should be compared with the experimental excitation energy of the first 2f state: 3.2 MeV 14. The strong peak at E = 22.3 MeV in the isoscalar strength function corresponds to the isoscalar giant quadrupole resonance (IS GQR). The isovector response, on the other hand, is strongly fragmented over the large region of excitation energies E N 18 - 38 MeV. The effect of pairing correlations on the isoscalar response is illustrated in the right panel of Fig. 3. As one would expect, the effect of pairing correlations is not particularly pronounced in the giant resonance region. The inclusion of pairing correlations, however, has a relatively strong effect on the low-lying 2+ state.

Summarizing, the relativistic QRPA formulated in the canonical basis of the RHB model represents a significant contribution to the theoretical tools that can be employed in the description of the multipole response of unstable weakly bound nuclei far from stability.

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References 1. J. Dobaczewski W. Nazarewicz J. Engel, M. Bender and R. Surman, Phys.

Rev. C60, 014302 (1999). 2. M. Matsuo, Nucl. Phys. A696, 371 (2001). 3. M. Grasso, E. Khan, N. Sandulescu and N. Van Giai, Phys. Rev. C66, 024309

4. D. Vretenar A. Wandelt, and P. Ring, Phys. Lett. B487 334 (2000). 5. J. Piekarewicz, Phys. Rev. C64, 024307 (2001). 6. Zhong-yu Ma, Nguyen Van Giai, A. Wandelt, D. Vretenar, and P. Ring, Nucl.

Phys. A686 173 (2001). 7. Zhong-yu Ma, A. Wandelt, Nguyen Van Giai, D. Vretenar, P. Ring, and

Li-gang Cao Nucl. Phys. A703 222 (2002). 8. P. Ring D. Vretenar, N. Paar and G. A. Lalazissis, Phys. Rev. C63, 047301

9. P. Ring D. Vretenar, N. P a r and G. A. Lalazissis, Nucl. Phys. A692, 496

10. T. Niksi’ c D. Vretenar, N. Paar and P. Ring, Phys. Rev., C65, 021301

11. G. A. Lalazissis T. Gonzalez-Llarena, J. L. Egido and P. Ring, Phys. Lett. B379, 13 (1996).

12. E. Tryggestad et al, Nucl. Phys. A687, 231c (2001). 13. A.Leistenschneider et al, Phys. Rev. Lett. 86, 5442 (2001). 14. M. Belleguic et al, Nucl. Phys. A682, 136c (2001).

(2002).

(2001).

(2001).

(2002).

SUPERALLOWED BETA DECAY OF 74Rb AND SHAPE COEXISTENCE IN 74Kr: A TEST OF THE

STANDARD MODEL

E. F. ZGANJAR AND A. PIECHACZEK

Department of Physics and Astronomy Louisiana State University

202 Nicholson Hall Baton Rouge, LA 70803

E-mail: [email protected], [email protected]

Precise measurements of the intensities for superallowed Fermi 0' 4 0' 0- decays have provided a powerful test of the CVC hypothesis at the level of 3 x

and have led to a disagreement with unitarity for the CKM matrix at the 98% confidence level. It is essential to address possible trivial explanations for the apparent non-unitarity such as uncertainties in the calculated isospin symmetry-breaking corrections. We have carefully studied the 74Rb to 74Kr p- decay in order to measure the total non-analog P-decay branching and especially the p decay branching to the 0' state at 509 keV in 74Kr. We observed experimentally a non-analog branch of 336(20) x lo-' and deduce with the aid of a recent shell model calculation an analog branch of 99.5(1)%. The branching to the 509 keV level is < 5 x which confirms a recent theoretical estimate of the isospin mixing in this level and its analog in 74Rb. We also show that high-precision, complete spectroscopy must be performed to obtain meaningful j3 decay branching ratios.

1. Introduction

The precise determination of half-lives, branching ratios and QEc values is absolutely essential to testing the Standard Model using superallowed Fermi P-decays. This radioactive decay data, together with muon-decay information, currently provides the most precise value for the up-down quark matrix element Vud in the Cabibbo-Kobayashi-Maskawa (CKM) matrix which connects the weak-interaction and mass eigenstates of the standard model's three-quark generations [l] . In that model, the CKh4 matrix is unitary. The most recent evaluation of CKM unitarity finds a 2.2-0 disagreement with the value of 1.0 for the sum of elements in the first row required for unitarity [l] . Since this result would have profound implications for the minimal Standard Model, it is essential to exclude trivial explanations for the apparent CKM non-unitarity such as uncertainties in the calculation of the radial-overlap and isospin-mixing corrections.

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Currently, precise studies of superallowed P-decay reach to 54C0. Substantial uncertainties in Vud are due to uncertainties in the calculated radial overlap and charge-dependent corrections [ 11. With this in mind, a program was initiated at the ISAC facility at TRIUMF to measure the half-lives and branching ratios for the T, = 0 (odd-odd) superallowed emitters with A L 60 where the charge dependent corrections are predicted to be larger than for the lighter emitters [ l , 21. The nucleus 74Rb was chosen for the first experiments. A half- life measurement on 74Rb at ISAC [3] has yielded 64.761(31) ms which has the requisite accuracy ( for the precise logft determination required in Standard Model tests. Information on P-delayed y emission in the 74Rb decay was also obtained [4] during these early measurements.

The objective of the present study was to determine the total non- analog branch of the 74Rb p decay, which is a prerequisite for tests of the standard model, and especially the non-analog P-branching to the O', level at 509 keV in 74Kr, which can be related to the magnitude of the Coulomb mixing corrections. This low-lying 0' level, whose existence was first suggested by J. H. Hamilton [S] in 1981, then verified [6] by indirect means in 1997, and finally, in 1999, was directly observed at 509 keV by an in-beam experiment [7]. It has a decay constant 7 = 18.8(10) ns [9] and decays primarily by an electric monopole transition (internal conversion) to the 74Kr 0'1 ground state. The early work of Hamilton [5] not only predicted the O'* to lie near 500 keV, but also described the shape coexisting structures and showed that the deformed shape was the ground state. Indeed, in that early paper Hamilton spoke of the 74Kr ground state as superdeformed. He then presented the idea of reinforcing shell gaps to explain why 74Kr, with both N and Z close to 38, can have such a large prolate deformation.

We first observed the 495 keV electrons arising from EO, K-shell conversion of the 509 keV O'& O', transition, as well as y rays at 456 and 1 199 keV in May, 2000. The earlier ISAC experiment [4] to look for P-delayed y rays also found the 456 and 1198 keV y rays in the 74Rb P decay. Around the same time, the ISOLDE collaboration observed 495 keV electrons, but they did not identify any y rays following the 74Rb P decay other than 5 11 keV annihilation radiation [ 81.

We repeated the branching-ratio experiment in May 2001 with an improved spectrometer, re-measured the 495 keV, K-shell, EO electron intensity and, for the first time, its L-shell conversion at 507 keV. We also observed the 39 keV, K-shell, E2 electron intensity (arising from the 52 KeV O', + 2+1 transition), and observed 74Rb P-delayed gamma rays at 456, 695, 748, 1199, 1233, 1286, and 4244 keV. While the intensity of the yrays other than the one at 456 keV are quite small (between 9 and 52 x they nonetheless contribute significant population to the 0' state at 509 keV and thus must be taken into

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account in the determination of non-analog P-feeding. Since the ISOLDE collaboration [8] did not observe any y rays (not even the 456 keV one with 0.25% intensity), their upper limit for the P-branch to the 0' state at 509 keV of 70 x is somewhat larger than our value of 54 x 10". Our results further suggest that still unobserved y rays may contribute additional intensity to the population of the 509 keV 0' level, an issue that can be resolved in the future using improved experimental equipment. Despite the imperfections of our experiments, we were able to experimentally identify a non-analog feeding of 0.336(20) % and the ground state P branch of the 74Rb decay was deduced to be 99.5(1) %. The latter result was achieved by comparing measured y ray intensities to results of a shell model calculation which reproduced well the observed relative y ray intensities following the 74Rb decay and was used to account for a small, unobserved, portion of non-analog feeding directly populating the ground state [ 1, lo].

2. Experimental methods

The 74Rb nuclei were produced in spallation reactions between the 500- MeV TRIUMF proton beam of 10-20 pA intensity and an electrically heated na"b foil stack target of 10 gm/cm2 (May 2000) and 22 gm/cm2 (May 2001) thickness in the ISAC target station. The products of the reaction were transported to a surface ion source followed by the ISAC on-line magnetic separator that selected the mass 74 isobars. Apart from the 74Rb decay products, the dominating contaminant was the 8.12 min 74Ga isobar. The mass separated activity was implanted onto a 1/4 inch wide (May 2000) or 1/2 inch wide (May 2001) tape in the spectrometer located at the GP2 position on the ISAC beamline. The implantation spot on the tape was viewed by three (May 2000) and two (May 2001) LN-cooled Si(Li) diodes of 200 mm2 area and 5 mm thickness for the detection of conversion electrons, and an 80% efficient HPGe detector. Two fast NE-102A and BC-403 plastic scintillators of 2 mm and 40 mm thickness registered leptons from the mass 74 P decays. These detectors triggered the list-mode data acquisition in coincidence with the Si(Li) or HPGe detectors, and the thicker of the plastic scintillators provided energy information of the registered leptons. A schematic of the spectrometer used in the May 2001 experiment is shown in Fig. 1.

During the measurements, the tape was moved every 4-6 s. A total of 6.9 x lo7 (May 2000) and 1.4 x 10' (May 2001) 74Rb atoms were counted in the plastic scintillators. The peak detection efficiencies of each Si(Li) and the HPGe detector were on the order of 1% at 500 keV. The most abundant contaminant was 74Ga.

129

-Light Guide

f /

HOIICIW tight Guide

I

Fig. 1. Sketch of the experimental set-up used in the May 2001 experiment. The transport tape and the Si(Li) detectors are in the center, surrounded by a germanium detector and two plastic scintillators. The scintillators are coupled through light guides to photomultiplier tubes (not shown) located outside the spectrometer vacuum. The tape is 112 inch wide and 2 pn thick. It can be moved periodically to prevent the buildup of long-lived contaminating activities. The light guide connected to the 2mm scintillator was made hollow in order to reduce the 511 keV gamma background that results from the annihilation of positrons.

3. Experimental results

Shown in Fig. 2 is part of the conversion electron spectrum measured with the Si(Li) diodes. The line at 495 keV corresponds to the emission of EO, K-shell conversion electrons from the 0'2 + O+, decay of the isomeric level (2 = 18.8(10) ns [9]) at 509 keV in 74Kr. The line at 507 keV corresponds to EO, L- shell conversion of the same transition, while the line at 39 keV corresponds to E2, K-shell conversion of the 52 keV O', + 2'1 transition. Total transition intensities are presented in the decay scheme (Fig. 4) and in tabular form in Ref. [lo].

Shown in Fig. 3 is part of the y-ray spectrum measured by the HPGe detector. This spectrum is dominated by the 5 11 keV annihilation radiation (not shown) and the 74Ga decay (several lines shown). The line at 456 keV is the 2+, + O+, E2 transition in 74Rb that is well known from in-beam data [7, 111. It has

130

"20 100 200 300 400 500 Energy [kew

Fig. 2. Parts of the conversion electron spectrum measured with the Si(Li) diodes. Energies of transitions are indicated. C.E. = Compton Edge. See the text for an explanation of the observed transitions.

5000

4000 74Rb 51 1 456

-.---* -- _.l .- 1000

420 440 460 480 5dO 520 4 0 40

1233 . . 74Rb 1286

O ' l r e o 1180 1200 1220 1240 12'60 1280 '

14 12 10 8 6 4 2

4120 4160 4200 4240 4280 4320 4360

Energy [kev]

Fig. 3. Parts of the gamma ray spectrum measured with the HPGe detector. Arrows and stars indicate contaminating radiation from the 74Ga and 74Br decays, respectively.

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an intensity of 0.00250( 14) per 74Rb P-decay [ 101. We have demonstrated that this line belongs to the 74Rb decay by the measurement of its half-life and by an exhaustive analysis of possible sources of contamination (see Ref. [lo] and references therein). Its observation clearly demonstrates that 74Rb undergoes Gamow Teller (GT) transitions to higher lying 1' states. Population of these states could also lead to y-ray population of the O', level at 508 keV. This will be discussed further in the next section.

The 1233 keV y ray line in Fig. 3 corresponds to the (2'3) + 0'2

transition that was also observed in-beam [7]. Look ahead to Fig. 4 to see its placement. We also observed coincidences between the 1233 keV y ray and the 495 keV electrons that depopulate the 509 keV level as in the in-beam data [7]. The lines at 1199, 1286, and 4244 are identified as (0'3) + 2'1 , (2'3) + 2'2, and (1') + 0'1 transitions [ 101. Coincidences between y rays would be helpful in that regard, but the spectrometer shown in Fig. 1 incorporated only one HPGe detector. Coincidence data with the 495 keV electrons, however, clearly show that the 1199 keV transition, unlike the 1233 keV transition, does not lead to feeding of the 509 keV level. Statistics on the 4244 keV transition were too low to draw any conclusions about its population or depopulation channels.

4. Discussion

The experimental 74Rb decay scheme is presented in Fig. 4. Total transition intensities, per 74Rb p decay, are given in units of In contrast to the results of the ISOLDE collaboration [8] our data clearly show that one cannot compute the non-analog p population of the 509 keV 0'2 level, and hence the Coulomb mixing probability, from the intensity of the 509 keV EO transition alone. One must include the 52 keV E2 transition that depopulates the level as well as the 695 and 1233 keV yray that populates it, not to mention any as yet unobserved transitions from high-lying states. The ISOLDE collaboration [8] did make an estimate for the 52 keV transition intensity, but they did not observe the 1233 keV or any other y-ray transitions. Additionally, since the total 456 keV 2' level depopulation is eight times that of its feeding by the 52 keV transition [lo], there must exist appreciable GT P-decay to higher lying 1' states. This is supported by our observation of y rays at 695, 748, 1233, 1286, and 4244 keV, and by the fact that energy gates on the P spectrum taken with the thick plastic scintillator suggest that the population of both the 2', (456 keV) as well as the 0'2 (509 keV) level comes about primarily from lower energy positrons. While this is expected for the 456 keV 2', level, it was not expected for the 509 keV O', state. This suggests that the non-analog p branch to the O', level at 509 keV could be even smaller than the upper limit of 54 x deduced from the decay scheme presented in Fig. 4.

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Fig. 4. The 74Rb decay scheme. See the text for a discussion of the transitions, their intensities, and implications for the determination of the non-analog p branch to the 0'2 state at 509 keV as well as the total non-analog branching.

The existence of a large number (- 400) of 1' states accessible to GT p decay was predicted in a Physical Review Letter [ 11 by Hardy and Towner and led them to the conclusion that the analog (ground state) branch of the 74Rb p decay cannot be determined using high resolution y ray spectroscopy. However, it is not necessary to measure the GT strength function, only the total amount of non-analog feeding must be determined. The decay scheme shows that the low lying levels in 74Kr act as collector states for intensity coming from unobserved 1' GT states and by observing their de-excitation, a large fraction of the non- analog feeding can be determined. The remaining component, which directly populates the ground state, can be deduced using a shell model calculation [ l ] which reproduces well relative y ray intensities. In this way, we obtained a ground state branch of 9 9 3 1) % [ 101.

5. Shape Coexistence in74Kr

Shape coexistence in the krypton isotopes has been reviewed in a larger context by J. L. Wood et al. [12]. The neutron deficient krypton isotopes are characterized by a spherical or prolate ground state and an oblate g2 level which descends in energy as neutrons are subtracted. The oblate configuration is most likely related to the g9/2 [404] Nilsson orbital, which is at the Fermi surface

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in this mass region [ 131. In 74Kr, the 0'2 level reaches its minimal energy at 509 keV. As more neutrons are subtracted, the O', level starts to rise again and is located at 671(2) keV in 72Kr [9]. In ref. [9], the energies of the unmixed 01,; states were determined by an extrapolation from the rotational structures built on the two 0' levels, assuming that no mixing is present at higher excitation energies. The resulting positions of the unperturbed 0' levels show that the oblate configuration descends below the prolate one in 72Kr [9]. Thus, 72Kr is one of the few nuclei with an experimentally demonstrated predominantly oblate ground state. The shape mixing in 74Kr must then be approximately 50:50, since the 0'2 level reaches here its minimal energy. The monopole strength for transitions between the two 0' states reaches a maximum with p2(EO) = 90(20) x 10" [12] in 74Kr. Such large values of p2(EO) have been shown to be a sure indicator of shape coexistence 1121.

6. Conclusions

We have undertaken for the first time a high-precision y and conversion electron decay study on 74Rb, one of the "heavy" superallowed Tz=O 0 emitters in the mass region beyond 54C0. Nine y and conversion electron transitions were observed, their intensities determined, and a partial decay scheme for the 74Rb superallowed decay was derived. In conjunction with a recent shell model calculation, the intensity of the 0'1 to 0'1 analog 0 transition was found to be 99.5( I)%. This result, together with the well know 74Rb half-life and a precise determination of the 74Rb 0 decay QEc value [14] will allow exacting tests of calculated correction terms which are applied to the experimental data in order to determine theft value of the decay. Ultimately, when these corrections are well understood, theft value of the 74Rb decay can serve as an additional datum for the determination of the average Pfvalue which is used in tests of the standard model.

We have shown that high-precision, complete spectroscopy is required to precisely determine the 0-decay branching ratios. In addition to measuring the EO (495 keV) and E2 (39 keV) conversion-electron intensity depopulating the 0'2 state at 509 keV, we determined y-ray feeding to this level and other levels from high-lying 1' states, which are weakly fed by GT 0 decay, and from the tentatively identified 0'3 level which is fed by non-analog 0 decay. Energy gates on the 0 spectrum (QEc = 10410 keV) suggest that appreciable feeding to the 0'2 level comes from high lying unobserved states. Observation of these levels will reduce the branching ratio to the O', even further, but should not significantly affect the total non-analog branch. Our identification of 7 yrays in the 74Rb 0 decay is in stark contrast to the results of the ISOLDE collaboration [8] where no gammas were observed.

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Improved experiments will be pursued at TRIUMFASAC using the greater statistics afforded by an order of magnitude increase in 74Rb production, a better isobaric separation, and by the use of the 8n spectrometer now in operation at the ISAC facility.

Acknowledgements

The authors would like to acknowledge the entire E823 collaboration of ISAC. Appreciation is also extended to the ISAC staff for the 74Rb beam, and to the personnel in the LSU machine and electronics shops who constructed the spectrometer. This work was supported in part by the Canadian funding agency NSERC and by the U. S. Department of Energy under contract DE-FG02- 96ER40978.

References 1. I. S. Towner and J. C. Hardy, Phys. Rev. C 66,035501 (2002).

J. C. Hardy and I. S. Towner, Phys. Rev. Lett. 88,252501-1 (2002). 2. W. E. Ormand and B. A. Brown, Phys. Rev. C 52,2455 (1995). 3. G. C. Ball et al., Phys. Rev. Lett. 86, 1454 (2001). 4. G. C. Ball et al., Applications of Accelerators in Research and Industry,

16" Int. Conf., ed. by J. L. Duggan and I. L Morgan, 2001, AIP, pp. 297ff. 5. J. H. Hamilton et al., Phys. Rev. Lett. 47, 1514 (1981). 6. C. Chandler et al., Phys. Rev. C 56, R2924 (1997). 7. F. Becker. et al., Eur. Phys. J. A 4, 103 (1999). 8. M. Oinonen et al., Phys. Lett. 511 B, 145 (2001). 9. E. Bouchez et al., Phys. Rev. Lett 90,082502 (2003). 10. A. Piechaczek et al., Phys. Rev. C 67,051305(R) (2003). 11. D. Rudolph et al., Phys. Rev. C 56,98 (1997). 12. J. L. Wood et al., Nucl. Phys. A651 323 (1999). 13. A. Piechaczek et al., Phys. Rev. C 61,047306 (2000). 14. K. Blaum et al., to be published.

VI. Computational Physics and Large-Scale Nuclear Models

COLLECTIVITY, CHAOS, AND COMPUTERS

CALVIN W. JOHNSON Department of Physics, S a n Diego State University

5500 Campanile Drive S a n Diego, CA 92182-1233, USA E-mail: [email protected]. edu

Two important pieces of nuclear structure are many-body collective deformations and single-particle spin-orbit splitting. The former can be well-described microscopically by simple SU(3) irreps, but the latter mixes SU(3) irreps, which presents a challenge for large-scale, a6 initio calculations on fast modern computers. Nonetheless, SU(3)-like phenomenology remains even in the face of strong mixing. The robustness of band structure is reminiscent of robust, pairing collectivity that arises from random two-body interactions.

1. Nuclear epistemology

The goal of nuclear structure theory is to understand experimental spectra, but what do we mean by understand?

There are two routes to understanding. One is the brute force, extreme reductionism of cab i n i t io computations': one starts from bare nucleon- nucleon scattering data and eventually computes many-body binding energies, spectra, transition rates, etc. This is of course very appealing to physicists, and I believe such ab i n i t io calculations are among the most important nuclear structure results of the past decade, but it has the very obvious danger of getting lost in the numerical details. Furthermore, brute- force calculations are limited by available computing power.

The alternate is back-of-the-envelope reasoning: simple, primarily analytic models that are easy to intuit. Algebraic models are prime example^^.^. The danger here is that the simple picture may not accurately represent the microscopic physics.

Ideally, one would like to combine these two: perform microscopically detailed ab in i t io calculations built upon basis state constructed from algebraic models. Such a hybrid approach would certainly allow one greater insight into the microscopic calculation, and would, one hopes, be much

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1 38

more efficient. More specifically, I wish to address the possibility of using SU(3) irreps

as a best basis for large-scale microscopic calculations. The biggest obstacle is spin-orbit splitting that arises from the nuclear mean field.

2. A brief guide to nuclear structure

Nuclear structure is driven by several competing degrees of freedom. First, the nucleus has a mean field, which allowed Haxel, Jensen, and Suess, and Mayer4 to propose the non-interacting shell model. One of the primary features of the nuclear mean field is a strong spin-orbit splitting. Spin-orbit splitting arises naturally from a non-relativistic reduction of the Dirac equation; as the nucleus is more relativistic than the atom, it is understandable that spin-orbit splitting is very small in atomic physics but is a large feature in nuclear physics. (In fact it becomes so large that it gives rise to pseudospin; see ref. 5 9 6 and the contributions by Ginocchio and van Isacker in this volume.) Eventually from the noninteracting shell model developed the interacting shell model, where one chooses a finite set of fermion shell model states and diagonalizes a Hamiltonian in that space.

The starting point of the shell model is the independent particle assumption: the component protons and neutrons interact primarily with the mean field. This is a simplification: there are correlations between the nucleons, and most important are the collective correlations. One well known form of collectivity is pairing, whereby fermions of opposite (angular) momentum couple to zero. This is a pervasive feature of cold, dense fermion systems, but it play only a peripheral role in this paper. Through the seniority model2 one can understand pairing in terms of microscopic fermion states.

Instead the collectivity that I will pay most attention to, and the one which has been the primary focus of Jerry Draayer’s work, is quadrupole deformation. One can have both quadrupole vibrations and “static” quadrupole deformations that lead to rotational bands. Quadrupole deformations arise naturally out of the semiclassical liquid drop model of the nucleus, and can be treated more formally in the Bohr-Mottelson model and its generalization to geometric-collective models7.

One of the great breakthroughs in nuclear stucture physics was discov- ering how to connect collective motion to the underlying fermion micro- physics. Elliot’s SU(3) model8 and its successors showed how one could map rotational motion easily onto the fermion shell model. Furthermore,

139

as Rowe3 has emphasized, SU(3) maps also onto the Bohr-Mottelson and similar models, thus providing a critical bridge between macroscopic and microscopic pictures. SU(3), at least as phenomenology, describes beauti- fully many features of nuclear spectra. But how well do microscopic SU(3) wavefunctions match ‘realistic’ microscopic wavefunctions? That is a question I will return to.

One of the most powerful tools for nuclear structure is the spherical interacting shell model. Here one starts by assuming a spherically symmetric mean field, so that all single-particle states have good j. The model space is partitioned into subspaces by single-particle configurations: one subspace might be, for example, all states with the configuration (Od5/2)3 ( l ~ ~ p ) ~ ( O d 3 / 2 ) ~ . In fact, because the Hamiltonian is rotationally invariant, one can restrict to the states with a fixed total M (that is, Jz) and hence programs that work in this basis are often referred to as M-scheme codes. The many-body Hamiltonian matrix elements are then computed in this basis.

Because the total angular momentum operator J2 does not connect across configurations, it is easy to construct a many-body model space for which angular momentum is a good quantum number. fithermore, spin- orbit splitting can be treated nearly trivially in such model spaces. What cannot be treated easily is deformation: deformation mixes many configurations, and typically one needs to add effective charges to get correct magnitudes for E2 transitions, etc.

Despite this important drawback, M-scheme and related codes are very popular today. Some of the older codes, such as the Glasgow codeg or OXBASHlO, store the many-body Hamiltonian on disk. This works for a basis size of up to about half a million basis states. Beyond that, more recent shell-model codes such as ANTOINElI or REDSTICK12 recompute the Hamiltonian many-body matrix elements on-the-fly. With hard work and clever coding, this can be very efficient.

The interacting shell model in a spherical basis is not the only possible approach. One of Jerry Draayer’s great achievements has been to construct, with a series of collaborators, SU(3) shell model C O ~ ~ S ~ ~ J ~ J ~ JAt the heart of these codes are Slater determinants in a cylindrical rather than a spherical single-particle basis. This allows one rather easily to get fermion representations of SU( 3).

A useful generalization of the SU(3) model is the symplectic model 18,

which unifies quadrupole operators with center-of-mass motion. This allows one to treat multi-hR shell spaces and to project out exactly spurious center-

140

of-mass motion. Futhermore one can generalize the SU(3) technology to symplectic calculation^^^.

Other properties, however, are not as easy. Projection of good angular momentum is not as straightforward as for spherical shell-model configurations. (This can probably be made more efficient.)

Unfortunately one remaining feature of nuclear structure remains a potential obstacle: spin-orbit splitting from the mean field, which I take up in the next section.

3. Mixed or pure SU(3)-that is the question

A number of well-known phenomenological interactions mix SU(3) irreps. One is pairing16. More germane to my discussion is singleparticle spin- orbit splitting which arises from the nuclear mean-field’7i20. The bottom line: in calculations in the sd- and lower pf-shells one finds that single- particle spin-orbit splitting is by far the most important source of mixing of SU(3) irreps. If one eliminates spin-orbit splitting, then mixing of SU(3) irreps is enormously reduced2Ol2l.

We investigated the role of spin-orbit splitting as follows2o. First, we took ‘realistic’ interactions: Wildenthal’s USD interaction in the sd-shell 22

and the monopole-modified KB3 interaction in the pf-shell 23. These interactions started life as exact G-matrix effective interactions reduced from nucleon-nucleon forces, with some empirical adjustments fit to hundreds of levels and decays. These interactions are by no means schematic and were derived blindly with respect to SU(3).

We computed for various nuclides the ‘exact’ wavefunctions for these complicated, messy interactions. Using a Lanczos moment method, similar to that developed to compute Gamow-Teller strength distribution^^^, we were able to compute the distribution of the exact wavefunction onto SU(3) irreps, without having to compute all the SU(3) eigenstates.

In the sd-shell and particularly in the pf-shell we found the wavefunctions to be fragmented over many SU(3) irreps. When we simply eliminated the spin-orbit splitting, and nothing else, then the fragmentation was enormously reduced, even for pf-shell nuclides. One can fairly interpret the fragmentation of SU(3) irreps due to realistic spin-orbit splitting to mean that wavefunctions of pure or nearly pure SU(3) irreps are not very realistic on a microscopic level.

Despite this fragmentation, the energy of deformation is larger than spin-orbit splitting. If one considers, for example, 24Mg in the sd-shell, the

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leading (8,4) irrep of SU(3) outperforms the simplest spherical configuration (~l5/2)~, in terms of binding energy, B(E2) values, e t ~ . ~ ~ .

While that is impressive, another calculation using a single Hartree-Fock state projected onto good angular momentum, showed that for a number of sd- and lower pf-shell nuclides the projected HF state outperforms a single SU(3) irrep21. Again the difference is driven almost entirely by the single-particle spin-orbit splitting: if one removes spin-orbit splitting, the difference between a projected HF state and the leading irrep is small.

So clearly one needs both deformation and spin-orbit splitting. Any route that neglects one over the other has to work hard to catch up. But which route? A shell-modeler faces a large number of choices for basis states: spherical shell model configuations; SU(3) irreps; configurations built upon deformed Hartree-Fock26; or a ’mixed-mode’ basis25 combining two or more of these. Which is best, and how can we tell which is ‘best’?

As discussed above, an SU(3) basis can be very illuminating in terms of the physics. For very-large-scale calculations, however, one must be concerned with computational effciency. For example, in full OhR shell- model calculations, codes working in a spherical basis are still much more efficient than an SU(3) basis; the former can compute 24Mg roughly ten times faster than the latter. Let me emphasize that is for the full space including all configurations; the motivation of using an SU(3) basis is the belief that one can truncate drastically to a smaller and more efficient basis and still get a very good description of the spectrum and wavefunctions.

Unfortunately SU(3) isn’t always as efficient as one would hope, due to mixing of irreps due to spin-orbit splitting (much as quadrupole deformation strongly mixes configurations in the spherical shell model). In a hybrid approach, Gueorgueiv et a125 showed that an obliquethat is, nonorthogonal- basis consisting of a few SU(3) irreps and a few spherical configurations could work very well, requiring only a few states: the SU(3) irreps encoded deformation and the spherical configurations encoded spin-orbit splitting.

While this sounds marvelous, the problem with such a statement is that those few states are not easy to represent in the computer. Therefore, in order to truly diagnose how eficient a basis is, I make the following observation and proposal. Most modern, large-scale, interacting shell-model codes use Slater determinants as the fundamental internal representation. The Slater determinant is in a single-particle basis: spherical, cylindrical, or other such as Hartree-Fock. We can discuss usefully the computational efficiency in terms of the number of Slater determanants needed to represent a state or t o project out a good quantum number.

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(Let me also note that there is a significant difference between a computationally efficient basis and one that illuminates the physics. A handful of states, while computationally inefficient, can still shed significant light upon the general structure of the state. But it is also important, when propos- ing a basis, to distinguish between computational efficiency and “physics efficiency.” )

One can project out states of good angular momentum in a spherical shell-model space relatively efficiently, requiring only a few tens or a few hundred Slater determinants, and all within a single configuration. Spin- orbit splitting is trivial; and if one crosses major harmonic-oscillator shells, it is possible to project out spurious center-of-mass motion as well, as long as one includes the right set of configurations. On the other hand, building in deformation requires many configurations.

In the cylindrical shell model, one can get the leading SU(3) irreps, and thus deformation, easily with just a few Slater determinants. Furthermore, if one has multi-shell calculations and uses the symplectic extension, it is possible to project out spurious center-of-mass motion. On the other hand, as presently written in the SU(3) shell-model codes, projection of good angular momentum is not very efficient, requiring several hundred or even more than a thousand Slater determinants. (Indeed, this is why, for exactly the same model space, spherical shell-model codes are faster than SU(3) codes; the latter can probably be sped up.) Finally, spin-orbit splitting can only be handled by mixing many irreps.

What about Hartree-Fock based states? They include deformation and spin-orbit splitting, and the amount of effort needed to project out good angular momentum is roughly comparable to SU(3) irreps. Unfortunately, I believe that consistent projection of spurious center-of-mass motion could be problematic (for reasons I do not have space to discuss here); and this is critical for large-basis calculations.

Ideally one would like to combine ab initio calculations with the lessons learned from algebraic models. The latter describe deformation, and in the symplectic extension can project out spurious center-of-mass motion, but are strongly mixed by spin-orbit splitting. I think some generalized approach is needed, such as the “optimal basis states’’27 which combine symplectic states with generator coordinate methods. Such a proposal seems very appealing, but needs further study.

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4. Persistence of collectivity

Despite mixing of SU(3) irreps by spin-orbit splitting, the resulting spectra -not only energy levels but also B(E2) ratios-can be described very well phenomenologically by SU(3). That is to say, if one numerically solves a Hamiltonian of the form Q .Q f l - s , the resulting wavefunctions will strongly mix SU(3) irreps, but one can fit the energy levels and B(E2)s to analytic SU(3) predictions. In other words, the mixing, while strong, appears to be coherent. Chairul Bahri (Draayer’s former student) and David Rowe term this ‘quasi-dynamical symmetry’28 and relate it to ‘adiabatic decoupling of colective motion along the lines of the Born-Oppenheimer approximation.’ They considered the symplectic shell model with the Davidson interaction, and found when the wavefunctions are fragmented over many SU(3) irreps, the spectra still look remarkably like SU(3) rotors. So even when microscopic wavefunctions are not good SU(3) states, the spectral properties still look like SU(3).

There are several lessons to take away from this. First, that SU(3) irreps are not intrinsically very good microscopic wavefunctions-and spin- orbit splitting is mainly responsible. Second, and paradoxically, SU(3)-like behavior is very robust, suggesting that one might be able to coherently mix SU(3) wavefunction to get both the microscopic description and the phenomenology correct. This, I believe, is critical for application to large- scale, ab ini to shell-model calculations.

The final lesson is that collective behavior does not appear very sensitive to the details of the Hamiltonian. This leads me to my next topic.

4.1. Collectivity and random interactions

The above results argue that collective behavior is robust even when one adds ‘messy’ or ‘noisy’ pieces to an algebraic Hamiltonian. One can take this to the extreme and ask: if one leaves off the algebraic Hamiltonian al- together and just have a ‘noisy’ or r a n d o m Hamiltonian, does any collective behavior remain?

Surprisingly, the answer is yes. If one has a random two-body interaction in a fermion shell model, one sees robust signatures of pairing co l le~t iv i tyIf one has a random two-body interaction in the interacting boson model, one sees robust rotational and vibrational band structures3’. (See also Roelof Bijker’s contribution.) To date, however, no one has found a convincing random ensemble for the fermion shell model that gives rise to robust band structure, despite some proposals31.

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There have been a large number of papers written analyzing and pur- porting to explain the pairing-like behavior in fermion models, but to my mind none of them are terribly convincing. We still need a general theory of how collective behavior can arise generically can arise, a more general version of Bahri and Rowe’s quasi-dynamical symmetry. The situation reminds me of quantum chaos: the quantum wavefunctions of classically chaotic systems display ‘scars’ of classical periodic, but unstable, orbits.

5. Conclusion

I lay out two challenges for the intersection of algebraic models with large- basis shell-model diagonalization:

First, one must account for both deformation and spin-orbit splitting. If SU(3)-symplectic wavefunctions are to be an efficient computational basis for large-scale ab initio calculations, we must generalize further to account for spin-orbit mixing a priori.

Second, we need to investigate further how collective behavior arises generically and how it remains robust even in the presence of messy interactions. A good explanation might help answer the first challenge.

Acknowledgements

It was a pleasure and an honor to be invited to give this overview talk at the celebration of Jerry Draayer’s 60th birthday. The bibliography below gives just a taste of his many contributions to nuclear structure theory.

The work described herein was funded by the U.S. Department of En- ergy and the National Science Foundation.

References 1. H. Kamada et al., Phys. Rev. C 64, 044001 (2001); R. B. Wiringa,

S. C. Pieper, J. Carlson, and V. R. Pandharipande, Phys. Rev. C 62, 014001 (2001); P. Navratii, J.P. Vary, and B.R. Barrett, Phys. Rev. Lett. 84, 5728 (2000); Phys. Rev. C 62, 054311 (2000).

2. I. Talmi, Sample Models of Complex Nuclea (Harwood Academic Publishers, Chur, Switzerland, 1993).

3. D. J. Rowe, Prog. Part. Nucl. Phys. 37, 268 (1996). 4. M. G . Mayer, Phys. Rev. 75, 1969 (1949); 0. Haxel, J. H. D. Jensen, and

H. E. Suess, Phys. Rev. 75, 1766 (1949). 5. K. T. Hecht and A. Adler, Nucl. Phys. A137, 129 (1969); A. Arima, M.

Harvey, and K. Shimizu, Phys. Lett. 30B, 517 (1969). 6. C. Bahri, J. P. Draayer, and S. A. Moszkowski, Phys. Rev. Lett. 68, 2133

(1992); J. N. Ginocchio, Phys. Rev. Lett. 78, 436 (1997).

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7. A. Bohr and B. R. Mottelson, Nuclear Structure, Vol I1 (W. A. Benjamin, Inc., Boston, 1975).

8. 3. P. Elliott, Proc. Roy. SOC. (London) A245, 128 and 562 (1958); M. Harvey, Adv. Nucl. Phys. 1, 67 (1968).

9. R. R. Whitehead, et al, Adv. Nucl. Phys. 9, 123 (1977). 10. A. Etchegoyen, et al, MSU-NSCL Report 524 (1985). 11. E. Caurier, computer code ANTOINE, CRN, Strasbourg, 1989 (unpub-

lished); E. Caurier, A. P. Zuker, and A. Poves, in Nuclear Structure of Light Nuclei Far from Stability, Proceedings of the Obernai Workshop, 1989, edited by G. Klotz (CRN, Strasbourg, 1989).

12. W. E. Ormand, private communication. 13. J. P. Draayer and Y. Akiymka, J . Math. Phys. 14, 1904 (1973); Y. Akiyama

and J. P. Draayer, Comp. Phys. Comm. 5, 405 (1973). 14. C. Bahri and J . P. Draayer, Comp. Phys. Comm. 83, 59(1994). 15. J. P. Draayer and K. J. Weeks, Ann. of Phys. 156, 41 (1984); 0. Castaiios,

J . P. Draayer and Y. Lebscher, Ann. of Phys. 180, 290 (1987). 16. C. Bahri, J. Escher, and J. P. Draayer, Nucl. Phys. A 592, 171 (1995). 17. J . Escher, C. Bahri, D. Troltenier, and J. P. Draayer, Nucl. Phys. A 633,

662 (1998). 18. G. Rosensteel and D. J . Rowe, Phys. Rev. Lett. 38, 10 (1977);

Ann. Phys. (N.Y.) 126, 343 (1980). 19. J . P. Draayer, K.J. Weeks and G. Rosensteel, Nucl. Phys. A413, 215 (1987);

J. Escher and J. P. Draayer, Phys. Rev. Lett. 82, 5221 (1999). 20. V. G. Gueorguiev, J. P. Draayer, and C. W. Johnson, Phys. Rev. C 63,

014318 (2000). 21. C. W. Johnson, I. Stetcu, and J. P. Draayer, Phys. Rev. C 66,034312 (2002). 22. B.H. Wildenthal, Prog. Part. Nucl. Phys. 11, 5 (1984). 23. T.T.S. Kuo and G.E. Brown, Nucl. Phys. A114, 235 (1968); A. Poves and

A.P. Zuker, Phys. Rep. 70, 235 (1981). 24. E. Caurier, A. Poves, and A. P. Zuker, Phys. Lett. B 252, 13 (1990);

Phys. Rev. Lett. 74, 1517 (1995). 25. V. Gueorguiev, W. E. Ormand, C. W. Johnson, and J. P. Draayer, Phys.

Rev. C 65, 024314 (2002). 26. T. Hjelt, K. W. Schmid, and A. Faessler, Nucl. Phys. A 697, 164 (2002);

E. Bender, K. W. Schmid and A. Faessler, Prog. Part. Nucl. Phys. 38, 159 (1997); V. Velhquez, J. G. Hirsch andY. Sun, Nucl. Phys. A 643, 39 (1998); R. Sahu and V. K. B. Kota, Phys. Rev. C 66, 024301 (2002).

27. M. J. Carvalho, D. J. Rowe, S. Karram, and C. Bahri, Nucl. Phys. A 703, 167 (2002).

28. C. Bahri and D. J. Rowe, Nucl. Phys. A 662, 125 (2000). 29. C. W. Johnson, G. F. Bertsch, and D.J.Dean , Phys. Rev. Lett. 80, 2749

(1998); C. W. Johnson, G. F. Bertsch, D. J . Dean, and I. Talmi, Phys. Rev. C 61, 014311 (2000).

30. R. Bijker and A. Frank, Phys. Rev. Lett. 84, 420 (2000). 31. V. VelAzquez and A. P. Zuker Phys. Rev. Lett. 88, 072502 (2002).

LARGE-SCALE COMPUTATIONS LEADING TO A

APPROACH TO NUCLEAR STRUCTURE FIRST-PRINCIPLES

w. E. ORMAND AND P. N A V ~ T I L

P.O. BOX 808, L-414, Lawrence Livermore National Laboratory,

Livermore, CA 94551 E-mail: [email protected]

We report on large-scale applications of the ab initio, no-core shell model with the primary goal of achieving an accurate description of nuclear structure from the fundamental inter-nucleon interactions. In particular, we show that realistic two-nucleon interactions are inadequate to describe the low-lying structure of log, and that realistic three-nucleon interactions are essential.

1. Introduction

An important god in the study of nuclear structure is to answer the question: Do we really understand how nuclei are put together? Towards this end, we wish to formulate a complete description of the properties of complex nuclei from first principles. In particular, we wish to determine if our knowledge of the fundamental interaction between pairs of nucleons is sufficient to describe the rich and complex structure observed in nuclei. This is an extremely difficult enterprise, and has really only been accomplished for the lightest of nuclei. Thus far, Faddeev-like1’2 approaches and the hyper- spherical f ~ r m a l i s m ~ > ~ have been applied to three- and four-body systems, while Monte Carlo methods5i6 have now been applied to systems with up to ten nucleons7. Also, the coupled-cluster expansion method8 has been applied to ISO. Here, we utilized new developments in many-body theory and the exceptional computational power of the ASCI system at LLNL to perform a study of nuclear structure from first principles, i.e., an ab initio approach, for nuclei throughout the pshell.

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147

2. Effective interactions and the shell model

The basic task at hand is to obtain solutions to the standard eigenvalue problem

( H - E,)P, = 0, (1) where E, is the desired eigenvalue, H is the Hamiltonian, and P, is the eigenfunction. One starting point for solving Eq. (1) is the interacting shell-modelg, where we introduce a set of orthogonal basis states $i to construct the exact solution, i.e., P, = Cicv i$ i . Solutions to Eq. (1) can then be obtained from a set of coupled equations that can be solved using matrix diagonalization techniques. The primary difficulty encountered is that because of the short-range repulsion in the nucleon-nucleon interaction, a basis of infinite dimension is required.

This infinite basis problem can, in principle, be circumvented by the use of effective-interaction theory. First, one chooses manageable subset of the original basis states, which is defined by the operator P , leading to the slightly different eigenvalue problem

(Herr - E,)PP, = 0, (2) where PP, is the projection of the exact solution onto the chosen model space, E, is again the eigenvalue, and Gejj is an effective Hamiltonian that yields the exact solution of Eq. (1). The excluded space is then usually defined by the operator Q, with @ + Q = 1, P2 = P , Q2 = Q, and

An important feature of H e j f is that it is composed of two-, three-, ..., n-body components even if the fundamental interaction is only pair-wise. The power of He jf is that it may provide a mechanism to carry out computationally tractable calculations while including the relevant physics. For most potentials, the dominant correlations in the effective interactions are at the two-body level, but for smaller P-space, the higher-body correlations are essential for a correct result.

Here, we utilized a unitary transformation due to Lee and SuzukilO to derive the effective interaction. This formalism is the foundation for the highly successful no-core shell model (NCSM)11J2. The procedure is based on finding the transformation, eS, to the Hamiltonian so that the P- and Q-spaces for the many-body problem are decoupled, i.e.,

PQ = QP = 0.

(3) Qe--SH e s P = O . ..

Strictly speaking, in this form, H,fj is not unitary, but can be made so. Explicit formulae for the n-body matrix elements are given by Eqs. (9) and

148

(10) in Ref. 12. Two important features are evident. First, in the limit that P + 1, the effective interaction tends to the bare interaction. Second, a subset of exact n-body solutions are required to determine the H$i . These exact solutions may be obtained by any method, e.g., large-basis shell-model calculations utilizing either the bare interaction or an (n - 1)-body effective interaction.

Our calculations begin with a two-body (also plus three-body) Hamil- tonian for the A-nucleon system, which depends on the intrinsic coordinates alone. We utilize realistic interaction potentials that are derived from nucleon-nucleon scattering data. To facilitate our calculations, we introduce an A-nucleon harmonic-oscillator Hamiltonian acting only on the center-of-mass, whose effect is subtracted from the many-body calculation. The primary advantages of the harmonic oscillator are that it acts as pseudo mean field providing a convenient basis for expanding the many-body wave function and that the relative motion of the center-of-mass can be separated from the intrinsic degrees of freedom exactly. Within the harmonic- oscillator basis, we specify the P-space, designated by the maximum number, N,,, , of oscillator quanta excitations, and construct the A-body basis. We then obtain the eigenvalues, E,, using a shell-model code. This amounts to diagonalizing a symmetric matrix, whose dimensions are given by the number of A-body basis states. Although the dimensions can be quite large, efficient numerical techniques, such as L a n c ~ o s ~ ~ , exist that yield the lowest eigenvalues. The parameters governing our convergence are then: N,,,, defining the model-space; n, the number of clusters in the effective interaction; and b = d m , the oscillator parameter setting the physical scale. Ideally, once convergence is achieved, the NCSM solution is independent of these parameters. In practice, however, our best solution is obtained for the largest N,,, that is computationally feasible and a value of the oscillator parameter where the binding energy is least sensitive.

In general, computational limitations impose a compromise in the choice of N,,, and H$i . This is due to the fact that for each increment in N,,, the number of A-body states increases dramatically. While for larger n, the number of interaction matrix elements increases and the sparsity of matrix decreases. In addition, the effective interaction itself is more difficult to evaluate for increasing n and/or N,,, . To illustrate the level complexity of the three-body calculations, for N,,, = 4, pshell nuclei require 39,523,066 three-particle interaction matrix elements. In this space, the number of M - scheme 10-body configurations for 1°B with J," = O+ is 581,740, and the resultant matrix to be diagonalized has over 2.2 x lo9 non-zero elements.

149

10; I+ I 2+ 1

@ ! , 8 t- AV8'

Figure 1. Comparison of the NCSM and GFMC spectra obtained for the Argonne AV8' potential. The NCSM spectra are shown as a function of the model size denoted by N,,,,tiR.

3. Nuclear Structure Calculations

Over the past few years, several extensive studies have been performed with the NCSM using realistic "-interactions such as the Argonne AV8' potentials6 and CD-Bonn14. These include first the ab initio applications15 for 12C, an extensive study of A=6 nuclei16, an examination of the nature of excited states in 8Be, and a recent large-basis application for A=10 nuclei18. The study with A = 6 provides an excellent example of the convergence and the utility of the no-core shell model16. In particular, in Fig. 1, we compare the NCSM spectrum for 6Li (as a function of the model space N,,,) using the Argonne AV8' potential with results obtained from the GFMC method. Overall, there is good agreement between the two methods. Also, as will be shown below, the converged NCSM value for the total binding energy 6Li is agreement to within 400 keV of the GFMC calculation.

The inclusion of higher-body clusters generally improves the overall con~ergence'~. The Binding energies of the nuclei 6Li, 8Be, and loB are shown in Fig. 2. On the left-side of the figure the binding energies are plotted as a function of the oscillator parameter, which effectively defines the

1 50

r

-18 -

20 -

F -22: -24-

w - m -26 -

-28 -

-25

30

-50

55

0

h a (MeV)

Figure 2. Calculated ground-state energy of 6Li (upper panel), 8Be (middle panel) and loB (lower panel) using the AV8’ NN potential with Coulomb. Results using the two- body effective interaction and the three-body effective interaction in basis spaces up to 6hR in the range of HO frequencies of hR = 8 - 28 MeV are shown and compared to the GFMC results from Ref. 6 , On the rhs, the energies at the HO frequency minima as a function of Nmaz are plotted.

size of the nucleus. The figure shows parabolas for the various model spaces, which are denoted by the N,,, value, and with two-body (Vzeff - dotted lines) and three-body (T/3,ff - solid lines) effective interactions. In general, the behavior on the oscillator parameter is lessened (flatter parabola) as either the model space size increases or when more clusters are included in the effective interaction. The “best” result for a given model space is chosen in the region exhibiting the least dependence on the oscillator parameter. These “best” values are then plotted on the right-side of the figure as a function of the model space Nmax and compared with the results from the

151

GFMC method (full solid lines with a dotted line band denoting the GFM uncertainty). In general, for any given value of N,,, faster convergence is achieved with higher clusters included in H , f f . In addition, we note that the NCSM calculation with the two-body effective interaction still differs from the GFMC result by M 1.8 MeV even for the largest model space. On the other hand, the three-body effective interaction results are in better agreement for smaller model spaces. Given that 8Be is actually unbound, and is two alpha-particle resonance, this suggests that the three-body effective interactions includes more correlations into the wave function. Overall, the results obtained with the three-body clusters in the effective interaction are in agreement with the GFMC calculations to within 400 keV.

8

7

6

5 9 g4 LLI

3

2

I

0

2+ 0 -- I+ 0 2+ o--- o+ 1-------. -

Figure 3. Comparison of low-lying spectrum of loB obtained with the AV8’ two-nucleon interaction alone (left side) and with the Tucson-Melbourne three-nucleon force (right side) with experiment.

152

With confidence in convergence, we now turn to a more systematic study of the structure of light nuclei. A particularly salient example is log. We show the spectrum obtained with the AV8’ in Fig. 3 (using a three- body effective interaction, V&ff) in comparison with experiment. The most striking feature is that the ground state (3+) and the first excited state (1+) are reversed in order. We must now conclude that realistic two-nucleon forces fail to describe the low-lying structure of log. Indeed, this is a feature that appears to be common to all the realistic nucleon-nucleon forces. This is the first direct evidence that, in addition to providing extra binding, three-nucleon forces also impact nuclear structure.

We are then forced to conclude that a proper description of nuclear structure must a include so-called “true” three-nucleon force. Note that these are quite different from the three-body clusters that we included in the effective interaction, as these term are actually induced because of the effect of the finite model space. We have recently carried out calculations including the Tucson-Melbourne three nucleon force20 for log. The results are shown in Fig. 3, where better agreement with the experimental spectrum is obtained. In particular, the ordering of the first two states is now correct. Overall, one finds that the three-nucleon interactions has spin-orbit components that play an important role in determining the structure of nuclei in the region 10 < A < 15. At this stage, investigations are underway to determine if the remaining disagreement in Fig. 3 is due to: i) convergence of the three-body effective interaction, ii) the many-body model space, and iii) the form of the three-nucleon interaction itself.

4. Conclusions

Substantial progress has been made towards an exact description of nuclear structure. In this work, we describe the ab initio, no-core shell model and recent results. In particular, we find that realistic NN interactions by themselves are inadequate and that three-nucleon forces play an important role in determining nuclear properties. Further research is currently underway.

Acknowledgements

This work was performed under the auspices of the US. Department of Energy by the University of California, Lawrence Livermore National Lab- oratory, under contract No. W-7405-Eng-48. This project received support through a Laboratory Directed Research and Development grant, tracking NO. 00-ERD-028.

1 53

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051301 (2001) 18. E. Caurier, P. NavrAtil, W.E. Ormand, and J.P. Vary, Phys. Rev. c66 ,

024314 (2002). 19. P. NavrAtil and W.E. Ormand Phys. Rev. Lett. 88, 152502 (2002) 20. S.A. Coon, M.D. Scadron, P.C. Mcname, B.R. Barrett, D.W.E. Blatt, B.M.J.

McKellar, Nucl. Phys. A317, 242 (1979).

COMPUTATIONAL CHALLENGES OF QUANTUM MANY-BODY PROBLEMS IN NUCLEAR STRUCTURE:

COUPLED-CLUSTER THEORY

D. J. DEAN Physics Division, Oak Ridge National Laboratory,

P. 0. Box 2008, Oak Ridge, TN 37831-6373 USA E-mail: deandjQornl.gov

Nuclear structure requires solutions to the complicated quantum many-body problem. I discuss an initial implementation of the coupled-cluster method for nuclear structure calculations and apply our method to a preliminary study of 4He.

1. The nuclear many-body problem and coupled cluster theory

Over the next 10-15 years nuclear structure will undergo a significant period of growth. We can optimistically state this due to the advent of new rare isotope accelerators such as those currently being proposed or pursued in various countries including Japan, Germany, Canada, and the US. These machines will further open doors to unstable nuclei and will present theoretical physics with new challenges. The challenges include a description of these systems and their low-lying modes of excitation.

One may follow various theoretical paths to obtain information about the properties of nuclear systems. One path originates from following a reductionist approach. One begins with some derivation of the nucleon- nucleon interaction (such as that built upon meson exchange, chiral perturbation theory, or phenomenology), and one develops computational tools for solving the many-body problem, as well as one can from this point of view. Examples of research efforts pursuing this approach include the Green’s Function Monte Carlo collaboration who begin with the Argonne interaction and supplement it with effective three-body interactions1, and the no- core shell-model collaboration who generate a G-matrix+folded-diagrams effective interaction and diagonalize in a given model space2. Both methods are ab initio from the many-body point of view: they begin with the bare nucleon-nucleon interactions. Another valid approach which has been

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155

successful requires the development of effective nuclear interactions at either the mean-field level through the use of Skyrme-like forces3, or with the shell model by using effective interactions derived from experimental level and transition information4.

The ab initio approaches, while difficult, allow one to study emergent phenomena such as deformation or vibrations of the nuclear systems from the fundamental level of the bare interactions; however, the applications of these methods are at the present time limited to light nuclear systems. The effective interactions (whether of the mean-field or shell-model variety) may be applied to various regions of the nuclear chart, but they often (especially in shell-model applications) rely on data-fitting within the region being calculated. The successful shell-model interactions, such as the lsOd interaction5 or the O f l p interactions6, all require large data sets in order to be adjusted appropriately to reproduce existant data and predict certain quantities within a given region. Herein lies the difficulty of relying on fitted interactions: the experimental data coming from the Rare Isotope Accelerator (RIA) and other facilities may not be dense enough to allow for a successful fitting of effective shell-model interactions in regions of interest. With this in mind, it becomes essential for nuclear theorists to explore methods that will allow for ab initio calculations of nuclei both near stability and in regions where RIA and other radio-isotopic facilities will probe.

In these Proceedings, I will discuss a many-body approach, known as coupled-cluster theory, that may prove quite useful in applications to nuclear structure7. Coupled-cluster theory was first introduced in nuclear physics by Coester8 and Coester and Kummelg. Initial nuclear structure applications came in the mid-1970s with several papers from the Bochum group1'. Since that time, nuclear physics applications have been rather sporadic. On the other hand, the first chemistry application was discussed by &Zek and Paldusll, and the method became computationally feasible due to work by Pople12 and Bartlett and Purvis13 and has become widely used and developed in computational chemistry14. The interesting and desirable theoretical properties of the coupled-cluster method within computational chemistry have made it the method of choice in computations of many- body correlation effects in atomic, molecular, and chemical systems. While it was originally developed for the many-body ground-state, applications of the coupled-cluster method in quantum chemistry now extend to excited states and open-shell systems.

Nuclear applications of the coupled-cluster technique include approaches

156

in coordinate-space being addressed by the Manchester group15. Recently, Heisenberg and Mihaila16 have suggested a somewhat different formulation than that espoused in quantum chemistry. The reasons for the sporadic pursuit of coupled-cluster methods in nuclear structure probably arise from the lack of a good bare nucleon-nucleon interaction. In the last 10 years this problem has been effectively eliminated due to excellent nucleon-nucleon interactions that give x' per degree-of-freedom of nearly one. These interactions include the phenomenological Argonne VI, potential17, the meson exchange potentials such as CD-Bonn18, and the very recent nucleon-nucleon potentials based on chiral perturbation theory 19. Another reason that the method was not pursued was certainly the lack of computational power available in the late 1970s as compared to today.

The Coupled-cluster method is a fully microscopic theory that can be used to obtain energies and eigenstates of a given Hamiltonian. Further- more, the theory is capable of systematic improvements through increasingly higher-order implementations of a well-defined scheme of hierarchical approximations. Coupled-cluster theory is size extensive, which means that only linked diagrams enter into a given comp~tat ionl~. The method is also size consistent. This latter property has vast implications for chemical reaction studies and is not a property of the shell model".

2. Approach to Coupled-Cluster Theory

The presence of a hard core in various channels of the nucleon-nucleon interaction (with repulsion on the order of 2-5 GeV) causes difficulty for theories that wish to use basis state expansion techniques. One way to overcome this difficulty is to use a renormalized effective interaction within the model space where one will actually perform computations. This model- space, dubbed the P space, is a subset of the full Hilbert space. The excluded space, dubbed the Q space, represents the remaining part of the Hilbert space and is, in principle, infinite in size. Brueckner" originally developed the G-matrix theory that allows for the solution of the full A- body problem in the reduced Hilbert space. The G-matrix is given byz2

where V is the bare nucleon-nucleon interaction, T is the kinetic energy operator, and w is a starting energy. Diagrammatically, the solution of this equation amounts to generating all particle-hole ladder diagrams, with intermediate two-particle states outside the P-space, to infinite order.

1 57

K

Ef

Figure 1. The choice of model space. Particle-hole excitations from the P-space (with energy cutoff K) to the Q-space are allowed during the computation of the G-matrix. Coupled-cluster computations occur only in the P-space where the Fermi energy, ~ f , is determined by the reference Slater determinant 1 a).

We begin our calculation by first choosing the P space, as shown in Fig. 1. Within that space, we compute the G-matrix elements of the renormalized interactions. We then define a reference Slater determinant from which we perform the coupled-cluster calculation. By performing the coupled-cluster calculations only in the. P-space, we insure that no double counting of many-body perturbation theory diagrams occurs. Those diagrams that we do not include in this expansion are those for which a particle below the Fermi energy in the reference Slater determinant moves to the Q-space; however, as one increases the P-space, the contribution of these diagrams to observable quantities such as the energy should become very small.

By implementing the G-matrix formalism, we obtain as our Hamiltonian H = Cpq K,,aia, + Cpq,.,(pq I G 1 rs)aiaia,a,. , where Kpq are the one- body matrix elements of the kinetic energy operator, Kpq = (q5p 1 K I &), and @q 1 G(w) 1 T S ) are the antisymmetrized two-body matrix elements of the effective nucleon-nucleon interaction. The single-particle wave functions are the basis states of the problem, and the labels p , q, T , s represent all single-particle quantum numbers. In the following, we use the labels z jk to represent single-particle states below the Fermi surface, and labels abc to indicate single-particle states above the Fermi surface.

The basic idea of coupled-cluster theory is that the correlated many- body wave function ] @) may be obtained by application of a correlation

1 58

operator, T, such that

where is a reference Slater determinant chosen as a convenient starting point. For example, we use the filled 0s state as the reference determinant for 4He. This exponential ansatz has been well justified for many-body problems using a formalism in which the cluster functions are constructed by cluster operators acting on a reference determinant23.

The correlation operator T is given by T = TI + T2 + . . . TA , and represents various n-particle-n-hole (np-nh) excitation amplitudes such as

a < E f , i > E f

and higher-order terms for T2 to TA. We are currently exploring the coupled-cluster method at the TI and T2 level. This is commonly referred to in the literature as Coupled-Cluster Singles and Doubles (CCSD).

We compute the expectation of the energy from

E = (@o 1 exp(-T)Hexp(T) I@o) . (4) The Baker-Hausdorf relation may be used to rewrite the similarity transformation as an expansion that terminates exactly at four nested commutators when the Hamiltonian contains, at most, two-body terms, and at six-nested commutators when three-body potentials are present. We stress that this termination is exact, thus allowing for a derivation of exact expressions for the amplitudes. The equations for amplitudes are found by left projection of excited Slater determinants so that for the l p - l h amplitudes we must solve

0 = (a: I exp ( -T) Hexp ( T ) I +) , (5)

and similar equations for higher-order amplitudes. The commutators also generate nonlinear terms within these expressions. To derive these equations is straightforward, but tedious, work14.

Because of the nonlinearity of the equations, one must have a good first guess for the np-nh amplitudes. We then solve the equations by iteration. For closed-shell nuclei, we use a Moller-Plesset-like approach to generate the first guess for the iteration. Shown in Fig. 2 is the convergence of the energy of the system as a function of iteration number. For our test example, 4He, we achieve convergence at the level by 30 iterations in a model-space that includes seven major oscillator shells. Notice from the figure that most of the convergence is obtained within 10 iterations.

1 59

Iteration #

Figure 2. ations.

The convergence of the ground-state energy as a function of the CCSD iter-

By investigating the different terms within the equations and their contributions to the energies, one is able to generate a correspondence between CCSD and many-body perturbation theory. One finds that CCSD iterates the lowest first-, second-, third-, and fourth-order many-body perturbation theory diagrams to all orders. It should be noted that the third-order diagrams are incomplete at the CCSD level of truncation, although third-order corrections may be included if they are desired24.

3. Initial Results

Our overall goal is to understand the structure of nuclei using coupled- cluster theory as our tool. We are at the very beginning of this effort and have a few preliminary results that we will report here. We are computing at the singles and doubles level of the coupled-cluster theory. At this level of truncation, we assume that all t 3 and higher-order amplitudes are zero. We also assume for the moment that only two-body potentials are present in the nuclear problem. We have not yet corrected these results for center-of-mass contamination, which means that they should be viewed as preliminary.

We employ the new class of chiral potentialslg as our bare nucleon- nucleon interaction starting point. The chiral effective Lagrangians employed include one- and two-pion exchange contributions up to chiral order three and contact terms that represent the short-range force. The chiral potential reproduces the N N phase shifts below 300 MeV laboratory en-

160

Figure 3. The CCSD energy for 4He as a function of w for various model spaces.

ergy and the properties of the deuteron with high precision. We use the Idaho-B potential throughout these Proceedings.

The oscillator parameter tiw is variational in our theory, and we find that the energy is minimal at tiw = 11 MeV for Idaho-B and the 4He nucleus. As was mentioned above, the G-matrix contains a starting-energy dependence, w. We show in Fig. 3 this dependence, along with the dependence of our results on the size of the P-space we are considering. Several interesting features emerge from this figure. The first is that as one increases the P-space, the resulting energy depends less on w. This is reasonable: if P were infinite, the solution would recover simply the bare V interaction which has no w dependence. The second interesting feature is the rapidity of convergence of the results. Already at seven major oscillator shells one sees the onset of convergence of the total energy. In this model space we obtain the energy E = -26.6 MeV. We are currently investigating various possibilities for including the center-of-mass corrections.

4. Perspectives

While the results presented above indicate our first steps toward coupled- cluster theory research, they show outstanding promise. Our 4He calculations show evidence of convergence using 7-8 major oscillator shells. Pre- liminary calculations of l60 also show convergence within this model space.

We are just at the beginning of this exciting endeavor, and we first want

161

to demonstrate the validity of the method for closed-shell systems such as 4He, l60, and 40Ca. If we transform the nuclear Hamiltonian from the oscillator basis to the Hartree-Fock basis, we will also be able to consider any nuclear system. For light systems, we will incorporate a center-of- mass correction. The CCSD does not include all third-order diagrams, but this deficiency can be alleviated a triples c ~ r r e c t i o n ~ ~ ? ~ ~ transforming our method into CCSD[T]. We will explore methods for computing excited states and open-shell systems within CCSD. We will extend CCSD[T] to include three-body interactions. We will also explore the applicability of CCSD[T] to open shell systems and excited-state calculations. We are confident that much can be learned from the many-body physics by moving along this direction of research. We are equally confident that we will eventually be able to extend the coupled-cluster techniques to very neutron- rich nuclei.

Acknowledgements

This work is performed in collaboration with M. Hjorth-Jensen. Research sponsored by the Laboratory Directed Research and Development Program of Oak Ridge National Laboratory, managed by UT-Battelle, LLC for the US. Department of Energy under contract No. DE-AC05-000R22725.

References 1. S. Pieper, and R. Wiringa, Annu. Rev. Nucl Part. S., 51, 53 (2001). 2. P. Navratil and B.R. Barrett, Phys. Rev. C, 57, 3119 (1998). 3. S. Mizutori, J. Dobaczewski, G. Lalazissis, W. Nazarewicz, and P.-G. Rein-

hard, Phys. Rev. C, 61, 044326 (2000). 4. M. Honma, T. Otsuka, B.A. Brown, andT. Mizusaki, Phys. Rev. C, 65,061301

(2002). 5. B.A. Brown and B.H. Wildenthal, Ann. Rev. Nucl. Part. Sci., 38, 29 6. A. Poves and A.P. Zuker, Phys. Rep., 70, 235 (1985). 7. D.J. Dean and M. Hjorth-Jensen, in preparation. 8. F. Coester, Nucl.Phys., 7 , 421 (1958). 9. F. Coester and H. Kummel, Nucl. Phys., 17, 421 (1960). 10. H. Kummel, K.H. Luhrmann, and J.G. Zabolitzky, Phys. Rep., 36, 1

and vreferences therein). 11. J. CiZek and J. Paldus, Int. J . Quantum Chem., 5, 359 (1971).

1988).

1978),

12. J.A. Pople, R. Krishnan, H.B. Schlegel, and J.S. Binkley, Int. J. Quantum

13. R.J. Bartlett and G.D. Purvis, Int. J . Quantum Chem., 14,561 (1978). 14. T.D. Crawford and H.F. Schaefer 111, Rev. Comp. Chem., 14, 33 (2000).

Chem. Symp., 14, 545 (1978).

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15. R.F. Bishop, The coupled cluster method, in Microscopic Quantum Many- Body Theories and their Applications, edited by J. Navarro and A. Polls, Springer-Verlag, Berlin, 1998, pp. 13-70.

16. B. Mihaila and J. Heisenberg, Phys. Rev. C, 61, 054309 (2000). 17. R.B. Wiringa, V.G.J. Stoks, and R. Schiavilla, Phys. Rev. C, 51, 38 (1995). 18. R. Machleidt, Phys. Rev. C, 63, 035202 (2001). 19. D.R. Entem and R. Machleidt, Phys. Lett. B, 524, 524 (2002). 20. R.J. Bartlett, Ann. Rev. Phys. Chem., 32, 359 (1981). 21. K.A. Bruekner, Phys. Rev., 97, 1353 (1955). 22. M. Hjorth-Jensen, T.T.S. Kuo, and E. Osnes, Phys. Reps., 261, 125 (1995). 23. F.E. Harris, H.J. Monkhorst, and D.L. Freeman, Algebraic and Diagrammatic

24. K. Raghavachari, G. Trucks, J.A. Pople, and M. Head-Gordon, Chem. Phys.

25. P. Piechuch, private communication.

Methods in Many-Fermion Theory, Oxford Press, New York, 1992.

Lett., 157, 479 (1989).

VII. Mathematical Physics

EMBEDDED REPRESENTATIONS AND QUASI-DYNAMICAL SYMMETRY*

D. J. ROWE Deparment of Physics, University of Toronto,

Toronto, Ontario, M5S 1A7, Canada E-mail: [email protected]

This presentation explains why models with a dynamical symmetry often work extraordinarily well even in the presence of large symmetry breaking interactions. A model may be a caricature of a more realistic system with a “quasi-dynamical” symmetry. The existence of quasi-dynamical symmetry in physical systems and its significance for understanding collective dynamics in complex nuclei is explained in terms of the precise mathematical concept of an “embedded representation”. Examples are given which exhibit quasi-dynamical symmetry to a remarkably high degree. Understanding this unusual symmetry and why it occurs, is important for recognizing why dynamical symmetries appear to be much more prevalent than they would otherwise have any right to be and for interpreting the implications of a model’s successes. We indicate when quasi-dynamical symmetry is expected to apply and present a challenge as to how best to make use of this potentially powerful algebraic structure.

1. Introduction

I intended to talk about vector coherent state theory. However, several examples shown by others of what Jerry Draayer appropriately referred to as an adiabatic coherent mixcing of representations, prompted me to change my topic to a description of the mathematical structure and physical significance of this potentially powerful and physically useful concept.

When a simple model is successful at describing a physical system, there is a temptation to infer that the model has a corresponding degree of reality. However, it is easy to be misled. This concern led us to investigate why systems frequently appear to hold onto a dynamical symmetry in spite of strong symmetry-breaking interactions. This is particular evident in systems which exhibit a Landau second order transition from a phase with

*Work supported by the Natural Sciences and Engineering Research Council of Canada.

165

166

one apparent symmetry to a phase with a different symmetry. The outcome was the discovery of quasi-dynamical symmet ry '~~ ,~ .

2. What is quasi-dynamical symmetry?

It is well known that states of different, but equivalent irreps, of a Lie algebra (or Lie group) can mix coherently to form new irreps. For example, if { IaLM)} are states of angular momentum L and z-component M , with a distinguishing different states of the same angular momentum, then the states

{ I Q K L M ) = C C K , l a L M ) , M = -L , . . . ,+L} (1) (Y

span another (equivalent) so(3) irrep of angular momentum L. What is remarkable is that, for some Lie algebras, there are linear combinations of states from similar, but inequivalent, irreps that actually form a basis for an irrep of the Lie algebra. Such an irrep is called an embedded representation. They may seem like bizarre mathematical oddities but, in fact, embedded representations are common in physics and underlie the adiabatic separation of variables. We say that a model has a quasi-dynamical symmetry if its states span a so-called embedded representation of a Lie algebra2.

Definition: If MI is the Hilbert space for a (generally reducible) representation U of a Lie algebra g and WO c W is a subspace then, if the matrix elements of g between states lying in MI0 are equal to those of a representation UO of g, then UO is said to be an embedded representation.

Subrepresentations and linear combinations of equivalent irreps are trivial examples of embedded representations. Non-trivial examples are found for semi-direct sum Lie algebras of the rotor model kind (semi-direct sums with Abelian ideals). Other Lie algebras contract to this kind of algebra in large quantum number limits and, consequently, have very good approximations to embedded representations. The 4 3 ) and symplectic model algebras are examples of the latter. This is important for the microscopic theory of collective motion because, although spin-orbit and other residual interactions break the dynamical symmetries of the su(3) and symplectic models, they mix representations in a highly coherent way that preserves the algebraic structures of these models as quasi-dynamical symmetries. This was predicted to happen as an algebraic expression of an adiabatic separation of rotational and intrinsic degrees of freedom' according to the Born-Oppenheimer approximation. Thus, it is exciting to discover how extraordinarily good quasi-dynamical symmetry is in practical situations.

167

3. The rigid rotor algebra as a quasi-dynamical symmetry of the soft-rotor model

Without vibrational degrees of freedom, the soft-rotor model is not an algebraic model. It nevertheless has a quasi-dynamical symmetry given by the dynamical symmetry of the (less realistic) rigid-rotor model.

A spectrum generating algebra for a rigid-rotor model4 is spanned by three angular momentum operators and five quadrupole moments. The angular momenta span an so(3) subalgebra; the quadrupole moments commute among themselves as elements of an Abelian subalgebra and transform under rotations as components of a rank two spherical tensor. This algebra, known as rot(3), has irreducible unitary representations characterized by rigid intrinsic quadrupole shape parameters ,tl and y, related to the rotational invariants by

[ Q @ Q 1 o m P 2 , [ Q @ Q @ Q Q l o m P 3 ~ ~ ~ 3 y . (2)

Rigid-rotor irreps have basis wave functions expressible in the language of coherent state theory in the form

*gPL(fl> = (P, ylR(fl)IKLM) . (3)

In the physical world, there is no such thing as a truly rigid rotor. Real rotor wave functions, have intrinsic wave functions that are linear superpo- sitions of rigid-rotor intrinsic wave functions with vibrational fluctuations;

@ ‘ K L M ( f l ) = / $(PI 7) (P, ylR(fl2)IKLM) W P , 7). (4)

Due to Coriolis and centrifugal forces, an intrinsic wave function $(P,y) will generally change with increasing angular momentum. However, if the rotational dynamics is adiabatic relative to the intrinsic vibrational dynamics, then $(Ply) will be independent of L as assumed in the standard (soft) nuclear rotor model; the rigid-rotor algebra is then an exact quasi- dynamical symmetry for the soft rotor. This is clear from the fact that the matrix elements between states of a soft-rotor model band are given by

( @ ~ ~ L ~ M ~ [ Q u I @ K L M ) = ( P c o s ~ ) .ofi’~;l:~’(o).o~~(o)do 1 s

+ -((Psiny) /.oL44fl)[.om + .o?,,,(fl)].ofiM(fl) df l , (5) Jz which is precisely the expression of the rigid-rotor model albeit with the rigidly-defined values of P cosy and ,d sin y replaced by their average values.

168

Note that there is no way to distinguish the states of a soft-rotor band from those of a rigid-rotor band without considering states of other bands. This is because an embedded irrep is mathematically a genuine representation of the rot(3) algebra; it is simply realized in a way that may seem contrived from a mathematical perspective but which is natural and very physical for a nuclear physicist. Moreover, it is useful to extract the essence of this simple structure because of its less-than-obvious implications for other dynamical symmetries which have rotor and vibrator contractions.

4. Effects of the spin-orbit interaction in the SU(3) model

In molecular physics, one can find near-rigid-rotor spectra of orbital angular momentum states weakly coupled by a spin-orbit interaction to the spins of the atomic electrons. In nuclear physics the spin-orbit interaction is much stronger. However, far from destroying the rotational structure of odd nuclei, the spin is usually strongly coupled to the rotor and participates actively in the formation of strongly-coupled rotational bands. Indeed, in the Nilsson model, one includes the spin-degrees of freedom explicitly in constructing unified model intrinsic states.

It is important to recognize that it is not the spin-orbit interaction that works against strong coupling; it is the Coriolis force. In other words, both a strong rotationally-invariant interaction between the spin and spatial degrees of freedom and adiabatic rotational motion (meaning weak centrifugal and Coriolis forces) are important for strong coupling. Thus, it was anticipated5 that a spin-orbit interaction might well modify the predictions of a simple su(3) model and even mix its irreps strongly. But, the underlying su(3) structure should nevertheless remain discernable and even be indistinguishable from strongly-coupling rotor model predictions in the limit of large-dimensional representations. In other words, the mixing of su(3) irreps should be highly coherent as expected for an embedded representation and give low-angular momentum states of the form

with C A ~ K coefficients essentially independent of L. Calculations3, cf. Fig. 1, confirm this to a high degree of accuracy. (Note that, since the SU(3) model does not include Coriolis interactions, the quasi-dynamical symmetry for this sitation becomes exact in the X + p -+ 03 rotor limit.)

169

- 8

6 -4 - 2

0

- -

-8

6 -4 -2

-

0 Mixed - (loo,o)x(s=o) + (lOl,l)x(S=l)

Figure 1. The figure shows two SU(3) irreps: a (100,O) irrep with spin S = 0 and a (101,l) irrep coupled to states of spin S = 1. The right figure shows the result of mixing these two irreps with a strong spin-orbit interaction In spite of the ground and a beta-like vibrational band being very strongly mixed (essentially 50-50) the resulting bands would be indistinguishable by experiment from pure su(3) bands. (The first calculations of this type were carried out by Rochford3 for lower-dimensional irreps.)

5. SU(3) quasi-dynamical symmetry and major shell mixing

We know, from Nilsson model calculations, that major shell mixing is essential for a reasonable microscopic description of rotational states. We also know that, while the symplectic model does not adequately account for the spin-orbit and short-range interactions, it contains the rigid-rotor and quadrupole vibrational algebras as subalgebras and, consequently, does well as regards the long-range rotational correlations. Thus, on the basis of many preliminary investigations, we are confidant that the symplectic algebra, sp(3, R), should be an excellent quasi-dynamical symmetry for a realistic microscopic theory of nuclear rotational states. At this time, I show results which demonstrate that, within a quite large symplectic model irrep with a Davidson interaction, both the su(3) and rigid-rotor algebras are extraordinarily good quasi-dynamical symmetries.

Fig. 2 shows the spectrum of 166Er fitted with three models7: the su(3), symplectic, and rigid-rotor models, The fitted results are barely distinguish- able; they are equally successful at fitting the lower levels and E2 transitions and equally unsuccessful at taking account of centrifugal stretching effects. In the symplectic model case, this is due to the Davidson potential.

170

4000

5000 1 experiment

-

3000-

kev

2000

1000-

0 -

16

14

- . 12

- 10

. 6

- 4

i

SU(3) 18

16

14

12

10

8

6

4 2 0

Davidson rotor model 1 18

16

14

12

10

8

6

4

8

18

16

14

12

10

8

6

4 2 0

Figure 2. rotor models7.

Fits to the ground state band of 166Er with the SU(3), symplectic, and rigid-

Fig. 3 shows what the symplectic model wave functions look like in an su(3) basis. They exhibit an extraordinary degree of coherence; i.e., the cofficients are independent of angular momenta for a large range of values and indicate the goodness of su(3) as a quasi-dynamical symmetry.

6. SU(3) quasi dynamical symmetry for a model with pairing interactions

Finally, we investigated what happens in a model that includes both pairing and Q . Q interactions with a Hamiltonian of the form

H ( a ) = Ho + (1 - 4 K u ( 2 ) + aK"(3) 7 (7)

where T/,,(Z) = -GS+S- is an su(2) quasi-spin pairing interaction and Vsu(s) = -xQ. Q is an su(3) interaction. When a is zero or one, H is easily diagonalized because of its respective 4 2 ) and 4 3 ) dynamical symmetries. However, for intermediate values of a, diagonalization of H is a notoriously difficult problem because of the incompatible nature of su(2) and su(3); they are incompatible in the sense' that, within a given harmonic oscillator shell model space, the only space that is invariant under

171

0.400 1

0.000

-0.400

- J=O J=O

I I I I I I I / I / I I I I I I I I 1 / I I I I I I I I I I / I 1 l I I / l I l / I I I I I I I I I I l I / / I I I I I I l I l I 1 / I I I

Figure 3. multi-shell SU(3) basis covering 12 major harmonic oscillator shells’.

Expansion coefficients of symplectic-Davidson model wave functions in a

both su(2) and su(3) is essentially the whole S = T = 0 subspace. We therefore considered a model having a unitary symplectic dynamical

symmetry, usp(6) (the smallest Lie algebra that contains both quasispin su(2) and su(3) as subalgebras) and generated large-dimensional usp(6) irreps by artificially considering particles of large pseudo sping~l0.

The lowest energy states of J = 0, . . . ,8 are shown in Fig. 4. The results exhibit a phase transition at a critical value of a! M 0.6 that becomes increasingly sharp as the number of particles is increased. However, the system does not flip from an su(2) to an su(3) dynamical symmetry at the critical point. In fact, it undergoes a second order phase transition in which the su(3) symmetry above the critical point is a quasi-dynamical symmetry. This is seen by looking at the extraordinary coherence of the wave functions shown for four values of Q in Fig. 5. When Q = 1 (not shown) the wave functions, of course, belong to a single su(3) irrep but, for smaller values of (Y > 0.6, they straddle large numbers of su(3) irreps with expansion coefficients that are essentially independent of angular momentum, as characteristic of a quasi-dynamical symmetry.

172

N=48, ~ = 2 0

a

Figure 4. Energy levels of the Hamiltonian (7) as a function of CY (taken from ref.I0).

7. Concluding remarks

What destroys rotational bands is not the residual interactions. It is the Coriolis and centrifugal forces. Thus, we can expect quasi-dynamical symmetry to be a characteristic of any realistic description of rotational states.

I conjecture that quasi-dynamical symmetry will prove essential for a realistic microscopic theory of the rotational states observed in nuclei and other many-body systems. My belief that this will be the case is a response to the fundamental question: why do physical many-body systems exhibit rotational bands? In spite of huge efforts to separate the variables of a many-body system into subsets of intrinsic and collective variables, the fact remains that the separation of collective dynamics is fundamentally due to the adiabaticity of collective motions (as understood long ago by the architects of the collective models). Thus, after years of grappling with the complexity of realizing collective states in microscopic terms, the conclusion emerges that unless we give the adiabatic principle a central place in the theory, there is no way we will ever succeed. The remaining question is: just how do we do this?

173

E0.20 o.800 1

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ~=0.40

0.400

~k0.60 o.800 7

.4"+--++4+

0 400

References

1. J. Carvalho, R. Le Blanc, M. Vassanji, D.J. Rowe and J. McGrory, 1986, Nucl. Phys. A452, 240 (1986).

2. D.J. Rowe, P. Rochford and J. Repka, J. Math. Phys. 29, 572 (1988). 3. P. Rochford and D.J. Rowe, Phys. Lett. B210, 5 (1988). 4. H. Ui, Prog. Theor. Phys., 44, 153 (1970). 5. J. Carvalho Ph.D. thesis (Univ. of Toronto, 1984). 6. R. Le Blanc, J. Carvalho, and D.J. Rowe, Phys. Lett. B140, 155 (1984). 7. C. Bahri and D.J. Rowe, Nucl. Phys. A662, 125 (2000). 8. D.J. Rowe, "Compatible and incompatible symmetries in the theory of nu-

clear collective motion", in New Perspectives in Nuclear Structure (ed. Aldo Covello, World Scientific) pp 169-183.

9. D.J. Rowe, C. Bahri and W. Wijesundera, Phys. Rev. Lett., 80, 4394 (1998). 10. C. Bahri, D.J. Rowe, and W. Wijesundera, Phys. Rev. C58, 1539 (1998).

Figure 5.Elgenfunctions of the Hamiltonian (7) for four values of or shown as histogramsin an SSU(3) basis (taken from ref.

SHAPE-INVARIANCE AND EXACTLY SOLVABLE PROBLEMS IN QUANTUM MECHANICS

A. B. BALANTEKIN University of Wisconsin, Department of Physics

Madison, W I 53706, USA E-mail: [email protected]

Algebraic approach to the integrability condition called shape invariance is briefly reviewed. Various applications of shape-invariance available in the literature are listed. A class of shape-invariant bound-state problems which represent twelevel systems are examined. These generalize the Jaynes-Cummings Hamiltonian. Co- herent states associated with shape-invariant systems are discussed. For the case of quantum harmonic oscillator the decomposition of identity for these coherent states is given. This decomposition of identity utilizes Ramanujan’s integral extension of the beta function.

1. Introduction

The technique of factorization is a widely-used method to find eigenvalues and eigenvectors of quantum mechanical Hamiltonians. The factorization method was most recently utilized in the context of supersymmetric quantum mechanics ‘i2. In this method the Hamiltonian, after subtracting the ground state energy, is written as the product of an operator A and its Hermitian conjugate, At:

fi - EO = A t d , (1)

where Eo is the ground state energy. With this definition the ground state wavefunction in supersymmetric quantum mechanics is annihilated by the operator A:

Al?#Io) = 0. (2)

(3)

The Hamiltonian in Eq. (1) is called shape-invariant if the condition

A(a1)At (q ) = A+(a2)A(a2) + R(a1)

is satisfied. In Eq. (3) a l , a2,. . . represent the parameters of the Hamil- tonian. (The original Hamiltonian has the parameter a l , the transformed

174

175

Hamiltonian has a2 and so on. The parameter a2 is a function of the parameter a1 and the remainder R(a1) is independent of the dynamical variables of the problem.

Shape-invariance problem was formulated in algebraic terms in Ref. [4]. To introduce this formalism we define an operator which transforms the parameters of the potential:

~ ( a l ) o ( u l ) P - l ( u ~ ) = O(a2). (4)

Introducing new operators

B, = At(a l )P(a l ) B- = 21 = lP't(a1)A(a1). (5)

one can show that the Hamiltonian can be written as

H - Eo = AtA = B+B-. (6)

Using the definition given in Eq. (5), the shape-invariance condition of Eq. (3) takes the form

[B-, B+1 = R(ao), 7)

R(a,) = T(al)R(an-l)P+(al) . (8)

B-l$o) = 0, (9)

(10)

(11)

where R(ao) is defined via

One can show that

[ H , 33 = (R(a1) + R(a2) + . . +R(a,))B&

[a, BE] = -B!!(R(ai) + R(a2) + . . +R(u,)) ,

and

i.e. 8; I&) is an eigenstate of the Hamiltonian with the eigenvalue R(a1) + R(a2) + . . +R(an). The normalized wavefunction is

The algebra is given by the commutators

176

and

[B+, (R(ai) - R(ao))B+] = ( (R(a2) - R(ai)) - (R(ai) - R(ao))}@, (15)

and so on. In general there are an infinite number of such commutation relations. If the quantities R(a,) satisfy certain relations one of the commutators in this series may vanish. For such a situation the commutation relations obtained up to that point plus their complex conjugates form a Lie algebra with a finite number of elements.

In the shape-invariant problem the parameters of the Hamiltonian are viewed as auxiliary dynamical variables. One can imagine an alternative approach of classifying some of the dynamical variables as “parameters”. An example of this is provided by the supersymmetric approach to the spherical Nilsson model of single particle states ’. The Nilsson Hamiltonian is given by

H = C atai - 2kL.S + kvL2. (16) i

The superalgebra Osp(1/2) is the dynamical symmetry algebra of this problem 6 . Introducing the odd generator of this superalgebra

F+ = C u i a i t

i

one can show that the “Hamiltonians”

and

can be considered as supersymmetric partners of each other 6 . The shape- invariance condition of Eq. (3) can be written as

FFt = F t F + R,

R = 0.L - 314,

(20)

(21)

where the remainder is

i.e. in this example the radial variables are considered as the main dynamical variables and the angular variables are considered as the “auxiliary parameters”.

177

A number of applications of shape-invariance are available in the literature. These include i) Quantum tunneling through supersymmetric shape- invariant potentials 7; ii) Study of neutrino propagation through shape- invariant electron densities s ; iii) Exploration of the relationship between algebraic techniques of Gaudin developed to deal with many-spin systems, quasi-exactly solvable potentials, and shape-invariance '; iv) Investigation of coherent states for shape-invariant potentials loill; and v) As attempts to devise exactly solvable coupled-channel problems, generalization of Jaynes- Cummings type Hamiltonians to shape-invariant systems l 2 > I 3 . In this article we focus on the last two applications.

2. A Generalized Jaynes-Cummings Hamiltonian For Shape-Invariant Systems

Attempts were made to generalize supersymmetric quantum mechanics and the concept of shape-invariance to coupled-channel problems 14*15. In general it is not easy to find exact solutions to coupled-channels problems. In the coupled-channels case a general shape-invariance is only possible in the limit where the superpotential is separable l5 which corresponds to the well-known sudden approximation in the coupled-channels problem 16. However it is possible to solve a class of shape-invariant coupled-channels problems which correspond to the generalization of the Jaynes-Cummings Hamiltonian l7 widely used in atomic physics to describe a two-level atom interacting with photons:

f i J C = woata + w03 + R (u+& + 0_&t ) . (22) The shape-invariant generalization of the Jaynes-Cummings Hamilto-

fisusyjc = AtA + f [A,"'] ( 0 3 + 1) + (o+A + ,-At) . (23)

To find the eigenvalues of the Hamiltonian in Eq. (23) we introduce the operator

nian is 12:

s = 0+A + 0-At (24) the square of which can be written as

We now introduce the states

178

where 1 m) is the eigenstate of the shape-invariant Hamiltonian AtA with eigenvalue E,. It can be shown that the states in Eq. (26) are the eigenstates of the operator S:

s I *,)% = JEm+l I *m)*.

HsUSYJC = S2 f ms,

(27)

Since the Hamiltonian of Eq. (23) can be written as

(28)

it has the eigenvalue spectrum

for all states except the ground state which is given by

I * o ) = [,;)I 1

where I 0) is the ground state of AtA. The Hamiltonian HSUSYJC has an eigenvalue 0 on the state given in Eq. (30). A variant of the usual Jaynes- Cummings Model takes the coupling between matter and the radiation to depend on the intensity of the electromagnetic field. This variant can also be generalized to shape-invariant systems 13.

3. Coherent States for the Quantum Oscillator and Ramanujan Integrals

3.1. Quantum Oscillator as a Shape-invariant Potential

One class of shape-invariant potentials are reflectionless potentials with an infinite number of bound states, also called self-similar potentials ''J9. Shape-invariance of such potentials were studied in detail in Refs. [20] and [21]. For such potentials the parameters are related by a scaling:

a, = p a l . (31)

For the simplest case studied in Ref. [21] the remainder of Eq. (3) is given by

R(Q) = ca1 , (32)

which corresponds to the quantum harmonic oscillator. Introducing the operators

s+ = JTiB+R(al)-1'2 (33)

179

and

s- = (S+) .. + - - @(ad -1PB-, (34)

one can write the Hamiltonian of the quantum harmonic oscillator as A A it - El) = R(a1)S+S-. (35)

This Hamiltonian has the energy eigenvalues 1 - qn

1 - q En = R(a1)-,

and the eigenvectors

In writing down Eq. (37) we used the q-shifted factorial defined as n- 1

( x ; q)o = 1, (2; q)n = (1 - zq') , 71 = 1 , 2 , . . . (38) j = O

3.2. Coherent States for Shape-Invariant Systems

Coherent states for shape-invariant potentials were introduced in Refs. [9] and [22]. (For a description of an alternative approach see Ref. [23] and references therein). Following the definitions in Eqs. (5) and (6) (with Eo = 0) we introduce the operator

H - I B + = 611, (B-BIl = 1). (39)

The coherent state can be defined as lo:

where f(t) is an arbitrary function. This state can explicitly be written as

(41) + ...

180

where we used the normalized eigenstates of the operator H :

In a similar way to the coherent states for the ordinary harmonic oscillator the coherent state in Eq. (40) is an eigenstate of the operator &:

B- I 4 = zf"a0)l 1.4. (43)

3.3. q-Coherent States

To derive the overcompleteness relation of q-coherent states here we follow the proof given in Ref. [ll]. An alternative, but equivalent, derivation was given in Ref. [24]. To introduce the coherent states for the q-oscillator we take the arbitrary function in Eq. (40) to be

f[R(an)] = R(an)- (44)

The resulting coherent states are

Further introducing the auxiliary variable

these coherent states take the form

The overcompleteness of these coherent states can easily be proven using the integral

This integral was proven by Ramanujan in an attempt to generalize integral definition of the beta function 25. (An elementary proof is given by Askey in Ref. [26]). Using Eq. (48) the overcompleteness relation of the coherent states in Eq. (47) can be obtained in a straightforward way:

181

This overcompleteness relation could be useful to write down coherent-state path integrals for quantum harmonic oscillator.

I would like to express my gratitude to my collaborators G. Akemann, A. Aleixo, J. Beacom, and M.A. Candido Ribeiro who contributed to various aspects of the work reported here. This work was supported in part by the U.S. National Science Foundation Grants No. INT-0070889, PHY-0070161, and PHY-0244384.

References

1. E. Witten, Nucl. Phys. B188, 513 (1981). 2. F. Cooper, A. Khare and U. Sukhatme, Phys. Rept. 251, 267 (1995)

[arXiv:hep-th/9405029]. 3. L. E. Gendenshtein, JETP Lett. 38, 356 (1983) [Pisma Zh. Eksp. Teor. Fiz.

38, 299 (1983)]. 4. A. B. Balantekin, Phys. Rev. A 57, 4188 (1998) [arXiv:quant-ph/9712018]. 5. A. B. Balantekin, 0. Castanos and M. Moshinsky, Phys. Lett. B 284, 1 (1992). 6. A. B. Balantekin, Annals Phys. 164, 277 (1985). 7. A. N. Aleixo, A. B. Balantekin and M. A. Candido Ribeiro, J. Phys. A 33,

1503 (2000) [arXiv:quant-ph/9910051]. 8. A. B. Balantekin, Phys. Rev. D 58, 013001 (1998) [arXiv:hep-ph/9712304];

see also A. B. Balantekin and J. F. Beacom, Phys. Rev. D 54, 6323 (1996) [arxiv: hep-ph/9606353].

9. A.B. Balantekin, to be published in the Proceedings of the Ettore Majorana Workshop on Symmetries in Nuclear Structure, March 2003, A. Vitturi and R. Casten, Editors (World Scientific, 2003); G. Akemann and A.B. Balantekin, in preparation.

10. A. B. Balantekin, M. A. Candido Ribeiro and A. N. Aleixo, J. Phys. A 32, 2785 (1999) [arXiv:quant-ph/9811061].

11. A. N. Aleixo, A. B. Balantekin and M. A. Candido Ribeiro, J . Phys. A 35, 9063 (2002) [arXiv:math-ph/0209033].

12. A. N. Aleixo, A. B. Balantekin and M. A. Candido Ribeiro, J. Phys. A 33, 3173 (2000) [arXiv:quant-ph/0001049].

13. A. N. Aleixo, A. B. Balantekin and M. A. Candido Ribeiro, J. Phys. A 34, 1109 (2001) [arXiv:quant-ph/O101024].

14. R.D. Amado, F. Cannata, and J.-P. Dedonder, Phys. Rev. A 38,3797 (1988); Int. J . Mod. Phys. A 5, 3401 (1990).

15. T.K. Das and B. Chakrabarti, J. Phys. A: Math. Gen. 32, 2387 (1999). 16. A. B. Balantekin and N. Takigawa, Rev. Mod. Phys. 70, 77 (1998)

17. E.T. Jaynes and F.W. Cummings Proc. IEEE 51 89 (1963). 18. A.B. Shabat, Inverse Prob. 8, 303 (1992). 19. V. Spiridonov, Phys. Rev. Lett. 69, 398 (1992) [arXiv:hep-th/9112075]. 20. A. Khare and U. P. Sukhatme, J. Phys. A 26, L901 (1993) [arXiv:hep-

[arXiv:nucl-th/9708036].

th/9212147].

182

21. D. T. Barclay, R. Dutt, A. Gangopadhyaya, A. Khare, A. Pagnamenta and

22. T. Fukui and N. Aizawa, Phys. Lett. A 189, 7 (1994). [arXiv:hep-th/9309153]. 23. V. P. Spiridonov, arXiv:hep-th/0302046. 24. C Quesne, J. Phys. A 35, 9213 (2002). 25. S. Ramanujan, Messenger of Math. 44, 10 (1915); reprinted in Collected

Papers of Srinivasa Ramanujan, Ed. by G.H. Hardy, P.V. Seshu Aiya, and B.M. Wilson (1927) (Cambridge University Press) [reprinted by Chelsea, New York (1962)l.

Askey, Amer. Math. Monthly 87, 346 (1980); Applicable Anal. 8, 125 (1978/79).

U. Sukhatme, Phys. Rev. A 48, 2786 (1993) [arXiv:hep-ph/9304313].

26. R.

NONLINEAR RESONANT STATES AND SCATTERING IN A ONE-DIMENSIONAL BEC-MODEL

A. LUDU Northwestern State University

Department of Chemistry and Physics, Natchitoches, LA 71497, USA

E-mail: luduaOnsula. edu

We investigate 1-D inelastic collision and resonance states in a strong localized potential by using nonlinear wave functions (solitons) in order to prevent wave function spreading. For small excitations energy the nonlinear terms can be neglected inside the potential region and we obtained exact solutions for the entire real axis: solitons in empty space region and Schrodinger states in the interaction region.

1. Physical motivation

The purpose of this paper is to investigate a nonlinear quantum mechanics puzzle, that is the collision of a Nonlinear Schrodinger (NLS) soliton wave function with a square potential well or barrier. The overall dynamics of the particle is provided by the Gross-Pitaevskii (GP) equation which describes a Bose gas with a two-body attractive &function interaction. We consider a free compound system outside the strong potential as being described by a NLS equation without potential term. For the NLS equation we have analytic solutions in the form of modulated solitons traveling with constan- t shape and constant speed. The advantage of solitons is that they are bounded and square integrable and stable in time. Inside the potential well the governing equation is the GP equation. However, if we consider a very deep potential valley, or very high potential barriers, and small excitation functions, we can neglect the nonlinear term inside the nonzero potential zone, and hence the GP equation reduces to the standard Schrodinger e- quation (SE) with exact time dependent, or stationary analytic solutions. We can match the boundary condition at the ends of the potential region and we can normalize the whole wave function. In this way one can estimate reflection and transmission coefficients and put into evidence resonant states between the classic region and the soliton waves.

183

184

2. Introduction

There are at least three different models guiding towards such nonlinear quantum approach, and moreover towards the same type of nonlinear dynamics described by the Gross-Pitaevskii equation (GP), or its reduced form (absence of external potential) i.e. the cubic Nonlinear Schrodinger equation (NLS). One model is provided by the nonlinear extension of the traditional geometric collective model, that is introduction of large amplitude collective excitation of nuclear surface as solitons (i.e. rotons). Second is provided by nonlinear Hamiltonian hydrodynamics in terms of mass and current densities with nonlocal potential. The last approach is provided by the recent studies in BEC dynamics with application in nuclear clusters and alpha resonances 2.

We introduce a one-dimensional nonlinear model for the collision between a very stable orthodox quantum system (target system) and a compound quasi-classical system (projectile). The target has just one internal degree of freedom, for example an alpha particle or other medium-heavy nucleus that can be described under some good approximation as a particle in a deep potential well. The projectile is described by a nonlinear wave function, i.e. a NLS soliton traveling in uniform motion towards the target. The many-body internal correlations in the projectile are taken in- to account by its nonlinear dynamics (via the Hartree-Fock, and via GP equation) and result in the shape and modulation of this soliton. Basical- ly, this process is similar with a quantum collision between a free particle and a finite potential well, which can be found in any elementary quantum mechanics text book 3. Contrary to orthodox QM case, we consider the projectile to be a soliton and not a plane wave. There are advantages (non dispersive effect of wavefunction, internal degrees of freedom of complex projectile and its coupling with the target degrees of freedom, normalizable wavefunctions, simpler exact calculations) as well as disadvantages (not orthodox QM approache: target is a traditional quantum system, projectile is a quasi-classical condensate approximation). In the following we recall how the GP equation is obtain from field theory models, and we present calculations of resonant states of these inelastic collision between a soliton and a particle in a quantum potential well.

185

3. Gross-Pitaevskii equation

For a system consisting of many bosons, one can define a boson field operator &r) = x k i i k u k , where k is the momentum, iik is the annihilation operator, u b f k is the single particle wave function. The many particle Hamiltonian is:

where m is the mass of the boson, V(r) is the external potential, e.g. the trapping potential. Because of Bose-Einstein statistics, at low temperature, there may be a finite density of bosons in the zero-momentum (k = 0) state. It is known that a general criterion for Bose-Einstein condensation is the off-diagonal long-range order of the one-particle density matrix pl(x, y, t ) =< @(x)G(y) >= Tr[pG+(x)$(y)], where p is the density matrix < G+(x)$(y) >-+ @(x)*@(y), where the "collective" wave function @(r) is the eigenfunction of p1 with the largest eigenvalue '. For convenience we set $(r) = @(r) / f i , so that J I@(r)12dr) is the fraction of condensed particles, and is 1 in the limit of zero temperature and zero interaction. In general, +(r) is governed by a nonlinear Schrodinger equation, known as Gross-Pitaevskii equation:

where g = JdrU(r) = 4xh2a/m, a is the s-wave scattering length of a binary collision. Therefore although the underlying fundamental quantum mechanics is linear, the "macroscopic wavefunction" of the system, as e- mergent entity, is governed by a nonlinear Schrodinger equation.

4. Wave functions and solitons

We consider a finite symmetric rectangular potential well of depth VO and length 1. The dynamics of the system is described by a nonlinear square integrable solution '$(x, t ) of the GP equation, Eq.(2), everywhere on the real axis. Eq.(2) becomes in the 1-dimensional case:

186

where we take p = g N . Outside the interval z E [-;, $1 the potential is zero and eq.(3) reduces to the NLS equation:

-$xx - PI$I2$ - iq$t = 0, (4)

where subscripts represent partial differentiation, and the constants are p = -w, and q = T. When p > 0 (that is gN = p < 0) we have stable traveling modulated soliton solutions for NLS equation in the form:

where the free parameters A amplitude, a wavelength of modulation, and c phase are real. They describe a family of solutions with different amplitude, group velocity V = F, and halfwidth L = fii. The norm of the soliton wavefunction (the outer region) is:

and it's constant in time. We stress that the NLS equation, eq.(4) provides norm conservation in time, so the quantum mechanical probabilistic formalism is preserved. Identical solitons move with the same speed so they never collide, while different solitons have different shapes ( A , L ) and travel with different speeds. If they collide the composed wave function is not given in terms of linear combination of such solutions, but a two- or many-soliton solution obtained by inverse scattering theory. One particular feature of the NLS eq.(5) is that both positive (soliton) or negative amplitude (anti- soliton) can move in both directions. Moreover there is no direct relation between the halfwidth of the soliton and its velocity like in the KdV, or MKdV cases. The geometry-dynamics connection is provided through the complex exponential through the wavelength of the modulation.

Inside the potential region, if the potential strength Vo is strong enough compared to the amplitude of the wavefunction, we can neglect the nonlinear term and reduce eq.(4) to the orthodox Schrodinger equation:

-$x2 - uo+ - iq$t = 0 , (7) 2mV) valind for 2 E [-a/2,a/2], where uo = F . The general separable vari-

ables stationary solution in the potential well is given by the linear combination of all stationary solution (x E [-a/2,a/2]):

00

in(^, t ) = Luo + B ( ~ ) e - ~ m ~ ] e - % ' ~ & . (8)

187

0 . 2

0 .8

0.6

0.4

0 . 2 0 . 2

0 5 0 150 2 5 0 0 5 0 1 5 0 2 5 0

Figure 1. Transmission and reflection coefficients (T and R) for a square barrier of depth Vo = 40% versus energy in orthodox quantum mechanics: resonances occur when the barrier width L matches the wavelength A. Left L = 1, center L = 3 larger barrier width, and right L = 0.3, that is classical limit of the penetration process (macroscopic situation).

0 . 5 lj(J 0 . 5

0 . 2 0.2 0 . 2

0 50 100 150 0 50 100 150 o 50 100 1 5 0

Figure 2. Transmission and reflection coefficients (T and R) versus energy for a NLS soliton wavefunction colliding with a square barrier of depth = 40%. From left to right we increase the barrier width L and keep the same soliton width Lsoliton = 1. For narrower barrier (left figure) the behavior is almost classical. For wider barrier (right) the transmission is strongly reduced and more resonances occur. If the soliton width is comparable to the barrier width resonances occur like in the orthodox quantum mechanics case (center).

The norm of this part of the solution is given by:

The first constrain between the parameters and the functions of this problem is to have the total wavefunction norm equal to unity, that is II$inIIto,~(e),~(c) + IId-'mtII%,a,c = 1, from eqs-(6, 9)-

Although eq.(B) represents the most general solution, we can look for special solutions, like travel solution which within the interval [ - a / 2 , a /2] are actually periodic solutions-in the form:

where d , B are arbitrary coefficients, v is the group speed of the wave, and the halfwidth of this periodic wavefunction is Li, = z. In the same way one can use instead of the NLS soliton solutions of eq.(5) the corresponding cnoidal waves which approach in the limit this soliton. This can

188

-0.5.

--

1.

0.5

-1

- 0 . 5

Figure 3. Interaction between a NLS soliton traveling from left of the potential well with the well. When the nonlinear wave interacts with the potential perturbation a time dependent state is excited inside the well, and a new soliton is emited in the same direction. After the inelastic collision the two solitons move apart (reflection and transmission) and the internal state fades out. At the boundaries of the potential the solution is continuous together with its first derivative, and the norm of the whole wave function is conserved to one all over the x-axis.

I

--

be an intermediate case between the pure nonlinear and the traditional plane wavefunction scattering case. Before presenting the h a 1 form of the wave function we want to stress one more time that this solution is only valid under the approximation of small excitations inside the potential trap cornpaxed to its depth, namely:

P ma~I$in(x, t)lz,[-a/z,a/21 << {uo,Q}- (11)

5. Resonant states

The wavefunction should be normalized to 1, eqs.(6,9), and smooth of class Ck. Consequently we have to fulfill the following restrictions:

II$inl12 + ll4JoutIl2 = 1, $i,(f&t) = $out(fa/2,t),

Eq . (12) introduces 5 restrictions among 8 free parameters (4 for each soliton to the left and to the right of the well: amplitude, velocity, phase, and initial

189

Figure 4 . Resonant inelastic collision between a left soliton and the well. After reflection and transmission, when the two nonlinear waves move apart, the well is still excited in a resonant stationary state. same second order continuity in wave function, and norm conservation are fulfilled.

position) and two arbitrary functions, A(c) ,B(c) , eq.(8). Two out of these constants are arbitrary phase factors which do not involve information, so we have three free parameters, and we choose them to define the left incoming soliton, and the initial position of the outgoing soliton to the right.

Within these restrictions, by using eqs.(6,7,9,10, and 13) we have for the internal potential trap the following solution:

03 /jLA sech[&(b + C a c ) l e ~ e - ~ ( 1 a + ~ + ~ + 2 a z z 0 ) 2v 2572

e 2 l d T Z T - 1

e-J-io?;rac e-ectdts, (13) 1 al+y+*+a(zo-21) 2 i L & G z i - + (e

2q , and zo,x1 are the initial coordinate

of the two solitons, respectively. The free parameters are A,a,Xo, and q. We introduce the nonlinear transmission coefficient to be the right soliton

190

1 1

Figure 5. Numeric simulation of soliton collision with a wider potential well. After a short instability the original soliton splits into a series of narrower solitons in both directions.

wavefunction norm and the reflection coefficient the left wave function soliton, calculated in the asymptotic limit, t -+ m. In Fig. 1 we present the traditional transmission coefficient behavior for a linear plane wave scat- tered by a rectangular barrier. The well-known result shows resonances in the transmission coefficient at specific energies when the wavelength of the wave function is a integer sub-multiple of the potential barrier (or well) width. In the limit of high energies (very small wavelength) the problem becomes quasi-classical and the transmission coefficient approaches asym- potic the value 1 for all energies.

In the case of the NLS soliton the transmission coefficient has a different behavior. For width of the soliton smaller or larger than the potential box we have very small transmission coefficient, see Fig. 2. It’s only the case of the width match when the transmission of soliton from left to right is enhanced. From this observation we note that such a quasi-classical system behaves rather like a band-pass filter for space scales. On the top of this restricted window of transmission we note a fine structure of resonances, Fig. 2, right. We assume these resonances are related to the radiative structure of the soliton-perturbation collision. In Fig. 3 we present analytic calculation of an example of a collision between a left-incoming soliton and a potential well of different width. During the collision some mixed states are excited in the potential valley and a right-emerging soliton (transmitted)

191

is generated. After this collision, both solitons move in opposite directions and the excited states inside the potential trap decay to zero. However, there are other possible situations. If the width of the incoming soliton is close to the potential trap width it is possible to excite stable states inside the trap, like the situation presented in Fig.4.

These calculation do not represent all channels of scattering of the NLS soliton with the potential box. In Fig.5 we present a numeric simulation of a collision between a widder NLS soliton with a narrower potential well. After the collision a series of smaller solitons is generated in the right domain and a smaller soliton is reflected to the left. This example shows that there is actually a huge spectrum of inelastic channels of nonlinear collision, namely the multi-soliton generation. This is also exact tractable by using the many- soliton solutions and a complex S-matrix can be generated. However, we consider very interesting the way a narrow but deep perturbation can break one soliton in more emerging solitons. This is very similar to the case of nonlinear dispersion when an initial wavefunction larger than the stable one (the solution of the dynamic equation) breaks spontaneously in a series of smaller pulses, like a cold fission process. This phenomenon, as well as a possible algebraic scattering approach based on q-deformed spaces for the nonlinear space of wavefunctions are the subject of a future study.

Acknowledgments

This work was supported by the National Science Foundation under NSF Grant 0140274.

References 1. A. Griffin, D. W. Snoke, and S. Stringari, Bose-Einstein Condensation (Cam-

bridge University Press, Cambridge, 1995). 2. M. Brenner e t al, Alpha-particle structure on the surface of the atomic nucleus

(Proceedings of the Intl. Workshop on Fission Dynamics, Luso, Portugal, May

3. A. Messiah, Quantum Mechanics, (Vol. I, any edition). 15-19, 2000).

VIII. Special Topics

VACUUM, MATTER, ANTIMATTER AND THE PROBLEM OF COLD COMPRESSION

W. GREINER Instatut f i r Theoretische Physak, J . W. Goethe- Universatat, D-60054 Frankfurt,

Germany

T. BUERVENICH Theoretical Davision, Los Alamos National Laboratory, Los Alamos, New Mexico

87545, USA

We discuss the possibility of producing a new kind of nuclear system by putting a few antibaryons inside ordinary nuclei. The structure of such systems is calculated within the relativistic mean-field model assuming that the nucleon and antinucleon potentials are related by the G-parity transformation. The presence of antinucle- om leads to decreasing vector potential and increasing scalar potential for the nucleons. As a result, a strongly bound system of high density is formed. Due to the significant reduction of the available phase space the annihilation probability might be strongly suppressed in such systems.

1. Introduction

In this proceedings article we would like to report on some recent exciting results that have been obtained together with our friends and collaborators I. N. Mishustin, L. M. Satarov, J. A. Maruhn, and H. Stocker I. Before embarking upon the physical discussion, we would like to introduce the ideas and the framework.

Presently it is widely accepted that the relativistic mean-field (RMF) model gives a good description of nuclear matter and finite nuclei 3 .

Within this approach the nucleons are supposed to obey the Dirac equation coupled to mean meson fields. Large scalar and vector potentials, of the order of 300 MeV, are necessary to explain the strong spin-orbit splitting in nuclei. The most debated aspect of this model is related to the negative-energy states of the Dirac equation. In most applications these states are simply ignored (no-sea approximation) or "taken into account" via the non-linear and derivative terms of the scalar potential. On the other

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hand, explicit consideration of the Dirac sea combined with the G-parity arguments leads to such interesting conjectures as the existence of deeply- bound antinucleon states in nuclei or even spontaneous production of nucleon-antinucleon pairs 5 1 6 .

Keeping in mind all possible limitations of the RMF approach, below we consider yet another interesting application of this model. Namely, we study properties of light nuclear systems containing a few real antibaryons. At first sight this may appear ridiculous because of the fast annihilation of antibaryons in the dense baryonic environment. But as our estimates show, due to a significant reduction of the available phase space for annihilation, the life time of such states might be long enough for their observation. In a certain sense, these states are analogous to the famous baryonium states in the N m system 7, although their existence has never been unambiguously confirmed. To our knowledge, up till now a self-consistent calculation of antinucleon states in nuclei has not been performed. Our calculations can be regarded as the first attempt to fill this gap. We consider first l60 and study the changes in its structure due to the presence of an antiproton. Then we discuss the influence of small antimatter clusters on heavy systems like 'OSPb.

2. Theoretical framework

Below we use the RMF model which previously has been successfully applied for describing ground-states of nuclei at and away from the ,&stability line. For nucleons, the scalar and vector potentials contribute with opposite signs in the central potential, while their sum enters in the spin-orbit PO- tential. Due to G-parity, for antiprotons the vector potential changes sign and therefore both the scalar and the vector mesons generate attractive potentials.

To estimate uncertainties of this approach we use three different parametrizations of the model, namely NL3 8 , NL-Z2 and TM1 lo. In this paper we assume that the antiproton interactions are fully determined by the G-parity transformation. We solve the effective Schrodinger equations for both the nucleons and the antiprotons. Although we neglect the Dirac sea polarization, we take into account explicitly the contribution of the antibaryon into the scalar and vector densities. For protons and neutrons we include pairing correlations within the BCS model with a &force (volume pairing) ll. Calculations are done within the blocking approximation l2 for the antiproton, and assuming the time-reversal invariance of the nuclear

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0.7 NL-Z2

......

0.4 $ 0 3

?- 0.5 0.4

0.1 0.0

-4 -2 0 2 4 r [fml

00

-0 I v- -

-02 E 2 27

-0 3

-300 -600

-900 \. I

*\./

r[fml r[fml r [fml -4 -2 0 2 4

-0.4 -0.4 -1200 1 -4 -2 0 2 4

Figure 1. The left panel represents the sum of proton and neutron densities as function of nuclear radius for l60 without (top) and with an antiproton (denoted by AP). The left and right parts of the upper middle panel show separately the proton and neutron densities, the lower part of this panel displays the antiproton density (with minus sign). The right panel shows the scalar (negative) and vector (positive) parts of the nucleon potential. Small contributions shown in the lower row correspond to the isovector (p meson) part. The densities and potentials are mirrored to negative r values.

ground-state. The coupled set of equations for nucleons, antinucleons and meson mean fields is solved iteratively and self-consistently. The numerical code employs axial and reflection symmetry, allowing for axially symmetric deformations of the system.

3. Structure of light nuclei containing antiprotons

As an example, we consider the nucleus l60 with one antiproton in the lowest bound state. This nucleus is the lightest nucleus for which the mean- field approximation is acceptable, and it is included into the fit of the effective forces NL3 and NL-Z2. The antiproton state is assumed to be in the s1/2+ state. The antiproton contributes with the same sign as nucleons to the scalar density, but with opposite sign to the vector density. This leads to an overall increase of attraction and decrease of repulsion for all nucleons. The antiproton becomes very deeply bound in the s1/2+ state. To maximize attraction, protons and neutrons move to the center of the nucleus, where the antiproton has its largest occupation probability. This leads to a cold compression of the nucleus to a high density.

Figure 1 shows the densities and potentials for l6O with and without the antiproton. For normal 0 all RMF parametrizations considered produce

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very similar results. The presence of an antiproton dramatically changes the structure of the nucleus. The sum of proton and neutron densities reaches a maximum value of (2 - 4) P O , where po e 0.15 fm-3 is the normal nuclear density, depending on the parametrization. The largest compression is predicted by the TM1 model. This follows from the fact that this parametrization gives the softest equation of state as compared to other forces considered here. Since nucleons feel a deeper potential as compared to the nucleus without the antiproton, their binding energy increases too. The total binding energy of the system is predicted to be 828 MeV for NL-Z2, 1051 MeV for NL3, and 1159 MeV for TM1. For comparison, the binding energy of a normal l60 nucleus is 127.8, 128.7 and 130.3 MeV in the case of NL-Z2, NL3, and TM1, respectively. Due to this anomalous binding we call these systems Super Bound Nuclei (SBN).

4. Doubly-magic lead with antiproton and anti-alpha

We would like to discuss here the structural effect of an antiproton or an anti-alpha nucleus in the doubly magic lead nucleus. Contour plots of the sum of proton and neutron densities are shown in figures 2 (lead with an antiproton) and 3 (lead with an anti-alpha nucleus). In these cases we encounter a quite different scenario: again, the complete system is affected, but not in the sense that the whole nucleus shrinks and becomes very dense. Here, a small and localized region of high density develops within the heavy system. Additionally, the lead nucleus deforms itself. This effect is largest for the case of lead with 6. The single-particle levels reflect this behaviour and indicate the cause for the deformation of lead: In a small region with a deep potential, only states with small angular momenta can be bound deeply. States with higher angular momenta do not have much overlap with the potential. The deformation effect probably has two reasons: firstly, a deformation might be energetically favourable to gain some binding for the higher lying states. Secondly, the distortion of the system due to the presence of antiparticles destroys the magicity of the system.

5. Life time, formation probability and signatures of SBNs The crucial question concerning a possible observation of the SBNs is their life time. The only decay channel for such states is the annihilation on surrounding nucleons. The mean life time of an antiproton in nucleonic matter of density p~ can be estimated as r =< C T A V , , ~ ~ B >-', where angular brackets denote averaging over the wave function of the antiproton

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0 35

0 30

0 2 5

L? 0 2 0 -z = 0 1 5

30

- 3, 010 I

0 05

0 00

Figure 2. with p .

Surface plot of the sum of proton and neutron densities for the system zo8Pb

and vrel is its relative velocity with respect to nucleons. In vacuum the N x annihilation cross section at low vpel can be parametrized as l4 UA = C+D/vrel with C=38 mb and D=35 mb. For < p~ >E 2po this would lead to a very short life time, T zlrl 0.7 fm/c (for vrel N 0.2). However, one should bear in mind that the annihilation process is very sensitive to the phase space available for decay products. For a bound nucleon and antinucleon the available energy is Q = 2 m ~ - BN - Bx , where BN and B x are the corresponding binding energies. As follows from our calculations, this energy is strongly reduced compared to 2 m ~ , namely, Q 2: 600 - 680 MeV

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0.4 1

0 1

0.0 i-

35 I"

Figure 3. with 6.

Surface plot of the sum of proton and neutron densities for the system zo8Pb

(TMl), 810-880 MeV (NL3) and 990-1050 MeV (NL-Z2) for the lowest antiproton states.

For such low values of Q many important annihilation channels involving two heavy mesons (p, w , 77, Q', ...) are simply closed. Other twebody channels such as 7rp, 7rw are considerably suppressed due to the closeness to the threshold. As is well known, the two-pion final states contribute only about 0.4% of the annihilation cross section. Even in vacuum all above mentioned channels contribute to (TA not more than 15% 15. Therefore, we expect that only multi-pion final states contribute significantly to antipro-

20 1

ton annihilation in the SBN. But these channels are strongly suppressed due to the reduction of the available phase space. Our calculations show that changing Q from 2 GeV to 1 GeV results in suppression factors 5, 40 and 1000 for the annihilation channels with 3, 4 and 5 pions in the final state, respectively. Applying these suppression factors to the experimental branching ratios l6 we come to the conclusion that in the SBNs the annihilation rates can be easily suppressed by factor of 20-30. There could be additional suppression factors of a structural origin which are difficult to estimate at present. This brings the SBN life time to the level of 15-20 fm/c which makes their experimental observation feasible. The corresponding width, I? - 10 MeV, is comparable to that of the w-meson.

Let us discuss now how these exotic nuclear states can be produced in the laboratory. We believe that the most direct way is to use antiproton beams of multi-GeV energy. This high energy is needed to suppress annihilation on the nuclear surface which dominates at low energies. To form a deeply bound state, the fast antiproton must transfer its energy and momentum to one of the surrounding nucleons. This can be achieved through reactions of the type PN + BE in the nucleus,

P + (A , 2) + B + B ( A - 1,Z’) , where B = n,p , A, C. The fast baryon B can be used as a trigger of events where the antibaryon B is trapped in the nucleus. Obviously, this is only possible in inelastic PN collisions accompanied by the production of pions or particle-hole excitations. One can think even about producing an additional baryon-antibaryon pair and forming a nucleus with two antibaryons in the deeply bound states. In this case two fast nucleons will be knocked out from the nucleus.

Without detailed transport calculations it is difficult to find the formation probability, W , of final nuclei with trapped antinucleons in these reactions. A rough estimate can be obtained by assuming that antiproton stopping is achieved in a single inelastic collision somewhere in the nuclear interior i.e. taking the penetration length of the order of the nuclear radius R . F’rom the Poisson distribution in the number of collisions the probability of such an event is

R wi = -exp Xin (-:) ,

where A,‘ = (TinPo and X - l = ((Tin + ( ~ ~ ) p o (here tin and (TA are the inelastic and annihilation parts of the p N cross section). The exponential factor in Eq. (2) includes the probability to avoid annihilation. For initial

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antiproton momenta plab 21 10 GeV we use oin = 25 mb, (TA N 15 mb l6

and get X N 1.6 fm which is comparable with the radii of light nuclei. For an oxygen target, using R N 3 fm leads to w1 N 0.17.

In fact we need relatively small final antiproton momenta to overlap significantly with the momentum distribution of a bound state, namely, Ap N x/RF, where Rp = 1.5 fm is the characteristic size of the antiproton spatial distribution (see Fig. 1). The probability of such a momentum loss can be estimated by the method of Refs. which was previously used for calculating proton spectra in high-energy pA collisions. At relativistic bombarding energies the differential cross sections of the + PX and p p + pX reactions are similar. The inelastic parts of these cross sections drop rapidly with transverse momentum, but they are practically flat as a function of longitudinal momentum of secondary particles. Thus, the probability of the final antiproton momentum to fall in the interval Ap is simply A p / p , b . For n a b = 10 GeV and Ap = 0.4GeV this gives 0.04. Assuming the geometrical fraction of central events - 20% we get the final estimate W 21 0.17 x 0.04 x 0.2 N 1.4- One should bear in mind that additional reduction factors may come from the matrix element between the bare massive antibaryon and the dressed almost massless antibaryon in a deeply bound state. But even with extra factors - 10-1 - lov2 which may come from the detailed calculations the detection of SBNs is well within the modern experimental possibilities.

6. Discussion and conclusions

Our main goal in this paper was to demonstrate that energetic antiproton beams can be used to study new interesting phenomena in nuclear physics. We discuss the possible existence of a completely new kind of strongly interacting systems where both the nucleons and the antinucleons coexist within the same volume and where annihilation is suppressed due to the reduction of the available phase space. Such systems are characterized by large binding energy and high nucleon density. Certainly, antinucleons can be replaced by antihyperons or even by antiquarks. We have presented the first self-consistent calculation of a finite nuclear system containing one antiproton in a deeply bound state. For this study we have used several versions of the RMF model which give excellent description of ordinary nuclei. The presence of an antiproton in a light nucleus like sBe or l60 changes drastically the whole structure of the nucleus leading to a much more dense and bound state. In heavy systems the presence of a few antinu-

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cleons distorts and deforms the nuclear system leading to a localized central region of highly increased density.

It is clear however that these structural changes can occur only if the life time of the antibaryons in the nuclear interior is long enough.

In summary, on the basis of the RMF model we have studied the structure of nuclear systems containing a few real antibaryons. We have demonstrated that the antibaryons act as strong attractors for the nucleons leading to enhanced binding and compression of the recipient nucleus. As our estimates show the life times of antibaryons in the nuclear environment could be significantly enhanced due to the reduction of the phase space available for annihilation.

References 1. T. Biirvenich, I. N. Mishustin, L. M. Satarov, J . A. Maruhn, H.

Stocker, and W. Greiner, Phys. Lett. B 542, 261 (2002) 2. B.D. Serot and J.D. Walecka, Adv. Nucl. Phys. 16, 1 (1985) 3. P.G. Reinhard, Rep. Prog. Phys. 52, 439 (1989) 4. N. Auerbach, A.S. Goldhaber, M.B. Johnson, L.D. Miller, and A. Pick-

lesimer, Phys. Lett. B182, 221 (1986) 5. I.N. Mishustin, Sov. J. Nucl. Phys. 52, 722 (1990) 6. I.N. Mishustin, L.M. Satarov, J. SchaEner, H. Stocker, and W. Greiner,

J. Phys. G19, 1303 (1993) 7. O.D. Dalkarov, V.B. Mandelzweig, and I.S. Shapiro, Nucl. Phys. B21,

66 (1970) 8. G. Lalazissis, J. Konig, and P. Ring, Phys. Rev. C55, 540 (1997) 9. M. Bender, K. Rutz, P.-G. Reinhard, J.A. Maruhn, and W. Greiner,

Phys. Rev. C60, 34304 (1999) 10. Y. Sugahara and H. Toki, Nucl. Phys. A579, 557 (1994) 11. M. Bender, K. Rutz, P.-G. Reinhard, and J.A. Maruhn, Eur. Phys. J.

A8, 59 (2000) 12. K. Rutz, M. Bender, P.-G. Reinhard, J.A. Maruhn, and W. Greiner,

Nucl. Phys. A634, 67 (1998) 13. Y. Akaishi and T . Yamazaki, Phys. Rev. C65, 044005 (2002) 14. C.B. Dover, T. Gutsche, M. Maruyama, and A. Faessler, Prog. Part.

Nucl. Phys. 29, 87 (1992) 15. C. Amsler, Rev. Mod. Phys. 70, 1293 (1998) 16. J . Sedkik and V. % m B , Sov. J. Part. Nucl. 19, 191 (1988) 17. R.C. Hwa, Phys. Rev. Lett. 52, 492 (1984) 18. L.P. Csernai and J.I. Kapusta, Phys. Rev. D29, 2664 (1984)

A TOY MODEL FOR QCD AT LOW AND HIGH TEMPERATURES

S. LERMA H., S. JESGARZ, P. 0. HESS Instituto de Ciencias Nucleares, Universidad Nacional Auto’noma de Mkxico,

Apdo. Postal 70-543, Mkxico 04510 D.F.

0. CIVITARESE’, 111. REBOIRO’ Departamento de Fisica, Uniuersidad Nacional de La Plata, C.C. 67 1900, La

Plata, Argentina

A simple model for QCD is presented. It is based on a Lipkin model, consisting of two levels for the quarks coupled to a boson level, representing pairs of gluons with color and spin zero. The basic ingredients are pairs of quark-antiquark coupled to combinations of flavor and spin, in addition to the gluon pairs. The interaction part of the Hamiltonian is a particle non-conserving interaction and commutes with spin, flavor, parity and charge conjugation. The four parameters of the Hamiltonian are adjusted to the meson spectrum at low energy, corrected for flavor mixing and Gell’man-Okubo terms of the two lowest meson nonets. The states exhibit mixture of quarks, antiquarks and gluons. In the second part of the contribution, the partition function is constructed and several observables calculated, like particle ratios and absolute production rates. The model exhibits a hint for a possible transition at temperature 0.170GeV from the Quark-Gluon Plasma to the hadron gas.

1. Introduction

Sometimes it helps to our understanding to use simplified models for the study of complicated, highly nonlinear systems, like the Quantum Chromo- dynamics (QCD), though they are not very realistic. One such example is the Lipkin model which shed light on the formation of collective states and the role of pairing. In ’ it was used to understand the formation of a pion condensate in nuclei. In complicated nuclear interactions were investigated also in the context of the ,O,O decay ‘. In a Lipkin model was applied to a many quark-system with particle conserving interaction. As can be seen, schematic models, though simple in structure, can describe rather complex phenomena. For gluons, as a further example, a simple

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model was developed in was rather successful.

and the agreement to lattice gauge calculations

It is also of interest to have a model with exact results. It can serve as a test for many-body techniques which are intended to be applied to realistic cases, where an estimation of an error is not possible.

In this contribution we will report on the present status of a toy model for QCD. It is analytic in the sense that the matrix elements can be obtained analytically but the Hamiltonian has to be diagonalized numerically. In section 2 we will shortly review the model and in section 3 applications are discussed. In section 3.1 the low energy meson spectrum is described and in section 3.2 some aspects of the high energy behavior. Due to a large number of data the presentation will be rather telegraphic and for details we refer to a future publication.

The relation to Jerry Draayer’s work is the use of SU(3) Clebsch-Gordan and recoupling coefficients and a work published in ’.

2. The toy model

One basic ingredient are two fermion levels, one at energy i -w f and the other at -w f , Each level has the degeneracy of 2 0 = n,nfn, = 18, where n, denote the three color degrees of freedom, nf the three flavors and n, the two spin components of a spin $ particle. The quarks can be distributed between the two levels, which corresponds to the Dirac picture. The other basic ingredient is a boson level at 1.6GeV (taken from Ref. ‘). The bosons represent gluon pairs coupled to spin 0 and color (0,O) (in SU(3) notation). There are more gluon states available but the interaction proposed will only connect the quarks to this particular gluon pair.

The states of the model can be classified according to the group chain

PN1 [h] = [hlh2h3] [hT1 U(4W 3 U ( $ ) @ U(12)

U U

( k , P C ) SUC(3) ( X f , P f ) SUf(3) 63 SUs(2) S , M , (1)

where N is the number of quarks and U ( $ ) = U ( 3 ) is the color group whose irreducible representation (irrep) is given by the Young diagram [h] = [hlh2h3]. The irrep of U(12) is complementary to U ( $ ) due to the antisymmetry of the irrep of U(4R) lo. SUc(3) is the unimodular color group, SUf(3) describes the flavor part and SUs(2) is the spin group. The

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reduction of U(12) to the subgroups was done using Ref. '. Multiplicity labels are not indicated.

The quark-antiquark pair operators B : Z z and B2:: will play a central role. The first pair operator creates a quark and an antiquark pair with color zero while the second annihilates it. The pair operators can be coupled to definite flavor and spin, i.e. BIX,X)f,SM = Bi f ,SM, with X = 0 , l and S = 0 , l . Instead of working in the fermion space it is more convenient to

b X f , S M . The price to pay for the simplicity of dealing with boson operators is the appearance of spurious states ll . The simplest way is to introduce an effective cutoff for the maximal allowed bosons, though it will not guarantee that all spurious states disappear. Further below we give an argument why the method chosen might work. The lowest weight state of U(12) is defined partially via the annihilation of the state through the application of a pair Bxf,sMlZw >= 0. Note, that there are several lowest weight state: If Jlw > contains any number of quarks (in the upper level), classified then via a baryon number, the application of a pair annihilation operator gives zero, because the pair operator contains an antiquark annihilation operator (hole in the lower level) which anticommutes with all quark operators and can act directly on the particle vacuum. A U(12) irrep [m] has in its lowest weight state c",=, m k quarks in the lower level and zk7 m k quarks in the upper level (Dirac picture!). In the highest weight state it is just the inverse. Therefore, the difference of the two numbers, given by 2 J = xkZl m k - X k = 7 m k , is equal to the maximal number of pairs (particle- hole) which can be excited within a given U(12) irrep. This is the cutoff used for the total number of bosons. For the irrep [3606] the J acquires the value R = 9 and the lowest weight state corresponds to the one with the lower level completely filled and the upper one empty. There are four different kinds of bosons, classified through [ X , S ] = [O,O] , [0,1],[1,0] and [1,1]. They correspond respectively to a one-, three-, eight- and a 24- dimensional oscillator. The individual cutoff is chosen such that in the order given above the maximal number of bosons is given by 25 , J, and $, in addition to the cutoff of the total number of quanta. For the U(l2) irrep [3606] unphysical states may appear starting from states with four quarks-antiquarks pairs. However, the cutoff chosen seems to give reasonable results.

map the pair operators to bosons 1 1 , i.e BIf,SM -+ bXf ,SM t and B A f , S M 4

6 12

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As the Hamiltonian of the model we propose H 2 W f n f + W b n b

The Hamiltonian is invariant under rotation, charge conjugation, flavor, color and G-parity. The factor (1 - 3) simulates part of the Pauli principle and limits the total number of bosons to 20. If we would restrict to the [0,0] bosons only, these factors would appear in the exact boson mapping. As a shorthand notation we used ( b I s ) 2 = ( b l , . bl,) where the dot indicates the scalar product, and the same for the other expressions. Because the Hamiltonian commutes with the flavor generators, all states belonging to a given flavor multiplet are degenerate. Before we adjust the parameters we have to correct the experimental mass values by taking into account flavor mixing and contributions from the Gel'man-Okubo mass formula. These effects will only be considered for the lowest multiplet (nonet) of the mesons with S p = 0- and 1-. The mixing angles are taken from Ref. 12.

The matrix elements are calculated in a seniority basis, each for the one-, three-, eight- and 24-dimensional harmonic oscillator. The parity is given by (-l)"f, where nf is the number of quark-antiquark pairs. The charge conjugation can also be determined and for one quark-antiquark pair it is given by CBLr,sMC-l = (-l)SB:f,sM, where f = -Y, T , -T,.

At low energy, the baryon states cannot yet be well described by the proposed Hamiltonian. Baryon states are described by putting three ideal quarks in the upper level. However, no interaction to the quark-antiquark pairs and to the gluons is taken into account yet.

In order to investigate the high temperature behavior the grand canonical partition function is calculated 13. We denote by Ei the energy of a state, as obtained by a diagonalization of the Hamiltonian, adding the flavor mixing and the Gel'man-Okubo interactions for the two lowest meson nonents and adding also the other gluon states which do not contain pairs of gluons coupled to spin and color zero '. The grand canonical partition function is given by 2, = Cie-P(E~-pBB-pss)l where pg and p, are the chemical potentials for the baryon number B and strangeness s and p =

(T is the temperature). The subindex a refers to c if all possible colored states are included and to (0,O) if only color singlet states are allowed. In studying the thermodynamical properties of the model we will consider two scenarios: one where the colored states are permitted to all energies and second when they are allowed only above a given critical temperature T,.

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The first scenario is unrealistic in the sense that low lying colored states appear, for example, when just one quark is put into the upper level. Nev- ertheless, it will help us to understand particular properties of the system.

The model does not contain explicitely a volume. Therefore, we have to introduce it by hand. We assume that the whole volume, comprised by the QGP, is divided into elementary cells of a radius of the order of 1 fm. This elementary cell represents a sample of the thermodynamic properties. For example, when we determine the expectation value of the number of a given particle species via < n k >= e-P(Ek-~BBk-~ss~)/Za, where the index k refers to the quantum numbers of that particle, then the total number of produced particles is obtained by multiplying with &, where V is the total and Vel the elementary volume. The elementary volume is adjusted to reproduce a bag pressure of @)a = 0.170 GeV and using p = FThe total volume is adjusted to the absolute rate of pions, i.e. if N,+ is the total number of 7r+ produced and < n,+ > the average number of T+ is an elementary cell then the relation is N,+ = < n,+ > T.

Tln(2)

V el

3. Applications

3.1. The low lying meson spectrum

In Fig. 1 the spectrum of spin 0 and 1 mesons is plotted up to the energy of 2.5 GeV. The parameters were adjusted to the lowest lying states in the sector of spin 0 and 1, while all others are a result of the calculation. In the fit the state with hypercharge Y = 0 and isospin zero within a multiplet is used, corrected by the effects of flavor mixing and Gell’man-Okubo terms. On the left hand side of each group the experimental spectrum is given and on the right hand side the theoretical states are listed. Solid boxes (or solid lines) on the experimental part refer to states in the summary table of Ref. l4 while the dashed boxes (or dashed lines) are not in the summary table. The theoretically obtained states are labeled by their respective flavor irrep and multiplicity (subindex).

Without interaction, there would be a large degeneracy of states at low energy, due to the various ways to obtain a given quantum number. The interaction lifts most of the states to higher energy and removes in such a way the high multiplicity. Due to the particle number mixing interaction each state will contain a mixture of quarks, antiquarks and gluons. We found that for the lowest states, including the physical vacuum, the average gluon content is of about 30%. The dominant contribution of quark-antiquark pairs are the bosons of type [ l , O ] (flavor (1,l) and spin 0). This is due to a

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’ ’ ’ 1 Z S W ( , , ,

I Exp 0- The0 O Exp o++ Then

Figure 1. obtained in a fit to experimental data14.

The meson spectrum for spin O p c (left panel) and spin lPc (right panel) as

strong interaction in the [1,0] channel while the parameters corresponding to the other types of bosons are small. For example, the pion contains about 2.7 pairs of the type [1,0] and 0.06 for the rest and the gluon content is about 1.2 gluon pairs. The qualitative agreement is acceptable. Note, that we do not claim that the model describes exactly QCD but rather that it should mimic the properties of QCD. A perfect agreement is therefore not essential. Giving the success in describing the meson spectrum, fixing the four parameters of the model, we proceed in looking at the thermodynamic properties of the model.

3.2. Thermodynamic properties

Calculating the expectation value of the color Casimir operator and the variation of it as a function of temperature, for the case when all color is allowed, we obtain the following: for large T the variation is smaller than the average value. In this situation the probability to form color singlet states in an elementary volume is small and color prevails over a large distance. However, at T = 0.170 GeV the variation is of the same order as the average and larger even for smaller temperatures. We can give the following interpretation: At T = 0.170 GeV there is a finite chance to form a color singlet state which can escape from the total volume, if it sits on the surface. At lower temperatures the probability increases. The point T = 0.170 GeV can be seen as a signature to start hadronization. The next step is to assume that below this point T, confinement sets in. We cannot, within our model, describe this process except putting it in by hand. Therefore, when we discuss the thermodynamical properties of the model below T,=0.170 GeV only color singlet states will be taken into account. The particle production will be calculated just below this transition temperature assuming chemical equilibrium to the state just above T,.

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Figure 2. Some particle production ratios for beam energies as taken from Ref. ''

Figure 3. Left panel: Total production rate of T+. The upper curve is for Npart = 394 (Au+Au) and the lower one is for Npart = 250. Data are taken from1*. Right panel: Total production of K+ (upper curve) and K - (lower curve) for Npart = 250. Data are taken from1*.

In Fig. 2 we plot some particle ratios for beam energies fi = 130AGeV. The experimental values were taken from 17. The chemical potential pa was adjusted to the Kaon ratio while the others are a consequence of the model. Also here we find a satisfactorily qualitative agreement to experiment.

In the left panel of Fig. 3 the absolute pion production rate in AutAu is plotted versus temperature. The experimental points are from Ref. I*.

The dashed curve is obtained under the assumption that the number of participants is Npart =394 while the solid line is for Npart =250. We

21 1

adjusted Npart to the experimental values of the pion production rate. Note, that the shape of the curve is a consequence of the model. In the right panel the K+ and K - production rate is calculated. The ratio was already fixed. However, the shape of the curve and the absolute production rate is not fitted but a consequence of the model. The number of participants Npart was already fitted in the total production rate of pions.

As can be seen, the toy model not only simulates the structure of QCD but also reproduces some important characteristics a t high energy, related to the possible formation of a QGP.

4. Conclusions

We presented a toy model which mimics the structure of QCD at low energy and in heavy ion collisions. The basic building blocks is a two-level Lipkin model for the quarks coupled to a boson level (gluon pairs with spin and color zero). The elementary operators are quark-antiquark and gluon pair creation and annihilation operators defined in a boson space. The model Hamiltonian has a non-interacting part where the effective masses are defined. The interaction mixes the number of quarks, antiquarks and gluons. It contains factors which mimic the Pauli principle for the quark part.

The model was applied first to the low energy meson spectrum. The four parameters were adjusted to some low lying meson states. The overall agreement was qualitatively good. The particle mixing interaction removed the large multiplicity of states a t low energy. The gluon content of the low lying meson states was in average 30% while the dominant quark-antiquark pair had flavor (1,l) (flavor octed) and spin 0.

In the second part we reported briefly on the thermodynamic properties of the model. It was shown that the model suggests a transition to the hadron gas at about T = 0.170GeV. Particle production ratios and absolute ratios for pions and kaons were calculated with a surprising agreement to experiment.

The results indicate that the toy model can mimic the structure of QCD. However, taking into account the simple structure of the model we did not expect such a good qualitative agreement to experiment. It suggests that either the model is more realistic than claimed or maybe the observables reported normally in literature are not sensitive enough to the detailed structure of the hadron gas and the QGP.

This contribution also shows that the work of Jerry Draayer is not only important to nuclear physics but also influences other fields of physics, and we are convinced that this influence is even increasing.

212

Acknowledgment

We acknowledge financial support through the CONACyT-CONICET agreement under the project name Algebraic Methods in Nuc lear and Subnuclear Physics and from CONACyT project number 32729-E. (S.J.) acknowledges financial support from the Deut scher Akademischer Aus- tauschdienst (DAAD) and SRE, (S.L) acknowledges financial support from DGEP-UNAM. Financial help from DGAPA, project number IN119002, is also acknowledged.

References 1. H. J. Lipkin, N. Meschkov and S. Glick, Nucl. Phys. A 62, 118 (1965). 2. D. Schiitte and J. Da Providencia, Nucl. Phys. A 282, 518 (1977). 3. J. Dobes and S. Pittel, Phys. Rev. C 57, 688 (1998). 4. J. G. Hirsch, P. 0. Hess and 0. Civitarese, Phys. Lett. B 390,36 (1997); 5. S. Pittel, J. M. Arias, J. Dukelsky and A. Frank, Phys. Rev. C 50,423 (1994). 6. P. 0. Hess, S. Lerma, J. C. Lbpez, C. R. Stephens and A. Weber, Eur. Phys.

Jour. C 9,121 (1999). 7. M. Peardon, Nucl. Phys. B (Proc. Suppl.) 63,22 (1998). 8. J. P. Draayer and Y. Akiyama, Jour. Math. Phys. 14, 1904 (1973);

J. Escher and J. P. Draayer, J. Math. Phys. 39, 5123 (1998). 9. R. Lbpez, P. 0. Hess, P. Rochford and J. P. Draayer, J. Phys. A 23, L229

10. M. Hamermesh, Group Theory and i ts Application to Physical Problems (Dover Publications, New York, 1989).

11. A. Klein and E. R. Marshalek, Rev. Mod. Phys. 63, 375 (1991). 12. F1. Stancu, Group Theory in Subnuclear Physics (Oxford University Press,

Oxford, 1996). 13. W. Greiner, L. Neise and H. Stticker, Thermodynamics and Statistical Me-

chanics, (Springer-Verlag, Heidelberg, 1994). 14. Particle Date Book, Phys. Rev. D 54, 1 (1996). 15. S. Lerma, S. Jesgarz, P. 0. Hess, 0. Civitarese and M. Reboiro, Phys. Rev.

C 66, 045207 (2002). 16. P. Senger and H. Strobele, J. Phys. G 25, R59 (1999). 17. J. Rafelski and J. Letessier, nucl-th/0209084 (2002) and references therein. 18. The NA49 Collaboration, nucl-ex/0205002 (2002).

(1990).

IX. Poster Session

ANALYSIS OF THE lg6Pt(& t)lQ5Pt TRANSFER REACTION IN THE FRAMEWORK OF THE IBA AND IBFA MODELS*

J. B A R E A ~ ~ ~ , C.E. ALONSO~ AND J.M. A R I A S ~ Instituto de Ciencias Nucleares, Universidad Nacional Autdnoma de Mkxico. Apartado

postal 70-543, 04510 Mkxico, D.E, Mkxico. Departamento de Fisica Atdmica, Molecular y Nuclear, Universidad de Sevilla,

Apartado 1065, 41080 Sevilla, Spain.

The pick-up reaction lg6Pt(& t)Ig5Pt has been recently the object of a detailed experimental study’. Its main interest is to test if the nuclear dynamical supersymmetry (SUSY) is a valid model to describe nuclei in this region. We compare the spectroscopic strengths obtained with different forms of the transfer operator and show that by introducing some parameters to weigh the different orders of their expansion we obtain better agreement with the experimental data.

In the context of the nuclear dynamical SUSY U(6/12) the states in lg5Pt correspond to wavefunctions obtained within the VBF(6) rg UF(2) 3 OBF(6) @ V F ( 2 ) dynamical symmetry of the IBFA model. The quantum numbers I “1, N2] (al , a2) (211 , 212) L J M ) for these wavefunctions are the labels associated to the irreducible representations of the groups within this dynamical symmetry, The core for lg5Pt is lg6Pt, which is described within the O(6) IBM dynamical symmetry and, consequently, its wavefunctions are labeled according to the irreducible representations of the groups in this chain. The ground state corresponds to I [6] (6) (0) 00).

We have used the following different forms for the transfer operator to calculate the spectroscopic strengths:

“Work supported in part by Spanish CICYT under Contract No. BFM2002-03315.

215

216

where v; (u; = 1 - vj") are occupation probabilities. We have used the values upl2 = 0.1 and v:,~ = vg12 = 0.6, which are consistent with the presence of the dynamical symmetry2. p j j ~ are given in Ref. 2, and K$ = cjj, pjj,. is a diagonal matrix whose elements are the square root of the eigenvalues of O J and C J is a matrix containing the eigenvectors of O J . For the defi nition 0' see Ref. 3. (z)jk represents the zero order of the transfer operator and is a tensor under the subgroups of the above mentioned dynamical symmetry of the IBFM. (x ) jk and (72) j k contain the fi rst and the second order corrections in the transfer operator, respectively. The parameters A and B allow to weigh the fi rst and the second order, respectively. We have calculated spectroscopic strengths using the unity for them, as prescribed by microscopic arguments, but also they have been fi tted using the MINUIT routine. The values obtained are A = 0.01093 for (T) jand A = 0.1157, B = 2.1454 for ('&')jk. We denote with primes the operators where these parameters are fi tted.

We compare the experimental spectroscopic strengths with the calculated ones in horizontal bars in fi gures l(a), l(b) and l(c) for & = $, jk = $ and jk = 5, respectively. Each fi gure shows in the y-axis the quantum numbers of the fi nal states in lg5Pt for each transfer. In the x-axis the spectroscopic strength is plotted in a logarithmic scale. We have kept the assignment of quantum numbers to experimental states given in Ref. 1. We can see that (70) j k can only connect the ground state of lg6Pt to a few states in lg5Pt. This is because we are dealing with a tensor with selection rules. The other operators are not tensors and can connect a larger number of states. In particular we see that (72)j, reproduces the observed fragmentation better than (x)j,, and conclude that the second order, which (z)incorporates, is important. Concerning (x)ik and (Z);, , the A parameters obtained from the fi t to the experimental data are small. For this reason (a);, gives similar results to the ones provided by ( % ) j k . However, the fi tting procedure for ( X ) i k gives B x 2 enhancing the importance of this term. Thus, in this particular case, the relevance of the second order in the microscopic transfer operator is clear. In general, the transfers for j = 4 and j = $ are better reproduced than the j = 3 ones. We have to mention that it is possible to get a better agreement by considering new assignment of quantum numbers for the experimental levels. Some work in this direction is in progress.

217

References

1. A. Metz et al., Phys. Rev. C61, 064313 (2000). 2. R. Bijker and 0. Scholten, Phys. Rev. C32, 591 (1985). 3. J. Barea, C. Alonso, and J. Arias, Phys. Rev. C65,34328 (2002).

Exp. T, T, T, T', T',

(a) j = $

Exp. To T, T, T', T',

(b) j =

EXD. To T, T, T', T',

( c ) j = 2 Figure 1. Spectroscopic strengths

NUCLEAR WAVE FUNCTIONS FOR SPIN AND PSEUDOSPIN PARTNERS*

P. J. BORYCKI,1'2 J. GINOCCH10,3 W. NAZAREWICZ,17475 AND M . STOITSOV1.6-7

'Department of Physics, University of Tennessee, Knoxville, Tennessee, USA Institute of Physics, Warsaw University of Technology, Warsaw, Poland

Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico, USA

Institute of Theoretical Physics, University of Warsaw, Warsaw, Poland Physics Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee, USA

'Jo in t Institute for Heavy Ion Research, Oak Ridge, Tennessee, USA 'Institute of Nuclear Research and Nuclear Energy,

Bulgarian Academy of Science, Sofia, Bulgaria

Using relations between wave functions obtained in the framework of the relativistic mean field theory, we investigate the effects of pseudospin and spin symmetry breaking on the single nucleon wave functions in spherical nuclei. In our analysis, we apply both relativistic and non-relativistic self-consistent models as well as the harmonic oscillator model.

1. Pseudospin/Spin Symmetry and the Dirac Hamiltonian

The Dirac Hamiltonian with external scalar Vs(r) and vector Vv(r) potentials, vanishing space components and non-vanishing time component, is invariant under an SU(2) algebra if the scalar potential Vs(r) and the vector potential Vv(r) are related up to a constant CpSls:

In the p s e u d o ~ p i n ~ ~ ~ ~ ~ (spin) symmetry limit4, the radial wave functions of the lower (upper) components are equal, while the upper (lower)

'This work was supported in part by the U.S. Department of Energy under Contract Nos. DE-FG02-96ER40963 (University of Tennessee), DE-AC05-000R22725 with UT- Battelle, LLC (Oak Ridge National Laboratory), W-7405-ENG-36 (Los Alamos), and by the Polish Comittee for Scientific Research (KBN) under Contract No. 5 P03B 014 21.

218

219

components gAjj=j*; ( T ) ( fne,j=i* ; ( T ) ) satisfy differential relations5:

Df&=j-; (49 f&- ; (.I = Df&=j+; ( T ) g A j j = j + ; (.I 7

Dnej=e-; (T)fnej=t-+ = Dnij=e+i (r)fnej=e+;(T), (3)

( 2 )

d 2+ 2 where DAjj=j-+(.) = 5 - y, DAjj=j+;(r) = d r + -. T

Relations (2,3) are strictly fulfilled only under condition (1). There- fore, comparing the differences between the left and right hand sides of the equations, one can learn about the pseudospin and spin symmetry-breaking effects.

2. Comparison within the Harmonic Oscillator Model

For the spherical harmonic oscillator potential, we take the analytical form of wave functions with an oscillator frequency fiw = 41/A1I3. Then, one can express DAjj (r)gAij ( T ) defined by Eq. ( 2 ) as:

where x = r22v, u = 2x2 7

(-l)a(2fi + I - 3 /2 - a)xa A-1

l(4) (f i - 1 - a)!a!r(a + l+ 3 /2 ) Xiid") c

a=O

x A ~ ( x ) = / (2(2u)+(n - l)!) p(ii +i- l/2)32ge-2/2. (6)

Expressions (4,5) can be expressed as products of the common envelope function xiij(r) and certain polynomials. These polynomials are of the same order (Ti- 1) independent of j , whereas the original harmonic oscillator eigenfunction with j = $ involves a polynomial of order f i + 1/2 f 1/2 in x.

Systematic self-consistent calculations of several doubly magic nuclei have shown that Eq. ( 2 ) holds better as f i increases or decreases, in agreement with a simple harmonic oscillator estimate.

3. Summary and Conclusions

In the pseudospin (spin) symmetry limits the radial wave functions of the upper and lower components of pseudospin (spin) doublets satisfy certain

220

differential relations. We demonstrated that these relations are not only approximately valid for the relativistic mean field eigenfunctions but also for the non-relativistic Hartree-Fock and harmonic oscillator eigenfunctions (see, e.g. Fig. 1). Generally, we expect them to be approximately valid for eigenfunctions of any non-relativistic phenomenological nuclear potential that fits the single-particle levels in nuclei.

Hence we seem to have both spin and pseudospin dynamic symmetry; that is, the energy levels are not degenerate but the eigenfunctions well preserve both symmetries.

0.1

0

0.1

0 . 2

0.1

0

0.1

0.2

2 4 6 8 2 4 6 8

Figure 1. Numerical check of identity (2). Comparison between D,ijjf- 4 gAij=i- 1

(dashed line) and Dnjj=j+;g,ij=i++l (solid line) for 2d pseudospin doublet in '08Pb obtained in different methods: HF, HO and RMF. The plot labeled '6f' shows 6 times scaled lower components of the RMF wave function (see Ref.

2

for details).

References 1. K.T. Hecht and A. Adler, Nucl. Phys. A137, 129 (1969) 2. A. Arima, M. Harvey, and K. Shimizu, Phys. Lett. 30B, 517 (1969) 3. R.D. Ratna-Raju, J.P. Draayer, and K.T. Hecht, Nucl. Phys. A202, 433

(1973) 4. J.N. Ginocchio, Phys. Rev. Lett. 78, 436 (1997) 5. J.N. Ginocchio, Phys. Rev. (266, 064312 (2002) 6. P.J. Borycki, J. Ginocchio, W. Nazarewicz, M. Stoitsov, to be published, nucl-

th/0301098

FINITE WELL SOLUTION FOR THE E(5) HAMILTONIAN

M. A. CAPRI0 Wright Nuclear Structure Laboratory, Yale University,

New Haven, Connecticut 06520-8124, USA E-mail: mark. caprio@yale. edu

The solution of the E(5) Hamiltonian for finite well depth is described, and the effects of finite depth on observables are discussed.

Nuclei in the spherical-deformed shape transitional regions have his- torically been among the most difficult to understand. A new family of models recently proposed by Iachello - E(5) for the spherical to deformed y-soft transition’ and X(5) for the spherical to axially symmetric rotor transition2 - yield analytic solutions with essentially parameter free predictions for nuclei near the critical points of these transitions. The models are based upon solution of the Bohr geometric Hamiltonian3 for potentials which are infinite square wells with respect to p. However, boson algebraic models suggest that a square well of finite depth may be more applicable to actual nuclei.’

For the y-soft five-dimensional finite square well potential, V(p)=Vo (Vo<O) for p<pw and 0 elswhere, the Hamiltonian is ~eparable .~ The eigenfunctions are of the form f(p)@(y,@), where the “angular” (y,@) wave functions5, common to all y-soft problems, are characterized by the quantum number T . Within each region of constant potential, the “radial” (p) equation reduces’ to the Bessel equation, leading to solutions of the form

(1) A[,TP-’ j7+1 [ (&[ ,T - w0)1/2p] p 5 P’UJ i BE,7P-1~T+1[ ( -EE,7)1 /2P] P > P W , f[,T(p) =

where E f S E and wo G SVo. The eigenvalues for the finite well are determined by the requirement that f(P) be continuous and smooth at p=PW. Further details of the solution procedure may be found in Ref. 6.

The solutions depend upon pw and wo only through the combination z o ~ ( - v o ) 1 ~ 2 ~ w , except for overall normalization of the energy scale and

22 1

222

4 -

2 -

................................ - - - 1 - ........... .... i T=O .....

- - - - i - : - ............ ...........

- - - - ............ .... , - : -

- - - - i - ............ . - ............ . - - - - - - ............ ............ ............ T= 1

Figure 1. Evolution of level energies as a function of well size parameter 10 for the y-soft well. In the inset, the absolute energies relative to the floor of the well, E - W O , are shown for fixed well width (pw=l). In the main graph, excitation energies are taken normalized to that of the first ~ = l state. The upper dashed line indicates the energy at which the system becomes unbound (the top of the well). (Figure adapted from Ref. 6 . )

dilation of all wave functions.6 All ratios of energies or electromagnetic matrix elements are thus the same for wells with the same value of zo.

Finite depth for the potential well has several consequences. Most obviously, there are only a finite number of bound states, and the wave functions penetrate the classically forbidden region (p>,&,). The eigenvalues are somewhat lowered relative to those for an infinite E(5) well of the same width, as shown in Fig. 1 (inset); the wave functions are given the freedom to spread into the region ,O>,&, which is analogous in its effect to a widening of the well, causing the energies to “settle” lower. Also, B(E2) strengths are larger than for the infinite well of the same width.

However, an examination of the solutions reveals that the level energies are nearly unzfomly lowered by the same factor for all levels in the well, and so energy ratios are essentially unchanged [Fig. 1 (main panel)]. The enhancement of B(E2) strengths is likewise nearly uniform for all transitions (Fig. 2). Only the very highest levels (e.g., the third T=O state for zo=lO), just short of being unbound, show appreciable deviations from the

223

79 76 75

Figure 2. Evolution of B(E2) strengths as a function of well size parameter "0 for the y-soft well. Values are calculated using DT(E2)c@ and are normalized to B(E2;2? ---t 0;) = 100. (Figure adapted from Ref. 6.)

E(5) normalized energies and B(E2) strengths. The uniform reduction of all energies and enhancement of all transition

matrix elements does not serve as a useful identifying feature of finite well depth, since the same effects are obtained for the infinite E(5) well with an increase in pw. There are thus few clear signatures of finite well depth. The signatures which are present consist of moderate modifications to energies or transition strengths for high-lying levels, but such levels are typically the most subject to contamination from degrees of freedom outside the collective model framework and also the least accessible experimentally. Although realistic potentials are expected to be of finite depth, these results suggest that the infinite depth of the E(5) potential is not a limitation in its application to actual nuclei.

Discussions with F. Iachello, R. F. Casten, and N. V. Zamfir are gratefully acknowledged. This work was supported by the US DOE under grant DE-FG02-91ER-40609.

References

1. F. Iachello, Phys. Rev. Lett. 85 , 3580 (2000). 2. F. Iachello, Phys. Rev. Lett. 87, 052502 (2001). 3. A. Bohr and B. R. Mottelson, Nuclear Deformations, Vol. 2 of Nuclear Struc-

ture (World Scientific, Singapore, 1998). 4. L. Wilets and M. Jean, Phys. Rev. 102, 788 (1956). 5. D. R. BBs, Nucl. Phys. 10, 373 (1959). 6. M. A. Caprio, Phys. Rev. C 65, 031304(R) (2002).

NEUTRINOLESS DOUBLE ELECTRON CAPTURE WITH PHOTON EMISSION*

VICTORIA E. CERON Centro d e Inuestigacidn Auanzada en Ingenieria Industrial, UAEH

km. 4.5 Carr. Pachuca-!klancingo C.P. 42184, Mdxico E-mail: uceronOuaeh.reduaeh.mx

JORGE G. HIRSCH Instituto d e Ciencias Nucleares, Universidad Nacional Autdnoma de MExico,

Apartado Postal 70-543 Mdxico 04510 DF, Mdxico E-mail: [email protected]

The observation of neutrinoless double beta decay would provide evidence about neutrino masses and their Dirac or Majorana character. We study the neutrinoless double electron capture accompanied by photon emission for the nucleus of ls6Dy using the pseudo SU(3) framework.

1. Introduction

Atmospheric and solar neutrino experiments have proven that neutrinos are massive, and the different flavors are largely mixed’. The existence of neutrino oscillations allows the determination of the difference between the square of the neutrino masses coming from different families2. However, the neutrino mass absolute scale must be obtained from direct kinematical measurements, or from the double beta decay.

The study of the neutrinoless double beta decay (pp )oV could contribute to answer two fundamental questions about neutrinos: if they are Dirac or Majorana particles, and what is their mass3. We are interested in the neutrinoless double electron capture with emission of a photon4

( A , Z ) + e + + e + -+ ( A , Z - 2 ) + 7 (1) In this decay a neutrino emitted when the first electron is captured must be absorbed together with the second electron, the emission of an additional

*This work was supported in part by CONACyT (Mexico)

224

225

particle being necessary due to energy-momentum conservation. The photon is the only particle effectively emitted in the process, making its study particularly appealing.

The evaluation of nuclear matrix elements associated with this process requires knowledge of the nuclear wave functions of the parent and daughter nuclei. To build these wave functions for heavy deformed PP emitters we use the Pseudo SU(3) shell-model5.

2. Model Scheme

The pseudo SU(3) model describes the collective rotational behavior of heavy deformed nuclei. As a first step valence nucleons are classified according to the parity of their single particle orbitals. Pseudo SU(3) wave functions are used to describe the normal parity sector, while nucleons in intruder orbits are restricted to seniority zero states. The normal parity orbitals are mapped into the Pseudo SU(3) space

i j = q - - l ,

j = j = l + S , B = l /2 ,

I = i j , i j - 2, .., 1 or 0,

I I

I

(2)

where (q) labels the harmonic oscillator shell, ( 1 ) and (s) the orbital angular moments and of spin, and (j) the total angular moment.

For 156Dy the most probable occupations of normal (N) and intruder, unique (U) valence parity orbitals are6

n: = 10, n," = 6 , n: = 6 , ny = 2. (3)

3. The decay rate

The decay probability amplitude can be expressed as a product of nuclear, leptonic and photonic factors7

I' = GOur IMOv12(mu/m,)21M712. (4)

In this expression Go"? is a kinematical factor, an integral in the phase space over the two captured electrons and the emitted photon, MY evaluates the probability of photon emission, and MoU denotes the nuclear matrix element. The second term is

226

where e is the electron charge, q is the photon momentum, and fE,M gives a corrective factor related with electric and magnetic transitions. The neutrinoless double electron capture (ECEC)o, nuclear matrix elements are

with

The index cy refers to Fermi or Gamow-Teller type charge exchange transitions. The operator O(a) includes the neutrino potential H(r,E), which arise from the exchange of the virtual neutrino, and has also information about finite nucleon size and short-range correlations'. The nuclear matrix elements will be evaluated following the same procedure employed in Ref.6 to calculate the double electron capture with neutrino emission.

4. Conclusions

This work is in progress. We plan to calculate the neutrinoless double electron capture with emission of a photon in 156Dy. While the phase space strongly restricts the decay, implying very long half-lives, it is worth to estimate how far from the present experimental limits is this exotic decay.

This work was supported in part by Conacyt and DGAPA-UNAM, M6xico

References 1. S. Fukuda et. al., Phys. Rev. Lett. 86 (2001) 5651; Q.R. Ahmad et. al., Phys.

Rev. Lett. 87 (2001) 071301. 2. J . N. Bahcall , P. I. Krastev, and A. Yu. Smirnov, J . High Energy Phys.05

(2001) 015; John N. Bahcall, M. C. Gonzalez-Garcia, Carlos Pena-Garay, J. High Energy Phys. 08 (2001) 014.

3. F. Boehm and P Vogel, Physics of massive neutrinos 2nd. Ed, Cambridge University Press (1992).

4. Z. Sujkowski and S. Wycech, Ac. Phys. Pol. B 33 (2002) 471. 5. R.D. Ratna Raju, J.P. Draayer, and K.T. Hecht, Nucl. Phys. A 202 (1973)

433; K.T. Hecht and A. Adler, Nucl. Phys. A 137(1969) 129; A. Arima, M. Harvey, and K. Shimizu, Phys. Lett. B 30 (1969) 517.

6. V. E. Cer6n and J. G. Hirsch, Phys. Lett.B 471 (1999) 1. 7. M. Doi, T Kotani and E. Takasugi, Prog. Theor Phys. Suppl 83 (1985) 1. 8. J.G. Hirsch, 0. Castaiios and P.O. Hess, Nucl. Phys. A 582 (1995) 124.

SYMPLECTIC MEAN FIELD THEORY

J.L. GRABER AND G. ROSENSTEEL Department of Physics, Tulane University

New Orleans, LA 70118

The algebraic mean field theory of the symplectic algebra sp(3,R) is studied for solutions to nuclear collective motion. This theory is a general method that can be applied to any dynamical symmetry model. Predicted energy levels agree most closely with experiment when the body is neither rigid nor irrotational. The theory is applied to nuclei with the 4 3 ) label p = 0.

1. Introduction

Algebraic mean field theory (AMFT) constructs a density matrix corresponding to a wave function from the expectations of the operators in the algebra. We restrict these densities to a coadjoint orbit of the group Sp(3,R), on which the Casimir functions are constant. The restriction to a level surface of the Casimirs in AMFT is analogous to the restriction to an irreducible representation in representation theory. Unlike the shell model, AMFT formulates the solutions in a physically transparent way. The character of the rotation and the shape of the nucleus are immediately clear in the solutions. Additionally, problems of very high dimension in the shell model reduce to three equations with three unknowns in AMFT. This paper applies the method to the symplectic algebra. The mean field theory for the sp(3,R) subalgebra su(3) has been explored previously'.

2. Coadjoint orbit theory

The dimensionless Cartesian components of the position and momentum vectors of particle Q in a finite system of particles are ( x a j , p a j ) . The operators of sp(3,R) are the hermitian one-body operators

227

228

a a

a

The sp(3,R) algebra in matrix form consists of the matrices

s= ( x v -XT -") where X , U, V are 3 x 3 real matrices and U, V are symmetric. The representation of the algebra of matrices is given by

The expectations of the sp(3,R) operators determine a

1 2

matrix p in sp(3,R) that satisfies

(p, S) = -tr(pS) = -i (QISIQ),

(2)

unique density

and thus the density matrix p corresponding to the state is

p = ( n T -q -n t ) where n, t and q are the expectations of their respective operators.

A rotationally invariant energy functional in units of fiw is E[p] = &[p] + V[p], the harmonic oscillator Eo[p] = (1/2)tr(t + q) , and V [ p ] , a functional of the quadrupole deformation. The energy calculated in the rotating frame,denoted by E , is this functional with the Riemann ellipsoid collective kinetic energy subtracted from it. Equilibrium solutions are found from the critical points on the coadjoint orbit of this energy functional. If ai denote the axes lengths and the potential tensor is Wi; = - u i g , then the AMFT equations are

3 4 c a 2

where P = N: + No (A + 2 p ) + 5X2/2 + Ap + p 2 - 9C2/2, Q = 9 (A2 - C2)((2No + X + 2 ~ ) ~ - 9 C2)/4, NO is the total number of oscillator quanta,

229

~ ~

r - 1 r = 0 25 experimental

Figure 1. the the AMFT values when the rigidity T = 1 (a rigid rotor) and when T = 0.3.

The experimental values of the yrast energy band of "Ne are compared to

(A, p) are the su(3) labels, and C is the Kelvin circulation. A full derivation is provided in a previous paper2.

3. Application to 20Ne

The Elliott su(3) representation for 20Ne is NO = 48.5, (Alp) = (8,0), and LO = 13.1MeV. Choose a potential V[p] = b2v2 + b4vi, where v2 =

itr(q(2))2 and bZ, b4 are dimensionless real constant^^)^. A good fit to the experimental data occurs when b2 = -0.026, and b4 = 2.4 . and the rigidity is 0.3.The rigidity is proportional to the ratio of the circulation to the angular momentum. The energy band found from this result is compared to a rigid rotor and to the experimental data in Fig. 1.

4. Conclusion

Algebraic mean field theory can be used on any group theoretical model. It approximates representation theory results without the large calculations required in the shell model. Very complicated many body operators like the potential energy and the Kelvin circulation are handled easily in AMFT. This method also has been successful in the heavy deformed r e g i ~ n

References 1. Ts. Dankova and G. Rosensteel, Phys. Rev. C, 63 154303 (2001). 2. G. Rosensteel and J.L.Graber, J. Phys. A 35, L1 (2002). 3. G. Rosensteel and D.J. Rowe, Phys. Rev. Lett. 38, 10 (1977). 4. D.J. Rowe, Prog. Part. Nucl. Phys. 37, 265 (1996) and references therein. 5. G. Rosensteel, Phys. Rev. C, 65 064321 (2002). 6. J.L. Graber and G. Rosensteel, submitted, (2003).

STRUCTURE OF 15'GD

CARLOS GRANJA Institute of Experimental and Applied Physics, Czech Technical University

Horska' 3a/22, 128 00 Prague 2, Czech Republic E-mail: Carlos. granja @Ute f. cuut. cz

DALIBOR NOSEK Faculty of Mathematics and Physics, Charles University

V HoleSouiCka'ch 2, 180 00 Prague 8, Czech Republic E-mail: [email protected]

The level scheme of well deformed nucleus 159Gd has been experimentally established by means of radiative neutron capture and single neutron transfer reactions. Previous identification of states is confirmed and expanded. Levels with vibrational components are investigated. The structure of this nucleus is described within the quasiparticle-phonon model.

1. The level scheme

Previous information about the level scheme of this nucleus1 has been expanded by results of recent experiments in radiative neutron ~ a p t u r eand neutron stripping and pick up reactions4. The (n,y) experiments were carried out at isolated and at averaged resonances providing extensive information on primary y-rays and levels with spin up to below 3 MeV. Investigating this reaction with thermal neutrons, it was possible to observe the decay of low-lying levels4. In the neutron tranfer experiments, unpolarised and polarised deuteron beams were used for the (d,p) and (&t) reactions to populate levels with spin up to below 3 MeV4. The proposed level scheme is considered complete below 1 MeV. Rotational bands identified in this nucleus are shown in Figure 1. Newly identified Nilsson assignments include ;[510], $[514], $[514], $[633] and +[651]. Several levels above 800 keV are suggested to have vibrational admixtures based on the observed population and decay in the (n,y) experiments4.

230

23 1

2087.65+ 2003.2 + i983:+

1/2[651] 1693.5 5- 1603 3- 1579.6 1- 1/2[510]

684.2 11-

558 1112[505]

1203.5 y- 1134.7 7-

7/2[5141 . .

11- 938.6 - 759.7

588.5

;- ,2- 2:

507.7 i- -

1/2[521]

1622.3 11-

9/2[514]

1365.9 ~- - 1110.5 -

9- 3/2[532] 1043.1 948.4 - 872.6 :I - - 5/2[5 1 21

ELL7- 212.6g- 146.35- - 121.9 7- 5/2[523] 5 0 . 6 5 -

0 -3- 3/2[521]

- 744.8 ' 3/2[402] "4"".5:

602 3 3/2[651]

13+ 372.7

273.9 - - 184.8 O+

7'- i l8 .9

1485.2 7+

7/2[404]

858.8 3+ 9+& 818,9=5+

781.6 633.67+ 11216601 - - 7/2[633]

Figure 1. Rotational bands in 159Gd. Level energy and spin given in keV and $ units.

2. The quasiparticle-phonon model

The quasiparticle-phonon model5 was adopted to describe the intrinsic structure of the observed levels. Quasiparticle excitations were approximated using a deformed average Nilsson field (6 = 0.25 and ,& = 0.08) and a standard pairing field (A x 1 MeV). Collective phonons were mod- elled microscopically with parameters of the residual multipole-multipole interaction chosen from p, y and octupole 0- , 1-, and 3- phonons in neighboring even-even nuclei. The calculated intrinsic states and their structures are compared with the experimentally determined levels in Figure 2. Anal- ysis of Coriolis mixing and comparison of calculated spectroscopic factors with experiment are in progress4.

Acknowledgments

Experimental work carried in cooperation with S. PospiSil from IEAP- CTU, L. Rub6Eek from CTU Prague, T. von Egidy and H. F. Wirth of TU-Munich, H. Borner from ILL Grenoble, S. A. Telezhnikov from JINR Dubna, R. E. Chrien from Brookhaven National Laboratory and A. Apra- hamian from Notre Dame University. The support by the Czech Committee for Cooperation with JINR Dubna and by the Czech Ministry of Education, Youth and Sport under Grant MSM 210 000 018 is acknowledged.

232

N

v! 7

z z - u

2

0

43/2-[521]Q32)47%+{3/2-[521])33% -(3/2-[521]Q30)96% -{5/2'[642]Q33)48%+ 1 /2-[770]31%+(5/2+[642]) 1 5% 4 / 2 - 1 5 1 2 65%+ 7/2-[514 Q22 13%

9/2-[514]- -{5/2 [52~]Q31~~Oo~+~3/~[52~]Q33)29n~+3/2~[642]10%

1/2-[51 o l ~ ~ ~ ~ ~ + { 5 / 2 ' [ 6 4 2 ] Q 3 3 ) 2 3 % + ( 5 / 2 C [ 6 4 2 ] ) 1 0%

1 /2+[400]- 70%+(3/2'[402]+Q22)13% 5/2-[512]-

1/21 660 -810'

5/2-[512](63%)+(5/2'[642]+Q30)( 14%)+[11/2-[505]+Q33)

3/2 ~ O ~ & F W -

experiment theory

Figure 2. Structure of instrinsic states in ls9Gd with dominant components.

References

1. R. G. Helmer Nuclear Data Sheets 72, 83 (1994) 2. C. Granja, S. PospiSil, J. KubaSta and S. A. Telezhnikov, Nuclear Physics A

(2003), in press. 3. C. Granja, S. PospiSil, S. A. Telezhnikov and R. E. Chrien, to be published in

J. Physics G (2003). 4. C. Granja, S. PospiSil, L. RubGek, T. von Egidy, H. F. Wirth, G. Graw,

H. Borner, A. Aprahamian and S. A. Telezhnikov, to be published in Phys. Rev. C (2003).

5. V. G. Soloviev, Theory of Atomic Nuclei: Quasiparticles and Phonons, Insti- tute of Physics Publishing, Philadelphia, 246 (1992).

OBLIQUE-BASIS CALCULATIONS FOR 44Ti *

V. G. GUEORGUIEV, J. P. DRAAYER, w. E. ORMAND~AND c. w. JOHNSON*

Department of Physics and Astronomy, Louisiana State University, Baton Rouge, LA 70803

The spectrum and wave functions of 44Ti are studied in oblique-basis calculations using spherical and SU(3) shell-model states. Although the results for 44Ti are not as good as those previously reported for 24Mg, due primarily to the strong spin- orbit interaction that generates significant splitting of the single-particle energies that breaks the SU(3) symmetry, a more careful quantitative analysis shows that the oblique-basis concept is still effective. In particular, a model space that includes a few SU(3) irreducible representations, namely, the leading irrep (12,O) and next to the leading irrep (10,l) including its spin S = 0 and 1 states, plus spherical shell- model configurations (SSMC) that have at least two valence nucleons confined to the f7p orbit - the SM(2) states, provide results that are compatible with SSMC with at least one valence nucleon confined to the f 7 p orbit - the SM(3) states.

Introduction. In a previous study we demonstrated the feasibility of the oblique-basis calculations.' The successful description of 24Mg followed from the comparable importance of single-particle excitations, described by spherical shell-model configurations (SSMC) , and collective excitations, described by the SU(3) shell model. An important element of the success is that SU(3) is a good symmetry in sd-shell nuclei.2 For the lower pf-shell nuclei, there is strong breaking of the SU(3) symmetry induced by the spin- orbit intera~tion.~ Therefore, it is anticipated that adding the leading and next to the leading SU(3) irreps may not be sufficient in lower pf-shell.

Here we discuss oblique-basis type calculations for 44Ti using the KB3 intera~tion.~ We confirm that the spherical shell model (SSM) provides a significant part of the low-energy wave functions within a relatively small

*This work was supported in part by U.S. National Science Foundation under grants (9970769 and 0140300) as well as a Cooperative Agreement (9720652) with matching from the Louisiana Board of Regents Support Fund. tpresent address: Lawrence Livermore National Laboratory, Livermore, CA 94551 $Present address: San Diego State University, San Diego, CA 92182

233

234

Table 1. Labels and MJ=O dimensions for various 44Ti calculations. The leading SU(3) irrep is (12,O); &(10,1) implies that the (10,l) irreps are included along with the leading irrep. SM(n) is a spherical shell-model basis with n valence particles anywhere within the full pf-shell; the remaining particles being confined to the f7I2.

Model space (12,O) &(10,1) SM(0) SM(1) SM(2) SM(3) FULL dimension 7 84 72 580 1908 3360 4000

1 dimension % 0.18 2.1 1.8 14.5 47.7 84 100

number of SSMC while a pure SU(3) shell-model with only few SU(3) irreps is unsatisfactory. This is the opposite of the situation in the lower sd-shell. Since the SSM yields relatively good results for SM(2), combining the two basis sets yields even better results with only a very small increase in the overall size of the model space. In particular, results in a SM(2)+SU(3) model space (47.7% + 2.1% of the full pf-shell space) are comparable with SM(3) results (84%). Therefore, as for the sd-shell, combining a few SU(3) irreps with SM(2) configurations yields excellent results, such as correct spectral structure, lower ground-state energy, and improved structure of the wave functions. However, in the lower sd-shell SU(3) is dominant and SSM is recessive (but important) and in the lower pf-shell one finds the opposite, that is, SSM is dominant and SU(3) is recessive (but important).

Model Space. 44Ti consists of 2 valence protons and 2 valence neutrons in the pf-shell. The SU(3) basis includes the leading irrep (12,O) with M J = 0 dimensionality 7, and the next to the leading irrep (10,l). The (10,l) occurs three times, once with S = 0 (dimensionality 11) and twice with S = 1 (dimensionality 2 x 33 = 66). All three (10,l) irreps have a total dimensionality of 77. The (12,0)&(10,1) case has a total dimensionality of 84 and is denoted by &(lO,l). In Table 1 we summarize the dimensionalities. As in the case of '*Mg, there are linearly dependent vectors within the oblique bases sets. For example, there is one redundant vector in the SM(2)+(12,0) space, two in SM(3)+(12,0) and SM(1)+(12,0)&(10,1) spaces, twelve in SM(2)+(12,0)&(10,1) space, and thirty-three in the SM(3)+(12,0)&(10,1) space. Each linearly dependent vector is handled as in the previous case.'

Ground-state Energy. The oblique-basis calculation of the ground-state energy for 44Ti does not look as impressive as for 24Mg. The calculated ground-state energy for the SM(l)+(l2,0)&(10,1) space is 0.85 MeV below the calculated energy for the SM(1) space. Adding the two SU(3) irreps to the SM(1) basis increases the size of the space from 14.5% to 16.6% of the full space. This is a 2.1% increase, while going from the SM(1) to SM(2) involves an increase of 33.2%. For SM(2), the ground-state energy is 2.2 MeV lower than the SM(1) result. However, adding the SU(3) irreps to the

235

SM(2) basis gives ground-state energy of -13.76 MeV which is compatible to the pure SM(3) result of -13.74 MeV. Therefore, adding the SU(3) to the SM(2) increases the model space from 47.7% to 49.8% and gives results that are slightly better than the SM(3) which is 84% of the full space.

Low-lying Energy Spectrum. In 24Mg the position of the K=2 band head is correct for the SU(3)-type calculations but not for the low-dimensional SM(n) calculations.' In 44Ti it is the opposite, that is, the SM(n)-type calculations reproduce the position of the K=2 band head while SU(3)-type calculations cannot. Furthermore, the low-energy levels for the SU(3) case are higher than for the SM(n) case. Nonetheless, the spectral structure in the oblique-basis caIcuIation is good and the SM(2)+(12,0)&(10,1) spectrum ( ~ ~ 5 0 % of the full space) is comparable with the SM(3) result (84%).

Overlaps with Exact States. The overlap of SU(3)-type calculated eigenstates with the exact (full shell-model) results are not as large as in the sd-shell, often less than 40010, but the SM(n) results are considerably better with SM(2)-type calculations yielding an 80% overlap with the exact states while the results for SM(3) show overlaps greater than 97%, which is consistent with the fact that SM(3) covers 84% of the full space. On the other hand, SM(2)+(12,0)&(10,l)-type calculations yield results that are as good as those for SM(3) in only about 50% of the full-space and SM(l)+(l2,0)&(10,1) overlaps are often bigger than the SM(2) overlaps.

Conclusion. For 44Ti, combining a few SU(3) irreps with SM(2) configurations increases the model space only by a small ( ~ 2 . 3 % ) amount but results in better overall results: a lower ground-state energy, the correct spectral structure (particularly the position of K=2+ band head), and wave functions with a larger overlap with the exact results. The oblique-bases SM(2)+(12,0)&(10,1) results for 44Ti ( ~ 5 0 % ) yields results that are comparable with the SM(3) results ( ~ 8 4 % ) . In short, the oblique-basis scheme works well for 44Ti, only in this case, in contrast with the previous results for 24Mg where SU(3) was found to be dominant and SSM recessive, in the lower pf-shell SSM is dominant and SU(3) recessive.

References 1. V. G. Gueorguiev, W. E. Ormand, C. W. Johnson, and J. P. Draayer, Phys.

2. J. P. Elliott and H. Harvey, Proc. Roy. SOC. London A272, 557 (1963). 3. V. G. Gueorguiev, J. P. Draayer, and C. W. Johnson, Phys.Rev. C63, 014318

4. T. Kuo and G. E. Brown, Nucl. Phys. A114, 241 (1968); A. Poves and A. P.

Rev. C65, 024314 (2002).

(2001).

Zuker, Phys. Rep. 70, 235 (1981).

APPLICATION OF GROUND-STATE FACTORIZATION TO NUCLEAR STRUCTURE PROBLEMS

T. PAPENBROCK~'~ AND D. J. DEAN] Physics Division, Oak Ridge National Laboratory,

Oak Ridge, T N 37831, USA Department of Physics and Astronomy,

University of Tennessee, Knoxville T N 37'996-1201, USA

We compute accurate approximations to the low-lying states of 44Ti by ground- state factorization. Energies converge exponentially fast as the number of retained factors is increased, and quantum numbers are reproduced accurately.

The nuclear shell model is difficult to solve for more than a few valence nucleons due to the large dimensions of the underlying Hilbert space. Furthermore, the complexity of the interaction makes it challenging to devise approximations that significantly reduce the size of the problem while still being sufficiently accurate. In recent years, various approximation schemes have been proposed1~2~3~4~5~6~7~s. We particularly mention the mixed-mode shell-model theory7, the approach based on a quasi- SU(3) truncation scheme5, and the very recently proposed ground-state factorization'. The first two of these approaches use a basis truncation scheme that is based on a small number of SU(3) coupled irreps and thereby includes important collective configurations. The third approach approximates the ground state in terms of a small number of products of optimally chosen proton and neutron states. Particularly promising results of the mixed-mode shell-model7 and the ground-state factorization8 have been reported for the sd-shell nucleus 24Mg. This is interesting since this nucleus exhibits competing single-particle and collective degrees of freedom. In this work we apply the ground-state factorization to the pf-shell nucleus 44Ti. This nucleus is an interesting test case as its SU(3) symmetry-breaking has recently been explored in detailg.

236

237

0 1000 2000 3000 4000

6

4

2

0

j

0 1000 2000 3000 4000 d

Figure 1. Low-lying states of 44Ti (KB3 interaction) computed from ground-state factorization. Top: Energy spectrum (data points connected by full lines) and exact results (dotted lines) versus the dimension d of the eigenvalue problem. Bottom: Angular momentum quantum number j from j ( j + 1) = ( J 2 ) (data points connected by full lines) versus d. An exact diagonalization has dimension d,,, = 4000.

The ground-state factorization is based on the ansatz

for the ground state I @ ) . Here, the unknown factors are the proton states Ipj) and the neutron states Inj) which are of dimension d p and d N , respectively. The truncation is controlled by the fixed input parameter R which counts the number of retained factors. Variation of the energy E = ($JIfi\$J)/{$J\$J) yields eigenvalue problems of dimension R d p (RdN) for the proton states (neutron states). Note that these dimensions are usually much smaller than the dimension d p d N of the full problem. For details, we refer the reader to Ref.8.

We apply the ground-state factorization p f -shell nucleus 44Ti and use

238

the KB3 interaction1'. In m-scheme, the Hilbert space has dimension d,,, = 4000, and the eigenvalue problem for the factorization has an R- dependent dimension d 5 dmax. Figure 1 shows the energies and angular momentum quantum numbers of the low-lying states versus the dimension d of the eigenvalue problem. Note the exponential convergence with respect to increasing dimension d of the eigenvalue problem. Very good energies are obtained once the dimension d exceeds d "N 0.2dm,,, while the angular momenta stabilize around d = 0.3dm,,. Note also that the angular momenta of the quasi-degenerate third and fourth excited states are accurately reproduced. The rapid convergence of the excited states suggests that they can be approximated by factors that are similar to those of the ground state. This similarity of the structure of low-lying states is also reflected in the strenght distribution of the SU(3) Casimir operator C2 '.

The results of this work and the results of Ref.8 demonstrate the accuracy and efficiency of the ground-state factorization for a variety of nuclei. This opens a promising avenue for large-scale nuclear structure calculations.

Acknowledgments

This research was partly supported by the US. Department of Energy under Contract Nos. DE-FG02-96ER.40963 (University of Tennessee) and DE-AC05-000R22725 with UT-Battelle, LLC (Oak Ridge National Labo- ratory).

References 1. G. H. Lang, C. W. Johnson, S. E. Koonin, and W. E. Ormand, Phys. Rev.

C48, 1518 (1993). 2. M. Horoi, B. A. Brown, and V. Zelevinsky, Phys. Rev. C50, R2274 (1994). 3. S. E. Koonin, D. J. Dean, and K. Langanke, Phys. Rep. 278, 1 (1997). 4. M. Honma, T. Mizusaki, and T. Otsuka, Phys. Rev. Lett. 75, 1284 (1995). 5. C. E. Vargas, J. G. Hirsch, P. 0. Hess, and J. P. Draayer, Phys. Rev. C58,

1488 (1998). 6. J. Dukelsky, S. Pittel, S. S. Dimitrova, and M. V. Stoitsov, Phys. Rev. C65,

054319 (2002). 7. V. G. Gueorguiev, W. E. Ormand, C. W. Johnson, and J. P. Draayer, Phys.

Rev. C65, 024314 (2002). 8. T. Papenbrock and D. J . Dean, eprint nucl-th/0301006, submitted to Phys.

Rev. C . 9. V. G. Gueorguiev, J. P. Draayer, and C. W. Johnson, Phys. Rev. C63, 014318

10. T. T. S. Kuo and G. E. Brown, Nucl. Phys. A114, 241 (1968); A. Poves and (2001).

A. P. Zuker, Phys. Rep. 70, 235 (1980).

MICROSCOPIC INTERPRETATION OF THE K" = 0; AND 2; BANDS OF DEFORMED NUCLEI WITHIN THE

FRAMEWORK OF THE PSEUDO-SU(3) SHELL MODEL

G. POPA', A. GEORGIEVA2'3, J. P. DRAAYER3 'Department of Physics, Rochester Institute of Technology

E-mail: gpopa@gxpsps. rit. edu 'INRNE, BAS, Sofia, Bulgaria

3Department of Physics and Astronomy, LSU

The behavior of the Kn = 0; and 2; bands in a sequence of deformed even-even rare earth nuclei, organized into F-spin multiplets of the Sp(4,R) scheme, is explored. The complex nature of these states and the collective bands built on them is interpreted in terms of the microscopic proton-neutron pseudo-SU(3) shell model.

The excited Kn = 0; and 2 ; bands of deformed even-even mass rare earth nuclei display systematic change as a function of nucleon mass number. An investigation into the structure of these nuclei, organized into F-spin multiplets of the Sp(4,R) scheme, shows some general trends. In particular, the energies of band-head states as a function of mass number suggest the need for a more comprehensive understanding of their microscopic structure. The purpose of this contribution is to provide an interpretation of this behavior through an application of the proton-neutron pseudo-SU(3) shell-model.

Pseudo-spin ~yrnmetry'~*.~ refers to the fact that single-particle orbitals of the shell with total angular momentum j = 1 - 1/2 and j = (1-2)+1/2 lie close in energy and can therefore be interpreted as pseudo-spin doublets with quantum numbers 7 = j , = 7- 1 and I = I - 1. The pseudo-orbital and pseudo-spin quantum numbers are assigned to single-particle states of the normal parity orbitals; the intruder level from the shell above, which has the opposite parity, is considered to host zero-coupled pairs and renormalize operators as appropriate but to be otherwise inactive and hence removed from further consideration in what follows.

Basic building blocks of the theory are pseudo-SU(3) proton and neutron states with pseudo-spin zero. The many-particle states are built as pseudo-SU(3) coupled states with pseudo-spin zero and well-defined particle number and total angular The Hamiltonian,

-

H = Hsp"+ Hspv + G,Hp"+ G,Hp" + a * Q + a L2 + b KJ' + asvm C2 + a3 C3,

includes spherical Nilsson single-particle proton and neutron terms (HspB and Hsp") and proton and neutron pairing interactions (Hp" and Hp") that mix SU(3) basis states. The quadrupole-quadrupole (Q*Q) as well as four 'rotor-like' terms (L2, K:, C2, and C,) preserve the SU(3) symmetry. The interaction strengths ( G , G, and x) are taken from systematics. The other parameters: a, b, usym,

239

momentum

240

and a3, are determined through fitting. Parameter values are close to those used in the description of the neighboring even-even as well as odd-A n u ~ l e i ~ . ~ ~ ' . As a result we obtain good agreement between the experimental and theoretical energies of the low-lying spectra. This is shown in Fig. 1 which compares the non-yrast Kn = 0; and 2; states in a sequence of deformed nuclei.

2

1.5 I . - -

* F 0.5

UI

0 t 2.

I -goso

2

1.5

1

0.5

0

12 14 16 18 20

Number of Neutron and Proton Pairs

Fig. 1 - The experimental and theoretical energies of the ground band 0, and 2, states and the non-yrast K' = 0; and 2; states of deformed nuclei in the F, = 0 multiplet of Sp(4,R). The experimental values are indicated with bars and the calculated numbers with various shapes.

A microscopic interpretation of the relative position of the collective band, as well as that of the levels within the band, follows from an evaluation of the primary SU(3) content of the collective states. The latter are closely linked to nuclear deformation4. If the leading configuration is triaxial (nonzero p), the ground and y bands belong to the same SU(3) irrep (see I6'Gd in Fig. 2); if the leading SU(3) configuration is axial (p=O), the 0; and y bands come from the same SU(3) irrep.

A proper description of collective properties of the first excited Kn = 0; and 2f states must take into account the mixing of different SU(3)-irreps, which is driven by the Hamiltonian. This study shows that pseudo-spin zero

241

neutron and proton configurations with a relatively few pseudo-SU(3) irreps with largest deformations (C2 values) suffices to obtain reasonable agreement with known experimental energies and to understand and reproduce the behavior of the low-lying non-yrast bands in deformed nuclei.

120 n 2 100

3 2 20

80

60 CI

- 40 c 0

I I6’Gd 172Y b 1 Oh

We Wd o c Wb Wa

I Fig. 2 - SU(3) content of wave functions of the collective ground and Kn = 0; and 2: bands in 16’Gd (the first three columns) and 172Yb (the last three columns). The different patterns label the following SU(3) irreps. in ’%d: ‘a’ --f (28,s) = (10,4) C3 (18,4), ‘b’ --f (30,4) = (10,4) €3 (18,4), ‘c’ + (30,4)’ = (10,4) €3 (20,0), ‘d’ 4 (30,4)’= (129) C3 (18,4), ‘e’ + (32,O) = (10,4) €3 (18,4), ‘f 4 (32,O)’ = (12,O) C3 (20,O); .and in 172Yb: ‘g’ + (36,O) = (12,O) C3 (24,0), ‘h’ 4 (28,lO) =(12,0) C3 (16,10), and ‘i’ --f (20,20) = (4,lO) €3 (16,lO).

Acknowledgments: GP and AG are grateful to J. P. Draayer for his guidance and support. This work was supported by the US. NSF through regular grants (9970769 and 0140300) as well as a cooperative agreement (9720652) that includes matching from the Louisiana Board of Regents.

References

1 . 2. 3.

4. 5 . 6. 7 .

K. T. Hecht and A. Adler, Nucl. Phys. A 137 (1969) 129. A. Arima, M. Harvey, and K. Shimizu, Phys. Lett. 30 (1969) 517. R. D. Ratna Raju, J. P. Draayer, and K. T. Hecht, Nucl. Phys. A 202 (1 973) 433. T. Beuschel, J. G. Hirsch, and J. P. Draayer, Phys. Rev. C 61 (2000) 54307. G . Popa, J . G. Hirsch and J. P. Draayer, Phys. Rev. C 62 (2000) 064313. J. P. Draayer, G. Popa and J. G. Hirsch, Acta Phys. Pol. B 32 (2001) 2697. C . Vargas, J. G. Hirsch, T. Beuschel and J. P. Draayer, C 61 (2000) 054307.

SP(4) DYNAMICAL SYMMETRY FOR PAIRING CORRELATIONS AND HIGHER-ORDER INTERACTIONS

IN ATOMIC NUCLEI

K. D. SVIRATCHEVA~, c. B A H R I ~ , A. I. GEORGIEVA~?~ , AND J. P. DRAAYER'

'Louisiana State University, Department of Physics and Astronomy, Baton Rouge, Louisiana, 70808-4001 USA

Institute of Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, Sofia 1784, Bulgaria

Properties of pairing and higher-order interactions are investigated with a Sp(?, (4) model. The q-deformation introduces an order parameter for a 'phase transition'.

1. Sp(4) pairing model and its quantum extension - The sp,(4) algebra' is realized in terms of creation/annihilation fermion operators { c y j , , , , , ( ~ ~ , ~ , , , } ~ f - ~ = q*%dm,rnl (u = f 1 / 2 (proton/ neutron), with j the total angular momentum (half-integer), m its projection, and 2R = Cj(2j + 1) the shell dimension). In addition to the number operator N = N1 + N-1 and the isospin projection TO = (Nl - N-1)/2, the generators of Spq(4) are T* = - a Cjm ~j~,*1/20jrn,71/2 (isos~in operators) 1 t

- C . ( - ~ ) j - ~ a ! ~ , p ~ , - ~ , , , t , A, = (AL)+ (ere- and A;=,+,, - d& J m

ate and annihilate a pair of total angular momentum J" = O+ and isospin T = 1). The isospin breaking model Hamiltonianl includes an isovector (T = 1) pairing interaction, a diagonal isoscalar (T = 0) force, and a symmetry term, H, = - Z ~ N - G , A ~ A ~ - F,(A:,A+~ + A ~ _ , A - ~ ) - +E,(% - [ 21 - Cq2fl [A] ( [% - Q] 2 2R - [fllk) - (D, - 3 ) O [i] [TO]&, where by

definition [ X ] k = 3+. The basis states are constructed as (T = 1)- paired fermions and model the O+ ground state for even-even and some odd-odd nuclei and the isobaric analog excited O+ state for odd-odd nuclei.

The interaction strength parameters are estimated in a fit with a small uncertainty for three groups of nuclei (ld3/2, lf7/2, lf5/22plgg,2)1. The values for the pairing strengths are found to lie on a curve that decreases

2

EX- -kX

242

243

with inass number 1 0 G = (R2 = 0.99, goodness-of-fit measure), $ = 23.921.1 (R2 = 0.96). This allows for their further prediction for the region of lg7~22d3s11~11/2 (where the energy spectrum of 2 M N and proton-rich nuclei is not yet measured). The available data in this region is suitable to be used in a fit (x = 1.92MeV) to determine the rest of the parameters, which also show a global dependence on A: & = (R2 = 0.75), D =

(R2 = 0.99). Among all the interactions in the model, only the p n isovector correla-

tions show a prominent peak in energy around Z = N. A simple Gaussian approximation, Ee-0.5(T0/")2, estimates the width to be (2 - N1/2 M 1 for the nuclei in the l f 7 p shell: W(E=2.0f0.4, ~ 2 = 0 . 8 4 ) = 1.0 f 0.2 for A = 44 and w ( ~ ~ 2 . 9 f 0 . 2 , ~ 2 = 0 , 9 4 ) = 1.37 f 0.09 for A = 48. The p n pairing also plays a significant role in reproducing the experimental data for the 5'2,

two-proton separation energy around 2 = N . The zero point of 5'2, determines the two-proton-drip line, which according to the Sp(4) model lies near the following nuclei: Ge28, Ga29, Se301 A s Q ~ , K~32, B7-33, Sr34] Rb35,

324-1 1.7*0.2 --37*5 A + (-0.24 f 0.09) (R2 = 0.97), C = (7)

ZT36, y37~ Zr381 y39~ M040, Nb41, RU42, TC43, Pd44~ Rh45, Cd46, Ag47.

2. q-Deformed Parameter and 'Phase Transition' - Higher-order interactions in nuclei can be investigated via the use of a local q-parameter that was found to vary smoothly with mass number2. The q-deformation not only accounts for many-body interactions] it also yields an order parameter] K (q = e x ) , for a 'phase transition' between a region of negligible higher-order correlations (I) and one where non-linear fields are required (11). In the first phase (I) the deformation parameter N is zero. This is observed for comparatively higher E4+/E2+ ratios (Fig. 1). Strong nonlinear effects are found for lower E4+ /E2+ ratios. Typically, K tends to zero

1.2 1.6 2 2.4 2.8 3.2 3.6 4 E(4')/E(2+)

Figure 1. The deformation K vs. E4+ /E,+ for isotopes of 2 = 30 - 38 nuclei.

244

around E4+ /E2+ M 2 - 2.5, where the ‘phase transition’ occurs (recall that E4+/E2+ M 1; 2; 3.33 for nuclei in the seniority/ vibrator/ rotator regime3).

The q-deformation, and hence the development of non-linear effects, is related in a non-trivial way to the underlying nuclear structure. The many-body interactions yield very complicated matrix elements and their analytical modeling is made possible due to the quantum extension of Sp(4). For even-even nuclei a functional dependence of K on the total number of particles N and the isospin projection TO is found in the form: N ( N , TO) =

where the step-function is Q(x) = 1(0), z 2 0 (x < 0). A fit (x = 0.13) to values of N for the even-even nuclei in the lf5p2plggp shell2 yields: a = -2.86, b = 2.46, c = 0.12, d = 0.21. An interesting observation is that the exponential dependence yields a width of b/2 = 1.23, which is close to the values found for the pn-energy peak width for the case of 1 f7/2.

A model Hamiltonian, written in terms of the (q-deformed) generators of Sp(,) (4), was used to describe pairing correlations and higher-order interactions. The lowest isovector-paired Of state energies were fit and the global interaction strength parameters were estimated. The general trend of the interaction parameters shows a smooth dependence on mass number. The pn isovector pairing is significant around 2 M N and decreases rapidly after 2 M N f 2. They play an important role in determining the 2p-drip line for medium nuclei. The q-deformation introduces an order parameter for a ‘phase transition’ in nuclei. The phase of strong higher-order interactions is observed for Eq+ JEz+ M 2.3 and lower; in the region of well shape- deformed nuclei these interactions may be negligible. A smooth non-trivial analytical dependence of the deformation parameter on the mass number and the isospin projection was found and prediction for the q-parameter in proton-rich nuclei could be made. Along with two-body p n correlations, the non-linear effects also contribute significantly to the energy peak at 2 = N .

- + d - 2 0 ( ~ - 2 ~ ) ) ~ - ~ . ~ ( 6 9 ~ + c ~ ( ~ - ~~)IT~I&-I ,

Acknowledgments - KDS, CB and AIG are grateful to J. P. Draayer for his constant help and support. This work was supported by the US National Science Foundation (Grants Numbers 9970769 and 0140300).

References 1. K.D. Sviratcheva et al., J . Phys. A : Math. Gen . 34, 8365 (2001); K.D. Svi-

ratcheva, A.I. Georgieva, and J.P. Draayer, arXiv . org/nucl-th/O20~0?’O0. 2. K.D. Sviratcheva, C. Bahri, A.I. Georgieva, and J.P. Draayer, in preparation. 3. R.F.Casten, N.V.Zamfir, and DSBrenner, Phys. Rev. Lett. 71, 227 (1993).

EXCITED BANDS IN ODD-MASS RARE-EARTH NUCLEI *

CARLOS E. VARGAS Instituto de Ciencias Nucleares, Universidad Nacional Autdnoma de Mkxico,

Apartado Postal 70-543 Mkxico 04510 DF, Mixico and

Facultad d e Fisica e Inteligencia Artificial Sebastidn Camacho No. 5

Centro, 91 000, Xalapa Ver., Mixico E-mail: cvargasOnuclecu.unam.mx

JORGE G. HIRSCH Instituto d e Ciencias Nucleares, Universidad Nacional Autdnoma de Mixico,

Apartado Postal 70-543 M6xico 04510 DF, M.4xico E-mail: [email protected]

JERRY P. DRAAYER Department of Physics and Astronomy, Louisiana State University,

Baton Rouge, Louisiana 70803, U.S. A . E-mail: draayerolsu. edu

Normal parity bands in 157Gd, 163Dy and ls9Tm are studied using the pseudo SU(3) shell model. Energies and B(E2) transition strengths between states belonging to low-lying, sameparity rotational bands in each nuclei are considered. The pseudo SU(3) basis includes states with pseudo spin 0 and 1, and and 4, for even and odd nucleon numbers, respectively. States with pseudo-spin 1 and 4 must be included for a proper description of some excited bands and M1 transition strengths.

In light deformed nuclei the dominance of quadrupole-quadrupole interaction led to the introduction of the SU(3) shell model '. However, the strong spin-orbit interaction renders the normal SU(3) truncation scheme useless in heavy nuclei, while at the same time pseudo-spin emerges as an

*This work was supported in part by CONACyT (Mbxico) and the US National Science Foundation.

245

246

[keV1

1600

1200

800

400

-11 -9- - -- 157Gd

(g .s. b. ) ..- 25- 25- -.. : i

23- -...- 23- band D

band E ;

band C (5 21- -11-

- 9- 11- - band B .21- -...-

(q...-11- - '7- - - 5- -.. 9- (9 l..- 9-

7- ( 5 2 ...- 5- ; . 17- ,...- 1'7- 11- 5- 3-= ... - 3-/,3- 3-i' I-- ...- 1- (3-)

: 3- ( 9 3 19- . 19- -...-

-7- ; =

1- 15- . 15- -."- g-- ...- 9-

13- 7--...- 7- 13- -'..- 5-- ...- 5- - 11- -."- 11-

9- -...- 9- 7-

3- -...-

Exp The0

Figure 1. Energy spectra of 157Gd. The integer numbers denote twice the angular momentum of each state. Experimental energies are plotted on the left hand size of each column, while their theoretical values are shown in the right hand side

approximately good symmetry 29314. Pseudo-spin symmetry refers to the experimental fact that single-particle orbitals with j = 1 - ; and j = (1 - 2) + $ in the shell q lie very close in energy, and can therefore be labeled as pseudo spin doublets with quantum numbers j = j , ij = q - 1, and I = I - 1.

The classification of many-particle states of n, active nucleons in a given normal parity shell q,, Q! = v (neutrons) or 7~ (protons) and the Hamiltonian employed can be found in the Refs. '. We only mention that the Hamiltonian includes spherical Nilsson single-particle energies for 7r and v as well as the pairing and quadrupole-quadrupole interactions, with their strengths taken from systematic^^>^.

-

247

In the present study the applicability of the pseudo-SU(3) model is broaden to include proton and neutron configurations with pseudo spin 3, = 1 and 3, = +. The final many-particle states in odd-mass nuclei have total pseudo spin +, or %. In this enlarged space, six or seven normal parity low-lying rotational bands in lS7Gd, 163Dy and 16’Tm are successfully described. Many of them have important pseudo spin 1 and $ components 5 . Intra and inter-band B(E2) and M1 transition strengths in the range between 2 and 4 MeV have been studied.

Fig. 1 shows the yrast and five excited normal parity bands in 157Gd (experimental data taken from Ref.8). These results should be compared with the three bands described in an earlier studyg, where the same Hamiltonian and parametrization was employed but the Hilbert space was restricted to 3, = 0 and 3, = $ states.

The new bands have predominantly 3 = i, but the most important contribution comes from the proton or neutron subspaces with 3 = 1. This implies that the pseudo-spin mixing in the wave function occurs mainly in the sub-space with an even number of particles. The B(E2) transition strengths confirm this band structure and are in good agreement with the measured values.

A quantitative microscopic description of normal parity bands, their B(E2) intra- and inter-band strengths and M1 transitions between the ground and those states in the range 2-4 MeV in odd-mass heavy deformed nuclei has been obtained using a realistic Hamiltonian and a strongly truncated pseudo SU(3) Hilbert space, including pseudo spin 1 and 3/2 states.

References 1. J. P. Elliott, Proc. Roy. SOC. London Ser. A 245, 128 (1958); 245, 562 (1958). 2. K. T. Hecht and A. Adler, Nucl. Phys. A 137, 129 (1969); A. Arima, et al.

Phys. Lett. B 30, 517 (1969). 3. J. P. Draayer, et. al., Nucl. Phys. A 381, 1 (1982). 4. J. G. Hirsch, C. E. Vargas, G. Popa, J. P. Draayer, “Pseudo + Quasi SU(3):

Towards a shell model description of heavy deformed nuclei” Contribution to this conference.

5. C. E. Vargas, J. G. Hirsch, J. P. Draayer, Phys. Rev. C 66, 064309 (2002); ibid, Phys. Lett. B 551, 98 (2003).

6. C. E. Vargas, J. G. Hirsch, and J. P. Draayer, Nucl. Phys. A 673, 219 (2000). 7. P. Ring and P. Schuck. T h e Nuclear Many-Body Problem Springer, Berlin‘

(1979); M. Dufour and A. P. Zuker, Phys. Rev. C 54, 1641 (1996). 8. National Nuclear Data Center, http://bnlnd2.dne.bnI.gov 9. C. E. Vargas, Ph.D. thesis, CINVESTAV, MBxico, 2001.

THE GEOMETRY OF THE PB ISOTOPES IN A CONFIGURATION MIXING IBM *

CARLOS E. VARGAS Instituto de Ciencias Nucleares, Universidad Nacional Auto'noma de Me'xico,

Apartado Postal 70-543 Mixico 04510 DF, Mixico and

Facultad de Fisica e Inteligencia Artificial Sebastia'n Camacho No. 5

Centro, 91000, Xalapa Ver., Me'xico E-mail: cvargasOnuc1ecu.unam.mx

ALEJANDRO FRANK Instituto de Ciencias Nucleares, Universidad Nacional Auto'noma de Mixico,

Apartado Postal 70-543 Mixico 04510 DF, Me'xico E-mail: frankOnuc1ecu.unam.mx

PIET VAN ISACKER Grand Acce'lirateur National d 'Ions Lourds, BP 5027, F-14076 Caen Cedex 5, France

E-mail: isackerOganil.fr

We apply a recently proposed matrix-coherent state approach for configuration mixing Hamiltonians in the context of the IBM l , to describe the evolving geometry of the neutron deficient Pb isotopes. The potential energy surface of 186Pb has three well developed minima, which correspond to spherical, oblate and prolate shapes, in close agreement with recent experimental measurements and deformation dependent mean-field calculations '. We find that the mixing between the three configurations is probably overestimated in the fit, since the oblate minimum is blocked in the full calculation. A slight modification of the mixing parameters, however, gives rise again to a remarkably similar shape for the potential surface of this neutron-deficient isoptope, when compared with the mean-field calculations. Moving away from mid-shell, towards the heavier Pb isotopes, the deformed minima tend to disappear immediately. Our analysis suggests that the method may be a reliable tool for the study of geometrical aspects of shape coexistence phenomena in nuclei.

'This work was supported in part by CONACyT (MBxico) and GANIL (France).

248

249

In the last few years, it has been accumulated strong evidence of the presence of intruder excitations at and near closed shell regions ' . These intruder excitations are belived to give rise to the interesting phenomenon of shape coexistence in the nuclei. New data has become available in several Pb isotopes, allowing to make the first observation of a set of three low-lying O+ states in lssPb 2 , which have been associated with the spherical, oblate and prolate shapes. The coexistence phenomenon in semi-magic nuclei has been associated with many-particle many-hole excitations across the closed shells. Twenty years ago 5 , it was suggested the possibility of include the most simple intruder configuration 2p-2h as the addition of two extra boson pairs to the IBM, allowing to mix them with the regular configurations. Nevertheless, the presence of the spherical, oblate and prolate shaped configurations suggest us the consideration of triple configuration mixing, namely the regular, 2p-2h and 4p-4h. In this work we consider the potential energy matrices formalism to study the shape coexistence phenomena in the 186-192Pb isotopes.

The Hamiltonian is H = Hreg + H2p-2h + H4p-4h + Hmix. Its different pieces are the functions Hi for each configuration Hj = c i i i d + K i Q i . Q i ,

the mixing between the different configurations.

1 1

1 . .

OpOh-2p2h 2p2h-4p4h and Hmix = Hmix + Hmix

The potential energy matrix can be defined as

fio-2 ( P ) 0 E(B, 7 ) = E2(N + 2, B, 7 ) + A2 fi2-4(B> ) ( Eo(N:'T) f i 2 - 4 ( P ) E4(N + 4,B,Y) + A4

where Ei(n ,B,y) , i = 0, 2, 4 denote the energy surfaces of the regular (N bosons), 2p-2h (N+2) and 4p-4h (N+4) configurations 7, respectively, and the parameter A2 (A,) essentially corresponds to the single-particle energy expended in rising 2 (4) particles from the lower shell, corrected for the gain in energy due to the pairing interaction and the increase in deformation energy made possible by allowing the proton bosons to be active. The s2°-2(/3) = ( N + 2,,B,ylHmixlN,P,y) (and f12-4(/3)) are the non-diagonal matrix elements, mixing the three surfaces.

Fig. 1 shows the Potential Energy Surfaces in lssPb as function of psin(y + 30") and ,!?cos(y + 30"), which is build mixing the three configurations. There is a very good agreement between the position of these three minima and their experimental partners, as all the O+ states are below 700 keV. The similarities between the mean field and the geometric interpre-

250

tation of the IBM show that the configuration mixing used in conjunction with matrix coherent state methods, are a reliable tool in the description of shape coexistence phenomena in nuclei.

L U

Figure 1. Potential Energy Surface in 186Pb

The result shows that the present framework is a very powerfull method to study the shape coexistence phenomena. The lowest PES predicted with the method of the eigenpotentials is in close agreement with that found using mean-field calculations, but the present result describes in addition several spectroscopic properties.

An more detailed study of the nuclei in the region including the binding energies will bring more arguments around the mixing parameters, which we believe are the responsable to block the oblate minimum.

Moving away from the mid-shell nuclei, the heavier Pb isotopes show a clear spherical dominance, in accord with experimental results.

References

1. A. Frank, 0. Castaiios, P. Van Isacker, E. Padilla, Wyoming Proccedings, “The Casten’s fiiangle”, 2002.

2. A. N. Andreyev, e t al. Nature 405 (2000) 430. 3. R. Fossion, K. Heyde, G. Thiamova, and P. Van Isacker, Phys. Rev. C 67

024306 (2003). 4. K. Heyde, et al., Phys. Rep. 102 (1983) 291; J. L. Wood, et al., Phys. Rep.

215 (1992) 101. 5. P. D. Duval and B. R. Barret, Nucl. Phys. A 376 (1982) 213. 6. C. E. Vargas, A. Frank, P. Van Isacker, To be published. 7. P. Van Isacker and Jin-Quan Chen, Phys. Rev. C 24 (1981) 684; E. L6pez

Moreno and 0. Castaiios, Phys. Rev. C 54 (1996) 2374.

The Banquet

List of Participants

A. Baha Balantekin JosC Barea Thomas Beuschel Roelof Bijker Andrey L Blokhin Piotr Borycki Mark Caprio Octavio Castaiios Richard F. Casten Victoria Cer6n David J. Dean Santiago Alfred0 Diaz Jerry P. Draayer Kalin Pavlov Drumev Jorge Dukelsky Tomas Dytrych Jutta Escher Alejandro Frank Ana I. Georgieva Vesselin Gueorguiev Joseph Ginocchio Jessica L. Graber Carlos Granja Walter Greiner Karl. T. Hecht Peter 0. Hess Jorge G. Hirsch Calvin W. Johnson Andy Ludu Marcos Moshinsky W. Erich Ormand Feng Pan Thomas Papenbrock Stuart Pittel Peter Ring David J. Rowe Gergana Stoitcheva Mario Stoitsov Kristina D. Sviratcheva Piet Van hacker Carlos E. Vargas Edward Zganjar

Univ. of Wisconsin ICN (UNAM) SAP ICN (UNAM) Hibernia N. Bank ORNL Yale Univ. ICN (UNAM) Yale Univ. UAE Hidalgo ORNL IA (UNAM) LSU LSU CSIC LSU LLNL ICN (UNAM) Bulgarian A.S. LSU LANL Tulane Univ. Czech TU Univ. Frankfurt Univ. of Michigan ICN (UNAM) ICN (UNAM) San Diego SU Northwestern SU IF (UNAM) LLNL Liaoning NULSU ORNL Univ. of Delaware TU Munich Univ. of Toronto ORNL ORNL LSU GANIL ICN (UNAM) LSU

[email protected] [email protected] thomas.beusche1 @sap.com [email protected] albl @bellsouth.net pborycki @utk.edu mark.caprio@ yale.edu [email protected] [email protected] vceron @uaeh.reduaeh.mx deandj @ornl.gov alf @ astroscu.unam.mx [email protected] [email protected] dukelsky @iem.cfmac.csic.es tdytryl @lsu.edu escherl @llnl.gov [email protected] anageorg @ inme.bas.bg [email protected] [email protected] jgraber@ tulane.edu granjaokf-alpha.fjfi.cvut.cz greiner@ th.physik.uni-frankfurt.de khecht @umich.edu [email protected] hirsch @ nuclecu .unam.mx [email protected] [email protected] moshi @ fisica.unam.mx ormandl @llnl.gov [email protected] [email protected] pittel @bartol.udel.edu ring@ph. tum.de [email protected] [email protected] stoitsovmv @ oml. gov kristina@ baton.phys.lsu.edu [email protected] vargas @ ganil. fr [email protected]

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Good luck, Jerry!

computational and group-theoretical methods in nuclear physics

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