THE SCIENCE AND CULTURE SERIES — ADVANCED SCIENTIFIC CULTURE Series Editor: A. Zichichi
HADRONS, NUCLEI AND APPLICATIONS PROCEEDINGS OF THE CONFERENCE: BOLOGNA 2000 STRUCTURE OFlHE NUCLEUS AT THE DAWN OF THE^ENTURY
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World Scientific
HADRONS, NUCLEI AND APPLICATIONS PROCEEDINGS OF THE CONFERENCE: BOLOGNA 2000 STRUCTURE OF THE NUCLEUS AT THE DAWN OF THE CENTURY
THE SCIENCE AND CULTURE SERIES — ADVANCED SCIENTIFIC CULTURE Series Editor: A. Zichichi, European Physical Society, Geneva, Switzerland Series Editorial Board: P. G. Bergmann, J. Collinge, V. Hughes, N. Kurti, T. D. Lee, K. M. B. Siegbahn, G. 't Hooft, P. Toubert, E. Velikhov, G. Veneziano, G. Zhou
1.
Nucleus-Nucleus Collisions Bologna 2000. Structure of the Nucleus at the Dawn of the Century
2.
Nuclear Structure Bologna 2000. Structure of the Nucleus at the Dawn of the Century
3.
Hadrons, Nuclei and Applications Bologna 2000. Structure of the Nucleus at the Dawn of the Century
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HADRONS, NUCLEI AND APPLICATIONS PROCEEDINGS OF THE CONFERENCE: BOLOGNA 2000 STRUCTURE OF THE NUCLEUS AT THE DAWN OF THE CENTURY
Bologna, Italy
29 May - 3 June 2000
Editors
Giovanni C. Bonsignori Mauro Bruno Dipartimento di Fisica dell' Universita di Bologna and INFN-Sezione di Bologna, Italy
Alberto Ventura Ente Nuove Tecnologie, Energia e Ambiente and INFN Bologna, Italy
Dario Vretenar Physics Department, University of Zagreb, Croatia
Series Editor
A. Zichichi
Vfe W o r l d Scientific « •
NewJersev Ne w Jersey • London* London • Sinaanore* Singapore • Hong Kong
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HADRONS, NUCLEI AND APPLICATIONS Proceedings of the Conference: Bologna 2000 Structure of the Nucleus at the Dawn of the Century Copyright © 2001 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, 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 981-02-4733-8
Printed in Singapore.
CONTENTS
Many B o d y Methods in Nuclear Structure New Microscopic Approaches to the Physics of Nuclei with A < 12 G. Orlandini
2
Interactions, Currents and the Structure of Few-Nucleon Systems R. Schiavilla
10
Quantum Chaos and Nuclear Structure V. Zelevinsky
20
Theories and Applications beyond Mean Field with Effective Forces R. R. Rodriguez-Guzman, J. L. Egido and L. M. Robledo
28
Relativistic Theory of Pairing in Infinite Nuclear Matter M. Serra, A. Rummel and P. Ring
34
Realistic Effective Interactions and Shell-Model Calculations for Medium- and Heavy-Mass Nuclei A. Gargano
38
Derivative Coupling Model Description of Nuclear Matter in the Dirac-Hartree-Fock Approximation P. Bernardos, R. Lombard, M. Lopez-Quelle, S. Marcos and R. Niembro
44
Generator Coordinate Method including Triaxial Angular Momentum Projection K. Tanabe, K. Enami and N. Yoshinaga
48
Effect of the Triaxial Angular Momentum Projection on the Potential Energy Surface K. Enami, K. Tanabe, N. Yoshinaga and K. Higashiyama
52
v
VI
Realistic Intrinsic State Densities for Deformed Nuclei E. Mainegra and R. Capote
56
Approximate Treatment of the Centre of Mass Correction for Light Nuclei M. Grypeos, C. Koutroulos, A. Shebeko and K. Ypsilantis
60
One-Body Density Matrix and Momentum Distribution in S-P and S-D Shell Nuclei C. C. Moustakidis and S. E. Massen
64
Correlation Induced Collapse of Systems with Skyrme Forces D. V. Fedorov and A. S. Jensen
68
Hadron Dynamics Hadron Dynamics: Present Status and Future Perspective T. Bressani Section I.
73
Strange Hadro-Dynamics
KNA and KNS Coupling Constants M. T. Jeong and I. T. Cheon
86
On The E Hypernucleus E. Satoh and M. Kimura
90
Non Mesonic Weak Decay of Hypernuclei A. Parreno
96
Study of Mesonic and Non-Mesonic Decay of A-Hypernuclei at DA3>NE L. Venturelli for the FINUDA Collaboration
100
Hypernuclear 7-Spectroscopy: Recent Results with Hyperball H. Tamura for the KEK E419, BNL E930 Collaborations
106
vii
On the Coalescence Production of Broad Resonances V. M. Kolybasov
110
Energetic Level Scheme of the Stable S = —2 Dihyperon P. Z. Aslanian and B. A. Shahbazian
114
Section II.
Mesons, Baryons and Antibaryons
Pionic Excitations of Nuclear Systems W. Weise
119
Status of Exotic Meson Searches M. Villa
127
A Study of the TT-TT Interaction in Nuclear Matter Using the 7r+ + A -+ n+ + IT* + A' Reaction P. Camerini for the CHA OS Collaboration
134
Pion-Pion Potentials by Inversion of Phase Shifts at Fixed Energy B. Bdthory, Z. Barman and B. Apagyi
140
Perspectives of the Antideuteron Physics at JHF F. Iazzi, J. Doornbos, T. Bressani and D. Calvo
146
Study of the TT+TT+ System in the Antineutron-Proton into Three Charged Pions Annihilation Reaction A. Filippi for the OBELIX Collaboration
150
Observation of an Anomalous Trend of the Antineutron-Proton Total Cross Section in the Low-Momentum Region A. Feliciello
154
A Study of the n Annihilation on Nuclei E. Botta for the OBELIX Collaboration
158
-np Scattering in the Coulomb-Nuclear Interference Region E. Fragiacomo for the CHAOS Collaboration
162
Vlli
Arguments against the Yukawa Concept of Nuclear Force at Intermediate- and Short-Ranges and the New Mechanism for NN Interaction V. I. Kukulin
166
Moving Triangle Singularities and Polarization of Fast Particles V. M. Kolybasov
170
Section III.
Hadron Structure and Electromagnetic Probes
Electron-Positron Pair Spectroscopy with HADES at GSI J. Friese for the HADES Collaboration
175
Precision Measurement of the Neutron Magnetic Form Factor from 3He(e, e') H. Gao for the E95-001 Collaboration
181
The Hypercentral Constituent Quark Model M. M. Giannini and E. Santopinto
187
Algebraic Model of Baryon Structure R. Bijker and A. Leviatan
193
Non-Perturbative vs Perturbative Nucleon Response to Electromagnetic Probes M. Traini
199
The LEGS Double Polarization Program M. Blecher for the LEGS Spin Collaboration
205
Hadrons in a Relativistic Many-Body Approach S. R. Cotanch and F. J. Llanes-Estrada
209
Light Meson Spectra and Strong Decays in a Chiral Quark Cluster Model L. A. Blanco, F. Fernandez and A. Valcarce
215
A Sketch of Two and Three Bodies H. W. Grief3hammer
219
ix
Realistic Study of the Nuclear Transparency and the Distorted Momentum Distributions in the Semi-Inclusive Process AHe{e, e'p)X H. Morita, C. Ciofi degli Atti and D. Treleani
224
Measurements of the Deuteron Elastic Structure Functions A(Q2) and B(Q2) at the Jefferson Laboratory M. Kuss for the Jefferson Lab. Hall A Collaboration
230
OZI Rule Violation in np Annihilations in Flight S. Marcello
236
Parity Violating Electron Scattering B. Mosconi and P. Ricci
240
Nuclear Astrophysics Nucleosynthesis in Supernovae and Neutron Star Mergers F.-K. Thielemann Section I.
246
Theoretical Aspects of Nuclear Astrophysics
Strange Hadronic Stellar Matter within the Brueckner-Bethe-Goldstone Theory M. Baldo, G. F. Burgio and H. -J. Schulze
257
Bubble Nuclei, Neutron Stars and Quantum Billiards A. Bulgac and P. Magierski
261
Microscopic Models for Nuclear Astrophysics P. Descouvemont
267
Towards a Hartree-Fock Mass Formula J. M. Pearson, M. Onsi, S. Goriely, F. Tondeur and M. Farine
273
Nuclear Aspects of Nucleosynthesis in Massive Stars T. Rauscher, R. D. Hoffman, A. Heger and S. E. Woosley
277
X
Weak Interaction Rates of Neutron-Rich Nuclei and the R-Process Nucleosynthesis /. N. Borzov and S. Goriely
283
Systematics of Low-Lying Level Densities and Radiative Widths A. V. Ignatyuk
287
Cooling of Neutron Stars Revisited: Application of Low Energy Theorems A. E. L. Dieperink, E. N. E van Dalen, A. Korchin and R. Timmermans
293
The Role of Electron Screening Deformations in Solar Nuclear Fusion Reactions and the Solar Neutrino Puzzle T. E. Liolios
299
Nuclear Masses and Halflives: Statistical Modeling with Neural Nets E. Mavrommatis, S. Athanassopoulos, A. Dakos, K. A. Gernoth and J. W. Clark
303
Quasi-Thermal Photon Bath from Bremsstahlung P. Mohr, M. Babilon, J. Enders, T. Hartmann, C. Hutter, K. Vogt, S. Volz and A. Zilges
308
Nuclear Structure Near the Neutron Drip-Line and R-Process Calculations W. B. Walters, K.-L. Kratz and B. Pfeiffer
312
Analysis of the Neutrino Propagation in Neutron Stars in the Framework of Relativistic Nuclear Models R. Niembro, S. Marcos, P. Bernardos and M. Lopez-Quelle
315
Hyperonic Crystallization in Hadronic Matter M. A. Perez-Garcia, J. Diaz-Alonso, L. Mornas and J. P. Sudrez
319
Radioactive Witnesses of the Last Events of Nucleosynthesis in the Neighbourhood of the Nascent Solar System V. P. Chechev Section II.
323
Experimental Aspects of Nuclear Astrophysics
Bound State Beta-Decay and its Astrophysical Relevance P. Kienle
328
Searching for Signals from the Dark Universe R. Bernabei, P. Belli, R. Cerulli, F. Montecchia, M. Amato, G. Ignesti, A. Incicchitti, D. Prosperi, C. J. Dai, H. L. He, H. H. Kuang and J. M. Ma
338
Experimental Studies Related to s- and r- Process Abundances K. Wisshak, F. Voss and F. Kappeler
346
Experimental Study of the Electron Screening Effect in the d( 3 He,p) 4 He Fusion Reaction S. Zavatarelli for the L UNA Collaboration
350
The Solar Neutrino Problem: Low Energy Measurements of the 7 Be(p,7) 8 B Cross Section F. Hammache, G. Bogaert, A. Coc, M. Jacotin, J. Kiener, A. Lefebvre, V. Tatischeff, J. P. Thibaud, P. Aguer, J. F. Chemin, G. Claverie, J. N. Scheurer, E. Virassamynaiken, L. Brilliard, M. Hussonois, C. Le Naour, S. Barhoumi, S. Ouichaoui and C. Angulo
354
Determination of the Astrophysical S'-Factors Sn and S\s from 7 Be(d,n) 8 B and 8 B(d,n) 9 C Cross-Sections D. Beaumel, S. Fortier, H. Laurent, J.-M. Maison, S. Pita, T. Kubo, T. Teranishi, H. Sakurai, T. Nakamura, N. Aoi, N. Fukuda, M. Hirai, N. Imai, H. Iwasaki, H. Kumagai, S. M. Lukyanov, K. Yoneda, M. Ishihara, T. Motobayashi and H. Ohnuma
360
A Measurement of the 13 C(a,o:) Differential Cross-Section and its Application on the 1 3 C(a,n) Reaction M. Heil, A. Couture, J. Daly, R. Detwiler, J. Gorres, G. Hale, F. Kdppeler, R. Reifarth, U. Giessen, E. Stech, P. Tischhauser, C. Ugalde and M. Wiescher
364
Neutron Cross Sections Measurements for Light Elements at ORELA and their Application in Nuclear Criticality and Astrophysics K. H. Guber, L. C. Leal, R. O. Sayer, R. R. Spencer, P. E. Koehler, T. E. Valentine, H. Derrien, J. Andrzejewski, Y. M. Gledenov and J. A. Harvey
368
The Stellar Neutron Capture of 2 0 8 Pb H. Beer, W. Rochow, P. Mutti, F. Corvi, K.-L. Kratz and B. Pfeiffer
372
The r-Process as the Mirror Image of the s-Process: How Does It Work? R. Gallino, M. Busso, F. Kdppeler and G. J. Wasserburg
376
The Neutron Capture Cross Section of 1 4 7 Pm at Stellar Energies C. Arlandini, M. Heil, R. Reifarth, F. Kdppeler and P. V. Sedyshev
382
Applications of Nuclear Physics Section I.
Fission, Spallation and Transmutation
The ENEA ADS Project G. Gherardi for the ENEA ADS Project
387
Heat Deposit Calculation in Spallation Unit F. I. Karmanov, A. A. Travleev, L. N. Latysheva and M. Vecchi
393
xiii
Nuclide Composition of Pb-Bi Heat Transfer Irradiated in 80MW Sub-Critical Reactor A. Y. Konobeyev and M. Vecchi
397
Radiological Aspects of Heavy Metal Liquid Targets for Accelerator-Driven System as Intense Neutron Sources E. V. Gai, A. V. Ignatyuk, V. P. Lunev and Yu. N. Shubin
401
Intermediate-Energy Nuclear Data for Radioactive Ion Beams and Accelerator-Driven Systems M. V. Ricciardi, P. Armbruster, T. Enqvist, F. Rejmund, K.-H. Schmidt, J. Taieb, J. Benlliure, E. Casarejos, M. Bernas, B. Mustapha, L. Tassan-Got, A. Boudard, R. Legrain, S. Leray, C. Stephan, C. Volant, W. Wlazlo, S. Czajkowski, J. P. Dufour and M. Pravikoff
407
Actinide Nucleon-Induced Fission Reactions up to 150 MeV V. M. Maslov and A. Hasegawa
413
The AUSTRON Spallation Source Project G. Badurek, E. Jericha, H. Weber and E. Griesmayer
418
Section II.
Other Applications of Nuclear Physics
Recent Model Developments for Nucleon Induced Reactions up to 200 MeV E. Bauge, J. P. Delaroche, M. Girod, S. Hilaire, J. Libert, B. Morillon and P. Romain
425
Multistep Description of Nucleon Production Spectra in Nucleon-Induced Reactions at Intermediate Energy E. Ramstrom, H. Lenske and H. H. Wolter
431
Hadron Cancer Therapy: Role of Nuclear Reactions M. B. Chadwick
437
XIV
Accelerator-Based Sources of Epithermal Neutrons for BNCT E. Bisceglie, P. Colangelo, N. Colonna, V. Variale and P. Santorelli
443
Study of the Light Ion Beam Fragmentation in Thick Tissue-Like Matters Using Tissue-Like Track Detector S. P. Tretyakova, A. N. Golovchenko, R. Ilic and J. Skvarc
447
Anisotropy Functions for Palladium Model 200 Interstitial Brachyterapy Source R. Capote, E. Mainegra and E. Lopez
451
Production of Radiopharmaceuticals Based on the 199 T7 and 2uAt for Myocardium Diagnostic and Cancer Therapy O. V. Fotina, D. 0. Eremenko, V. O. Kordyukevich, S. Yu. Platonov, E. I. Sirotinin, A. V. Tultaev and O. A. Yuminov
455
Horizontal Compilations of Nuclear Data Z. N. Soroko, S. I. Sukhoruchkin and D. S. Sukhoruchkin
459
Tuning Effect in Nuclear Data S. I. Sukhoruchkin
463
List of Participants
468
Author Index
495
Many Body Methods in Nuclear Structure
N E W M I C R O S C O P I C A P P R O A C H E S TO T H E P H Y S I C S OF N U C L E I W I T H A < 12
Dipartimento
G. O R L A N D I N I di Fisica, Universita di Trento, 1-38050 Povo (Trento) and I.N.F.N. Gruppo Collegato di Trento E-mail:
[email protected]
Italy
The important progresses achieved in recent years in describing nuclei with A < 12 within microscopic theories are reviewed. In particular both results for bound states and for continuum states are presented. It is inferred that, because of these progresses, few-body physics is playing an increasingly important role in modern nuclear physics. The microscopic knowledge of light systems represents in fact the necessary bridge between our understanding of nuclear structure and QCD, the fundamental theory of strong interaction.
1
Introduction
Few-body systems are playing an increasingly i m p o r t a n t role in modern nuclear physics to the extent t h a t ab initio calculations performed within different approaches are able to produce very accurate results. T h e possibility of comparing these results with experimental d a t a unambiguously gives valuable information about the properties of the forces governing nuclear dynamics, challenges to a more fundamental comprehension of their origin and sheds some light on the many-body mechanisms generating typical properties of heavier systems. In this talk I intend to review briefly the recent progresses m a d e in treating both static and dynamical properties of few-nucleon systems.
2
Few-body bound states
Various methods are used to calculate binding energies and low lying spectra of light nuclei. They can be grouped as follows: l)Montecarlo (stochastic) methods. T h e Green Function Montecarlo Method ( G F M C ) 1 is based on the use of the evolution propagator in imaginary time and is designed to calculate expectation values of the Hamiltonian and other operators. Combined with the Variational Montecarlo ( V M C ) l technique G F M C is able to give very accurate results for nuclei with A = 3 and 4 and has been extended up to systems with A = 8. T h e accuracy of this m e t h o d is governed by the "statistical" errors. T h e Stochastic Variational Method (SVM) 2 is based on expansions of the variational wave functions on correlated gaussians basis and subsequent stochastic selection of the most important components. 2
3
Because of the "statistical sampling" also the SVM is able to treat a large number of variables and therefore to treat systems with A > 4.
Figure 1: VMC, GFMC and experimental energies of nuclear states for A < 4 < 8. From Ref. 3 .
2)The Faddeev-Yakubosky (FY)4 or equivalently the Alt-Grassberger-Sandhas5 (AGS) methods. They are based on solutions of coupled integral equations. The A-body wave function is obtained. Up to now A is limited to 3 and 4. They are naturally formulated in momentum space, but also treated in configuration space 6 . The accuracy of these methods is driven by the numerical errors in the solutions of the integral equations. $)The no core shell model (NCSM) method7. The Hamiltonian is diagonalized on the harmonic oscillator basis. Effective interaction operators calculated within either the Bloch-Horowitz 8 or the Lee-Suzuki unitary transformation methods 9 are used. The accuracy is driven by the size of the model space and by the number of particles in the subsystems where the effective interaction is built. This method has been applied to systems up to A=12. 4)The Hyperspherical Harmonics (HH) methods. They are based on expansions of the wave functions over the HH basis. They are formulated in configuration
4
space. The accuracy is driven by the number of basis functions one is able to treat and therefore by the rate of convergence. Two methods exist to improve the convergence rate. One of them, known as the Correlated Hyperspherical Harmonics method (CHH) 10 introduces correlation functions in the HH basis. The other very recent one, which will be called the Effective Interaction Hyperspherical Harmonics (EIHH) method 1 1 , makes use of the effective interaction operator built with the unitary transformation method 9 . While the CHH approach has been used only for A=3,4 the EIHH has extended calculations to A=5,6 nuclei. In Fig.l VMC and GFMC results for the energies of the low-lying nuclear states of nuclei with A = 4 -r- 8 3 are reported together with experimental values. In Fig.2 the spectrum of positive parity states of 12 C obtained within the NCSM approach 12 is shown. From the quality of the agreement between theory and experiment, and considering that these are ab initio calculations with realistic NN interactions (in some cases even with inclusion of three-body forces) one can probably conclude that we are close to understanding the microscopic origin of those spectra. If on the one hand the remaining disagreement challenges us to get a better control on the few- (many-)body techniques, especially for the heavier systems, on the other hand it focuses our interest on the off shell properties of the NN potential and/or on the origin and role of the many-body forces. Of course the issue of a better and better control on the
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5
few (many) - body techniques is an important one, before definite statements can be made about the origin of the dynamics. In this respect the continuous improvements of computational facilities will continue to have a major impact. But also the progresses in finding new algorithms and new ideas how to generalize or combine different methods are going to be very important. In this respect I would like to mention an example of how one can make important progresses by combining ideas originated in the many body field with a technique typical of few-body physics. I refer to the results obtained in Ref. n . As it was said in the previous subsection the limits of the HH expansion approach lies in the rate of convergence to the exact eigenvalue as more and more HH functions are considered. The problems of convergence may become serious especially when one has to do with strong core potentials like the NN potential. In fact this generates high momentum components in the wave function which then require HH functions of high order to be described. The CHH approach tries to incorporate those high momentum components in the basis functions by modifying them in a "physically" sensible, though rather arbitrary, way. This is done by multiplying the basis functions by a product of "correlation" functions. An alternative way (well known in the many body field as well as
Figure 3: Binding energies and r.m.s. radii of nuclei with A = 4 4- 6 systems as a function of the hyperangular quantum number K. For A = 4 (right) EIHH results (full thick line) are compared to NCSM results and to bare interaction results (full light line). From Ref. x l .
6
in field theory) of incorporating in the model space effects coming from the neglected space is the "effective operator" approach. In this framework one is able to build systematically the " effective interaction" to use in the Schrodinger equation instead of the bare interaction. The effective interaction generates in the wave functions large parts of those effects which would otherwise be left in the neglected basis states. The NCSM method 7 makes large use of this concept within the harmonic oscillator basis. The EIHH method n instead uses it within the HH expansion. The convergence is improved considerably and at the same time some of the well known drawbacks of the h.o. basis are cured. In Fig. 3 (left) the rate of convergence of ground state energy and radius of the A=4 system are plotted in function of the hyperspherical quantum number K and compared to NCSM results. One notices the striking improvement in the convergence rate due to the use of the effective interaction. The comparison with NCSM results puts in evidence the additional advantage of the HH formalism in that it does not require any additional parameter (like the h.o. frequency) affecting the rate of convergence. In Fig. 3 (right) rate of convergences of energies and radii of A=5,6 systems are also shown. From these results one can conclude that the EIHH is a very promising alternative method to study bound state properties of light systems and allows the HH formalism to be applied beyond A=4.
3
Few-body problem in the continuum
Finding exact solutions of the Schrodinger equation in the continuum is a very difficult task, even if the number of degrees of freedom is small. The difficulty lies in the definition and treatment of the boundary conditions. While solutions are easily obtained in the two-body case the problem alread becomes very involved going to A=3. It is clear that when A > 2 an increasing number of break up channels opens up, each of them requiring different boundary conditions. At present only few groups are able to comply with this difficult task. This is done within two different approaches: in one case the continuum Faddeev-Yakubosky integral equations are solved 4 ' 5 ' 6 ' 13 , in the other the HH expansion method is coupled to the use of the complex form of the Kohn variational principle 14 . A third unconventional, but very powerful approach, the Lorentz Integral Transform Method 17 , which is able to reduce the problem of calculating transition matrix elements to continuum states into bound state problems, will be the topics of the next section. In Figs. 4 and 5 results for n-d and p-d scattering cross sections are shown. One notices the remarkable agreement between theory and experiment for differential cross sections at different energies. This might imply that the
7
three-body problem is very well understood within a non relativistic framework with realistic potentials. However, the properties and origin of these potentials is just the i m p o r t a n t issue opened by this comparison. In fact, as it is shown in Fig. 4, it is the inclusion of three-body forces t h a t brings theory t o agree with data. O n the other hand the vector analyzing powers of b o t h p-d and
E„
90
120
150
160
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60
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120
150
180
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Figure 4: n-d differential cross sections at two different neutron energies. Long dashed curves: results with only Vjyjv- Solid curve: Vjvw + VNNN results. From Ref.13.
90 135 180
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45 90 135 180
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Figure 5: p-d (solid) and n-d (dashed) differential cross sections, nucleon analyzing power, deuteron analyzing power and tensor analyzingpowers at two different energies. From Ref. l i .
n-d scattering are not reproduced by the same two and three-body potentials. This fact brings u p the issue of our understanding of the three-body force. It is clear t h a t it is i m p o r t a n t t o find more observables which are sensitive t o it and can serve as test ground for the construction of this force in the same way as NN scattering d a t a have been determining the two-body force.
4
Calculation of inclusive reactions and the role of N N N interaction
In this section it will be shown that total photonuclear cross sections are sensitive to the NNN interaction and can serve to study its properties. From what has been said in the previous section one could infer that such inclusive observables are tremendously difficult to calculate. In fact they require the knowledge of continuum states in all break up channels. It turns out, however, that this is not the case if one uses the LITM. The LITM can be listed among the most t\
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powerful methods to treat problems involving the knowledge of continuum states since it reduces the problem of calculating transition matrix elements to continuum states into bound state problems. It can be formulated and tested both for inclusive as well as for exclusive reactions. In Ref. 16 the inclusive version of the method has been used to calculate the total photodisintegration cross section of three body systems. The results are shown in Fig.6. One can notice the important role that NNN interaction plays in lowering the resonance peak towards experimental results. This effect is very promising in view of explaining the origin of the strong disagreement between theory and experiment in 4 H e ? . Even more interesting effects of the three body force are found in the higher energy tail of the cross section 16 . Unfortunately this energy range
9
is poorly known experimentally. We hope that these results will create some interest for modern experimental investigations of such observables having as principal aim the better knowledge of three body forces. Acknowledgment s An important part of the results shown here have been obtained in fruitful collaborations with N. Barnea, V.D. Efros, W. Leidemann and E.L. Tomusiak. References 1. J. Carlson and R. Schiavilla, Rev. Mod. Phys.70 ,743 (1998) and references therein. 2. V.I. Kukulin and V.M. Krasnapol'sky, /. Phys. G3, 795 (1977); K. Varga and Y. Suzuki, Phys. Rev. C 52, 2885 (1995). 3. R. B. Wiringaet al., Phys. Rev. C 62, 014001 (2000). 4. A. Nogga, H. Kamadaand W. Gloeckle, Phys. Rev. Lett. 85, 944 (2000) and references therein. 5. W. Sandhas et al.,iVttc/. Phys. A 631, 210c (1998) and references therein; A.C. Fonseca, Phys. Rev. Lett. 83, 4021 (1999). 6. F. Ciesielski and J. Carbonell, Phys. Rev. C 58, 58 (1998). 7. P. Navratil, G. P. Kamuntavicius, B. R. Barrett, Phys. Rev. C 61, 044001 (2000) and references therein; W. C. Haxton and C.-L. Song, Phys. Rev. Lett 84, 5454 (2000). 8. C. Bloch and J. Horowitz, Nucl. Phys.8, 91 (1958). 9. K. Suzuki and S.Y. Lee, Progr. Theor. Phys. 64,2091, (1980). 10. Yu.I Fenin and V. Efros, Sov. J. Nucl Phys. 15, 497 (1972); A. Kievsky, M. Viviani and S. Rosati,iVue/. Phys. A 577, 511 (1994). 11. N. Barnea, W. Leidemann and G. Orlandini, Phys. Rev. C 61, 054001 (2000). 12. P. Navratil, J. P. Vary and B. R. Barrett, Phys. Rev. Lett. 84, 5728 (2000). 13. H. Witalaet al. Phys. Rev. Lett. 81, 1183 (1998). 14. A. Kievsky, S. Rosati, M. Viviani, Phys. Rev. Lett. 82, 3759 (1999). 15. V.D Efros, W. Leidemann and G. Orlandini, Phys. Lett. B 338, 130 (1994). 16. V.D Efros et al., Phys. Lett. B 484, 223 (2000). 17. V.D Efros, W. Leidemann and G. Orlandini, Phys. Rev. Lett. 78, 4015 (1997).
INTERACTIONS, CURRENTS, A N D THE STRUCTURE OF FEW-NUCLEON SYSTEMS
R. S C H I A V I L L A Jefferson
Lab, Newport and Old Dominion University,
News, Norfolk,
VA
23606 VA
23529
Our current understanding of the structure of nuclei with A < 8, including energy spectra, electromagnetic form factors, and weak transitions, is reviewed within the context of a realistic approach t o nuclear dynamics based on two- and threenucleon interactions and associated electro-weak currents. Low-energy radiative and weak capture reactions of astrophysical relevance involving these light systems are also discussed.
1
Introduction
Few-nucleon systems provide a unique opportunity for testing the simple, traditional picture of the nucleus as a system of point-like nucleons interacting among themselves via effective many-body potentials, and with external electro-weak probes via effective many-body currents. Through advances in computational techniques and facilities, the last few years have witnessed dramatic progress in numerically exact studies of the structure and dynamics of systems with mass number A < 8, including energy spectra of low-lying states, momentum distributions and cluster amplitudes, elastic and inelastic electromagnetic form factors, /3-decays, radiative and weak capture reactions at low energies, inclusive response to hadronic and electro-weak probes at intermediate energies. In the present talk, I will review the "nuclear standard model" outlined above, and present the extent to which it is successful in predicting some of the nuclear properties alluded to earlier. Of course, given the limited time, some of the theoretical and experimental developments will be treated cursorily. Nevertheless, I still hope to be able to convey a broad view of the intriguing and important studies in few-nucleon physics today.
2
Potentials and Energy Spectra
The Hamiltonian in the nuclear standard model is written as
10
11
i
i<j
i<j
where the kinetic energy operator Ki has charge-independent and chargesymmetry-breaking components due to the difference in proton and neutron masses, and Vy and V%jk are two- and three-nucleon potentials. The two-nucleon potential consists of a long range part due to pion exchange, and a short-range part parameterized either in terms of heavy meson exchanges as, for example, in the Bonn potential : , or via suitable operators and strength functions, as in the Argonne v\% (AV18) potential 2 . The shortrange terms in these potentials are then constrained to fit pp and np scattering data up to energies of ~ 350 MeV in the laboratory, and the deuteron binding energy. The modern models mentioned above provide fits to the Nijmegen data-base 3 characterized by \ 2 P e r datum very close to one, and should therefore be viewed as phase-equivalent. The AV18 model is most widely used; it has the form
= E ""MOy.
(2)
p=l,18
where the first fourteen operators are isoscalar, Of^ 1 " 14 = [l,
• S ) y ,L2,L2<7i • a^ (L • S) 2 ,] ® [1,7* • r,-] , (3)
while the last four isospin-symmetry-breaking operators have isovector and isotensor character, °%
= Tij > °i • a3Tij i SijTij , (n + Tj)z .
(4)
Here Sy is the tensor operator, and Ty is defined as Ty = 3TiZTjZ — TJ • Tj. Unique among the modern potentials, the AV18 includes a fairly complete treatment of the electromagnetic interaction, since it retains, in addition to the leading Coulomb term, also contributions from magnetic moment interactions, vacuum polarization and two-photon exchange corrections. These terms, while typically very small (for example, in the deuteron the magnetic dipole-dipole interaction gives 18 keV extra repulsion 2 ) , need to be taken into account when very accurate predictions are required, as in the case, for
12
example, of studies of energy differences of isomultiplet states 4 , or the cross section for proton weak capture on proton at keV energies 5 . It is now well established that two-nucleon potentials alone underbind nuclei 4 : for example, the AV18 and Bonn models give 6 , in numerically exact calculations, binding energies of 24.28 MeV and 26.26 MeV respectively, which should be compared to the experimental value of 28.3 MeV. Moreover, 6 Li and 7 Li are unstable against breakup into ad and at clusters, respectively, and that energy differences are not, in general, well predicted, when only two-nucleon potentials are retained in the Hamiltonian. Important components of the three-nucleon potential arise from the internal structure of the nucleon. Since all degrees of freedom other than the nucleon have been integrated out, the presence of virtual A resonances, for example, induces three-nucleon potentials. They are written as Vijk=V&
+ Vgk,
(5)
2lr
where V is the "long-range" term, resulting from the intermediate excitation of a A with pion exchanges involving the other two nucleons, known as the Fujita-Miyazawa term 7 . This term is present in all models, such as the Tucson-Melbourne potential 8 or the series of Urbana models 9 . The Urbana models parameterize VR as VR
= AR
£
TZ(rij)TZ(rjk)
,
(6)
cyclic ijk
where T^{r) is the strength function of the pion-exchange tensor interaction. This term is meant to simulate the dispersive effects that are required when integrating out A degrees of freedom. The strengths of the Fujita-Miyazawa and dispersive terms are then determined, in the Urbana models, by fitting the triton binding energy and the saturation density of nuclear matter. The Hamiltonian consisting of the AV18 two-nucleon and Urbana-IX three-nucleon potentials (AV18/UIX) predicts reasonably well the low-lying energy spectra of systems with A < 8 nucleons in "exact" Green's function Monte Carlo calculations 4 . The experimental binding energies of the a particle is exactly reproduced, while those of the .4=6-8 systems are underpredicted by a few percent. This under binding becomes (relatively) more and more severe as the neutron-proton asymmetry increases. An additional failure of this Hamiltonian model is the underprediction of spin-orbit splittings in the excitation spectra of these light systems. These failures have in fact led to the development of new three-nucleon interaction models 10 . These newly developed models, denoted as Illinois models, incorporate the Fujita-Miyazawa and
13
dispersive terms discussed above, but include in addition multipion exchange terms involving excitation of one or two A's, so-called pion-ring diagrams, as well as the terms arising from S-wave pion rescattering, required by chiral symmetry. 3
The Nuclear Electromagnetic Current
The nuclear current operator consists of one- and many-body terms that operate on the nucleon degrees of freedom:
j(q) = Ei*a)(q) + E & W + E © i ) > i
i<j
(y)
i<j
where q is the momentum transfer, and the one-body operator j ^ has the standard expression in terms of single-nucleon convection and magnetization currents. The two-body current operator has "model-independent" and "model-dependent"components (for a review, see Ref. n). The modelindependent terms are obtained from the charge-independent part of the AV18, and by construction satisfy current conservation with this interaction. The leading operator is the isovector "7r-like" current obtained from the isospin-dependent spin-spin and tensor interactions. The latter also generate an isovector "p-like "current, while additional model-independent isoscalar and isovector currents arise from the central and momentum-dependent interactions. These currents are short-ranged and numerically far less important than the 7r-like current. Finally, models for three-body currents have been derived in Ref. 12 , however the associated contributions have been found to be very small in studies of the magnetic structure of the trinucleons 12 . The model-dependent currents are purely transverse and therefore cannot be directly linked to the underlying two-nucleon interaction. Among them, those associated with the A-isobar are the most important ones in the momentum-transfer regime being discussed here. These currents are treated within the transition-correlation-operator (TCO) scheme 12>13; a scaled-down approach to a full N+A coupled-channel treatment. In the TCO scheme, the A degrees of freedom are explicitly included in the nuclear wave functions by writing
* N+A
:
* , i<j
(8)
14
where * is the purely nucleonic component, 5 is the symmetrizer and the transition correlations UjjR are short-range operators, that convert NN pairs into TV A and A A pairs. In the results reported here, the * is taken from CHH solutions of the AV18/UIX Hamiltonian with nucleons only interactions, while the U?R is obtained from two-body bound and low-energy scattering state solutions of the full N-A coupled-channel problem. Both 7-ATA and 7AA Mi couplings are considered with their values, (J,^NA — 3 n.m. and ^ 7 A A = 4.35 n.m., obtained from data 13 . 4
The pd Radiative Capture
There are now available many high-quality data, including differential cross sections, vector and tensor analyzing powers, and photon polarization coefficients, on the pd radiative capture at c m . energies ranging from 0 to 2 MeV 14>15.16,!7 These data indicate that the reaction proceeds predominantly through S- and P-wave capture. The aim here is to verify the extent to which they can be described satisfactorily by a calculation based on a realistic Hamiltonian (the AV18/UIX model) and a current operator constructed consistently with the two- and three-nucleon interactions 18 . The predicted angular distributions of the differential cross section a (6), vector and tensor analyzing powers Ay(0) and T2o(0), and photon linear polarization coefficient P 7 (0) are compared with the TUNL data below 50 keV from Refs. 14,16 in Fig 1. The agreement between the full theory, including many-body current contributions, and experiment is generally good. However, a closer inspection of the figure reveals the presence of significative discrepancies between theory and experiment in the small angle behavior of <x(0) and 18 T2Q(0), as well as in the S-factor below 40 keV . The S-wave capture proceeds mostly through the Mi transitions connecting the doublet and quartet pd states to 3 He-the associated reduced matrix elements (RMEs) are denoted by m 2 and m 4 , respectively. The situation for P-wave capture is more complex, although at energies below 50 keV it is dominated by the E\ transitions from the doublet and quartet pd states having channel spin 5 = 1 / 2 , whose RMEs I denote as pi and p±. The E\ transitions involving the channel spin 5 = 3/2 states, while smaller, do play an important role in T2O(0). The TUNL 16 and Wisconsin 17 groups have determined the leading Mi and Ei RMEs via fits to the measured observables. The results of this fitting procedure are compared with the calculated RMEs in Table 1. The phase of each RME is simply related to the elastic pd phase shift 17 , which at these low energies is essentially the Coulomb phase shift. As can be seen from Table 1, the most significant differences between theoretical and experimental RMEs
15
d+p Capture 0.30
/
x E p = 8 0 - 0 keV
-f I 1 1 J /
/\
y
30
60
90 120 150 180
O
\ - 0.20 \
0.15®
V
^
0.10
i7-*-
0
~ 0.25
0.05 50
100
150
1.20 1.00 0.80 0.60 S_ 0.40 0.20
0
50
100
150
O
50
100
150
O.OO
Figure 1. T h e energy integrated cross section a(0)/ao (47rao is the total cross section), vector analyzing power Ay(0), tensor analyzing power T2o(0) and photon linear polarization coefficient P-y(0) obtained with the AV18/UDC Hamiltonian model and one-body only (dashed line) or both one- and many-body (solid line) currents are compared with the experimental results of Ref. 1 4 .
are found for |p4|. The theoretical over prediction of p$ is the cause of the discrepancies mentioned above in the low-energy (< 50 keV) S-factor and small angle cr(6). It is interesting to analyze the ratio TEI = \PA/P2\2- Theory gives TEI — 1, while from the fit it results that TEX * 0.74 ± 0.04. It is important to stress that the calculation of these RMEs is not influenced by uncertainties in the two-body currents, since their values are entirely given by the long-wavelength form of the E\ operator (Siegert's theorem), which has no spin-dependence (for a thorough discussion of the validity of the long-wavelength approximation in E\ transitions, particularly suppressed ones, see Ref. 1 8 ). It is therefore of interest to examine more closely the origin of the above discrepancy. If the interactions between the p and d clusters are switched off, the relation TE\ — 1 then simply follows from angular momentum algebra. Deviations of this ratio from one are therefore to be ascribed to differences induced by the interactions in the 5 = 1 / 2 doublet and quartet wave functions. The AV18/UIX interactions in these channels do not change the ratio above significantly. It should be emphasized that the studies carried out up until now ignore, in the continuum states, the effects arising from electromagnetic interactions beyond the static
16
Coulomb interaction between protons. It is not clear whether the inclusion of these long-range interactions, in particular their spin-orbit component, could explain the splitting between the p-2 and p^ RMEs observed at very low energy. This discrepancy seems to disappear at 2 MeV 18 . Table 1. Magnitudes of the leading M i and E\ RMEs for pd capture at Ep = 40 keV.
RME |ra 2 | |m 4 |
H M
IA 0.172 0.174 0.346 0.343
FULL 0.322 0.157 0.371 0.378
FIT 0.340±0.010 0.157±0.007 0.363±0.014 0.312±0.009
Finally, the doublet m.2 RME is underpredicted by theory at the 5 % level. On the other hand, the cross section for nd capture at thermal neutron energy is calculated to be 578 fib with the AV18/UIX model, which is 15 % larger than the experimental value (508±15) /xb 19 . Of course, Mi transitions, particularly doublet ones, are significantly influenced by many-body current contributions. Indeed, an analysis of the isoscalar (/xs) and isovector (nv) magnetic moments of the trinucleons 12 suggests that the present model for the isoscalar two-body currents, constructed from the AV18 spin-orbit and quadratic-momentum dependent interactions, tends to overestimate fis by about 5 %. The experimental value for fiy, however, is almost perfectly reproduced. It appears that the present model for two-body currents needs to be improved. 5
The Nuclear Weak Current and the p 3 H e Weak Capture
The nuclear weak current and charge operators consist of vector and axialvector parts, with corresponding one- and many-body components. The weak vector current and charge are constructed from the corresponding (isovector) electromagnetic terms, in accordance with the conserved-vectorcurrent hypothesis, and thus have 2 0 "model-independent" and "modeldependent" components. The former are determined by the interactions, the latter include the transverse currents associated with A excitation. The leading many-body terms in the axial current, in contrast to the case of the weak vector (or electromagnetic) current, are those due to A excitation, which are treated within the TCO scheme, discussed above. The axial charge operator includes the long-range pion-exchange term 2 1 , required by low-energy theorems and the partially-conserved-axial-current relation, as
17
well as the (expected) leading short-range terms constructed from the central and spin-orbit components of the nucleon-nucleon interaction 22 . The largest model dependence is in the weak axial current. The NA axial coupling constant g*A is not well known. In the quark-model, it is related to the axial coupling constant of the nucleon by the relations gA = ( 6 V 2 / 5 ) P A This value has often been used in the literature in the calculation of A-induced axial current contributions to weak transitions. However, given the uncertainties inherent to quark-model predictions, a more reliable estimate for gA is obtained by determining its value phenomenologically. It is well established by now 5 that one-body axial currents lead to a ~ 4 % underprediction of the measured Gamow-Teller matrix element in tritium /?-decay. This small 4 % discrepancy can then be used to determine gA 20 . While this procedure is inherently model dependent, its actual model dependence is in fact very weak, as has been shown in Ref. 5 . The calculated values for the astrophysical S-factor in the energy range 0-10 keV are listed in Table 2 20 . Inspection of the table shows that: (i) the energy dependence is rather weak, the value at 10 keV is only about 4 % larger than that at 0 keV; (ii) the P-wave capture states are found to be important, contributing about 40 % of the calculated 5-factor. However, the contributions from D-wave channels are expected to be very small, as explicitly verified in 3 Di capture, (iii) The many-body axial currents associated with A excitation play a crucial role in the (dominant) 3 Si capture, where they reduce the 5-factor by more than a factor of four; thus the destructive interference between the one- and many-body current contributions, obtained in Ref. 13 , is confirmed in the study of Ref. 2 0 , based on more accurate wave functions. The (suppressed) one-body contribution comes mostly from transitions involving the D-state components of the 3 He and 4 He wave functions, while the manybody contributions are predominantly due to transitions connecting the Sstate in 3 He to the D-state in 4 He, or viceversa.
Table 2. The hep S-factor, in units of 1 0 - 2 0 keV b, calculated with CHH wave functions corresponding to the AV18/UIX Hamiltonian model, at p 3 H e c m . energies E=0, 5, and 10 keV. The rows labelled "one-body" and "full" list the contributions obtained by retaining the one-body only and both one- and many-body terms in the nuclear weak current. The contributions due the 3 S i channel only and all S- and P-wave channels are listed separately.
E=0 keV Si S+P 26.4 29.0 6.38 9.64 3
one-body full
E=5 keV Si S+P 25.9 28.7 6.20 9.70
3
E=10 keV S! S+P 26.2 29.3 6.36 10.1
3
18
The chief conclusion of Ref. 20 is that the hep S- factor is predicted to be ~ 4.5 times larger than the value adopted in the standard solar model (SSM) 2 3 . This enhancement, while very significant, is smaller than that first suggested in Ref. 24 . Even though this result is inherently model dependent, it is unlikely that the model dependence is large enough to accommodate a drastic increase in the value obtained here. Indeed, calculations using Hamiltonians based on the AV18 two-nucleon interaction only and the older AV14/UVIII two- and three-nucleon interactions 25 predict zero energy 5-factor values of 12.1 x 1(T 20 keV b and 10.2 x 10" 2 0 keV b, respectively. It should be stressed, however, that the AVI 8 model, in contrast to the AV14/UVIII, does not reproduce the experimental binding energies and low-energy scattering parameters of the three- and four-nucleon systems. The AV14/UVIII prediction is only 6 % larger than the AV18/UIX zero-energy result. This 6 % variation should provide a fairly realistic estimate of the theoretical uncertainty due to the model dependence. The precise calculation of the S-factor and the consequent absolute prediction for the hep neutrino flux should allow much greater discrimination among proposed solar neutrino oscillation solutions 20 . 6
Conclusions and Acknowledgments
Improvements in the modeling of two- and three-nucleon interactions and nuclear electro-weak currents, and the significant progress made in the last few years in the description of bound and continuum wave functions, make it now possible to perform first-principle calculations of interesting nuclear properties of light nuclei. Experimentally known electromagnetic and weak transitions of systems in the mass range 2 < A < 8 provide powerful constraints on models of nuclear currents. I wish to thank J. Carlson, A. Kievsky, L.E. Marcucci, V.R. Pandharipande, S.C. Pieper, D.O. Riska, S. Rosati, M. Viviani, and R.B. Wiringa for their many important contributions to the work reported here. This work was supported by DOE contract DE-AC05-84ER40150 under which the Southeastern Universities Research Association (SURA) operates the Thomas Jefferson National Accelerator Facility. References 1. R. Machleidt, F. Sammarruca, and Y. Song, Phys. Rev. C 53, R1483 (1996); R. Machleidt, nucl-th/0006014. 2. R.B. Wiringa, V.G.J. Stoks, and R. Schiavilla, Phys. Rev. C 5 1 , 38 (1995).
19
3. J.R. Bergervoet et al, Phys. Rev. C 41, 1435 (1990); V.G.J. Stoks, R.A.M. Klomp, M.C.M. Rentmeester, and J.J. de Swart, Phys. Rev. C 48, 792 (1993). 4. R.B. Wiringa, S.C. Pieper, J. Carlson, and V.R. Pandharipande, Phys. Rev. C 62, 014001 (2000). 5. R. Schiavilla et al, Phys. Rev. C 58, 1263 (1998). 6. A. Nogga, H. Kamada, and W. Glockle, Phys. Rev. Lett. 85, 944 (2000). 7. J. Fujita and H. Miyazawa, Prog. Theor. Phys. 17, 360 (1957). 8. S.A. Coon et al, Nucl. Phys. A317, 242 (1979). 9. B.S. Pudliner, V.R. Pandharipande, J. Carlson, and R.B. Wiringa, Phys. Rev. Lett. 74, 4396 (1995). 10. J. Carlson, V.R. Pandharipande, S.C. Pieper, and R.B. Wiringa, private communication. 11. J. Carlson and R. Schiavilla, Rev. Mod. Phys. 70, 743 (1998). 12. L.E. Marcucci, D.O. Riska, and R. Schiavilla, Phys. Rev. C 58, 3069 (1998). 13. R. Schiavilla, R.B. Wiringa, V.R. Pandharipande, and J. Carlson, Phys. Rev. C 45, 2628 (1992). 14. G.J. Schmid et al, Phys. Rev. Lett. 76, 3088 (1996). 15. L. Ma et al, Phys. Rev. C 55, 588 (1997). 16. E.A. Wulf et al, Phys. Rev. C 61, 021601(R) (1999). 17. M.K. Smith and L.D. Knutson, Phys. Rev. Lett. 82, 4591 (1999). 18. M. Viviani, A. Kievsky, L.E. Marcucci, S. Rosati, and R. Schiavilla, Phys. Rev. C 61, 064001 (2000). 19. E.T. Jurney, P.J. Bendt, and J.C. Browne, Phys. Rev. C 25, 2810 (1982). 20. L.E. Marcucci, R. Schiavilla, M. Viviani, A. Kievsky, and S. Rosati, Phys. Rev. Lett. 84, 5959 (2000); L.E. Marcucci, R. Schiavilla, M. Viviani, A. Kievsky, S. Rosati, and J.F. Beacom, nucl-th/0006005, Phys. Rev. C. in press. 21. K. Kubodera, J. Delorme, and M. Rho, Phys. Rev. Lett. 40, 755 (1978). 22. M. Kirchbach, D.O. Riska, and K. Tsushima, Nucl. Phys. A542, 616 (1992). 23. J.N. Bahcall, S. Basu, and M.H. Pinsonneault, Phys. Lett. B 433, 1 (1998). 24. J.N. Bahcall and P.I. Krastev, Phys. Lett. B 436, 243 (1998). 25. R.B. Wiringa, R.A. Smith, and T.L. Ainsworth, Phys. Rev. C 29, 1207 (1984); R.B. Wiringa, Phys. Rev. C 43, 1585 (1991).
Q U A N T U M CHAOS A N D N U C L E A R S T R U C T U R E Vladimir Z E L E V I N S K Y Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University, East Lansing, 48824-1321
USA
After nuclear physics has successfully served as a testing ground for many-body quantum chaos, the inverse process of penetration of ideas of quantum chaos into studies of nuclear structure is now opening new perspective avenues. Two examples, interplay of chaos and thermalization and geometrical chaoticity, show how the two areas enrich each other and advance our understanding of underlying physics.
1
Chaos and thermalization
The presence of quantum chaos is revealed by the local spectral statistics (level spacing distribution and spectral rigidity). In a description of a stable finite many-body system, we start typically from the mean field (MF) which determines the symmetry of the ground state and elementary excitations. The spectrum of eigenstates \k) of independent quasiparticles becomes very dense due to the combinatorial growth of a number of possibilities for distributing available energy between quasiparticles. Each state \k) is a specific configuration of quasiparticles occupying certain MF orbitals; the level statistics are at this stage Poissonian. However, the residual interactions between the quasiparticles, being effectively strong because of the dense spectrum, induce multiple avoided level crossings and convert "simple" configurations \k) into very complicated eigenstates \a) which are superpositions of N ^> 1 basis states. The web of energy terms drawn as a function of the interaction strength is the first qualitative manifestation of quantum chaos. Quantitatively, the spectral statistics very fast go to the limit of random matrix theory (RMT) with the Wigner spacing distribution and high spectral rigidity 1,2 . The structure of eigenfunctions continues to evolve even after that being much more sensitive to the strength of the residual interaction 3 ' 4 . The emerging complexity can be measured by the number N of significant basis components, or by information entropy
S? = -5>£ln<,
(1)
k
where w% = |C^| 2 are weights of simple states \k) in the wave function of the eigenstate |a). At the realistic interaction strength in complex nuclei 3 , Sf is a 20
21
smooth function of energy Ea since the eigenstates in a narrow energy window "look the same" (have the same degree of complexity 5 ). [Still, the magnitude of Sa does not reach the RMT limit of ln(0.48JV) even in the middle of the spectrum; this would require a stronger interaction inconsistent with the MF since both are derived from the same original hamiltonian.] Information entropy (1) depends on the choice of the reference basis |A;). One can argue 6 that the MF basis is the optimal one because it concentrates the most regular features of the system separating them from the random fluctuations and correlations measured by information entropy. In the randomly taken reference basis, 5f is close to the RMT limit for nearly all eigenstates revealing no spectral evolution 3 while in the MF basis Sf behaves as a thermodynamic variable being a weakly fluctuating function of excitation energy. This brings us to the hot topic of thermodynamics for a closed system. Standard statistical mechanics use the microcanonical ensemble with equal probabilities of all microstates in a small energy interval AE. The level density p(E) in this interval determines thermal entropy Sth(E), and the temperature scale, T = (dSth/dE)-1. A small subsystem of a large system can be described with the aid of the canonical ensemble where the surrounding plays the role of a heat bath, and temperature of the subsystem is that of the heat bath. For a small system, as an excited nucleus, there is no heat bath, but the microcanonical approach is still possible. Certainly, it is impossible to equiprobably populate all allowed microstates. The validity of the statistical approach is based on the fact that the observable macroscopic quantities are insensitive to the precise details of the population. But this can be the case only if available microstates are indeed very similar, "look the same" for a macroscopic observer. The mechanism for such unanimity is provided by the strong mixing of neighboring states, i.e. by quantum chaos. Even if the initial state has almost exactly defined energy, as in the case of an isolated neutron resonance in a heavy nucleus, the macroscopic measurements pick specific basis components of the wave function which are present in another resonance function nearby with a similar amplitude. [The exceptional case is the observation of parity nonconservation in polarized neutron scattering 7 which may depend on the proximity of another resonance of opposite parity.] In a more frequent situation, the initial state is a wave packet of many stationary components with random phases which irregularly depend on the history of the state. In a closed system, the amplitudes of the components are constant while the phase dynamics lead to decoherence without any external agent. Then the system is described by the density matrix which, as was shown explicitly for special cases 8 ' 9 of weakly interacting particles, leads to the standard Fermior Bose-distributions.
22
In a strongly interacting self-sustaining system, as a nucleus, the MF forms a regular skeleton covered by the incoherent interactions. In a sense, the mediators of those interactions (mesons, photons, ...) play the role of the thermostate 10,3 ' 11 . The von Neumann entropy S = — Tr(p]np) calculated with the actual density matrix pap = CaCP* is a quantity averaged over all such incoherent interactions. The same result for p can be achieved by the averaging over few adjacent eigenstates; since the trace is basis-independent, we can take the MF basis where the amplitudes Cg are close to the GOE and uncorrelated so that the averaging over few states selects the diagonal elements of p. Then we come to information entropy (1). Indeed, large-scale shell model calcualtions invariably show that thermal entropy (which is in fact von Neumann entropy for an equilbrium thermal ensemble) is very close to information entropy in the MF basis 3 . This result bridges the gap between the local behavior (RMT) and spectral evolution, i.e. between quantum chaos and equilibrium statistical mechanics: chaos becomes a driving force for equilibration. Moreover, since the MF basis corresponds to independent quasiparticles, and the fraction w% shows the smoothly changing degree of participation of the quasiparticle mode |fc) in an eigenstate |a), the third entropy, namely that of the quasiparticle gas, S
F = - £ [ < L n < + (1 - < ) l n ( l - < ) ] ,
(2)
a
carries the same physical information. Here n " are occupation numbers of quasiparticle orbitals \a) in the many-body state \a). The equivalence of three entropies, S, Sf and SF, which was shown in the analysis of the shell model wave functions 3 , takes place in the self-consistent case only when the interaction strength is adjusted to the MF. For artificially weak interactions, Sf does not work because there is no considerable mixing of quasiparticle configurations, whereas in the case of too strong interactions, both thermometers using individual eigenstates, Sf and Sf, are useless because they are unable to resolve the spectral evolution if practically all states "look the same". In the self-consistent regime, the equivalence of three entropies shows that the Fermi-liquid theory is actually valid beyond its traditional area of low excitation and long lifetime of quasiparticles. Even at a small particle number, the quasiparticle distribution in nuclei and complex atoms is close to the FermiDirac one with temperature determined by a normal thermodynamic scale 3 ' 11 . The quasiparticle thermometer does not work correctly near the ground state where the correlations destroy the sharp Fermi-surface 6 . The plausible scenario of the transition to the Fermi-Dirac distribution in a strongly interacting system was developed n with the use of the general properties of the strength
23 functions of simple states and level d e n s i t y 1 2 ' . T h e concept of entropy is not unique. A family of entropies was studied and applied t o emphasize different aspects of uncertainty in physics 1 4 ' 1 5 . A p a r t of entropies discussed above, Kolmogorov-Sinai entropy of classical chaos, and entropy related t o q u a n t u m measurements, we suggested t o study entropy associated with the response of a complex q u a n t u m system to a r a n d o m noise 15 . This can be defined with the aid of the density matrix p£, — C^C"* for a given energy t e r m Ea where the components of the wave function are averaged over an interval of values of a r a n d o m parameter in the hamiltonian. T h e corresponding noise entropy S% is invariant and reflects the correlation properties of the system. In application to shell-model systems, 5 " shows the complexity of the eigenfunction in t e r m s of the exciton classes mixed by the perturbation. Using the so-called pointer basis (eigenstates of p) we can bring t h e eigenvalues of p t o a thermal form ~ exp(-^H) with an effective hamiltonian H and an effective noise t e m p e r a t u r e 1//3. This leads t o a distinction between apparent and real t e m p e r a t u r e of radiation in the multiple production processes 1 5 and to new signatures of phase transitions in finite s y s t e m s 1 6 . It is interesting to mention t h a t the detail analysis of the pairing correlations in exact shell model wave functions shows a non-standard behavior close to a second order macroscopic phase transiton with a long tail of the order p a r a m e t e r above critical temperature 3'17.
2
Geometrical chaoticity
Above we ignored the fact t h a t finite self-sustaining systems are rotationally invariant and therefore the t o t a l spin is an exact integral of motion even in the case of chaotic intrinsic dynamics. Hilbert space of the system is decomposed into classes labeled by the q u a n t u m numbers J, M of the t o t a l spin. Since all classes are governed by the same hamiltonian, we expect the presence of interclass correlations 1 8 . [Another source of inevitable correlations is connected 1 9 with the rank of the interaction (usually two-body)]. A possible manifestation of such correlations would b e a similarity of chaotic mixing in adjacent classes. In this case one could expect the appearance of predicted by Mottelson "compound", or "ergodic", b a n d s where the energy and angular m o m e n t u m flows in the deexcitation process proceed along specific chains of compound states, without significant branching. This idea is supported by shell model calculations of nuclear spectra and transition probabilities as well as the behavior of pairing in different J classes 1 7 . A similar effect is possible in pairing rotation where the "band" consists of ground states of neighboring nuclei. Recently a new aspect of the problem of correlations between classes of a
24
hamiltonian system became a subject of a theoretical discussion 20 . It turns out that the ground state of a rotationally invariant shell-model-like system with a random hamiltonian prefers to have the ground state spin Jo equal to zero although the states J = 0 comprise a small fraction of the total space. A detail study shows 21 ' 22 that the probability to have the ground state spin at its maximum possible value, Jo = Jmax, is also enhanced against statistical expectations. The situation reminds the Hund rule in atomic physics although the "magnetic ordering" in the case of random interactions looks enigmatic. Such an ordering is present in the interacting boson model as well 2 3 . Until recently, no plausible explanation of this effect was provided; possible candidates included pairing, Bose-condensation of fermion pairs, and time-reversal invariance of J = 0 states but neither works, in particular for J = JmaxWe suggested a semi-quantitative theory based on the idea of geometric chaoticity 3 ' 4 , 1 8 . In a shell-model system, the total spin is a vector sum of individual particle angular momenta, J = Ylia- The majority of possible vector coupling schemes leading to the same value of J are essentially random. We can mimic this chaoticity by a random walk in the quantum number of projection M. This is a well known approach to the level density p(E, J ) of a Fermi-gas. Assuming that random interactions do not create well pronounced coherent phenomena, the only direction which can be singled out to characterize a mean field is that of the total spin. Consider the state M = J. The most probable distribution rijm of particles over individual orbitals \jm) is dictated in this limit by pure statistical arguments of the entropy maximum under constraints of the conservation laws. The expectation value of the hamiltonian for such a distribution can be presented as a series E0{J) = h0 + h232 + hi3i + ...
(3)
The coefficients h2k depend on the parameters of random interactions. Averaging over this ensemble we obtain the mean value EQ{J) which is an approximation to the actual average yrast-line. If the expansion is valid, and the coefficients h2k converge relatively fast, the region of the parameters where both h2 and /14 are positive, corresponds to the zero ground state spin Jo (antiferromagnetism). If they both are negative and the higher order terms are negligible up to the J = J m 0 x, the system is ferromagnetic. Being more specific, we have considered the simplest model of a single j level where the most general two-body hamiltonian is determined by the elastic scattering matrix elements VL, L = 0, 2,..., 2j — 1, according to H=
£ L(even)A
VLPIAPLA,
(4)
25
10
15
20
Figure 1: The probabilities of the ground state spins Jo = 0, left, and Jo = Jmax, right, for a system of N = 4 fermions on different j-Ievels; ensemble results (solid lines) and statistical theory (dotted lines; the line for Jmax indicates an upper theoretical limit).
where PLA are annihilation operators for the pair of spin L. The amplitudes VL are random quantities taking equiprobably positive and negative values; their exact distribution functions, as Gaussian or uniform on the range [-1,1], are of minor importance and the results are qualitatively similar. The sensitivity to the assumed L-dependence of the variances V£ is also weak. For the trial distribution function fully determined by the entropy consideration, exp(7rn — /x) + 1'
(5)
where the cranking frequency 7 and the chemical potential fi are defined by
£<
N,
E<
M,
(6)
the ground state energy EQ is linear in VL- Therefore, at h^ -C ^2, we immediately predict the probability around 50% to have JQ = 0. The more accurate predictions for the uniform distribution function of all VL are shown in Fig. 1 in comparison with the ensemble data for Jo = 0 and Jo = Jmax- Of course, this "zero order" statistical approach explains only the general trend but not the fine irregular details. The actual distribution n m behaves similarly to the Fermi-Dirac function (5) but reveals an oscillatory component related to the specific m-splitting of the single-particle levels in the effective mean field. (Alternatively, the dynamics can be considered with the aid of the boson expansion techniques). The dynamics are responsible for an approximately constant shift of actual ground state energy at Jo = 0 from the pure statistical result, Fig. 2; for JQ = Jmax the ground state wave function is
26
10 5
a$T-
r
0
I
-10
-15
/
"
; E 0 =h 0 ^.46
-15
15-15
N,0 =
h
0
< H >
-5
5
NJ
Figure 2: Energy of the ground states for Jo = 0, left, and Jo = Jmax, right, vs statistical theory; dots correspond to simulations for different sets of random interaction parameters. The straight line on the left panel shows a constant shift.
unique and the dynamical mixing plays no role. The random character of the wave function for Jo = 0 is confirmed by the direct analysis of the distribution of its components in the seniority basis. The fully paired component of seniority zero is distributed according to the RMT predictions. We come to the conclusion that elements of the ordered spectra of finite Fermi-systems are in fact stipulated by the rotational invariance and the presence of many competing quasi-random paths of angular momentum coupling. The geometrical chaoticity should be included as an important ingredient in microscopic nuclear models of collective phenomena. In particular, it can serve as a criterion for the selection of the coherent contributions to large amplitude collective motion. Another interesting area is the study of quantum chaos in a rotationally invariant mesoscopic system. One can envisage new applications to nuclei, atoms, atomic aggregates and traps. Acknowledgments The author is indebted to B.A. Brown, P. Cejnar, M. Horoi, D. Mulhall, V. Sokolov and A. Volya for fruitful collaboration. The work was supported by the NSF grants 96-05207 and 0070911.
27
References 1. T.A. Brody et. ah, Rev. Mod. Phys. 53, 385 (1981). 2. T. Guhr, A. Miiller-Groeling and H.A. Weidenmuller, Phys. Rep. 299, 189 (1998). 3. V. Zelevinsky et ai., Phys. Rep. 276, 85 (1996). 4. V. Zelevinsky, Ann. Rev. Nucl. Part. Sci. 46, 237 (1996). 5. I.C. Percival, J. Phys. B6, L229 (1973). 6. V.G. Zelevinsky, Nucl. Phys. A555, 109 (1993). 7. O.P. Sushkov and V.V. Flambaum, Sov. Phys. Usp. 25, 1 (1982). 8. L. Van Hove, Physica 2 1 , 517 (1955); 23, 441 (1957); 25, 268 (1959). 9. M. Srednicki, Phys. Rev. E 50, 888 (1994). 10. M. Horoi, V. Zelevinsky and B.A. Brown, Phys. Rev. Lett. 74, 231 (1995). 11. V.V. Flambaum and F.M. Izrailev, Phys. Rev. E55, R13 (1997); 56, 5144 (1997). 12. B. Lauritzen et ai., Phys. Rev. Lett. 74, 5190 (1995). 13. N. Frazier, B.A. Brown and V. Zelevinsky, Phys. Rev. C54, 1665 (1996). 14. M. Ohya and D. Petz, Quantum Entropy and Its Use (Springer, Berlin, 1993). 15. V.V. Sokolov, B.A. Brown, and V. Zelevinsky, Phys. Rev. E58, 56 (1998). 16. P. Cejnar, V. Zelevinsky and V. Sokolov, to be published. 17. M. Horoi and V. Zelevinsky, BAPS 44, No. 1, 397 (1999). 18. V. Zelevinsky, in Recent Progress in Many-Body Theories, Eds. D. Neilson and R.F. Bishop (World Scientific, Singapore, 1998) p. 225. 19. V.V. Flambaum, in Parity and Time Reversal Violation in Compound Nuclear States and Related Topics, eds. N. Auerbach and J.D. Bowman (World Scientific, Singapore, 1996) p. 41. 20. C.W. Johnson, G.F. Bertsch, and D.J. Dean, Phys. Rev. Lett. 80, 2749 (1998); C.W. Johnson et ai., Phys. Rev. C 6 1 , 014311 (2000). 21. M. Horoi et ai. BAPS 44 No. 5, 45 (1999). 22. D. Mulhall, A. Volya and V. Zelevinsky, to be published. 23. R. Bijker, A. Frank and S. Pittel, Phys. Rev. C 60, 021302 (1999).
THEORIES A N D A P P L I C A T I O N S B E Y O N D M E A N FIELD W I T H EFFECTIVE FORCES R.R. RODRIGUEZ-GUZMAN, J.L. EGIDO AND L.M. ROBLEDO Departamento de Fisica Tedrica C-XI, Universidad Autonoma de Madrid, 28049-Madrid, Spam. Techniques beyond the mean field are used to describe the properties of the lowlying states of the light nuclei 30<32'3iMg. The theoretical framework is the angular momentum projected Generator Coordinate Method using the quadrupole moment as collective coordinate and the Gogny force as the effective interaction.
1
Introduction
In nuclear physics the mean field approximation is always the first step to understand the properties of the ground and lowest-lying excited states. The mean field approximation provides the concept of magic numbers as well as the concept of spontaneous symmetry breaking. For nuclei with proton and/or neutron numbers close to the magic ones one expect symmetry conserving (i.e. non superconducting and spherical) ground states and the elementary excitations are therefore of vibrational character. On the other hand, for nuclei away from the magic configurations one expect strong symmetry breaking and the appearance of deformed ground states that generate bands (as the rotational bands). The experimental studies of light nuclei away from the stability line N = Z seem to imply that for those (usually neutron rich) nuclei some of the properties associated to magic numbers are not preserved. The most striking example is the experimental evidence towards the existence of quadrupole deformed ground states for the neutron-rich nuclei around the magic number ./V = 20. In addition, the extra binding energy coming from deformation can help to extend thereby the neutron drip line in this region far beyond what could be expected from spherical ground states. Among the variety of available experimental data, the most convincing evidence for a deformed ground state is found in the 32Mg nucleus where both the excitation energy of the lowest lying 2 + state * and the B(E2,0+ -» 2 + ) transition probability 2 have been measured. Both quantities are fairly compatible with the expectations for a rotational state. At the mean field level, the ground state of 32Mg is spherical. However, when the zero point rotational energy correction (ZPRE) is considered, the energy landscape as a function of the quadrupole moment changes dramatically and 32Mg becomes deformed 3,4,5,6,7,8,9
28
29 A more careful analysis of the energy landscape with the ZPRE correction included reveals that, in fact, there are two coexistent configurations (prolate and oblate) with comparable energy indicating thereby that configuration mixing of states with different quadrupole intrinsic deformation has to be considered. Here, we report on the results obtained 10 for the nuclei 30-34 M g in an angular momentum projected (AMP) configuration mixing calculation using the constraint on the quadrupole moment as the device to generate the configurations to be mixed. We have used in the calculations the Gogny force n (with the D1S parameterization 12 ) which is known to provide reasonable results for many nuclear properties like ground state deformations, moments of inertia, fission barrier parameters, etc, all over the periodic table. 2
Theoretical
framework
The angular momentum projected Generator Coordinate Method (AMPGCM) with the mass quadrupole moment as generating coordinate is used as the theoretical description. As we restrict ourselves to axially symmetric configurations, we use the following ansatz for the K = 0 wave functions of the system
H)=
J dq2OfUl20)P^ M
(1)
In this expression |
(2)
In the equation above we have introduced the projected norm AfI(q2o, q'2o) = (vfeo)! -Poo \f(l;2o))' an< ^ t n e projected hamiltonian kernel TiI(q20,q2o) = (yfeo)\HPQQ \tp(q2o))- As the generating states P^ \
30
states |A;7) = (nl)-1/2 f dq2ou{(q2o)Poo I'Pfeo)) which are defined in terms of the eigenstates u{(q2o) and eigenvalues n\. of the projected norm, i.e. f dq!20AfI(q2o,q2o)u{(cl2o) = nkul(lw)The correlated wave functions | $ 7 ) are written in terms of the natural states as \$i) = J ^
Discussion of the results
In figure 1 the collective wave functions squared |
31
-1.0 0
1.0 2.0 -1.0 0
1.0 2.0 -1.0 0
b
b
920 ( )
920 ( )
1.0 2.0
920 ( b )
Figure 1. The collective amplitudes |g^(
32 Table 1. Calculated and experimental results for excitation energies and B(E2, 0J~i —>• 2 j 2 ) transition probabilities. The columns marked a, b and c correspond to 0^" — 2^", 0+ — 0+ and 2^" — 2+ respectively. In the experimental data columns values marked with an (*) correspond to Monte Carlo Shell Model results taken from Ref. 1 6 . The experimental data for the excitation energies have been taken from : for the 32Mg nucleus and from 1 9 for 30 Mg. The B(E2) transition probability has been taken from 2 .
30
Mg Mg 3i Mg 32
Energies (MeV) b c a 2.15 2.30 1.60 1.46 1.77 3.35 1.02 2.35 3.31
Exp. a 1.482 0.885 0.75(*)
B(E2)e2fm4 b c a 229 3 218 395 3.4 199 525 0 290
Exp. a 300(*) 454±78 580(*)
In table 1 the energy splittings between different states and the E2 transition probabilities among them are compared with the available experimental data. Concerning the B(E2,0f —> 2+) transition probabilities we find a very good agreement with the only known experimental value and with the theoretical predictions of Utsumo et al. 16 using the Monte Carlo Shell Model (MCSM). The 2f excitation energies rather nicely follow the isotopic trend but they are larger than the experimental values by a factor of roughly 1.5. This discrepancy is the result 10 of using angular momentum projection after variation (PAV) instead of the more complete projection before variation (PBV). Usually, the PBV method yields to rotational bands with moments of inertia larger than the PAV ones 17>18. In Ref 10 we estimated the effect of considering PBV on the present results. The conclusion was that the moment of inertia gets enhanced by a factor 1.4 and therefore the excitation energies have to be quenched by a factor 0.7. This quenching factor brings the theoretical excitation energies in much closer agreement with experiment. 4
Conclusions
In conclusion, we have performed angular momentum projected Generator Coordinate Method calculations with the Gogny interaction D1S and the mass quadrupole moment as generating coordinate in order to describe rotational like states in the nuclei 30Mg, 32Mg and 34Mg. We obtain a very well deformed ground state in 34Mg, a fairly deformed ground state in 32Mg and a spherical ground state in 30Mg. In the three nuclei, states with spins higher or equal I — 4h are deformed. The intraband B(E2) transition probabilities agree well with the available experimental data and results from shell model like calculations. The 2 + excitation energies follow the isotopic trend
33
but come out a factor 1.5 too high as compared with the experiment. We attribute the discrepancy to the well known deficiency of Projection After Variation calculations of providing small moments of inertia. Acknowledgments One of us (R. R.-G.) kindly acknowledges the financial support received from the Spanish Instituto de Cooperacion Iberoamericana (ICI). This work has been supported in part by the DGICyT (Spain) under project PB97/0023. References 1. D. Guillemaud-Miiller et al. Nucl. Phys. A426, 37 (1984). 2. T. Motobayashi et al. Phys. Lett. B346, 9 (1995). 3. X. Campi, H. Flocard, A.K. Kerman and S. Koonin, Nucl. Phys. A251 (1975) 193. 4. M. Barranco and R.J. Lombard, Phys. Lett. B78 (1978) 542. 5. R. Bengtsson, P. Moller, J.R. Nix and J. Zhang, Phys. Scr. 29 (1984) 402. 6. J. F. Berger et al., Inst. Phys. Conf. Ser. 132 (1993) 487. 7. P.-G. Reinhard et al., Phys. Rev. C60 (1999) 014316. 8. P.-H. Heenen, P. Bonche, S. Cwiok, W. Nazarewicz and A. Valor, nuclth/9908083. 9. R. Rodriguez-Guzman, J.L. Egido and L.M. Robledo, Phys. Lett. B 474 (2000) 15. 10. R. Rodriguez-Guzman, J.L. Egido and L.M. Robledo, arXiv.nuclth/0001020 (to appear in Phys. Rev. C). 11. J. Decharge and D. Gogny, Phys. Rev. C21 (1980) 1568. 12. J.F. Berger, M. Girod and D. Gogny, Nucl. Phys. A428 (1984) 23c. 13. L. M. Robledo, Phys. Rev. C50 (1994) 2874; J.L. Egido, L.M. Robledo and Y. Sun, Nucl. Phys. A560 (1993) 253. 14. K. Hara and Y. Sun, Int. J. Mod. Phys. E4 (1995) 637. 15. P. Ring and P. Schuck, The Nuclear Many Body Problem (Springer, Berlin, 1980). 16. Y. Utsumo, T. Otsuka, T. Mizusaki and M. Honma, Phys. Rev. C60 (1999) 054315. 17. K. Hara, A. Hayashi and P. Ring, Nucl. Phys. A385 (1982) 14. 18. K.W. Schmid and F. Griimmer, Rep. Prog. Phys. 50 (1987) 731. 19. P.M.Endt, Nucl. Phys. A521, (1990) 1.
RELATIVISTIC T H E O R Y OF P A I R I N G I N I N F I N I T E NUCLEAR MATTER M. SERRA, A. RUMMEL, AND P. RING Physik-Department der TU Miinchen, 85748 Garching,
Germany
The 1So pairing gap at the Fermi surface for symmetric nuclear matter at zero temperature is calculated in the framework of a fully consistent relativistic model. The relativistic Bonn potential is used to investigate the pairing properties and the resulting gap is compared with a non-relativistic calculation based on a phenomenological density dependent force. Good agreement between the relativistic and the non-relativistic solution of the gap equation is found.
Although it is well known for decades that pairing correlations are essential for astrophysical 2 and nuclear structure 1 calculations, a number of questions is still open on the subject. According to the fact that in principle the effective force in the pairing channel should be the particle-particle Kmatrbc, we present a solution of the relativistic gap equation using as force in the pp-channel a relativistic version of the Bonn potential 3 , a realistic bare nucleon-nucleon interaction. As shown in Ref.6, the relativistic equation for the pairing gap reduces to the non-relativistic BCS-equation, therefore it reads oo
A(fc)
A(p) = - - ^ / vpp{p, k)
k2dk
(1)
in which e(k) is now the eigenvalue of the Dirac hamiltonian h, and A the chemical potential. As we investigate pairing in infinite nuclear matter, we may consider, up to a good approximation, only the a and the w fields in the p/i-channel. This leads to the following expressions for e(k) and A
e(k) = V + \ A 2 + M*2 2
A = V + ^Jk
(2) 2
F
+ M*
(3)
where the vector field and the effective mass are given by V = guw and M* = M + g^a respectively. As already mentioned, as force in the pairing channel vpp(j), k) we use a relativistic version of the Bonn potential 3 , which originates mainly from the exchanges of the a- and u mesons 4 . The resulting gap at the Fermi surface AF is shown in Fig. 1 as function of the Fermi momentum fcf for the Bonn-B potential and compared with the corresponding quantity 34
35
0.0
0.5
L0
ISO
0.5
1.0
Figure 1. The gap parameter at the Fermi surface AF as a function of the density represented by the Fermi momentum kp for the relativistic Bonn-B potential and the Gogny force D l . Figure 2. Contributions of the different meson-exchange potentials to the gap parameter at the Fermi surface A p as a function of the density represented by the Fermi momentum HF- The thick full line corresponds to the total gap as shown in Fig.l, while the thin lines refer to the different meson exchange contributions. The total gap mainly results form the cancellation between the large positive term of the
obtained in a non-relativistic calculation 7 based on the Gogny force Dl 5 . We find excellent agreement between the two solutions up to a fcp of roughly one fifth of the nuclear matter density, i.e. ]ZF — 0.8 f m - 1 , where pairing correlations are maximal with A ^ w 2.8 MeV. This is in agreement with the usual statement that pairing is a surface phenomenon. At larger densities the relativistic solution drops to zero faster than the non-relativistic one and this is due to the fact that at large fc^ the repulsive contribution of u becomes stronger than the attractive contribution of a. Whether nuclear matter is superfluid at saturation, kF = 1-35 f m - 1 , is hard to decide as it seems to depend critically on the details of the interaction: for the Bonn potential there is no pairing at this density, whereas for the Gogny force a small gap of roughly 0.5 MeV is left. Fig.2 shows the contributions to the gap parameter
36
Mfm)
Figure 3. Coherence length as function of the Fermi momentum ICF for the relativistic Bonn-B potential and for the Gogny force.
at the Fermi surface of the different one-meson exchanges which define the Bonn potential. As for the potential, we notice that the gap Ap results from the difference between two large contributions: the large and positive contribution from the attractive -exchange. Nevertheless we point out that, although small, the contribution of the TT- and /9-meson exchanges in the pairing channel does not vanish. Another important quantity that we investigate for a better understanding of the pairing properties is the coherence length £ that, from a microscopic point of view, represents the squared mean distance of two paired particles on top of the Fermi surface. In terms of the Cooper pair wave functions it is defined as c2
_ /rf3r|x(r)|V _ £>dk#)px{k)ldk\* /d3r|X(r)|2
/"dfctaW*)!2
(4)
and for our calculation we have chosen the coordinate space representation
37
since it is more convenient on the numerical point of view. In Fig.12 we show the resulting f as a function of the Fermi momentum ftp- for the Bonn potential and for the Gogny force, plotted with a solid line and with a dashed line respectively. In both cases, we observe that in the interval 0.4 < fcjp(fm_1) < 0.9, £ has a minimum of the order of 5.0 — 6.0 MeV and it is an almost constant function of the Fermi momentum. This is in agreement with the fact that in this interval pairing correlations are maximal, namely we find Af > 1.5 MeV as it may be seen from Fig. 3.1. For low densities (kp < 0.25 fm _ 1 ) and high densities (kp > 1.0 fm _ 1 ) the strength of the coherence length increases rapidly, meaning that the two nucleons becomes more and more separated, i.e. do not form a Cooper pair. The difference of the coherence length obtained with the Bonn potential and the coherence length calculated with the Gogny force at larger densities agrees with the observation that the pairing gap drops faster for the relativistic interaction than for the non-relativistic force. In conclusion, we have shown that pairing properties of symmetric nuclear matter at the Fermi surface calculated using the relativistic Bonn potential are in good agreement with phenomenologically adjusted Gogny's results, which are supposed to describe correctly pairing correlations in finite nuclei. Although further terms of the if-matrix should be taken into account for a fully microscopic description of nuclear superfluidity, the fact that the application of a bare nucleon-nucleon interaction in the BCS-equation leads to very reasonable pairing correlations at the Fermi surface gives us confidence to believe that the renormalization effects of higher order terms in the AT-matrix can be neglected in the 1SQ pairing channel. To confirm this statement the effect of the polarization diagram into the pairing potential should be investigated. References 1. 2. 3. 4. 5. 6. 7.
A. Bohr, B.R. Mottelson, and D. Pines, Phys. Rev. 110 (1958) 936 T. Takatsuka, Prog. Theor. Phys. 48 1517 (1972) R. Machleidt, Adv. Nucl. Phys. 19 (1989) 189 M. Serra, A. Rummel, and P. Ring, Submitted for Publication. J. Decharge and D. Gogny, Phys. Phys. Rev. C21 (1980) 1568. H. Kucharek and P. Ring, Z.Phys A339 (1991) 23 H. Kucharek, P. Ring, P. Schuck, R. Bengtsson, and M. Girod, Phys. Lett. B 216, 249 (1989)
REALISTIC EFFECTIVE I N T E R A C T I O N S A N D SHELL-MODEL CALCULATIONS FOR M E D I U M - A N D HEAVY-MASS NUCLEI A. GARGANO Istituto Nazionale di Fisica Nucleare, Complesso Universitario di Monte S. Angela, Via Cintia, 1-80126 Napoli, Italy E-mail: [email protected] The aim of this paper is to evidence the merit of modern realistic effective interactions in nuclear structure calculations. To this end, we report on some achievements of a study we have performed in the last few years in the framework of the shell model by making use of effective interactions derived from the Bonn A free nucleon-nucleon potential. In particular, we present here results of our calculations for several medium- and heavy-mass nuclei having or lacking two nucleons with respect to double shell closures and show that a very good agreement with the experimental data is obtained for all the nuclei considered.
1
Introduction
A basic ingredient of nuclear shell-model calculations is the model-space effective interaction Veff. In the last decade the stage has been set for a newgeneration of realistic effective interactions following substantial improvements in both the development of high-quality nucleon-nucleon (AW) potential and the many-body methods which provide a microscopic derivation of V^fF starting from the bare potential. A review of recent developments in the field of NN potentials is given in Ref. 1, while the main apects of the modern derivation of V'eff are discussed in Ref. 2. In recent years we have performed a number of shell-model calculations 3 - u making use of these modern realistic Veff aimed at assessing their role in the description of nuclear structure properties. To this end, we have studied all the medium- and heavy-mass nuclei having or lacking few identical particles with respect to double shell closures by making use of effective interactions derived from the meson-theoretic Bonn A potential. 12 In all cases considered we have found a good overall agreement between theory and experiment. Thus we have come to the conclusion that realistic effective interactions provide a description of nuclear structure phenomena at least as accurate as that provided by traditional, empirical interactions. The aim of this paper is to illustrate our conclusion by presenting results of our work for some nuclei in different mass region. They are 98 Cd, 134 Te, and 2 1 0 Po. A brief description of our calculations is given in Sec. 2, while 38
39
Sec. 3 contains the comparison of our results with experimental data, Some conclusions are drawn in Sec. 4. 2
Outline of Calculations
Our effective interaction was derived from the Bonn A potential using a Gmatrix folded-diagram formalism, including renormalization from both core polarization and folded diagrams. The two-hole effective interaction for 98 Cd was derived by considering 100 Sn as inert core, while the doubly closed 132 Sn and 2 0 8 Pb were assumed as core in the calculation of the effective interaction for 134 Te and 2 1 0 Po, respectively.
Figure 1: Experimental and calculated spectrum of
98
Cd.
As regards the model space, we let the two proton holes in 98 Cd occupy the four sigle-hole orbits 0519/2, lpi/2, 1^3/2, a n d 0/ 5 / 2 , while for 134 Te we include the five orbits Ogr/2, ld5/2, 2si/2, ld3/2, and 0hu/2. For 2 1 0 Po the two valence particles are distributed in the 0/i 9 / 2 , I/7/2, 0ii3/ 2 , l / s / 2 , 2p 3 / 2 and 2pi/2 orbits. A brief description of the derivation of our effective interaction, including earlier references, can be found in Refs. 10 and 11. The single-particle energies required in the caculations of 134 Te and 2 1 0 Po have been taken from the observed spectra of 133 Sb and 209 Bi, respectively. All the experimental data in the present paper are taken from Ref. 13. As
40
regards the single-hole energies for 98 Cd, no information on the spectra of " i n is available. In Ref. 11 we describe in detail how they have been determined. 134Te 8" 6 5+
8' 6" 5" 7 5+
7" 9~
4
9"
2+
3
-
> •3
.
2+
-
5+
0+ 5+ 3+ 4+ 2+ 1+ 6+
: i+ -
2+
-
-
6+
-
6+ 4+
-
2+
-
0+
2
1
0 Expt.
Calc.
Figure 2: Experimental and calculated spectrum of
3
134
Te.
Results and Comparison with Experiment
The calculated spectra for 98 Cd, 134 Te, and 21D Po are compared with the experimental ones in Figs. 1 to 3. The theoretical results presented in this section have been obtained by using the OXBASH shell-model code 14 . The few excited states recentely identified in 98 Cd by means of in-beam spectroscopic experiments are well reproduced by the theory, only the calculated and experimental energies of the 2 + state differing by about 150 keV.
41
More complete spectra are presently available for 134 Te and 2 1 0 Po. In Fig. 2 the comparison between theory and experiment for 134 Te is made up to 5 MeV, however in the energy region above 3.2 MeV only for those observed states which have received a firm spin-parity assignment. 210
Po
m
/••4+ .•'.'• i o
_
.'.-•.•..()1
:•••••:•••"..
7 ~
•:'••-'..
4~
-••-.2 + ' - • • • • 3
'•'•::•
i l ~
"•.(31'
,--x
'S-1+ '••••:•-.
'
= =g wS
? +
•-•It
'••8 +
It
8+
St
4+ 2+
—
2+
0+
Expt.
Calc.
Figure 3: Experimental and calculated spectrum of
210
Po.
From Figs. 2 and 3 we see that, with very few exceptions, a one-to-one correspondence between experimental and theoretical spectra can be established for both nuclei. Concerning 134 Te, the theory predicts the existence of two unobserved states. They are the 3 + and the 0 + state at 2.65 and 2.78 MeV, respectively, whose existence, however, is strongly supported by the experimental information available for the two heavier N = 82 isotones. As for 210 Po, the 3~, 4~, and 5 - states at 2.4, 2.9, and 3.1 MeV, respectively, have no theoretical counterpart. The first one, however, reflects the collective nature of the 3~ state in 2 0 8 Pb, while the other two states originate from neutron particle-hole configurations. It should also be noted that in 2 1 0 Po two levels with no angular momentum and parity assignment have been observed at 2.66
42
and 2.87 MeV, respectively. As it was suggested in Ref. 15, we identify our calculated 2 J state at 2.95 MeV with the experimental one at 2.87 MeV. A measure of the quality of the results is given by the rms deviation a,16 whose values are 107, 106, and 87 keV for 98 Cd, 134 Te, and 2 1 0 Po, respectively. While the first value may not be very meaningful due to the very small number of observed levels, we note that the calculation of a for 134 Te and 2 1 0 Po includes 15 and 24 levels, respectively. Finally, in Table I the calculated ground-state binding energies, which are all relative to the closest doubly closed core, are compared with the experimental ones. The mass excess values for " i n , 133 Sb, and 209 Bi needed for absolute scaling of the single-hole or single-particle levels were taken from Ref. 13. For all the three nuclei we calculate the Coulomb contribution by diagonalizing the Coulomb force for a system of two protons in the appropriate model space. From Table I we see that a very good agreement with experiment is obtained, the calculated values falling within the error bars in all the three nuclei considered. Table 1: Experimental and calculated ground-state binding energies (MeV) for and 2 1 0 P o . Nucleus
98
Cd
134Te 210p0
4
98
Cd,
134
Sn,
Binding energy. Expt.
Calc.
- 3 . 9 8 ± 0.48 20.56 ± 0.04 8.78 ± 0 . 0 0
-4.55 20.57 8.79
Closing R e m a r k s
We have shown that effective interactions derived from the meson-theoretic Bonn A potential by means of the G matrix folded-diagram approach lead to a quite accurate description of nuclei with two valence particles or holes in the region of doubly magic 100 Sn, 132 Sn, and 2 0 8 Pb. At the present stage of our investigation, the main conclusion is that we may be confident in the reliability of realistic shell-model calculations, at least as regards the T = 1 matrix elements of the the effective interaction. A test of T = 0 interaction is of course equally important. To this end we have studied the doubly odd nucleus 1 3 2 Sb 8 and have recently started calculations for nuclei with protons and neutrons outside 2 0 8 Pb. I would like to close this paper by emphasizing that it is highly important now to carefully examine the results obtained so far to try to learn something about possible improvements in the derivation of the effective interaction.
43
Acknowledgments The results presented in these paper are part of a research project carried out in collaboration with L. Coraggio, A. Covello, N. Itaco, and T. T. S. Kuo. References 1. R. Machleidt, in Highlights of Modern Nuclear Structure, ed. A. Covello (World Scientific, Singapore, 1999), and references therein. 2. T.T.S. Kuo, in New Perspectives in Nuclear Structure, ed. A. Covello (World Scientific, Singapore, 1996), and references therein. 3. F. Andreozzi, L. Coraggio, A. Covello, A. Gargano, T.T.S Kuo, Z.B. Lee, and A. Porrino, Phys. Rev. C 54, 1636 (1996). 4. F. Andreozzi, L. Coraggio, A. Covello, A. Gargano, T.T.S Kuo, and A. Porrino, Phys. Rev. C 56, R16 (1997). 5. A. Covello, F. Andreozzi, L. Coraggio, A. Gargano, T.T.S Kuo, and A. Porrino, Prog. Part. Nucl. Phys. 38, 165 (1997). 6. A. Covello, L. Coraggio, and A. Gargano, Nuovo Cimento A 111, 803 (1998). 7. L. Coraggio, A. Covello, A. Gargano, N. Itaco, and T.T.S Kuo, Phys. Rev. C 58, 3346 (1998). 8. F. Andreozzi, L. Coraggio, A. Covello, A. Gargano, T.T.S Kuo, and A. Porrino, Phys. Rev. C 59, 746 (1999). 9. A. Covello, L. Coraggio, A. Gargano, N. Itaco, and T.T.S Kuo, in Nuclear Structure 98, ed. C. Baktash, AIP Conf. Proc. 481 (1999). 10. L. Coraggio, A. Covello, A. Gargano, N. Itaco, and T.T.S Kuo, Phys. Rev. C 60, 064306 (1999). 11. L. Coraggio, A. Covello, A. Gargano, N. Itaco, and T.T.S Kuo, J. Phys. G , in press. 12. R. Machleidt, Adv. Nucl. Phys. 19, 189 (1987). 13. National Nuclear Data Center, Brookhaven National Laboratory. 14. B A . Brown, A. Etchegoyen, and W.D.M. Rae, The computer code OXBASH, MSU-NSCL, report number 524. 15. L.G. Mann et al, Phys. Rev. C 38, 74 (1988). 16. We define a = {{1/Nd) £ j £ e * P ( i ) -E^ii)]2}1'2, where Nd is the number of data.
DERIVATIVE C O U P L I N G M O D E L D E S C R I P T I O N OF N U C L E A R MATTER IN THE DIRAC-HARTREE-FOCK APPROXIMATION
Departamento E.T.S.I.I.T.,
P. B E R N A R D O S de Matemdtica Aplicada y Ciencias de la Computacion, Universidad de Cantabria, E-39005 Santander, Spain E-mail: [email protected] R. L O M B A R D
Groupe de Physique Theorique , Institut de Physique F-91406 Orsay Cedex, France E-mail: [email protected]
Nucleaire,
M. L O P E Z - Q U E L L E Departamento
de Fisica Aplicada, Facultad de Ciencias, Universidad E-39005 Santander, Spain E-mail: [email protected]
Departamento
de Fisica Moderna, Facultad de Ciencias, Universidad E-39005 Santander, Spain E-mail: [email protected] E-mail: [email protected]
de
Cantabria,
de
Cantabria,
S. M A R C O S , R. N I E M B R O
Properties of nuclear matter are calculated in the Dirac-Hartree-Fock approximation within the Zimanyi and Moszkowski model, which considers a derivative coupling in the scalar field. The present model improves the previous a-tu version, by the inclusion of the charged mesons 7r and p. The 7rN interaction is a mixing of pseudoscalar and pseudovector couplings. The introduction of the 7v degrees of freedom has a strong influence on the nucleon effective mass, while keeping the compression modulus in a range of values in agreement with experiments. Effects on the spin-orbit splitting are also discussed.
1
Introduction
The relativistic model of Zimanyi and Moszkowski (ZM)*, including a scalar, nonlinear a field with derivative coupling and the linear u field with Yukawa coupling to nucleons 2 , is able to successfully describe bulk properties of nuclear matter, in particular the compressibility. However, this model gives a nucleon scalar effective mass (M*) close to the unity and, consequently, very small spin-orbit splittings. 44
45
The aim of this work is to show that this problem can be solved by the inclusion of the 7r and p mesons in the Hartree-Fock (HF) approximation 3 . 2
Basic features of the model
The interacting part of the original ZM Lagrangian is made of a derivative coupling between the scalar and fermion fields instead of the usual linear form. The crN coupling entering the Lagrangian takes the following form (see Ref. 3 for more details): CaN = ipmgaaip , (1) where
™=[l-^]-\
(2)
We linearize the scalar field equation with respect to the cr operator by approximating the cr field in rh by its ground state expectation value do. Then, the cr field equation takes the following form = -g<7rh2^,
(3)
*=[>-••£]"•
<4>
(d,d"+ml)a where
(To is obtained, by taking the static limit in Eq. (3), from the following equation: ( - A + ml)a0 - -garh2
< ipip > .
(5)
The expectation value < ipip > is calculated in the ground state. The
(6) 3
The to field is considered in the usual linear form . Concerning the wN interaction, we have considered a mixture of pseudoscalar (PS) and pseudovector (PV) couplings in the Lagrangian density
£wN=xC™+(l-x)£Z,
(7)
where x is the mixing parameter. The presence of this mixed coupling improves the description of the NN scattering data, in models involving only the cr, ui, it and p meson exchanges, and pionic atoms 4 . For the p meson, we have taken vector and tensor components 3 .
46 From the Lagrangian density, it is straightforward to obtain the Hamiltonian and the potential energy densities as it is done in Ref. 3 . A nonrelativistic expansion of the NN potential generated by the a, to, •K and p mesons through order p2/M2 has been made by Bryan and Scott 5 . For instance, the pseudoscalar contribution of pions to the Lagrangian density yields in turn a contribution to the NN potential, which, in the coordinate space, takes the form 2 2
n y(PS) _ XT_J JL 4TT
12M 2
ml
4 7 r ^ 3 ) ( r ) a1.a2+...>.
(8)
Similarly, the pseudovector and mixing contributions, as well as the cr, u> and p meson contributions in the nonrelativistic limit also exhibit a contact interaction 6^3\r), which is suppressed in realistic many body calculations by short-range correlations, due to the repulsion of the NN potential at short distances (mainly produced for the u> meson exchange). A genuine prescription has been given in Ref. 3 to subtract the 6^3\r) contact interaction contribution t o the potential energy. This procedure, which has been also adopted here, does not determine univocally the terms to be added, except in the nonrelativistic limit, but does not spoil the good properties of the model at large densities. 3
Numerical results and conclusions
T h e results clearly underline the ability of the present model to match the known properties of nuclear m a t t e r . Improvement over the original ZM model is notable (see Table 1). We have found t h a t the model is able to increase the scalar E s and timelike So self-energies in a significant way. Moreover, the (So — S 5 ) quantity inside the nucleus has increased enough to reproduce, presumably, the overall trends of the experimental spin-orbit splittings. The inclusion of the 7r and p mesons is crucial for obtaining values of M* in the range 0.60 < M*/M < 0.75, which is compatible with the experimental values of the spin-orbit splittings. T h e nuclear m a t t e r compressibility modulus (K) results indicate t h a t , although the Fock terms slightly increase the K value, it remains in an acceptable range [255 MeV (x=0) < K < 295 MeV (x=0.5)] according with the experiments. It exhibits a kind of saturation effect. This behaviour radically differs from t h a t exhibited by other models where a quasilinear increase of K is predicted when M* decreases 6 , with no sign of saturation.
47
The exchange terms, even those coming from the a and u mesons, give a considerable contribution to the symmetry energy. Thus, the experimental value of this parameter a^ can be reproduced with a relatively small value of gp like gJ,/(4:Tr) = .17, in comparison with the one necessary in the ZM or Walecka models. Finally, we would like to stress that the derivative coupling model has a better high density behaviour than other nonlinear relativistic models, extensively considered, including terms in a3 and
Model H((7 + w) 1 HF HF HF
x
gl/4ir
gl/4ir
M*/M
K (MeV)
a 4 (MeV)
0.0 0.25 0.5
5.50 4.99 4.27 3.03
2.44 4.54 3.76 2.36
0.85 0.76 0.72 0.60
225 254 268 283
13.7 34.7 33.6 29.6
References 1. J. Zimanyi and S. Moszkowski, Phys. Rev. C 42, 1416 (1990). 2. B.D. Serot and J.D. Walecka, Adv. Nucl. Phys.16, (1986). 3. P. Bernardos, R. Lombard, M. Lopez-Quelle, S. Marcos and R. Niembro, Phys. Rev. C 62, (2000). 4. P.F.A. Goudsmit, H.J. Leisi and E. Matsinos, Phys. Lett. B 271, 290 (1991). 5. R. Bryan and B.L. Scott, Phys. Rev. C 177, 1435 (1969). 6. W. Koepf, M.M. Sharma and P. Ring, Nucl. Phys. A 533, 95 (1991). 7. S. Kubis and M. Kutschera, Phys. Lett. B 399, 191 (1997).
G E N E R A T O R COORDINATE M E T H O D INCLUDING TRIAXIAL A N G U L A R M O M E N T U M P R O J E C T I O N
Department
K. E N A M I , K. T A N A B E f a n d N. Y O S H I N A G A of Physics, Saitama University, Uraiva, Saitama 338-8570, ' E-mail: [email protected]. ac.jp
Japan
The generator coordinate method (GCM) calculation combined with the triaxial angular momentum projection has been carried out for the single j-shell model with quadrupole-quadrupole interaction. Validity of the GCM is demonstrated by the exact reproduction of the level energies and the E2-transition rates for the excited states in the 4-hole systems, and also by the precise reproduction of these quantities for the states belonging to several low-lying collective bands for the halffilled systems. The whole states turn out to be more or less of triaxial deformation, which can not be anticipated from the mean field approximation.
1
Introduction
The microscopic and quantum mechanical description of various nuclear excitations has been one of the main purposes of nuclear structure study. As such a theoretical framework, the generator coordinate (GCM) method was proposed by Hill and Wheeler in 1953 1, but its application to realistic cases has been put off until recent development of fast computer system. In the present paper, we investigate in detail the effectiveness and the feasibility of the GCM combined with the triaxial angular momentum (or simply spin) projection in an exact manner 2 ' 3 , by comparing the GCM results with the exact solutions of the shell model for the single j-shell model. In the present series of calculations 4 , (i) we employ, for simplicity, only the quadrupole-quadrupole interaction (QQI) for the rotationally invariant microscopic Hamiltonian, and we do not include pairing interaction to avoid the particle number projection, (ii) We start from the single j-shell model in which like-nucleons are distributed in the degenerate single-particle levels of angular momentum j , and the system is to be closed by the occupancy of 2Q(= 2j + 1) nucleons. (iii) Our trial wave function for the GCM calculation are generated from the conventional Nilsson model of triaxial deformation. For the single j-shell model with degenerate spherical single-particle levels, the deformation can be effectively described only by the single parameter 7. Therefore, the number of relevant GCM parameters is reduced to four, i.e. the deformation parameter 7 in addition to three Euler angles necessary for the spin projection which may be regarded as a part of GCM. We expect that the GCM is capable of describing both collective excitations, i.e. the rotaional states which are 48
49
well accounted for by the spin projection, and the vibrational states by the generator coordinate 7. 2
Model and Formalism
In case of the single j-shell, the energy of the degenerate spherical singleparticle levels can be set zero so that the Nilsson Hamiltonian becomes /iNii =
-* sin 'Y ~ " i -D C O S 7 Q 0 — • / = - (Q2 + Q - 2 ) ,
(i)
where Q^ (/u = 0, ±2) is the dimensionless mass quadrupole operator, D = hu)o(3, U>Q the oscillator frequency for the spherical nucleus, and Q, 7 are the deformation parameters. The intrinsic state becomes independent of the scaling factor D, and specified by a unique parameter 7 which is regarded as a generator coordinate in the present approach. The spin projection is meaningful only when it is applied to the rotationally invariant Hamiltonian. We employ the QQI as such a Hamiltonian corresponding to Eq. (1), i.e. H = —x/2X) / J = _ 2 QtQv w n e r e x(> 0) is the force strength whose magnitude is not important like the scaling factor D in Eq. (1). The wave function with exact spin / is obtained by applying the triaxial spin projection operator PMK to the Nilsson state |$(7)) 2 , i.e. |*W7)> = £
FkPin)PMKmi))
•
(2)
K=-I
The spin-projected potential energy surface (PES) £^(7) and a set of the coefficients FlK (7) are determined from the generalized eigenvalue equation
J2 {HKK'{l,l)-EIp{i)NIKK,(1,1)}FIK,p{1)
=Q
(3)
K' = -I
with the norm and the Hamiltonian kernels {l,l')
= my)\PKK>\*h%
H ^ ( 7 , 7 ' ) = <*(7)|£^*'|*(7')>
(4)
x
We have solved the Hill-Wheeler (HW) equation to determine the GCM energy Ep for the p-th excited state of spin /, i.e.
f ^°
H
J2 {H'KK'il, 7') - Ep < ^ , ( 7 , 7')}^„(7') = 0 , K'--I
(5)
50
where the Hamiltonian and the norm kernels HKK'{I, calculated between the GCM wave functions
l*7MCpM)) = f3^ Jo
l') and
NKK'(I,
E tfcpWHiKmi)) •
l') are
(6)
K=-I
In calculating the spin-projected non-diagonal matrix elements like HKK1 (7,1 1 ), the extended form of the generalized Wick's theorem 5 is useful. 3
Numerical Analysis
In Fig. 1, we can see that all the exact solutions of shell model up to spin I ~ 8 (the levels of even spin are represented by the circle, and the levels of odd spin by the square) are perfectly reproduced by the 50 GCM levels (crosses) for the 4 hole system (2Q,iV) = (20,16). This indicates that not only the GCM solutions are axact, but also the number of the allowed states restricted by the /^-symmetry coincides with that of the shell model determined through the fractional parentage. As our original expectation, all the collective level sequences are identified with the ground band, a /3-band starting from spin 7 = 02 and 5 7-bands. It is also confirmed that both the B(E2) values and the selection rules are exactly reproduced for the (2fi, N) = (20,16) system. It is quite noticeable that the PES's plotted vs. 7 are rather flat, but for the excited state their minima are located in the triaxial region. On the other hand, the mean field solution for the ground state (the upper and the lower dotted lines are without and with the exchange contributions, respectively) predicts the location of its minimum at the prolate deformation. The same analysis shows that the GCM results are axact also for the other 4 hole systems of (2f2,A0=(12, 8), (14, 10), (16, 12) and (18, 16). As observed in Fig. 2, the GCM results are not exact for the half-filled system of (2Q, N) = (20,10) in which the number of possible configurations takes its maximum and the calculation becomes the most difficult for a given 2Q = 2j + 1. However, the GCM results are almost exact, or quite precise for the 26 low-lying levels. Each PES associated with the 4 low-lying levels has 2 minima near 7 = 30°, while the PES's for the other excited levels have their unique minima at 7 = 30°. Thus, the states with triaxial deformation contribute much to the physical states. In this respect, the mean field PES having its two minima at 7 = 0° and 60° fails to predict correct nuclear shapes. The success of the GCM is in part attributed to this simplified model, for which the necessary sets of the generator coordinates are apparent, and their integral ranges are determined without ambiguity. However, the excellent reproduction of the exact results by the GCM is far from trivial since there is
51
u0 i®2345678
1=0 2345678
or
(2fl,N)=(20,16) •
*® EI
>
-10 8®
J ®
:
IH «
-20
a
s, J,
8«sj
^ >,
V
^rrrSJ ^ T -
S -30 a
1S1»--
*
8, <Sj^'_
l-l-Trgg^".
Mt
!'
W
--7,
r^xTT
r ^ ^ V
--'-..5,-
^^-y?--''/"
*-..\
-40
/
.•'
^^JJh. ,'• ---'"'
-50
4
,
,
0
ii^"
'
i
1
2,
i.
i'
10
20
_^f/
_—--^/ __—-^^ .
i
30 40 y(degrees)
.
i
50
Figure 1: The (20,16) system.
.
-
60
20
30 y(degrees)
Figure 2: The (20,10) system.
no common aspect in the formalism between the shell model and the GCM. The calculation of the PES's provides an important information of the deformation which is not directly given in the shell model results. The importance of the triaxial spin projection cannot be overemphasized since the triaxiality of system cannot be described in terms of the superposition of axially symmetric states. Our results encourage further application of the GCM with the triaxial spin projection which will be advantageous in treating heavy mid-shell nuclei for which the shell model calculation is far beyond its ability on the existing computer. References 1. 2. 3. 4.
D. L. Hill and J. A. Wheeler, Phys. Rev. 89, 1106 (1953). K. Enami, K. Tanabe and N. Yoshinaga, Phys. Rev. C59, 135 (1999). K. Enami, K. Tanabe and N. Yoshinaga, Phys. Rev. C61, 027301 (2000). K. Enami, K. Tanabe and N. Yoshinaga, Generator coordinate method combined with exact triaxial spin projection, Saitama University preprint (2000). 5. K. Tanabe, K. Enami and N. Yoshinaga, Phys. Rev. C59, 2494 (1999).
EFFECT OF T H E TRIAXIAL A N G U L A R M O M E N T U M P R O J E C T I O N O N T H E POTENTIAL E N E R G Y SURFACE K. E N A M I , K. T A N A B E * , N. Y O S H I N A G A a n d K. H I G A S H I Y A M A Department of Physics, Saitama University, Urawa, Saitama 338-8570, Japan ^E-mail: [email protected] The triaxial spin projection is applied to the ground state expectation value of the Hamiltonian with the monopole pairing plus quadrupole interaction. It is turned out that the restoration of exact spin value has strong triaxiality-driving effect for all of three nuclei 1 4 2 Nd, 1 6 8 E r and 1 8 8 Os, which have been regarded as the spherical, the prolate and the 7-unstable nuclei, respectively.
1
Introduction
When the angular momentum (or simply spin) projection is applied to the mean field solution for a given microscopic Hamiltonian, it takes two parts of projecting out a correct spin component and taking account of the correlations which are not included in the mean field approximation. In the present paper, we investigate the latter function of the spin projection specifically. Hayashi et. al.1 have applied the triaxial spin projection to the potential energy surface (PES) of the ground state for the 7 unstable transitional nucleus 188 0s, and found that 7-soft PES is modified to have its minimum at 7 ~ 30° due to the energy gain from the spin projection. Our interest is focused on the problem whether the triaxiality-driving tendency of the projection is dependent on the specific shell filling or not. Therefore, we will calculate the projected PES also for 142 Nd and 168 Er which are usually regarded as spherical nucleus and the one of stable prolate deformation, respectively. The deformation parameters, (£,7), are introduced through the singleparticle Hamiltonian Ho with the spherical modified oscillator (MO) potential. The MO parameters K and \x of the I s - and {l2 — (1 2 )JV} -terms are the same as those given in Ref. 2. For comparison, we perform the numerical analysis based on the two types of the single-particle space, i.e. "the single-particle space I" spanned by two major shells of N = 4, 5 (5,6) for protons (neutrons) and "the single-particle space II" spanned by three major shells of N — 3,4,5 (4, 5,6) for protons (neutrons). 2
Microscopic Hamiltonian and Triaxial Spin Projection
We prepare the deformed mean-field solutions from the pairing plus quadrupole model with some modifications presented below. The intrinsic states deter52
53
mined in this scheme are essentially the same as those obtained from the Nilsson BCS model. The microscopic Hamiltonian is composed of the quadrupolequadrupole interaction (QQI) and monopole-pairing interaction (MPI), i.e.
H = Ho~lY,
XTT-QUQW
~ £G
T,T',II
(1)
T
Here, the operator QTfl (fi = 0, ± 1 , ±2) denotes the dimensionless mass quadrupole operator and PT the monopole pair operator with r = proton, or neutron. The oscillator energy fuv and he MPI force strength change depending on the isospin, i.e. ftu,p/n = fc* ( l =F I ^
)
,
G^n = (n ± g2 ^
)
1 (MeV)
(2)
with hu0 = 41.2>l~1/3(MeV). The adopted values of these force strength are given in Ref. 3. The spin projected energy for any spin / is determined from the generalized eigenvalue equation 4
E
{(HPKK>)
- EiiPIcK^Fk* = 0,
PUK
=
2J
^ JdQD^K(Q)R(H)-
(3) In the above spin projection operator, R(Q) and DlMK(Q) are the rotation operator and Wigner's D-function, respectively; and Q stands for three Euler angles. For the purpose of calculating the projected ground state PES, we put I = K = K' = 0. Using the solution of the eigenvalue equation in Eq. (3), we calculate the energy gain due to the spin projection defined by
Eproj(£,1)={JI0^±-{H). \M50
(4)
/
The calculation of Eproj(£,7) is repeated at various points in the (5,7) plane to complete the contour plot. 3
Numerical Results
In Fig. 1, we show the contour maps of the energy gain calculated based on two types of the single-paricle space for the spherical nucleus 142 Nd. Also in Fig. 2, we show the contour maps of the energy gain calculated based on two
54 (a) Single-particle space I
(b) Single-particle space II
60
Figure 1: The contour plot of the energy for (b) the single-particle space II.
60
142
(a) Single-particle space I
Figure 2: The contour plot of the energy for (b) the single-particle space II.
Nd with (a) t h e single-particle space I, and
(b) Single-particle space II
168
Er with (a) the single-particle space I, and
55
types of the single-paricle space for the nucleus with prolate deformation 168 Er. In both figures, at the spherical shape (e = 7 = 0), the energy gain from the spin projection is automatically zero, i.e. EpToj(0,0) = 0, since the fluctuation of the angular momentum does not exist (note that -Potf0! ) = | ) at e = 0). Moreover, the energy gain becomes always negative for e ^ 0 since the pairing plus quadrupole model includes only attractive interactions. Accordingly, all the contour lines in the energy gain plots represent only negative values. The contour line separation is 200 keV. The clear minima of the energy gain indicated by the crosses (x) are located in the triaxial deformation region. The maximum difference of the energy gain between the axially symmetric and the triaxial shape for a fixed value of e (> 0.1) amounts to about 1.5^2 MeV. We recognize some common features in these figures as follows. In the region of small triaxiality (7 = 0° ~ 10° or 7 = 50° ~ 60°), the contour lines of the energy gain are rather parallel with a straight line of 7 = 0° or 7 = 60°, i.e., the energy gain varies with 7, but it is not much dependent on e. This implies that the qualitative properties of the unprojected PES is not much affected by the spin projection as far as the projected PES is analyzed along the line of the prolate (oblate) deformation with 7 = 0° (7 = 60°). The maximum energy gain is always realized at an explicit triaxial shape irrespective of the nuclei. If the energy gain is analyzed for a fixed e, the shape favored by the spin projection is not axially symmetric but triaxial. Such a feature is more pronounced for the single-particle space II. In short, the spin projection tends to stabilize the triaxial shape. This triaxiality-driving tendency of the spin projection is common in all kinds of nuclei and not limited to the 7unstable nuclei. It is remarked that this effect can not be described by the mean field approximation with the a priori assumption of the axially symmetric deformation since it originates from the correlations disregarded in the mean field approximation. References 1. A. Hayashi, K. Hara and P. Ring, Phys. Rev. Lett. 53, 337 (1984). 2. S. G. Nilsson and I. Ragnarsson, Shapes and Shells in Nuclear Structure (Cambridge University Press, New York, 1995). 3. K. Enami, K. Tanabe, N. Yoshinaga and K. Higashiyama, Strong triaxiality-driving effect extracted by the spin projection, Saitama University preprint (2000). 4. K. Enami, K. Tanabe and N. Yoshinaga, Phys. Rev. C59, 135 (1999); Phys. Rev. C61, 027301 (2000).
REALISTIC INTRINSIC STATE DENSITIES FOR DEFORMED NUCLEI ERNESTO MAINEGRA f AND ROBERTO CAPOTE* CEADEN, Calle 30 # 502, e/5ta y 7ma, Miramar, Habana 11400, Cuba A microscopic method for calculation of total io(U) state densities based on a combinatorial Monte Carlo evaluation from a Woods-Saxon single-particle level (SPL) scheme at given deformation has been implemented. Residual pairing interaction can be considered in the calculations. A fast Monte Carlo algorithm is used, allowing large shell-model spaces to be studied. Intrinsic level densities for 162Dy nucleus are calculated for various model spaces. Discussions of results relevant for Shell Model Monte Carlo calculations are presented.
1
Introduction
Most of the semi-empirical approaches for the calculation of nuclear level densities are based on drastic approximations and their shortcomings at matching experiments are often overcome only by parameter adjustments, limiting their practical use to the region of excitation energy to which most of the experimental knowledge of level densities is confined. The advent of high-speed computers has made it possible to use methods of calculations, which do not depend on closed formulae, such as the combinatorial method. This method yields an exact level density, but it is very time consuming and becomes intractable at high excitation energies or for large shell model spaces as occur in medium and heavy nuclei. Thus it is natural to resort to a Monte Carlo technique based on the Metropolis algorithm as proposed by Cerf [1] in order to avoid an exhaustive counting of the excited levels. 2
Methods and Materials
2.1
The Monte Carlo method
The Metropolis method is based on a guided random walk, which proceeds through configuration space according to a given matrix of transition probabilities. Detailed information about the Metropolis sampling algorithm [2] can be found elsewhere in the literature. This Metropolis prescription has been proved to be rigorous for a discrete space, which is just the case for our nuclear configuration space.
[email protected],cu [email protected]: [email protected] 56
57 The Monte Carlo simulation is based on a random sampling of a very small fraction of the excited states for a given range of excitation energy. The resulting sample is assumed to be representative of the whole configuration space, in analogy with what is done when estimating a multidimensional integral by a Monte Carlo procedure. Thus, the properties of the whole spectrum of excited states (i.e., energy, spin, parity, etc.) can be simply derived by extrapolating from the sample, applying the appropriate scale factor. To determine the scale factor needed to obtain an absolute state density we calculated the total number of states by means of Williams's recursive method [3]. Total and partial state densities are obtained from realistic single particle level schemes, using a recursive method, which is mathematically exact. This method was validated against a direct counting combinatorial code included in the RIPL [4]. Both methods yielded exactly the same results. For a definition or more detailed description of the formalism the reader is referred to the extensive work of Cerf [1] and references therein. Although the method is exact in principle, it is subject to a statistical error, which scales like \/J~N~ • The accuracy of the results can be imposed by choosing an appropriate size N for the sample, and do not depend on the actual number of states in the considered energy range. 2.2
Single particle levels and pairing interactions
The computer code CASSINI [5] has been used to compute double degenerated SPL of an axiafly deformed Woods-Saxon potential with cassinian ovals shape parameterization [6]. The universal Woods-Saxon parameters proposed by Dudek et al [7] were used in the calculations, except that smaller values (r0=7.25 fin for both particles) of the central potential radius parameter were employed [8]. Ground state deformation parameters for 162Dy were taken as e= 0.28. To estimate differences due to the use of truncated shell spaces, we calculated the state density up to 10 MeV for both a reduced and the full shell space neglecting pairing interactions. Pairing interaction is taken into account in the frame of the BCS theory [9], but implementation of any approximate number projection method should be straightforward. The strength of the pairing interaction is determined from experimental odd -even mass differences. For each sampled excited configuration we solve the BCS equation in order to obtain its pairing energy. We also use the so-called blocking method thus orbits occupied by unpaired nucleons are not considered in BCS sums. The energy of each sampled configuration is shifted by the amount of pairing energy.
58
3 Results 3.1
Comparison with the recursive method and influence of pairing
Nuclear state density for 162Dy has been calculated by the Monte Carlo method and compared in figure 1 with calculations performed with a recursive relation. The latter is obtained under the independent particle assumption and no appreciable difference is found with the Monte Carlo results without pairing validating the implemented algorithm. As was expected the independent particle model overestimates the state density. In fact from figure 1 we see that inclusion of pairing depresses the state density for neutrons in a major grade than for protons. The minimum energy needed to excite a particle over the Fermi level is also shifted towards higher energies finding a 1 MeV and 2 MeV gap for neutrons and protons respectively. Those effects reflect in the same way on the total state density, which present a 1 MeV gap imposed by the neutron state density.
0
1
2
3
4
5
6
7
8
9
10
Excitation energy E[MeV]
3.2
State density in a reduced shell model space
The Shell Model Monte Carlo (SMMC) method is capable of providing exact results for a range of observables in model spaces where the dimensions are prohibitive for direct diagonalization. SMMC storage scales like A^2 + N* , where Nv and Nn are the number of SPL for neutrons and protons respectively. This scaling is seemingly fairly rapid, and limitations in computer space and time have to be considered limiting the calculations to a finite model space. Consequently the shellmodel level density will always underestimate the true level density due to the presence in the later of states representing excitations outside of the SMMC model space.
59 " D y F U L L S P A C E vs. R E D U C E D S H E L L
2
3
4
5
6
Excitation energy
7
8
SPACE
9
E[MeV]
Full shell space was arbitrarily limited to 272 double degenerated levels. This limitation consequently imposes a maximum energy for which all of the levels could be determined (the energy for which one nucleon is promoted to the highest included orbital). For neutrons this energy is 31.3MeV and for protons 39.2MeV. The truncated space encompasses 32 proton levels and 44 neutron levels [10]. Within this space, 62Dy has sixteen valence protons so that the proton shell is half-filled and fourteen neutrons. Figure 2 shows the results for both cases. There exists a clear underestimation of the neutron state density of 55% at 5 MeV and 80% at 10 MeV. For protons this effect is somewhat weaker finding still no differences at 5 MeV but increasing differences arise at higher energies reaching 35% at lOMeV. References 1. 1. 2. 3. 4. 5. 6. 7. 8. 9.
N. Cerf, Phys.Rev.C49(1994) 852 N. Metropolis etal, J.Chem.Phys.21(l953) 1087 F.C.Williams, Nucl.Phys.A133 (1969) 33 CAPOTE_MICRO.FOR code, RIPL CD version, IAEA, Vienna, 1999 E.Garrote, R.Capote, R.Pedrosa, Comp.Phys.Comm.92{\995) 267 V.V.Pashkevich, Nucl.Phys.A169(l97l) 275 J.Dudek , Z.Szymanski and T.Werner, P/iys./?ev.C23(1981) 920 Z.Lojewski et al, Phys.Rev.C5l(l995) 601 J.Bardeen , L.N.Cooper and J.R.Schrieffer, Phys.Rev.lOS(\957) 1175 J.A.White, Ph.D. Thesis, California Institute of Technology, 1998
APPROXIMATE TREATMENT OF THE CENTRE OF MASS CORRECTION FOR LIGHT NUCLEI M. GRYPEOS, C. KOUTROULOS,A. SHEBEKO* AND K. YPSILANTIS Department of Theoretical Physics, Aristotle University of Thessaloniki, Greece and NCS Kharkov Institute of Physics and Technology, 310108 Kharkov, Ukrain
1
Introduction
The treatment of the centre of mass (CM) motion has been an attractive subject of earlier and recent studies in nuclear theory (see e.g. ref. 1 and 2 and references therein). The aim of the present investigation is to adopt the "fixed CM approximation" 3 ' 4 to restore translational invariance of a manybody wave function (WF) which does not have this property and use it for the evaluation of the elastic form factor in Born approximation, F(q) and of the nucleon momentum distribution rj(p) of light nuclei, and more specifically of 4He in its ground state. The study of r](p) has attracted much interest in the last two decades or so but its CM correction does not appear to have been properly treated except in certain studies in which harmonic oscillator (HO) wave functions were used (see e.g. ref. 5). The treatment of 4He is strongly advisable not only because of its relative simplicity but also because the CM correction is very important for such a light system. Furthermore, the values of F(q) have been extracted 6 in a wide range of q. In the following section the WFs used are given. Section 3 is devoted to the relevant formalism, while our numerical results are reported and discussed in the final section. 2
The single-particle wave functions
The single-particle (non-translationally invariant) WFs, used here are the following which have desirable properties, as is clear from earlier work: First, those of the "modified harmonic oscillator" (MHO) potential V(r) = -V0 + (h2/2mb4)r2
+ B/r2
,
V0 > 0, B > 0
(1)
The energy eigenvalues and eigenfunctions for this potential are given analytically. The (normalized) ls-radial (cp(r) = rR(r)) state needed for AHe is: &H°(r) = [2/&r(2A0 + l/2)] 1 /2(r/&) 2 *° e -'- 2 /2& 2 ) A0 = (1/4)[1 + (1 + (SmB/h2))1/2}. Analytic expressions have also been given for the single (point) 60
61
particle (the "body") density ps{r), the corresponding elastic form factor Fs(q) and the nucleon momentum distribution rjs{p)7 . Secondly, the Radhakant, Khadkikar and Banerjee3 (RKB) normalized radial WF for the lowest single-particle state of AHe : cf>RKB(r) = (1 + ^ ) - 1 / 2 ( ^ o ( r ) + /tyio(r))
(2)
where oo and cpio are the normalized HO radial orbitals with parameter bn for the states with n = 0,1 = 0 and n = 1,Z = 0, respectively and fl is the mixing parameter. This WF leads to simple analytic expressions for the quantities of interest, such as: Vs(p) = 3
TT-3/2^(1
+ P2)-I[l
+ P(3/2)^2(-l
+ (2/3)(bHP)2)fe-(b«tf
(3)
Expressions for F(q) and r](p) corrected for the CM motion.
In the "fixed CM approximation", the Ernst, Shakin and Thaler (EST) prescription 3 ' 4 the nuclear many-body wave function is written
* = (27r)3/2|P)$fniT
(4)
A round bracket is used to represent a vector in the space of the CM coordinate only so that e.g. \P) means the eigenstate of total momentum operator P. The EST intrinsic WF: $f„fj = {R = 0 | * . > [< $S\R = 0)(R = 0|$ s >)-1'2
(5)
is constructed from an arbitrary (in general, non-translatinally invariant) WF $ s , by requiring that the CM coordinate R be equal to zero. Use of a Slater determinant for $ s , leads to the following expression of F(q), the elastic form factor for 4He corrected for the CM motion 3 ' 8 : F(q) = jFs(\q
+ ti\)F?(u)du /JF*(u)du
(6)
This expression may be used to calculate F(q) numerically. A convenient way to do this with the RKB WF has been considered in 8,9 where such a calculation is reduced to one dimensional integrals from 0 to 1 of polynomials and of other well-known functions of suitable arguments. Pertaining to rj(p) with the "fixed CM approximation": r?(p) = < $s|(203<5(-R)
4
> / < f s\(2ir)36(R)\s >
(7)
, it can be shown that for He with the RKB WF, r)(p) can be calculated again semi-analytically (after a lengthy procedure) by reducing it to onedimensional integrals of structure similar to those derived for F(q).
62
In view of the valuable advantages of $RKB(r), the approximation of QMHO by §RKB k a s b e e n investigated. This was achieved through a best approximation in the mean, that is by requiring 62 = / \(j)MHO{r) — cf>RKB{r)\2dr to be minimum, which leads to expressions of /3 and &# in terms of b and A13. 4
Results and discussions.
In this section we give first the results of the charge form factor of iHe with the MHO model and the RKB WF by fitting to the known experimental 6 values using mainly the "fixed CM approximation" and considering for the finite proton size fp(q) the Chandra and Sauer prescription 7 . The two pa- HO (*Hi TB (actor) - RKB {with the OS transformation) - RKB (wtlti the tixed CM correction) experimental values - M H O (with tm fixed CM correction)
Fig.1
Figure 1. The log\Fcfl(q2)\
q'Cfm'2)
versus q2 for various cases. For the abbreviations see text.
rameters in each case are determined by least-squares fit. The results are shown in fig. 1. The most satisfactory fit is with the MHO model. Also the fit with the RKB wave function is very good apart from the higher q values where a second diffraction minimum is predicted at q2 ~ 37 fm~2, which does not seem to be indicated there by the data. The results with <j)RKB and the Gartenhaus-Schwartz (GS) transformation 10 for the CM motion and those with the HO (and the Tassie and Barker (TB) factor) are shown as well. The quality of the fit is very poor in these cases. Having determined the parameters in the described way, the values of 77(p) were calculated with the (j>RKB{r) wave function in certain cases in fig. 2. It is seen that a considerable improvement is mostly observed in comparison with the HO case, if we consider the "experimental points" (which are model dependent). It should be also noted that if cpRKB is determined by the minimization of £2, it approximates very well the (f>MHO since the minimum value of €2 is very small (62 ~ 0.00163).
63 • «xp«rimental •— RKB (with th» fixed CM correction) — RKB (without CM correction) - — HO (without TB factor) - - HO (with TB (actor)
Fig. 2
P (fm')
Figure 2. The log\ri(p)\ versus p for various cases. For the abbreviations see text. The parameters in the (f>RKB used were determined directly from the fit to the experimental Fch(q)-
In conclusion, the present analysis shows that the approximate treatment of the CM motion with the fixed CM method for the r](p) of 4He is feasible, through the RKB wave function, although quite cumbersome. References 1. J.L. Friar, Nucl. Phys. A173, 257 (1971) 2. Bogdan Mihaila and Jochen Heisenberg, Phys, Rev. C60 054303 (1999) 3. S. Radhakant, S.B. Khadkikar and B. Banerjee, Nucl. Phys. A142 81 (1970); S.B. Khadkikar, private communication to M. Grypeos 4. D. Ernst, C. Shakin and R. Thaler, Phys. Rev. C7 925 (1973); ibid 1340 5. S. Dementiji, V. Ogurtzov and A. Shebeko, Sov. J. Nucl. Phys. 22, 6 (1976); P.Stringari (Private communication to M. Grypeos) 6. R.F. Frosch et al, Phys. Rev. 160, 1308 (1966); R.G. Arnold et al, Phys. Rev. Let. 40, 1429 (1978) 7. M. Grypeos and K. Ypsilantis, J. Phys. G, Nucl. Part. Phys. 15, 1397 (1989); K. Ypsilantis and M. Grypeos ibid. 21 1701 (1995) 8. A. Shebeko, Lectures on selected topics of nuclear theory, (Thessaloniki 2000) 9. M. Grypeos, C. Koutroulos, A. Shebeko, K. Ypsilantis, to be submitted 10. S. Gartenhaus and C. Schwartz, Phys. Rev. 108, 482 (1957)
ONE-BODY DENSITY MATRIX AND MOMENTUM D I S T R I B U T I O N I N S-P A N D S-D S H E L L N U C L E I
C H . C . M O U S T A K I D I S A N D S.E. M A S S E N Department
of Theoretical
Physics, Aristotle Thessaloniki,
University Greece
of Thessaloniki,
GR-54006
Analytical expressions of the one- and two- body terms in the cluster expansion of the one-body density matrix and momentum distribution of the s-p and s-d shell nuclei with N = Z are derived. They depend on the harmonic oscillator parameter b and the parameter /? which originates from the Jastrow correlation function, b and (3 have been determined by least squares fit to the experimental charge form factors. The inclusion of short-range correlations increases the high momentum component of the momentum distribution, n(k) for all nuclei we have considered while there is an A dependence of n(k) both at small values of k and the high momentum component.
1
INTRODUCTION
The momentum distribution (MD) n(k) is of interest in many research subjects of modern physics, including those referring to helium, electronic, nuclear, and quark systems. In the last two decades, there has been significant effort for the determination of the MD in nuclear matter and finite nucleon systems 1 _ 9 . MD is related to the cross sections of various kinds of nuclear reactions. The experimental evidence obtained from inclusive and exclusive electron scattering on nuclei established the existence of a high-momentum component for momenta k > 2 fm _ . It has been shown that, in principle, mean field theories cannot describe correctly MD and density distribution simultaneously 5 . The reason is that MD is sensitive to short-range and tensor nucleon-nucleon correlations which are not included in the mean field theories. Usually, the MD of the closed shell nuclei 4 He, 1 6 0 and 4 0 Ca as well as of 2 0 8 Pb and nuclear matter is studied. There is no systematic study of the one body density matrix (OBDM) p(v,v') and MD which include both the case of closed and open shell nuclei. For that reason, in the present work, we attempt to find some general expressions for p(r,r') and n(k) which could be used both for closed and open shell nuclei. The expression of p(r,r') was found, first, using the factor cluster expansion of Clark and co-workers 1 0 , n and Jastrow correlation function which introduces SRC for closed shell nuclei and then was extrapolated to the case of N = Z, s-p and s-d open shell nuclei. n(k) was found by Fourier transform of p(r,r').
64
65
2
CORRELATED O B D M A N D MD
A nucleus with A nucleons is described by the wave function ^ ( r i , ^ , •••,TA) which depends on 3A coordinates as well as on spins and isospins. The evaluation of the single particle characteristics of the system needs the one-body density matrix p ( r , r ' ) = N(*\0„,{A)\9')
= N{0rr,{A))
(1)
r
where \f ' = ^(r'^r^, •••,r'A) and N is the normalization factor. The one-body "density operator" Orr'(A), has the form A
A
%,--rWr;-r')n%i-ri)
Orr,(A)=J2
(2)
If we denote the model operator, which introduces SRC, by T, an eigenstate $ of the model system corresponds to an eigenstate $ = T<& of the true system. In the present work the model operator T(r\i) is taken to be the Jastrow correlation function, /(r,-j) = 1 — e~/3(r'~ri> . In order to evaluate the correlated one-body density matrix pcor(r,r'), we consider the generalized integral 1(a) = (\F| exp[a/(0)O rr /(A)]|^ r '), corresponding to the one-body "density operator" O r r /(A), from which we have (Orr,(A))
= [dlnI(a)/da)a=0
(3)
For the cluster analysis of equation (3), following the factor cluster expansion of Clark et al 1 0 , n and neglecting three and many-body terms, we have / w ( r , r ' ) * iV[(O rr /)i - 0 2 2 ( r , r ' , g i ) - 0 2 2 ( r , r ' , g 2 ) + 0 2 2 ( r , r ' , g3)]
(4)
where (O r r /)i is the one-body contribution to the OBDM and 0 2 2 (r, v',gi) (£= 1, 2, 3) is given by the following general expression, 022(r,r',g/)=4
J2
^,;,»7n J ./ J (2/ i +l)(2/ J + l ) x
riili ,rijlj
(5) fc=o
where rjni are the occupation probabilities (0 < rjni < 1) and ^^"^'"(r.r'.gO = - ^ O )
^
3
( 0
exp[-/?r 2 ] P/ 3 (cos W r r 0 x
/•OO
/ Jo
KJM^IM)
exp[-/?r22] ik{2(3rr2) r\ dr 2 (6)
66
The matrix element ^ V n * * ' (r, r', g2) can be found from (6) replacing r f-> r' and n\l\ «->• 713/3 while the matrix element corresponding to g 3 can be found from (6) replacing e _ / 3 r —>• e - / 3 ( r +r ), P; 3 (cosw rr /) -> £lf l (wrr#) (afunction which depends on the directions of r and r') and ik^lfirr^) —» ife(2/?|r + r'|r2). It is noted that Eq. (5) is also valid for the cluster expansion of the density distribution and the form factor as it has been found in ref. 12 and also in the cluster expansion of the MD which is the Fourier transform of p(r,r'). The only difference is the expressions of the matrix elements A. In the case of the HO wave functions, analytical expressions of the correlated OBDM and MD for the s-p and s-d shell nuclei have been found. These expressions depend on the HO parameter 6 and the correlation parameter /?. The values of b and /? have been determined by fit of the theoretical charge form factors, Fch(q) to the experimental ones. It is found that the inclusion of SRC's improves the fit of Fch(q) of the above mentioned nuclei and all the diffraction minima are reproduced in the correct place 12»13. The values of the parameter j3 (see Fig. 1) is almost constant for the closed shell nuclei and takes larger values (less correlated system) in the open shell nuclei. The behaviour of the parameter /? has an effect on the MD of nuclei as it is seen from Fig. 2a, where the MD, of the various s-p and s-d shell nuclei have been plotted. It is seen that the inclusion of SRC's increases considerably the high momentum component of n(k), for all nuclei we have considered. Also, while the general structure of the high momentum component of the MD for A = 4,12,16,24, 28,32,36,40, is almost the same, in agreement with other studies 1]4 , there is an A dependence of n(k) both at small values of k and in the region 2 f m - 1 < k < 5 fm _ . In the previous analysis, the nuclei 24 Mg, 28 Si and 32 S were treated as Id shell nuclei, that is, the occupation probability of the 2s state was taken to be zero. The formalism of the present work has the advantage that the occupation probabilities of the various states can be treated as free parameters
Figure 1. The correlation parameter 0 versus the mass number A. The solid line correspond to the case when 24 28 32 36 the nuclei Mg, Si, S, Ar were treated as Id shell nuclei while the dashed line to the case when these nuclei were treated as Id-Is shell nuclei. 0
10
20
30 A
40
67
in the fitting procedure of Fch{q)- For that reason we considered the case in which the occupation probability rj2S of the nuclei 24 Mg, 28 Si and 32 S was taken to be as free parameter together with the parameters b and /?. We found that the A dependence of the parameter j3 and of the high momentum component of the MD are not so large as it was before (see Figs. 1 and 2b).
Figure 2. (a) The correlated MD for various s-p and s-d shell nuclei calculated with the parameters b and /? of the case when the nuclei 2 4 Mg, 2 8 Si, 3 2 S and 3 6 A r were treated as Id shell nuclei. The normalization is f n ( k ) d k = 1. (b) The same as in Fig. (a) but when the above mentioned nuclei were treated as ld-2s shell nuclei.
References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.
J. G. Zabolitzky, W. Ey, Phys. Lett. 76B, 527 (1978). 0 . Bohigas, S. Stringari, Phys. Lett. 95B, 9 (1980). M. Dal Ri, S. Stringari, O. Bohigas, Nucl. Phys. A376, 81 (1982). M. Traini, G. Orlandini, Z. Phys. A 321, 479 (1985). M. Jaminon, C. Mahaux, H. Ngo, Nucl.Phys. A440, 228 (1985); M. Jaminon, C. Mahaux, H. Ngo, Nucl.Phys. A452, 445 (1986). O. Benhar, C.Ciofi degli Atti, S. Liuti, G. Salme, Phys. Lett. 177B, 135 (1986). S. Stringari, M. Traini, and 0 . Bohigas, Nucl. Phys. A516, 33 (1990). M.V.Stoitsov, A.N.Antonov, S.S.Dimitrova, Phys.Rev. C47, 2455 (1993). F. Arias de Saavedra, G. Co', M.M. Renis, Phys. Rev. C55, 673 (1997). M.L. Ristig, W.J. Ter Low, J.W. Clark, Phys. Rev. C3, 1504 (1971). J.W. Clark, Prog. Part. Nucl. Phys. 2, 89 (1979). S.E. Massen, Ch.C. Moustakidis, Phys. Rev. C 60, 024005 (1999). Ch.C. Moustakidis, S.E. Massen, nucl-th/0005009 and Phys.Rev C (2000) in press.
Correlation induced collapse of systems with Skyrme forces D.V. Fedorov and A.S. Jensen Aarhus University, DK-8000 Aarhus C, Denmark We show that a many-body system with (possibly density dependent) zero-range two- and three- body forces collapses due to three-body correlations.
Introduction. A direct application of a zero-range force V = — 7o<5( r i~ r 2) in a three-body system results in a collapse of the system known as the Thomas effect 1. Correspondingly, the forces of this type, including the two-body Skyrme force, also lead to a collapse of infinite symmetric nuclear matter in the sense that the energy per particle E/A as a function of density n has no minimum: E h2 3 / 3 7 r 2 n \ 2 / 3 3^
A = 2^s{—)
-8%n-
(1)
However in the case of infinite nuclear matter the collapse can be seemingly removed by extending the Skyrme force with either a zero-range three-body force 73<5(ri— r2)S(ri —TZ) or a density dependent zero-range force 73na(ri)S(ri —r2) which results in an additional term in Eq. (1) proportional to n 2 or na+1. Notwithstanding, these additional forces must also remove the collapse of the three-body system, otherwise the many-body system will still be unstable under clusterization into collapsed three-body subsystems. We shall show in this contribution that neither the three-body zero-range force nor the density dependent zero-range force is actually able to remove the collapse of the three-body system. The Skyrme forces therefore can only be used with the wavefunctions where the three-body correlations are excluded. Three-body systems with zero-range forces. We start with the general hyperspherical adiabatic expansion 2 of the three-body wavefunction "J *(p,ft) = ^ ^ / „ ( / 0 ) $ „ ( p , f i ) , "
(2)
n
where the hyperradius p and the set of hyperangles fi are defined in the appendix. If we truncate the infinite sum in this expansion we shall, according to the variational principle, obtain the upper bounds on the discrete spectrum of the system. If then the truncated expansion provides a collapse the full wavefunction will collapse as well. It is then sufficient to consider only the lowest 68
69
term in the expansion - the so called hyperspherical adiabatic approximation, \?(p,fi) = p~5/2f(p)$(p,ft),where the angular coordinates fi correspond to the "fast" subsystem while the hyperradius p represents the "slow" subsystem. Within this approximation the lowest eigenvalue X(p) of the "fast" subsystem simply serves as the effective potential for the "slow" hyperradial subsystem: d2 \(p) + 15/4 2| 2 + dp p„22 '
2mE p h -]/(p)
= 0.
(3)
The Schrodinger eigenvalue equation for the "fast" angular subsystem is U2-\{p)
+ ^^Vi{p,il))*(p,il)
= 0,
(4)
where Vj is the potential between particles j and k (i,j,k is the cyclic permutation of 1,2,3), A2 is the hyperangular part of the three-body kinetic energy operator and m is the mass in the definition of the hyperradius p (Eq.(14)). For the short-range potentials, however, the Faddeev equations provide a more convenient basis for analytical insights into the properties of the system. The Faddeev decomposition of the angular wavefunction is
where at is the hyperangle, and where only s-waves are included in each of the three Faddeev components <j>i{p,ai). This also implies that only the momentum independent term of the full Skyrme force is important in the following. The Faddeev equations for the three components of the wavefunction A
A
( * - (P))
2mp2 —£bx sin(2c*i) +' ± hX2 v ; *
= o, (i = i, 2,3),
(6)
are equivalent to the original Schrodinger equation. The zero-range potentials vanish identically except at the origin and we are therefore left with the free Faddeev equations and their solutions fa 1/2
(P) J 4>i{p,ai) = 0 , (j>i{p,ai) = A;sin ]u (on -
-j
where v2 = A + 4 and the solutions obey the boundary condition i(p,%) = 0. For three identical particles we can drop the index i completely. The generalization for non identical particles is straightforward.
70
The zero-range potential appears as a boundary condition at a = 0 1 <9$\
—
p 1
(8)
where p, = ( l / m ) ( m i m 2 / ( m i +m 2 )) is the reduced mass of the two particles in units of the mass m, and a is the two-body scattering length. Expressing the total angular wavefunction $ for small angles within a given Jacobi system 2 sin(2a)$( Q ) = 4>(a) + 4 = a 0 ( J ) + 0 ( Q 2 ) ,.
(9)
where 4>(a) = sin[i/(a - 7r/2)], we obtain the eigenvalue equation 2 for v -i/cos(i/f) + ^ s i n ( i / f ) _
p
i
v^0
sin(i/|)
(10)
The solution v(p) of this equation defines the needed adiabatic potential (v2 — l/4)/p2 for the hyperradial equation (3) from which one obtains the total energy and the radial wavefunction of the system. Collapse. In the small distance region, p
2ms h2
f(p) = 0 , p « a.
(11)
For a given negative energy E = —h2K2/(2m) the energy is negligible compared to the potential when p <£ « _ 1 , i.e
which has solutions of the form f(p) ~ pn, where n = \± \\/\ C > 1/4 the exponent n acquires an imaginary part ±ib, i.e.
f{p)~Jpe±mn».
— 4C. For
(13)
For any given energy this wavefunction has infinitely many nodes at small distances which means that there is actually no ground state - the system collapses. In a three-body system with zero-range interactions this collapse at small distance is called the Thomas effect1.
71
Both the three-body zero-range force 7i(5(ri -T2)6(TI - r 3 ) and the density dependent zero-range force 73n"(ri)<5(ri-r2), applied to a three-body system, are non-vanishing only when all three particles are located at the same point in space. This configuration corresponds to p = 0 or Inp = —oo. Apparently, independent of whatever might happen at Inp = — oo, the infinitely many nodes of the wavefunction (13) mean that the system still collapses. Conclusion. We have shown that the Skyrme forces, directly applied to a many-body system in coordinate space, imminently lead to a collapse of the system driven by the three-body correlations. Any three-body correlation must therefore be excluded from the wavefunction of the many-body system. To remove the collapse one has to regularize the zero-range force by introducing a finite length scale R, which will alter the —C/p2 behaviour of the effective potential at p ~ R and consequently remove the collapse. The three-body system will then have approximately ln(a/R) bound states with the ground state having a finite binding energy of the order of h2 /(2mR2). It is also possible to remove the collapse by abandoning the local density approximation in the density dependence of the forces. This will also introduce a physical length scale which will regularize the three-body system. Appendix. If rrii and r; refer to the z-th particle then the hyperradius p and the hyperangle a, are defined in terms of the Jacobi coordinates x^ and y* as J psin(ai) = Xi , Xi = W (r, -rk) V mrrij +mk
pcos(aj) = |yi| , yi =
I 1 m j (rrij + mk) mrrii + rrij + m,k
,
(14)
_ rrijTj + mkVk
rrij + mu
The set of angles fl, consists of the hyperangle a^ and the four angles and yi/|yi|. The kinetic energy operator T is defined as 1
-Ip
+
2mpiA
' lp~
2m\P
dp29
p> 4 J '
XJ/|XJ|
(15)
*2 1 d2 . (n x A l\. II. A2 = - . , z*' + — £2 — , 2 sin(2a < - 4 + sui(2aj) daf sin (aj) cos (ai) where 1^; and lyi are the angular momentum operators related to x, and yj. 1. L.H. Thomas, Phys. Rev. 47 (1935) 903 2. D.V. Fedorov and A.S. Jensen, Phys. Rev. Let. 25 (1993) 4103
Hadron Dynamics
H A D R O N DYNAMICS: P R E S E N T STATUS A N D F U T U R E PERSPECTIVES*
TULLIO BRESSANI Dipartimento
di Fisica Sperimentale. Universita di Torino and I.N.F.N. - Sezione di Torino Via P. Giuria 1. 10125 Torino Italy, e-mail: [email protected] Present status and future perspectives in Hadron Dynamics are discussed. Examples of how significant experiments can shed light on topics like Nuclear Medium effects, Exotic Meson and Quark description of low energy Hadron interactions are given.
1
Introduction
In Nuclear Physics the item "Hadron Dynamics" covers a vast domain of nuclear and subnuclear phenomena, with not well defined borders. Very roughly we may set an upper limit of distances at 1 fm, and that corresponds to the description of nuclei in terms of colourless particles, nucleons and mesons, in which case Hadron Dynamics has to be interpreted as Dynamics of Hadrons inside Nuclei. But if we interpret Hadron Dynamics as Dynamics of Elementary Constituents inside Hadrons, we deal with much shorter distance scales, where Hadrons are composed of interacting quarks and gluons. In the last two decades QCD has emerged as the theory for the strong force with quarks and gluons as the building blocks of nuclear matter at large densities and high temperatures. One of the most exciting challenges for nuclear physics is the study of the non-perturbative regime of QCD. It is this regime which is relevant for understanding how the elementary fields of QCD, quarks and gluons, build up particles such as protons and neutrons. A basic theoretical difficulty is the non-existence of asymptotic, isolated, coloured objects. This is a feature of the richness of the vacuum structure of QCD. Understanding the different QCD phases and the transitions among them is the challenge of the modern study of strong interactions. At low energy, chiral symmetry can be used to build an effective theory of hadron interactions. At higher energies the parton model uses non-perturbative quark and gluon distributions to describe hadronic scattering processes. *I WISH TO DEDICATE THfS PAPER TO THE MEMORY OF MY FRIEND GIULIANO PREPARATA, WHO PASSED AWAY PREMATURELY 24 APRIL 2000.
73
74
In parallel to this unified theoretical approach to describe so different phenomena, I have noticed also an interesting unification in the experimental attitude. In the past years, experiments using e.m. probes and hadronic probes were carried out, obviously, at different machines by different communities which, instead of being in close communication and sharing the benefits (cleanliness of the e.m. probe, flavour richness of the hadronic one), tried perhaps to stress the advantage of one approach against the other. Thanks to a new generation of machines (TJNAF, DA$NE) and of very performing general-purpose detectors, approaching 47r angular acceptance and featuring the best technology, experiments combining the benefits peculiar to each probe are more frequent. I shall review some items of Hadron Dynamics that seem to me particularly interesting, also in perspective, and bearing in mind the contributions that were presented at this Conference. 2
Nuclear Medium effects
This topic may include many different phenomena, in particular if we consider both aspects implicitly comprised in the definition: modifications of the Hadron properties by the Nuclear Medium or modifications of the Nuclear Medium (the Nucleus) by a Hadron. To my opinion the more spectacular example of how the basic properties of a hadron are modified when it is embedded in a nucleus is given by the non-mesonic (NM) decay of A -Hypernuclei. The effect is known since long I1) and is very often forgotten. The free A decays principally into a AT and a 7r, the Af having a momentum of approximately 100 MeV/c or about 5 MeV of kinetic energy. Even neglecting the binding of the A , a A at rest in nuclear matter cannot decay into a Af of this momentum, because the nucleon Fermi momentum is about 270 MeV/c. That is, the process is Pauli blocked. It has long been recognized that, because the mesonic decay channel would be strongly inhibited in all but the lightest Hypernuclei, the primary decay channel would be the weak non-leptonic reaction AAf —> AfAf. The NM decay of A -Hypernuclei has received a growing attention in the last few years, in particular from the theoretical point of view. Models based on one-meson exchange or on a quark description were developed in order to explain the existing data, scarce and affected by large errors. More details will be given in the contribution of Parreno ( 2 ). In spite of the difficulty of the experiments (low rates with existing machines and experimental facilities) recent very interesting new data were published by the SKS Collaboration at KEK ( 3 ).
75
An unprecedented step forward in the experimental study of NM decays of A -Hypernuclei is expected from the facility FENTUDA at the recently commissioned ^-Factory DA$NE at Frascati. The A -Hypernuclei are produced by stopping the K~ from $~decay in solid nuclear targets surrounding the interaction region. Fig. 1 shows a sketch of the apparatus.
Figure 1. Schematic view of t h e PINUDA apparatus.
A description of the expected performances of the detector will be given by Venturelli ( 4 ). With the new data from FINUDA I may expect that important answers will be given to crucial issues like the validity of the A I = l / 2 rule for weak decays, that could be violated in NM decays and the study of the parity non-violating term in the four-baryon weak interaction (not accessible in A/W scattering experiments due to the overwhelming strong interaction). The detector is ready but the data taking has not yet been started due to delays of commissioning the superconducting solenoid and, mainly, to the low luminosity up to now achieved by DA$NE. Remaining in the field of Hypernuclear Physics, a beautiful example of how we may modify a nucleus by adding to it a hadron in a well defined state is given by the experiment on 7-spectroscopy of \Li ( 5 ). The experiment is very nice since it features the best state-of-art technologies in mediumand low-energy Nuclear Physics. \Li is prepared by means of the reaction 7T++ rLi -> \Li + K+ at 1.05 GeV/c using the SKS spectrometer at KEK. \Li may be formed in the ground or low-lying excited states (see fig. 2 a)). When a A in a Is orbit is added to a loosely—bound nucleus such as6 Li, the nucleus is expected to shrink into a more compact system due to the attractive force between A and nucleons ("glue-like" role of the A ) which results from a property of the A being free from the Pauli principle in a nucleus ( 6 ). This effect can be verified from the E2 transition probability B(E2), which contains information of the nuclear size.
76
a)
IOSGCV/C
ySt-JU'±
-1/2 T-I
ff (6-0-15°) 0.60
Ml
z±
7/2*
out
S/2* 2.05
1.23
J/2
I*:--— (MeV) ,
0.69
-1/2* 7Lj
(McV)j " UJ,) I " " " " jHiymuail.
2000
2050
2100
2150
2200
Figure 2. a) Expected level scheme and 7 transitions of \Li b) ~y^ay spectrum of the E2 (5/2+ -» 1/2+) transition of \Li (from( 5 )).
The E2 (5/2+ -> 1/2+) transition of \Li is essentially the E2 (3+ -> 1+) transition of the core nucleus 6Li, but the existence of a A in the Is orbit is expected to shrink the 6Li core. Experimentally, B(E2) is derived from the lifetime of the 5/2+ state. The expected lifetime (~ 10 ps) is of the same order of the stopping time of the recoil \Li in Lithium in the case of the (n+, K+) reaction at 1.05 GeV/c. This condition is ideal for measuring the lifetime using the Doppler shift attenuation method (fig. 2 b)). B(E2) experimentally found by means of this very clever experiment indicates a shrinkage of the \Li size from 6Li by about 20%. Hadronic degrees of freedom appear in an effective way in the Nuclear Medium. Significant modifications are predicted by QCD-inspired models for the density and temperatures regimes encountered in heavy ion reactions. Sizeable mass changes (10 — 20%) are predicted already at nuclear matter densities and should be accessible experimentally in pion induced reactions in nuclei. These modifications of hadron properties may be associated with a decrease in the chiral condensate with increasing density and temperature. They could be a precursor phenomenon for chiral symmetry restoration. An indication for the existence of such an effect is perhaps provided by the study of 7T+A -> -K+IT^X reactions by the CHAOS Collaboration at TRIUMF. A marked and increasing with A near threshold enhancement appears in the (7r+,7r^) invariant mass distributions, not in the (TT+,TT+) ones. It could be related to a mass shift of the quite elusive a-meson (7) More clear evidence should come from Nuclear Medium modifications of the mass and width of very much well known mesons, namely the vector mesons p, w, . Well grounded predictions exist in this case. However the main decay channels of vector mesons are into hadrons (TT, K) and experimentally the signal for a possible medium modification could be completely
77
p;(MeV/c)
Figure 3. fi annihilation cross section on C, Al, Cu, Ag, Sn, P b as a function of the n momentum (from ( 9 )).
washed out by final state interactions. A possible way to overcome such a difficulty is to look at the very rare (B.R.~ 10~4) dielectron (e + e~) decay of these resonances, immune from final state interaction. Experiments up to now performed (DLS at Berkeley) suffered from reduced statistics. A dedicated large acceptance and good energy resolution dielectron spectrometer (HADES) has been recently built at GSI and will start soon data taking with the pion and ion beams from SIS. An updated report on this facility will be given at this Conference by Priese ( 8 ). Let me conclude this item with a speculation of studying possible modifications of well understood hadron interactions in a Nuclear Medium with temperature/density different than the normal ones. Antinucleon annihilation in nuclei may be considered as a very simple and well understood interaction. As shown by the systematic study of n-nucleus annihilation over the full A range and for p n in the range (50 - 400) MeV/c, presented for the first time by Botta ( 9 ) at this Conference (see fig. 3) the full set of data can be parametrized by the simple formula
78
in nuclei. The first M annihilates on one nucleon and heats a small piece of nuclear matter, the second M annihilates in such a piece and a modification of the observables of the annihilation (multiplicity, single particle energy spectra,..) would be a signal for a modification of the interaction. Present high energy proton accelerators do not have enough primary beam current to allow for such an experiment. If the Japan Hadron Facility will be built, a d beam could be installed, with an intensity (~ 100 J/pulse) suited for the study of J annihilations in nuclei ( 10 ). 3
(T,K,
quasi)—nuclear states
The simpler phenomenological/theoretical description of hadron-nucleus and hadron-hadron interactions is provided by the potential models. As a first step these models give an overall picture of the experimental data, relating possibly the various terms appearing in the potentials to the appropriate description of the elementary interaction (hadron-nucleon, quark-quark). A second step is that of utilizing these models to predict new effects that can be looked for experimentally. A nice example of this approach was given recently by the experimental discovery of the deeply bound 7r-Nuclear states by means of the (d, 3He) reaction on 20aPb at GSI ( n ) , whose existence was predicted by Toki and Yamazaki ( 12 ) using 7r-Nucleus potential models. Stimulated by this circumstance, it was speculated (see, e.g. ( 13 )) that also K~ and p could be found in deeply bound atomic states, relatively narrow, in heavy nuclei. Experimentalists are looking whether it is possible to observe also these states in some dedicated experiments. Turning to the potential model description of hadron-hadron interactions, a well studied case is that of the M~M interaction in terms of the exchange of mesons. It can be linked to the M~M interaction by a G-parity transformation, which changes the sign of some of the contributions from the exchanged mesons. As a result the MM potential is, on the average, deeper than the MM one and may accomodate a variety of MM bound states near threshold (~ 2mff), called "quasi-nuclear" states. Their prediction and their claimed observation in some low-statistics experiments was one of the major physics motivations for the construction of the low-energy p accumulator ring (LEAR) at CERN. However the first dedicated experiments at LEAR did not confirm the previous observations and the case of the "quasi-nuclear" MM states was given off. A possible "resurrection" of these states was provided by the experiment FENICE at Frascati, which reported an anomalous behaviour of the e+e~ —> fin and e+e~ —+ multihadron cross section near the MM threshold ( 14 ) and,
79
0
FTI I
0
i 1 i i i i li i i 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 i i i 50
100 150 200 250 300 350 400
450
n momentum [MeV/c] Figure 4. Experimental values of the total and annihilation np cross sections (from ( 1 5 )).
very recently by a measurement of the ftp total and annihilation cross section down to a laboratory momentum of 50 MeV/c. The annihilation cross section exhibits a smooth behaviour, well reproduced by an effective range expansion approximation, whereas the total cross section shows a dip-bump structure below 100 MeV/c (see fig. 4), attributable to a dip-bump structure of the elastic cross section ( 15 ). Such a structure would exist in the pp elastic cross section too, never measured down to these low momenta. It would be possible perhaps ( 16 ) to perform this measurement at the new facility AD of CERN, even though not optimized for this kind of experiments. 4
Exotic mesons
Our understanding of the meson spectrum is largely based on the so-called Constituent Quark Model. In its simplest form it describes mesons as colour neutral (qq) pairs, bound by a relativistic or relativized potential, where a linear behaviour of the potential at great distances accounts for the Confinement. Considering only the lightest quark flavours (u,d,s) the model assumes flavour independence of the quarks. This leads to the prediction that mesons will appear grouped in nonets, with specific combinations of spin (J), parity (P) and C-parity (C) which are those of a two—fermion system with total spin 5 and orbital angular momentum L. Actually, the success of the Constituent Quark Model is not understood from the point of view of QCD. Indeed, why complicated bound states of dressed quarks can be described in terms of few constituents at all? Moreover, why do we need to consider only the simplest states (qq), (qqq) out of them?
80
In effect QCD predicts the existence of entire classes of hadrons, which do not exist in the Constituent Quark Model. For instance, since QCD is a non-Abelian gauge theory, the gluons also carry colour charges and are able to interact with each other. Gluon-gluon interaction is the distinctive feature of QCD and is responsible of its most spectacular consequence: the Confinement. Additionally it gives origin to the most striking prediction of QCD: the existence of bound states of gluons (gg, ggg) called glueballs. Furthermore, a gluon may stick on a (qq) pair forming an hybrid meson (qqg). Four quark (qqqq) states or even more complicated quark combinations may exist as bound states. All these expected states are generally named as "exotics". An outlook on this subject will be given by Villa (1T) at this Conference. A general consensus exists about the quantum numbers of the ground state glueball that should be a scalar (Jpc = 0 + + ) at a mass around 1.5—1.7 GeV (a region full of normal (qq) states); glueballs with quantum numbers 2 + + and 0 h are expected at masses above 2 GeV. Being flavour blind, the decay pattern of a glueball should be definitely different than that of a (qq) meson. The most interesting prediction concerning the hybrid mesons is that they should appear in nonets with quantum numbers combinations not accessible to a (qq) system (JPC = 1 l", 0"1 , ...). This evidence would provide the most clean experimental signature for an exotic state. The Jpc = 1 ^ hybrid state is a feature of all models; in particular the flux-tube model predicts it to be relatively narrow, at a mass around 1.9 GeV. The experimental situation can be summarized as follows. A few years ago the /o(1500) state, first observed by the Crystal Barrel Collaboration ( 18 ) seemed to have the right mass and decay pattern compatible with that predicted for a glueball. Several other experiments confirmed the existence of such a scalar meson, relatively narrow ( r = 120 MeV), and implemented the study of the decay pattern. At present, there is a growing consensus that / 0 (1500) is a mixture of (gg) and (qq) configurations, as well as the neighbouring ao(1450 or 1300?) and /o(1710), leaving still open the question of which is the ground state glueball. The existence of an object with exotic Jpc(l '"J was established by the BNL-E852 Collaboration ( 19 ) and by Crystal Barrel (m), with a mass of 1370 MeV and a width of ~ 400 MeV. Afterwards, two further states with exotic Jpc were reported by the same BNL Collaboration. The assessment of the quantum numbers ensures the exotic nature of these mesons, but the mass of the lightest of them lies ~ 300 MeV lower than all the theoretical predictions for hybrids, leaving open the possibility for a (qqqq) nature. An 1=2 resonant state resonant would be certainly a (qqqq) system. How-
81 log(L)
tog(L)
7725 7720 7715 7710
—
r = 100 MeV
* T
r=200MeV r=3O0MBV
A
•
o> 7740
O r = 400 MeV D r = 500 MeV A * *
7730
r=600Mev
» r=7oowev
7705 7700
•
r-BOOWeV r=900MeV
I
7720
• M
7710
-
7695 10 7690 7685
: -
r = 100 M«V A r = 200 M»V T r = 30OMeV o r = 400MaV D T = 500 MeV A !" - 500 MeV
700 MeV « rr == eooMev r = goo Mev : ** r
b)
Jj J\ Jr
7700
i-
7690
,,!,,,, TI(JT*TT*),
GeV
1T l , , , , I , ,
.,!,,,, m(7T*TT*), GeV
Figure 5. -log(likelihood) values for best fit solutions as a function of 7T+7T+ invariant mass and for fixed values of the width of the possible isospin two resonant state, in the two hypotheses a) of a scalar resonance, or b) of a tensorial one (from( )).
ever, there is only one entry in the Particle Data Book for such a system, named X (1600), not firmly established, observed in the pp channel. An attempt to see this object in the 7r+7r+ channel, though expected to be depressed by two orders of magnitude with respect to the pp one, was reported at this Conference by Filippi ( 21 ). The analysis was performed through a grid scan of the invariant (7r+7r+) mass spectra obtained ( 22 )in the exclusive annihilation channel np —»7r+7r+7r~ . The -Zog(likelihood) values obtained as a function of the (7r+7r+) invariant mass, for fixed values of T, are reported in fig. 5, in the two hypotheses of a scalar 1=2 resonance (a) and a tensorial one (b). A quite clear maximum is achieved for m = (1.42 ± 0.02) GeV and T = (0.16 ± 0.01) GeV for the spin 0 hypothesis. 5
Quark description of low energy Hadron interactions
An old-standing problem of Hadronic Physics is how to disentangle, for a given low energy interaction, a hadron-exchange description from a Quark Line description. In all the cases that I know it was impossible to give a definite answer in favour of a quark based mechanism. However, recent results on (JTAf) annihilations at LEAR, in particular from OBELIX, showed unambiguous evidence for the failure of the application of the Okubo-Zweig-Iizuka (OZI) rule for some annihilation channels with a naive nucleon wave function. A detailed discussion on this subject will be presented at this Conference by Marcello ( 23 ). Two explanations were put forward to explain such an anomaly. Locher et al. ( 24 ) proposed rescattering diagrams with OZI-allowed transitions in the intermediate state, e.g. ATjV —> K*K —> ($m. Ellis et al. (25) assumed that the nucleon wave function contains polarized (ss) pairs.
82
Then the observed OZI violation is only apparent since there are additional classes of connected quark-line diagrams. This dilemma, quite typical of low energy hadronic physics (do we need quarks or simply hadrons to describe the phenomena?) convinced the OBELIX Collaboration to undertake a study of the following two-body reactions, with n momentum from 50 to 400 MeV/c: (1) np —• <jm+ [ 3 5:, ^ J ; (2) np - u>n+ pSi, ^ J ; (3) np - K*°K+ p 5 i , ^ J ; (4) np -> W+ pPo, 3P2}; (5) np ->77V+ [ 3 P 0 , 3 P 2 ]In the brackets I have indicated the initial states from which the reactions may proceed, at low energies. For reaction (1) the new data confirm the previous observation of a dynamical selection rule for which the annihilation proceeds only from S-wave. This is clearly demonstrated both by the angular distribution of <j> decay products and of cj> production angle and by the momentum dependence of the cross-section, shown in fig. 6a). From this complete set of data it is concluded that production in A/W —> <jm annihilation occurs at 100% from 3Si state, with a ~ 2% incertitude. The results for reaction (2) are quite interesting. From a very careful analysis of the w-decay Dalitz plots it appears that there is a sizeable P-wave contribution (~ 20% at low p„, ~ 40% at high p ft ). The P-wave contribution manifests itself also in the momentum dependence, shown by fig. 6b). A departure from the S-component behaviour represented by the Dover-Richard fit ( 26 ) is quite clear. Reaction (3) is important in this discussion since it is the intermediate state in the rescattering process suggested in ( 24 ). The cross section as a function of pn exhibits a flat behaviour, shown in fig. 6c). This is in contrast with the S-wave scaling trend required by the rescattering model, by hypothesis. Given the success of the mechanism based on the polarized intrinsic strangeness in the nucleon (PISN) in explaining the data on <j>/uj production by n, it seemed useful to analyze other reactions that could test other predictions. One of them was that the formation of an (ss) system with Jpc = 0 h should be enhanced from spin-singlet states, n and n' have a strong (ss) component but the annihilation channels (4) and (5) proceed from spin triplet states only, namely 3 Po and zPi. From this measurement it was concluded that no strong violations of the predictions of the Quark Line Rule were observed, in agreement with the PISN picture. This result strengthens also the statement by Dover and Fishbane 2T that the 0 v quantum numbers for the (ss) admixture in the nucleon may be discarded. The conclusion from this nice series of measurements is that the only way to explain all the data is that of assuming a polarized (ss) content in the nucleon wave function, of the order of 10%. This value is consistent with
83 0.16 \ 0.14 0.12 0.1 0.08 0.06 ~ 0.04 0.02 L a ) 0
[ "+•
f'V *
N. " - * . .
l-c) E
0.2 0 p(n) (GeV/c)
p(n) [GeV/c]
.
" .
i
.
I
.
0.2 0.4 p(n) [GeV/cl
Figure 6. Trend of cross sections as a function of pn for the reactions a) rip —» <^r+, b) np —» unr+ and rip —» K°* A"+. T h e dashed curve represent the trend of S-wave annihilations as delivered by Dover-Richard's model.
that inferred from the irM tr-term, but higher than that deduced from deep inelastic lepton scattering. If on one side it is conceptually very appealing the idea that quite different experimental observations could be linked by the "strangeness in the nucleon", it is not surprising that the quantitative estimations are different. The oversimplifications inherent to the Naive Quark Model lead to theoretical evaluations that cannot compete with those of QCD, which in the regime of deep inelastic scattering is ideally suited to analyze quark and gluon dynamics in Hadrons. The cross sections factorize into non-perturbative, or soft parts, and perturbatively calculable hard parts. I will not discuss these items, covered by other speakers at this Conference. 6
Conclusions
My feeling is that the field of Hadron Dynamics is in good shape since: • some of the expected effects were verified experimentally, at least at a qualitative level (glue-like role of the A in Nuclei, narrow bound states of Hadrons predicted by potential models, Exotic Mesons, Quark description of selected low-energy Hadron interactions). • the bridging between the high energy regime (perturbative QCD) and low energy regime (chiral perturbation theory) is under way both experimentally and theoretically. • powerful dedicated facilities will enter soon into operation (FINUDA at Frascati, HADES at GSI). • in the long range perspective, the approval of the proposed projects (JHF
84
in Japan, ELFE and Glue/Charm Factory in Europe) could allow a real breakthrough in the field. I am grateful to Dr. E. Botta for the precious help in the preparation of this manuscript. References 1. see, e.g., the review by R.H.Dalitz in Nuclear Physics, ed. C. de Witt and V. Gillet (Gordon and Breach, New York, N.Y.,1969), p. 701 2. A. Parreno, these Proceedings 3. see, e.g., H. Bhang et al., Phys. Rev. Lett. 81 4321 (1998) 4. L. Venturelli, these Proceedings 5. H. Tamura et al, Nucl. Phys. A663&664 481c (2000); H.Tamura, these Proceedings 6. T. Motoba, H. Bando and K. Ikeda, Prog. Theor. Phys. 80 189 (1983) 7. F. Bonutti et al, Phys. Rev. Lett. 77 103 (1966); P. Camerini, these Proceedings 8. J. Friese, these Proceedings 9. E. Botta, these Proceedings 10. T. Bressani et al., these Proceedings 11. T. Yamazaki et al, Z. Phys. A355 219 (1996); T. Yamazaki et al, Phys. Lett. B418 246 (1998) 12. H. Toki and T. Yamazaki, Phys. Lett. B213 129 (1988) 13. E. Friedman and A. Gal, Nucl Phys. A663&664 557c (2000) 14. A. Antonelli et al, Nucl. Phys. B517 129 (1998) 15. F. Iazzi et al, Phys. Lett. B 475 378 (2000) 16. A. Feliciello, these Proceedings 17. M. Villa, these Proceedings 18. C. Amsler et al, Phys. Lett. B291 347 (1992) 19. D.R. Thompson et al, Phys. Rev. Lett. 79 1630 (1997) 20. A. Abele et al, Phys. Lett. B423 175 (1998) 21. A. Filippi, these Proceedings 22. A. Bertin et al, Phys. Rev. D 5 7 55 (1998) 23. S. Marcello, these Proceedings 24. M.P. Locher, Y. Lu and B.S. Zou, Z. Phys. A347 281 (1994) 25. J. Ellis, M. Karliner, D.E. Kharzeev and M.G. Sapozhnikov, Phys. Lett. B353 319 (1995) 26. C.B. Dover et al, Progr. Part. Nucl. Phys. 29 87 (1992) 27. C.B. Dover and P.M. Fishbane, Phys. Rev. Lett. 64 3115 (1990)
Section I. Strange Hadro-Dynamics
KNA A N D K N E C O U P L I N G C O N S T A N T S IL-TONG CHEON Department of Physics, Yonsei University,Seotil 120-749 ,Korea MOON TAEG JEONG Department of Physics, Dongshin University, Naju 520-714, Korea Coupling constants of KNA and KNT, vertices have been calculated in the Chiral Bag Model. Their form factors were also estimated.
1
Introduction
In order to investigate processes containing strangeness such as kaon-nucleon scattering, kaon photoproduction and hypernuclear processes, KNA and KNT, form factors and their coupling constants should well be determined. However, their available values are very rough. For example, kaon photoproduction gives - 0 . 0 > ffig > - 4 . 0 and 1.82 > *^g* > -0.24[lj. In this paper, we will calculate these form factors in the Chiral Bag model with vector mesons by extending the SU(2) model of wNN vertex to the SU(3) model where the pseudoscalar mesons are it and K and vector mesons are p,w and K*. And their coupling constants are determined in the limit of vanishing momentum.
2
Theory
By the chiral transformation of the chirally invariant Lagrangian given by Chodos at.a/[2], Kalbermann at.al derived an effective Lagrangian which does not contain a fictitious sigma field[3]. In order to introduce vector mesons, we must take a local gauge transformation of the above effective Lagrangian, i.e. the minimal coupling,^ ->• 9M — ^gv{r • /?M -I- wM), in which rho mesons are coupled to the SU(2) isospin and omega mesons are coupled to the U(l) baryon number. The anomaly term, i.e. Wess-Zumino term, (3gngl/4:Tr2)et"'aPdlicjv(%ll-dpTr is added by hand to this resultant effective Lagrangian [4], in which the interactions of pion-quark and vector meson-quark are contained. The kaon-quark interaction Lagrangian can be obtained by extending from the basic doublet (u,d) to a triplet (u,d, s), substituting the Xt of SU(3) generators for the r, of SU(2), and taking into account pseudoscalar
86
87
mesons (ir,K) and vector mesons (u,p,K*). They are /"d3xg(x)7"75A • DliK(x)q{x)Qv, CK-„ = 9v J #xq(xyf\
• K;(af)g(aj)ev,
(1)
where gk = l/2fk and gkq = mkqk with fk = 114MeV. The KNA and KNT, form factors are obtained by evaluating Feynman diagrams shown in Figs.l based on the interactions given in eqs.(l) and (2) and the Wess-Zumino term. The results are expressed as
^
P
= A^9aU(K) x
/
dkpkp
+
^ s r V ^ U i k l ) B(si)u\kp)
U0(U)0 + OJNS)(U0 + Ura + UNS')
+ a2 f dk k2U(krM D
™'Si(kvr\)
+
•
k^)U(K)
4a)si(*Sa))
M fdk 4
SS'Si(kv)
+ 9,9, IdkAK
y (
+
C
W
k
, 4a)So(kia)))
(k»xki a ) )K+^ a ) +^ s )
" Vwvuia\uv+t-<-~V'--'-<—v>-< -*->->{a)>
x[G(sa) Soik^S, (*(«)) + ^ a ) ,Sb(*i a ) )5i (*„)]},
(2) (a}
where a and Y denote K and (A, S), respectively. We also define as k), kv + A£, u>v = (ml + ki
U(k) = N2J
y/2
and LJNS —ms
r2dr(j0(kr)\j2(ar/R)
Sn(kv) = ( i + |)iV 2 J
r2+ndrEl^0(Sab
- ^(flr/fl)] -
=
— mN.
-h(kr)j 2 (ilr/R))
n)2ja(OMOJo(kvr),
where n = 0 or 1 and £ = flr/R. ji(x) is the spherical Bessel function and ft = 2.043 is determined by the bag boundary condition jo(^) = jt(Q). All coefficients are given as AW = -y/S^fi^ = -125^(100), B ^ = -640^3(500) ,£#£? = -160^3(250) ,B^] = -128>/5(76),
88
Table I. T h e form present k(MeV) /KN\ 0 1 200 0.73 400 0.50 600 0.33
factors. work ref.(5) JKNY.
1 0.61 0.38 0.25
firNN
1 0.79 0.55 0.36
ref.(6) /TTJVA
1 0.70 0.47 0.32
firNN
1 0.75 0.48 0.33
firNA
1 0.69 0.45 0.29
B
A£? = -256V5(76),£Jft,° = -525^3(270) .tfjgf0 = -1050^3(60), B ] N^ = -210v/3(24),^ K) = -64V3(36),B^V = -256\/3(72), r>(*0 _ D(^T) r-W - n(Xir) r-W - /?(**) ^ ) - *?(**"> r W °iVA
— % A >°JV£
— ^ i V E >°iVX:*
—
^AfE* ' °AE
~
D
4£
'°AE*
~
K)
B ^ ' ^ M = - 5 4 0 ^ ( 0 ) , D™ = 0 ( 1 8 0 ) , ^ = -160v/3(40),E^ = -60 V / 3(20),F| X ) = - 6 4 > / 3 ( 1 2 ) , F ^ ) = - 1 0 8 7 5 ( 3 2 ) , ^ = -48v/3(-16), Gffi = -320 v / 3(-24),if| / f ) = -128^3(0) and H ^ = 0(216) where values in parentheses are for the KNT, form factor.
3
Numerical results and discussion
The results of numerical calculation with eq.(3) are shown in Table I. We used the bag radius R = 0.9fm and gv = 5.21. Our /KN\ and /R-JVE form factors are harder than f„NAIn our calculation, we used values of coupling constants, f„ = 93MeV, gv = l/2/ f f = 0.75/m ff , /* = 114MeV and gv = 5.21 for the bag radius R = 0.9fm. This gv = 5.21 was actually determined to obtain the coupling constant glNN/4n = 14.3 with R = 0.9/m. The value gv = 5.21 is coincident with (gp + gJ)/2 = (5.8 + 4.62)/2 = 5.21. Using the current-field identity and the limit of vector meson dominance, we also calculated the coupling constant for K* -¥ Kj decay , GK*K-r = ^1$^ = 0.0732, while the measured decay width (115 ± ll)keV gives GK.Kl = 0.0533. The coupling constants of KNA and KNT, couplings are obtained in our present framework as gKN\/V4n = —3.77 and gKNz/V^Tr = +1.19 with R = 0.9/m and contributions from each diagram are given in Table II. Our results are certainly in the predicted range [1] and agree reasonably with those extracted by analyses of the kaon photoproduction[7],—4.17 ±0.75 and +1.18 ±0.66.
89
Table II. Contributions from each diagram. diagram 9KN\/\/4M /V4TT
(la) -2.69 0.52
(lb) -0.24 0.11
(lc) -0.09 0.03
(Id) -0.19 0.07
./'
Y N \
S
(H)
(lb)
S
K 3
/
S
Y N
(Id)
Y N
(le) -0.13 0.08
(If) -0.43 0.38
sum -3.77 1.19
'\
(lc) K \
/
S
•
Y N
(le)
(II)
Figure 1. Feynman diagrams for the KNA and KNT, form factors.
4
Conclusion
The results of the form factors and the coupling constants derived in the chiral bag model are successful and comparable to those obtained by conventional methods. Our values of coupling constants, gxN\ and gxNi: are in the range of those determined by analysis of production and different models. It should be stressed that the Wess-Zumino term makes an important contribution to those quantities calculated in this work. Acknowledgements Support came from BK21 Project of the Korean Ministry of Education. References 1. 2. 3. 4. 5. 6. 7.
M.K.Cheoun, B.S.Han, B.G.Yu and Il-T.Cheon, PRC 54, 1811 (1996). A.Chodos and C.B.Thorn, PRD 12, 2733 (1975). G.Kalbermann and J.M.Eisenberg,PRD 28, 71 (1983). Il-T. Cheon and M. T. Jeong, J. Phys. Soc. Japan 6 1 , 2726 (1992). M. T. Jeong and Il-T. Cheon, Eur. Phys. J. A4, 357 (1999). C.Schutzet.a/.,PRC 49, 2671 (1994). R.A.Adelseck and B.Saghai, PRC 42, 108 (1990); PRC 45, 2030 (1992).
ON THE £ HYPERNUCLEUS
E. SATOH Kanto Gakuin University, 1162-2 Ogikubo, Odawara, 250-0042, Japan E-mail: [email protected] M. KIMURA Department of Electronics Engineering, Suwa College, 5000-1 Toyohira, China, 391-0292, Japan E-mail: [email protected] We calculate the £ well depth with the YN potential reproducing all the YN scattering data, the observed binding energy (3.12 MeV) of ^He and the empirical A well depth (30 MeV). We point out that the 2 hypernuclear state would almost impossible except the particular nuclear system.
1
Introduction
First observation of sigma-hypernucleus was made by collaboration of HeidelbergSaclay-Strasbourg 1 , that is f;Be. After this report, several "sigma hypernuclei" were reported by several experimental groups. However, it was realized that simple structure assignments based on narrow, particle-hole substitutional state were not possible 2 . The £ particle should be converted to the A particle inside the nucleus by the strong interaction (EiV —*• AN), therefore the £ hypernucleus would not exist. Therefore, the existence of f;Be was quite strange. If there is the £ hypernucleus, a mechanism for suppressing the TUN —• AN transition must be existing. We realize that the mechanism is due to the Pauli principle connected with the isospin conservation 3 . The isospins of A and E particles are 0 and 1, respectively. The 8 Be is the nucleus with total isospin zero, the infinite nuclear matter is also the system with total isospin zero, let us investigate the E hypernucleus by using the infinite nuclear matter (NM) instead of 8 Be. Assuming the transition: initial state (NM+E) —• intermediate state (NM+A) —> final state (NM+E), the isospin conservation of the system (NM+F) is broken. However the isospin conservation is guaranteed by imposing the Pauli principle on the nuclear matter, we verify the suppression mechanism by a simple model calculation. 90
91
2
YN
Potential
In order to make a model calculation, we set up the hyperon-nucleon potential. For simplicity, we assume that all the hyperon-nucleon potential are represented by the separable potential of Yarnaguchi's type. Since we treat the following transitions: AN ^ AN, AN ^ EN and EN ?=* EN, the isospin state 1—1/2 and 7=3/2 for YN pairs must be considered. 2.1
YN Potential viith 1=1/2
Since we take account of A and E components explicitly, we start with the coupled channel equation: (T - E)V = - VV
-(&)-(£)•
(1)
<2»
The two channel YN potential is represented by / ft I V I T3 \ - ( VAA {P IV I p )- ^ ^
FAS
^ - _ (X*3h(p)9*(j/) ^ J {xxg^gAb/)
K9Kip)9n{jf)1 \ XvgzWgxip ) J
(o\(6)
here, gA(p) = gK{ph) = (pX+PlT1, Ps(p) = 9v(Pz) = {PZ+PT)~\ PA a n d p r are the relative momenta of AN and EN pairs respectively. The range parameters are related by /XA/3| = ME/^A + 2 ^ A M E ( T O £ - % ) , HA and /is are reduced masses of AN and £JV respectively. Then, the parameters of YN potential are -^A> ^x, AE and /fe (or /?A), SO the number of parameters are 8, 4 parameters for spin singlet state and 4 parameters for spin triplet state. On the other hand, the available experimental data are the scattering length and the effective range: a^s = - 1.8 fm, TA« = 2.8 fm, a^t = - 1.6 fm and r^t — 3.3 fm, the scattering cross sections: a(Ap —» E°p), cr(E~p —> An) and CT/=1/2(SiV —> EN) = [3{cr(E-p -> E~p)+a(E~p -» E°n)}-a(E+p -» E+p)]/2. It turns out that the available experimental data are 7. The parameters of Y N potential are not uniquely determined. Therefore we make many sets of YN potentials. Then we calculate the binding energy of A He and the A well depth D A , for investigating the YN potentials. The binding energy of A He has been calculated by the shell model of Law and Nguyen 4 . The formalism for calculating the A well depth can be found in our previous work 5 . The observed binding energy of A He is BA = 3.12±0.02 MeV, and the empirical A well depth is about 30 MeV. Let us show the BA and the DA calculated with various potential sets ( setl to set 7).
92
Table 1: Calculated values of B\ and DA (>n MeV). BA DA
setl 0.89 27.98
set2 1.04 28.50
set3 1.19 29.98
set4 1.81 30.97
set5 2.46 32.96
set6 2.80 33.96
set7 3.15 34.96
Prom these results let us adopt set 7 as the YN potential for calculating the E well depth with isospin 7=1/2 state. Let us show the parameters of set7 in Table 2. Table 2: Set7 of YN potential (7=1/2). ^(fm-1) ^(fm-1) A A (fm- 2 ) A*(fm- 2 )
Mfm- 2 )
2.2
spin singlet 1.56630 2.14390 0.0423998 0.0291312 0.0232190
spin triplet 1.27390 1.94178 0.00695881 0.0589969 0.168926
YN Potential with 1=3/2
The YN potential with 7=3/2 is set up so as to reproduce the experimental data E + p) and da(E+p —> T,+p)/dQ. The E + p potential is described by { Ps | V I Px ) = - As^sQ^Jfifs^). The E + p potential has been set up in our previous work 6 . Table 3: E+p potential (1=3/2). /^(fm-1) AE(fm-2)
3
spin singlet 1.24579 0.0229809
spin triplet 1.24579 -0.0191179
Nuclear Matter
The formalism for calculating the E well depth can be given in our previous work 6 . Let us write only the necessary formalism. The ground state energy Ez(I) with the isospin I is described by
£ E ( / ) = - 4 " [tPWikE,
kv)9{avkF - At)
(4)
93
herefcsis the relative momentum of £ and N, a E = raE/(mE +mjv) and kp = 1.36 fin-1. The G-matrix is obtained by solving Bethe-Goldstone equation in the independent pair approximation. G/=1/3(*E,
C
ks) = - ^ ( f c X A s - (AAAE - \2X) JA}/D
/=3/2
( f e , A*) = -A E 5 E (fc E )/(l - AE J E )
D = 1-A
A
JA-A
S
J
S
(5) (6)
+ (AAAE-A2)JAJS
(7)
with
JK=f*P&4&± 7
(8)
eA
JS=/^QMIM
(9)
where Q A and QY, stand for the Pauli principle for the nucleons which interact with the A and £ particles respectively. Energy denominators are represented by eA=eA + UN = —-(pl-k 2 A -ie) + UN, (10) 2/xA e E = e E + f/jv + A s = — - ( p | - Ag. - »e) + UN + A E ,
(11)
•'ME
where A E stands for the single particle energy spectrum of a £ in nuclear matter. UN is the single particle energy spectrum of a nucleon in nuclear matter, which is obtained in nuclear matter calculation by Bhoragava and Sprung, UN = S5.17 - 12MK% (MeV) (12) here KN is the nucleon momentum. The well depth 7>E(7) and width r E ( 7 ) are defined by £ E (7) = D E (7) + ^ r E ( 7 ) .
(13)
Thus total energy E-z is obtained by £ 2 = \{E^{I=l/2)
+ 2£s(/=3/2)}.
(14)
Therefore the total well depth D E and the total width r E are obtained by 7?E = R e £ E
(15)
rE=2Im£E.
(16)
The results calculated with YN potentials (7=1/2, 7=3/2) are as follows: D s = 20.16 MeV, r E (total) = 1.34 MeV while r E ( 7 = l / 2 ) = 4.02 MeV and r E (7=3/2) = 0.
94
4
Effect of Effective Masses
Let us summerize the calculated results: B\(AHe) = 3.15 MeV, DA = 34.96 MeV, Z?s = 20.16 MeV, r E (total) = 1.34 MeV and r E ( / = l / 2 ) = 4.02 MeV. The calculated A well depth DA is slightly larger than the empirical value of about 30 MeV. Next we consider the effctive masses of the hyperons. The effctive masses should be chosen so as to make D J minimum (stable value) by changing {mA/mA, m^/my) under the condition mA/mA < 1, m^/m^ > mA/mA. Where D^ stands for the S well depth calculated with the set of effective masses. The Z?E, r ^ ( / = l / 2 ) and r E (total) calculated with the effective masses are shown in Table 4. Table 4: The E well depth and width with effective masses. TOA/rnA 0.7 0.7 0.7 0.7 0.7 0.7 0.8 0.8 0.8 0.8 0.8 0.9 0.9 0.9 0.9 1.0 1.0 1.0
mf,/TO£ 0.8 0.9 1.0 1.1 1.2 1.3 0.9 1.0 1.1 1.2 1.3 1.0 1.1 1.2 1.3 1.1 1.2 1.3
Dh
17.0 14.7 14.8 14.9 15.2 15.4 16.7 14.76 14.83 15.1 15.3 16.2 14.8 14.9 15.1 15.7 14.9 15.0
rf.(J=l/2) 27.0 17.4 11.0 7.8 5.8 4.6 29.6 16.8 10.6 7.4 5.6 30.8 16.0 10.2 7.2 29.8 15.2 9.8
I^total) 9.0 5.8 3.7 2.6 1.9 1.5 9.9 5.6 3.5 2.5 1.9 10.3 5.3 3.4 2.4 9.9 5.1 3.3
Next we recalculate the A well depth DA with the effective masses which make £>£ minimum. Then let us list up all together in Table 5. Table 5: D A calculated with the selected effective masses. mA/mA 0.7 0.8 0.9 1.0
TO^/TOE
0.9 1.0 1.1 1.2
Of 14.7 14.8 14.8 14.9
rj,(/=i/2) 17.4 16.8 16.0 15.2
r £ (total) 5.8 5.6 5.3 5.1
D* 32.8 32.0 31.2 30.5
95
The D A = 30.5 MeV is acceptable, so we choose (m A /m A =1.0, mj,/m£=1.2), £>£ = 14.9 MeV, r £ ( 7 = l / 2 ) = 15.2 MeV and rf,(total) = 5.1 MeV. As the effect of effective masses, the S well depth becomes shallower and the width becomes broader. 5
Effect of Pauli Principle
The effect of Pauli principle can be seen by discarding the Pauli principle and making the energy free in the intermediate state. This means that we recalculate the total ground state energy ET. after taking QA = 1 and replacing the energy denominator <=A by the free energy eA in Eq. (8). The results are as follows: D£(Q A =l,e A - • e A ) ~ 10 MeV, r £ ( I = l / 2 , Q A = l , e A -» e A ) ~ 150 MeV, r|;(total,Q A =l,e A —• e A ) ~ 50 MeV. This result means that the £ hypernuclear state can not be realized without the Pauli principle. An example is the experiment performed by Tang et al 2 . 6
Concluding Remarks
We made a model calculation on nuclear matter. The nuclear matter is the total isospin zero system as 8 Be. We show that the Pauli principle plays an essential role for suppressing the transition EAT —• AN in nuclear matter. The S hypernuclear state would not be easily observed 7 . The £ hypernucleus would be only |.Be sor far. As the analogy, possible candidates are 1 2 C + £ , 16 0 + £ , 2 8 S i + £ and so on. Finally, we make a comment on f;He system 8 . We show that the binding energy of A He is very sensitive to the YN potential. Harada should investigate his potential arranged from Nijmegen's potential by calculating other real system like A He before the calculation of f;He. References 1. 2. 3. 4. 5.
R. Bertini et al, Phys. Lett. B 90, 375 (1980). L. Tang et al, Phys. Rev. C 38, 846 (1988). E. Satoh and M. Kimura, Prog. Theor. Phys. 94, 561 (1995). J. Law and T.D. Nguyen, Nucl. Phys. B 24, 579 (1970). Y. Nogami and E. Satoh, Nucl. Phys. B 19, 93 (1970). M. Kimura and E. Satoh, Prog. Theor. Phys. 88, 605 (1992). 6. M. Kimura and E. Satoh, Prog. Theor. Phys. 9 1 , 319 (1994). 7. S. Bart et al, Phys. Rev. Lett. 83, 5238 (1999). 8. T. Harada et al, Nucl. Phys. A 507, 75 (1990). T. Harada et al, Few Body Sys. Suppl. 5, 341 (1992).
N O N M E S O N I C W E A K DECAY OF H Y P E R N U C L E I A. PARRENO Institute for Nuclear Theory, University of Washington, Seattle, WA 98195, USA The nonmesonic decay (NMD) of single- and double-A hypernuclei is approached within the framework of a One-Meson-Exchange (OME) model. Realistic baryonbaryon (BB) potentials are used in order to account for the strong interaction. Comparison with the available experimental data for s- and p-shell A-hypernuclei is presented and predictions for the decay of J^He are given.
1
Introduction
Using the change of strangeness as a signature, the AN —> NN reaction has been very useful to extract simultaneous information about the parityconserving and parity-violating (PV) NN weak interaction. For the time being, the lack of stable hyperon beams makes hypernuclear decay the only experimental source of information about this interaction. Hypernuclei are bound systems of nonstrange baryons plus one or more hyperons. Being the A the lightest among the hyperons, various A-hypernuclei in a wide range of masses have been experimentally produced. Data on AA-hypernuclei is more scarce but promising efforts are being invested at different labs (as BNL and Da$ne). The hypernucleus is tipycally created by using hadronic (using K or 7T beams) or leptonic (e,e' K+) reactions. Once the new system is stable against strong decay it decays via weak interaction mechanisms. For the very light A-hypernuclei, the dominant process is the mesonic decay mode, by which the hyperon decays into pions and nucleons. As the mass number increases the less important this mode becomes due to the Pauli blocking on the outgoing nucleon, and the one-nucleon induced-decay - the nonmesonic decay mode, AN —> NN - becomes the dominant one. 2
Hypernuclear Decay
Assuming the initial hypernucleus to be at rest, the NMD rate is given by 1 :
r
"." = / 0 / 0 M E <*>«*.-*.-*-*> pjVi)l«,.r. (1) 96
97
where the quantities MJJ, En, E\ and E2 are the mass of the hypernucleus, the energy of the residual (A — 2)-particle system, and the total asymptotic energies of the emitted baryons, respectively. The integration variables k\ and k2 stand for the momenta of the two baryons in the final state. The sum, together with the factor 1/(2J + 1), indicates an average over the initial hypernucleus spin projections, Mi, and a sum over all quantum numbers of the residual (A — 2)-particle system, {R}, as well as the spin and isospin projections of the exiting particles, {1} and {2}. The hypernuclear transition amplitude, Mfi, has to be writen in terms of the possible elementary \AS = 1| two-body amplitudes, B\B2 -> -B3-B4. To do so, we work in a shell-model framework, assume a weak coupling scheme for the hyperons and use spectroscopic factors to decouple the interacting nucleons from the core. We assume spherical configuration for the initial hypernucleus. Deviations from this configuration were explored in Ref. 3 where we applied the Nilsson model with angular momentum projection to the decay of A Be, and found that deformation effects could change the decay observables by at most a 10 % from the spherical limit. Assuming the A to decay from the l\ = 0 state, one can write the total NMD rate as T n m = TNN + TYN , where the nucleon-induced decay can be split into the neutron-induced, Tn : An —y nn and the proton-induced decays, T p : Ap -* np, such that TNN = T n + T p . The A-induced decay, on the other hand, contributes to the final An, S°n and S~p hyperon-nucleon (YN) states, i.e.,
r Y N = TAn + Tsora -i- r s - p . 2.1
Weak transition potential
Our OME potential includes the exchange of mesons up to a mass value of 1 GeV. The long-range behaviour of the interaction is given by the IT meson while shorter ranges are described by T],K,p,ui and K*. A detailed derivation of the weak transition potential can be found in Ref. x . The final expression of the potential in momentum space involving pseudoscalar mesons is
V„M = -GrmlJL. ( i + JL* ^ _ f ^
,
(2)
where GFmn2 = 2.21 x 1 0 - 7 is the (Fermi) weak-coupling constant, q is the momentum carried by the meson directed towards the strong vertex, p, the meson mass and M (M) is the average of the baryon masses at the strong (weak) vertex (the other way around for the exchange of strange mesons). For
98
vector mesons the potential reads: Vv(
-
{& +
^%+F*\a1 4MM
x q){52 x
-1—2M—(
(3)
with F\ and F2 strong coupling constants. In Eqs. (2,3) the operators A, B, d, $ and e contain, apart from the weak coupling constants, the specific isospin dependence of the potential, i.e., f\ f2 for IT and p, 1 for TJ and u> and a combination of both operators for the isodoublet K and K*. For K and K* exchanges, where the weak and strong vertices are switched with respect to the other exchanges, there is an additional minus sign to the ( A / = 1/2) PV potential. This sign was not included in previous works and therefore, the results presented here will differ from the ones of Ref. x. At the strong vertex, we use the coupling constants given Nijmegen potentials 2 . In the weak sector, we use the experimentally known couplings involving pions, while for the rest of the mesons, where there is not enough phase space for their free production, SU(3)/SU(6)w4 has to be used. Ref. * gives the values of the couplings at the strong and weak vertices for the AN —> NN reaction. The values for the AA —>• YN channel will be presented elsewhere in the future. In order to account for the strong interaction between the baryons, we solve a scattering T-matrix equation for the outoing pair using the last version of the Nijmegen potentials 2 . For the two-baryon system in the initial hypernucleus we use a spin independent parametrization which simulates the effect of solving a microscopic G-matrix equation. 3
Results
As I have pointed out before the numbers presented here for single-A hypernuclei differ from our previous results 1 , due to a mistake in the inclusion of strange mesons in the mechanism. All those numbers had a common feature: even if they reproduced fairly well the total decay rates for different s- and p-shell hypernuclei, the neutron-to-proton (n/p) ratio was poorly estimated, approximately a factor 10 smaller than the central experimental value. Once the K and K* contributions have been corrected, we still get numbers for the total decay rate in agreement with the experiment, while the n / p ratio has been increased by a factor of 3-4 (depending on the mass number). Unfortunately, the theoretical n / p is still smaller than the measured one and more effort is being invested in order to address this issue. Our results for the
99 Table 1. Weak decay observables for various single-A hypernuclei in units of the free A decay, r A = 3.8 x 1 0 - 9 s _ 1
AHe I2r<
A ^
A A He
r n m (EXP:) 0.436 (0.41 ± 0.14 5 ) 0.735 (1.14 ± 0.2 5 ) (0.89 ± 0.18 6 ) AN -» NN 1.077
Tp (EXP:) 0.340 (0.21 ± 0.07 5 ) 0.590 (0.3llg;U 5 ) AA-> AN 0.257
r n /r p (EXPO 0.283 (0.93 ± 0.5 5 ) 0.247 (1.33±J-|? 5 ) (1.87i?;959 6) AA -+YN 0.108
s-shell A He and the p-shell A2C hypernuclei are listed in Table 1. It has been argued many times that due to the large energy release in the AN —>• NN reaction, one can safely neglect Final State Interactions (FSI) between the outgoing nucleons and the residual (A-2) particles in the evaluation of the total decay rate. This could be not the case for the evaluation of the partial rates and calculations of FSI effects could easily alter the theoretical n / p value due to charge-exchange interactions. Preliminary results for AA He show a non negligible contribution from the AA —> YN channels. Although it is not displayed here, the contribution of those new channels is specially enhanced by the final YN strong interaction which produce large YN wave functions in the low momentum region. Acknowledgments The author acknowledges helpful discussions with K. Sasaki and Prof. M. Oka which led to the correction of a mistake in the sign of the contribution of strange mesons of Ref.1. References 1. 2. 3. 4.
A. Parreho, A. Ramos, and C. Bennhold, Phys. Rev. C 56, 339 (1997). V.G.J. Stoks and Th.A. Rijken, Phys. Rev. C 59, 3009 (1999). K. Hagino and A. Parreno, nucl-th/0004059. J.F. Dubach, G.B. Feldman, B.R. Holstein, L. de la Torre, Ann. Phys. (N.Y.) 249, 146 (1996). 5. J.J. Szymanski et al., Phys. Rev. C43 (1991) 849. 6. H. Noumi et al., Phys. Rev. C52 (1995) 2936.
S T U D Y OF MESONIC A N D N O N - M E S O N I C D E C A Y OF A-HYPERNUCLEI AT D A $ N E L. V E N T U R E L L I * Dipartimento di Chimica e Fisica per I' Ingegneria e per i Materiali, Universitd di Brescia and I.N.F.N, sez. di Pavia, via Valotti 9, 1-25133 Brescia, Italy E-mail: [email protected] The expected performances of the FINUDA experiment in hypernuclear decay measurements are presented. Lifetimes and relevant observables of mesonic and non-mesonic decay of light and medium A-hypernuclei can be measured with good precision at the DA<&NE design luminosity C
1
Introduction
Measurements of hypernuclear physics will be performed by the FINUDA experiment x at the DA$NE machine 2 starting from next year. FINUDA is essentially a Nuclear Physics experiment for studying the formation and decay of the A-hypernuclei obtained by stopping the K~ from the <j> decay in a suitable target through the reaction: K~ + AZ
-> £Z +
IT-
(1)
where in the elementary process the K~ transforms a neutron into a A. The FINUDA physics items include, besides an extensive program of highresolution spectroscopy of A-hypernuclei over all the nuclides that can be machined into solid targets, high statistics studies of the decay and lifetime of the p-shell A-hypernuclei. 2
Decay of Hypernuclei
A hypernucleus is a bound system composed of nucleons and one or more hyperons. Hypernuclei can be produced in an excited state or directly in the ground state through hadronic reactions (like, for example, (1)). In the former case, they can reach their ground state through electromagnetic 7 emission or the ground state of a fragment hypernucleus through nuclear emission. Eventually, the hypernucleus will decay through weak interaction processes which *FOR THE FINUDA COLLABORATION
100
101
involve the emission of pions or nucleons. The elementary processes of the weak decay of a A-hypernucleus are: A ->• p + 7r~ A + p ->• n + p
A -» n + 7T° A + n ->• n + n
(2) (3)
The processes of Eq.2 are called mesonic decays while the processes of Eq.3 are called non-mesonic decays. When the A is embedded in a non-light nucleus, the mesonic decay is strongly inhibited by the Pauli principle due to the reduced phase space (PN ~100 MeV/c). The prevailing decay modes in medium and heavy hypernuclei are the non-mesonic ones since the momentum of the final-state nucleons (mean value~420 MeV/c) is higher than the Fermi momentum. From the study of hypernuclear decay important information can be obtained 3 . From the mesonic channel, it is possible to discriminate between different A-nucleus potentials and to extract information about the 7r-nucleus potentials. From the non-mesonic channel, the investigation of the four-fermion weak vertex and AN —• NN weak interaction can be realized. Moreover, the large momentum transfer in non-mesonic decays permits to probe short distances of the order of 0.5 fm, allowing the role of the quark degrees of freedom to emerge. In spite of the great scientific interest of hypernuclear physiscs, the present experimental information is very poor. 3 3.1
Measurement of Hypernuclear decays with F I N U D A Apparatus
The FINUDA apparatus, see Fig.l, looks like a typical collider detector. It is a non focusing spectrometer of cylindrical geometry with large solid angle acceptance (~ 2ir), excellent momentum resolution and good trigger capabilities. It consists of three main parts: the interaction-target region — the innermost part devoted to select the (K + ,K~) pairs from -decay and to provide the first level trigger. There is a barrel of 12 scintillators (tofino) placed around the beam pipe for selecting the (K + ,K~) pairs at trigger level by AE/Ax measurements and back-toback correlation. In addition, an octagonal array of silicon microstrips, with thin nuclear target modules on the external side, surrounds the tofino with the purpose of measuring the position of the stopping point of the K~ in the nuclear target and to allow for a further AE/Ax measurement for particle identification.
102 interaction/target region end cap clepsydra
compensating magnets 1m
Figure 1. Overview of the FINUDA apparatus. the external tracking system — devoted to measure the 7r~ momentum of the reaction of Eq.l with a Ap/p of 0.3% FWHM and to detect and measure the protons coming from the hypernuclear non-mesonic decay (see Eq.3). It consists of four arrays of position sensitive detectors: a decagonal array of silicon microstrips surrounding the target modules, two arrays of planar low mass drift chambers and a system of longitudinal and stereo straw tubes. The whole tracking volume is immersed in a helium atmosphere in order to minimize the effect of the multiple Coulomb scattering. the outer detector — used for trigger purposes and to detect the neutrons coming from hypernuclear non-mesonic decays (see Eq.3). It is a barrel of 72 scintillators (TOFONE), 10 cm thick. 3.2
Lifetime
measurement
FINUDA will measure directly the hypernuclei lifetime from the spectrum of the differences between the arrival time on TOFONE of the prompt 7r~ from (1) and of the proton from (3) (which is delayed by the hypernucleus lifetime) after correcting these hit times for the times of flight of the particles. This technique has already been exploited succesfully at BNL and KEK 4 . In the case of FINUDA, since the momenta and the trajectories are very well estimated, the instrumental resolution is restricted only by the instrumental time resolution of the TOFONE. Considering that from light to heavy hypernuclei the measured mean lifetimes are THY = (0.5 — 1)T\ 5 and assuming an overall time resolution of 500 ps FWHM, the lifetime of a hypernucleus can be measured with a statistical error of less than 2% with 3000 selected events
103
achievable in one week of data-taking at £ = 1032 cm rate of 10~ 3 . 3.3
2
s
1
with a capture
Mesonic decay measurement
FINUDA can detect the TT~ coming from the mesonic decay allowing the 7r~ decay rate (1^-) to be measured. Due to their low momenta (~ 100 MeV/c), these pions are out of the acceptance of the whole spectrometer; however they can be efficiently measured by means of the 2 arrays of silicon microstrips. From Monte Carlo simulations we extimate 20% in acceptance and 40% in momentum resolution limited by the multiple scattering. The correlation between the measurement of the momentum and of the energy lost in the mictrostrip detectors allows to distinguish these pions from the other particles and to count them succesfully, see Fig.2.
100
200
300
Particle momentum
Figure 2. Energy lost in the microstrip detectors per unit lenght versus momentum of the charged particles for hypernuclear events. The n~ 's from mesonic decay can be easily distinguished also from TT+ 's and fi+ 's since their vertices are very far from each other.
3.4
Non-mesonic decay measurement
The possibility to detect in coincidence the final products (neutron and proton or two neutrons) of Eq.3 makes FINUDA a very powerful tool for studing the non-mesonic decay. The proton can be measured by the external tracking system with 1.3 MeV resolution at kinetic energy of 80 MeV and an estimated acceptance of 24%. The neutron will be measured by the TOFONE with a
104
measured efficiency of 10% and 10 MeV resolution through time of flight measurement, while the acceptance will be 71%. The main background comes from gammas (from the decay of TT° coming from the decay of K+) faking neutrons. We have simulated the apparatus performances for the non-mesonic decay measurement with the decay event generator coming from the work of Ref.6. Without any selection, the ratio between signal and background events results to be 0.60 for the reaction with the proton in the final state and 0.17 when there are two neutrons. Cutting on the time of flight of the candidate particles to neutrons, on the angular correlation and on the total energy of the candidate particles to non-mesonic decay products, only 20% of signal events are lost in the former case and less than 10% in the latter case, increasing contextually the signal-to-noise ratio to the value of 20 and 11 respectively.
Kinetic energy
Kinetic energy
Kinetic energy
Figure 3. Kinetic energy of non-mesonic decay products: a) generated events at arbitrary integrated luminosity; b),c) reconstructed momentum of p' s and n' s at integrated luminosity of 250 p6 - 1 for events with both final products detected 4
Conclusions
The FINUDA performances to obtain world class results in decay measurements for ^2C with the needed integrated luminosity and data taking time are reported in Table 1, assuming a DA$NE luminosity of £ = 1032 cm"^"" 1 and the rate of hypernuclear formation in the ground state of 10~ 3 . Few days of data taking could allow the lifetimes of light and medium hypernuclei to be measured, whereas, in one month, relevant observables of the mesonic and non-mesonic decay could be measured with a per cent error, one order of magnitude better than for the existing data. In particular, the measurement of the neutron over proton induced decay ratio r n / r p could solve the existing disagreement between most of the theoretical predictions and the available experimental data (affected by an error of 100%), helping in understanding the four-baryon weak interaction.
105 Table 1. FINUDA expected performances on A 2 C versus data taking time at the reference luminosity C = 10 32 c m _ 2 s _ 1 assuming 10~ 3 rate of the hypernuclear ground state
Observable Thyp = l / r t 0 t
rP/rA p seen
ryr A p and n seen
r„/rA n and n seen
E v./hour in coinc. with prompt w~ 18p/h 18 p/h not in coinc. with n 2 (pn)/h p, n in coinc. 0.4 (nn)/h n, n in coinc.
Data taking time
Collected events
One week (~ 50 pb" 1 )
3 • 103 p ~ 2 % error 3 • 103 p ~ 2 % error 1.3 • 103 (pn) ~ 3 % error 3 • 102 (nn) ~ 6 % error ~ 6 % error 1.2 -10 3 7T~ 3 % error
» One month (~ 250 pb" 1 ) 55 55
•L n/-*- p
rV/r A B.R.(TT-)~11%
1.7 7r-/h
»
References 1. The FINUDA Collaboration, LNF-93/021(IR), (1993) The FINUDA Collaboration, LNF-95/024(IR), (1995) T. Bressani in Workshop on Physics and Detectors for DA&NE, Frascati, April 9-12, 1991; ed. G. Pancheri (INFN-LNF, 1991) p.475. T. Bressani in Common Problems and Ideas of Modern Physics eds. T. Bressani, B. Minetti and A. Zenoni (World Scientific, Singapore, 1992) p.211 A. Zenoni in Second Workshop on Physics and Detectors for DA$NE, Frascati, April 4-7, 1995; eds. R. Baldini, F. Bossi, G. Capon, G. Pancheri (INFN-LNF, 1995) p.293. 2. G. Vignola in Workshop on Physics and Detectors for DA&NE, Frascati, April 9-12, 1991; ed. G. Pancheri, (INFN-LNF, 1991) p . l l G. Vignola in DAFCE Workshop, Nucl. Phys. A 623, (1997). 3. J. Cohen, Prog. Par. Nucl. Phys. 25, 139 (1990) T. Bressani Nuovo Cimento A 108, 649 (1995) 4. J.J. Szymanski et al, Phys. Rev. C 43, 849 (1991) H. Outa et al, Nucl. Phys. A 547, 109c (1992) 5. H. Park et al, Phys. Rev. C 61, 054004 (2000) 6. A. Ramos et al, Phys. Rev. C 55, 735 (1997)
HYPERNUCLEAR 7 SPECTROSCOPY RECENT RESULTS WITH HYPERBALL
-
H. TAMURA 1 , D. ABE 1 , S. AJIMURA 2 , H. AKIKAWA 3 , K. ARAKI 1 , H.C. BHANG 4 , R.E. CHRIEN 5 , T. ENDO 1 , P. EUGENIO 6 , G.B. FRANKLIN 6 , J. FRANZ 7 Y. FUJII 1 , T. FUKUDA 8 , L. GAN 9 , O. HASHIMOTO 1 , H. HOTCHI 1 0 ' 5 , K. IMAI 3 , Y. KAKIGUCHI 8 , P. KHAUSTOV 6 , J.H. KIM 4 , Y.D. KIM 11 , H. KOHRI 2 , M. MAY5, T. MIYOSHI 1 , T. MURAKAMI 3 , T. NAGAE 8 , J. NAKANO 10 , H. NOUMI 8 , H. OUTA 8 , K. OZAWA1, P.H. PILE 5 , B.P. QUINN 6 , A. RUSEK 5 , T. SAITO 12 , J. SASAO 1 , Y. SATO 1 , S. SATOH 1 , R.I. SAWAFTA 13 , R.A. SCHUMACHER 6 , M. SEKIMOTO 8 , H. SCHMITT 7 T. TAKAHASHI 1 , T. TAMAGAWA 10 , L. TANG 9 , K. TANIDA 10 , H.H. XIA 14 , L. YUAN 9 , S.H. ZHOU 14 , L.H. ZHU 14 ' 3 , X.F. ZHU 14 1) Department of Physics, Tohoku University, Sendai 980-8578, Japan 2)Department of Physics, Osaka University, Toyonaka 560-0043, Japan 3)Department of Physics, Kyoto University, Kyoto 606-8502, Japan 4)Department of Physics, Seoul National University, Seoul 151-742, Korea 5)Brookhaven National Laboratory, Upton, NY 11973, USA 6)Department of Physics, Carnegie Mellon University, Pittsburgh, PA 15213, USA 7) Faculty of Physics, University Freiburg, D-79104 Feiburg, Germany 8) Institute of Particle and Nuclear Physics, KEK, Tsukuba 305-0801, Japan 9) Department of Physics, Hampton University, Hampton, VA 23668, USA 10) Department of Physics, University of Tokyo, Tokyo 113-0033, Japan 11) Department of Physics, Sejong University, Seoul 143-747, Korea 12) Laboratory of Nuclear Science, Tohoku University, Sendai 980-0826, Japan 13) North Carolina A&T State University, Greensboro, NC 27411, USA 14) China Institute of Atomic Energy, Beijing 102413, China We succeeded in observing hypemuclear 7 transitions with germanium detectors for the first time. Overview of the Hyperball project and recent experimental results for J^Li and ^Be are described.
1
Introduction
One of the most important roles of hypemuclear physics is to extract information on hyperon-nucleon (YN) interactions from hypemuclear structures. However, the energy resolution of hypemuclear levels is limited to a few MeV ( F W H M ) in the present method of the (K~ ,ir~) and (ir+,K+) reaction spectroscopy using magnetic spectrometers. It is not at all satisfactory for investigation of AN interactions, in particular, the spin-dependent (spin-spin, spin-orbit, and tensor) parts of the interactions. It has been a d r e a m of
106
107
hypernuclear physicists to introduce high-resolution 7-ray spectroscopy with germanium (Ge) detectors to the field of hypernuclear research. However, it was almost impossible because of technical problems due to huge backgrounds inherent in high energy secondary meson beams. We have recently solved the technical problems and constructed "Hyperball", a large-acceptance Ge detector array dedicated to hypernuclear 7-ray spectroscopy. Then we have started a series of experiments at K E K (E419) and at BNL (E930) l . T h e AN interaction may be expressed as 2 ; VAN
= V0(r)
+ V„(r) sjvs A + VA(r)
1 N A S A + VN(r)
IJVASJV + VT(r)S12
(1)
T h e strength of the spin-averaged central force Vo is well known, but the other terms are almost unknown. T h e strength of those terms can be experimentally investigated from hypernuclear level structure; when a A in Qs orbit is coupled to a core nucleus with spin J, a doublet with spin J—1/2 and J + l / 2 appears of which energy spacing is determined by the spin-dependent AN interactions. Since such spacings are expected to be very small (typically less t h a n 100 keV), high-resolution 7-ray spectroscopy with Ge detector is almost the only method to investigate them. 2
Hyperball and experimental setup
Hyperball consists of fourteen N-type Ge detectors and holds a photo-peak efficiency of 2.5% at 1 MeV in total. It is equipped with fast electronics, namely, low-gain transistor-reset preamplifiers and fast shaping amplifiers with gated integrators. Each Ge detector is surrounded by six B G O scintillation counters. They were used not only for Compton suppression but for rejection of high-energy photons from w° which is the most serious background for our experiments. More descriptions are found in Ref. 1. In the first experiment at K E K (E419), we produced A Li bound states using the (TT+,K+) reaction on 7 Li target employing the K6 b e a m line and the SKS spectrometer. Each of TT~ b e a m at 1.05 G e V / c was m o m e n t u m analyzed by the b e a m line spectrometer, and the secondary K+ was analyzed by Superconducting Kaon Spectrometer (SKS) having a high resolution (Ap/p = 1 x 10~ 3 ) and a large acceptance (100 msr). A 7-ray spectrum was taken in coincidence using Hyperball installed around the target. In the next experiment (BNL E930), we use the (K~ ,ir~) reaction utilizing a high intensity K~ beam. Much larger cross sections of the (K~ ,ir~) reaction t h a n the (7r+ ,K+) reaction is desirable in view of radiation damage of Ge detectors. T h e data-taking for A Be was finished, and more beam time for 6 2 A 0 , A Li and A C is scheduled in 2001.
108
fri
AN spin-spin force
In E419 we succeeded in observing well-identified hypernuclear 7 transitions using Ge detectors for the first time 3 . Figure 1 is the 7-ray spectrum when the bound-state region of ^Li is selected. We observed four 7-ray peaks at 691.7±0.6±1.0 keV, 2050.4±0.4±0.7 keV, 3186±4±6 keV, and 3877±5±7 keV. The 692 keV peak is assigned as the spin-flip M l ( | + — + | + ) transition, and the 2050 keV peak is assigned as the £'2(| + —>| + ) transition. The shapes of these peaks are consistent with Doppler broadening estimated from expected lifetimes of those states and stopping time of the recoil JyLi in the target material. The 692 keV peak becomes sharp after an event-by-event Doppler-shift correction as shown in the left inset of Fig. l(left). The peaks at 3186 keV and 3877 keV, which were observed in the Doppler-shift corrected spectrum, are assigned as the M l transitions from the | + ( T = 1 ) state to the ground state doublet (5 + , f + ) - We have thus confirmed the level scheme of \Li as shown in Fig. 2, where our observed transitions are shown in solid arrows and the measured level energies are also shown. The relative yields of these 7-rays are consistent with the calculated cross sections shown in Fig. 2. The energy spacing of the ground state doublet ( § + , f + ) is determined almost only by the AN spin-spin interaction. Therefore, the observed M l energy unambiguously gives the strength of the spin-spin interaction, corresponding to A (a radial integral of V„{r) in Eq.(l)) of 0.50 MeV. In addition, the ground state spin of ^Li was confirmed to be | from weak-decay branching ratio of ^Li—>7Be* + ir~ which was measured from the observed yield of 429 keV 7-ray from 7 Be*. 7
Li (TT + , K + 7 )
E419 Doppler shift corrected JLI M l ( l / 2 \ . ^ l / 2 * y Ml(l/2\,,-3/2*)
Ml(3/2+-l/3+)
\
,
E2(5/2 + ^l/2*)/
VMWV»»hwVTjrtoj;
y*Y^*^fii^MM*V^^ 1000
1500 E 7 (keV)
2500
2000
3000 E, (keV)
4000
Figure 1. 7-ray spectrum of J^Li measured with Hyperball. Four hypernuclear transitions, M l ( | + - •| + ),E2(f + -. i + ) , a n d J t f l ( i + ( T = l ) - • 2 "*", 5 "*"), were observed.
109
iLi a(9=0°-15°) Ex (n+,K+) (MeV) Hiyama et al
^5.37 (Mi (MeV) T=l /
T=l .He+d ^-3.94 r + 3.88 ' 'i/2 MIL. 7/2 + \
ir.
5/2+ 1
Ml
0.08 2.05
1.23
0.69
0.13
0
1.21
T=0 3/2 ••l/2+
^Jj
present
Figure 2. Level scheme and 7 transitions of ^Li. "Present" shows level energies measured in the present experiment.
5
\Be : AN spin-orbit force
shrinking effect
The £ 2 ( § + ^ i + ) transition at 2050 keV was previously observed at BNL with Nal counters 5 . Taking advantage of our high resolution, we have derived the lifetime of the | + state using Doppler shift attenuation method 6 . The right inset of Fig. 1 (left) shows the fitted peak shape for the optimum lifetime. B(E2) was then derived to be 3.6±0.5ig'4 e2fm4. This result, compared with the B(£2)=10.9±0.9 e2fm4 of the core nucleus 6Li(3+—>1+), indicates a shrinkage of the ^Li size from the 6 Li size by 19±4%. It is an evidence of the "glue-like role" of A predicted by cluster-model calculations 7 ' 4 .
In BNL-AGS E930, we aim at determining the AN spin-orbit and tensor interactions from 7-spectroscopy of several light hypernuclei. In the ^Be spectrum, we successfully observed twin peaks at 3.05 MeV which are assigned as the E2(| + —>-| + , !+—*| + ) transitions. Since the ( | + , | + ) doublet corresponds to the Be(2 + ) state with a A in 0s orbit, their energy spacing is determined almost purely by the AN spin-orbit interaction. Our observed small spacing of the doublet (a few tens of keV) will comfirm a very small but finite size of the AA^ spin-orbit interaction, and a comparison with theoretical calculations 8 will provides us with a clue to understand the origin of the spin-orbit interaction in the nuclear force. References 1. 2. 3. 4. 5. 6.
H. Tamura, Nucl. Phys. A639, 83c (1998). D.J. Millener et al, Phys. Rev. C31, 499 (1985). H. Tamura et al., Phys. Rev. Lett. 84, 5963 (2000). E. Hiyama et al, Phys. Rev. C59, 2351 (1999). M. May et al., Phys. Rev. Lett. 51, 2085 (1983). K. Tanida et al., submitted to Phys. Rev. Lett.) K. Tanida, Ph.D thesis, University of Tokyo, 2000. 7. T. Motoba, H. Bando and K. Ikeda, Prog. Theor. Phys. 70, 189 (1983). 8. E. Hiyama et al., Phys. Rev. Lett. 85, 270 (2000).
ON T H E C O A L E S C E N C E P R O D U C T I O N OF B R O A D RESONANCES
Lebedev Physical
V . M. K O L Y B A S O V Institute, Leninsky prospekt 53, 117924 Moscow, E-mail: kolybasvQsci.lebedev.ru
Russia
The situation is considered when a particle, produced in the initial interaction of a projectile with one of nucleons, then coalesces with a residual nuclear system forming a nearthreshold resonance with the width of the order of several MeV. It is shown that a sharp variation of the amplitude, associated with triangle mechanism, can appreciably influence visible parameters (the width and position) of final resonance system. The case of |,He production is described in detail.
Recently several examples have appeared of unusual situation for coalescence mechanism of nearthreshold production of broad resonances in nuclear interactions. If the amplitude of resonance production is sharply varying function of the resonance mass and changes substantially on the mass interval of the order of resonance width, it can lead to the appreciable difference of visible and true resonance parameters (the position, shape and width). Let a projectile firstly interacts with one of the nucleons producing a different particle (i.g. S-hyperon). Then this particle coalesces with residual nuclear system with formation of a resonance (E-hypernucleus in our example). If the resonance mass is close to the threshold of the channel "particle + residual nucleus" (the distance from the threshold must be of the order of nucleon separation energy, 5-j-20 MeV), the amplitude of resonance production would be a sharp function of the mass of resonance system due to the peculiarities of the triangle mechanism [1]. If the resonance width is also of the order 5 -=- 20 MeV it would lead to the noticeable effects. Some distinct features appear associated with the shape and position of the resonance peak and with the magnitude of the cross section (additional enhancement etc.) [2]. In spite of rather exotic character of above-stated conditions, there are several such examples: bound and excited S-hypernuclear states [3,4], S-hypernuclei [5], tentative 77-nuclei [6], deeply bound states of the Pb pionic atoms [7]. The above-mentioned effects must be mostly clear seen when the initial nucleus has the structure "nucleon + cluster" as in this case the dominant contribution would be made by the triangle graph with two-particle intermediate state for which the cusp behaviour is mostly distinct. Now we will illustrate these statements by the example of j,He production. The recent study of the missing mass spectrum from the reaction 4 He(K~, TT~ ) 110
111
at 600 MeV/c has revealed a peak in the region close to S production [3]. The peak corresponds to the bound |,He state with parameters Eex — —7 MeV and T = 7 MeV (Eex is the missing mass to a pion measured from the sum of masses S°+ 3 He). The origin of the main part of Ref.[3] spectrum in general is clear [1,2] (so called "quasi-free" S° production and the tail of A production). The most probable mechanism of |,He production is presented in Fig. la and b. At first, E-hyperon is born on one of neutrons, and then it coalesces with residual nuclear system.
Figure 1: Graphs for ^.He production in the reaction 4 He(K ,ir ).
The difference between Fig. 2a and b is that in the first case we have the two-particle intermediate state 3 He - E°, and in the second case the threeparticle state p d - E° is present (there could also be four-particle state p p - n - S°). An amplitude of Fig. la has singularities of two kinds: the root threshold singularity at Eex — 0, and the triangle logarithmic singularity located in complex plane. The latter is also situated near Eex = 0 for kinematical conditions of Ref. [3]. Modulus squared of the triangle graph amplitude M A for this case is shown (without Breit-Wigner factor) by solid curve in Fig. 2. Fig. 2 shows that | M A | 2 strongly varies on the resonance width. It can noticeably influence the result of j,He parameters estimation from the experimental data. Sharp behaviour of | M A | 2 is characteristic only for the triangle graph of Fig. l a with a two-particle intermediate state. The graph of Fig. lb with a three-particle intermediate state leads to the smooth amplitude whose maximum is shifted rightward. It is shown by dotted curve in Fig. 2. Therefore the comparative contribution of Fig. l a and b graphs is rather important. It is
112
possible to assert [2] t h a t the contribution of Fig. l a graph in any case should be noticeable against a "background" of Fig. l b graph. A rapid variation of 5I
1
1
1
1
1
1
1
E„ (MeV) Figure 2: |Af^| 2 for the triangle graphs of Fig. 1 with two—particle (solid curve) and t h r e e particle (dotted curve) intermediate states.
the I;He production amplitude as function of Eex was not taken into account in the analysis of Ref. [3]. It m a d e the procedure not quite correct. Let's look what are the results of the correct account for the production mechanism, corresponding to the graphs of Fig. l a and b . We shall begin from a t t e m p t to describe Eex spectrum with the parameters from Ref. [3], t h a t is, the binding energy 4.4 MeV (it corresponds to Eex = —7 MeV) a n d the width 7 MeV. T h e best description for this case is shown in Fig. 3a. It is necessary to accept here t h a t the ratio of Fig. l a and b contributions is not more t h a n 1:5. Otherwise there would be too large enhancement at Eex — 0 in obvious contradiction with the d a t a . It is possible t o see t h a t the peak is described not so well, especially the left wing. T h e situation can be improved by a modification of J He parameters. Various versions of the fitting procedure have shown t h a t the best description of Eex spectrum could b e obtained with t h e binding energy 5.4 MeV (it corresponds to Eex = —8 MeV) a n d the width 8.5 MeV. This fit is shown in Fig. 3b. T h e smaller width would lead to a poor description for the left wing of the resonance peak. T h e larger width would lead t o too strong peak at Eex = 0. T h e latter is also essential in another respect. T h e technique for inclusion of A production tail [1] is incomplete. In other version the resonance left wing would be broader. It would d e m a n d the larger value of the width. However, as it appears, the width more t h a n 8.5 4- 9.0 MeV is forbidden as it would too strengthen the peak near Eex — 0. Besides, to keep the m a g n i t u d e of this peak in reasonable limits, it is necessary t o suppose t h a t the contribution of multi-particle intermediate states in Fig. 2 is several times more t h a n the
113
contribution of two-particle states. From here follows that the probabilities of virtual |;He decays to three and four-particle channels are much larger than to two-particle ones.
Figure 3: Theoretical description of Eex spectrum for the reaction 4 He(K , x ): (a) with j H e parameters from Ref. [3]; (b) with the binding energy 5.4 MeV and the width 8.5 MeV.
Acknowledgments This investigation was partly supported by RFBR grant 99-02-17263. References 1. O.D.Dalkarov and V.M.Kolybasov, e-print nucl-th/9901040. 2. V.M.Kolybasov, Physics of Atomic Nuclei 62, 1134 (1999); Phys. Rev. C 60, 037001 (1999). 3. T.Nagae et al, Phys. Rev. Lett. 80, 1605 (1998). 4. T.Nagae, in Nuclear and Particle Physics with High-Iniensity Proton Accelerators, ed. T.Komutsubara et al (World Scientific, Singapore, 1998), p. 265. 5. P.Khaustov et al, e-print nucl-ex/9912007. 6. Q.Haider, L.C.Liu, Phys. Lett. B 172, 257 (1986). 7. T.Yamazaki et al, Phys. Lett. B 418, 246 (1998).
ENERGETIC LEVEL SCHEME OF THE STABLE S=-2 DIHYPERON
Joint Institute
P. Z. Aslanyan, B.A.Shahbazian for Nuclear Research, LHE, Dubna, p.o. E-mail: [email protected]
141980,
Russia
The quark and soliton Skyrme-type models predict the two different sets of S=-2 stable dibaryon states. The lowest state of the quark model set is an 1=0, Jn = 0+, MHa < 2M\ isosinglet dibaryon, whereas t h a t of solyton Skyrmetype model set is an 1 = 1 , . / " = 0, MH- Ho H+ fs 2370 MeV/c 2 isotriplet dibaryon. On photographs of the JINR LHE 2m propane bubble chamber exposed to 10 GeV/c proton beam two groups of events interpreted as S=-2 stable dibaryons were observed [1-7]. Quasi-diffractive process plays the decisive role in dihyperon production.The average life for weak decay of stable dihyperons exceeds 3.3 1 0 ~ 1 0 s . Several H— hyperons have been registrated in these collisions with an effective cross section of 1300-600 nb. The formally estimated effective cross section for H dibaryon production in propane at a momentum of 10 GeV/c is 100 nb .
1 1.1
The observation of S = - 2 stable dibaryons Evidences for S=-2 light dibaryons
It has been observed that the S=-2 stable light dibaryon does exist in the following energetic level scheme ( Fig. 1)[1-3]. The ground state H° of M ff o=(2146.3±1.0) MeV/c 2 mass and of most probable spin-parity J^0 = 0 + (see Fig.l)[l]. H° suffered weak decay H° —»• p + £-(x 2 (lV-lC)=1.08,C.L. =29.8%). The earliest Jafe predictions of a light H°(MHO < 2M\) stable S=-2 dibaryon of a mass of 2150MeV/c2 meet the measured one well. 1. This is the first candidate for the neutral S=-2 stable dibaryon, both weak decay particles of which stopped in propane(Fig.l). A 10 GeV/c beam proton colliding with the 12C nucleus produces a four-prong star of the total Q = + 4 electric charge. The incident proton hits T + + + , fuses with it forming a highly excited four-baryon fireball B4 which however does not leave the nucleus as if being confined to it but suffers explosive phase transition to an: p + T+++(Z07b.0±100.0)-^ B4^ H°(2203.0±5.9) + A°+p + I
114
115
two detected light S=-2 stable dibaryons [2,3], (2172.7±15.2)MeV/c2[2] and (2218.0±12.0) MeV/c2[3] , equal to (2195.4±9.7) MeV/c 2 . 2. The second candidate for the H° emitted from two-prong star, weakly decays forming an 8 GeV/c V° [2]. Only the weak decay hypothesis H° —* pT,~ fits the V° with x 2 (lV-2C)=0.014, C.L.=99.3, MHo = (2172±5.4) MeV/c 2 . The hypotheses on H° creation in pp collisions failed to the event. The dibaryonic fluctuon D+ mass as a free parameter, the sequence of processes pD+ —• K°K+pH°, H° -+pL- fits the event with x 2 (2V-2C)=0.014, C.L.=99.3%. 3. The third candidate for the H° emitted from a six-prong star, weakly decays forming a 1.10 GeV/c Vr°,both decay particles of which produce kinked tracks [3]. Only the sequence of processes H° —• pT,~, pl2C —* p12C, S~ —* mr~ fits the event with x 2 (2V-2C)= 0.73, C.L.=69.3%,Mffo = (2218.0± 12.0) MeV/c2 The second excited state H% of M^o=(2203.0±5.9) MeV/c 2 mass suffers electromagnetic transition H\ —• H° +j (see Fig.l)[l]. Its spin is bounded by the limits | / — s |< JHo <\ I + s |, where s = l is the 7 -quantum spin and 1 is its orbital momentum. The weighted average over the three masses from exited candidates gives MHo = (2200.9 ±A.\)MeV/c2. The light S=-2 stable dibaryon does exist at least in two quantum states. 1.2
Evidences for S=-2 heavy dibaryons
The third excited state heavy neutral H of MH= (2396.9±17.3) MeV/c 2 average mass has been detected two events [4](see Fig.2). There exists as well its charged counterpart H+. Three events were detected as heavy positive charge H+ dibaryons [4-7](see Fig.3-Fig.4). 4. The first candidate for the heavy neutral H dibaryon, emitted from a one prong star marked as Pr in weakly decays forming a 1.97GeV/c V° both decay particles of which produce kinked tracks. Only the sequence of processes H -* p S - , p 1 2 C — p12C,12- -> nw- fits the event with x 2 (2V-3C)=1.97, C.L.=57.9%, MH - (2408.9 ± 11.2)MeV/c 2 . 5. The second candidate for the heavy H dibaryon, emitted from a two prong star weakly decays forming a 2.50GeV7cVA°, both decay particles of which produce kinked tracks(Fig.2). Only the sequence of processes H —> p £ ~ , p12C —• p12C, XT -+ nit- fits the event with x 2 (2V-3C)=3.05, C.L.=38.4%,M H = (2385.8 ±31.0) MeV/c2. The functional target D+ hypothesis, possible production reactions pD+ - • HKfK%B°, HKfK*°p (both followed by the reaction sequences H —• p £ ~ , p12C —> p12C,T, —* mr~, Kf —*• i/fi) were tried. 6.The first candidate for the H+ in [4]( Fig.3) is presented by very slow, heav-
116
Fig.l.
ilOGM.
"Ci Fig.2..
^Pl„
Fig.3.
FW
-9>% Fig.4.
117
ily ionizing positively charged massive particle, suffering violent scattering in propane and forming a kinked track. The part of the track after the kink is certainly due to a proton, stopping in propane. The hypothesis on weak decays H+ — 7r°pA, A - • pir~ fits the event with x 2 (2V-3C)=0.94, C.L. =87.1%, MH+ = (2375.8 ± 9.3) MeV/c2. Neither of hypothesis on reactions, initiated by particles of positive and negative electric charges fits the event. The time of flight of the H+ is 2.6 x lO" 1 0 . 7. The second detected event [6] ( Fig.4) is produced from an eight-prong star in collisions of beam protons with propane. This ionization allows one to selected only two hypothesis of H+ or deuteron, which can imitate this track. The second part is certainly due to a slow thick track identified as a E + —> p + 7T° because there is a break on this track 1.5 cm long. The first hypothesis of inclusive three-body weak decay H+ —> p+ir° + A°, A —> p+7r~ fits the event with x 2 (2V - 3C) = 1.95, CL = 86%, MH+ = 2580 ± 108MeF/c 2 [6]. The success hypothesis of H+n —>• E + + A + n,np —• pn is probable with the best fits y?(2V - 3C) = 1.38, C.L. = 98%, MH+ = 2410 ± 90. Thus, the kinematic parameter of the neutron from first vertex was determined using the (lV-lC)fit of reaction np —• pn (Fig.4). The kinematic does not permit imitating the reaction with deuteron d\ and also with fermi motion included. 8.The ionization of third detected event also allows one to selected only two hypothesis of H+ or deuteron, which can imitate this track [7]. The second part is due to the proton 33.95 c m . long. The hypothesis of inclusive three-body weak decay H+ ->• p + j + A0, A -+ p + w~ fits the event with x 2 (2K - 3C) = 2.85, C.L. = 73%,MH+ = 2448 ± 47MeF/c 2 (the H+ - • p7r°A°,A0 ^ pir' hypothesis is also possible fits with x2(2V - 3C) = 2.86, C.L. - 72%, MH+ = 2488 ± 48MeV/c 2 ). The kinematic threshold does not permit imitating the reaction with deuteron d\ and also with fermi motion included. References 1. B.A. Shahbazian et al., JINR Rapid Communications, No 1[69]-95,1995, p.61.. 2. B.A. Shahbazian et al., Z.Phys., 1988, C39, p.151. 3. B.A. Shahbazian et al.,Phys. Lett. B235(1990)208;B238,p.452(E), 1990; B244, p.580(E). 4. B.A. Shahbazian et al.,Phys. Lett. B316(1993)593. 5. P.Zh.Aslanyan et al., JINR Rapid Communications, N 1(87)-98,1998. 6. P.Zh.Aslanyan et al., Nucl. Phys.B(Proc.SuppL) 75B (1999)63-65. 7. P.Z. Aslanian et al., Int. Workshop HIT99, 17-20 May, CERN;Int. Conference PANIC99, 10-16 June , Uppsala University.
Section II. Mesons, Baryons and Antibaryons
P I O N I C EXCITATIONS IN N U C L E A R S Y S T E M S * W. WEISE Physik-Department Technische Universitdt Miinchen D-85747 Garching, Germany We discuss recent developments in our understanding of the lightest quarkantiquark excitations in a nuclear medium and focus on two selected topics: the thermodynamics of the chiral condensate and in-medium s-wave interactions of pions with special emphasis on the recently observed deeply bound pionic atom states.
1
Symmetries and symmetry breaking patterns in Q C D
The QCD ground state, or vacuum, is characterized by the presence of a strong condensate (qq) of scalar quark-antiquark pairs (the chiral condensate) which represents the order parameter for spontaneous chiral symmetry breaking in QCD. The light hadrons are quasi-particle excitations of this condensed ground state. Pions and kaons are of special importance in this context, as they are identified with the pseudoscalar Goldstone bosons of spontaneously broken chiral symmetry. The pion decay constant, fn ~ 92.4 MeV, determines the chiral scale 47r/^ ~ 1 GeV (refered to as the "chiral gap") which governs the low-mass hadron spectrum. For example, the lightest vector mesons (p, to) can be interpreted as the lowest resonant qq "dipole" excitations of the QCD vacuum. Current algebra combined with QCD finite energy sum rules The deviation of the physical pion mass, m T ~ 0.14 GeV, from zero reflects weak explicit chiral symmetry breaking by the small masses of uand d-quarks, mu^d < 10 MeV. Spontaneous and explicit chiral symmetry breaking imply the PCAC or Gell-Mann, Oakes, Renner (GOR) relation, m\fl
- --(mu+md)(uu
+ dd) ,
(1)
to leading order in the quark masses mu^. One of the basic issues in strong interaction physics is to explore the QCD phase diagram as it evolves with increasing temperature and/or baryon chemical potential. A key element in this discussion is the chiral transition from the • P R E S E N T E D AT "BOLOGNA 2000". WORK SUPPORTED IN PART BY BMBF, GSI AND DFG
119
120
Nambu-Goldstone realization of chiral symmetry (with non-zero condensate {qq)) to the "restored" Wigner-Weyl realization in which the chiral condensate vanishes. In QCD, chiral restoration is probably linked to the transition between composite hadrons and deconfmed quarks and gluons. Lattice QCD : locates the critical temperature for the chiral transition at Tc ~ (150-200)MeV. The leading dependence of {qq) on baryon density p at zero temperature is controled by the pion-nucleon sigma term, ON — 0.5 GeV: ( « > o -
1 _
^ ^
P +
-'
(2)
indicating a rapidly decreasing magnitude of the chiral condensate in cold compressed nuclear matter 2 . The GOR relation (1) continues to hold 3 in matter at finite temperature T < Tc and at finite density, when reduced to a statement about the time component of the axial current AM. One finds
ff(p,T)
= -^(qq)p,T
+ ...
(3)
to leading order in the average quark mass mq = | ( m „ + m
Thermodynamics of the Chiral Condensate
Suppose we are given a chiral effective Lagrangian, £eff, with Goldstone bosons (pions) coupled to baryons (nucleons). Let Z be the partition function derived from this theory, and p, the baryon chemical potential. The pressure as a function of n and T is P(n,T) — ylnZ, where V is the volume. The Hamiltonian which determines Z depends on the pion mass mn, or equivalently, on the quark mass mq through the GOR relation (1). Given the
121
equation of state P{p,, T), a variant of the Hellmann-Feynman theorem (with the quark mass treated formally as an adiabatic parameter) together with (1) leads to the following expression for the density and temperature dependent chiral condensate: mP,T (qq)o
1 dP<jx,T) f* dm* '
=
K
>
whith the baryon density p = dP/d/j. The task is therefore to investigate how the equation of state, at given temperature and baryon chemical potential, changes when varying the squared pion mass (or the quark mass). Consider now the following effective Lagrangian as an approximation to the hadronic phase of QCD: £eff
= £jv + ^ir + £TTN + C-NN -
(5)
The free nucleon Lagrangian is £jv = N(i'y-p — M)N, where M is the nucleon mass in vacuum. The pion sector with inclusion of TTTT interactions is described by the non-linear sigma model plus pion mass term. The chiral pion-nucleon coupling to leading order in pion momentum is £*N = i^Nlfll5rN
- 9"i? - _ L j v 7 / 1 f AT • 7f x d"n ,
(6)
and the short-distance dynamics is absorbed in NN contact terms, CNN
= ~ ( N N )
2
+ ^(N7»N)2
+ ...,
(7)
with the coupling strength parameters Gs,v fixed by the ground state properties of normal nuclear matter. In essence, this is a variant of relativistic mean field theory combined with "soft" pion fluctuations treated within the framework of chiral perturbation theory. We have used two-loop thermal field theory to perform a self-consistent calculation of P(/J,,T) and then deduced the chiral condensate as a function of temperature and baryon density using Eq. (7). This calculation 4 generates temperature dependent mean fields for the nucleons at the same time as it treats thermal pion fluctuations with inclusion of leading 7T7T interactions. The result for (qq)p,T is shown in Fig. 1. One notes that the temperature dependence at p = 0 is quite similar to the result of lattice QCD 1 . The critical temperature in the present calculation is Tc ~ 180 MeV. At T = 0 and low baryon density the linear behaviour as in Eq. (3) is recovered. Naive extrapolation of this linear density dependence with UN — 45 MeV would find the condensate dropping to zero at about three
122
Figure 1. Dependence of the chiral condensates {qq)PtT o n temperature T and baryon density p (in units of p0 = 0 . 1 7 / m - 3 ) , calculated 7 in two-loop thermal field theory with the effective Lagrangian (10-13).
times nuclear matter density. However the scalar density becomes significantly smaller than p at high density, so that (qq)p still keeps almost half of (qq)o at p - 3p03
Pionic s-waves in the nuclear medium
The investigation of pion-nucleus interactions has a long history 5 . The reasons for revisiting this topic are two-fold: first, the recent observation of deeply bound pionic atom states in Pb isotopes 6 has sharpened the quantitative constraints on the local (s-wave) part of the pion-nuclear optical potential, and secondly, there is renewed interest in the theoretical foundations of this optical potential from the point of view of chiral dynamics 7 . Pions as Goldstone bosons interact weakly at low momentum. Their s-wave interactions with nucleons in leading order are determined by the pion decay constant /^ as the relevant scale of spontaneously broken chiral symmetry. In the nuclear medium, this scale changes, and the obvious question is whether accurate data, such as those from deeply bound pionic atoms, are a sensitive measure for the expected density dependence of f„. The spectrum of pionic modes with energy u> and momentum q in nuclear matter at density p is determined by solutions of the wave equation [to2 - f
- m l - U(LJ, q; p)] = 0 ,
where the self-energy II is often expressed in terms of the optical potential U
(8)
123
as n = 2uU. Consider a low-energy w~ interacting with matter at low proton and neutron densities pp,n. To leading order in these densities, II = —T(n~p)pp — T(ir~n)pn, where T denotes the itN T-matrix (at threshold, its relation to the corresponding scattering length is T{q = 0) = 47r(l + mn/M)a). The low-energy behaviour of T is ruled by theorems based on chiral symmetry. Consider the isospin even and odd amplitudes, T^ = ^[T(ir~p) ± T(ir~n)}. The Tomozawa-Weinberg theorem gives
T < + W = 0 ) = 0, T < - W = 0 ) = ^
(9)
to leading order in w. In next-to-leading order an attractive scalar term proportional to cjv//^ combines with a repulsive term of order u2 so as to reproduce the observed very small isospin-averaged scattering length. For the case of T(~\ chiral perturbation theory gives corrections of order w3 which close the 15 % gap between the lowest order result (20) and the empirical isospin-odd scattering length. In the actual calculations we use the threshold amplitudes T^ = 0 and T^ = f^-(l + 0.066^-), compatible with the empirical scattering lengths. The s-wave pion-nucleus optical potential involves more than just the amplitudes in leading order. Double scattering terms are known to be important, and absorptive corrections of order p2 must be added 5 . The s-wave in-medium self-energy for a 7r~ becomes II(
_
_3PFT(-)=
( _)
_
(_}
/
3p£r(-A
where PF is the nuclear Fermi momentum. Note that, with T^ = 0, the leading term in T L I now involves the squared isospin-odd amplitude proportional to / ~ 4 . In Tgjl the double scattering correction (about —10% at PF ~ 2m x ) is often ignored, but we prefer to take it into account. The p 2 -term has a complex, phenomenological constant B0. Its imaginary part is fitted to reproduce pion absorption rates. Apart from the B0p2 term, the s-wave TT~ potential at u = m^ and for an N/Z ratio of 1.5 gives about 16MeV of repulsion at p = p0. From the analysis of pionic atom data it is known that this repulsion is too weak by about a factor of two. It has been common practice to choose the phenomenological Re B0 such that the missing repulsion is accounted for. This requires a large
124
negative Re B0 for which there is little theoretical foundation. Also, the real part of the Tr~d scattering length is perfectly well reproduced just by the single and double scattering terms already present in Teff, suggesting Re B0 ~ 0. We should look for an alternative way to generate the "missing repulsion". The previous considerations were based on the chiral low-energy theorem (9), expressed in terms of the vacuum pion decay constant. However, the nuclear medium defines a new vacuum, with the magnitude of the chiral condensate (qq) reduced (see section 2). Following the in-medium GOR relation (3) for the time component of the axial current, we h&ve(qq)fi/ (qq) 0 — f*2(p)/f2. The shift of the vacuum therefore implies a density dependent pion decay constant,
f?(p) = Z-%p
(")
to leading order in the baryon density. While the minimum of the effective potential is shifted at p > 0, the chiral low-energy theorem for TTN scattering still holds, but now with respect to the new vacuum with its reduced condensate and reduced pion decay constant /^(p). The isospin-odd in-medium TTN amplitude becomes *-> = $ *
(12)
to leading chiral order, while the isospin-even amplitude still has T^ = 0. This immediately implies that the s-wave optical potential, with fn replaced by f*, will be substantially more repulsive. In fact, this potential becomes about twice as large at p = p0 and N/Z = 1.5 when replacing fn by f*(p0) — 0.82/TT according to Eq. (11). We will now investigate the consequences of this assertion for the understanding of deeply bound pionic atom states in heavy nuclei. 4
Deeply bound pionic atom states
The existence of narrow Is and 2p states in heavy pionic atoms results from a subtle balance between the attractive Coulomb potential and the repulsive s-wave optical potential 8 . The net attraction is localized at and beyond the nuclear surface. Under these special conditions the overlap of the pion densities with the nuclear density distribution is sufficiently small so that the absorptive width is reduced and the deeply bound states have a chance to be observed as narrow structures. This is the case in the GSI measurements 8 of Is and 2p pionic states in 207Pb and, most recently, in 205Pb, using the (d, 3He) reaction for their production.
125 1.2 1.1
1
-
i
207
[—r*s-»=
Pb,
0 9
0.8
^0.7 0.6 0.5 0.4 0.3
,,—.- -. 4V* ! Ji
" "
•u
,r-iiMi-„J
~ -
- •»/ir
'1 u»*' i*' \«
1.0 _ |
r—r
"•-"LU1.
A^-w *w*\
<^ak ItfSfsisi fwM£§& ^^MMf
w
mp), t 5.5 2p
i Ji , * * i v '
I . * M r. I - 1 | m. «•». J»
•
6
7.5
6.5
B[MeV]
Is
Figure 2. Binding energy B and width T of I s and 2p pionic atom states in 20TPb. Points (/„•) are obtained 9 using the chiral s-wave optical potential with vacuum pion decay constant {f,, — 92AMeV) and Re B0 = 0. Dark ellipses ( / ; ) are results 9 when replacing / „ by the in-medium decay constant (11) with o-jv = (45 ± S)MeV. Light shaded areas: empirical range of B, V from ref. 6 .
We have performed detailed calculations for pionic Is and 2p states in Pb and other isotopes 9 . The aim is to explore, in particular, the sensitivity of the widths of these states with respect to the density dependence of the pion decay constant as it enters in the chiral s-wave potential. We have combined this swave potential, treated in the local density approximation, with the non-local p-wave potential which systematically reproduces the binding energies and widths of the higher-lying pionic atom states previously measured in stopped 7r~ experiments 1 0 . Our results for pionic 207Pb are shown in Fig. 2. The points denoted " / ^ " are obtained using the vacuum value of the pion decay constant in T ' - ' as it enters the s-wave optical potential. We have used Re B0 = 0 in our "standard" set and Im B0 ~ 0.06m~ 4 . Clearly, the "vacuum fn" scenario is quite far off the lightly shaded areas which give the range of I s and 2p binding energies and widths as deduced from the 20SPb(d,3He)207PbK data 6 . The missing s-wave repulsion can of course be generated by simply adjusting Re B0. This would require a large negative value, Re B0 ~ — 0.07m~ 4 , which would be at odds with theoretical many-body calculations and also with the 7r~-deuteron scattering length as mentioned earlier. On the other hand, replacing the vacuum pion decay constant in r ( _ ) by f*(p) as given by Eq. (11), with p treated as local density distribution, the missing repulsion in the s-wave optical potential is easily supplied. The calculated results are shown by the dark ellipses in Fig. 2. The data are now well reproduced using Re B0 = 0.
126
Our predictions for the case of pionic 205Pb are also in good agreement with recent (preliminary) data n . This is a particularly interesting example because here the Is state has been established as a well isolated peak in the (d,3He) spectrum. 5
Concluding remarks
Our results clearly demonstrate that detailed high precision studies of deeply bound pionic atoms do provide strong additional constraints on the s-wave pion-nucleus optical potential, especially when these studies are carried systematically through isotopic chains of neutron-rich nuclei. We have taken the position here that the repulsion in the s-wave pion-nuclear interaction required to generate narrow Is and 2p states, is naturally linked to the density dependence of the pion decay constant which in turn reflects the change of the QCD vacuum structure in dense matter. Thanks to N. Kaiser, R. Leisibach, M. Flaskamp and Th. Schwarz whose work has contributed substantially to this paper. Thanks also to P. Kienle, H. Gilg and T. Yamazaki for many fruitful discussions. References 1. G. Boyd et al., Phys. Lett. B 349 (1995) 170; F. Karsch, Nucl. Phys. B (Proc. Suppl.) 8 3 - 8 4 (2000) 14 2. E. G. Drukarev and E. M. Levin, Nucl. Phys. A 511 (1990) 679; M. Lutz, S. Klimt and W. Weise, Nucl. Phys. A 542 (1992) 521; T. D. Cohen, R. J. Furnstahl and D. K. Griegel, Phys. Rev. C 45 (1992) 1881 3. V. Thorsson and A. Wirzba, Nucl. Phys. A 589 (1995) 633; M. Kirchbach and A. Wirzba, Nucl. Phys. A 604 (1996) 395; G. Chanfray, M. Ericson and J. Wambach, Phys. Lett. B 388 (1996) 673 4. M. Flaskamp, N. Kaiser, Th. Schwarz and W. Weise, in preparation; 5. M. Ericson and T. Ericson, Ann. of Phys. (NY) 36 (1966) 383; T. Ericson and W. Weise, Pions and Nuclei, Oxford (1988) 6. H. Gilg et al., Phys. Rev. C 62 (2000) 025201; K. Itahashi et al., Phys. Rev. C 62 (2000) 025202; T. Yamazaki et al., Z. Physik A 355 (1996) 219; Phys. Lett. B 418 (1998) 246 7. T. Waas, R. Brockmann and W. Weise, Phys. Lett. B 405 (1997) 215 8. E. Friedman and G. Soff, J. Phys. G 11 (1985) L 37; H. Toki and T. Yamazaki, Phys. Lett. B 213 (1988) 129 9. R. Leisibach and W. Weise, in preparation 10. C. J. Batty, E. Friedman and A. Gal, Phys. Reports 287 (1997) 385 11. H. Gilg et al., private communication
STATUS OF EXOTIC M E S O N
SEARCHES
M A U R O VILLA INFN, Sezione di Bologna, via Irnerio, J)2, 1-40126 Bologna, E-mail: [email protected]
Italy
An overview of the current experimental status on exotic meson searches is presented. Particular attention is devoted to the results obtained by LEAR experiments where some exotic candidates have been found. The most important one, from the theoretical point of view, is the / o (1500), which is a candidate for the ground state scalar glueball and which has been observed in different decay channels. One of the most interesting exotic candidates found up to now is the p(1405), observed decaying into 7)7T°, which, having J = 1 ^ quantum numbers is clearly a non qq meson. Other exotic candidates can be clearly identified only when the corresponding meson nonet is established. This is the case, for example, of the a o (980) meson, which has lost its position in the lowest scalar meson nonet in favour of the recently discovered
1
Introduction
The fundamental theory of quark interactions, namely the Quantum Chromodynamics (QCD), is an SU(3) gauge theory which has few experimental facts and hypotheses at its foundations and it has a strong predictive power expecially for high energy phenomena. In the low energy region, since the QCD equations cannot be fully solved neither explicitly nor pertubatively, but only numerically, basic information such as mass spectra and properties of bound objects have to be determined from experiments. In addition to ordinary qq mesons, QCD models 1,2 ' 3,4,5 foresee other bound states which collectively are called exotic mesons. They can be multiquark mesons (such as qqqq), quarks with valence gluons called hybrids (qqg) or pure glue matter called glueballs, such as gg or ggg combinations. Masses, widths, decay patterns and other properties of these exotic states vary widely from model to model; it is then extremely important to measure them experimentally, since this is the only way to get an indication of how well we understand QCD at these energies. 2
Ordinary and exotic mesons
In the Constituent Quark Model 1 , which is one of the simplest low energy QCD ones, ordinary light mesons are composed by quark-antiquark pairs of light flavors only (u, d, s). This simple assumption together with the flavor 127
128
independent interaction and the neglection of any detail of the Q C D vacu u m and sea structures, lead t o the remarkable explanation of the observed mass s p e c t r u m 1 . Mesons will appear to be grouped in 577(3)/ nonets, with q u a n t u m numbers given by n2S+lLj(Jpc). Inside each nonet it is possible to define a hierarchy of masses (due to the breaking of the exact 5 £ / ( 3 ) / s y m m e t r y ) and a hierarchy of decay widths. Among other interesting properties of the ordinary mesons, for what is strictly related to the exotic searches, there are definite rules for the allowed q u a n t u m numbers. Any ordinary meson, m a d e by a quark-antiquark pair, must have: isospin / < 2, charge Q < 2 and strangeness S < 2. For what concerns the spin-parity numbers, some combinations are forbidden for any fermion-antifermion system: Jpc = 0~~, 0 + ~ , 1 _ + , 2 + ~ , 3 _ + . . . 2.1
Exotic meson
properties
In order to identify exotic mesons unambiguously at least the approximate knowledge of their properties is needed. Some of the most i m p o r t a n t ones are listed here: • exotic candidates should not fit into SU(3)f
nonets.
• hybrid and m u l t i q u a r k mesons might have very exotic q u a n t u m numbers such as the total charge or the strangeness q u a n t u m number equal to 2. • exotic Jpc • 7T7T and KK yield.
q u a n t u m numbers are a clear identification of exotic mesons. decay channels of glueball should have roughly the same
• glueball production should be very low in 7 7 collisions since there is no electric charge in glueballs and very high in gluon rich environments. It is clear t h a t the hunting for exotic mesons can be easier from the experimental point of view when mesons with really unusual properties are searched for. However, m a n y Q C D exotics, called cryptoexotics, hide themselves among the ordinary ones, having the same q u a n t u m numbers. In order t o isolate this type of mesons, it is m a n d a t o r y to get a full knowledge of 5C/(3)/ nonets. 3
The LEAR experiments
From the point of view of completeness of mesons production in the low energy region the nucleon-antinucleon annihilation environment is one of the
129 best. Since annihilations here mean quark-antiquark annihilations with production of gluons, the environment is gluish. Therefore ordinary mesons, hybrids and glueballs should have the same chances to be produced. One of the most i m p o r t a n t features of this environment is t h a t there are many techniques to control the properties of the initial and the final state in which the annihilations take place. For example changing the beam from antiproton to antineutron or changing the target from pure hydrogen to deuterium it is possible to vary the distribution of the initial isospin and G-parity, while by changing the density of the target from liquid to very low pressure gas or moving from at rest to in flight annihilations it is possible to select different initial angular m o m e n t u m states. All these techniques were extensively exploited at the Low Energy Antiproton Ring (LEAR) at C E R N by different experiments 6 . T w o of them, Crystal Barrel and Obelix, were general purpose detectors with high acceptance, charge and neutral detection capabilities. They acquired complementary approaches to the meson spectroscopy 6 , 7 . The first experiment preferred to study mainly LH2 and in flight annihilations into all neutral final states. The second took huge statistics of different charged channels in different target conditions, reaching more initial and final states and a better control on the results of the spin parity analyses 8 .
4 4.1
Exotic meson candidates Jpc
exotics
There are several candidates seen by different experiments for the J = 1_ + state only. T h e most studied one, named p(1405), has been seen as a r/n P wave first by G A M S 9 in the charge exchange reaction irp —> niT0n with a mass of 1400 MeV and later questioned 1 0 . All the scattering experiments t h a t studied the rjn P wave actually found some activity clearly seen in the forwardbackward cross section asymmetry. Only with high statistics it is possible to establish unambiguously if the corresponding phase motion is resonant or not. T h e latest results from BNL d a t a 1 1 on 7rp —> rjnp at 18 G e V / c show t h a t the P wave is actually resonating with mass 1370 ± 16 and a width of 385 ± 40 MeV, in rough agreement with the first G A M S observation. T h e Crystal Barrel collaboration, analyzing pd annihilations into r)ir~Tr°p, found evidence 1 2 for an i]iv P wave interfering with the p°(770) with a mass 1400 and width 310 MeV. T h e same state was observed in the reactions pp —>• r?7r°7r° in gas, produced mainly from protonium P waves 1 3 . In the BNL d a t a on the scattering reaction Tr~p —> 7r + 7r~7r _ p, there is a
130
clear evidence11 for an exotic and resonating pir 1 + wave, called /S(1600), at a mass of 1593 ± 8atatt\^sys MeV and with a width affected by large systematic Me uncertainties: T = 168 ±20$tatt\l°sys ^'• From the theoretical point of view, all exotic Jpc states mentioned might be associated with the ground state hybrid, since their mass is not too far below the lattice QCD 3 ' 4 and flux tube model 5 predictions (1.8 - 1.9 GeV). 4-2
The scalar glueball and the scalar nonet
Scalar mesons are the most difficult to classify since they show a rich dynamics: they have large widths, many final channels and the isoscalar ones show in addition mixing phenomena. In principle they can be of any type: ordinary mesons, hybrids, multiquark or mesonic molecules. A huge experimental effort has been put in this sector since here the ground state glueball is expected 2,4 in the mass range 1.5 — 1.7 GeV. The surer qq scalar meson is the well known A'Q(1430). The isovector component of the scalar nonet was for long time the ao(980) mainly for a lack of other candidates, but many indications pointed to other non-95 interpretations: its mass proximity to the KK threshold (—> threshold effect or KK molecule), its low width (T = 50 (=a 100 MeV) compared to the A'Q one (—> multiquark), and its low 77 coupling (IX77) = 0.3 ± 0.1 keV), compared with the naive QM expectations T(«o —>• 77) ^ 2r(a2 —> 77) = 1.5 keV. At LEAR a second scalar isovector, named ao(1450) was found 7,14 in pp —> 7r°7r°?y. Although some inconsistencies still exist on mass and width values 10>15; they fit much better than those of the a o (980) in the scalar nonet. Another confirmation of the qq nature of the ao (1450) comes from the comparison of the measured branching ratios with the SU(3)f predictions 14 . This adds further evidence to the conclusion that the problematic ao(980) might not be a qq meson. In the isoscalar sector, below 2 GeV 5 states have been clearly identified and disentangled with the last generation of coupled channel spin parity analyses: / o ( 4 0 0 - 1200), / O (980), / O (1370), / o (1500), /O(17'20). Only two of them can enter into the ground state scalar nonet. Guided by the properties of the K*, a de-facto member of scalar nonet, and of the a o (1450), a natural isovector candidate, the isoscalar candidates can be the / O (1370), / o (1500) and / O (1720). The /O(980) has been ruled out having similar problems 2 of the a o (980): mass close to KK threshold, low total and 77 widths. The / o (400 — 1200), which is also called a meson, has been excluded since its mass and width properties 15 , although not yet well understood, are in clear disagreement with a qq interpretation. The /O(1370) and / o (1500) mesons where seen firstly in NN annihilations, from Crystal Barrel 14 and Obelix 6,8 experi-
131 Table 1. Decay branching fractions for / o (1500) in % from Crystal Barrel 7T7T
29.0 ± 7 . 5
VV 4.6 ± 1 . 3
vv' 1.2 ± 0 . 3
KK 3.5 ± 0.3
4?r excl. pp 61.7 ± 9 . 6
ments in different decay modes. The closeness of the mass (1360 ± 23 MeV) and width (350±41 MeV) of the / O (1370) with those of the K* and ao(1450) and the strong coupling with the u, d sector make the /O(1370) an ideal candidate for the nh member of the scalar nonet. The / o (1500) meson 15 (mass: 1500 ± 10 MeV, width: 112 ± 10 MeV, see fig. 1) does not fit well with the ss assignment since its decay pattern show a reduction of kaonic or hidden strangeness channels (see table 1). Other interesting properties of this meson, which is one of the main glueball candidates, is that it appears in all gluon rich processes like NN annihilations, central production and J/ip radiative decays, while it is not seen in 77 collisions16 (1X77) < 0.17 keV at 90 % c.l.). The /O(1720) meson, known formerly as /j(1710), has been recently studied by the WA102 collaboration in the glue-rich environment of Double Pomeron Exchange. In a coupled channel fit17 of the TT+TT~ and K+K~ final states, they determined unambiguously the spin of the meson being J = 0 and the KK /TTTT decay ratio being 5.0 ± 0.6 ± 0.9, which is a clear suggestion that its internal state is mainly ss. If this is the case, then the /o(1720) fills the ground state scalar nonet and the / o (1500) is definitely left out as an exotic meson (mainly as a glueball candidate). Mixing phenomena between ordinary and exotic mesons can alter this simple picture. In table 2, a possible assignment 15 of the ground state scalar nonet, together with a gg mixing estimation 4 is presented. Other assignments and mixing estimations can be found in extended reviews 2,4,18,19 . 4-3
A tensor glueball candidate?
The first 2 + + glueball is expected4 from Lattice QCD at a mass around 2.3 GeV. A narrow state has actually been seen (f« 4cr effect) by different experiments 2 in J/ip radiative decays in the final states 7T+TT~ , w0ir0, K+K~, KsI<s and pp at a mass of 2230 ± 25 MeV and width 23 ± 8 MeV, with Table 2. A possible scheme for the scalar meson nonet Name Mass (MeV) Width (MeV) Isospin gg admixture 4
a o (1450) 1465 ± 2 5 310 ± 6 0 1=1
A"»(1430) 1429 ± 6 287 ± 2 3 1=1/2
/ O (1370) 1360 ± 2 3 350 ± 4 1 1=0 (nn) 24 %
'
with the gg candidate . /o(1720) 1715 ± 7 125 ± 1 2 1=0 (ss) 18 %
/o(l500) 1500 ± 10 112 ± 10 1=0 (gg) 59%
132
Jpc = (even)++. A similar state J = 2 has been seen at L E A R by J E T S E T in pp —>• 4>, but it has not been seen by Crystal Barrel in their high-statistics formation study in the channels pp —> 7r°7r°, r\r\ in flight. Therefore although the state seen in J/ip radiative decays has some glueball properties (produced in glue-rich environments, not seen in 7 7 collisions, right mass and width too narrow for a qq system), its mere existence is presently in doubt, being not confirmed by other experiments 2 , 7 .
4-4
The first radial pseudoscalar
excitations
Historically the one of the first glueball candidates was the pseudoscalar meson 77(1440) seen mainly in J/ip —> fKK-K. T h e latest analyses in this field actually found out t h a t the broad enhancement seen in KKir around 1.4 — 1.5 GeV contains at least two pseudoscalar and an axial-vector meson. This was confirmed by a high statistics study of Obelix 2 0 where the analysis of pp —> KKirwTr final state shows t h a t the lighter pseudoscalar (mass: 1405 ± 5 MeV, width: 50 ± 5 MeV) decays mainly into KKir, while the heavier (mass: 1500 ± 10 MeV, width: 100 ± 2 0 MeV) prefers the A'A'* final state. If the 7/(1295) meson, seen in peripheral production only 1 5 , will be confirmed, there are three isoscalar candidates for the first radial excitation of the pseudoscalar nonet, where only two can be placed. Clearly the extra state is a pseudoscalar glueball candidate (pure or mixed). For the nonet properties 1 5 and the production and decay characteristics the favoured gg candidate is the 7/(1440) having a stickiness parameter two orders of m a g n i t u d e higher t h a n a clearly qq meson: S[rj(U40)] > 4 5 5 [ T 7 ( 5 4 8 ) ] .
5
Conclusions
T h e latest generation of experiments in meson spectroscopy systematically searched for exotic as well as ordinary mesons. There is now a wide knowledge on meson dynamics, mass spectra and decay properties. Few reliable exotic candidates have been singled out: / o (1500) (gg candidate, possibly with some qq admixture), 7/(1400), p(1405) and /3(1600) (Jpc exotic, qqg candidate). Some states, left out from the scalar nonet, await classification and might t u r n out to be the older Q C D exotics known: ao(980) and /o(980). Acknowledgments T h e author wishes to thank the organizers of the Conference for the kind invitation and the members of the Obelix-Bologna group for their help.
o
175
125
1111 111111111111111
133
WUIKA
100 —
1
1.2
1
1
1
1
1.4
1
1
1.8
1
1
I
1
1
1
1
1
1 1
2.2 GcV/c 2
m(TiTi)
Figure 1. The T\T] invariant mass spectrum for the reaction pp —>• ir°riri at 1.94 GeV/c p momentum obtained by Crystal Barrel 7 . The peak in the real data (points with error bars) and in the Spin Parity fit (continuous line) around 1.5 GeV/c2 is due to the / o (1500).
References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17.
S. Godfrey and N. Isgur, Phys. Rev. D 32, 189 (1985). S. Godfrey and J. Napolitano, Rev. of Mod. Phys. 7 1 , 1411 (1999). P. Lacock et al., Phys. Lett. B401, 308 (1997). D. Toussaint, Nucl. Phys. Proc. Supp. 83, 151 (2000) and ref. therein. T. Barnes et al., Phys. Rev. D52, 5242 (1995). A. Zenoni, Nucl. Phys. A 654, 96c (1999) and ref. therein. K. Braune, Nucl. Phys. A 655, 3c (1999) and ref. therein. N. Semprini Cesari et al., Nucl. Phys. A 655, 82c (1999) and ref. therein. D. Aide et al., Phys. Lett. B205, 397 (1988). S.A. Sadovsky, Nucl. Phys. A 655, 131c (1999) and ref. therein. K. K. Seth, Nucl. Phys. A663&664, 113c (2000) and ref. therein. A. Abele et al., Phys. Lett. B423, 175 (1998). A. Abele et al., Phys. Lett. B446, 349 (1998). C. Amsler, Rev. of Mod. Phys. 70, 1293 (1998). D. E. Groom et al., The European Physical Journal C15, 1 (2000). G.D. Lafferty et al., AIP conference proceedings 432, 80 (1998). D. Barberis et al., Phys. Lett. 453, 305 (1999), D. Barberis et al., Phys. Lett. 453, 316 (1999). 18. R. Landua, Annu. Rev. Part. Sci. 46, 351 (1996). 19. P. Blum, Int. J. Mod. Phys. A l l , 3003 (1996). 20. C. Cicalo et al., Phys. Lett. B 462, 453 (1999).
A S T U D Y OF T H E n-w I N T E R A C T I O N I N N U C L E A R M A T T E R U S I N G T H E vr+ + A -> TT+ + TT± + A' R E A C T I O N P. CAMERINI Dipartimento di Fisica-Universita di Trieste and INFN. via Valerio 2 34127 Trieste - ITALY E-mail: [email protected] THE CHAOS COLLABORATION The pion-production 7T+J4 —> TT+TT A' reactions were studied on nuclei 2H, i0 Ca and 20SPb at an incident pion energy of Tn+ =283 MeV.
1
12
C,
The 7r27r reaction and the mr interaction in the nuclear medium
The influence of the nuclear medium on the TTTT interaction was studied at TRIUMF by means of the pion induced pion-production reaction ir+A —> ir+ir:izA' (TT2TT). The initial study was directed to the deuterium, that is, to the 7r+n(p) —> Tr+Tr~p(p) and 7c+p(n) —» 7r + 7r + n(n) reactions, in order to understand the TT2TT behaviour on both a neutron and a proton. The ir2ir process was then examined on complex nuclei 12C, ^Ca and 20SPb in order to derive possible 7T7T medium modifications by direct comparison of the 7r27r data. To perform such a study, the 7r27r data were collected under the same kinematical conditions. The final ir+n:i: pairs were detected in coincidence to ensure a reliable identification of the 7r27r events, and the pion pairs were analyzed down to 0° 7T7T opening angles to determine the TTTT invariant mass at the 2m T threshold. The experiment was carried out at the TRIUMF Meson Facility using the M i l pion beam at a central momentum p=398.5 MeV/c. The targets used in this experiment were either solid self-supporting plates of carbon, calcium and lead, or a vessel for the liquid deuterium. The outgoing pions were detected with CHAOS, which is a magnetic spectrometer which was designed for the detection of multi-particle events in the medium-energy range -1. Some TT2TT articles were recently published by the CHAOS collaboration at TRIUMF. The novel data highlighted some properties of the in-vacuum TTTT interaction 2 ' 3 , 4 as well as the in-medium modifications of the TTTT interaction 5 6 7 ' ' . The appearance of the CHAOS results has renewed theoretical interest, which now grounds the interpretation of the 7r27r and 7T7T data on some common features: a) the mr interaction is strongly influenced and modified by the presence of the nuclear medium when the -KIT interaction occurs in the scalarisoscalar channel, conventionally called the a—channel8,9,10'11'12; b) nuclear 134
135
matter weakly affects the (7T7r)r=2,j=o interaction 11 ' 13 ; c) models which only include standard many-body correlations, i.e. the P—wave coupling of IT'S to p — h and A — h configurations, are able to explain part of the observed M^+v- yield near the 2mv threshold 9,11,13 . In recent theoretical works on the (irir)i=j=o interaction in nuclear matter 14 , the effects of standard many-body correlations are combined with those deriving from the restoration of chiral symmetry in nuclear matter. As a result, the M^+v^ distributions are shown to gain strength near the 2mff threshold as A (thus the average p) increases. Such a property was earlier outlined in some theoretical works, which demonstrated that the M^+7r_ enhancement near threshold is a distinct consequence of the partial restoration of the chiral symmetry at p < pn 12 . 2
Results
A general property of the 7r27r process on nuclei in the low-energy M„v regime was outlined by previous experimental works: it is a quasi-free process both when it occurs on deuterium 15 and on complex nuclef3. Thus the study of the 7r+ 2H —* TT+IT^NN reaction is dynamically equivalent to studying the elementrary ir+n —» 7r+7r~p and ir+p —•*• 7r+7T+n reactions separately. In the present measurement, the TT+A —* ir+TT±A' reactions were studied under the same experimental conditions. Thus for a given observable the distributions are directly comparable. Fig. 1 shows the single differential cross sections (diamonds) as a function of the 7T7T invariant mass ( M „ , MeV) for the two TV+ —»• TV+TC~ and n+ —» TT+TT+ reaction channels. The horizontal error bars are not indicated since they lie within symbols. The irA —> TTTTN[A — 1] phase space simulations (dotted histograms for A : 2H, 1 2 C, ^Ca and 208Pb) are also provided and are normalized to the area subtended by the experimental distributions. Regardless of the nucleus mass number, the invariant mass for the 7r+ —> TT+TT+ distributions closely follow phase space and the energy maximum increases with the increase of A. The n+ —> 7r+7r~ channel discloses a different behaviour; as compared to phase space, the 2H invariant mass displays little strength from 2mff to 310 MeV while, on the same energy interval, the 1 2 C, i0Ca and 208Pb 7r+7r~ invariant mass distributions increasingly peak as A increases. In order to explain the nature of the reaction mechanism contributing to the peak structure, the c o s O ^ * distributions were examined, where 0 £ j f is the angle between the direction of a final pion and the direction of the incoming pion beam in the n+ir~ rest frame. A partial wave expansion limited to 2mw < Mir+n- < 310 MeV, indicates that the TT+IT~ system predominantly couples S—wave (~ 95%) and a remaining 5% is spent in a D—wave state.
136 TT TT
->
7T 7T
Pb
Pb
S$
72 48
Pill*
24
2
4o
0
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60
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Ca
55
*»
45
CD
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0 4.5 3.0
15
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0.0
>
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^
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-&H
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260
300
340
3B0
M
420 260
300
340
^••io.O
390
420
(MeV)
Figure 1: Invariant mass distributions (diamonds) for the ir+ —» 7r+7r and 7r+ —> x+ir+ reactions on 2H, 1 2 C , 4 0 C a and 20aPb. Diagrams (dots) are the result of phase-space simulations for the pion-production nA —> -inrN{A — 1] reaction.
We introduce now the observable C^v in comparison with recent theoretical predictions. C^ is denned as the composite ratio -^Y-f—ff-, where a^ {&T) is the measured total cross section of the 7r27r process in nuclei (nucleon). This observable has the property of yielding the net effect of nuclear matter on the (7T7r)j=j=o interacting system regardless of the TT2TT reaction mechanism used to produce the pion pair 7 . Therefore, C£„ can be compared with the predictions of 13 and n which explicitly calculate both M%£ and Mj%, but also with the theories described in 12 and 1 4 because they calculate the mass distribution of an interacting (w7r)i=J=O system (i.e. ImD^) both in vacuum and in nuclear matter. Since the above calculations are reported either in arbitrary units n ' 1 3 or in units which are complex to scale 1 4 ' 1 2 ) theoretical predictions are normalized to the experimental distributions at M, rir =350±10
137
340
380 260
M
300
(MeV)
Figure 2: T h e composite ratios C^ for 1 2 C , 40Ca and 20sPb. T h e curves are taken from 1 1 (full), 1 2 (dashed), 1 3 (dotted) a n d 1 4 (dash-dotted). Further details are reported in the text
MeV, where C^. present a flat behaviour Fig. 2. For both reaction channels, the fulP1 and dotted 13 curves in Fig. 2 are obtained by simply dividing M%£/ M , f . It is worthwhile recalling that for 13 the option p=0.5pn is used while for n the mean density is p=0.24/? n . Furthermore, for both approaches the underlying medium effect is the P—wave coupling of 7r's to p — h and A — h configurations, which accounts for the near-threshold enhancement. When applied to the C^£,13 and u predict the same result; in fact, they well describe the behaviour of C++ throughout the M-mr energy range, while for C^t only part of the near-threshold strength is reproduced. The models of12 and 1 4 examine the medium modifications on the scalarisoscalar meson, the a—meson. Nuclear matter is assumed to partially restore chiral symmetry and consequently mCT to vary with p. Both models are capable of yielding large strength near the 2m„. threshold. In Fig. 2 the predictions of 12 and 14 for p = pn, are reported with dash-dotted line and dashed line, respectively. Ref. 14 provides a larger near-threshold strength, which is due to the combined contributions of the in-medium P—wave coupling of pions to p—h and A—h configurations, and to the partial restoration of chiral symmetry in nuclear matter. This model, however, is still too schematic for a conclusive comparison to the present data, therefore full theoretical calculations are called for. 3
Conclusions
In this article the results of an exclusive measurement of the pion-production n+A —* •K+TT^A' reactions on 2H, 12C, i0Ca and 2 0 8 P6 at an incident pion
138
energy 2^+=283 MeV were presented. The primary interest was directed to the study of the irir—dynamics in nuclear matter. The reaction was initially examined on deuterium to understand the elementary pion-production mechanism, then on nuclei to determine the effects of medium modification on the 7T7T— system. Some of latter effects could be obtained by direct comparison of the TT2TT distributions since the data were taken under the same kinematical conditions. C ^ was found to yield the net effect of nuclear matter on the TTTT system regardless of the ir2ir reaction mechanism used to produce the pion pair. These distributions display a marked dependence on the charge state of the final pions: (i) the C^+7r_ distributions peak at the 2m x threshold and the yield increases as A increases thus denoting that pion pairs form a strongly interacting system; furthermore, the 7T7T system couples to the I = J = 0 channel, the a—meson channel, (ii) In the 7r+ —» 7T+7T+ channel, the C^„ behaviour barely depends on both A and energy thus indicating that nuclear matter weakly affects the (7r7r)/jr=2,o interaction.
The C^+7r_ observable was compared to theories studying the (mc)i=j=0 in-medium modifications associated to the partial restoration of chiral symmetry in nuclear matter, and with model calculations which only include standard many-body correlations, i.e. the P—wave coupling of IT'S to p — h and A — h configurations. It was found that both mechanisms are necessary to interpret the data, although chiral symmetry restoration yields the near-threshold larger contribution. Whether this conclusion is correct, the TT2TT CHAOS data would indicate an example of a distinct QCD effect in low-energy nuclear physics. Montecarlo simulations of the n+A —• ir+ir±N[A — 1] reaction phase space revealed useful to interpret some of the 7r27r data. In the case of Mv+r+, 7r+7r+ pairs distribute according to phase space. In addition, simulations are able to describe the high-energy part of the distributions which are sensitive to the nuclear Fermi momentum of the interacting 7r+p[^4 — 1] —> 7r+7r+n[.A — 1]' proton. For the Mv+V- distributions the IT-IT dynamics overwhelms the dipion kinematics: unlike phase space, the near-threshold 7r+7r~ yield is suppressed in the elementary production reaction, 7r+ 2H —> ir+ir~pp in the present work, while in the same energy range medium modifications strongly enhance Mr+„-. A guideline to the interpretation of the M\ ± behaviour should combine the effects of the chiral symmetry restoration in nuclear matter and standard many-body correlations. Such an approach would exclude high-density nuclear matter for both the production reaction to take place and the irir system to undergo medium modification.
139
References 1. G.R. Smith et al, Nucl. Instr. and Meth. in Phys. Res. A362, 349 (1995). 2. F. Bonutti et al, Nucl. Phys. A638, 729 (1998). 3. M. Kermani et al, Phys. Rev. C58, 3419 (1998). 4. M. Kermani et al, Phys. Rev. C58, 3431 (1998). 5. F. Bonutti et al, Phys. Rev. Lett. 77, 603 (1996). 6. F. Bonutti et al, Phys. Rev. C55, 2998 (1997). 7. F. Bonutti et al, Phys. Rev. C60, 018201(1999). 8. P. Schuck et al, Z. Phys. A330, 119 (1988); G. Chanfray et al, Phys. Lett. B256, 325 (1991); 9. P. Schuck et al., 36th International Winter Meeting on Nuclear Physics, Bormio, Jan. 1998. 10. H. C. Chiang et al, Nucl Phys. A644, 77(1998); 11. M. J. Vicente-Vacas et al, Phys. Rev. C60, 064621(1999). 12. T. Hatsuda et al., Phys. Rev. Lett. 55 158(1985); T. Hatsuda et al, Phys. Lett. B185 304(1987); T. Hatsuda et al, Phys. Rep. 247 221(1994); T. Kunihiro, Prog, of Theor. Phys. 120 75(1995); T. Hatsuda et al, Phys. Rev. Lett. 82 2840(1999). 13. R. Rapp et al., Phys. Rev. C59, R1237 (1999). 14. Z. Aouissat et al, Nucl-th/9908076 v2 31 Aug 1999; D. Davesne et al, Nucl-th/9909032 15 Sept 1999. 15. R. Rui et al., Nucl. Phys. A517, 445 (1990), C. W. Bjork et a l , Phys. Rev. Lett. 44, 62 (1980), J. Lichtenstadt et al., Phys. Rev. C33, 655 (1986), V. Sossi et al., Nucl. Phys. A548 562 (1992).
PION-PION POTENTIALS B Y INVERSION OF PHASE SHIFTS AT FIXED ENERGY B. BATHORY, Z. HARMAN, AND B. APAGYI Technical
University
of Budapest, E-mail:
H-llll Budapest, Budafoki [email protected]
ut 8,
Hungary
Isoscalar pion-pion scattering potentials are derived by applying fixed energy quantum inversion theory using experimental phase shifts as input. Attempts are being made to supplement experimental information with phase shifts belonging to nonphysical partial waves of odd angular momenta. The potentials obtained at energies below the threshold of kaon production are real and exhibit a Coulomb-like attraction with strength and range similar to those found by inversion at fixed angular momentum. At energies above threshold the potentials are complex valued and have in general a larger spatial extent than those describing pion-pion scattering at lower energies.
1
Introduction
Recent lattice QCD calculations 1 indicate that both the free and interacting isoscalar pion-pion systems posses an s—wave projected effective local interaction potential that is attractive with a strength of several GeV and a range of less than one fermi. Similarly, quantum inversion calculations 2 at fixed angular momentum I also resulted in an attractive and short ranged shape for the s— wave isoscalar 7T7T potential, but with a strength of several hundreds GeV and a range of less than a half fermi. Since both types of calculations use approximations and theoretical simplification the above results represent preliminary information on the spatial extent of the mr (four-quark) system which plays an important role in the theoretical understanding of the interaction between mesons. Because of this importance it is interesting to explore further the pion-pion system by using a third theoretical tool which provides us with independent information on the interaction properties (the strength and the range) of this very basic lightest 'elementary' particle system.
2
Input data for the inversion
We shall start with the isoscalar (/ = 0) ir — IT scattering data derived by Protopopescu et al.3 from experimental irN —> TTTTN cross section by applying 140
141
the Chew-Low formula (with standard notations) avw = lim t—*m%
z\z
dtdi/s
af%
(1)
ty/sk
for getting the 7T7T cross sections. From this a partial wave decomposition {21 + l)Pt [i#e 2 a ? - l]
V ^ oc £
(2)
1=0,2,..
yields the input data for our quantum inversion calculation, the phase shifts 6% and the elasticities ?j° at several fixed energies expressed in terms of invariant mass y/s = Mnn or wave number A; = \ / s / 4 — m\. The phase shift data used in the inversion calculation is listed in Table 1. Because of isospin conservation only phase shifts belonging to even partial waves are accessible to measurements. However, at low energies this means that only a low number (2-6) of experimental data is disposable to perform quantum inversion. This lack of information may cause a problem 4 in the interpretation of the inversion potential. To minimize this ambiguity we shall apply different techniques of quantum inversion at fixed energy developed by Newton and Sabatier (NS) 5 and by Munchow and Scheid (mNS).6 As a third method, we also perform calculations with inclusion of interpolated phases at nonphysical odd partial waves. Those interpolated values used are also listed in Table 1. Table 1. Experimental phase shifts 8® (in degree) and elasticities 77° derived by Protopopescu et al.1 for isoscalar TTTX scattering at several invariant masses MW7r (in GeV) in various partial waves corresponding to I = 0,2,4,6. Some nonphysical values applied are also indicated at t = 1,3,5 waves. Mnn
6°
«¥
0.55
43 56 81 88 99 134 194 215 208
0 0 0
0.625 0.795 0.85 0.91 0.965
1.0 1.075 1.150
60
°2 0 0 0 1.6 4.4 8.9 12 27 44
«s
20
«2
0 0 0 0 0 0
n%
1?
1 1 1 1 1 1
1 1 1
4
4
1
1 1 1 1 1 1
1 1 1 1 0.99 0.94
0.39
0.78
0.48 0.57
41
0.35
0.94
142
3
Theoretical methods
The NS inversion method amounts to write the inversion potential as a sum o
J
i °°
V{r) = — — - Y, ceMr)3i(kr) r ar r *-—'
(3)
i
with ji being the Riccatti-Bessel function and <j>i the solution function calculated from the Regge-Newton equation oo
Mr) = ji(kr) - ] T cvLu'{r)4>v(r) v
(4)
with LW(T) = / 0 r ^(A;r')j>(A;r')(r')- 2 dr'. In the NS method the coefficients ci(k) are calculated from Eq. (4) when the limit r —> oo is taken. Then one obtains the solution a = (1 + M - 1 tan A M tan A ) - 1 ( M _ 1 tan Ae + av)
(5)
for the vector ai = Aici cos Si with (tan A)«< = 6a> tan Si, ei = 1, and V( and Mw are defined in the works of Newton and Sabatier 2 where also another equation for determining the normalisation constants Af is given. In Eq. (5) the phase shifts Si = In 2iS® + 77° are taken from Table 1 and the real number a plays the role of a technical parameter. In the mNS method the coefficients ci(k) are calculated also from Eq. (4) but with using the supposition that beyond a finite distance ro the potential is known [V(r > ro) = 0 say] and thus the solution function is related with the input phase shifts as 4>t(r > ro) = Ai[ji{kr)
- tan^n^Ar)]
(6)
with Si = In 2iS° + rft taken from Table 1. When this assumption is inserted in the Regge-Newton Eq. (4) one obtains the expansion coefficients ci (and the normalization coefficients Ai) as the solution of the matrix equation: <Mr«) = 3l{kri) -^2ci-LU'(ri)(l>i'(ri),
i = 1,2
(7)
l'
where the outer radius ri > r 0 (and Ti = T\ + 0.05 fm) plays the role of the technical parameter. As it has been shown in case of the NS method, 5 the resulting inversion potentials posses a physical asymptotics only at a special choice of the technical parameter a if an infinite set of phase shifts is used in Eq. (5). It has been proved also for the mNS method 6 that the coefficients ci (and the inversion
143
potential resulted) do not depend on the choice of the outer radius n if the condition V(r > ro) = 0 is true and infinite many phase shifts are used in the procedure (even though they are zero). A stability statement of similar type holds also in the case of the third procedure where interpolated (nonphysical) phase shifts are involved in the inversion procedures discussed above. However, in the practice we always take a finite number £max +1 of phase shifts so that the special values of a should be corrected for getting physically acceptable (relatively smooth) potential. On the other hand the stability of ce's with respect to change of the technical parameter n (at fixed £max) is only approximately fulfilled. Nevertheless, by carefully investigating the magnitude of ce's, we have found that it is possible to get comparable c/'s in all the three methods if the technical parameters are chosen properly. Therefore we accepted results presented below only if two of the three inversion procedures sketched above provide the same expansion coefficients in the relevant partial waves. 4
Results
200
550 625 795 850 910
150 100 50
_ u >
MeV — MeV MeV —• ' MeV MeV
.
-50
•
-100 -150
i 13 i 1 ft
.
-200 -250 -300
1 .
. r
(fm)
Figure 1. Isoscalar im potentials at several energies below the KK
threshold.
In Fig. 1. the inversion potentials obtained at several invariant masses M XT below the threshold energy of the kaon production are presented. The potentials are in general real and exhibit a Coulomb-like attraction at smaller distances between the pions. Although not directly comparable with earlier calculations dealing with TTTT interactions, the strengths and ranges of our
144
potentials are similar to those found by quantum inversion at fixed (I = 0) angular momentum. 2 However no kind of soft core repulsion at small distances is seen at this low energy which is in harmony with the lattice QCD findings.1 Tl
'
\ \
150 100 50
965 1000 1075 1150
MeV MeV MeV — • MeV —
\ ?
0
•
.
V
-50
n
-100
1 ^" ^
-150 -200 -250
if/
-
w
•
; *
0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 r (fin) 200
\A '
150 •
/
100
/
;
• ii, i 1 0
50
-50 -100 -150
-i / III '
1 !/ .'/
'
/
\l\
'
965 1000 1075 1150
MGV MGV MGV
MeV-—
"
" "
vKix xy
-200 -250 -300
•
:
/ 0.05
0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 r (fin)
Figure 2. Isoscalar 7T7T potentials at several energies above the KK
threshold.
The potentials at energies above threshold of the KK channel are shown in Fig. 2. They exhibit a more sophisticated structure being complex valued and of larger extent than those below threshold. Except for the potential close to the threshold energy (where the absorption comes only from the 1 = 1 partial wave) these potentials are characterized by an over-all repulsion at smaller distances and by oscillations in the intermediate (0.1 < r < 0.3 fm) region. The former property has also been found by quantum inversion at fixed (t = 0) angular momentum 2 and by phenomenologically re-analysing
145
the relevant 7T7T data. 7 The latter characteristics (the oscillations) may however originate from the deficiency of the data information.4 It should be noted however that in spite of more input data, the stability analysis was successful only in a narrower domain of values of technical parameter r\ than in case of calculations at lower energies. 5
Conclusion
It has been demonstrated that quantum inverse scattering at fixed energy can be applied to isoscalar irn phase shift data to derive pion-pion interaction in coordinate space. The results of the present calculation are in a qualitative agreement with those of other calculations. The isoscalar irir potentials are strongly energy dependent. Below the KK production threshold energy they exhibit a very short range of less than 0.2 fm and a strong Coulomb-like attraction of several hundreds GeV. Above the KK threshold the potentials become complex and of extended shape giving rise to a repulsion at smaller distances and oscillations in the intermediate (0.1 < r < 0.3 fm) region. Acknowledgments This work has been supported in part by OTKA T29884, T25019. References 1. H. R. Fiebig, H. Markum, A. Miliary, and R. M. Woloshyn, Nucl. Phys. B. (Proc. Suppl.) 63 A-C, 188 (1998); H. R. Fiebig, H. Markum, A. Mihaly, and K. Rabitsch, Nucl. Phys. B. (Proc. Suppl.) 53, 884 (1997); H. R. Fiebig, O. Linsuain, H. Markum, and K. Rabitsch, Nucl. Phys. B. (Proc. Suppl.) 473, 695 (1996). 2. M. Sander and H.V. von Geramb, Phys. Rev. C 56, 1218 (1997). 3. S. D. Protopopescu, M. Alston-Garnjost, A. Barbaro-Galtieri, S. M. Flatte, J. H. Friedman, T. A. Lasinski, G. R. Lynch, M. S. Rabin, and F. T. Solmitz, Phys. Rev. D 7, 1279 (1973). 4. M. Eberspacher, K. Amos, B. Apagyi, Phys. Rev. C. 6 1 , 64605 (2000). 5. R.G. Newton, J. Math. Phys. 3, 75 (1962); P.C. Sabatier, J. Math. Phys. 7, 1515 (1966). 6. M. Miinchow and W. Scheid, Phys. Rev. Lett. 44, 1299 (1980). 7. S. Ishida, M. Ishida, T. Ishida, K. Takamatsu, and T. Tsuru, Progr. of Theor. Physics, 98, 621 (1997).
PERSPECTIVES OF T H E A N T I D E U T E R O N PHYSICS AT JHF
F . IAZZI Dipartimento
di Fisica
del Politecnico, C.so Duca degli Abruzzi E-mail: [email protected]
24, Torino,
Italy
J. D O O R N B O S TRIUMF.
Vancouver,
Canada
V6T
2A3
T , B R E S S A N I A N D D. CALVO Dip.
di Fisica Superiore
dell'Universita'
e INFN,
10125 Torino,
Italy
The future Japanese Hadronlc Facility (JHF) can provide, in addition to antiprotons and other particles, also a quite intense flux of antideuterons a t low momentum. In the present work some aspects of the antideuteron physics are illustrated, the main features of a magnetic line for the transport of the antideuterons are summarized and some possible experiments are briefly sketched.
1
Introduction
Few years ago the Japanese Hadron Facility (JHF) was proposed at KEK 1 with the appealing feature of an expected intensity of the 50 GeV proton beam, which is about 1 0 1 4 / J A Such an intensity suggested that the production of antideuterons (d) on a nuclear target could be much higher than in the past and enough to explore new fields of physics involving d's 2. The motivations for using such a kind of heavy antiparticle as projectile, reviewed in Sec. 2, cover several fields of physics, from the d's production mechanism to the antinucleon interaction with the excited nuclear matter and to the symmetry properties of the matter-antimatter. A dedicated magnetic line for an d beam at JHF has been designed5 and its performancies, recalled in Sec. 3, overcome the problems related to the d transport. Finally, for two experiments involving J's, a scheme of a possible setup is suggested in Sec. 4. 2
Physics with antideuterons
The d can be used for studying its own properties as well as a probe inside the nuclear matter. Some ideas about the study of gross properties of the d production, the symmetry properties of d with respect to d, the properties of the J-nucleus systems and the interaction of two antinucleons with a nucleus are listed below. 146
147
2.1
Antideuteron
production
The d's can be produced from a high energy proton beam impinging on a nuclear target. A rough interpolation of the scarce available data gives the following laws for the production rate Rj/p of J with respect to the p (produced in the same conditions) as a function of the d (p) momentum p, the proton energy Ep and the emission angle 6: Rs/p(p,Ep = 70GeV,9 ~ 0°) = i0-4.i8+o.o22P-o.ooo93.p2 Rd/p(P, Ev = 200 GeV, 6 ~ 0°) = i0-4.3+o.044.P-o.ooo77.P2
(1)
A rich amount of new data at Ev = 50Gel/ could be taken at JHF in order to improve the above expressions and to clarify the d production mechanism which is still largely unknown 3 . 2.2
antideuteron-deuteron symmetry properties
The first symmetry d — d is the equality of the masses, required by the CPT invariance; moreover, the d mass is correlated with the d binding energy, i.e. with the NN potential: the d mass could be measured with high precision both through the elastic J scattering and detecting the X-rays in the "exotic d atoms" transitions (like for the £ mass 4 ) Another symmetry should appear in the n momentum distribution inside d, which is directly correlated with the NN wave function (expected equal to the NN one): some details of a possible experiment are reported in Sec. 4. Last, the dp —> TTN reaction should have the same cross section (very low) of the symmetric Pontecorvo reaction but a dedicated experiment seems penalized by the required high d flux. 2.3
Antideuteron-nucleus
system
The antinucleus d can be used also to study the interactions with protons an d deuterons at low energy (d is charged and can be slowed down in LH-i and LD2), looking, for instance, at the Coulomb effects with a very massive projectile; brought at rest, J e a n form the system d—p (let's call it "deutonic anti-atom") which is symmetric to the "anti-deutonic atom" and a wide set of measurements, similar to those of the p — p system, become possible. 2.4
Antideuteron-nucleus
interaction
An d travelling through a nucleus can deposit therein twice the energy of p and the measurement of the multiplicity of the annihilation products could be
148
an interesting test for the classical Intranuclear Cascade and Fireball models. Another nuclear process involving d is: one of the antinucleons of d annihilates inside a nucleus, excites the residual nuclear matter that becomes an hot matter target for the second survived antinucleon. With a low energy d the detection of the reaction products will not present problems of forward boost: the feasibility is shown in Sec. 4. 3
Design of an antideuteron line at J H F
The main requirements of a magnetic line for transporting the d's from the production target to the experimental areas are: a) a high transmission rate and b) reduction of the overwhelming background due to the pions (~ 2 • 1077r~/
Features of some experiments with antideuterons
Each experiment with d's needs a dedicated design: few remarks are reported below about the measurements of the n momentum distribution in d and of the n annihilation in the excited matter. 4-1
Antineutron momentum
distribution
The technique for this measurement is based on the reaction d + p —> n + mesons, that can be produced by an d impinging onto a LH? target. Taking into account that: pn(inLAB)
~ pn(in d) + (m^/m^)
• pj(inLAB)
(2)
by measuring pn(inLAB) and pj(inLAB) one gets easily pn(in d). Both quantities can be measured with a quite good accuracy by means of a setup like in fig. 1 , where BM indicate a bending magnet and ANC an n detector of the type used at LEAR (see experiments PS178 and PS199). Assuming p j ~ 1 GeV/c, a flux ~ 210 d/s, a LH2 target ~ 1.3m long and approximating &ann(d + P —*• n + i^s) ~ o-ann(dp) one can expect ~ 15 n/s distributed over a wide momentum range.
149
Figure 1. antineutron n momentum distribution: sketch of an apparatus.
4-2
Antinucleon interaction with excited matter
A setup similar to fig. 1 has been already sketched for measuring the annihilation of the n of d in a nucleus excited by the annihilation of the p. Suitably dimensioned and segmented targets have been demonstrated in Ref. 6 to overcome the difficulty of distinguish the annihilations of both p and n in the same nucleus from those ones in 2 different nuclei. 5
Conclusions
The high intensity of the JHF primary proton beam, offers the possibility to study d's and their interaction with the nuclear matter. Future experiments with d's at low energy can investigate the production of d, the d-d symmetry, the J-nucleus systems and the excited nuclei. A preliminary design of a J beam at 1 GeV/c shows an expected flux of ~ 210 d/s practically without contaminants. The measurements of the n momentum in d and of the n annihilation in excited nuclei seem, at a first insight, feasible. References 1. 2. 3. 4. 5. 6.
Proposal for JHF, JHF Project Office, KEK Report JHF-97-1 (May, 1997) Nucl. Phys. A 665, 371c (1999) Phys. Rev. Lett. 5,6, 276 (1960) and Phys. Rev. Lett. 7,2, 69 (1961). Phys. Rev. Lett. 60,3, 186 (1988). J. Doornbos and F. Iazzi, in press on Nucl. Instrum. Methods , (2000). F. Iazzi, "First experiments with antideuterons at JHF", in "Proc. of JHF98", Vol.11, Eds. J. Chiba, M. Furusaka, H. Miyatake and S. Sawada, KEK Proceedings 98-5
S T U D Y OF T H E 7r+7r+ S Y S T E M I N T H E ANTINEUTRON-PROTON INTO THREE CHARGED PIONS ANNIHILATION R E A C T I O N A. FILIPPI * Istituto Nazionale di Fisica Nucleare, sez. di Torino, Via P. Giuria, 1, 10125 Torino, Italy The results of a search of a possible evidence of a 7r+7T+ resonant state in the np —• 7r+7r+7r— annihilation reaction with data collected by the OBELIX Experiment are reported.
Search for a resonant state in the np reaction
annihilation
So far. no n+Tr+ resonant structure has been observed, in any reaction. An early analysis of the pd —> Tr+ir~Tr~ps annihilation channel by means of a final state interaction approach 1 tried to check the effect of a 1=2 amplitude in the description of the Dalitz plot, suggested by an accumulation of events along its diagonal edge. However, the evidences found for a possible resonant state in the 7r+7r+ channel were not conclusive.
5 2 2,5 3 mV?T), GeV2 Figure 1. Symmetrised m (TT ") Dalitz plot, for the np —* n+n+n~ annihilation reaction (35118 events).
In the present analysis a statistics one order of magnitude larger is exploited to study the effective presence of such a state. The symmetrized Dalitz plot for the data sample under study (35118 selected events) is shown in Fig. 1. Besides the horizontal/vertical bands due to the p°(770) and the / 2 (1270) resonances, and a density enhancement near the plot corners at about 1500 MeV, one can note a broadening along the diagonal edge, in a similar fashion as observed by Anninos et al. 1. A first complete spin-parity analysis of this channel had been performed to primarily study the nature of the enhancement around 1500 MeV 2 .
*FOR THE OBELIX COLLABORATION
150
151
In this analysis no 7r+7r+ amplitude was inserted. The />(1450) signal was the major ingredient, together with / 2 (1270) and / 2 (1565), for a satisfactorydescription of the enhancement along the Dalitz plot rim, as shown in Fig. 2. S
3
P,
's.
P2
interference
''iiV*»i fi\T>0)
3E f,\1270)
sffcK _>
I
i
t
p"(1+50)
2 -
a*>; _,
L
ttm®&0&..„_ Uiisoo)
\ 2
0
na(n+ff-),GeV*
2
0
mV^lGeV*
2
°
m^O-OV2
2
mVVO.GeV
Figure 2. Contribution of each resonant state from each partial wave to the symmetrised Dalitz plot.
With a basic set of n+n~ resonant states the fit was acceptable, with 1213/1264 — 0.96. However, some slightly critical points emerged in XDP the description of the Dalitz plot in the region of the /a(1270) crossing (where X2 — 1.45) and of the 7r+7r+ invariant mass spectrum, where a x2 — 1-69 reflected an imperfect reproduction of the data especially near 1.3 GeV and 1.5 GeV. The details of the fitting methods are fully reported in Ref. 2. Every amplitude for each partial wave is given by a coherent sum of ?r+7r~ isobars contributions. An eventual isospin 2 amplitude has to be inserted adding it coherently as well, properly weighted by the isospin Clebsh-Gordan factor. For its dynamical description a Breit-Wigner parameterization has been chosen
152
to reproduce the contribution of a possible resonance; as an alternative, a non resonant amplitude has been used to estimate the -K+-K+ scattering length. The fit strategy consisted in performing several series of fits using the basic set of 7r+7r~ amplitudes, starting from the best fit solution of Ref. 2 plus the isospin 2 one for two different hypotheses of its spin (0 or 2), with the mass and width varying on a discrete grid. The best fit solution achieved in each grid step is based on the minimization of the (reversed sign) log(likelihood) —£. Inserting the contribution of a spin 0 resonant 7r+7r+ state a clear maximum of £, 7729, is achieved atTO= (1.42 ± 0.02) GeV and T = (0.16 ± 0.01) GeV; the production branching ratio of such a state, integrated over the full volume of the Dalitz plot, is (3.22±0.15) x 10~ 3 - the error is statistical and is derived by the spread of values obtained in many fits. The x 2 o n the global Dalitz plot improves rather sensibly, passing to 1058/1268. The trend of likelihoods as a function of the 7r+7r+ mass for given widths of the resonant state is shown in Fig. 3. log(L)
log(L) 7750
7730 7725 E7720 7715 7710 7705
I k r > : i > !• '
r - 1 0 0 MeV r=2DOMeV r=30OMeV r-40OMeV r=500Mev r=60OM«V r-70OM*V r-M0M*V r=-900MeV
7700 7695 f r * 7690 7685 76S0
'it
•Jl
"ftflft f
:
i r=ioou«v
: T~ I I_
A r=°2WMeV T r = 300 MeV O r = 400MeV D r = 500MsV
:
A r=60ou«v
I ~ Z
<> r a 700 MeV Q r - 8 0 0 MeV * r = SOOMeV
-
*
b)
a
ft
S s, •* U
«$*»*
£\jfl lr
(*
•
: *.*'*> H i " .
-
f *,SJ
* Wi *.••
m ( ? r V ) , GeV
m(n*if), GeV
Figure 3. log(likelihood) values for best fit solutions as a function of TT+W+ invariant mass and for fixed values of the width of the possible isospin two resonant state, in the two hypotheses a) of a scalar resonance, or b) of a tensor one.
The test of the second hypothesis, an isospin two tensor resonant state, delivered a less distinct likelihood maximum atTO— (1500±100) MeV and T — (400 ± 100) MeV. These values are formally in agreement with an observation by ARGUS and CELLO of a 1=2 J = 2 state in the reaction 77 -> p°p° 3 , that could be due to the formation of an exotic q2q2 state 4 ' 5 . The weight obtained by the best fit solution is around 6.5 x 10~ 3 . It is worth noticing that the MIT bag model expects the two pseudoscalar decay mode of an exotic q2q2
153
state to be heavily suppressed as compared to the vector-vector mode. 2
Evaluation of the 7r+7r+ scattering length
A non resonant isospin two contribution can be described by the S wave scattering length formula Tg — 02(7/(1 — ia%q). where q is the momentum of one 7r+ in the 7T+TT+ c m . system, and a® is the S wave 7r+7r+ scattering length. A grid procedure was again applied scanning an interval 0.1m" 1 wide. Theoretical predictions of the value of 7T+7r+ scattering length suggest it to be negative and small, in the range —(0.027-=- O.OGQ^m'1. The best fit to the np —» n+n+7r~ data occurs at a® — (—0.025 ± 0.005)m~1. However, the changes in I through the different steps of the grid are too small to allow to discriminate among different choices for a\. 3
Conclusions
The results of a large number of fits performed applying different hypotheses about the spin, the nature (resonant or diffusive), the mass and the width of a possible isospin two contribution to the amplitude show that its introduction has the general effect of improving, though slightly, the fits quality. Its production fraction, over the full Dalitz plot, does not exceed 0.4% in the scalar hypothesis, and 0.7% in the tensor one. Its effect cannot be considered as essential in the description of the total amplitude, even if several checks have shown that it cannot anyway substitute other minor 7r+7r~ contributions. Nonetheless, the statistical significance of the observed (, maxima and their height is rather striking, especially in the scalar hypothesis. If confirmed, this could be the first evidence for an exotic q2q2 scalar state, in a region of mass where other exotic structure have been observed, namely the 1 h state at BNL and by Crystal Barrel 6 and the 2++ state by ARGUS and CELLO 3 . References 1. P. Anninos et al, Phys. Rev. Lett. 20, 402 (1968). 2. The OBELIX Collaboration, A. Bertin ei al, Phys. Rev. D 57, 55 (1998). 3. H. Albrecht et al, Phys. Lett. B 217, 205 (1989); H.-J. Behrend et al, Phys. Lett. B 218, 493 (1989). 4. N.N. Achasov et al, Z. Phys. C 27, 99 (1985). 5. B.A. Li and K.F. Liu, Phys. Rev. D 30, 613 (1984). 6. D.R. Thompson it et al., Phys. Rev. Lett. 79, 1630 (1997). A. Abele et al., Phys. Lett. B 423, 175 (1998)
OBSERVATION OF A N ANOMALOUS T R E N D OF T H E A N T I N E U T R O N - P R O T O N TOTAL CROSS SECTION IN THE LOW-MOMENTUM REGION
A. F E L I G I E L L O Istituto
Nazionale di Fisica Nucleare - Sezione di Via P. Giuria 1, 1-10125 Torino, Italy
Torino,
The final result of the antineutron-proton total cross section measurement in the 50 -f- 400 MeV/c range is presented. The region below 100 MeV/c, explored for the first time, turned out to be of great interest since the experimental points exhibit an anomalous trend. A possible explanation for this unexpected behaviour could be the existence of a quasi-nuclear bound state, just above the antinucleonnucleon threshold. The idea of looking for such an effect by means of a dedicated measurement of the antiproton-proton elastic cross section will be discussed.
1
Introduction
The study of the antinucleon (AT) - nucleon (.A/") interactions represents one of the most interesting fields that was possible to investigate at the CERN LEAR machine. The motivations to push down the measurements in the lowenergy region are basically two: to determine the isospin dependence of the AT — Af interactions, deeply related to the meson exchange mechanism in the medium and long range part of the strong interaction, and to check whether the AT — Af cross sections really show a smooth rise near the AT — Af threshold, as predicted by different, well established, potentials models 1. Actually the real question, underlying the last point, is about the existence of the so called quasi-nuclear bound states, raised several years ago 2 . If one looks at the literature, one finds a large set of data concerning the antiproton (p) - proton (p) total cross section, but limited at the region above 200 MeV/c 3 ; on the contrary the pp annihilation cross section has recently measured down to very low-energy values by the OBELIX Collaboration 4 : the final analysis confirms a smooth rise. The situation is more or less the same for the antineutron (n) - proton interactions: the total cross section measurement stops at 100 MeV/c 5 , while the experimental points for the annihilation one exhibit a regular trend 5 , 6 , r , even if the lowest energy data 6 are affected by huge errors. The measurement described in this paper was carried out with the aim of filling one of these experimental gaps. However the measured rip total cross section showed an anomalous, and to some extent unexpected, behaviour below 100 MeV/c, adding a new chapter to the very long and controversial 154
155
history of the quasi-nuclear bound states 8 . 2
Experimental results and discussion
The rip total cross section (
J, 700 \ \
•
600 L • ojfip) 500 400 300 200 100 1 1 1.... 0 t..,.,..l 100 200 300 400 n momentum [MeV/c]
Figure 1.
1700 ° 600 \ h 500 400 300 200 100 0 iJh
•
<Wnp)
°
^elamin^P)
i i i.„. 100 200 300 400 n momentum [MeV/c]
Figure 2. aeia(ftp) versus p ^ : it has been obtained as difference between crtot(np) (from Ref. n ) and aann(np) (from Ref. 7 ) . The diamonds represent the lower limits of
by Armstrong et al. 5 : the two data sets agree well in the range where they overlap. A second comparison was made with the tip annihilation cross section ((Tann(np)), previously measured by the OBELIX Collaboration 7 : the tricking thing is that the trend of the <J ann (np) does not exhibit any structure (see Fig. 1). In fact there was no problem in fitting a a n n (np) in terms of scattering length approximation 7 ; on the contrary it was not possible to find a satisfactory way of reproducing (ftot{np) (see Fig. 1). At this point it appeared clear that the anomalous behaviour of crtot(np) is actually due to the rip elastic cross section (<je;a(np)): Fig. 2 shows that the aeia(np) exhibits a clear dip in the low-energy region n . The explanation for such anomaly is not unambiguous: it could be due to the threshold in the
156
pp —* nn channel 12 or it could be explained by the s-wave dominance, in the frame of the coupled channel analysis approach 13 or it could be related to the existence of a resonant state near the AfAf threshold 2 . Concerning this last hypothesis it is interesting to note that the FENICE Collaboration reported a surprising behaviour of the a(e+e~ —> hadrons) 14 . Fig. 3 shows a comparison of the FENICE and the OBELIX results. Even if the mass values of the hypothetical states are very close, the two anomalies are not necessarily related to the same object: this is in agreement with some potential models, which predict a rich spectrum just near the AfAf threshold. •fr
a(e + e" -» mh) [nb] (world average)
^ m n SH Z
100 1000 X ii n threshold ji n p threshold 900 90 n * a(e + e" —> mh) • | °tot( P) 800 m 80 §—- fit to a(e + e" -» mh) 70 700 o 600 S1 60 500 £, 50 400 40 Si IB 300 S 30 200 t 20 o 100 10 Illlimiini I \i\ mill I I o 0 0 1.84 1.85 1.86 1.87 1.88 1.89 1.9 1.91 1.92 0 E fGeV]
vs
hadrons) (from Figure 3. Comparison between
As far as the AfAf interaction isospin dependence is concerned, it is worth to flash that the ratio between the extrapolated crtot(pp) 3 and the measured values of atot (np) is strongly in favour of a dominance (more than a factor 2) of the 1 = 0 amplitude over the the 1 = 1 one, at low energy. 3
Future perspectives
In order to confirm the observed effect and to discriminate among the different theoretical interpretations it would be necessary to perform a new measurement. Unfortunately both the LEAR machine and the OBELIX spectrometer are no longer available. The unique source of p is now the CERN AD machine;
157
however the characteristics of this p beam are not suitable for the production of an n beam (maximum p momentum ~ 100 MeV/c, threshold for pp —> fin reaction ~ 98 MeV/c). Nevertheless it must be reminded that the np interaction proceeds from a pure 1 = 1 initial state, while in the pp case both the I = 0 and 1 = 1 amplitudes are involved. Hence also the <Jeia(pp) should exhibit an anomalous behaviour, at worst weaker if the effect is due only to the I = 1 component. For this reason it has been proposed to directly measure the 106 particles in a time < 1 //s, with a repetition time of ~ 10 2 Hz), which prevents the use of any beam detector for coincidence or timing purposes. The proposed detector is made by four identical modules (~ 6 x 60 x 60 mm 3 ), surrounding the beam axis; each of them consists of six layers of scintillating fibers (1 x 1 x 60 mm 3 ), oriented alternatively along and orthogonally to the beam axis and coupled to multianode photomultipliers. Their signals are read out by a 100 MHz FADC system, able to give up to 100 event snapshots per burst. On the basis of this information and thanks to the high level of granularity of the detector it it will be possible to distinguish the p annihilation from the p elastic scattering events, which will take place in a very thin (1 -9- 10 /xm) CH2 target, placed few mm upstream the detector. References 1. 2. 3. 4. 5. 6. 7. 8.
9. 10. 11. 12. 13. 14. 15.
C.B. Dover et al, Prog. Part. Nucl. Phys. 29, 87 (1992). I.S. Shapiro, Phys. Rep. C 35, 129 (1978). D.V. Bugg et al, Phys. Lett. B 194, 563 (1987). A. Zenoni et al, Phys. Lett. B 461, 405 (1999). T. Armstrong et al, Phys. Rev. D 36, 659 (1987). G.S. Mutchler et al, Phys. Rev. D 38, 742 (1989). OBELIX Collaboration, Nucl. Phys. B 56A, 227 (1997). F. Iazzi, in Proc. of Workshop on Hadron Spectroscopy, Frascati, March 8-12, 1999, Eds. T. Bressani, A. Feliciello, A. Filippi (Frascati Physics Series, Vol. XV, 1999) p. 659. M. Agnello et al, Nucl. Instrum. Methods A 339, 11 (1997). OBELIX Collaboration, Sov. J. Nucl. Phys. 55, 1732 (1992). F. Iazzi, A. Feliciello et al, Phys. Lett. B 475, 378 (2000). W. Bruckner et al, Phys. Lett. B 158, 180 (1985). A.E. Kudryavtsev and B.L. Druzjinin, ITEP-23-94. A. Antonelli et al, Nucl. Phys. B 517, 3 (1998). T. Bressani et al, Letter of Intent, 1 (2000).
A S T U D Y OF n ANNIHILATION ON NUCLEI
ELENA BOTTA Dipartimento
Via P. Giuria
for the OBELIX experiment di Fisica Sperimentale, Universita di Torino and I.N.F.N. - Sezione di Torino 1, 10125 Torino Italy, e-mail: [email protected]
T h e ra—nucleus annihilation cross section
1
Introduction
An accurate knowledge of the features of the J7Af and A7"-Nucleus interaction is crucial for a correct description of the strong/nuclear interaction. Up to now, only a limited set of experimental data is available for the A/"-nucleus annihilation at low momenta. In particular, for the n -Nucleus annihilation cross section, measurements were performed at ANL (*) on 1 2 C and subsequently at CERN LEAR on Fe (2, 3 ) and on C, Al, Cu, Sn, Pb (4) in the 100-800 MeV/c momentum range. In (4) the results of a first determination of the n -Nucleus annihilation cross section by the OBELIX experiment were given; in this paper preliminary results of the analysis of a more consistent data sample are reported. 2
Measurements and analysis of the data
The OBELIX experiment, mainly dedicated to A/"A/ interaction and meson spectroscopy investigations with p , was provided with a n beam facility to complete the information coming from the pp system. A complete description of the OBELIX n beam and of its performance can be found in ( 5 ). Measurements on nuclei were performed "parasitically" by placing the nuclear targets behind the central LH2 one, used for A/OV measurements, considering that only ~ 30% of the incoming n beam interacted inside it. During the data taking periods of 1994 and 1995, to which the preliminary results presented here refer, the n momentum range was 50-400 MeV/c and C, Al, Cu, Ag, Sn and Pb targets (natural isotopic composition) of suitable thickness were used. Annihilation events have been selected by requiring that the coordinates of the reconstructed vertex lie in the fiducial volume of the target: strong cuts 158
159
were imposed in order to have a clean sample of events. Thanks to the amount of the available data, the selected annihilation events have been divided, for each target, in 7 n momentum bins and for each bin the corresponding cross section has been evaluated as: A Na(pn,A) cra(Pn,A) (1) PANAVUCA Nn(pn) where NAV is the Avogadro number, £ ^ A is the target constant, CA is the total efficiency (geometrical plus reconstruction), that turns out to be constant in the considered momentum range, Na(pn, A) is the number of selected annihilation events in the momentum bin, Nn(pn.) is the number of incoming n 's in the same bin. In all results the statistical errors only are reported; a preliminary estimate of the systematic error is ~ 4%, while for the absolute normalization an error ~ 8% can be preliminarily assumed. 3
Discussion of results
Figure 1 shows the values of the annihilation cross section obtained for the various targets as a function of the mean value of each momentum bin. •?
s
I
14000
-
12000
10000
• •
t
t>
A
-
Pb Sn
ft
*> Cu
D
Al
O
C
8000
• *
tiOOO
t
: 4000
ft
o
n o
•
2000
n
-
* 4
•
....150 200 ....
. i ..,, 100
O
•
•
•
•
I
1
ft
ft
I
4
• o
n o
* 8
I i
250
i >
. 1 300
•o:
0
•
i .
350
i .
1 1 1 1 1
400 450 p; (MeV/c)
Figure 1. n annihilation cross section on C, Al, Cu, Ag, Sn, Pb as a function of the n momentum.
To determine the cross section dependence on A, the points corresponding to each bin have been fitted to the formula: 0a{pn,A) = ao(pn)Ax
(2)
160
where the contributions due to p% and A are factorized and ooipn) indicates the mean value of the "elementary" annihilation reaction in the particular momentum bin. The weighted mean of the x values turns out to be < x >= 0.6507 ± 0.0068. It is evident that the obtained scaling law is compatible with A2/3 within 2%, thus extending the previous results (4) to a very low n momentum range for all targets. To determine the dependence on the incoming n momentum, the weigthed mean values of the A2/3 scaled cross section of the six nuclei have been successively considered, for each momentum bin. A fit to the parametrization function: co(Pn) = a + b/pn 2
(3)
3
as suggested in literature ( , ), has been performed, but it doesn't succeed in reproducing the experimental data at low momenta; indeed, the best parametrization can be obtained with the function: <70(Pn) = O, + b/pfi +
c/pl
(4)
as shown in figure 2, where the continuous curve represents the best fit with function (3) and a = 62.3 ± 1.7 mb, b = 18790 ± 490 mb MeV/c, x2/DF = 2.178, the dashed curve represents the best fit with function (4) and a = 82.4 ± 3.3 mb, b = (9.3 ± 1.4) x 10 3 mb MeV/c, c = (9.3 ± 1.5) x 105 mb MeV 2 /c 2 and x2/DF = 0.743. Then, it is possible to conclude that a term proportional to p^ is necessary to fit the data satisfactorily. .—.
E S e
600
. a/A** = o + b/p . a/A** - a + b/p + c/p*
. a/A** = a c„ + (a-1) am *-'
500
•
present data (oil nucM)
O PAN data (1997) 400
*
OBOJX pnmiou* data (1994)
A ANTIN data (1988) A
Cundw»on«tal.(t981)
Figure 2. Scaled n annihilation cross section values and best fit parametrizations (see text for details). Previous data are also shown.
161
The results can be directly compared with the data available in literature, divided by A2!3 in turn, as shown in figure 2, where data from ( 1 ), ( 2 ), (4) and (3) are reported with statistical errors only (( 4 ) and ( 3 )) and with statistical and systematic errors ((*) and ( 2 )). It is possible to observe that present preliminary data are in quite good agreement with both previous OBELIX results (4) and results from (2, 3 ) . The confirmation of the validity of the A2!3 scaling law suggests that the n annihilation is actually a well localized process. A confirmation of this hypothesis can come from the comparison between the scaled cross section data and an appropriate sum over the elementary hn and pn cross sections. To this purpose, the parametrization of aa np(Pfh) from (6) and the best fit to the pp annihilation data from (7) (pp > 180 MeV/c) and from (8) (65 < pp < 180 MeV/c), to parametrize aa nn(p), have been used; charge independence is obviously assumed. By adding: aaa np(p) + (1 — a)aa nnip), where a = Z/A, the dotted curve in Figure 2 has been obtained for a = 0.4 (Lead). The curve shows a qualitative agreement with the data, better with increasing pn, as it may be expected considering that the Fermi momentum of the nucleon has been disregarded. It is concluded that the process of n annihilation on nuclei involves single nucleons only. 4
Conclusions
From parasitic measurements performed by the OBELIX experiment a complete systematics of h -Nucleus annihilation cross section has been obtained in the low momentum region (50 < pn < 400 MeV/c). The preliminary results confirm completely the A2'3 scaling law previously suggested in literature, while the incoming n momentum dependence seems to show a not negligible oc p^2 term. The annihilation process shows a completely local behaviour. References 1. 2. 3. 4. 5. 6. 7. 8.
B. Gunderson et al, Phys. Rev. D23 587 (1981). M. Agnello et al, Europhys. Lett. 7 13 (1988). C. Barbina et al, Nucl. Phys. A612 346 (1997). V.G. Ableev et al., N. dm. 107A 943 (1994). M. Agnello et a/., Nucl. Instr. Meth. A 399 11 (1997). A. Bertin et al, Nucl. Phys. B 56A 227 (1997). W. Bruckner et al, Z. Phys. A 335 217 (1990). A. Zenoni et al, Phys. Lett. B 461 405 (1999).
Trp scattering in the Coulomb-nuclear interference region
E. Fragiacomo Universita degli Studi di Trieste and INFN Trieste, via Valerio 2, 34127, Trieste, Italy E-mail: [email protected] THE CHAOS COLLABORATION The presentation will review both the experimental and the theoretical challenge of the CNI (Coulomb Nuclear Interference) experiment and discuss preliminary results which demonstrate that the proposed measurement can be performed with the requested accuracy (~ 5%).
1
Introduction
CNI is presently running at the low-energy pion channel at TRIUMF, and aims at measuring 7r±p absolute differential cross sections in the very forward angle region (the CNI region) where Coulomb scattering interferes destructively (constructively) with Tt+p (TT~P) nuclear scattering amplitudes 1 . The proposed measurements will determine absolute differential cross sections with a precision of < 5% for angles from 5° to 180° (laboratory frame) and for incident pion energies from 15 to 70 MeV. The experimental challenge is the separation of the muon cone from scattered pions in the angle region 5° — 20° with a precision below 3%. In general, muons are rather difficult to separate from pions because they have similar masses, thus a novel tecnique was employed, which is based on a neural network approach, that is capable of achieving an efficiency of identification > 97%. Section 2 is devoted to the experimental issues. The theoretical goal is a measurement of the real part of the isospin-even forward scattering amplitude D* at t = 0 (t is the Mandelstam variable), which will improve the determination of the irN scattering lengths and the irN Eterm, which are crucial observables to test predictions of chiral perturbation theory. The amplitude ReD^(t = 0) will be determined directly from the absolute differential cross sections with a precision of ~ 5%. The method to estract both the amplitude ReD^(t = 0) and the scattering length at+ is independent from both partial wave analysis and dispersion relations. Section 3 is devoted to introduce the theoretical goals and to illustrate this method. 162
163
2
Experimental issues
The experimental apparatus consists of the magnetic spectrometer CHAOS2 and of a stack of scintillator detectors 3 . CHAOS records trajectories for incoming and outgoing particles and allows the determination of the momenta and the interaction vertex. The stack of scintillator detectors has an angular acceptance of ~ 30° and is used as a ir/fi identifier3. The incident pion beam strikes a liquid hydrogen target contained in a vessel 5 cm in diameter and 5 cm in height and situated in the centre of the CHAOS spectrometer. Fig. 1 shows the geometry of CHAOS and the 7r//i identifier.
Magnetyoke ' Wire chamber Incoming beam
H
f 1 Q^ "ZFT telescope
Target
sJ //
Outgol tigbeam
'/A
/ 1
3C*
1 nip. stack
Figure 1: Top view of the experimental setup
The n^p differential cross section can be written 3 in terms of the ^p ential cross section do_( ± s_N™att{6)Rli(kj dtt
±
differ-
(1)
where N£ffi*(8) is the number of pions (muons) scattered at 8 and - R ^ ) is the fraction of pions (muons) in the incident beam. The quantity dcr^±p/dO can be calculated from measured scattering amplitudes to an accuracy of ~ 1 % . An accurate 7r//x separation is required. Muons can either originate from the incoming pion beam or from decay of pions in the spectrometer region by pion decay. For most of them, the tracking capability of CHAOS is sufficient for an efficient recognition (> 99%). The
164
muons which cannot be recognised by CHAOS are those from pion decay in the target region. These have interaction vertices similar to scattered pions, and there is a region in the momentum versus angle plane which overlays to that of pions scattered from protons. These muons and pions of identical momentum have to be identified. This is why a new telescope was added to the pre-existing CHAOS spectrometer3. Previous measurements4 as well as GEANT simulations suggest the use of a stack of plastic scintillators since it can simultaneously determine time-of-flight, energy loss and range of a n or a /j, of known momentum. This information is analysed by a neural network algorithm implemented in the analysis program which turns out to be capable of achiving the required efficiency (97%) of identification. 3
Theoretical issues
Absolute differential cross-sections will be used to determine, to a precision of a few percent, the real part of the isospin-even scattering amplitude D^, at t = 0. This information is missing in the current partial wave analyses. Its inclusion will greatly improve the determination of the irN scattering amplitudes. As a result, it will improve the determination of the ivN S-term, which is a direct measure of chiral symmetry breaking and which can be used to determine the strange-quark content of the proton. The irN S-term can be obtained experimentally from the isospin-even S and P-wave scattering lengths a'Q+ and a'1+. A systematic set of measurements in the region of Coulomb-nuclear interference can be used to measure these scattering lengths. The method to estract both the amplitude ReD^(t = 0) and the scattering length at+ is independent from both partial wave analysis and dispersion relations. The sensitivity of the method was tested using cross sections obtained from the existing database. Fig. 2 compares the extrapolations of ReD^ at t = 0: the curve labelled as "Exact" refers to the ReD^ calculated from partial waves, while the curve labelled as "Calculated" to the ReD^ calculated from cross sections. This method is applied to the -KN cross sections for each value of the energy, thus obtaining several points ReD^(t = 0,T„). Finally, ReD^(t = 0, Tv) is studied as a function of Tv. A polynomial fit is used to extrapolate to Tn = 0. The extrapolation to T„ — 0 provides the scattering length a,Q+. 4
R e s u l t s a n d conclusions
In order to test the experimental apparatus, ^p elastic cross sections have been measured and compared with theoretical predictions. Muons, in fact, are
165
a
i
s. - -
I
:\ :
'•
I
,
I
Exact
|!
% - """ ~^
N r
I
I
!
Calculated
I—
-
: 1
2
3
+
5
6
7
-t{!&V"ZMnooo^°
Figure 2: ReD^ derived from the partial waves ("Exact") and calculated from the cross sections ("Calculated")
easier to identify than pions since they have a longer mean path and represent a better benchmark to test the reliability of the experimental setup. Fig. 3 shows the results. The theoretical curve labelled as "Rutherford" takes into account proton form factor and relativistic effects. The theoretical curve fit 10'' o
n\
rutherford
-p
10*
a
,o'H
^11
r
-
*i
-a
~
;j--p
10
0
10
20
3<
fell!
it fe
40
t0
Figure 3: / i ± p elastic cross sections compared with theoretical predictions.
to the data up to three order of magnitude, over the angular range of interest (5° - 20°). References 1. 2. 3. 4.
G.R. Smith, Triumf-Research Proposal (1996) G.R. Smith et al., Nucl. Instr. and Meth. 362 (1995) 349 E. Fragiacomo et al., Nucl. Instr. and Meth. A 439 (2000) 45 Ch. Joram et al., Phys. Rev. C 51 (1995) 2142
A R G U M E N T S A G A I N S T T H E YUKAWA C O N C E P T OF N U C L E A R FORCE AT INTERMEDIATE- A N D S H O R T - R A N G E S A N D THE N E W M E C H A N I S M FOR NN INTERACTION VLADIMIR I. KUKULIN* Institute of Nuclear Physics, Moscow State University, 119899 Moscow, Russia E-mail: [email protected] Instead of the Yukawa mechanism for intermediate- and short-range interaction some new approach based on formation of the symmetric six-quark bag in the state |(0s)6[6]xj L = 0) dressed due to strong coupling to -K, a and p fields are suggested. This new mechanism offers both a strong intermediate-range attraction which replaces the effective u-exchange (or excitation of two isobars in the intermediate state) in traditional force models and also short-range repulsion. Simple illustrative model are developed which demonstrate clearly how well the suggested new mechanism can reproduce NN data.
It was found in recent years that the traditional models for NN forces, based on the Yukawa concept of one- or two-meson exchanges between free nucleons even at the quark level lead to numerous disagreements with newest precise experimental data for few-nucleon observables (especially for spinpolarised particles) 1>2,3. There are also various inner inconsistencies and disagreements between the traditional AW force models and predictions of fundumental theories for meson-baryon interaction (e.g. for meson-nucleon cut-off factors). All these disagreements stimulate strongly new attempts to develop alternative force models based either on chiral perturbation theory or a new quark-meson models. Our recent studies in the field 1>2'3 have led us to a principially new mechanism for intermediate- and short-range NN forces - the so called "dressed" bag mechanism which is able to explain the failure of the traditional Yukawa exchange models and also to solve many long-standing puzzles in the field. This mechanism can also shed some light to the puzzles in baryon spectroscopy (e.g. normal ordering in A-sector and inverse ordering in nucleon sector for excited negative and positive parity states). On the microscopic level, two possible space symmetries |s6[6]a;Z/ = 0) and |s4p2[42]j;, L = 0,2) are allowed for NN system in s- and d-partial waves. "THE RESPECTIVE ORIGINAL WORK INCLUDED IN THE TALK WAS DONE JOINTLY WITH DRS. I.T.OBUKHOVSKY, V.N.POMERANTSEV AND PROF. A. FAESSLER
166
167
The new model is based on the important observation 4 that two possible sixquark space symmetries in even NN partial waves (for illustration we consider here the S-wave only), viz. |s6[6]L = 0) and \s4p2[42]L = 0) correspond to the states of different nature. The first states have rather small projections into proper NN channel and corresponds to bag-like intermediate states while the states of second type are projected into NN channel with a large weight and thus can be presented as clusterised NN states with nodal NN relative motion wavefunctions. In the present work we develop this picture much further on the quark-meson microscopic basis and derive the microscopic NN transition amplitudes through six-quark +27r intermediate states in s-channel (see Fig. 1).
J^0T(r) ST=01(10) L=0,2
J = 0(1) J = 0(1) J = 0(1) r=oT(D ST=01(10) ST= 10(01) ST=10(01) ST=01(10) L=0,2 L=l L=l L=0 F I G . 1. The graph illustrates two sequential 7r-meson emissions and absorptions via an intermediate a- (or p-) meson cloud and the generation of a symmetric six-quark bag.
The transition is accompanied by a virtual emission and subsequent absorption of two tightly correlated pions by diquark pairs or, alternatively, by two lp-shell quarks when they jump from the lp- to the Os-shell orbit or vice versa. These two pions can form both the scalar a and vector p mesons which surround the symmetric six-quark bag. It follows from previous studies (see e.g. 5 ) for chiral symmetry restoration effects in multiquark systems or in high density nuclear matter that some
168
phase transition happens when the quark density or the temperature of the system is increased, which leads to a restoration of the broken chiral symmetry. The most probable consequence of the above restoration should be strengthening of the sigma-meson field in the NN overlap region. This could be modelled by "dressing" of the most compact six-quark configurations |s 6 [6]x£ = 0) and |s 5 p[51]x£ = 1) inside the NN overlap region with an effective sigma-meson field. The scalar- and vector-meson cloud will stabilize the multi-quark bag due to a partial chiral symmetry restoration effect in the dense multi-quark system and thus enhances all the contributions of such a type. Thus, the picture of NN interaction emerged from the model can be referred to as the 6q "dressed" bag (DB) model for baryon-baryon interaction. The "CT" or a similar " scalar-isoscalar meson" is assumed to exist only in a high density environment and not in the vacuum, contrary to the 7r and p mesons. This mechanism, being combined with an additional orthogonality requirement 6 , can describe both the short-range repulsion and the medium range attraction and can replace the i-channel exchange by a- and w-mesons in the conventional Yukawa-type picture of the NN force. The direct calculation of the multiloop diagram on Fig. 1 1'2 using quark pair-creation model results for S- and D-partial waves (in iViV-channel) in a separable operator of form: VL'L(ri E
r ) =
92oGoo(E)\2s(r'))(2s(r)\
(
V 929oG20(E)\2d(r')){2s(r)\
g0g2G02{E)\2s{r')){2d{r)\
\
) ' (1) where the generalised propagators Gw (E) are related to the DB intermediate state l'2. The interaction given by Eq.(l) can be interpreted as an effective NN potential in our model. In accordance with this, the contribution of mechanism displayed in the diagram in FIG. 1 to the NN interaction in the S and D partial waves can be expressed through the matrix element: ALNk^do+^NN
g*G22(E)\2d(r')){2d(r)\
= Jtpr'iPr9%N\E;T')VgL(T',T)*hN(E;T),
(2)
where ^%N and 9J!fN are the "proper" nodal NN scattering wave functions in initial and final state respectively. The interaction operator (1) mixes S- and .D-partial waves in the triplet NN channel and thus it leads to a specific tensor mixing with the range ~ 1 fm (about that of the intermediate DB state). Thus the proposed new mechanism for NN interaction induced by the intermediate dressed six-quark bag |s 6 + 2TT) results in a specific matrix separable form of interaction with
169 nodal (in S- in P-partial waves) form factors and a specific tensor mixing of new type 7 . These nodal form factors make it possible to explain within this mechanism the origin of the NN repulsive core by the nodes in the transition form factors (see Eq.(l)). Moreover, the proposed model will lead to the appearance of strong 3N and 4N forces mediated by 2ir and p exchanges 3 . The new 3N forces include both central and spin-orbit components. Such a spin-orbit 3N force is extremely desirable to explain the low energy puzzle of the analyzing power Ay in N-d scattering and also the behavior of Ay in the 2>N system at higher energies E^ — 250 -i- 350 MeV at backward angles. The central components of the 3N and 4N forces are expected to be strongly attractive and thus they must contribute to 3N and (may be) 4N binding energies possibly resolving hereby the very old puzzle with the binding energies of the lightest nuclei. Future studies must show to what degree such expectations can be justified. Acknowledgments The author thanks the Russian Foundation for Basic Research (grant RFBRDFG No.92-02-04020) and the Deutsche Forschungsgemeinschaft (grant No. Fa-67/20-1) for partial financial support. References 1. A. Faessler, V. I. Kukulin, I. T. Obukhovsky and V. N. Pomerantsev, E-print:nucl-th/9912074. 2. V. I. Kukulin, I. T. Obukhovsky and V. N. Pomerantsev,A. Faessler, Phys. Atom. Nucl. in press. 3. V. I. Kukulin, Proceeds, of the V Winter School on Theoretical Physics PIYaF, Gatchina, S.-Petersburg, 8-14 Febr. 1999., p. 142; 4. A. M. Kusainov, V. G. Neudatchin, and I. T. Obukhovsky, Phys. Rev. C 44, 2343 (1991). 5. T. Hatsuda and T. Kunihiro, Phys. Rep. 247, 221 (1994); T. Hatsuda, T. Kunihiro and H. Shimizu, Phys. Rev. Lett. 82, 2840 (1999). 6. V. I. Kukulin, V. N. Pomerantsev, and A. Faessler, Phys. Rev. C 59, 3021 (1999); V.I. Kukulin and V.N. Pomerantsev, Nucl. Phys. A 631, 456c (1998). 7. V. I. Kukulin, V. N. Pomerantsev, S. G. Cooper and R. Mackintosh, Few-Body Systems, Suppl. 10, 439 (1998).
MOVING TRIANGLE SINGULARITIES A N D OF FAST PARTICLES
Lebedev Physical
POLARIZATION
V. M. KOLYBASOV Institute, Leninsky prospekt 53, 117924 Moscow, E-mail: kolybasvQsci.lebedev.ru
Russia
It is shown that, due to sharp variation of the amplitude of a triangle graph near its singularity, the polarization of a fast final particle can noticeably changes in the interval of residual system excitation energy of the order of characteristic nuclear binding energy. It gives an additional possibility for experimental observation of moving singularities.
Basic postulate of the reaction theory is that a reaction amplitude must be an analytic function of proper kinematical variables. The properties of an analytic function are determined by its singularities. The simplest kinds of singularities are pole (corresponding to a pole graph), branch points (threshold singularities) and triangle (corresponding to the graph of Fig.l). A
x
¥, M
Figure 1: A generic triangle graph.
The triangle mechanism is very frequently used in the description of nuclear reactions. The experimental determination of its contribution as well as separation of the kinematical regions of its domination would permit us to extract valuable and reliable information on nuclear structure and the off-shell amplitude of the "elementary" process in the lower vertex. Until now very few cases are known when the dominant contribution of the triangle mechanism is seen. On the other side, it is well known that the characteristic feature of the triangle graph in nuclear physics consists in so called moving complex triangle 170
171
singularity. I t s position in the invariant mass M of the lower group of final particles depends on the value q ( q = p x - P y ) of three-momentum transfer from initial t o final fast particles in the right vertex. T h e distribution with respect t o M depends on the value of q. It is a model-independent criterion of the dominant role of the graph [1]. U p t o now this test was not used for the identification of the triangle mechanism. T h e purpose of present report is threefold: (i) t o show t h a t we can find rather distinct experimental indication of moving complex triangle singularities; (ii) t o demonstrate t h a t t h e experimental picture would be more clear for the cases when the lower vertex (secondary interaction) are not s-wave, b u t p-wave and (iii) t o show t h a t polarization phenomena can b e of nontrivial nature in the region where triangle singularity is situated near physical region. Predicted pictures for two cases of reactions on the deuteron (the process with intermediate A-isobar and the process pd —> pdrj with intermediate pion) was demonstrated in ref.[2,3]. One can see distinct moving of b u m p s in the distributions with respect t o the invariant masses with m o m e n t u m transfer. Unfortunately, there are rather broad m a x i m a in t h e M distributions with widths of the order of 100 -=- 200 MeV. T h e most distinct picture could be expected in the region of the excitation energy £?ex = M — ( m i +WI2) of the order of nuclear binding energies and m o m e n t u m transfer q of the order of nuclear F e r m i - m o m e n t a . Here the complex triangle singularity comes close t o the physical region and narrow peaks and background suppression could be anticipated. Moreover, the cross sections in this region are much larger. It is hindered by the fact t h a t Fig.l triangle graph has also the threshold root singularity a t Eex = 0 a n d usually it determines a picture of the spectrum a t small i?ex- However, the situation sharply varies if not the s-wave b u t p - or higher waves d o m i n a t e in the amplitude of secondary interaction. T h e shape of the cusp singularity changes a n d it ceases t o dominate a t small i?ex- As can be seen from comparison of upper and lower p a r t s in Fig. 2, the situation with observation of the moving triangle singularity is essentially improved and we can see very clear and unambiguous picture [4]. Here we use the notation e = m i + m 3 — m A , n2 — 2m13e and two dimensionless variables £ and A, associated with M a n d q:
^
W g ) =
J^-(^-+Jfa+^7a-2TOl\, 2
2m3e yM + q
2
2
,/M + q
2
(1)
J
In terms of these variables the triangle graph has a square root branching point
172
at £ = 0 and logarithmic singularities at £ = A — 1 ± ivA.
nr
°-5
i 0
i 5
1 10
15
5 Figure 2: |.M| 2 as a function of £ for A = 1 (solid line), 3 (dotted line), 6 (dashed line), 9 (dash-dotted line) in the case of the deuteron with Paris wave function: a) for s-wave lower vertex, b) for p-wave lower vertex.
Nontrivial behaviour of real and imaginary parts of the triangle graph amplitude (for example, see Fig.3 for the triangle graph with intermediate £ [4]) enables us to guess interesting and characteristic features in polarization phenomena. Fig.4 shows the results of toy calculations for the case when we take into account the triangle graph plus some smooth background. Mf = A + iB + (ReM + UmM)(l
+ an).
(3)
Different curves correspond to different magnitudes and phases of background amplitudes: full curve A = 2, B = 0, dotted curve A — 0, B = 2, dashed curve A = 2, B = 2, dash-dotted curve A = 1,B = 1. All curves have a common property: the polarization sharly varies on rather narrow interval (several tens of MeV) of invariant mass of residual nuclear system. It gives an additional possibility for experimental observation of moving singularities. Acknowledgments This investigation was partly supported by RFBR grant 99-02-17263.
173
ReM, IraM 4|a.u.
20
-20
40
60
Eex,MeV Figure 3: Real and imaginary parts of the triangle graph.
_=^=_5.
/n
•'/i
0.5
i
i
^ * v
""•-•>
N \
y/i //
V\^ \
0 -
-0.5
\ ,L
20
20
X
/
\
40
60
Eex (MeV) Figure 4: Possible behaviour of polarization as a function of invariant mass of residual nuclear system. See text for details.
References 1. 2. 3. 4.
E.I.Dubovoj and I.S.Shapiro, Sov. Phys. JETP 24, 1251 (1967). V.M.Kolybasov, Nucl. Phys. A 626, 97c (1997). V.M.Kolybasov, Phys. Lett. B 439, 251 (1998). V.M.Kolybasov, Physics of Atomic Nuclei 62, 932 (1999); e-print nuclth/9901071.
Section III. Hadron Structure and Electromagnetic Probes
ELECTRON - P O S I T R O N PAIR S P E C T R O S C O P Y W I T H H A D E S AT GSI J. FRIESE, FOR THE HADES COLLABORATION Physik-Depariment E12, Technische Universitdt Miinchen, James Franck Str. 1 D-8574-8 Garching, Germany E-mail: [email protected] The High Acceptance DiElectron Spectrometer HADES is presently set up at GSI, Darmstadt, for the systematic study of medium modifications of hadron properties. Short lived hadrons will be produced in nuclear matter from normal up to medium baryon densities utilizing pion, proton and heavy ion induced reactions. Electron pair spectroscopy will be used to measure resonance widths and effective masses of vector mesons as well as transition formfactors of neutral mesons and baryon resonances in the low mass region M < \ GeV/c2. The spectrometer is designed for an invariant mass resolution AMi„v/Minv ~ 1%(
1
Introduction
The investigation of the basic properties of hadrons embedded in a surrounding of strongly interacting matter has increasingly become of interest. Experimental verification of possible changes of these properties will shed additional light on our understanding of non perturbative quantum chromodynamics (QCD) and may give hints to a complete or partial restoration of chiral symmetry. Various theoretical approaches (see 1>2'3 and references therein) predict that the hadron spectral functions are modified inside the medium according to its temperature and density. In earthbound laboratories such an environment can be provided in nuclear collisions, however with finite space-time volume only. Experimentally, the particle spectral functions can be obtained from a complete momentum measurement of their lepton pair decays. Hence, detailed measurements of in medium e + e~ (or ^ + ^t ) pair production and of electromagnetic decay properties of free hadrons are needed. The new High Acceptance Di-Electron Spectrometer HADES at GSI (Darmstadt, Germany) aims at precision measurements 4 of e+e~ pair production in elementary and heavy ion induced reactions with unprecedented invariant mass resolution, event statistics, and signal to background discrimination.
175
176
2
e+e
Pair Spectroscopy in the GSI Energy Regime
In central heavy ion collisions of 1 — 2AGeV incident energy a hadronic fireball is formed, with temperatures and baryon densities ranging up to T = 50 — 90 MeV and 3 times normal nuclear matter density, respectively. The formation of the reaction volume lasts for as long as 10 fm/c and expands with comparatively moderate changes in temperature and density 5 . The light and short lived vector mesons p and w are particularly attractive probes, as they are produced with sufficient yield, offer a direct e + e~ decay channel, and decay at least partly within the reaction volume. Moreover, the u> meson has a narrow resonance width and its modification6,7 can probably be observed already in cold nuclei when produced in n- induced reactions under adjusted kinematical conditions 8 . Dielectron (e + e~) invariant mass spectra have already been measured in this energy regime 9,10 and show for light and heavy ion reactions a remarkable excess of yield in the low mass range 200 MeV/c2 < Minv < 600 MeV/c2 as compared to current theoretical calculations 11 . However, the limited invariant mass resolution and background separation has prevented so far from unambiguous conclusions with respect to medium modifications. 3
The Spectrometer Setup
The HADES experiment is designed to cope with the high-multiplicity environment of heavy ion collisions and the small branching ratios for the dielectron production channels. The key features are a large acceptance flPair — 40% combined with a high resolution for invariant mass reconstruction AMinv/Minv ~ 1% (a) and a signal to background ratio > 1 for invariant masses up to M ~ 1 GeV/c 2 . The instrument (see left panel in Fig. 1) is azimuthally segmented into 6 sectors and covers polar angles 18° < 9 < 85°. Four planes of 6 trapezoidal Multi-wire Drift Chambers (MDC) 12 together with a superconducting toroid form a magnetic spectrometer for charged particle momentum measurement. A hadron blind RICH counter with a gaseous decafluorobutan (C4F10) radiator around the target is used for electron identification. Rings of Cherenkov photons with nearly constant diameter are detected in a MWPC type photon detector with Csl pad cathode 13 and an expected figure of merit No = 108. A set of electromagnetic PreShower detectors covering polar angles up to 45° and a Time-of-Flight (TOF) wall with 648 scintillators provide additional lepton identification and a multiplicity signal to trigger on central collisions. The Time of Flight (TOF) wall has a time resolution lOOps < otof < 150psu and allows to discriminate electrons from
177
Figure 1. The new e+e"" spectrometer HADES. Left panel: Schematic cross section with t h e main components and detectors. The system has a sixfold symmetry around the beam axis. T h e magnet coils are projected into t h e plane. Right panel: Upstream view of t h e spectrometer with t h e outer detectors pulled back t o service position and three PreShower detectors removed. T h e inner driftchambers and t h e RICH are mounted inside t h e superconducting toroid in the center.
pions up to 0.5 GeV/c and from protons up to 2 GeV/c. The low effective radiation length pd/X® cz 10" 2 of the inner detectors keeps multiple scattering and combinatorial background from external pair production low. The data acquisition is based on a switched ATM network 15 with a sustained transfer rate of 14 Mbyte/'s and uses a three level trigger system 16 to reduce the primary data rate of 4 Gbyte/s from all 73, 000 detector channels to a maximum of about 4 Mbyte/'s. While the Irst level trigger selects central collisions, e+e"" pair candidates are selected at the second level by position matching of ring centers found with hardware image processing in the RICH 17 and hits in the TOF and PreShower detectors. The third level suppresses background by an online tracking analysis with driftchamber hits. 4
Status, a n d First Commissioning R e s u l t s
The installation of the spectrometer is nearly completed (Fig. 1), with the exception of the outer driftchambers, where the i r s t sector is presently installed. Several subsystems have been successfully tested in commissioning runs with beam intensities up to 107 s""1. As an example, the electron identification capability of the RICH is illustrated in Fig. 2. In a preliminary analysis of 2 • 105 central Ar 4- Ti collisions at E = 1.75 AGeV about 2500 candidates for
178
TOF only
30k h
f 20kl
e ,e~,re,p
I
^lOkj-
m TOF & RICH
1&*,.
s»
m
• i^CW^BriBaffihaihfc
10
15
20
25 30 35 Time of flight [ns]
Figure 2. Hadron suppression with t h e RICH in Ar + Ti collisions a t E = 1.75 AGeV. Left panel: Ring image of a single e * from external pair conversion in t h e target or radiator gas with a coincidence track signal in the drift chamber. Right panel: Time of flight distribution for all particles (upper) and for electrons (lower) with rings found in t h e RICH. Note t h e suppression in yield.
Cherenkov rings were found, in reasonable agreement with simulation results. The corresponding electrons and positrons come mostly from external pair conversion of 7-s in the target and the radiator gas. In almost all events the ring candidates (left panel of Fig. 2) were not obscured by electronic noise or signals from charged particle background. The yield of ring centers declines homogeneously with increasing polar angle and is in good agreement with simulation results. The time of flight spectra in the right panel of Fig. 2 demonstrates the achieved suppression of pions and protons. 5
Simulation Results
The expected yield of detected e + e~ pairs has been studied by detailed Monte Carlo simulations for heavy systems 18,19 , -K- induced u>- production in nuclei 8 and elementary pp reactions. For central Au + Au collisions the low invariant mass range (100 MeV/c2 < Minv < 600 MeV/c2) is dominated by contributions from p-n Bremsstrahlung, Tro—,rj- and -less pronounced- A- Dalitz decays (see Fig. 3). In the upper accessible mass range (600 MeV/c2 < Minv < IGeV/c2) only p/w decays are expected with little contributions from w- Dalitz and 4> decays. The good suppression of combinatorial background from multiple external pair production {-KQ —* 77 —> rye+e~~) and the high mass resolution will allow to detect any significant mass shift or
179
Figure 3. Simulated e+e"~~ invariant mass distributions for heavy left and light right collision systems. T h e yield of e+e~~ pairs in central An + An collisions comes from various sources according t o a BUU transport calculation without taking any medium modifications into account. Not t h e low combinatorial background. T h e yield of e+e~" pairs from ~ 10 6 rjDalitz decays (line) corresponds to ~ 100 h of pp collisions at Ep = 2GeV. Background from pp2iro (dotted) reactions can be reduced by a missing mass of t h e pprf- reaction.
resonance broadening. The large and flat acceptance gives access to systematic investigations as function of transverse momentum and rapidity. individual components of the heavy ion induced e+e~~ cocktail can be studied in elementary wp or pp reactions in which the electron background from secondary sources can be kept at minimum. As an example we have investigated the rj- Dalitz decay in p p collisions (pp —• piV*(1535) -~>-ppti-—^ ppe+e"~(7)) at energies around the rf~ production threshold. This channel is of particular interest, since it dominates the yield in the invariant mass region M*nt, ~ 300 — 500 MeV/c2, where the excess in the DLS data has been observed. With both protons and electrons identified the reconstruction of the Dalitz decay is Mnematically complete without requiring an additional photon detector for 7 detection. The expected electron pair invariant mass distribution is shown in the right panel of Fig. 3. Rrom the achievable statistics the electromagnetic formfactor of the rj meson can be determined to a much better precision than from the two measurements published so far 2 0 ' 2 1 . 6
Conclusions a n d O u t l o o k
HADES is a high rate, high acceptance apparatus for electron pair spectroscopy with < 1 % invariant mass resolution and a fast and selective multi
180
level trigger scheme. The spectrometer is meanwhile ready for first runs and results obtained in commissioning runs demonstrate the efficient and redundant lepton identification. The physics program is broad and includes the study of lepton pair emission in relativistic heavy ion collisions, dielectron production in elementary reactions and experiments aimed at studying the structure of hadrons. Particular attention will be paid to the decay properties of vector mesons with respect to mass shifts and resonance broadening. The shape of the invariant mass spectrum will give access to electromagnetic transition form factors of neutral mesons. The long-range aim is to understand the dynamical properties of hot and compressed hadronic matter, an environment in which partial restoration of chiral symmetry is expected to show up. The experimental program of HADES will focus first on pp collisions and light heavy ion systems to verify the yield enhancements found in earlier measurements. Dedicated investigations of the u) meson decay are foreseen utilizing both, pion and heavy ion induced reactions. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21.
R. Rapp and J. Wambach, hep-ph/9909229 W.Cassing and E.L. Bratkovskaya, Phys.Rep.308 (1999) 65 C M . Ko, V. Koch and G.Q. Li, Annu. Rev. Nucl. Part. Sci. 47 (1997) 505 P.Salabura et al., Nucl. Phys.B 44, (1995) 701c B. Friman, W. Norenberg and V.D. Toneev, Eur. Phys. J. A3 (1998) 165 F.Klingl and W. Weise,hep-ph/9802211 M.Effenbergeret al., Phys. Rev. C60 (1999) W.Schon et al., Acta Phys. Pol. B27 Vol. 11 (1996) R.J.Porter et al., Phys.Rev.Lett.79 (1997) 1229 W.K. Wilson et al., Phys. Rev. C57(4) (1998) 1865 E.L. Bratkovskaya et al., Nucl. Phys. A634 (1998) 168 C. Garabatos et al., Nucl. Instr. Meth. A 412 (1998) 38 K. Zeitelhack et al., Nucl. Instr. Meth. A 433 (1999) 201 Agodi et al., IEEE Trans. Nucl. Sci.45No.3 (1998) 665 M. Munch et al., XXXVII Meet. Nucl.Phys., Bormio,Italy (1999), 509 M.Petri et a l , XXXVIMeet.Nucl.Phys,Bormio,Italy (1998), 622 J.Lehnert et a l , Nucl. Instr. Meth. A 433 (1999) 268 H.Neumann et a l . Acta Phys.Slov.44 (1994) 195 R. Schicker et a l , Nucl. Instr. Meth. A 380 (1996) 586 M.R. Jane et a l , Phys. Lett. 59B (1975) 103 G. Agakichiev et a l , Eur. Phys. J. C4 (1998) 231
Precision Measurement of the Neutron Magnetic Form Factor from 3He(e,e') H. Gaoa Laboratory
for Nuclear Science and Department of Physics, Massachusetts Institute of Technology, 77 Massachusetts Ave., Cambridge, MA 02139, U.S.A.
A precision measurement of the inclusive quasielastic transverse asymmetry Aj-i from 3 He(e,e') was completed recently in Hall A at Jefferson Lab (E95-001). In this talk, I review the existing data on the neutron electromagnetic form factor in the low Q2 region and present the preliminary results on the neutron magnetic form factor from the JLab experiment E95-001.
1
Introduction
Electromagnetic form factors are of fundamental importance for an understanding of the underlying structure of nucleons. Knowledge of the distribution of charge, magnetization within the nucleons provides a sensitive test of models based on Quantum Chromodynamics (QCD), as well as a basis for calculations of processes involving the electromagnetic interaction with complex nuclei. The understanding of the nucleon structure in terms of quark and gluon degrees of freedom of QCD will provide a basis to understand more complex strongly interacting matter at the level of quarks and gluons. While the proton form factors are known with good precision over a large range of four-momentum transfer squared, Q2, the corresponding data for the neutron are of inferior quality due to the lack of free neutron targets. The most precise information on G1^ at low Q2 prior to any polarization experiment is from the elastic electron-deuteron scattering experiment by Platchkov et al. 1 . However, the extracted G-g values are extremely sensitive to the deuteron structure, and the absolute value of GJJ contains a systematic uncertainty of about 50% from the measurement by Platchkov et al. x. In quasielastic scattering, the spin degrees of freedom introduce new response functions into the differential cross section, thus providing additional information on nuclear structure 2. Experiments with longitudinally polarized electron beams and recoil neutron polarimeters have been carried out at MITBates 3 and Mainz 4 ' 5 and G^ has been extracted from the d(e, e'n) process. Recently, the neutron electric form factor was extracted for the first time 6 from the d(e, e'n) reaction. Currently, our best knowledge of G7^ from these "For the Jefferson Lab E95-001 collaboration. 181
182
polarization measurements is ~ 30% for Q2 < 0.6(GeV/c) 2 . Until recently, most data on G%[ had been deduced from elastic and quasielastic electron-deuteron scattering experiments. For inclusive measurements, this procedure requires the subtraction of a large proton contribution and suffers from large theoretical uncertainties due to the deuteron model employed and corrections for final-state interactions (FSI) and meson-exchange currents (MEC). The precision of G\f from the inclusive measurements is limited to ~20% at low Q2 in the existing data. The proton subtraction is avoided in coincidence d(e,e'n) experiments 7 , and the sensitivity to nuclear structure can be greatly reduced by measuring the cross section ratio d(e,e'n)/d(e,e'p) at quasielastic kinematics. Several recent experiments 8,9 ' 10 have employed the latter technique to extract G7^ with uncertainties of <2% in the Q 2 range of 0.1 to 0.8 (GeV/c) 2 . While this precision is excellent, the results of these experiments 7,8 ' 9 ' 10 are not fully consistent.
2
Inclusive Quasielastic Scattering of Polarized Electrons from Polarized 3 He Targets
The polarized 3 He nucleus is a good candidate for an effective neutron target because its ground state wave function is dominated by the S-state in which the proton spins cancel and the nuclear spin is entirely due to the neutron. Following the earlier work by Blankleider and Woloshyn 11 and Friar et al.12, the plane wave impulse approximation (PWIA) calculations 13 ' 14 using spindependent spectral functions show that the spin-dependent asymmetries are very sensitive to the neutron electric or magnetic form factors at certain kinematics near the top of the quasielastic peak. Recently, Faddeev calculations have been carried out which include FSI 1 5 , FSI and MEC 1 6 have confirmed this picture. The differential cross section for 3He(e', e') in the scattering plane can be written in terms of four nuclear response functions RK(Q2, V) 2 as rft
. ^ . p , = OMott
[VLRL
+ vTRT
-
II(COS6*VT>RT>
+ 2sin0* cos 4>*VTL'RTL')]
,
(1) where 9* and (ft* are the polar and azimuthal angles defining the direction of the target spin with respect to the momentum transfer vector q, the VK are kinematic factors, v is the electron energy transfer, h is the helicity of the incident electron beam, and Q2 = q2 — v2, RT> and RTV are two response functions arising from the polarization degrees of freedom. The spin-dependent
183
asymmetry is defined as cr+ - cr_ _
COS9*VT'RT'
a+ +
+2sm.6* VTRT
+
COS(J)*VTL
(r>\
VLRL
where the subscript + (-) refers to the electron helicity h. By orienting the target spin at 9* = 0° corresponding to the spin direction along q, one can select the transverse asymmetry AT' (proportional to RT> ) • For the quasielastic 3 ife(e,e') process, AT, is most sensitive to (G%,)2 I3,i4,i5,i6_ 3
JLab Experiment E95-001
The experiment was carried out in Hall A at JLab in early 1999 using a longitudinally polarized continuous-wave electron beam of 10 ^A current. A high pressure polarized 3 He gas target based on spin-exchange optical pumping of rubidium at a density of 2.5 x 10 20 nuclei/cm 3 was employed. The average beam (target) polarization for the experiment was approximately 70% and 30%, respectively. Six kinematic points were measured corresponding to Q 2 = 0.1 to 0.6 (GeV/c) 2 in steps of 0.1 (GeV/c) 2 . An incident electron beam energy of 0.778 GeV was employed for the two lowest Q2 values of the experiment and the remaining points were completed at an incident beam energy of 1.727 GeV. The scattered electrons were detected in the two Hall A High Resolution Spectrometers, HRSe and HRSh. Both spectrometers were configured to detect electrons in single-arm mode. The HRSe was set for quasielastic kinematics while the HRSh detected elastically scattered electrons. Since the elastic asymmetry can be calculated very well at low Q2 using the well-known elastic form factors of 3 He 1 7 , the elastic measurement allows precise monitoring of the product of the beam and target polarizations, PtPtThe preliminary results for AT' as a function of u are shown in Fig. 1 for the two lowest Q2 settings of the experiment. The error bars on the data are statistical only, and the total experimental systematic error is indicated as an error band in each figure. PWIA calculations 18 using the AV18 for the NN interaction potential and the Holder nucleon form factor parametrization 19 are shown as dashed lines. The Faddeev calculations 16 with FSI only and with both FSI and MEC using the Bonn-B potential and the Hohler form factor parametrization are shown as dash-dotted lines and solid lines, respectively. All theory results were averaged over the spectrometer acceptances using a Monte Carlo simulation. The overall systematic uncertainty of AT> is 2% for Q2 values of 0.1 and 0.2 (GeV/c) 2 dominated by the uncertainty in determining PtPb-
184
The state-of-the-art three-body calculation treats the 3 He target state and the 3N scattering states in the nuclear matrix element in a consistent way by solving the corresponding 3N Faddeev equations 2 0 . The MEC effects were calculated using the prescription of Riska 21 , which includes 7r- and p-like exchange terms. While the agreement between the data and full calculations is very good at Q 2 = 0 . 1 and 0.2 (GeV/c) 2 , the full calculation is not expected to be applicable at higher Q 2 because of its fully non-relativistic framework. A full calculation within the framework of relativity is highly desirable.
BO
SO
100
110
120
130
140
ti) ( M e V )
Figure 1: The preliminary results on the transverse asymmetry KJ-I at Q 2 = 0 . 1 - 0.2 (GeV/c) 2 .
To extract G%j for the two lowest Q2 kinematics, the transverse asymmetry data were averaged over a 30 MeV bin around the quasi-elastic peak. The full Faddeev calculation including MEC 22 was employed to generate AT' as a function of G%j in the same u) region. By comparing the measured asymmetries with the predictions, the G%j values at Q2 = 0 . 1 and 0.2 (GeV/c) 2 were extracted. The extracted preliminary values of G7^ are shown in Fig. 2 as solid circles. The uncertainties shown are the quadrature sum of the statistical and experimental systematic uncertainties.
185
4
Summary
The inclusive transverse asymmetry AT> from the quasi-elastic 3 He(e, e') process has been measured with high precision at Q2-values of 0.1 to 0.6 (GeV/c) 2 from JLab experiment E95-001. Using a full Faddeev calculation which includes FSI and MEC the neutron magnetic form factor G7^ at Q 2 values of 0.1 and 0.2 (GeV/c) 2 was extracted. Full calculations are at present not available for Q2 > 0.3 (GeV/c) 2 to allow the extraction of G1^ with high precision at higher Q 2 . Theoretical efforts are currently underway to extend the full calculation to higher Q 2 29 .
0.0
0.2
0.4 0.6 Q2 ( G e V / c ) 2
Figure 2: The neutron magnetic form factor G^ in units of the standard dipole parameterization, finGu, in the low Q2 region, as determined in several recent measurements: Markowitz et al. 7 (diamonds) using d(e, e'n); Anklin et al. 8 (star), Bruins et al. 9 (squares), and Anklin et al. 1 0 (triangles) using the ratio d(e,e'n)/d(e,e'p); and Gao et al. 2 3 (circle) using 3 He(e, e'). The preliminary results from JLab experiment E95-001 are shown as solid circles with total experimental uncertainties. The solid curve is a cloudy bag model calculation 2 4 and the dotted curve is the minimal vector dominance model calculation 2 5 . The short and long dashed curves are the non-relativistic and relativistic quark model calculations 2 6 . 2 7 , respectively. The dash-dotted curve is a calculation 2 8 based on a fit of the proton data using dispersion theoretical arguments.
186
Acknowledgments This work is supported in part by the U.S. Department of Energy under contract number DE-FC02-94ER40818. 1. S. Platchkov et al, Nucl. Phys. A510, 740 (1990). 2. T.W. Donnelly and A.S. Raskin, Ann. Phys. (N.Y.) 169 (1986) 247. 3. T. Eden et al, Phys. Rev. C 50, R1749 (1994). 4. M. Ostrick et al, Phys. Rev. Lett. 83, 276 (1999). 5. C. Herberg et al, Eur. Phys. Jour. A5, 131 (1999). 6. I. Passchier et al, Phys. Rev. Lett. 82, 4988 (1999). 7. P. Markowitz et al, Phys. Rev. C 48 (1993) R5. 8. H. Anklin et al, Phys. Lett. B336 (1994) 313. 9. E.E.W. Bruins et al, Phys. Rev. Lett. 75 (1995) 21. 10. H. Anklin et al, Phys. Lett. B428 (1998) 248. 11. B. Blankleider and R.M. Woloshyn, Phys. Rev. C 29, 538 (1984). 12. J.L. Friar et al, Phys. Rev. C 42 (1990) 2310. 13. R.-W. Schulze and P.U. Sauer, Phys. Rev. C 48 (1993) 38. 14. C. Ciofi degli Atti, E. Pace and G. Saline, Phys. Rev. C 51 (1995) 1108; C. Ciofi degli Atti, E. Pace and G. Salme, in Proceedings of the 6th Workshop on Perspectives in Nuclear Physics at Intermediate Energies, ICTP, Trieste May 1993, (World Scientific); C. Ciofi degli Atti, E. Pace and G. Salme, Phys. Rev. C 51, 1108 (1995). 15. S. Ishikawa et al, Phys. Rev. C 57 (1998) 39. 16. J. Golak, private communication. 17. A. Amroun et al, Nucl. Phys. A579 (1994) 596. 18. A. Kievsky, E. Pace, G. Salme, M. Viviani, Phys. Rev. C 56, 64 (1997). 19. G. Hohler et al, Nucl. Phys. B114, 505 (1976). 20. J. Golak et al, Phys. Rev. C 51, 1638 (1995). 21. D.O. Riska, Phys. Scr. 31, 471 (1985). 22. V.V. Kotlyer, H. Kamada, W. Glockle, J. Golak, Few-Body Syst. 28, 35 (2000). 23. H. Gao et al, Phys. Rev. C 50 (1994) R546; H. Gao, Nucl. Phys. A631, 170c (1998). 24. D.H. Lu, A.W. Thomas, A.G. Williams, Phys. Rev. C 57, 2628 (1998). 25. U.-G. Meifiner, Phys. Rep. 161, 213 (1988). 26. E. Eich, Z. Phys. C 45, 627 (1988). 27. F. Schlumpf, J. Phys. G 20 237, 1994. 28. P. Mergell, U.-G. Meifiner, D. Dreshel, Nucl. Phys. A596, 367 (1996). 29. W. Glockle, private communication.
T H E H Y P E R C E N T R A L C O N S T I T U E N T Q U A R K MODEL
Dipartimento
M. M. G I A N N I N I , E. S A N T O P I N T O dell'Universita di Genova, via Dodecanese) 33, Genova, and Nazionale di Fisica Nucleare, Sezione di Genova, Genova, Italy di Fisica
Istituto
Italy
We report on t h e results of a Constituent Quark Model based on a hypercentral approach (hCQM), including three-body force and standard two-body potential contributions. T h e model has three free parameters, which have been fixed by a fit of t h e baryon spectrum. T h e model has then been used for the calculation of the electromagnetic transition amplitudes and the elastic form factors. T h e effect of introducing relativistic corrections a t a kinematical level is also briefly discussed. T h e results are compared to t h e experimental data, with particular attention t o t h e ratio fip GE/GM recently measured at JLab.
1
Introduction
Constituent Quark Models have been recently widely used for the description of the internal structure of baryons, leading to a satisfactory account of the spectrum 1,2 ' 3,4 . However, in order to distinguish among the various forms of quark dynamics, one has to analyze in a consistent way all the physical observables of interest and such a systematic study is better performed within a general framework. In this respect a hypercentral approach to quark dynamics can be used 5 , which is sufficiently general to investigate new dynamical features, such as three-body mechanisms, and also to reformulate and/or include the currently used two-body potential models. 2
The hypercentral Constituent Quark Model
The internal quark motion is described by the Jacobi coordinates p and A: P = T/f^1-^2) '
^
=
771(^1 + ^ 2 - 2 ^ 3 ) ,
(1)
or equivalently, p, Qp, A, Q\. It is convenient to introduce the hyperspherical coordinates, substituting the absolute values p and A by x = J? + A2 ,
i = arctg{^),
(2)
where x is the hyperradius and £ the hyperangle. In this way one can use the hyperspherical harmonic formalism 6 . The quark potential, V, is assumed to depend on the hyperradius x only, that is to be hypercentral, therefore, V = V(x) is in general a three-body potential. In this case one can factor out in the three-quark wave function the hyperangular part, which is given by the known hyperspherical harmonics 6 and the Schrodinger equation is reduced to a single hyperradial equation: [
^
+
l l -
l h
p
1 ]
^
{ x )
187
=
-2m[£-V(*)] Vr,](*) ,
(3)
188 where ip[y] (x) is the hyperangular wave function; 7 is the grand angular quantum number and it is given by 7 = 2v 4- lp + />,, where lp and lx are the angular momenta associated with the /fand A variables and v is a non negative integer; m is the quark mass. The hypercentral equation can be solved analytically at least in two cases, that is for the h.o. potential, which turns out to be exactly hypercentral, and the hypercoulomb one 5 ' 7 , 8 . In our model 5 we assume a confining hypercentral potential of the form V(x) =
+ ax, (4) x plus a standard hyperfine interaction 1 , treated as a perturbation. The hypercentral equation is solved numerically, using a computer code, which has been tested by comparison with the known analytical solutions. The spectrum is described with T = 4.59 and a = 1.61 fm~2 and the standard strength of the hyperfine interaction needed for the N — A mass difference 1 . 3
The form factors
With the parameters fixed by the spectrum, the model has been used to calculate the helicity amplitudes for the photoexcitation of the nucleon resonances 9 , the transition form factors to the negative parity resonances 10 and the elastic nucleon form factors u . The results for the helicity amplitudes of some of the negative parity nucleon resonances, calculated in the Breit frame, are given by the dot-dashed curves in figures 1, 2 and 3 10 . The medium Q2 behaviour is very well reproduced, while serious problems are present at small Q2, specially in the A^,2 amplitude of the transition to the Di3(1520) state. These discrepancies could be ascribed to the non-relativistic character of the model. The kinematic relativistic corrections can be introduced in the quark model calculation of the transition form factors by means of an expansion of the current matrix elements in terms of inverse powers of the quark mass 15 . As shown in Figures 1,2 and 3 the relativistic corrections 15 do not modify strongly the medium <52-behaviour, which remains in agreement with data. On the contrary they give a significant contribution at low-Q2, as already observed in 16, but there still remains a strong discrepancy with the experimental data at low-Q2. In the elastic case, a similar procedure x l , expanding the matrix elements in quark momenta, leads to charge and magnetic form factors with an analogous factorized form GE(Q2)
= FgG%(q/g)
,
GM(Q2) = FefG^(q/g),
(5)
where G™', and G^j are the electric and magnetic form factors as given by the non relativistic quark model and F^ , F^f are kinematical factors. For g the Breit system value g — E/M is used. The elastic form factors of Eq. (5), calculated using as input the nucleon form factors obtained in the hCQM, lead to an improvement of the theoretical description
189
\ 200
100
7
1 A",
•
l
Is
-
\
"** .-
s
^ i £ i
K=3/2
.
J:
LLI
:# -100
1 ;
* *
N(1520)D 1 3
-it.
- ";-
V£
-
*•=!/« 1 , , , . ! , , ,
" i ,
8
Q (GeV/c)
i
_
8
Figure 1: Comparison between t h e experimental data for t h e helicity amplitudes A%.,2,A?!. for t h e Di3(1520) resonance and t h e calculations with t h e potential of Eq.6. T h e full curve is t h e calculation including t h e relativistic corrections described in t h e text; t h e dashed and dot-dashed curves are t h e non relativistic results in t h e Equal Velocity and t h e Breit frame, respectively (from refs.[5j!5]). T h e d a t a are from t h e compilation of ref. [14].
. 1 200
~
150
7
100
\i 50
'
•
'
1
i
|
1
N(1B35)S„
A'l/,
1.
~ -
~~ -""* ^
f 1^
- i
"_--
,
,
ii^J^
i 1 q« (GeV/c)*
1 . . .
I~
.
1
•
2
Figure 2: T h e same as in Fig. 1, for t h e helicity amplitude Ap./2 of t h e S'uflBSS) resonance.
190 . I '
'
'
'
I '
—
'
'
'
I '
'
'
'
I .
A»1/a N(1650)S„
—
-
i
.
,
.
.
i
,
.
,
,
I
,
,
.
.
i
•
q* (GeV/<0*
Figure 3: T h e same as in Fig. 1, for t h e helicity amplitude A^,„ of the Sii(1650) resonance.
1
1.2
1
•
•
•
•
i
•
•
•
'
i
'
•
•-
,
,
,
,
i
,
,
,-
-
^C «
^
1 0.6
, i
,
,
i
,
1
i 2
Figure 4: T h e ratio R = ftp GE/GM calculated with t h e hCQM, taking into account t h e relat i v i s t s kinematical corrections (full line, ref. [18]). T h e horizontal full line represents t h e ratio for CQM without neither hyperfine nor relativistic corrections. T h e dashed curve is the fit of ref. [19], t h e dot-dashed curve is t h e dispersion relation fit of ref. [20]. T h e points are t h e data of t h e recent Jefferson Lab experiment Ref. [17].
191 11 , but the Q2 behaviour is still different from the experimental data. However one should note that the model of Eq. (8), which reproduces the spectrum and the helicity amplitudes, corresponds to a confinement radius of the order of 0.5 fm which is just what is needed in order to describe the photocouplings i2.13.s.9>n We consider finally the ratio
R
= ">• ^ m
(6)
for the proton which has been just measured in a direct way in a polarization experiment at the Jefferson Lab (for details see Ref. 1 7 ). The data are characterized by much lower errors with respect to previous experiments and show a significative deviation from the scaling behaviour. In Fig. 7 we report the ratio calculated with the model presented in the previous section 18 . The non relativistic calculation gives R = 1, and it remains 1 within 1% even if the hyperfine mixing is included. The relativistic corrections lead to a significant deviation from the scaling behaviour, which is close to the new Jlab data up to Q2 of the order of l(GeV/c)2. We should remind that we have used the first order relativistic corrections and so the range of their applicability is limited, but certainly below l(GeV/c)2. In Fig. 7 we report other two theoretical curves. The first one, given by the VMD model 19 deviates from the scaling behaviour in fair agreement with the data; however such fit predicts a dip of the proton form factor for Q2 of the order of 8.5(GeV/c) 2 , whereas present data on the proton form factor do not exhibit such zero. The second curve shown in Fig. 7 has been obtained by means of a dispersion relation calculation including two-pion effects 20 ; this curve gives a reasonable description of data up to 1.5(GeV/c)2 and is very close to the results evaluated using the hCQM with relativistic corrections. It should be stressed that the hCQM calculations are parameter free.
4
Conclusions
We have presented the results for the nucleon form factors predicted by the hypercentral Constituent Quark Model 5 . The transition form factors are well described in the medium-high Q2 region, while the elastic ones are strongly affected by the confinement radius of 0.5 fm needed for the fit of the spectrum. The introduction of relativistic kinematic corrections is beneficial for the elastic form factors but not sufficient to reproduce the data. In particular, the relativistic corrections are responsible for a significant deviation from the scaling behaviour of the proton form factors and the calculated ratio R is not far from the recent Jefferson Lab data, at least at low Q2, that is within the range of validity of the 1/ro expansion used in our approach. As for the transition form factors the relativistic corrections modify only slightly the medium Q/2-behaviour, leaving the agreement with data unchanged, while at low Q2 there remain problems. As already noted elsewhere 10 8 4 ' ' , some fundamental dynamical mechanism (effective at large distance, that is at low Q2) is still lacking, such as the explicit inclusion of quark-antiquark pairs both in the baryon states and in the electromagnetic transition operator.
192
References
1. N. Isgur and G. Karl, Phys. Rev. D18, 4187 (1978); D19, 2653 (1979); D20, 1191 (1979); S. Godfrey and N. Isgur, Phys. Rev. D32, 189 (1985); S. Capstick and N. Isgur, Phys. Rev. D 34,2809 (1986) 2. M.M. Giannini, Rep. Prog. Phys. 54, 453 (1991). 3. L. Ya. Glozman and D.O. Riska, Phys. Rep. C268, 263 (1996). 4. R. Bijker, F. Iachello and A. Leviatan, Ann. Phys. (N.Y.) 236, 69 ( 1994) 5. M. Ferraris, M.M. Giannini, M. Pizzo, E. Santopinto and L. Tiator, Phys. Lett. B364, 231 (1995). 6. G. Morpurgo, Nuovo Cimento 9, 461 (1952); Yu. A. Simonov, Sov. J. Nucl. Phys. 3, 461 (1966); J. Ballot and M. Fabre de la Ripelle, Ann. of Phys. (N.Y.) 127, 62 (1980). 7. E. Santopinto, M.M. Giannini and F. Iachello, in "Symmetries in Science VII", ed. B. Gruber, Plenum Press, New York, 445 (1995); F. Iachello, in "Symmetries in Science VII", ed. B. Gruber, Plenum Press, New York, 213 (1995). 8. E. Santopinto, F. Iachello and M.M. Giannini, Nucl. Phys. A623, 100c (1997); Eur. Phys. J. A l , 307 (1998) 9. M. Aiello, M. Ferraris, M.M. Giannini, M. Pizzo and E. Santopinto, Phys. Lett. B387, 215 (1996). 10. M. Aiello, M. M. Giannini, E. Santopinto, J. Phys. G: Nucl. Part. Phys. 24, 753 (1998) 11. M. De Sanctis, E. Santopinto, M.M. Giannini, Eur. Phys. J. A l , 187 (1998). 12. L. A. Copley, G. Karl and E. Obryk, Phys. Lett. 29, 117 (1969). 13. R. Koniuk and N. Isgur, Phys. Rev. D 2 1 , 1868 (1980). 14. V. Burkert, private communication. 15. M. De Sanctis, E. Santopinto, M.M. Giannini, Eur. Phys. J. A2, 403 (1998). 16. S. Capstick and B.D. Keister, Phys. Rev. D51, 3598 (1995) 17. V. Punjabi, Proceedings of the 2 n d International Conference on Perspectives in Hadronic Physics, ICTP Trieste 10-14 may 1999, (S. Boffi, C. Ciofi degli Atti and M. M. Giannini eds.), World Scientific, Singapore 1999, p. 214; C. F. Perdrisat and V. Punjabi, private communication; M. K. Jones et al., Phys. Rev. Lett. B 84, 1398 (2000) 18. M. De Sanctis, M. M. Giannini, L. Repetto and E. Santopinto, Phys. Rev. C 62, 025208 (2000). 19. F. Iachello, A. D. Jackson and A. Lande, Phys. Lett. B43, 191 (1973). 20. H.-W. Hammer, U.-G. Meissner and D. Drechsel, Phys. Lett. B385, 343 (1996); P. Mergell, U.-G. Meissner and D. Drechsel, Nucl. Phys. A596, 367 (1996)
ALGEBRAIC M O D E L OF B A R Y O N STRUCTURE
ICN-UNAM,
Racah Institute
A.P.
of Physics,
R. BIJKER 70-543, 04510 Mexico, A. LEVIATAN The Hebrew University,
D.F.,
Mexico
Jerusalem
91904,
Israel
We discuss properties of baryon resonances belonging to the N, A, S, A, E and H families in a collective string-like model for the nucleon, in which the radial excitations are interpreted as rotations and vibrations of the string configuration. We find good overall agreement with the available data. The main discrepancies are found for low lying S-wave states, in particular iV(1535), IV(1650), £(1750), A*(1405), A(1670) and A(1800).
1
Introduction
The development of dedicated experimental facilities to probe the structure of hadrons in the nonperturbative region of QCD with far greater precision than before has stimulated us to reexamine hadron spectroscopy in a novel approach in which both internal (spin-flavor-color) and space degrees of freedom of hadrons are treated algebraically. The new ingredient is the introduction of a space symmetry or spectrum generating algebra for the radial excitations which for baryons was taken as U(7) 1. The algebraic approach unifies the harmonic oscillator quark model with collective string-like models of baryons. In this contribution we present an analysis of the mass spectrum and strong couplings of both nonstrange and strange baryon resonances in the framework of a collective string-like qqq model in which the radial excitations are treated as rotations and vibrations of the strings. The algebraic structure of the model enables us to obtain transparent results (mass formula, selection rules and decay widths) that can be used to analyze and interpret the experimental data, and look for evidence of the existence of unconventional (i.e. non qqq) configurations of quarks and gluons, such as hybrid quark-gluon states qqq-g or multiquark meson-baryon bound states qqq-qq2
Mass s p e c t r u m
We consider baryons to be built of three constituent parts which are characterized by both internal and radial (or spatial) degrees of freedom. The internal degrees of freedom are described by the usual spin-flavor (sf) and color (c) algebras 5[/ s f(6) ® SUC(3). The radial degrees of freedom for the relative motion of the three constituent parts are taken as the Jacobi coordinates, which are 193
194
treated algebraically in terms of the spectrum generating algebra of ^7(7) 1 . The full algebraic structure is obtained by combining the radial part with the internal spin-flavor-color part G = U(7)®SUat(6)®SUc(3)
,
(1)
in such a way that the total baryon wave function is antisymmetric. For the radial part we consider a collective (string-like) model of the nucleon in which the baryons are interpreted as rotational and vibrational excitations of an oblate symmetric top 1. The spectrum consists of a series of vibrational excitations labeled by (v\ ,V'i), and a tower of rotational excitations built on top of each vibration. The occurrence of linear Regge trajectories suggests to add, in addition to the vibrational frequencies Ki and K2, a term linear in L. The slope of these trajectories is given by a. For the spin-flavor part of the mass operator we use a Giirsey-Radicati form 2 . These considerations lead to a mass formula for nonstrange and strange baryons of the form 3 M 2 = M$ + KI Vl + K2 v2 + a L + a [
^}
+b [(C2(SU((3))) - 9] + c [S(S + 1) - | ]
+d[y-i]+e[r2-i]+/[/(i + i ) - | .
(2)
The coefficient MQ is determined by the nucleon mass M§ = 0.882 GeV 2 . The remaining nine coefficients are obtained in a simultaneous fit to 48 three and four star resonances which have been assigned as octet and decuplet states. We find a good overall description of both positive and negative baryon resonances of the N, A, S, A, E and fi families with an r.m.s. deviation of S = 33 MeV 3 to be compared with 6 = 39 MeV in a fit to 25 N and A resonances 1 . There is no need for an additional energy shift for the positive parity states and another one for the negative parity states, as in the relativized quark model 4 . The three resonances that were assigned as singlet states (and were not included in the fitting procedure) show a deviation of about 100 MeV or more from the data: the A*(1405), A*(1520) and A*(2100) resonances are overpredicted by 236, 121 and 97 MeV, respectively. An additional energy shift for the singlet states (without effecting the masses of the octet and decuplet states) can be obtained by adding to the mass formula of Eq. (2) a term A M 2 that only acts on the singlet states. However, since A* (1405) and A* (1520) are spinorbit partners, their mass splitting of 115 MeV cannot be reproduced by such a mechanism. In principle, this splitting can be obtained from a spin-orbit interaction, but the rest of the baryon spectra shows no evidence for such a
195 Table 1: Masses of the first three P u states in MeV
PDG6
Zagreb 5
RQM 7
present 3
iV(1440) JV(1710)
1439 ± 19 1729 ± 1 6 1740 ± 1 1
1540 1770 1880
1444 1683 1713
large spin-orbit coupling. A more likely explanation is the proximity of the A*(1405) resonance to the NK threshold (see next section). A common feature to all q3 quark models is the occurrence of missing resonances. In a recent three-channel analysis by the Zagreb group evidence was found for the existence of a third P u nucleon resonance at 1740 ± 1 1 MeV 5 . The first two Pu states at 1439 ± 19 MeV and 1729 ± 16 MeV correspond to the AT(1440) and N(17W) resonances of the PDG 6 . It is tempting to assign the extra resonance as one of the missing resonances 7 . In the present calculation it is associated with the 2 8i/2[20,1+] configuration and appears at 1713 MeV, compared to 1880 MeV in the relativized quark model (RQM) 4 (see Table 1). A recent analysis of new data on kaon photoproduction 8 has shown evidence for a D13 resonance at 1895 MeV 9 . In the present calculation, there are several possible assignments 3 . The lowest state that can be assigned to this new resonance is a vibrational excitation (^1,^2) = (0,1) with 2 8 3 / 2 [56,1~] at 1847 MeV. This state belongs to the same vibrational band as the ^(1710) resonance. In the relativized quark model a Di3 state has been predicted at 1960 MeV 4 . 3
Strong couplings
Decay processes are far more sensitive to details in the baryon wave functions than are masses. Here we consider strong decays of baryons by the emission of a pseudoscalar meson B -> B' + M ,
(3)
in an elementary emission model 3 . The calculation of the strong decay widths involves a phase space factor, a spin-flavor matrix element and a spatial matrix element which is obtained in the collective model by folding with a distribution
196
function over the volume of the nucleon. The calculations are carried out in the rest frame of the decaying resonance. The transition operator that induces the strong decay is determined in a fit to the Nir and ATT channels which are relatively well known 10 . It is important to stress that in the present analysis the same transition operator is used for all resonances and all decay channels. The calculated decay widths are primarily due to spin-flavor symmetry and phase space. TV and A resonances decay predominantly into the IT channel, and strange resonances mainly into the 7r and K channel. Phase space factors suppress the 77 and K decays. The use of the collective form factors introduces a power-law dependence on the meson momentum, compared to an exponential for harmonic oscillator form factors. In general, our results for the strong decay widths are in fair overall agreement with the available data, which shows that the combination of a collective string-like qqq model of baryons and an elementary emission model for the decays can account for the main features of the data. As an example, in Table 2 we present the strong decays of three and four star A resonances. There are, however, a few exceptions which could indicate evidence for the importance of degrees of freedom which are outside the present qqq model of baryons. The 77 decays of octet baryons show an unusual behavior: the 5-wave states iV(1535), S(1750) and A(1670) are found experimentally to have a large branching ratio to the 77 channel with partial decay widths of 74 ± 39, 39 ± 28 and 9 ± 5 MeV, respectively 6 . In our calculation these resonances are assigned as octet partners and only differ in their flavor content. The smallness of the calculated 77 widths (< 0.5 MeV) is mainly due to the available phase space. The results of this analysis suggest that the observed 77 widths are not due to a conventional qqq state, but may rather indicate evidence for the presence of a state in the same mass region of a more exotic nature, such as a quasimolecular S-wave resonance qqq-qq just below or above threshold, bound by Van der Waals type forces 11 (for example Nrj, £77 or A77). The decay of 48[70, Lp] A states into the NK channel is forbidden by a spin-flavor selection rule 3 which is similar to the Moorhouse selection rule in electromagnetic couplings. Therefore, the calculated NK widths of A(1800), A(1830) and A(2110) vanish, whereas all of them have been observed experimentally 6 . The A(1800)50i state has large decay width into N'K*(892) 6 . Since the mass of the resonance is just around the threshold of this channel, this could indicate a coupling with a quasi-molecular S wave. The NK width of A(1830) is relatively small (6 ± 3 MeV), and hence in qualitative agreement with the selection rule. The situation for the the A(2110) resonance is unclear. The A* (1405) resonance has a anomalously large decay width (50 ± 2 MeV) into E7r. This feature emphasizes its quasi-molecular nature due to the prox-
197 Table 2: Strong decay widths of three and four star delta resonances in MeV. For the t; mesons we introduce a mixing angle Op = —23° between the octet and singlet mesons. The experimental values are taken from 6 . Decay channels labeled by - are below threshold.
Baryon
A(1232)P 33 A(1600)P 33 A(1620)5 3 i A(1700)£>33 A(1905)F 35 A(1910)P 3 i A(1920)P 33 A(1930)D 35 A(1950)F 37 A(2420)ff 3 ,n
Nir
116 119 ± 5 108 61 ± 3 2 16 38 ± 1 1 27 45 ± 2 1 9 36 ± 2 0 42 52 ± 1 9 22 28 ± 1 9 0 53 ± 2 3 45 120 ± 14 12 40 ± 2 2
£#
ATT
ATJ
E*(1385)tf
—
—
—
—
—
—
—
—
—
—
—
1
0
2
25 193 ± 76 89 68 ± 2 6 144 135 ± 64 45 < 45 ± 45 4
0
0
1
29
1
0
0
0
0
0
6
36 80 ± 1 8 11
2
0
2
1
— 0 1
4
imity of the NK threshold. It has been shown 12 that the inclusion of the coupling to the NK and T,-K decay channels produces a downward shift of the qqq state toward or even below the NK threshold. In a chiral meson-baryon Lagrangian approach with an effective coupled-channel potential the A* (1405) resonance emerges as a quasi-bound state of NK n . 4
Summary and conclusions
In this contribution we have analyzed the mass spectrum and the strong couplings of both strange and nonstrange baryons. The combination of a collective
198
string-like qqq model of baryons in which the orbitally excited baryons are interpreted as collective rotations and vibrations of the strings, and a simple elementary emission model for the strong decays can account for the main features of the data. The main discrepancies are found for the low-lying 5-wave states, specifically JV(1535), JV(1650), £(1750), A*(1405), A(1670) and A(1800). All of these resonances have masses which are close to the threshold of a meson-baryon decay channel, and hence they could mix with a quasi-molecular S wave resonance of the form qqq — qq. In contrary, decuplet baryons have no low-lying S states with masses close to the threshold of a particular decay channel, and their spectroscopy is described very well. The results of our analysis suggest that in future experiments particular attention be paid to the resonances mentioned above in order to elucidate their structure, and to look for evidence of the existence of exotic (non qqq) configurations of quarks and gluons. Acknowledgments This work was supported in part by DGAPA-UNAM under project IN101997 and by CONACyT under project 32416-E. References 1. R. Bijker, F. Iachello and A. Leviatan, Ann. Phys. (N.Y.) 236, 69 (1994). 2. F. Giirsey and L.A. Radicati, Phys. Rev. Lett. 13, 173 (1964). 3. R. Bijker, F. Iachello and A. Leviatan, Ann. Phys. (N.Y.) 284, 89 (2000). 4. S. Capstick and N. Isgur, Phys. Rev. D 34, 2809 (1986). 5. M. Batinic, I. Dadic, I. Slaus, A. Svarc, B.M.K. Nefkens and T.-S.H. Lee, Physica Scripta 58, 15 (1998). 6. Particle Data Group, Eur. Phys. J. C 3, 1 (1998). 7. S. Capstick, T.-S.H. Lee, W. Roberts and A. Svarc, Phys. Rev. C 59, R3002 (1999). 8. M.Q. Tran et al., Phys. Lett. B 445 (1998), 20. 9. T. Mart and C. Bennhold, Phys. Rev. C 6 1 (2000), 012201. 10. R. Bijker, F. Iachello and A. Leviatan, Phys. Rev. D 55, 2862 (1997). 11. N. Kaiser, T. Waas and W. Weise, Nucl. Phys. A 612, 297 (1997). 12. M. Arima, S. Matsui and K. Shimizu, Phys. Rev. C 49 (1994), 2831.
N O N - P E R T U R B A T I V E VS P E R T U R B A T I V E N U C L E O N R E S P O N S E TO E L E C T R O M A G N E T I C P R O B E S MARCO TRAESfl Dipartimento di Fisica Universita degli Studi di Trento 1-38050 POVO (Trento), Italy and INFN G.G. Trento E-mail: [email protected] A partonic description of the nucleon can be generated from gluon radiation off a purely valence quark system used to model the non perturbative input in the Operator Product Expansion (OPE) approach to lepton-hadron scattering in QCD. In this paper I discuss a systematic analysis based on Light-Front Hamiltonian dynamics, consistently developed at Next-to-Leading Order both for polarized and unpolarized structure functions. Non-perturbative effects survive evolution and can be observed in present and future experiments on polarized scattering.
1
Light-Front constituent quark model
In the light-front quark model 1 the nucleon state factorizes into \N, J, Jn) \P) where P is the total light-front nucleon momentum P = (P+, Pj_) = px+p2 + Pz. P+ = P ° + n - P and the subscript _L indicates the perpendicular projection with respect to the h axis. In order to achieve the ordinary composition rules, the intrinsic light-front angular momentum eigenstate \N, J, Jn) must be obtained from the canonical angular momentum eigenstate \N,j,jn) by means of a unitary transformation which is a direct product of generalized Melosh rotations. Finally the intrinsic part of the nucleon state, \N,j,jn) is eigenstate of the mass operator (M 0 + V) \N, j,jn) = M \N,j,jn), where the interaction term V must be independent on the total momentum P t o t and invariant under spatial rotations. Results of a confining mass equation of the following kind (Mo + V)
>J^ + rril- j + *t)
i M O = M i M f l , (1)
have been obtained in ref.2. (here £ = y p 2 + A 2 is the hyperradius). Solutions for non-relativistic reductions of Eq.(l) have been discussed by Ferraris etal.3. The mass equation (1) is solved numerically and the parameters of the interaction determined phenomenologically in order to reproduce the basic 199
200
Figure 1. The proton polarized structure function at Q2 = 3 GeV2. The full curve represents the NLO (MS) results of a complete light-front calculation within a scenario where no gluons are present at the hadronic scale (scenario A); the corresponding non-relativistic calculation are shown by the dot-dashed line. A scenario including negative polarized gluons ( f AG = —0.7) at the hadronic scale is summarized by the dotted line; dashed line shows the case of positive gluon polarization ( f AG = +0.7).
features of the (non strange) baryonic spectrum up to w 1600 MeV. The relevant effects of the relativistic covariance are particularly evident looking at the polarized distributions 2 . In that channel the introduction of Melosh transformations results in a substantial suppression of the responses at large values of x and in an enhancement of the response for ar < 0.15. The consequences can be appreciated looking at the results at the experimental scale after a Next-to-Leading order evolution (see Fig. 1). The Melosh rotation dynamics introduce the basic new ingredient in the calculations and its effect is quite sizeable in suppressing the proton response in the region x < 0.4. 2
Relativistic spin effects in Drell-Yan processes
A complete description of the spin degrees of freedom of quarks and antiquarks in the nucleon requires, at leading twist, the definition of two sets of parton distributions. The helicity distribution gi(x,Q2), has been intensively investigated in the last few years and discussed in the previous section, while the so called transversity distribution, hi (x, Q2), has come to the attention of theorists and experimentalists more recently in the analysis of Drell-Yan spin asymmetries. In fact transversity is strongly suppressed (by powers of mq/Q) in deep inelastic lepton-nucleon scattering and in general in any hard pro-
201
Figure 2. Ratio between transverse and longitudinal parton distributions, Eq. (2), as a function of the invariant mass of the produced lepton pair (Q2) at a center of mass energy corresponding to HERA-iV (-^/J = 39.2) GeV. Dashed line shows results neglecting Melosh rotations, the dotted line corressponds to the non-realtivistic model. Error bars have been calculated at LO and include acceptance corrections. Error bars in the lower curve have been slightly shifted to appreciate the overlap.
cess that involves only one parton distribution. In hadron-hadron collisions the chirality of the partons that annihilate is uncorrelated and the previous restrictions do not apply. At the hadronic scale the equality hi (x, Q%) = gi (x, Q\) is a typical outcome of non-relativistic models of the nucleon4, in which motion and spin observables are uncorrelated. In other words, any departure from the previous identity is a signature of relativity in the employed hadronic model. A complete theoretical study of fti and g\ has to account for the relativistic effects which distinguish h\ from
(C2,
™
v
_ I(EaelK{xl,Q2)h^x2,Q2) 2
+ ( g l +> x2))dy 2
J(E.«fctf(*i,Q K(*a,Q ) + (*i *+ *2))dy '
[)
where p°(ar, Q2) (ft"(x, Q2)) are the lngitudinally (transverse) polarized parton distributions with flavor a and charge e 0 ; the arguments xi and x2 are related,
202
for Drell-Yan processes, to the center of mass energy y/s, the invariant mass of the produced lepton pair Q2, and the rapidity y — arctan(Q 3 /Q°): x\ = y/Q2/se* and x2 = y/ipjie-v. The results for this ratio and for the kinematics of HERA-iV are shown in Fig. 2. The relative insensitivity to the details of the chosen potential is also evident in this representation. The error bars shown take into account the limited acceptance of the detectors. While a measurement in the region Q > 5 GeV cannot distinguish the importance of Melosh Rotations, in the low mass region (Q « 3 GeV) it would be possible to single out which is the right spin-flavor basis, though some overlap between the error bars still persists. For RHIC the acceptance corrections are too large to appreciate the differences. 3
Orbital angular m o m e n t u m distribution
The measurement of the integrated helicity parton distributions triggered the interest in a deeper understanding of how the total angular momentum of the nucleon is shared among its constituents. In fact the spin sum rule should read: \M1{Q2) + Ag(Q2) + Lq(Q2) + Lg{Q2) = | , where | AE(Q 2 ) (A.g(Q2)) is the spin carried by the quarks and antiquarks (gluons) and Lq(Q2) (Lg(Q2)) is the orbital angular momentum (OAM) contribution of the quarks (gluons) 7 . The consequences of a light-front treatment of relativistic spin effects on the helicity distributions previously discussed can be enlarged to OAM investigating in detail the predictions of the light-front covariant quark model 8 . In Fig. 3 the OAM distribution at the hadronic scale Q%, are shown within various approximation: i) the complete results obtained solving Eq. (1) as shown in Fig. 3.a. ii) In Fig 3.b the MR factors are combined with a wave function ($') obtained from the non-relativistic Schrodinger reduction of the Eq. (1) to test the sensitivity to the details of the momentum density. In this case one appreciates the effect of the lack of high momentum components generated by the presence of relativistic kinetic energy operator, iii) The comparison with the bag model results 9 is also provided in Fig. I.e. It is clear that the LF CQM, regardless of the details of the spatial wave function, provides OAM distributions which are comparable (even bigger by a factor 2) to the bag model. From the comparison between Figs, l.a and l.b one can see that the MR (and not the specific shape of the spatial wave function) is responsible for this sizeable OAM. In non-covariant quark models such as the Isgur-Karl model, where MR is omitted, the OAM distributions is almost flat 9 .
203
if .i -
Figure 3. Quark orbital angular momentum distributions calculated in light-front dynamics with the wave function * (a), with the modified wave function * ' (see text) (b) and in the bag model (c). Solid lines correspond to the initial hadronic scale fi2,, short-dashed lines to Q2 = 10 GeV2 and long-dashed ones to Q 2 = 1000 GeV2.
In order to bring the OAM distributions to the high-energy experimental scale, we use the recently obtained evolution equations at LO 1 0 . In the process the OAM distributions for the gluons will be generated. In fig. l.a and l.b we also present the evolved OAM distributions up to Q2 = 10 GeV2 (shortdashed line) and Q2 = 1000 GeV2 (long-dashed line). By comparing again the LFCQM with the bag model (Fig. l.c) it is
204
also clear that a non-vanishing OAM persists in the large x region and this is a distinctive feature of relativistic treatments of the nucleon. Indeed, in I-K models, the OAM is entirely concentrated at low x. This may constitute a clear signature of relativity in the low-energy models of the nucleon if Lz(x,Q2) is measured. Acknowledgments The terminal condition of my wife's illness prevented an oral presentation of this work. I dedicate this paper to her memory. I am grateful to Florencio Cano, Pietro Faccioli, Sergio Scopetta and Vicente Vento for their collaboration and useful suggestions. References 1. S.J. Brodsky, H.C. Pauli and S.S. Pinsky, Phys. Rep. 301, 299 (1998) J. Carbonell, B. Deplanques, V.A. Karmanov and J.-F. Mathiot, Phys. Rep. 300, 215 (1998). 2. P. Faccioli, M. Traini and V. Vento, Nucl Phys. A656, 400 (1999). 3. M. Ferraris, M.M. Giannini, M. Pizzo, E. Santopinto and L. Tiator, Phys. Lett. B364, 231, (1995). 4. R.L. Jaffe and X. Ji, Phys. Rev. Lett. 67, 552 (1991); Nucl. Phys. B375, 527 (1992). 5. M. Traini, V. Vento, A. Mair and A. Zambarda, Nucl. Phys. A614, 472 (1997); S. A. Kulagin, W. Melnitchouk, T. Weigl, W. Weise, Nucl.Phys. A597, 515 (1996); R. Jakob, P. J. Mulders, J. Rodrigues, Nucl. Phys. A626, 937 (1997); D. I. Diakonov, V. Yu. Petrov, P. V. Pobylitsa, M. V. Polyakov, C. Weiss, Phys. Rev. D 56, 4069 (1997); V. Barone, T. Calarco and A. Drago, Phys. Lett. B390, 287 (1997); H. Weigel, L. Gamberg, H. Reinhardt, Phys. Rev. D 55, 6910 (1997). 6. F. Cano, P. Faccioli, M. Traini, Phys. Rev. D 62, 094018 (2000). 7. For a recent review: B. Lampe and E. Reya, Phys. Rept. 332,1 (2000). 8. F. Cano, P. Faccioli, S. Scopetta and M. Traini, Phys. Rev. D 62 (2000) 054023. 9. S. Scopetta and V. Vento, Phys. Lett. B460, 8 (1999); Phys. Lett. B474, 235 (2000). 10. X. Ji, Phys. Rev. Lett. 78, 610 (1997); P. Hoodbhoy, X. Ji and W. Lu, Phys. Rev. D 59, 014013 (1999). O. Martin, P. Hagler and A. Schafer, Phys. Lett. B448, 99 (1999).
T H E LEGS D O U B L E POLARIZATION P R O G R A M M. BLECHER l a , K. ARDASHEV 2 , A. CARACAPPA 3 , C. COMMEAUX 4 A. D'ANGEL0 5 ,J-P. DIDILEZ 4 , R. DEININGER 2 , K. HICKS 2 , S. HOBLIT 3 , A. HONIG 6 , M. KHANDAKER 7 , O. KISTNER 3 , A. KUCZEWSKI 3 , F. LINCOLN 3 , R. LINDGREN 8 A. LEHMANN 9 , M. LOWRY 3 , M. LUCAS 2 , H. MEYER 1 , L. MICELI 3 , B. M. PREEDOM 9 , B. NORUM 8 , T. SAITOH 1 , A. M. SANDORFI 3 , C. SCHAERF 5 , H STROHER 1 0 , C. E. THORN 3 , K. WANG 8 , X. WEI 3 , and C. S. WHISNANT 9 (The LEGS Spin Collaboration) 6 l)Va Tech, 2)Ohio U., 3)Brookhaven National Lab., 4)U. Roma Tor Vergata & INFN-Roma II, 5)Orsay, 6)Syracuse [}., 7)Norfolk St. U., 8)U. Virginia, 9)1]. So. Carolina, and 10)Juelich The LEGS facility provides a tagged, polarized (linear or circular) gamma ray beam. A target consisting of frozen HD that can have independent polarizations for the free and bound nucleons is under production. Double polarization experiments of pion photo-production and nucleon-Compton scattering will allow measurements of various nucleon structure parameters and a test of the GDH sum rule.
1 1.1
The LEGS Facility Gamma Ray Beam and Target
The Laser Electron Gamma Source (LEGS) facility 1 at the National Synchrotron Light Source (NSLS) of the Brookhaven National Laboratory (BNL) produces gamma ray beams by Compton backscattering laser photons from the electrons circulating in the NSLS ring. In back scattering there is no spin flip, so the gamma rays preserve the polarization of the laser photons. LEGS typically operates in the region where the polarization, linear or circular, exceeds 80%. The struck electrons lose a small fraction of their energy and are magnetically diverted out of the beam into a tagging spectrometer. Knowledge of the ring energy and the tagged energy yields the gamma ray energy of an event and all tagged (FWHM< 3 MeV) gamma ray energies, now 2 185-470 MeV, are measured under the same conditions. LEGS has developed a strongly polarized target of HD molecules in the solid state (SPHICE) that has many desireable features.3 Either H or D or both can be polarized and simultaneous measurements on protons and bound neutrons are possible. The polarizations achievable are comparable to those of existing technologies (80%H, 50% D), but are not significantly diluted by "mailing address: Phys. Dept., Va Tech, Blacksburg, VA 24061 USA; email: [email protected] ''http://www.legs.bnl.gov 205
206
non-polarizable material. These can be known to an accuracy of ± 1 % because no paramagnetic impurities are present enabling a cleaner line shape. Targets of reasonable size, 5 cm length by 2.5 cm OD, in which HD is 90% of the target by mass are under production. In the in-beam condition of 0.8 T and 1.2 K the relaxation times are 2(H) and 6(D) weeks. In nuclear physics experiments the target quality factor depends on P2D2, where P, D are the target polarization and dilution factor, respectively. Compared to a C4H9OH target 4 currently in use at Mainz and Bonn, where similar physics is studied, the LEGS quality factor in the in-beam condition is at least 4 times larger for free protons. In the case of a deuterated target (MainzBonn) for polarized D the LEGS quality factor can easily be more than 30 times larger. 1.2
Spectrometer
A large acceptance spectrometer (SASY) has been installed in the LEGS beam line. It has medium energy resolution and can identify all final state particles from photo reactions in this energy range: 7,7r"1 °,p, and n. Surrounding the target are aerogel counters to veto events with atomic electrons. Surrounding the aerogel are four faces of ninety Nal crystals. In front of each face is a scintillator to distinguish neutral from charged particles. The segmented crystals can measure the vector momentum of gamma rays and thus identify 7T°. At LEGS energies the largest contributor to the cross section is pion production. In these cases the final state nucleons are emitted in the forward direction. To detect these and forward going pions, three layers of 10 cm thick scintillating bars, are placed downstream of the target. In front of the bars is a thin scintillator to distinguish charged from neutral particles. Phototubes on each end of a bar allow the energy, time of flight, and position to be determined. Downstream of the bars is a wall of lead glass blocks to enhance the gamma ray acceptance. In the beam hole is a gas Cerenkov counter to veto events with atomic electrons. The spectrometer is run in the wide open condition. An event is a coincidence with the gamma beam tag and any bar, Nal, or lead glass block. All energies and angles are simultaneously measured. 2
The Physics Program
The amplitude for the photoproduction of pions requires a minimum of eight independent quantities.5 Up to now only unpolarized cross sections and asymmetries with linearly polarized photons on protons are abundantly available
207
in the energy range of the A. LEGS has provided most of the asymmetry data.6 Several double polarization measurements with linearly and circularly polarized gamma rays on polarized targets and one recoil nucleon polarization, to completely specify the amplitude, are planned. One double polarization measurement is particularly important. It involves left and right circularly polarized gamma rays on target nucleons polarized parallel to or opposite the gamma polarization. The quantity of interest is the difference in the reaction cross sections as a function of gamma ray energy u>, 5a =
— du,
GDH = /
— duj = -2a{nK/m)2.
(1)
In the GDH prediction, a is the fine structure constant, UJQ is the photoproduction threshold energy, and m, K are the nucleon mass and anomalous magnetic moment, respectively. Both of these quantities are related to the nucleon spin structure at Q2 = 0, where Q is the 4-momentum transfer. 7 comes from the forward nucleon-Compton amplitude, which is related to photoproduction by unitarity. It is given by, 4(w) = /(w)e'-?+iw<7(w)CT-(e'xe) )
g(u) = g{0) + ju2 + ...,
(2)
where e, e* are the initial and final photon polarizations, respectively. For 7 there is rapid convergence due to the to3 denominator in Eq. 1. LEGS covers about 88% of the integral. 7 has been evaluated by using the present photoproduction multipoles (MP) and from chiral perturbation theory (ChPT). The latter calculation is made to the one loop level including a large correction due to the A. Both calculations agree. One should note that the A effect is identical for neutrons and protons so that a measurement of 7 P — -yn is independent of this large model dependent correction. The GDH integral converges less rapidly because of the OJ denominator in Eq. 1. LEGS covers 64% of the integral from threshold across the A region. DGH is the Q2 = 0 limit of an integral over x of the form factor gi(x,Q2) measured in high energy lepton nucleon scattering experiments at CERN and SLAC, where x = Q2/2mui and here u) is the energy of the virtual photon. The lepton scattering data and the GDH prediction require a very sudden turn over at small Q2 that is at present unexplainable. The present multipole calculations disagree with the GDH prediction. When the calculations for 7 are taken into account it would indicate that either the GDH prediction is
208
incorrect or that the present MP are wrong and that 2-loop corrections to 7 are large.8 LEGS also intends to measure the electromagnetic polarizabilities a, /? of the neutron via neutron-Compton scattering using a deuterium target. These are the first order coefficients of the energy expansion of the spin-independent Compton amplitude. They are as basic as the static nucleon properties such as the magnetic moment and are calculable by ChPT. The present measurements for the neutron largely disagree with each other.9 For these measurements a high resolution spectrometer similar to that of reference 6 will be used. However, these polarizabilities also obey a sum rule that is measureable with SASY, 1
[°°
a
i(oj)
unpo a + ^—j^—j-du,.
(3)
Vastly improved measurements of the neutron reaction cross sections are expected from LEGS. Acknowledgements The LEGS Spin Collaboration is supported by the U.S. National Science Foundation, the U.S. Department of Energy, and the Instituto Nazionale di Fisica Nucleare, Italy. References 1. C. E. Thorn, et al, Nucl. Instrum. Methods A45, 447 (1989). 2. A. J. Kuczewski et al, SPIE LASE'99, Talk 3610(1999) and BNL Rep. 66320. 3. A. Honig, et al, Nucl. Instrum. Methods A356, 39 (1995). 4. A. Thomas, et al, PANIC'99 Proceedings Nucl. Phys. A663& 664, 393c (2000). 5. W. Chiang and F. Tabakin, et al, Phys. Rev. C55, 2054 (1997). 6. G. Blanpied, et al, Phys. Rev. Lett. 79, 4337 (1997) and BNL Rep. 67526 - Phys. Rev. C (submitted) . 7. S. D. Drell and A. C. Hearn, et al, Phys. Rev. Lett. 16, 908 (1966) and S. B. Gerasimov, et al, Sov. J. Nucl. Phys. 2, 430 (1966). 8. A. M. Sandorfi, et al, Phys. Rev. D 50, R6681 (1994). 9. J. Schmiedmayer, et al, Phys. Rev. Lett. 66, 1015 (1991) and L. Koester, et al, Phys. Rev. C51, 3363 (1995).
H A D R O N S IN A RELATIVISTIC M A N Y - B O D Y
APPROACH
S T E P H E N R. C O T A N C H A N D F E L I P E J . L L A N E S - E S T R A D A Department
of Physics,
North
Carolina
State USA
University,
Raleigh NC
27695-8202
Results from a relativistic, field theoretical QCD analysis are reported for the low lying meson, glueball and hybrid spectra. Alternative many-body techniques are utilized to approximately diagonalize an effective QCD Hamiltonian containing a linear confining interaction with slope (string tension) determined from lattice gauge calculations. The ground state vacuum properties (condensates and dressed/constituent masses) are calculated using the BCS approach with spontaneous dynamical chiral symmetry breaking and a non-linear (similar to the DysonSchwinger) gap equation. The excited meson states are then predicted using the Tamm-Dancoff (TDA) and random phase (RPA) approximations (analogous to the Bethe-Salpeter equation). With only one predetermined interaction parameter, the string tension, and standard u, d, s and c current quark masses, the low mass meson states in the different spin and flavor channels are reproduced. Significantly, new insight is obtained concerning the condensate structure of the vacuum, meson decay constants, spin/orbital and flavor mass splitting contributions and the chiral symmetry governance of the pion. Substantial TDA-RPA differences are found in the light quark sector with the pion emerging as a Goldstone boson only in the RPA. This comprehensive approach also encompasses the gluon sector and, with the same string tension parameter, reproduces the gluon condensate value from QCD sum rules and, most importantly, the quenched lattice glueball spectrum. Finally, the exotic 1 •" hybrid meson mass is calculated to be above 2 GeV and in rough agreement with lattice and flux tube model results. This suggests the recently observed 1 •" exotic states have an alternative, perhaps four quark, structure.
The theory of strong interactions, QCD, is now well accepted and, as a relativistic quantum field theory, is clearly a many-body formulation, especially for nonperturbative hadronic studies. It is therefore an ideal arena for many body methods successfully applied in condensed matter, atomic and nuclear physics1. In this communication we report a many-body study providing new hadronic structure information, especially concerning the important issues of chiral symmetry and gluonic degrees of freedom. In particular we detail a comprehensive promising approach which unifies the quark and gluon sectors and, with only one predetermined dynamical parameter, yields a semiquantitative description of the known meson, glueball and hybrid properties.
209
210
Our effective Hamiltonian in the Coulomb gauge is H = / " d x t f t ( x ) ( _ j a . v + /?m)*(x) - 1 J dxdypa(x)VL(\x -g,
fM*(x)a-A(x)$(x)+rr
- y|)p°(y)
/dx(Iin +BB)
containing both quark, $ , and gluon, A, fields with color density pa = $ t ^ - \ | j + fabcAb • I F . We adopt the standard current quark mass, m, values for the u, d, s and c flavors; m u = m<j = 5 MeV, ms = 150 MeV and m c = 1200 MeV. The linear interaction, VL = or, is obtained from lattice measurements and Regge phenomenology yielding a = 0.18 GeV2. For certain observables we supplement this with the canonical Coulomb potential 2
Vc = — ^r with as = %£. Unless specifically mentioned we exclude Vc from our calculations. We begin by calculating the QCD ground state (vacuum) using the BCS variational procedure. This is similar to the Schwinger-Dyson method and leads to quark and gluon (uncoupled) gap equations. The generated dynamical mass for the u/d quarks 2 is of order 100 MeV and for the gluons 3 800 MeV. The vacuum constructed in this way contains quark-antiquark Cooper pairs interacting to form 3P0 condensates. With these quasiparticle degrees of freedom we construct the excited states from this vacuum. For the quark sector we represent mesons as quasiparticle/anti-particle pairs and angular mometum couple to form states of good JPC. Next we invoke the TDA at the lp-lh level and diagonalize the model Hamiltonian in this truncated space. A subset of our extensive calculations is displayed in Figure 1. In general there is broad agreement with the data, the most notable exception being the light pseudoscalar mesons. The insufficient « 200 MeV, w/ p mass splitting and related issue of chiral symmetry motivated our improved RPA treatment. Now the pion creation operator generalizes to
3t = E ( x ^ ] - y y < M ; ) > ij
with pion state \w) = Q^\RPA) and improved vacuum satisfying Q\RPA) = 0. Here qi, q{ are the BCS rotated quasiparticle operators and Xij, l y are the RPA wavefunction components obtained from the coupled equations of motions generated via (v\[H,Q*]\RPA)
= M„(w\Qi\RPA)
.
211
ice J E(MeV) mfc,
.
• 3t •
"P,(B '
p,W '
2000 11(1800)
15001
tl(1440)
j[
. .t. .
71(1300)
1000
if (958)
ri(S47)
^
500
n (138)
PDG98 TDA RPA
Figure 1. TDA (dots), RPA (dashes) and data (bars) for the light pseudoscalar and vector meson spectrum.
In the chiral limit (m = 0) the chiral condensate operator commutes with Q t and we rigorously compute Mn = 0 consistent with Goldstone's theorem. For m = 5 MeV explicit chiral symmetry breaking yields M* = 294 MeV, significantly better than the TDA value (note, the physical pion mass is reproduced with m = 3 MeV). Since both TDA and RPA produce comparable spin splittings, we therefore conclude that chiral symmetry, which only the RPA implements, is responsible for most of the ir/p mass difference. Similarly, we generate the glueball spectrum for two quasiparticle gluons. Now chiral symmetry is not an issue and the RPA and TDA spectra agree to within a few percent. Our calculations, using the same string tension as above, reproduce the lattice measurements as illustrated in Figure 2.
212
Figure 2. TDA glueball spectrum (bars) and lattice measurements (rectangles).
A fundamental test for QCD is the existence of exotic mesons (quantum numbers not possible in any qq model). In particular, it has been speculated that two recently observed4 states with JPC = 1~+ at 1.4 and 1.6 GeV contain explicit glue. Of several gluonic scenarios glueballs should be eliminated since the transversality Coulomb gauge condition, V • A = 0, combined with Bose symmetry forbids constituent J = 1 glueball states for two gluons (Yang's theorem) (note four or more gluons would be too massive). Another conjecture is that one or both states are hybrid mesons, containing a qq pair and at least one constituent gluon. We address this issue by considering Fock states in the combined quark/glue sector of the type qqg\BCS) and diagonalize the Hamiltonian in this basis. Now the quarks are in an octet color state which is not governed by chiral symmetry since the octet chiral charge does not commute with the Hamiltonian. Therefore for hybrids the TDA is sufficient and should be essentially identical to the RPA.
213 Lattice
TDA
Flux Tube Bag Model Experimental
1l
5000
FW
3
^vnmvvwq
[
)
1
|
] |
40001
Charm Sector
Mass
(MeV)
3000,
Light Sector
I .=.1
PC
20001 I
1000
fNK«§§«s?
-+
J == 1
I .=. I 1
1
i-
Figure 3. Light and charmed exotic 1 + hybrid mesons. The TDA calculation is comparable to the lattice results but both disagree with observed resonances.
Because a three-body (TDA or RPA) formulation is numerically quite formidable (nonlocal equations in 6 dimensional momentum space) we generate the spectrum variationally. In the hybrid cm frame there are two independent momentum vectors, conveniently chosen as q+ (gluon momentum) and q_ (relative momentum of the quark-antiquark pair) and two corresponding orbital angular mometa, L+ and L_ We use the lowest orbital waves compatible with a given set of quantum numbers (neglecting couplings with higher orbital states) and incorporate an exponential variational radial wavefunctions for each of the two momenta. For s-waves there are no exotic hybrids as the quantum numbers are 1H , 0 + + , 1 + + , 2 + + . For a combination of s and p-waves there are three exotic states, 1 _ + , 3 ~ + , 0 .
214
The variational hybrid meson spectrum is displayed in Figure 3. Note that our model hybrids are predicted to be quite massive with no state below 2 GeV. Related, the first exotic 1~ + appears around 2.4 GeV, significantly above the BNL data 4 . However, our general agreement with lattice 5 and flux tube models, in contrast to the somewhat dated bag model prediction, suggests that the observed states are not hybrids but have an alternative, perhaps four quark or meson molecular, structure. This is further affirmed by our good agreement with the other theories for the charmed exotic hybrid state. In summary, our many body approach provides a unified, comprehensive framework for hadron structure. Using only one predetermined dynamic parameter, a semi-quantitative description has been obtained for the known meson, glueball and hybrid properties. Our results further support the conclusion that the recently observed exotics are not hybrids. It is quite possible that these states are molecular mesons (e.g. qq singlets) and model calculations are in progress. Also in progress are applications to hadronic decays, IT — n scattering and full implementation of our recent renormalization program 6,7 . Acknowledgments This work is partially supported by grants DOE DE-FG02-97ER41048 and NSF INT-9807009. NERSC is also acknowledged for supercomputer time and F. J. Llanes-Estrada is grateful for a SURA-Jefferson Lab graduate fellowship. References 1. P. Ring and P. Schuck, The Nuclear Many-Body Problem (SpringerVerlag, New York, 1980). 2. F. J. Llanes-Estrada and S. R. Cotanch, Phys. Rev. Lett. 84, 1102 (2000). 3. A. P. Szczepaniak, E. S. Swanson, C.-R. Ji, and S. R. Cotanch, Phys. Rev. Lett. 76, 2011 (1996). 4. G. S. Adams et al. (E852 Collaboration) Phys. Rev. Lett. 8 1 , 5760 (1998); D. R. Thompson et al. (E852 Collaboration) Phys. Rev. Lett. 79, 1630 (1997). 5. C. McNeile, hep-lat/9904013 (1999). 6. D. G. Robertson, E. S. Swanson, A. P. Szczepaniak, C.-R. Ji, and S. R. Cotanch, Phys. Rev. D 59, 074019 (1999). 7. E. Gubankova, C.-R. Ji, and S. R. Cotanch, Phys. Rev. D 62, in press (2000).
LIGHT M E S O N S P E C T R A A N D S T R O N G D E C A Y S IN A CHIRAL QUARK CLUSTER MODEL L. A. BLANCO, F. FERNANDEZ AND A. VALCARCE Grupo de Fisica Nuclear, Universidad de Salamanca, E-37008 Salamanca,
Spain
The light meson spectra and their strong decays are studied within a chiral quark cluster model with a minimal set of parameters.
QCD at intermediate energies remains an unapproachable theory. As a consequence, one has to resort to models in order to study the hadron phenomenology in terms of quark degrees of freedom. Such models, with a common theoretical origin, should be able to account for a huge a m o u n t of experimental d a t a based on a few fundamental parameters. Among the basic properties of QCD t h a t any model approach should satisfy, confinement is the most relevant one from the hadron spectra point of view. Quarks and antiquarks do not show themselves as free particles, but they appear confined inside hadrons. Therefore, baryon and meson spectra should be governed by the confining interaction to a large extent. This can be observed in the heavy meson spectra, showing a pattern t h a t can be approximately fit by a linear rising potential. Although a lot of evidence exists t h a t the non-abelian nature of Q C D leads to confinement of color charge, little is known about it. Most of the d a t a come from lattice-QCD simulations, establishing a short linear-rising force between two static color sources, in agreement with the behavior of the heavy quark systems mentioned above. However, the linearly rising potential between static quarks in pure non-abelian gauge theories is altered due to the spontaneous quark-antiquark pair creation when the color-electric flux t u b e breaks at large separations between the color sources 1. Therefore, at larger distances qq pair creation screens the color charge and therefore the potential becomes flat. Besides, confined quarks seem to be very different from the current quarks which appear in the Q C D lagrangian. At scales below 1 GeV chiral symmetry appears spontaneously broken and quarks acquire a constituent mass. T h e chiral symmetry breaking energy scale is larger t h a n confinement scale, and therefore Q C D can be modeled as a low energy theory consistent of interacting constituent quarks through the elementary Goldstone bosons of the chiral symmetry, and eventually gluons which mimic the perturbative behavior of QCD.
215
216
In this model, which we will refer to as chiral quark cluster model (a model of confined (cluster) quarks with a constituent (chiral) mass), there should be a close connection between the meson and baryon dynamics. Confinement should be very similar in both cases, since color forces at large distances are only sensitive to the net color charge of the interacting sources. Thus, whether a quark is bound to an antiquark (meson) or to a diquark (baryon) in a color singlet should make no difference. The model presented here differs from those, like Nambu-Jona-Lasinio, which do not incorporate confinement or assume that confinement effects can be neglected from most hadron structure purposes. They identify the physical pion with a quark-antiquark Goldstone boson bound by some elementary interaction, but not confined, and the heavier mesons (cr, p, etc.) appear as resonances of the theory. On the contrary, in the present model confinement provides the most important contribution to the binding energy and the rest of the interaction only provides fine details of the spectra. Primary ingredients of the chiral quark cluster models are the confining potential and an one-gluon exchange term. This last piece is taken from De Riijula et al. 2 , whereas the screened static potential is given by 1 Vconinj) = -aXiXj (1 - e - " r « )
(1)
In the intermediate region between the chiral symmetry breaking scale and confinement scale, constituent quarks interact through pseudoscalar and scalar fields. This interaction has been derived elsewhere 3 and applied successfully to the NN phenomenology 3 ' 4 . The parameters of the qq interaction are taken from the study of the NN sector, except for confinement, because the WW interaction is not sensitive to it. A suitable way to fix the confinement parameters is to reproduce the quarkonia spectra. In this case chiral symmetry is explicitly broken. Therefore, Goldstone bosons do not appear and the one-gluon exchange is suppressed by the high value of the quark masses. In Fig. 1 we show the spectra for the heavy sector and the parameter-free prediction for the light-mass sector. A stringent observable for the calculated meson wave functions is given by the strong decays. We use a modified version of the 3 Po model 5 in which the radial amplitude is momentum-dependent f(j>) = 7 i + 7 2 exp (-73P 2 )
(2)
The amplitude parameters are fixed to a representative set of light S and P-
217 cc
t>5
4400 —
4000
I
_ _
:
^ - —
-
2. 10000
LU
LU
3600 —
3200
-
L=0
L=1
L=0
L=1
L=0
L=1
Figure 1. Comparison of experimental (shadow boxes and thin solid lines quoted with t h e name of the state) and calculated (solid lines) spectra of heavy L = 0 S = l (left) and light (right) mesons. The lines labeled with a "*" represent states where t h e calculated and the experimental d a t a cannot be distinguished. The experimental d a t a for the Pj states correspond to the centroid of the multiplet.
wave qq meson decay rates (see Table 1). Using these parameters, we show in Fig. 2 the prediction of our model for the rest of well-established meson decays. Table 1. Decay widths used to fit the s PQ model amplitudes. T in MeV. 1 p —> 7T7T /2
—> 7T7T
02 —*• pn ai
—>• pn
6l
—>• W7T
hi —> pn Kfi ->• Kn
Exp
1 Theory
151 ± 1 157 ± 5 72 ± 3 400 142 ± 8 360 ± 40 287 ± 23
112 142 95 438 132 314 299
D / S (Exp.)
-0.09(2) +0.260(35)
b
D / S (Theory)
-0.215 +0.363 +0.295
Finally, it can be measured the relative phase as well as the magnitude of decays where more than one partial wave contribute, constituting a sensitive test of the decay model. We have calculated the D/S amplitude ratio for the decays b\ —> con and a\ —> pn, the only measured ones. The results are given in the last two columns of Table 1.
218
§
6 g
ti* " t i l l
5 6
& "r & w |\ (vj !y e^
to
*i
<•> OJ
-Q
r^
<»
< 6
I H U
t
=• =• =• *-
Oi
t
K
S
^
O
K Q ? a Q ? * - " * l « : ^ i C ^ i C
^ Sc~ J T «~
Figure 2. Ratio between the calculated and the experimental strong decay widths
In summary, we have presented a chiral quark model that allows to understand simultaneously the meson spectra and their strong decays with the same set of parameters that explains the NN interaction. References 1. E. Laermann, F. Langhammer, I. Schmitt, and P. M. Zerwas, Phys. Lett. B 173 (1986), 443. 2. A. de Rujula, H. Georgi, and S.L. Glashow, Phys. Rev. D 12 (1975), 147. 3. F. Fernandez, A. Valcarce, U. Straub, and A. Faessler, J. Phys. G 19 (1993), 2013. 4. D. R. Entem, F. Fernandez, and A. Valcarce, Phys. Rev. C62 (2000), 034002. 5. R. Bonnaz and B. Silvestre-Brac, Few-Body Syst. 27 (1999), 163. 6. Particle Data Group, Eur. Phys. J. C 3 (1998), 1.
A S K E T C H OF T W O A N D T H R E E B O D I E S HARALD W. GRIEfiHAMMER Institut fur Theoretische Physik, Physik-Department der Technischen Universitat Mimchen, 85748 Garching, Germany Email: [email protected] This presentation is a concise teaser for the Effective Field Theory (EFT) of two and three nucleon systems as it emerged in the last three years, using a lot of words and figures, and a few cheats. For details, I refer to the exhaustive bibliographies in 1 , and papers with J.-W. Chen, R.P. Springer and M.J. Savage2, P.F. Bedaque 3,4 , and F. Gabbiani 4 . Effective Field Theory methods are largely used in many branches of physics where a separation of scales exists. In low energy nuclear systems, the scales are, on one side, the low scales of the typical momentum of the process considered and the pion mass, and on the other side the higher scales associated with chiral symmetry and confinement. This separation of scales produces a low energy expansion, resulting in a description of strongly interacting particles which is systematic, rigorous and model independent (meaning, independent of assumptions about the non-perturbative QCD dynamics). Three main ingredients enter the construction of an EFT: The Lagrangean, the power counting and a regularisation scheme. First, the relevant degrees of freedom have to be identified. In his original suggestion how to extend EFT methods to systems containing two or more nucleons, Weinberg 5 noticed that below the A production scale, only nucleons and pions need to be retained as the infrared relevant degrees of freedom of low energy QCD. The theory becomes non-relativistic at leading order in the velocity expansion, with relativistic corrections systematically included at higher orders. The most general chirally (and iso-spin) invariant Lagrangean consists hence of contact interactions between non-relativistic nucleons, and between nucleons and pions, with the first few terms of the form CNN = N*(idQ + —)N+
i £ tr[(d^)(d^)}
+ gAN^A-aN
- C 0 (iV T F i iV) t (JVTPjJV) + + Y
(1)
[ ( A ^ P ' A O t ^ P ^ - d)2N) + H.c] + . . . ,
where N — (£) is the nucleon doublet of two-component spinors and Pl is the projector onto the iso-scalar-vector channel, Pl'^13 — ^ ( C T 2 0 ' * ) Q ( T 2 ) O - The 219
220
iso-vector-scalar part of the NN Lagrangean introduces more constants Ct and interactions and has not been displayed for convenience. The field £ describes the pion, £(ar) = VT, = em/f«, A„ = fad^ - gd^)All short distance physics - branes and strings, quarks and gluons, resonances like the A or a - is integrated out into the coefficients of the low energy Lagrangean. The most practical way to determine those constants is by fitting them to experiment. The EFT with pions integrated out (formally, gA = 0 in (1)) is valid below the pion cut and was recently pushed to very high orders in the two-nucleon sector6 where accuracies of the order of 1% were obtained. It can be viewed as a systematisation of Effective Range Theory with the inclusion of relativistic and short distance effects traditionally left out in that approach. As the second part of an EFT formulation, predictive power is ensured by establishing a power counting scheme, i.e. a way to determine at which order in a momentum expansion different contributions will appear, and keeping only and all the terms up to a given order. The dimensionless, small parameter on which the expansion is based is the typical momentum Q of the process in units of the scale A at which the theory is expected to break down. Values for A and Q have to be determined from comparison to experiments and are a priori unknown. Assuming that all contributions are of natural size, i.e. ordered by powers of Q, the systematic power counting ensures that the sum of all terms left out when calculating to a certain order in Q is smaller than the last order retained, allowing for an error estimate of the final result. Even if calculations of nuclear properties were possible starting from the underlying QCD Lagrangean, EFT simplifies the problem considerably by factorising it into a short distance part (subsumed into the coefficient of the Lagrangean) and a long distance part which contains the infrared-relevant physics and is dealt with by EFT methods. EFT provides an answer of finite accuracy because higher order corrections are systematically calculable and suppressed in powers of Q. Hence, the power counting allows for an error estimate of the final result. Relativistic effects, chiral dynamics and external currents are included systematically, and extensions to include e.g. parity violating effects are straightforward. Gauged interactions and exchange currents are unambiguous. Results obtained with EFT are easily dissected for the relative importance of the various terms. Because only 5-matrix elements between on-shell states are observables, ambiguities nesting in "off-shell effects" are absent. On the other hand, because only symmetry considerations enter the construction of the Lagrangean, EFTs are less restrictive as no assumption about the underlying QCD dynamics is incorporated. In systems involving two or more nucleons, establishing a power counting is complicated by the fact that unnaturally large scales have to be accom-
221
modated: Given that the typical low energy scale in the problem should be the mass of the pion as the lightest particle emerging from QCD, fine tuning seems to be required to produce the large scattering lengths in the 1 S 0 and 3 Si channels ( l / a l s ° = - 8 . 3 MeV, l / a 3 S l = 36 MeV). Since there is abound state in the 3 Si channel with a binding energy B = 2.225 MeV and hence a typical binding momentum 7 = \/MB ~ 46 MeV well below the scale A at which the theory should break down, it is also clear that at least some processes have to be treated non-perturbatively in order to accommodate the deuteron. A way to incorporate this fine tuning into the power counting was suggested by Kaplan, Savage and Wise 7 : At very low momenta, contact interactions with several derivatives - like p2C-2 and the pion-nucleon interactions - should become unimportant, and we are left only with the contact interactions proportional to Co- The leading order contribution to nucleons scattering in an S wave comes hence from four nucleon contact interactions and is summed geometrically, and the coefficient of the four-nucleon interactions scale as Co ~ JJQ , C2 ~ M^ni > Dimensional regularisation preserves the systematic power counting as well as all symmetries (esp. chiral invariance) at each order in every step of the calculation. Even at NNLO in the two nucleon system, simple, closed answers allow one to assert the analytic structure. The deuteron propagator has the correct pole position and cut structure. One surprising result arises from this analysis because chiral symmetry implies a derivative coupling of the pion to the nucleon at leading order, so that the instantaneous one pion exchange scales as Q° and is smaller than the contact piece Co ~ Q - 1 - Pion exchange and higher derivative contact terms appear hence only as perturbations at higher orders. In contradistinction to iterative potential model approaches, each higher order contribution is inserted only once. In this scheme, the only non-perturbative physics responsible for nuclear binding is extremely simple, and the more complicated pion contributions are at each order given by a finite number of diagrams. In this formulation, the elastic deuteron Compton scattering cross section2 to NLO is parameter-free with an accuracy of 10%. Contributions at NLO include the pion graphs that dominate the electric polarisability of the nucleon from their ^ - behaviour in the chiral limit. The comparison with experiment in Fig. 1 shows good agreement and therefore confirms the HBxPT value for oj£. The deuteron scalar and tensor electric and magnetic polarisabilities are also easily extracted 2 . In the three body sector, the equations that need to be solved are computationally trivial, as opposed to many-dimensional integral equations arising in other approaches. The absence of Coulomb interactions in the nd system ensures that only properties of the strong interactions are probed. In
222 w»49 MeV
w«69 MeV
<2[rad]
«[rad)
Figure 1. The differential cross section for elastic -y-deuteron Compton scattering at incident photon energies of JS7 = 49 MeV and 69 MeV in an E F T with explicit pions 2 , no free parameters. Dashed: LO; long dahed: NLO without the graphs that contribute to the nucleon polarisability; solid curve: complete NLO result. Accuracy of calculation at NLO (±10%) indicated by shaded area.
the quartet channel, the Pauli principle forbids three body forces in the first few orders. In the S wave, spin-doublet (triton) channel, an unusual renormalisation makes the three-body force large and as important as the leading two-body forces8. More work is needed there. A comparative study between the theory with explicit pions and the one with pions integrated out was performed3 in the spin quartet S wave for momenta of up to 300 MeV in the centre-of-mass frame (Ecm fa 70 MeV). As seen above, the two theories are identical at LO: All graphs involving only Co interactions are of the same order and form a double series which cannot be written down in closed form. Summing all "bubble-chain" sub-graphs into the deuteron propagator, one can however obtain the solution numerically from the integral equation pictorially shown in Fig 2.
Figure 2. The Faddeev equation for the three body system.
The calculation with/without explicit pions to NLO/NNLO shows convergence. Pionic corrections to nd scattering in the quartet S wave channel - although formally NLO - are indeed much weaker. The difference to the theory in which pions are integrated out should appear for momenta of the order of mw and higher because of non-analytical contributions of the pion cut, but those seem to be very moderate, see Fig. 3. Finally, the real and imaginary parts of the higher partial waves I =
223
50
100
150 kinMeV
200
250
300
ts
to
7.B «_W«1
10.0
vu
Figure 3. Real parts in the quartet S and doublet D wave phase shift of nd scattering versus the centre-of-mass momentum 3 ' 4 . Dashed: LO; solid (dot-dashed) line: NLO with perturbative pions (pions integrated out); dotted: NNLO without pions 4 . Realistic potential models: squares, crosses, triangles. Stars: pd phase shift analysis.
1 , . . . ,4 in the spin quartet and doublet channel were found4 in a blablameterfree calculation, see Fig. 3. Within the range of validity of this pion-less theory, convergence is good, and the results agree with potential model calculations (as available) within the theoretical uncertainty. That makes one optimistic about carrying out higher order calculations of problematic spin observables like the Ay problem where the EFT approach will differ from potential model calculations due to the inclusion of three-body forces. References 1. U. van Kolck, M. Savage and R. Seki, eds., "Nuclear Physics with Effective Field Theory", Proceedings of the INT-Caltech Workshop at Caltech (1998), World Scientific; P. Bedaque, U. van Kolck, M. Savage and R. Seki, eds., "Nuclear Physics with Effective Field Theory II", Proceedings of the INT-Caltech Workshop at the INT (1999), World Scientific. 2. J. Chen, H.W. Griefihammer, M.J. Savage and R.P. Springer, Nucl. Phys. A644, 221 (1998); Nucl. Phys. A644, 245 (1998). 3. P.F. Bedaque and H.W. Griefihammer, Nucl. Phys. A675, 601 (2000). 4. F. Gabbiani, P.F. Bedaque and H.W. Griefihammer, nucl-th/9911034. 5. S. Weinberg, Nucl. Phys. B363, 3 (1991). 6. J. Chen, G. Rupak and M.J. Savage, Nucl. Phys. A653, 386 (1999). 7. D.B. Kaplan, M.J. Savage and M.B. Wise, Nucl. Phys. B534, 329 (1998). 8. P.F. Bedaque, H.W. Hammer and U. van Kolck, Phys. Rev. Lett. 82, 463 (1999); P.F. Bedaque, H.W. Hammer and U. van Kolck, Nucl. Phys. A646, 444 (1999); P.F. Bedaque, H.W. Hammer and U. van Kolck, Nucl. Phys. A676, 357 (2000).
REALISTIC S T U D Y OF T H E N U C L E A R T R A N S P A R E N C Y A N D T H E DISTORTED M O M E N T U M D I S T R I B U T I O N S IN i T H E SEMI-INCLUSIVE P R O C E S S HE{E,E'P)X H. MORITA Sapporo Gakuin University, 11-Bunkyodai, Ebetsu 069-8555, Japan E-mail: [email protected] C. CIOFI DEGLI ATTI Department of Physics, University of Perugia, and INFN, Sezione di Perugia, Via A. Pascolo, 1-06100 Perugia, Italy E-mail:claudio. [email protected] D. TRELEANI Department of Theoretical Physics, University of Trieste, Strada Costiera 11, INFN, Sezione di Trieste, and ICTP, 1-34014 Trieste, Italy E-mail:daniel@ts. infn.it The nuclear transparency and the distorted momentum distributions of iHe in the semi-inclusive process 4He(e,e'p)X are calculated within the Glauber multiple scattering approach using for the first time realistic four-body variational wave function. The contributions from NN correlaions and from Glauber multiple scattering are taken into account exactly to all orders. It is shown that the net effect of nucleon-nucleon correlation on the nuclear transparency is small (« 3%), and the effect of Glauber final state interactions on the momentum distributions is reduced by the inclusion of tensor correlations.
1
Introduction
In spite of many fruitful studies before,1 the role played by ground-state nucleon-nucleon (NN) correlations [ or initial state correlations (ISC)] on the nuclear transparency in semi-inclusive processes A(e, e'p)X is still not clear. All of these works adopt the Glauber multiple scattering approach for the description of the final state interaction (FSI), which seems to work well. On the other hand the ISC are described on the basis of different schems, which are rather model dependent and therefore lead to different conclusions. In this sense the calculation of nuclear transparency on the basis of the realistic NN correlation is awaited. Recently we have accomplished such calculation 2 including also the distorted momentum distributions nr)(k) and in this paper we will show both of them. The latter quantity has been recently calculated in Ref. 3, where it has been argued that the high momentum part of no{k)
224
225
which could be measured by semi-inclusive processes, is almost entirely dominated by FSI, leaving little room for the investigation of ISC. Undoubtedly this pointing out is very important, however since their argument is based on the simple central NN correlation (state-independent Jastrow-type correlation) their claim should be reexamined in more realistic way. This paper is organized as follows: in Sec. 2 the formulation of nuclear transparency and distorted momentum distributions are summarized; the structure of the 4He wave function used in the calculations is briefly described in Sec. 3; the results of calculations of the nuclear transparency and the distorted momentum distributions are presented in Sees. 4 and 5 respectively; the Summary and Conclusions are given in Sec. 6. 2
F o r m u l a t i o n of t h e d i s t o r t e d m o m e n t u m d i s t r i b u t i o n s a n d nuclear t r a n s p a r e n c y w i t h i n t h e G l a u b e r a p p r o a c h
The distorted momentum distribution no{k) of 4He is defined by nD(k) PD(r',r)
= (27r)- 3 / d r d r ' e - i k < r - r ^ p D ( r ' , r ) , = J'dR1dR29*{Ri,R2,R!!l=r')
&'S
(1) * ( f l i , B 2 , H 3 = r ) , (2)
where * is the four-body intrinsic wave function and ilj's are the Jacobi coordinates explicitly given by JRi = Ti — n , R
S = J ] G(4i),
G(4t) = 1 - 9(zi - Zi)T{bi - b4),
(3)
i=l
where T(4i) is the Glauber profile function and "4" refers to the knocked out proton. In the Glauber approach, coordinates Tj's are expressed as T*J = bi + Zip. Here p is the direction of the knocked proton momentum. In the following calculation, we take it as the direction of momentum transfer q, because we are confined ourselves at high momentum transfer region, which allows us such replacement. 3 We adopt a standard parameterizations for T =
crtot(l-ia) 471-6*
b2/ibl
(4)
226
where otot is the total proton-nucleon cross section, and a is the ratio of real to imaginary part of the forward elastic pN scattering amplitude. In the following calculation we use the value, a = —0.33, 60 = 0.5(/m) and &tot = 43 (mb), which was used in Ref. 3. These values were taken from the PN scattering data at the condition where the knocked out proton's kinetic energy Tp » l(GeV). The nuclear transparency T is obtained by integrating nn(A;), that is , / dfcri£)(fc). 3
(5)
The Realistic Four-Body Wave Function
As for the four-body wave function * we applied the ATMS method, wave function is written as *ATMS
= F $
0
,
4
whose (6)
where F represents a proper correlation function, and $0 is the mean field (uncorrected) wave function. Thus, one can ascribe any difference between the result with ^ATMS and $ 0 to the effect of the correlation F. The correlation function F has the following form 4 :
F = D-1 5 > « j ) - &£—*lv(ij)) JJ u(kl),
(7)
kl^ij
ij
P
kl^ij
where nP = A(A—1)/2 is the number of pair, w(ij) and u(ij) are on-shell and off-shell two-body correlation functions, respectively. Here the realistic NN interaction generates a state dependence of NN correlations, which is taken into account by introducing the following state dependence for the on-shell correlation function: w(ij) =' ws(ij)P1B(ij) 1B
ZE
+3 ws(ij)P3B(ij)
+ 3 wD(ij)SijP3E(ij),
(9)
where p (p ) [s a projection operator to the singlet-even (triplet-even) state and Sij is the usual tensor operator. We also includes the tensor-type off-shell correlation function and the explicit form of wave function is given in Ref. 5. The best set of correlation functions {«} = {w's, u's} are determined by the Euler-Lagrange equation. 5
227 Table 1. The results of the nuclear transparency T T0 0.754
TATMS
0.778
Tj astrow
TS
0.806
0.780
In the following calculation the Reid soft core V 8 model potential 6 is used as the realistic NN force. The calculated binding energy is E4 = —21.2 MeV, the rms radius is < r 2 > 1 / 2 = 1.57 frn, and the probabilities of the various waves are Ps = 87.94%, P s - = 0.24% and PD = 11.82%, respectively. 4
Results of the Nuclear Transparency
The results of the nuclear transparency are summarized in Table 1. Here the result shows that the effect of correlation on the transparency, given by the difference between TATMS and T 0 , is very small (~ 3%). We have also calculated T with the Jastrow-type wave function 3 * Jastrow = [ J fMj)
fjiTij)
= 1 - e- r ?i '2r' ,
(10)
ij
with rc = 0.5 fm. It can be seen from Table 1 that in such a case correlation effects amount to ~ 7%, which is about a factor of 2 larger than the realistic one. Since the Jastrow-type correlation function in Eq. (10) takes only into account short-range repulsive correlations, the difference between the realistic and the Jastrow cases should be ascribed to the intermediate-range attractive correlation. In order to clearly illustrate this point, we show the realistic correlation function 3ws(r) =3 ws(r) — (5/6)w(r) comparing with the Jastrow one in Fig. 1. Here the strong overshooting in the realistic correlation function, which is induced by the intermediate-range attractive correlation and is lacking in the Jastrow wave function, can be noticed. The attractive correlation enhances FSI effects, which reduces the nuclear transparency. Thus the cancellation between the short-range repulsive and intermediate-range attractive correlations occurs, and the net effect of ISC becomes small. 5
Result of the Distorted M o m e n t u m Distributions
Fig. 2 shows the distorted momentum distributions at three kinematics. Where 0 is the angle between the three-momentum transfer q and k. From
228
Figure 1. The realistic correlation function.
this figure we can see that at perpendicular kinematics {9 = 90°), FSI dominates the high momentum component. 3 On the other hand at parallel (9 = 0°) and antiparallel (6 = 180°) kinematics, effect of FSI becomes smaller, which suggests the posibility of the investigation of ISC in such kinematics. Here it should be noted, that is, the effect of FSI becomes smaller than the case of Jastrow-type (state-independent central) correlation. This is due to the existence of the Z?-wave component. Since the D-wave component, being produced by the tensor-type correlation, is more peripheral than the S-wave component, it is less affected by FSI. From these points of view an interesting quantity is the forward-backward asymmetry ApB{k):
AFB(k)
(a)Realistic
=
nD(k : 9 = 0°) - nD(k : 9 = 180°) nD(k : 0 = 0°) + nD(k : 9 = 180°)'
(b)Realistic
FSI(9=0°)
- FSI(e=90°)
FSI(0=1S0")
- No FSI
No FSI
0.6 0.5 0.4 0.3 0.2 2 0.1 0.0 -0.1 -0.2 -0.3
(11)
-Realistio(P =11.8%) -Realistic(P =10.0%) -Realistic(P = 1 5 . 0 % ) / • Jastrow
k lfm-1
Figure 2. FSI effect on the nD(k).
Figure. 3 The forward-backward asymmetry
AFB
229
which has been first introduced in Ref. 7. Because the D-wave component has little asymmetry as is explained above, AFB (k) gets contributions almost entirely from the S-wave. This suggests that this quantity might filter out the 5-wave component being little affected by the existence of the D-wave component. To demonstrate such a point, we have calculated ApB(k) by changing the D-wave probability Pp within a reasonable range. The results are presented in Fig. 3, which indeed shows that AFB(k) slightly depends upon the D-wave probability. 6
S u m m a r y and Conclusions
Our findings are summarized in the following: 1. The effect of NN correlation on nuclear transparency T amounts to ~ 3%. This small value is due to the cancellation between short-range repusive correlation and intermediate-range attractive correlation. 2. The FSI effect dominates the high momentum component at perpendicular kinematics, though its magnitude is reduced if one takes into account the tensor-type correlation which induce the D-wave component in iHe. 3. At (anti-) parallel kinematics FSI effect becomes rather smaller than the case of perpendicular kinematics. And forward-backward asymmetry Affl(fc) filters out the S-wave component of wave function. The same kind of calculations for heavier nuclei by using cluster expansion technique 8 are now in progress and will be published elsewhere. References 1. for example, N.N. Nikolaev et al., Phys. Lett. B 317, 281 (1993), and see the reference in 2. 2. H. Morita, C. Ciofi degli Atti and D. Treleani, Phys. Rev. C60, 34603(1999). 3. A. Bianconi et al., Nucl. Phys. A608, 437 (1996). 4. M. Sakai, et al., em Prog. Theor. Phys. Suppl. 56, 32 (1974). 5. H. Morita, Y. Akaishi, O. Endo and H. Tanaka et al., Prog. Theor. Phys. 78, 1117 (1987). 6. I.E. Lagaris and V.R. Pandharipande, Nucl. Phys. A359, 331 (1981). 7. A. Bianconi et al., Phys. Lett. B 343, 13 (1995). 8. C. Ciofi degli Atti and D. Treleani, Phys. Rev. C60, 24602(1999).
M E A S U R E M E N T S OF T H E D E U T E R O N ELASTIC S T R U C T U R E F U N C T I O N S A(Q2) A N D B(Q2) AT T H E J E F F E R S O N LABORATORY M. KUSS* FOR THE JEFFERSON LABORATORY HALL A COLLABORATION1 Results from the Jefferson Laboratory E91-026 Hall A experiment are reported. The aim of this experiment was to extract the deuteron structure functions A(Q2) and B(Q2) from coincidence elastic electron-deuteron cross section measurements. A squared four-momentum transfer (Q 2 ) range of 0.7 to 6.0 (GeV/c) 2 was covered for A(Q2), and of 0.7 to 1.4 (GeV/c) 2 for B(Q2), respectively. The results are compared to conventional meson-nucleon calculations based on either non-relativistic or relativistic impulse approximation. For A(Q2), they are also compared to predictions of dimensional scaling and perturbative quantum chromodynamics.
1
Introduction
Electron scattering from the deuteron is an ideal testing ground to study the nucleon-nucleon (NN) interaction at short distances and the role of meson exchange currents (MEC). Models based on the non-relativistic impulse approximation 1 (IA) augmented by MEC 2 and isobar contributions 3 ' 4 (IC) are expected to describe the low momentum transfer data 5 ' 6 ' 7 ' 8 . At higher momentum transfers, relativistic calculations 9 ' 10 ' 11 ' 12 ' 13 (RIA) might become necessary giving way to quantum chromodynamics 14 (QCD) at very large momentum transfers. There are problems in this simple picture: in the case of the IA and RIA calculations, the coupling constants and form factors of MEC like the jnp are poorly determined. In the case of QCD or the calculationally simpler perturbative QCD (pQCD), several authors have argued 15 ' 16 that pQCD is not applicable at the momentum transfers reached by this experiment. Finally, there are hybrid models 17 ' 18 ' 19 which incorporate both descriptions: at large NN separations, the deuteron is treated in terms of nucleons and mesons while for internucleon distance smaller than lfm, it is described as a six-quark object. Here, we present a brief description of experiment E91-026 in Hall A at the Thomas Jefferson National Accelerator Facility (JLab) and report results * E-mail: [email protected] tBlaise Pascal, California State LA, Duke, Florida International, Florida State, Gent, Georgia, Grenoble, Hampton, Harvard, INFN Roma, Jefferson Laboratory, Kent State, Kentucky, Kharkov, Maryland, MIT, New Hampshire, Norfolk State, North Carolina Central, Old Dominion, Orsay, Princeton, Regina, Rutgers, Saclay, New York Stony Brook, Syracuse, Temple, Tohoku, Virginia, William and Mary, Yamagata, Yerevan.
230
231
on the A(Q2) and B(Q2) structure functions. The results for A(Q2) have been published 20 . The B(Q2) results are preliminary. 2
Form Factors
The cross section for unpolarized elastic electron scattering of the deuteron is given by d
dn=aM
A(Q2) + B(Q2)Un
l 2
(1)
with 0M the Mott cross section, Q2 the squared four-momentum transfer and 6 the electron scattering angle. The electric and magnetic structure functions A(Q2) and B(Q2) are expressed in terms of the charge monopole, magnetic dipole and charge quadrupole form factors FQ, -FM and FQ A(Q2)
= F2 + | T * & + ^ T 2 F 2
B(Q2) = |T(1+T)F&,
(2) (3)
with r being a kinematical factor. A(Q2) and B(Q2) can be extracted by means of a Rosenbluth separation where the differential cross section da/dCl is measured at different electron scattering angles, keeping Q2 constant. A separation 21 of all three form factors can be achieved by measuring a polarization observable, e.g. the deuteron tensor polarization i2o223
Experimental Setup
The experiment used the Continuous Electron Beam Accelerator and the Hall A facilities of JLab. Electrons with energies of 0.54 to 4.4 GeV and beam currents of 5 to 120/uA were scattered off a high power (700 W) deuterium and hydrogen cryogenic target. Scattered electrons and recoiling deuterons were detected in coincidence using the two 4GeV/c High Resolution Spectrometers (HRS) in Hall A. In both HRS's two planes of scintillators for triggering and timing and a drift chamber system for particle tracking were used. The electron HRS was also equipped with a gas Cerenkov counter and a lead-glass calorimeter for electron identification. Elastic electron-proton scattering in coincidence was used to check the double-arm system acceptance. Incomplete triggers (one of the two scintillator planes missing) were prescaled and recorded, for detector efficiency studies.
232
4
Discussion of the Results
Figure 1 shows the A{Q2) data of this experiment as filled circles. The left panel focuses on the low Q2 region. One of the long-standing issues is, despite large experimental errors, an apparent discrepancy between the SLAC6 and the CEA 5 and Bonn 7 data sets. The E91-026 data show excellent agreement with the SLAC results. The IA predictions 1 , after inclusion of MEC, seem to provide a good description of the data. The right side of Fig. 1 shows A(Q2) over the entire Q2 range studied. Again, there is excellent agreement between our results and the SLAC6 data in the range of overlap. Our results continue with a smooth fall-off to larger Q2. RIA calculations 9 ' 10 are able to describe this behaviour. The agreement, however, depends on the jnp MEC coupling constants and form factors chosen: those of Hummel and Tjon 10 are based on a vector dominance model (VDM) while those of Van Orden et al.9 are based on a quark model. Figure 2 shows preliminary B{Q2) data of this experiment as filled circles. The data agree with the previous world data, with error bars improved
0.5
0.7
0.9
Q2
1.1
1.3
1.5
[(GeV/c)2]
1.7
1.9
0
1
2
Q2
3
4
5
[(GeV/c)2]
6
7
Figure 1. A(Q2) data from this experiment, compared to data taken at SLAC 6 , Saclay 8 , Bonn 7 , CEA 5 , and Hall C 2 3 , as function of squared four-momentum transfer. In the left half [Q 2 <1.9 (GeV/c) 2 ], they are compared to IA predictions 1 without (dashed curve) and with inclusion of MEC (solid), in the right [Q 2 <7 (GeV/c) 2 ] to RIA models of van Orden et al.9 without (solid) and with (dashed) pn'y MEC, and of Hummel and Tjon 1 0 without (dotted), with pirj MEC (dotted short dashed), and with pn'y and coaj MEC (dotted long dashed).
233 :
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V
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,
A Saclay
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|
:
:
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w\U
... ji]'^••-. j ~ - - .
/jT--.
:
- - -."""^-^ -^ n
~ - - . ;_•:
!
i '1'/
r
, . ! .«!,"
[(GeV/c)2]
:
:
-
%
1 AW
r
I
-=
° Bonn
''.'Am
r
I
° SLAC NE4
:
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Q2
=
[(GeV/c)2]
Figure 2. Preliminary B{Q2) data from this experiment, compared to previous data taken at SLAC 6 , Saclay 8 and Bonn 7 , as function of squared four-momentum transfer. In the left half, they are compared to IA predictions 1 , in the right to RIA models. The notation is the same as in Fig. 1.
significantly. It is remarkable that the IA model 1 (left side) that was able to describe the A(Q2) data (see Fig. 1) now fails. Two problems are apparent: there is a lack of precise data around and above the diffraction minimum, and IA and RIA calculations give various results for the location of the minimum and the absolute value of B(Q2). Dimensional scaling24 and pQCD 14 predict ^A{Q2) (the deuteron form factor Fd) to fall off as (Q 2 )" 5 at large Q2. In this case, the product F d -(Q 2 ) 5 should be independent of Q2 (scaling). Figure 3 shows this product as function of Q2. Our data seem to indicate an approach to scaling. Despite the validity of pQCD being questioned 15,16 at moderate values of Q2, such a scaling behaviour has been also reported from deuteron photo-disintegration experiments 25 . 5
Summary
The deuteron elastic structure function A(Q2) has been measured for squared four-momentum transfers up to 6(GeV/c) 2 . The results have clarified inconsistencies in previous data sets for Q2 < 1.4 (GeV/c) 2 . The data are consistent with calculations based on RIA including MEC, with specific choices for the
234
0.500
• JLab Hall A • SLAC E101
o.ioo
•
0.050
• „ m
.
• :
Deuteron Form Factor F d (Q 2 )[(Q 2 ) 5 ]
» _,
"
"
-
^
0.005
*
•m
• 0.010
*
,
""0
,
,
i
,
,
2
Q2
,
,
i
i
4
6
,
._
[(GeV/c) 2 ]
Figure 3. ( F d ) (i.e. y/A(Q2)), multiplied by ( Q 2 ) 5 , as function of Q2. The filled circles represent data from this experiment, the open squares those taken at SLAC 6 .
MEC form factors. At large Q2 A(Q2) seems to approach scaling, as predicted by pQCD. To extract B(Q2), a Rosenbluth separation was performed for a small range of Q2, below the diffraction minimum. Future precise data at and above the minimum are needed, to guide theoretical calculations. Additionally, A(Q2) and B(Q2) data as well as measurements of the form factors of the helium isotopes at larger Q2 will be crucial in verifying the above mentioned scaling behaviour. The Rosenbluth technique is, however, limited to small Q2 due to B(Q2) becoming much smaller than A(Q2). Backward electron scattering (180° scattering) at higher Q2 values allows us to extend the knowledge of B(Q2), because the contributions from F Q and F Q vanish at 180°. Similarily, the separation of i*c and F Q is restricted to small Q2, for experimental reasons. It will be difficult to extend the Q2 range of 0.65-1.85 (GeV/c) 2 spanned in Ref. 22 to higher Q2. Acknowledgments This work was supported by the Department of Energy contract DE-AC0584ER40150 under which the Southeastern Universities Research Association operates the Thomas Jefferson National Accelerator Facility. This work was supported in part by the National Science Foundation, the Kent State University Research Council, the Italian Institute for Nuclear Research, the French Atomic Energy Commission and National Center of Scientific Research, the
235
Natural Sciences and Engineering Research Council of Canada, and the Fund for Scientific Research-Flanders of Belgium. The author's participation in the Bologna 2000 conference was possible due to the free accomodation provided by the organizers. References 1. R. Schiavilla and D. 0 . Riska, Phys. Rev. C 43, 437 (1991), and references therein. 2. V.V. Burov, V.N. Dostovalov, and S.E. Suskov, Sov. J. Part. Nucl. 23, 317 (1992), and references therein. 3. P.G. Blunden, W.R. Greenberg, and E.L. Lomon, Phys. Rev. C 50, 1541 (1989). 4. R. Dymarz and F.C. Khanna, Nucl. Phys. A 516, 549 (1990), and references therein. 5. J.E. Elias et al, Phys. Rev. 177, 2075 (1969). 6. R.G. Arnold et al, Phys. Rev. Lett. 35, 776 (1975). 7. R. Kramer et al, Z. Phys. C 29, 513 (1985). 8. S. Platchkov et al, Nucl. Phys. A 510, 740 (1990). 9. J.W. Van Orden, N. Devine and F. Gross, Phys. Rev. Lett. 75, 4369 (1995), and references therein. 10. E. Hummel and J.A. Tjon, Phys. Rev. C 42, 423 (1990). 11. J. Carbonell and V.A. Karmanov, Nucl. Phys. A 663, 361 (2000). 12. D.R. Phillips, S.J. Wallace, and N.K. Devine, Phys. Rev. C 58, 2261 (1998). 13. F.M. Lev, E. Pace, and G. Salme, submitted to Phys. Rev. C, nuclth/0006053. 14. S.J. Brodsky et al, Phys. Rev. Lett. 51, 83 (1983). 15. N. Isgur and C.H. Llewellyn-Smith, Phys. Lett. B 217, 535 (1989). 16. G.R. Farrar, K. Huleihel, and H. Zhang, Phys. Rev. Lett. 74, 650 (1995). 17. V. Kukulin, proceedings of this conference; A. Faessler et al, Dec. 1999, nucl-th/9912074. 18. T.-S. Cheng and L.S. Kisslinger, Phys. Rev. 35, 1432 (1987). 19. H. Dijk and B.L.G. Bakker, Nucl. Phys. A 494, 438 (1989). 20. L.C. Alexa et al, Phys. Rev. Lett. 82, 1374 (1999). 21. D. Abbott et al, Eur. Phys. J. A 7, 421 (2000). 22. D. Abbott et al, Phys. Rev. Lett. 84, 5053 (2000). 23. D. Abbott et al, Phys. Rev. Lett. 82, 1379 (1999). 24. S.J. Brodsky and G.R. Farrar, Phys. Rev. Lett. 31, 1153 (1973). 25. C. Bochna et al, Phys. Rev. Lett. 8 1 , 4576 (1998).
OZI RULE VIOLATION IN np ANNIHILATIONS IN FLIGHT S. M A R C E L L O INFN Sezione di Torino, via P. Giuria 1, 10125 Torino, ITALY E-mail: [email protected] Antineutrons at LEAR have been shown to be very useful to study the NN annihilation. A sound result on annihilation dynamics concerns the strong deviation from Okubo-Zweig-Iizuka rule, which has been observed comparing the channels np —> 4>w+ and np —> u>ir+. These features could be explained with the presence of a ss quark component in the nucleon wave function. Results about the isoscalar singlet/octet mesons in the vectorial and pseudoscalar sectors are reported.
1
Introduction
In the last years the analysis of LEAR data on NN annihilation into channels containing a -meson 1'2>3'4 has shown a significant deviation (between 30 -f- 60 times) from the expected value of the Okubo-Zweig-Iizuka (OZI) rule 5 , strongly dependent on the channel and on the quantum numbers of the initial states. This semi-phenomenological rule had been confirmed in many experiments at different projectile energies. In fact for itN, NN and pp interactions the violation for vector mesons is not more than 10%. In fig. 1 the cross section ratio, ~R(4>X/LOX), for NN annihilations is shown as a function of the mass of the X particle recoiling against <$> and ui, the expected values by the OZI rule are close to R = 0. The OZI rule in its different formulations states the suppression of reactions with disconnected quark lines. If the <\> meson were a pure ss state, it coulnd't be created with hadrons composed only by u and d quarks. However since real 4> has a small admixture of light quarks in its wave function, it can be still created. This admixture can be parametrised by means of the difference between the physical and the ideal mixing angles: 5 = 6y — Oid = 3.2°. Then the OZI rule may be written in terms of 8 as T(A + B^C + (p)_ Z + tan5 T(A + B -^C + UJ) ~~1ZtanS'
^'
if Z = 0 the rule is satisfied: a (A + B -> C + ) 2 r R = -)_—ZL = tan2 6, a(A + B -> C + UJ) RQZI = 0.003 is expected for vector mesons. 236
(2) w
237
-^350
300
250
200
100
50
:=*i: 0
200
400
600
+" 800
1000 U (MeV)
Figure 1: The R(4>X/uiX) x 10 3 ratio as a function of the mass of the X particle.
A possible interpratation of the observed violation in NN annihilations was put forward by Ellis et al. 6 , assuming that the nucleon wave function contains a negatively polarised ss pair, the "strangeonia", even at low energies and in well defined quantum numbers. Two possible mechanisms could describe the abundant production observed in NN annihilations: the shake-out and the rearrangement ones. For the first mechanism the same deviation from OZI rule for production is expected in any channel, that is not experimentally verified. On the other hand an enhancement of 0 production at increasing fraction of S-wave annihilation is predicted if the rearrangement mechanism were dominant. An alternative approach 7 , based on meson exchange description, which has been successfully used at low energies, takes into account two or three step rescattering diagrams, where K*K is first created. This model is able to reproduce the violation within a factor 2 of accuracy, but it requires a suppression of K * K from P-wave.
2
Experimental results in np annihilations with OBELIX
The OBELIX experiment 8 performed measurements of NN annihilations at different initial states. Indeed two main features distinguished it among LEAR experiments. Firstly the unique facility producing n beam 9 of very good quality, with many advantages such as the possibility to fix the isospin 1 = 1 by studying the np system 10 . Secondly the possibility of using hydrogen targets
238
with different conditions of pressure and temperature, which allowed to select the angular momentum of the initial state. The annihilation cross sections of np —> 4>TT+ and Tip —> u)it+ were measured 4 with n in flight with momentum between 50 4- 405 MeV/c. The allowed initial states for these two channels are 3 Si a n d 1 ^ . It was found that the d>rr+ cross section follows a trend as a function of momentum, which is the same of the S-wave annihilation fraction predicted by Dover-Richard's model 11 , as shown in fig. 2(a). Besides for the u!ir+ channel an almost flat trend has been found(fig. 2(b)), showing that P-wave contribution is still present. The fact .0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 z 0 -
i'X
a)
X Hi. , , 0.2 0.4 p(n) (GeV/c)
0.2 0.4 p(n) [GeV/c]
0.2 0.4 p(n)(GeV/c)
Figure 2: np —• (j>ir+ (a), unr+ (b) cross sections and R(7T+/(xi7r+) (c) vs n momentum.
that (f>-n+ production is coming only from S-wave is also confirmed in the 0 decay angular distribution where, fitting the data at different momentum ranges, it is clear that P-wave is suppressed by a factor 20 at least. This anambigously indicates the presence of a dynamical selection rule for this channel. On the contrary in the omega decay angular distribution at higher momenta the Swave component doesn't exceed 60%. The behaviour of the ratio of the two cross sections, R(4>ir+/ujTr+), is shown in fig. 2(c), as a function of n momentum. The R value at lowest momentum is in agreement with that one found in pp annihilations 2 in liquid H2: R = 0.112 ± 0.007, confirming a deviation from Rozi of a factor 36 for pure S wave. Such a scaling of <j> production as spin triplet S-wave is the same predicted by the intrinsic strangeness model 6 , whether the rearrangement mechanism were the most effective one. On the other hand to test the validity of the alternative interpratation by means of KK* scattering, OBELIX performed the measurement of the cross section np —> K°*K+ as a function of the momentum, using the same n data set. The results exclude the main hypothesis of the model about a suppression of P-wave annihilation fraction. Moving to the pseudoscalar mesons, rf and 77 also contain a sizeable ss component in their wave function, then the existence of "strangeonia" in the
239 nucleon could give important effects, especially when the annihilation is from spin singlet s t a t e s 6 . T h e annihilation cross sections of the channels np —> r\ ir+ and 7771-+, which have 3 Po and 3 P 2 initial states, have been m e a s u r e d 1 2 . For the first channel a few statistics was available therefore the cross section, a = (0.128 ± 0.043) mb, was determined for the full range of momenta. Besides the channel TJTT+ showed a nice trend as P - w a v e as a function of the m o m e n t u m . T h e R(r]lir+ /nir+) ratio t u r n e d out to be 0.63±0.16. Then from eq. 2 is possible to determine the pseudoscalar mixing angle: 9ps — (—17.59 ± 3.39)°. It was found \Z\ < 0.083, close to zero as required by OZI rule. Therefore n' and n productions don't claim for an extra content of ss in the nucleon to be justified. T h a t fact is not in disagreement with the intrinsic strangeness model, it simply means "strangeonia" does not have 0 _ + q u a n t u m numbers. Conclusions For the first time an extensive study of specific channels in np annihilations has been done using the unique n beam facility of OBELIX. One of the most striking result concerns the dramatic deviation from OZI rule observed measuring the np —> 4>TT+ and np —-> um+ annihilation cross sections. A dynamical selection rule has been found in the <j> production, which scales as 3 5 i partial wave as a function of n momentum. Such a rule should be explained by the presence of a polarised ss component in the nucleon wave function. No violation has been observed for pseudoscalar mesons measuring the np —> 7] 7r+ and np —> «7r + annihilation cross sections. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.
V. G. Ableev et al, Phys. Lett. B 3 3 4 , 237 (1994). C. Amsler et at, Phys. Lett. B 3 4 6 , 363 (1995). V. G. Ableev et al., Nucl. Phys. A 594, 375 (1995). A. Filippi et al, Nucl. Phys. A 6 5 5 , 453 (1999). S. Okubo, Phys. Lett. B 5, 165 (1963), I. Iizuka, Prog. Theor. Phys. Suppl. 3 7 - 3 8 , 21 (1966). J. Ellis et al., Phys. Lett. B 3 5 3 , 319 (1995), J. Ellis et al., h e p - p h / 9 9 0 9 2 3 5 (1999). M. P. Locher et al., Z. Phys. A 347, 281 (1994). A. Adamo et al., Sov. J. Nucl. Phys. 55, 1732 (1992). M. Agnello et al, Nucl. Instrum. Methods A 3 9 9 , 11 (1997). S. Marcello, Nucl. Phys. A 6 5 5 , 107c (1999). C. B. Dover et al, Prog. Part. Nucl. Phys. 29, 87 (1992). A. Filippi et al., Phys. Lett. B 4 7 1 , 263 (1999).
P A R I T Y VIOLATING ELECTRON SCATTERING B. MOSCONI AND P. RICCI Dipartimento di Fisica, Universita di Firenze I.N.F.N., Sezione di Firenze Largo E. Fermi 2, 1-50125 Firenze, ITALY E-mail: [email protected] , [email protected] We study the parity-violating (PV) components of the target asymmetry A. in elastic electron-proton scattering discussing their sensitivity to the strangeness proton form factors. In particular, we show that the component of A. along the momentum transfer and the transverse component could be used for constraining the strangeness magnetic moment and the strangeness radius, respectively. Moreover, both components could give experimental information on the strangeness axial charge.
1
Introduction
Parity violating (PV) electron scattering can provide very interesting information on the electroweak structure of the nucleon, and, in particular it could shed light on the possible strange quark contributions to the nucleon properties. A first measurement of the PV beam asymmetry {ALR) in e— p elastic scattering was performed at Bates/MIT Laboratory by the SAMPLE Collaboration 1 giving the first experimental determination of the proton strangeness magnetic form factor at Q2 = 0.1(GeV/c)2 (/zs = 0.23±0.37±0.15±0.19/MBecause of the difficulties inherent in the PV electron scattering experiment an independent determination of fis, of the strangeness radius r2 and of other strangeness properties of the proton could be extremely useful. In this contribution we report on the results of our study 2 on the asymmetry A of the elastic e — p scattering cross section (in the low Q2 range) arising from the polarization of the proton target. In principle, this asymmetry is even more versatile than ALR for disentangling the different weak form factors because the polarization of the proton target can be freely chosen whereas the electron beam can be polarized only along the beam momentum.
2
Results
The only nonzero components of the target asymmetry are those in the scattering plane, i.e. the transverse (Ax) and the longitudinal (Az) ones (we 240
241
10 9 8 7 — 6 5 z 4 3 2 1 0 -- , I , 20
E CL CL (f) CD 'i_ •+->
E E <
^-'
IAu,l ' A, : ^
< :
y
A T ^ ,
,
I
I
40
I
,
I
,
60
,
,
I
, , ! , , , ! ,
,
80
, ! , , , !
100 120 140
160
tf.(deg) Figure 1. Angular distribution of the target asymmetry Ax{fte') (full line), A ^ i V ) (dashed line) and of the modulus of the helicity asymmetry ALR{^C') (dotted line) at Q 2 =0.1(GeV/c) 2 , with n„ = 0.23 fiN 1 and r 2 = 0.16 fm 2 3 .
assume the z-axis along the momentum transfer): gev^(l
Ax =V2 9eff
+ r)e(l+e)GAGE
+ gA
(GEGM
+ GMGE) s/re{\ - e)
eGl + rGl (1) e
Az = 2 geff
g v^T(l
+ T)GAGM
+
e
g AGMGMTVT^
sGl + rGl 2
where r - Q /AM
2
, geff
= Q2G/(8V2-Ka)
stant and a the fine-structure constant), e = GE,M,A
(G being the weak Fermi con1 + 2(1 + r) tan 2 (i? e -/2)
;
are the nucleon weak electric, magnetic and axial form factors;
242
gA = 1 , gv = (—1 + 4 sin2 $w) are the neutral vector and axial-vector electron couplings in the Standard Model, dw being the Weinberg angle. In Fig.l
a
«r o
0.8
0.6 0.4 0.2 r \
I^T'I
0
i i i i i i i i i i i i i i i i i i t i i i i i i i i i i i i i i i i i i i i i i i i i i i i
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Q 2 [(GeV/c) 2 ] Figure 2. Dependence of the target asymmetry -AZ(Q2) on the strangeness magnetic moment ns, at tfe/ = 170° and for r\ = 0.16 fm 2 3 . The solid line is for fj,s = - 0 . 7 5 ixN 5 , the dashed line for fi3 = 0.40 fix 6 , the dotted line for /j,s=0 and the dot-dashed line for Us = 0.23 [iN 1.
it is shown the angular distribution of Ax, Az and of the modulus of ALR for <32=0.1 (GeV/c) 2 , calculated with Jaffe's value 3 r 2 =0.16 fm2 and the central experimental value l fis = 0.23 ^ w Ax and Az show a remarkably different angular dependence as i?e' increases: Ax —> 0 at backward angles while Az reaches its maximum. Az does not depend on GE (and then on G^') and its dependence on GA is lowered with respect to that on GM because gv <£. geA, in particular at backward angles (e -> 0). In principle, a measurement of GM in the target asymmetry is more convenient than in the helicity asymmetry making Az an useful quantity for a determination of fis complementary to
243
o o
0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Q 2 [(GeV/c) 2 ] Figure 3. Dependence of the target asymmetry AX{Q2) on the strangeness radius r 2 , at 0 e / = 170° and for /J,S = 0.23 /x w *. The solid line is for r 2 =-0.32 fm 2 7 , the dashed line f o r r ? = 0 . 2 1 fm 2 6 , the dotted line for r 2 = 0 and the dot-dashed line for r?=0.16 fm 2 3 .
that coming from ALR- In Fig.2 it is shown the effect on Az of variations in ns at i9 e '=170° (r 2 =0.16 fm 2 ). On the contrary, Ax could be an useful quantity to constrain r 2 as shown in Fig.3 where it is plotted for several values of r 2 (/is = 0.23/x;v) and for backward scattered electrons (i?e» = 170°) in order to minimize the impact of GA- Finally, as it is easily seen from Eq.l, the terms containing GE,GM are suppressed in Ax and Az at forward angle (e —)• 1) making substantial the impact of GA- Then, a determination of GA (and of 9A )i alternative to that deriving from vfv scattering experiments 4 could be carried out in this kind of PV electron scattering experiments.
244
3
Conclusions
The asymmetry A of the elastic e — p scattering cross section arising from the polarization of the proton target may be a possible PV observable for an experimental determination of the proton weak form factors. The most convenient decomposition of A is obtained considering the proton polarization along and perpendicular to the momentum transfer. The longitudinal asymmetry Az is independent of GE allowing an experimental determination of the proton strangeness magnetic moment JJLS . The transverse asymmetry Ax is rather sensitive to the proton strangeness radius r2s in the case of backward detected electrons. A peculiarity of Ax and Az with respect to ALR, is that their dependence on GA can be enhanced over that on GE,GM- In fact, in the strict forward scattering (i?e/ = 0°) Ax and Az are determined by GA only. References 1. 2. 3. 4. 5. 6.
B. Mueller et al., Phys. Rev. Lett. 78, 3824 (1997). M. Moscani, B. Mosconi and P. Ricci, Phys. Rev. C 59, 2844 (1999). R.L. Jaffe, Phys. Lett. B 229, 275 (1989). L.A. Ahrens et al., Phys. Rev. D 35, 785 (1987). D.B. Leinweber, Phys. Rev. D 53, 5115 (1996). H.-W.Hammer, Ulf-G.Meissner and D.Drechsel, Phys.Lett. B 367,23 (1996). 7. H-C. Kim, T. Watabe and K. Goeke, preprint RUB-TPII-11/95 (unpublished).
Nuclear Astrophysics
NUCLEOSYNTHESIS IN SUPERNOVAE A N D STAR MERGERS
NEUTRON
FRIEDRICH-K. THIELEMANN Department
of Physics
& Astronomy, Univ. of Basel, Klingelbergstrasse CH-4056 Basel, Switzerland E-mail: [email protected]
82,
Astrophysical nucleosynthesis sites are the big bang and stellar objects. Stars contribute to galactic nucleosynthesis via hydrostatic burning phases in stellar evolution and explosive stellar events. Here we concentrate on type II supernova explosions - SNe II, the endpoints in the evolution of massive stars, and some events in binary stellar systems, e.g. type la supernovae - SNe la, and binary neutron star mergers. Emphasis is given to discuss the major nuclear physics issues involved.
1
Type II Supernovae
Stars with masses M > 8 M Q develop an onion-like composition structure, after passing through all hydrostatic burning stages, and produce a collapsing core at the end of their evolution, which proceeds to nuclear densities 27>44.3.7.42_ The size of the homologous core, turning into nuclear matter during bounce 33 , is dependent on the amount of prior electron captures on pf-shell nuclei 20 . The total energy released, 2-3xl0 53 erg, equals the gravitational binding energy of a neutron star. Because neutrinos are the particles with the longest mean free path, they are able to carry away that energy in the fastest fashion as seen for SN1987A in the Kamiokande, 1MB and Baksan experiments. The apparently most promising mechanism for supernova explosions is based on neutrino heating beyond the hot proto-neutron star via the dominant processes ve + n —> p + e~ and Pe + p —>• n + e + with a (hopefully) about 1% efficiency in energy deposition 12 - 22 . The neutrino heating efficiency depends on the neutrino luminosity, which in turn is affected by neutrino opacities 2,32 . The explosion via neutrino heating is delayed after core collapse for a timescale of seconds or less. Aspects of the explosion mechanism are still uncertain and depend on Fe-cores from stellar evolution, electron capture rates of pf-shell nuclei, the supranuclear equation of state, as well as the details of neutrino transport 22 and Newtonian vs. general relativistic calculations 23,18 rphe observational fact that many core collapse supernovae show polarized light emission hints also towards a nonspherical explosion mechanism 14
246
247
1.65
1.7
1.75
1.8
1.85
1.9
1.95
2
2.05
2.1
M/X,
Figure 1. Isotopic composition for a core collapse supernova from a 2OM0 progenitor star with a 6M© He-core and a net explosion energy of 10 5 1 erg, remaining in kinetic energy of the ejecta. The exact mass cut in M(r) between neutron star and ejecta depends on the details of the delayed explosion mechanism.
1.1
Composition of Ejecta
As long as uncertainties are still existing in self-consistent models, but typical kinetic energies of 10 51 erg are observed in supernova remnants, light curve as well as explosive nucleosynthesis calculations have been performed by introducing a shock of appropriate energy in the pre-collapse stellar model 44,40,10,24,42 g u c n induced calculations, making use of strong and weak reaction nuclear rates 13 . 34 . 35 > 17 j still lack self-consistency and cannot predict the ejected 56Ni-masses from the innermost explosive Si-burning layers, powering supernova light curves by the decay chain 56 Ni- 56 Co- 56 Fe, due to the missing knowledge of the mass cut between the neutron star and the supernova ejecta. This relates also to the neutron-richness (or Ye = ) of the ejected composition and the weak interactions during stellar evolution and the explosion. Fig. 1 shows the composition after explosive processing 8 due to the shock wave causing a supernova explosion. The outer ejected layers are unprocessed by the explosion and contain results of prior H-, He-, C-, and Ne-burning in
248
stellar evolution. The interior parts of SNe II contain products of explosive Si, O, and Ne burning. In the inner ejecta, which experience explosive Siburning, Ye changes from 0.4989 to 0.494. The Ye originates from the preexplosive hydrostatic fuel in these layers. Huge changes occur in the Fe-group composition for mass zones below M(r)=1.63M Q . Then the abundances of 58 Ni and 56 Ni become comparable. All neutron-rich isotopes increase ( 57 Ni, 58 Ni, 59 Cu, 61 Zn, and 62 Zn), the even-mass isotopes ( 58 Ni and 62 Zn) show the strongest effect. One can also recognize the increase of 40 Ca, 4 4 Ti, 48 Cr, and 52 Fe for the inner high entropy zones, but a reduction of these N=Z nuclei in the more neutron-rich layers. More details can be found in extended discussions 40 > 41 ' 24 . A correct prediction of the amount of Fe-group nuclei ejected (which includes also one of the so-called alpha elements, i.e. Ti) and their relative composition depends directly on the explosion mechanism and the size of the collapsing Fe-core. Three types of uncertainties are inherent in the Fe-group ejecta, related to (i) the total amount of Fe(group) nuclei ejected and the mass cut between neutron star and ejecta, mostly measured by 56 Ni decaying to 56 Fe, (ii) the total explosion energy which influences the entropy of the ejecta and with it the amont of radioactive 44 Ti as well as 48 Cr, the latter decaying later to 4 8 Ti and being responsible for elemental Ti, and (iii) finally the neutron richness or Ye=< Z/A > of the ejecta, dependent on stellar structure, electron captures and neutrino interactions. Ye influences strongly the ratios of isotopes 57/56 in Ni(Co,Fe) and the overall elemental Ni/Fe ratio. The latter being dominated by 58 Ni and 56 Fe. The pending understanding of the explosion mechanism also affects possible r-process yields for SNe II. Some recent calculations seemed to be able to reproduce the solar r-process abundances well in the high entropy neutrino wind, emitted from the hot protoneutron star after the SN II explosion 39 45 ' . However, present-day supernova models have difficulties to reproduce the entropies required for such abundance calculations and in addition face problems in abundance features in the mass range 80-120 5 . The inclusion of non-standard neutrino properties may perhaps achieve low enough Ye 's for intermediate entropies to correct for such unwanted features 21 . However, recent observations shed some doubts on the supernova origin. On average SNe II produce Fe to intermediate mass elements in ratios within a factor of 3 of solar. If they would also be responsible for the r-process, the same limits should apply. But the observed bulk r-process/Fe ratios vary widely in low metallicity stars by more than a factor of 100 38 .
249 2
Type la Super novae
There are strong observational and theoretical indications that SNe la are thermonuclear explosions of accreting white dwarfs in binary stellar systems 9,30,29,19 High rates of H-accretion cause high temperatures at the base of the accreted matter and lead to quasi-stable H-burning and subsequent Heburning in shells surrounding the white dwarf, possibly related to supersoft X-ray sources. This increases the mass of the white dwarf consisting of C and 0 towards the maximum stable Chandrasekhar mass and leads to contraction. 2.1
Ignition and Burning Front Propagation
Contraction causes carbon ignition in the central region and a thermonuclear runway with a complete explosive disruption of the white dwarf 28 ' 45 . High accretion rates cause a higher central temperature and pressure, favoring lower ignition densities. A flame front then propagates at a subsonic speed as a deflagration wave due to heat transport across the front 25 . Here the most uncertain quantity is the flame speed which depends on the development of instabilities of various scales at the flame front. Multi-dimensional hydro simulations of the flame propagation have suggested that a carbon deflagration wave might propagate at a speed v^e{ as slow as a few percent of the sound speed vs in the central region of the white dwarf. The nucleosynthesis consequences of such slow flame speeds witness the actual burning front velocities and can thus serve as a constraint. Electron capture affects the central electron fraction Ye, which determines the composition of the ejecta from such explosions. The amount of electron capture depends on (i) the electron capture rates of pf-shell nuclei, (ii) v^ef, influencing the time duration of matter at high temperatures (and with it the availability of free protons for electron captures, and (iii) the central density of the white dwarf pign (increasing the electron chemical potential i.e. their Fermi energy) 11>1>17. After an initial deflagration in the central layers, the deflagration can turn into a detonation (supersonic burning front) at lower densities 26 . The transition from a deflagration to a detonation (delayed detonation model) leads to a change in the ratios of Si-burning subcategories with varying entropies. This also leaves an imprint on the Fe-group composition. Nucleosynthesis constraints can help to find the "average" SN la conditions responsible for their contribution to galactic evolution, i.e. especially the Fe-group composition. SNe la contribute essentially no elements lighter than Al, about 1/3 of the elements from Si to Ca, and the dominant amount of Fe group nuclei (Ti to Ni). In addition, the average Fe-group yields of SNe II and SNe la differ.
250 .51 .50 .49 .48 ^".47 .46 .45 .44 .43 .00
.05
.10 M(r)/MQ
.15
.20
Figure 2. Ye after freeze-out of nuclear reactions measures the electron captures on free protons and nuclei. Small burning front velocities lead to steep Ye-gradients which flatten with increasing velocities (see the series of models CS15, CS30, and CS50 or WS15, WS30, and W7). Lower central ignition densities shift the curves up (C vs. W), but the gradient is the same for the same propagation speed. Only when the Ye from electron captures is smaller than for stable Fe-group nuclei, subsequent /3 _ -decays will reverse this effect (WSL and WLAM).
2.2
Nucleosynthesis Details
Fig. 2 shows the influence of central ignition densities pign 1.37 (C) and 2.12x 109 g c m - 3 (W) at the onset of thermonuclear runaway and slow (S) deflagration speeds of vde{/vs = 0.015 (WS15, CS15), 0.03 (WS30,CS30) or 0.05 (CS50) on the resulting Ye due to the different amount of electron capture. Ye values of 0.47-0.485 lead to dominant abundances of 54 Fe and 58 Ni, values between 0.46 and 0.47 produce dominantly 56 Fe, values in the range of 0.45 and below are responsible for 58 Fe, 54 Cr, 5 0 Ti, 64 Ni, and values below 0.43-0.42 are responsible for 48 Ca. The intermediate Ye-values 0.47-0.485 exist in all cases, but the masses encountered which experience these conditions depend on the y e -gradient and thus Vdef- Whether the lower vales with Ye<0.45 are attained, depends on the central ignition density Pign. Therefore, 54 Fe and 58 Ni are indicators of vdef while 58 Fe, 54 Cr, 5 0 Ti, 64 Ni, and 4 8 Ca are a measure of pign. The conclusions to be drawn from these results are that: (i) Vdef in the range 1.5-3% of the sound speed is preferred (cases 15 and 30 over 50) n , and (ii) the change in pf-shell electron capture rates 1T made it possible to have ignition densities as high as pign=2 x 109 g c m - 3 without
251
distroying the agreement with solar abundances of very neutron-rich species ii,i
If a deflagration turns into a detonation, the transition density ptr affects the total amount of 56 Ni, the intermediate mass elements Si-Ca, and the ratios of different explosive Si-burning regimes, being responsible for elements like 55Mn a n ( j 52£ r ^ s g N e j a a n ( j SNe II have a different abundance pattern for these nuclei, observational constraints can help to limit such quantities as ptr as long as self-consistent 3D modeling does not give clear answeres, yet u . 3
The r-Process
Site-independent classical analyses, based on neutron number density nn, temperature T, and duration time r, led to the conclusion that the r-process experienced a fast drop from (n, 7) — (7, n) chemical equilibrium in each isotopic chain. The combination of nn and T is related to an r-process path in the nuclear chart along nuclei with a neutron separation energy Sn(nn,T). Thus, the r-process and its abundance features probe nuclear structure far from stability via mass properties and the beta-decay half-lives along contour lines of constant Sn 15 . This gives some indication for the need of quenching of nuclear shell effects far from stability 31>16. The possible role of fission will be addressed later. A continuous superposition of components with neutron separation energies in the range 4-1 MeV on timescales of 1 - 2.5 s, provides a good overall fit 4 . For the heavier elements beyond A=130 this reduces to about 5 n = 3 - l MeV. These are predominantly nuclei not accessible in laboratory experiments to date. Exceptions exist in the A = 80 and 130 peaks and continuous efforts are underway to extend experimental information in these regions of the closed shells N=50 and 82 with radioactive ion beam facilities 16 . A recent detailed analysis of the A=206-209 abundance contributions to Pb and Bi isotopes from alpha-decay chains of heavier nuclei permitted for the first time also to predict abundances of nuclei as heavy as Th with reasonable accuracy 4 . A different question is related to the actual astrophysical realization of such conditions. The observations of stellar spectra of low metallicity stars, stemming from the very early phases of galactic evolution, are all consistent with a solar r-abundance pattern for elements heavier than Ba, and the relative abundances among heavy elements do apparently not show any time evolution 4 38 ' . This suggests that all contributing events produce the same relative rprocess abundances for the heavy masses, although a single astrophysical site will still have varying conditions in different ejected mass zones, leading to a superposition of individual components.
252
However, from meteoritic abundances and observations in low metallicity stars we also know by now that at least two r-process sources have to contribute to the solar r-process abundances 4 3 . The observed non-solar r-process pattern for e.g. Ag, I, and Pd in some objects indicate the need for a second r-process component in the nuclear mass range A«80-120, in addition to the main process which provides a solar r-process pattern beyond Ba 38 . It is not exactly clear which of the two processes is related to SNe II and which one is related to possible other sources.
3.1
Possible Stellar r-Process Sites
An r-process requires 10 to 150 neutrons per seed nucleus (in the Fe-peak or somewhat beyond) which have to be available to form all heavier r-process nuclei by neutron capture. For a composition of Fe-group nuclei and free neutrons that translates into a Ye ==0.12-0.3. Such a high neutron excess is only possible for high densities in neutron stars under beta equilibrium (e~~ + p O n + v, fie + Up = /z n ), based on the high electron Fermi energies which are comparable to the neutron-proton mass difference. Neutron star mergers which eject such matter are a possible (low entropy) site and have been debated in the past. Recent calculations show that on average about 1 0 - 2 M o of neutron-rich matter are ejected 36>37. This amount depends on the central high density equation of state 33 encountered in these events. Present calculations show densities up to four times nuclear matter density and temperatures of up to 50 MeV 36>37. First nucleosynthesis calculations with assumpotions on Ye predict a solar-type r-process pattern for nuclei beyond A=130 6 . The smaller masses are depleted due to a long duration r-process with a large neutron supply in such neutron-rich matter, which also leads to fission cycling. This seems (accidentally?) in accordance with the main observed r-process component. Fig. 3 shows the abundance pattern expected from such an event Given the frequency (10~ 5 y _ 1 per galaxy) and amount of ejected matter, this component alone could be responsible for the heavy solar r-process pattern and also explain the large scatter of r/Fe elements found in low metallicity stars. Neutron star - black hole mergers have not yet been analyzed with the same accuracy, but bear similar options. Another option is an extremely alpha-rich (i.e. high entropy) freeze-out in complete Si-burning with moderate y e >0.40, which however faces some of the problems already mentioned in the section on SNe II.
253
Figure 3. Calculated r-process distribution for different Ye's. In general one obtains useful contributions for 0.08
4
Conclusions and Outlook
This overview concentrated on nucleosynthesis processes in supernovae (type II and la) and analyzed the options and sites of r-process nucleosynthesis. These are the major contributions to galactic evolution. Nucleosynthesis calculations have a right on their own to predict abundance patterns for many stellar events, but they can also serve as a tool to test the correctness of model descriptions, either in comparison to direct observations or indirect information from galactic evolution. We tried to show especially for SNe la and II, how specific isotopic abundances can test ignition densities, burning front velocities or explosion energies, entropies, and temperatures. These are the astrophysical model constraints. But it was also clearly demonstrated how advances in nuclear physics (weak interaction rates in the pf-shell and decay properties far from stability, nuclear structure far from stability, fission properties and yields, the nuclear equation of state at high densities and
254
temperatures) are essential for the outcome and correct modeling of these events. References 1. Brachwitz, F. et al. 2000, ApJ, 536, 934 2. Burrows, A., Sawyer R.F. 1999, Phys. Rev C59, 510 3. Chieffi, A., Limongi, M., Straniero, 0 . 1998, Ap. J. 502, 737 4. Cowan, J.J. et al. 1999, ApJ 521, 194 5. Freiburghaus, C. et al. 1999, ApJ 516, 381 6. Freiburghaus, C , Rosswog, S., Thielemann, F.-K. 1999, ApJ 525, L121 7. Heger, A., Langer, N., Woosley, S.E. 2000, ApJ, 528, 368 8. Hix, W.R., Thielemann, F.-K. 1999, J. Comp. Appl. Math. 109, 321 9. Hoflich, P., Khokhlov, A. 1996, Ap. J., 457, 500 10. Hoffman, R.D. et al. 1999, ApJ 521, 735 11. Iwamoto, K. et al. 1999, ApJS 125, 439 12. Janka, H.-T., Miiller, E. 1996, A&A, 306, 167 13. Kappeler, F., Thielemann, F.-K., Wiescher, M. 1998, Ann. Rev. Nucl. Part. Sci. 48, 175 14. Khokhlov, A.M. et al. 1999, ApJ 524, L107 15. Kratz, K.-L. et al. 1993, Ap. J. 402, 216 16. Kratz, K.-L., Pfeiffer, B., Thielemann, F.-K., Walters, W.B. 2000, Hyperfine Interactions, in press, astro-ph/9907071 17. Langanke, K., Martinez-Pinedo, G. 2000, Nucl. Phys. A673, 481 18. Liebendorfer, M. et al. 2000, astro-ph/0006418, subm. to PRL 19. Livio, M. 2000 in Type la Supernovae: Theory and Cosmology, Cambridge Univ. Press, in press 20. Martinez-Pinedo, G. et al. 2000, ApJS, 126, 493 21. McLaughlin, G.C. et al. 1999, Phys. Rev. C59, 2873 22. Mezzacappa, A., Messer, O.E.B. 1999, JCAM 109, 281 23. Mezzacappa, A. et al. 2000, astro-ph/0004059, PRL, in press 24. Nakamura, T. et al. 1999, ApJ 517, 193 25. Niemeyer, J.C., Bushe, W. K., Ruetsch, G.R. 1999, ApJ 524, 290 26. Niemeyer, J.C. 1999, ApJ 523, L57 27. Nomoto, K., Hashimoto, M. 1988, Phys. Rep. 163, 28. Nomoto, K., Thielemann, F.-K., Yokoi, K. 1984, Ap. J. 286, 644 29. Nomoto, K. et al. 2000, in Type la Supernovae: Theory and Cosmology, Cambridge Univ. Press, eds. J. Niemeyer & J.W. Truran, in press (astroph/9907386) 30. Nugent, P. et al. 1997, Ap. J. 485, 812
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31. Pfeiffer, B., Kratz, K.-L., Thielemann, F.-K. 1997, Z. Phys. A357, 235 32. Pons, J.A. et al. 1999, ApJ 513, 780 33. Prakash, M. et al. 1997, Phys. Rep. 280, 1 34. Rauscher, T., Thielemann, F.-K. 2000, ADNDT, 75, 1 35. Rauscher, T. et al. 2000, Nucl. Phys. A675, 695 36. Rosswog, S.K. et al. 1999, A & A 341, 499 37. Rosswog, S.K. et al. 2000, A&A, 360, 171 38. Sneden, C. et al. 2000, Ap. J. 533, 139 39. Takahashi, K., Witti, J., Janka, H.-T. 1994, A&A, 286, 857 40. Thielemann, F.-K., Nomoto, K., Hashimoto, M. 1996, ApJ 460, 408 41. Thielemann, F.-K. et al. 1998, in Nuclear and Particle Astrophysics, eds. J. Hirsch, D. Page, Cambridge Univ. Press, p. 27 42. Umeda, H., Nomoto, K., Nakamura, T. 2000, in The First Stars, Springer, eds. A. Weiss et al., in press (astro-ph/9912248) 43. Wasserburg, G., Busso, M., Gallino, R. 1996, Ap. J. 466, L109 44. Woosley, S.E., Weaver, T.A. 1995, ApJS 101, 181 45. Woosley, S.E. et al. 1994, ApJ 433, 229
Section I. Theoretical Aspects of Nuclear Astrophysics
STRANGE HADRONIC STELLAR MATTER WITHIN THE BRUECKNER-BETHE-GOLDSTONE THEORY
INFN
Departament
M. B A L D O , G. F . B U R G I O di Catania, 51 Corso Italia, 1-95129 Catania, Italy and ECT*, 286 Strada delle Tabarelle, 1-38050 Trento, Italy E-mail: [email protected], burgioQct.infn.it Sezione
H.-J. S C H U L Z E d'Estructura i Constituents de la Materia, Universitat Av. Diagonal 6\1, E-08028 Barcelona, Spain E-mail: [email protected]
de
Barcelona,
In the framework of the non-relativistic Brueckner-Bethe-Goldstone theory, we derive a microscopic equation of state for asymmetric and /J-stable matter containing E~ and A hyperons. We mainly study the effects of three-body forces (TBF's) among nucleons on the hyperon formation and the equation of state (EoS). We find that, when TBF's are included, the stellar core is almost equally populated by nucleons and hyperons. The resulting EoS, which turns out to be extremely soft, has been used in order to calculate the static structure of neutron stars. We obtain a value of the maximum mass of 1.26 solar masses (1 solar mass M0 ~ 1.99 • 1033
1
Neutron stars within the B B G approach
The nuclear matter equation of state (EoS) is the fundamental input for building models of neutron stars. These compact objects, among the densest in the universe, are indeed characterized by values of the density which span from the iron density at the surface up to eight-ten times normal nuclear matter density in the core. Therefore a detailed knowledge of the equation of state over a wide range of densities is required 1 . This is a very hard task from the theoretical point of view. In fact, whereas at densities close to the saturation value the matter consists mainly of nucleons and leptons, at higher densities several species of particles may appear due to the fast rise of the nucleon chemical potentials. In our work we perform microscopic calculations of the nuclear matter EoS containing fractions of A and E~ hyperons in the framework of the Brueckner-Hartree-Fock (BHF) scheme 2 . The BHF approximation, with the continuous choice for the single particle potential, reproduces closely the manybody calculations up to the three hole-line level. In this approach, the basic input is the two-body interaction. We chose the Paris and the Argonne vis potential for the nucleon-nucleon (NN) part, whereas the Nijmegen soft-core 257
258
model has been adopted for the nucleon-hyperon (NY) potential. No hyperonhyperon interaction is taken into account, since no robust experimental data are available yet. For more details, the reader is referred to ref. 3 and references therein. However, as commonly known, all many-body methods fail to reproduce the empirical nuclear matter saturation point po = 0.17 fm~3. This drawback is commonly corrected by introducing three-body forces (TBF's) among nucleons. In our approach we have included a contribution containing a long range two-pion exchange attractive part and an intermediate range repulsive p a r t 4 . This allows the correct reproduction of the saturation point. In figure 1 we show the chemical composition of /^-stable and asymmetric nuclear matter containing hyperons (panel (a)) and the corresponding equation of state (panel (b)). The shown calculations have been performed using the Paris potential. We observe that hyperon formation starts at densities p ~ 2 — 3 times normal nuclear matter density. The S~ baryon appears earlier than the A, in spite of its larger mass, because of the negative charge. The appearance of strange particles has two main consequences, i) an almost equal percentage of nucleons and hyperons are present in the stellar core at high densities and ii) a strong deleptonization of matter, since it is energetically convenient to mantain charge neutrality through hyperon formation than /3-decay. The equation of state is displayed in panel (b). The dotted line represents the case when only nucleons and leptons are present in stellar matter, whereas the solid line shows the case when hyperons are included as well. In the latter case the equation of state gets very soft, since the kinetic energy of the already present baryonic species is converted into masses of the new particles, thus lowering the total pressure. This fact has relevant consequences for the structure of the neutron stars. 2
E q u i l i b r i u m configurations of n e u t r o n s t a r s
We assume that a star is a spherically symmetric distribution of mass in hydrostatic equilibrium. The equilibrium configurations are obtained by solving the Tolman-Oppenheimer-Volkoff (TOV) equations 1 for the pressure P and the enclosed mass m, 1 J_ EM.]
dP(r) _ dr
Gm(r)p(r) r2
\l _i_ 47rr 3 P(r)l
[x + P(r) J [l + m(r)_\_ i _ 2Gm(r) r
^
= 4*r'p(r) ,
(2)
being G the gravitational constant (we assume c = 1). Starting with a central mass density p(r = 0) = pc, we integrate out until the pressure on the surface
259 500 10"
n \ ^ ^
(a): 400
P
3 10"
/ 10 '
300 -
e
200 -
/V A \ 0
0.2
0.4
\ .
0.6_
nB (frrf3)
0.S
1.2
Figure 1: In panel (a) we display the equilibrium composition of asymmetric and /3-stable nuclear matter containing E~ and A hyperons. In panel (b) the solid(dotted) line represents the EoS obtained in the case when nucleons plus hyperons (only nucleons) are present. 2.4
1
(b)
(a) -
2 -
\ 1.6
PSR1913+16
\ \
/
:
'
/ Q==QK / / . ^--"'
\
w If
•
if Jl
\ 0.4 10
11
12
"
fl / ^NY 11/
>
o
:
1/
-
NY~N[
-
/
/
03
§ 1.2
/
\
Radius R (km)
11 13
14
0.0
0.3
0.6
0.9
1.2
1.5
_3
Central density nc (fm )
Figure 2: In panel (a) the mass-radius relation is shown in the case of beta-stable matter with hyperons (solid line) and without hyperons (dashed line). The thick line represents the measured value of the pulsar PSR1913+16 mass. In panel (b) the mass is displayed vs. the central density. The dotted line represents the equilibrium configurations of neutron stars containing nucleons plus hyperons and rotating at the Kepler frequency QK •
260
equals the one corresponding to the density of iron. This gives the stellar radius R and the gravitational mass is then fR
MG = m(R)=4n
drr2p{r).
(3) Jo For the outer part of the neutron star we have used the equations of state by Feynman-Metropolis-Teller 5 and Baym-Pethick-Sutherland 6 , and for the medium-density regime we use the results of Negele and Vautherin 7 . For density p > 0.08 frn^3 we use the microscopic equations of state obtained in the BHF approximation described above. For comparison, we also perform calculations of neutron star structure for the case of asymmetric and /3-stable nucleonic matter. The results are plotted in Fig.2. We display the gravitational mass MQ (in units of the solar mass M0) as a function of the radius R (panel (a)) and central baryon density nc (panel (b)). We note that the inclusion of hyperons lowers the value of the maximum mass from about 2.1 M0 down to 1.26 M0. This value lies below the value of the best observed pulsar mass, PSR1916+13, which amounts to 1.44 solar masses. However the observational data can be fitted if rotations are included, see dotted line in panel (b). In this case only equilibrium configurations rotating at the Kepler frequency Q,K are shown. In conclusion, the main finding of our work is the surprisingly low value of the maximum mass of a neutron star, which hardly comprises the observational data. This fact indicates how sensitive the properties of the neutron stars are to the details of the interaction. In particular our result calls for the need of including realistic hyperon-hyperon interactions. References 1. S. L. Shapiro and S. A. Teukolsky, Black Holes, White Dwarfs and Neutron Stars (John Wiley & Sons, New York, 1983) 2. M. Baldo, Nuclear Methods and the Nuclear Equation of State (World Scientific, Singapore, 1999) 3. M. Baldo, G. F. Burgio, and H.-J. Schulze, Phys. Rev. C 6 1 , 055801-1 (2000). 4. M. Baldo, I. Bombaci, and G. F. Burgio, Astron. Astrophys. 328, 274 (1997). 5. R. Feynman, F. Metropolis, and E. Teller, Phys. Rev. C 75, 1561 (1949); 6. G. Baym, C. Pethick, and D. Sutherland, Astrophys. Journ. 170, 299 (1971). 7. J. W. Negele and D. Vautherin, Nucl. Phys. A 207, 298 (1973).
B U B B L E NUCLEI, N E U T R O N STARS A N D BILLIARDS
Department
of Physics,
Max-Planck-Institut
Institute
of Physics,
QUANTUM
Aurel B U L G A C University of Washington, Seattle WA 98195-1560, E-mail: [email protected] fur Kernphysik, Postfach GERMANY
10 39 80, 69029
Piotr M A G I E R S K I Warsaw University of Technology, PL-00662, Warsaw, POLAND E-mail: [email protected]
USA
Heidelberg,
ul. Koszykowa
75,
We briefly review the significance of quantum corrections in the total energy of systems with voids: bubble nuclei, atomic clusters and the inhomogeneous phase of neutron stars
It was suggested a long time ago t h a t very large nuclei might not undergo a Coulomb explosion if they acquire a new topology, t h a t of a bubble or a torus 1 . W h e n a void is formed, while the density and therefore the total volume is kept unchanged, the surface area of such a nucleus naturally increases and t h a t leads to an increased surface energy and less binding.. However, at the same time the average distance between protons increases as well and the total Coulomb energy then decreases. T h e balancing of these two types of energy and the fact t h a t configurations with larger binding energy t h a n the familiar compact geometries exists is the reason why bubble-like and torus-like nuclei could in principle be someday observed. It was realized however t h a t shell effects play a crucial role in stabilizing these new shapes 2 . During the last decade many experimentalists have tried to manufacture highly charged metallic clusters, but, again, Coulomb repulsion prevented their creation. T h e idea t h a t objects with a different topology, in particular bubble-like charged metallic clusters could be a possible route to create highly charged metallic clusters was recently put forward 3 , and again the stabilizing role of the shell corrections was noted as playing a decisive role. There was an aspect of bubble systems, which for mysterious reasons never caught the attention of previous authors: Where should one position a bubble inside a nucleus? Symmetry considerations seem to suggest t h a t a spherical bubble should be placed at the center of a spherical system. A closer look will show however t h a t there is something more t h a n mere symmetry and 261
262
that Coulomb energy plays perhaps the most important role in stabilizing the bubble position. It is relatively straight forward to show that if one were to displace a bubble from the center of a nucleus the Coulomb energy would increase. When considering Coulomb effects, one can think of a bubble as being a charged object, having the same charge density as the rest of the matter, but of opposite sign. One can then easily evaluate the Coulomb force acting on a bubble. Inside a spherical uniformly charged object the electric field is radial and can be easily be evaluated using Gauss law:
where po is the charge density. Thus the force acting on a bubble is simply the integral over the bubble "effective charge" times the electric field F. = - / Jbubble
d
a
r
4 ^ •J
=
_/iHR ]
(2)
«*
where V& is the volume of the bubble and R is the position vector of the bubble center, with respect to the nuclear center. When the suggestion was made to make cavities inside charged metallic clusters it became clear to us that the above argument is incomplete 4 ' 5 and there is no apparent physical candidate responsible for determining the optimal bubble position inside a homogeneous fermi system. As freshmen physics students know, there is no electric field inside a metal in the absence of electric currents. If in the case of nuclei one could invoke, either symmetry arguments (for some not totally clear reasons) or, better yet, the stabilizing role of Coulomb force, it was not obvious what made a bubble system stable in the case of a metal cluster. None of the "usual suspects" (volume, surface, curvature or Coulomb energies) seem to play any significant role and one might naturally expect that if there is something happening in a metal cluster, a similar mechanism should most likely be operative in a nucleus as well. (In metal clusters one has of course to deal with additional ionic degrees of freedom, however, many cluster properties are determined mostly by the electrons alone and the ions are merely spectators.) The solution to the above puzzle was rather simple, but at the same time to a large extent unexpected as well: the physics of a bubble is governed by pure quantum effects, known in nuclear physics as shell corrections and in quantum filed theory as Casimir energy 6 . Instead of presenting formulas and results of numerical calculations we shall limit ourselves here to a general discussion of some of the novel aspects of these systems and refer the interested readers to the available references. When one mentally starts pushing a bubble around inside a finite fermi system one obviously excites such a system, if initially the bubble was in its
263
optimal position and therefore the entire system in its ground state. Since the displacement of a bubble will affect many particles, bubble displacements are naturally collective excitations. Apart from collective pairing excitations, perhaps no other collective mode in a fermi system is purely quantum in nature. Since shell corrections effects scale with particle number as oc N1'6, see Ref. 7 , and other collective modes involve some degree of surface deformation, and therefore their effects scale with particle number as oc N2!3, one can expect that bubble displacements would correspond to perhaps the softest collective modes possible. Our vast experience seem at this point to lend support to the idea that symmetry should play a major role in determining the optimal position of a bubble, since shell correction effects are largest for spherical systems. To some extent this is true, see Ref. 5 , however with many provisos. Even if a system is "magic", once one would displace a bubble off center significantly, the potential energy surface becomes rather flat. One would also expect that the amplitude of the shell corrections will become smaller when the bubble is significantly off center, since classically the motion of a particle in the corresponding single-particle potential is chaotic to a large degree 8 and the single-particle spectrum is expected to have no large gaps. As our detailed numerical results show this expectation is hardly ever true. In all our numerical analyses so far we have used hard wall potentials (which thus partially explains the origin of the term quantum billiards in our title), for which there is significant evidence that they do reproduce the realistic spectra with sufficient accuracy for the purpose of computing the gross shell structure 9
. A particular feature of the shell correction energy evaluated for hard wall potential, and which we do not expect to survive entirely in a selfconsistent calculation, is particularly interesting however, as it underlines a general trend. We have observed in Ref. 5 , and later confirmed as a general feature in Refs. 10 , that the amplitude of the shell correction energy increases as the bubble approaches the boundary of the system. This is particularly puzzling, since the closer the bubble is to the system boundary the classical motion is more chaotic and one would naturally expect then the shell energy to decrease, but not to increase. Part of the explanation is that a particular periodic orbit becomes prominent and leads to a significant "scarring" of the single particle density of states. This is the orbit bouncing between the points of closest approach. The relative size of the bubble also plays a major role. If the fractional volume of the bubble is small, then the shell correction energy oscillates with a relatively small amplitude when compared with a bubble with a larger fractional volume. Sidestepping the question of bubble stability and of the energy cost of bubble formation, one can reasonably ask a number of quite relevant questions as well: "Why not have a system with two or more bubbles?" Neutron stars
264
have been predicted a long time ago to have a locally inhomogeneous phase, often referred to as "the pasta phase" n . Due to the same type of interplay between the surface and Coulomb energies, at depths of about 0.5 km below the surface of a neutron star and at densities just below nuclear saturation density a new phase is favored, where spherical and rod-like nuclei embedded in a neutron gas, plates, cylinders and bubbles exist. Almost all previous analyses of this phase have been performed in the liquid drop or Thomas-Fermi approximations. It was determined that on the way inside a neutron star, while the average density is increasing, there is a well defined sequence of phases: nuclei —• rods —• plates —> tubes —• bubbles —»• uniform matter. The energy of each of these phases is significantly below the energy of the uniform phase at the same average density, irrespective of nuclear model used l x . The energy differences between various phases even though are very small, of the order of keV's per fm3, are apparently independent of the model for the nuclear forces used. The various models for nuclear forces can lead to significant variations in the values of the interface surface tension. Shell correction energy on the other hand is known to be of geometric origin essentially. Since in infinite matter the presence of various inhomogeneities does not lead to the formation of discrete levels, one might call the corresponding energy correction for neutron matter the Casimir energy 6 . The inhomogeneous phase of a neutron star is basically nothing else but a Sinai billiard, a model which is widely popular in classical and quantum chaos studies. In a first approximation one can treat various objects in the inhomogeneous phase as spherical, cylindrical or plate like voids in a neutron gas. In order to better appreciate the nature of the problem we are addressing here, let us consider the following situation. Let us imagine that two spherical identical bubbles have been formed in an otherwise homogeneous neutron matter. For the sake of simplicity, we shall assume that the bubbles are completely hollow. We shall ignore for the time being the role of the Coulomb interactions, as their main contribution is to the smooth, liquid drop or Thomas-Fermi part of the total energy. Then one can ask the following apparently innocuous question: "What determines the most energetically favorable arrangement of the two bubbles?" According to a liquid drop model approach (completely neglecting for the moment the possible stabilizing role of the Coulomb forces) the energy of the system should be insensitive to the relative positioning of the two bubbles. Using Gutzwiller trace formula one can show that pure quantum effects lead to an approximate interaction energy of the following form
int
Ti2k2F R2 f cos[2kF(a - 2R)] ~~ 2m ira(a - 2R) { 2kF(a - 2R)
sm[2kF(a - 2R)] \ 4k2F(a - 2R)2 ' J
{
'
265 Bubble radius R = 10 fm
Figure 1: The interaction energy between two bubbles as a function of the distance between their "tips".
where R is the bubble radius, a is the distance between the bubble centers and kF is the Fermi wave length, see Fig. 1. It came as surprise to us to find that two bubbles have a long range interaction. In condensed matter physics a similar type of interaction is known for about a half of a century, the interaction between two impurities in a fermi gas 12 . The fact that this interaction oscillates suggest the intriguing possibility of forming di-bubble molecules with various radii. However, until one will determine the inertia of a di-bubble system it is not obvious whether such a molecule could indeed exist. It can be shown that in the case of three or more bubbles the interaction among them contains besides the expected pair-wise interaction we have just described, also genuine three-body, four-body and so forth interactions. The interaction Rel. (3) has its origin in the existence of the periodic orbit bouncing between the two bubbles. In the case of three or more bubbles there are distinct periodic orbits bouncing between three or more objects, which are the reason these genuine three and more body interactions arise. Using semiclassical methods (Gutzwiller trace formula), we have analyzed the structure of the shell energy as a function of the density, filling factor, lattice distortions and temperature 10 . The main lesson we have learned is that the amplitude of the shell energy effects is comparable with the energy differences between various phases determined in simpler liquid drop type models. Our results suggest that the inhomogeneous phase has perhaps an extremely complicated structure, maybe even completely disordered, with several types of shapes present at the same time.
266
At higher densities in neutron stars one expects that quarks and mesons will lead to similar structured mixed phases 13>14. The formation of either quark-gluon droplets embedded in a hadron gas or of hadron droplets embedded in a quark-gluon plasma has been studied and predicted for almost a decade. One naturally expects that similar quantum corrections are relevant in these cases as well. References 1. H.A. Wilson, Phys. Rev. 69, 538 (1946); J.A. Wheeler, unpublished notes; P.J. Siemens and H.A. Bethe, Phys. Rev. Lett. 18, 704 (1967); W.J. Swiatecki.PAysico Scripta 28, 349 (1983); W.D. Myers and W.J. Swiatecki, Nucl. Phys. A 601, 141 (1996). 2. C.Y. Wong, Ann. Phys. 77, 279 (1973). 3. K. Pomorski and K. Dietrich, Eur. Journ. Phys. D 4, 353 (1998). 4. A. Bulgac et al, in Proc. Intern. Work, on Collective excitations in Fermi and Bose systems, eds. C.A. Bertulani and M.S. Hussein (World Scientific, Singapore 1999), pp 44-61 and nucl-th/9811028. 5. Y. Yu et al, Phys. Rev. Lett. 84, 412 (2000). 6. M. Kardar and R. Golestanian, Rev. Mod. Phys. 71, 1233 (1999) and references therein. 7. V.M. Strutinsky and A.G. Magner,Soi>. /. Part. Nucl. Phys. 7, 138 (1976). 8. O. Bohigas et al., Phys. Rep. 223, 43 (1993); O. Bohigas et al, Nucl. Phys. A 560, 197 (1993); S. Tomsovic and D. Ullmo, Phys. Rev. E 50, 145 (1994); S.D. Frischat and E. Doron, Phys. Rev. E 57, 1421 (1998). 9. M. Brack and R.K. Bhaduri, Semiclassical Physics, Addison-Wesley, Reading, MA (1997); M. Brack, Rev. Mod. Phys. 65, 677 (1993) and references therein. 10. A. Bulgac and P. Magierski, astro-ph/0002377, Nucl. Phys. A, in print; astro-ph/0007423, Physica Scripta, in print; A. Bulgac. P. Magierski and A. Wirzba, unpublished. 11. C.J. Pethick and D.G. Ravenhall, Annu. Rev. Nucl. Part. Sci. 45, 429 (1995) and references therein. 12. M.A. Ruderman, C. Kittel, Phys. Rev. 96, 99 (1954). 13. H. Heiselberg et al, Phys. Rev. Lett. 70, 1355 (1992). 14. M.B. Christiansen and N.K. Glendenning, astro-ph/0008207. 15. G. Neergaard and J. Madsen, Phys. Rev. D 62, 034005 (2000) and earlier references therein.
MICROSCOPIC MODELS FOR N U C L E A R A S T R O P H Y S I C S P. DESCOUVEMONT Physique Nucleaire Theorique et Physique Mathematique, CP229, Universite Libre de Bruxelles, B1050 Bruxelles, BelgiumE-mail: [email protected] We report on a microscopic description of low-energy nuclear reactions, in the framework of the Generator Coordinate Method. The model is briefly presented and illustrated by an application to the 39 Ca(p,7) 40 Sc reaction. The reaction rate is found much larger than previously assumed.
1
Introduction
Current studies of stellar evolution require an ever increasing number of nuclear reaction rates 1. At stellar temperatures, typical energies between charged particles are so low with respect to the Coulomb barrier that the cross sections are minute. In addition, several interesting scenarios involve short-lifetime nuclei, which can not be used as targets. The smallness of the cross sections and the radioactive character of some nuclei make it experiments in laboratories very difficult. In spite of new detection techniques, and of the availability of radioactive beams, a theoretical support is often necessary to derive nuclear reaction rates at stellar temperatures. Theoretical models can be roughly classified into three categories: (i) Models containing adjustable parameters, like the .R-matrix 2 or the if-matrix 3 methods; in this case, parameters are fitted to available experimental data and are used to extrapolate the cross sections at astrophysical energies, usually inaccessible to experiment, (ii) Statistical models, where the relevant cross sections are calculated from the level properties in the compound nucleus 4 . (Hi) "Ab initio" models where the cross sections are determined from the wave functions of the system. Here, we present the Generator Coordinate Method (GCM) which belongs to the last category 5 ' 6 . This cluster model offers a number of important advantages for the description of low-energy reactions. In particular, it presents some predictive power since it only depends on a nucleon-nucleon interaction, and on a few assumptions concerning the cluster structure of the nuclei. Many reactions important in astrophysics have been studied in the GCM s 6 ' ; here we present recent results on the 39 Ca(p,7) 40 Sc reaction 7 . In the rp process 8 , the proton capture sequence 39
Ca(p, 7 ) 4 0 Sc( P ) 7 ) 4 1 Ti 267
268
determines the flow of nucleosynthesis beyond mass 40 for temperatures above 3 x 108K. If the lifetime of 3 9 Ca is well known (0.86 s), the 39 Ca(p,7) 40 Sc cross section relies on theoretical models and on indirect experimental data 8 ' 9 . In this mass region, statistical models, such as the Hauser-Feshbach theory, are widely used, and known to be in general accurate enough for network calculations. However, reactions involved in the rp process usually present a low level density and a low Q value, owing to the proton excess. In those conditions, statistical models are not applicable, and more detailed theories must be used. 2
The microscopic model
In a microscopic model 5 ' 6 , the whole information concerning a A-nucleon system is deduced from the hamiltonian A
H
A
T
= E * + E^-
(^
i<j
i
where Tj is the kinetic energy of nucleon i and Vij the two-body nucleonnucleon interaction involving the Coulomb and nuclear contributions. In most of our applications, the central part of the nuclear term is chosen as a Volkov interaction 10 , which is given by a superposition of gaussian terms, and a spin-orbit force; up to now, the tensor force has not been investigated. The Schrodinger equation associated to hamiltonian (1) can not be solved exactly as soon as A is larger than three. Accordingly, some approximations must be done for the determination of the wave functions. In the cluster model, the A nucleons of the system are assumed to be divided into two clusters with A\ and A2 nucleons, spins and parities (Iiiri) and (/2T2) a n d internal wave functions 0 / l 7 r i and /27r2 respectively. In RGM notations, the total wave function of the system, with spin and parity {Jit), reads
*JM" = E
A hni
^
® ^'"V
® Ydp)]JM9u(p)
(2)
11
where p is the relative coordinate between the clusters, and A the A-body antisymmetrizor. Here the 3 9 Ca wave functions are defined in the harmonic oscillator model (b — 1.8 fm), with one neutron hole in the sd shell. In addition to the ground state I\ = 3/2+, we also include the l / 2 + and 5/2+ excited states. In Eq. (1), the radial functions gJlT are determined from the Schrodinger equation. One of the colliding nuclei can be itself defined in a multicluster n . This extension of the "natural" two-cluster approach to a more refined multi-cluster
269
model enables to study reactions involving a deformed nucleus, such as 8 Be or 12 C for example. However, the number of clusters is limited by computer times since, because of projection on good quantum numbers, the calculation of matrix elements between GCM basis functions requires a numeral integration whose dimensions increase with the number of clusters. A drawback of the GCM is the asymptotic behaviour of the basis wave functions. Because of the gaussian expansion of the radial functions, their asymptotic part is not physical neither for bound nor for scattering states. We solve this problem by using the Microscopic R-matrix Method (MRM) which is detailed in ref. 12 . The use of the GCM enables a common determination of bound and scattering states of the system. In particular, the asymptotic behavior of resonant wave functions is treated exactly. Many applications of the GCM have been dedicated to nuclear astrophysics. The ability of the method to describe bound and scattering states provides tests of the cross functions through the spectroscopy of the unified nucleus. Also, the Coulomb symmetry provides useful constraints. In the 39 Ca(p,7) 40 Sc reaction, 40 K and 40 Sc spectroscopy will be used to test the model with known properties, such as quadrupole moments or electromagnetic transition probabilities. The accuracy of the model for these quantities can be transposed to capture cross sections, unknown experimentally.
3 3.1
The
39
Ca(p, 7 ) 4 0 Sc reaction
Introduction
Recently, Hansper et al. 9 investigated the 40 Sc spectrum just above the 39 Ca+p threshold (0.539 MeV). Resonance energies represent the main input of the reaction rate, but the calculation also involves proton and gamma widths which are not available directly. To derive these quantities, approximations such as mirror symmetry, and the use of spectroscopic factors, are necessary 8 . In the 39 Ca(p,7) 40 Sc reaction, it is quite important to reproduce the experimental 2j" energy 9 since this resonance dominates the reaction rate 8 . We adopt here a nucleon-nucleon interaction which provides the experimental value £ cm (2j~) = 0.232 MeV, and a reasonable agreement with experiment for the other low-lying states. This interaction is used for spectroscopy and reaction calculations, for the 3 9 K+n as well as for the 3 9 Ca+p systems. With these conditions, the excitation energies in 39 K are 2.85 MeV and 3.53 MeV for the 1/2+ and 3/2+, respectively. These results are consistent with the
270
experimental data (2.52 and 4.51 MeV). 3.2
40
K and 40Sc spectroscopy
In Figure 1, we present the 4 0 K and 40 Sc spectra. Energies are given in the c m . frame, i.e. with respect to the 3 9 K+n and 3 9 Ca+p thresholds, respectively. The theoretical spectra are of course not perfect since the interaction is determined on the 2f energy of 40 Sc only, but the overall agreement is fairly good. The lowest states are 4~ and 3~, a 5 _ resonance appears near the experimental energy, and the model predicts two 2~ resonances at low excitation energies. This result is important and represents a significant improvement with respect to the potential model 8 which can account for one 2~ state only. Two states within the same partial wave and separated by about 1 MeV have rather different properties since they are orthogonal to each other. Since both 2~ energies are well reproduced by the model, we may expect realistic wave functions, and consequently reliable spectroscopic properties. Finally, it must be pointed out that the interaction has been adjusted on the 2~ state of 40 Sc only. However, the energy of the mirror state in 40 K is fairly close to experiment, which shows that the charge symmetry is well described by the present model.
°Sc - (3-.4-) III-.*) (1-,*-)
exp
Figure 1. Experimental and GCM indicated as a dashed line.
GCM
40
K and
exp
40
GCM
Sc spectra. The proton threshold in
40
Sc is
271
3.3
The
39
Caftvy) 40 Sc cross section
Table 1 gives electromagnetic transition probabilities in 40 K, where experimental data are available. Again, this test calculation enables us to probe the model before applying it to the mirror system. As for spectroscopic properties 7 , the agreement with experiment is not perfect, but indicates that the model is realistic. In particular, transition probabilities involving the 2j~ resonance are supported by experiment. We also provide the corresponding values in 40 Sc, unknown experimentally. Table 1. Electromagnetic transition probabilities. B(Ml) in e 2 .fm 4 Ji -y Jf
4UR
a\
3 - -> 4 2~ -> 4 " 2~ -> 3 " 5 - -> 4 5 - -> 3 -
Ml E2 E2 Ml E2 Ml E2 E2
are expressed in fij^ and 4Ug c
exp.
GCM
0.35
0.12 6.2 5.6 0.15 13.3 0.03 15.6 9.5
9.3 0.315 0.06 11.5 19.2
B{E2)
GCM 0.24 15.8 26.4 0.29 35.7 0.10 4.5 6.5
100
o
/
Co
1 0.01
0.1
1
10
T9(K) Figure 2. Ratio of the present
39
Ca(p,7) 4 0 Sc reaction rate with the results of Iliadis et al. 8
The reaction rates are compared to those of Iliadis et al.8 in Figure 2.
272
Below T8 = 0.7 the reaction rate is mainly non-resonant and the difference with Iliadis et al. comes from the difference in the S factors. Between T8 = 0.7 and T8 ss 6, the reaction rate is essentially given by the contribution of the 2" resonance. We find a proton width much larger than Iliadis et al., and this difference appears in the reaction rate. A factor of 80, as found here, should affect the conclusions of Iliadis et al. about nucleosynthesis in novae and X-ray bursts. Beyond T8 ss 6, high-lying resonances are the main input to the reaction rate. For these states, the microscopic model and the model of Iliadis et al. provide similar spectroscopic properties, and the reaction rates are very close to each other. Details can be found in Ref.7 4
Conclusions
The main advantage of a microscopic model is its predictive power since the physical quantities only depend on the nucleon-nucleon interaction. The 39 Ca(p,7) 40 Sc reaction is one of the the heaviest systems considered so far in a microscopic cluster model. The reaction rate is found much larger than previously assumed, and a direct measurement using a 3 9 Ca radioactive beam seems necessary to derive more reliable conclusions. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.
M. Arnould and T. Takahashi, Rep. Prog. Phys. 62, 393 (1999). F.C. Barker and T. Kajino, Aust. J. Phys. 44, 369 (1991). B.W. Filippone et al., Phys. Rev. C 40, 515 (1989). F.K. Thielemann, M. Arnould J.W. and Truran, in "Advances in Nuclear Astrophysics", 1987, eds. Vangioni-Flam et al., 525 P. Descouvemont, J. Phys. G 19, S141 (1993). K. Langanke, Adv. In Nuclear Physics 2 1 , 85 (1994). P. Descouvemont, Astrophys. J., in press. C. Iliadis et al., Astrophys. J. 524, 434 (1999). V.Y. Hansper et al., Phys. Rev. C 6 1 , 028801 (2000). A.B. Volkov, Nucl. Phys. 74, 33 (1965). M. Dufour and P. Descouvemont, JVuci. Phys. A 672, 153 (2000). D. Baye and P. Descouvemont, Nucl. Phys. A 407, 77 (1983).
TOWARDS A HARTREE-FOCK MASS F O R M U L A
J. M. P E A R S O N , M. ONSI Dept.
de Physique, Universite de Montreal, Montreal (Qc) H3C 3J7, Canada E-mail: [email protected] S. G O R I E L Y , F . T O N D E U R
Institut
d'Astronomie
et d'As trophy sique, CP 226 Universite B1050 Brussels, Belgium
Libre de
Bruxelles,
M. F A R I N E Ecole Navale,
29240 Brest Naval,
France
We first describe our nuclear mass formula HFBCS-1, the first to be based on the Hartree-Fock-BCS method. The fitted force is a conventional 10-parameter Skyrme force (to, t\ , ti, and £3 terms), along with a 4-parameter^-functionpairing force and a 2-parameter phenomenological Wigner term. The rms error of our fit to the 1888 measured nuclei with Z, N > 8 is 0.738 MeV. The value of the effective mass M*/M that emerges from these fits is 1.05, which falls within the range of values long known to be necessary for a fit to single-particle (s.p.) level densities in the vicinity of the Fermi surface. On the other hand, realistic nuclear-matter calculations lead to M*/M ~ 0.8, a result that is confirmed experimentally. By adding to the Skyrme force a term with simultaneous momentum and densitydependence, M* becomes a non-monotonic function of density, and we find that it is possible to impose the realistic value of M*/M — 0.8 at nuclear-matter densities, i.e., at the centre of the nucleus, while maintaining an effective mean value over the nucleus of around 1.0, so that heavy-nucleus s.p. level densities are well reproduced in the Fermi surface. In this way we find acceptable mass fits even under the constraint of M*/M = 0.8 at nuclear-matter densities. Such forces should be useful in describing the nuclear processes that occur in stellar collapse and in decompressing nuclear matter.
1
T h e H F B C S - 1 Mass Formula
We take a Skyrme force with the conventional form 1 Vij = to{l + x0P
1 1 + a2-P
tfW°(ai
+ a
i)-Pij
x
Hrij)PiJ 273
+ h.c.} X3P(T)pa8(Tij)
(!)
274 and a (^-function pairing force Vpairinj)
= Vvq6{rij)
.
(2)
All nuclei are calculated with the Skyrme force handled in the Hartree-Fock (HF) method, and the pairing force in the BCS approximation (see Ref. 1 for all details). Deformation is allowed, but with the constraint of axial and reflection symmetries. To all calculated energies we add a Wigner correction Ewigner
= VW e X P ( - A | i V - Z\/A)
.
(3)
Fitting the parameters to the 1888 nuclei with Z, N > & whose masses have been m e a s u r e d 2 gives an rms error of 0.738 MeV. The parameters of the force MSk7 emerging from this fit are given in the first column of Table 1 (also Vw = -2.35 MeV and A = 35). A complete mass table based on this force, going from one drip line to the other up to Z — 120, has been constructed. Table I.
t3a t3b U
Parameters of the forces presented in this paper.
t0 (MeV.fm 3 )
W0
(MeV.fm 5 ) <*a (*b
P
V+n V+p V~n V-v
(MeV.fm 3 ) (MeV.fm 3 ) (MeV.fm 3 ) (MeV.fm 3 )
MSk7 -1828.23 259.400 -292.840 13421.7 0.0 0.0 0.576761 -0.500000 -0.500000 0.785290 118.807 0.333333 -227.0 -242.0 -236.0 -251.0
MSk5 -1827.96 254.326 -287.766 13419.5 0.0 0.0 0.605152 -0.500000 -0.500000 0.827182 115.932 0.333333 -220.0 -228.0 -224.0 -232.0
MSk5* -1728.73 362.859 -126.139 11084.8 0.0 0.0 0.647000 -0.100000 -0.145864 0.994321 121.987 0.333333 -266.0 -246.0 -274.0 -258.0
Skt4.1 -1507.60 -632.324 -483.786 23288.0 -28415.5 2332.61 0.533774 1.68964 -0.750000 0.933215 1.00000 1.00000 122.552 0.550000 1.10000 0.40000 -248.0 -274.0 -260.0 -286.0
275 T h e value of the nuclear-matter symmetry coefficient J corresponding to the force MSk7, as determined by our mass fits, is 27.95 MeV, which is to be compared with the values of 32-35 MeV t h a t are found in macroscopicmicroscopic mass formulas 3 ' 4 . This discrepancy has implications for the rate of neutron-star cooling. A very recent calculation of asymmetic nuclear m a t t e r with realistic interactions leads to J = 28.7 M e V 5 .
2
S k y r m e Force w i t h S u r f a c e - P e a k e d Effective M a s s
The nuclear-matter value of M*/M corresponding to the force MSk7 t h a t emerges from the mass fits of the previous section is 1.05, which is consistent with the observation t h a t unless M* /M ~ 1.0 the single-particle (s.p.) level density in the vicinity of the Fermi surface will be w r o n g 6 , whence it would be impossible to fit the masses of open-shell nuclei. On the other hand, all nuclear-matter calculations with realistic forces indicate t h a t at the equilibrium density M*/M ~ 0.8 7 ' 8 ' 9 : this is the "Jfc -mass". Experimental confirmation of this result comes from deep-lying s.p. states, and from the giant isoscalar quadrupole and isovector dipole resonances (see Ref. x for a s u m m a r y ) . The discrepancy between these two values of M*/M was explained by Bernard and G i a i 1 0 , who showed t h a t one can obtain reasonable s.p. level densities in finite nuclei with realistic values of M*/M, i.e., of around 0.8, provided one takes into account the coupling between s.p. excitation modes and surface-vibration RPA modes. Since the good agreement with measured s.p. level densities found in Ref. 6 was obtained without making these corrections it must be supposed t h a t the resulting error is being compensated by the higher value of M*/M, i.e., M*/M ~ 1.0, which may thus be regarded as a phenomenological value t h a t permits considerable success with straightforward HF, or other mean-field calculations, without the complications of Ref. 1 0 : this is the "w-mass". Thus the use of our force MSk7 in pure HF calculations of isolated nuclei is quite legitimate, but a difficulty will arise in the context of stellar collapse: one begins with isolated nuclei, for which force MSk7, or some other force with M* /M ~ 1.0, would be optimal, but as the collapse progresses the nuclei are squeezed together, and one finishes with (neutron-rich) homogeneous nuclear matter, for which one should take rather M*/M ~ 0.8. Adiabatic processes run at a significantly higher temperature if the smaller value of M* /M is taken: see Ref. n and references quoted therein. A similar problem is faced in decompressing nuclear matter, where the reverse sequence of events is traced. To form an idea of just how bad the mass fits would be if we took 0.8 for the nuclear-matter value of M*/M, we took a conventional force of the
276 form (1) with this condition imposed, and fitted to the masses of 416 spherical (or near-spherical) masses. T h e rms error of the resulting force (MSk5*) is 1.141 MeV, while if M* JM is allowed to vary freely the rms error for the same sample is 0.709 MeV (force MSk5). Thus acceptable mass fits are impossible with conventional Skyrme forces if this condition on M*/M is imposed. Now with conventional Skyrme forces M* jM varies linearly with density, and thus will change monotonically from the imposed nuclear-matter value at the centre to 1.0 outside the nucleus, so t h a t if we set M* jM = 0.8 the average value over the nucleus will be less t h a n 1.0. However, if to the conventional form we add a term with simultaneous m o m e n t u m and density dependence, ^5-^4(1 + xnPa){pijfP6{rij) + /i.e.}, a non-linearity is introduced into the density dependence of M*/M, and it becomes possible for the average value over the nucleus to be close to 1 1 2 . Proceeding in this way we have been able to fit the above restricted sample of 416 masses with an rms error of 0.764 MeV (force Skt4.1), while maintaining the nuclear-matter value of 0.8 for M*/M. This improvement made possible by the presence of the ^4 term is sufficient to make it worthwhile to extend the calculations to deformed nuclei. In any case, we find t h a t with increasing density adiabats (see Ref. n ) of force Skt4.1 follow the adiabats of force MSk5 {M* / M = 1.05) in the inhomogeneous phases and then rise more steeply in the homogeneous phase, reaching the adiabats of force MSk5* (M* /M = 0.8) at the equilibrium density. References 1. F . Tondeur, S. Goriely, J. M. Pearson, and M. Onsi, Phys. Rev. C 62 (2000) 024308. 2. G. Audi and A. H. Wapstra, Nucl. Phys. A 5 9 5 (1995) 409. 3. P. Moller, J. R. Nix, W. D. Myers, and W. J. Swiatecki, At. D a t a Nucl. D a t a Tables 59 (1995) 185. 4. W. D. Myers and W. J. Swiatecki, Nucl. Phys. A 6 0 1 (1996) 141. 5. W. Zuo, I. Bombaci, and U. Lombardo, Phys. Rev. C 6 0 (2000) 024605. 6. M. Barranco and J. Treiner, Nucl. Phys. A 3 5 1 (1981) 269. 7. K. A. Brueckner and J. L. Gammel, Phys. Rev. 109 (1958) 1023. 8. B. Friedman and V. R. Pandharipande, Nucl. Phys. A 3 6 1 (1981) 502. 9. R. B. Wiringa, V. Fiks, and A. Fabrocini, Phys. Rev. C 38 (1988) 1010. 10. V. Bernard and Nguyen Van Giai, Nucl. Phys. A 3 4 8 (1980) 75. 11. M. Onsi, H. Przysiezniak, and J. M. Pearson, Phys. Rev. C 55 (1997) 3139. 12. X. Campi and S. Stringari, Z. Phys. A 309 (1983) 239.
N U C L E A R A S P E C T S OF N U C L E O S Y N T H E S I S IN M A S S I V E STARS T. R A U S C H E R 1 ' 2 , R . D . H O F F M A N 3 , A. H E G E R 2 , S.E. W O O S L E Y 2 1
2
Departement
Department
3
Nuclear
fur Physik
und Astronomie, Universitat Basel, Switzerland E-mail: [email protected]
CH-4056
of Astronomy and Astrophysics, University of California Cruz, Santa Cruz, CA 95064, USA Theory
and Modeling Group, L-414, Lawrence Livermore Laboratory, Livermore, CA 94551, USA
Basel,
at
Santa
National
Preliminary results of a new set of stellar evolution and nucleosynthesis calculations for massive stars are presented. These results were obtained with an extended reaction network up to Bi. The discussion focuses on the importance of nuclear rates in pre- and post-explosive nucleosynthesis. The need for further experiments to study optical a+nucleus potentials is emphasized.
1
Introduction
Nuclear reactions play a major role not only in the nucleosynthetic processes determining the elemental abundances in the solar system and the Galaxy but also for determining structure and final fate of a star. Massive stars (> 8M Q ) experience a number of burning phases before they explode as type II supernovae after the collapse of the Fe core. Important nuclear reactions in the late burning stages and in the explosion proceed on isotopes experimentally not sufficiently well investigated or on unstable nuclei which cannot be studied in the laboratory. Thus, astrophysics requires a sound theoretical understanding of nuclear reactions and tests our knowledge at the extremes. Numerous studies have been devoted to the evolution of such stars and their nucleosynthetic yields. However, our knowledge of both the input data and the physical processes affecting the development of these objects has improved dramatically in recent years. Thus, it became worthwhile to attempt to improve on and considerably extend the previous investigations on preand post-collapse evolution and nucleosynthesis. Here we present first results for a 15 MQ stellar model with improved stellar and nuclear physics. In this report we mainly concentrate on a few of the nuclear physics issues involved, a more extended report including all details of the simulation will be published elsewhere 1 ' 2 . Below, we discuss the prediction of nuclear rates 277
278
in the statistical model and, specifically, the treatment of a-particle capture on isospin conjugated targets. The importance of obtaining more information on a+nucleus potentials is emphasized separately.
2
Nuclear Reactions
The nuclear reaction network during the explosive phase (as the most extreme case in our calculation) contains about 2350 isotopes and is shown in Fig. 1. It is evident that there are many isotopes far off stability included for which experimental information is scarce. This is even more true in more exotic scenarios such as the r- and rp-processes which involve isotopes close to the neutron- and proton-dripline, respectively. (In our calculations we do not follow the proposed r-process in the z/-wind emanating from the protoneutron star shortly after the collapse of the Fe core.) Important are weak reactions and nuclear reactions with nucleons and a particles. Decay data is also available further off stability whereas nuclear reaction cross sections involving nucleons and light ions are practically known only for stable nuclei. The majority of the latter reactions can be described in the framework of the statistical model (Hauser-Feshbach theory) which describes the reaction proceeding via the formation of a compound nucleus and averages over resonances 3 . Many nuclear properties enter the computation of the HF cross sections: mass differences (separation energies), optical potentials, Giant Dipole Resonance widths, level densities. The resulting transmission coefficients can be modified due to pre-equilibrium effects which are included in width fluctuation corrections (see also a previous paper 3 and references therein) and by isospin effects. It is in the description of the nuclear properties where the various HF models differ. In astrophysical applications usually different aspects are emphasized than in pure nuclear physics investigations. Many of the latter in this long and well established field were focused on specific reactions, where all or most "ingredients" were deduced from experiments. As long as the reaction mechanism is identified properly, this will produce highly accurate cross sections. For the majority of nuclei in astrophysical applications such information is not available. The real challenge is thus not the application of well-established models, but rather to provide all the necessary ingredients in as reliable a way as possible, also for nuclei where no such information is available.
279 - ' ' ' I ' ' ' I ' ' ' I ' ' ' I ' ' ' I ' ' ' I '. 80 —
;
~
' —
60 —
40 —
—
«
'expl'
20 -
0
—
20
40
60
80
100
120
140
N
Figure 1. Reaction network for explosive burning.
2.1
Statistical Model Rates
As the basis for the creation of our reaction rate set we used statistical model calculations obtained with the NON-SMOKER code 3 ' 4 . A library of theoretical reaction rates calculated with this code and fitted to an analytical function — ready to be incorporated into stellar model codes — was published recently 4 . It includes rates for all possible targets from proton- to neutron-dripline and between Ne and Bi, thus being the most extensive published library of theoretical reaction rates to date. It also offers rate sets for a number of mass models which are suited for different purposes. For the network described here we utilized the rates based on the FRDM set. 2.2
a Particles: Isospin Effects
The consideration of isospin effects has two major effects on statistical cross sections in astrophysics 8 : the suppression of 7 widths for reactions involving self-conjugate nuclei and the suppression of the neutron emission in protoninduced reactions. Here, we only discuss the former. In the case of (0,7) reactions on targets with N = Z,the cross sections will be heavily suppressed because T = 1 states cannot be populated due to isospin conservation. A suppression will also be found for capture reactions leading into self-conjugate
280
nuclei, although somewhat less pronounced because T = 1 states can be populated according to the isospin coupling coefficients. In previous reaction rate calculations 7 ' 8 the suppression of the 7-widths was treated completely phenomenologically by employing arbitrary and mass-independent suppression factors. In the NON-SMOKER code, the appropriate 7 widths are automatically obtained, by explicitly accounting for T< and T > states 5 . The astrophysical importance of a capture on target nuclei with N = Z is manifold. In the Ne- and O-burning phase of massive stars, alpha capture reaction sequences are initiated at 24 Mg and 28 Si, respectively, and determine the abundance distribution prior to the Si-burning phase. Nucleosynthesis in explosive Ne and explosive 0 burning in type II supernovae depend on reaction rates for a capture on 20 Ne to 36 Ar. An a capture chain on such self-conjugate nuclei actually determines the production of 44 Ti 9 , which contributes to the light curve by its j3 decay to 4 4 Ca via 44 Sc. 2.3
a Particles: Optical a+Nucleus
Potentials
A further complication in the treatment of reactions on intermediate and light nuclei involving a particles is the limited success in defining an appropriate optical potential, especially for the low energies typical for astrophysical environments. Early astrophysical studies (e.g.7) made use of simplified equivalent square well potentials and the black nucleus approximation. It is equivalent to a fully absorptive potential, once a particle has entered the potential well and therefore does not permit resonance effects. This leads to deviations from experimental data at low energies, especially in mass regions where broad resonances in the continuum can be populated 9 . An additional effect, which is only pronounced for a particles, is that absorption in the Coulomb barrier 9 is neglected in this approach. Improved calculations have to employ appropriate global optical potentials which make use of imaginary parts describing the absorption. In the case of a-nucleus potentials, there were only very few global parametrizations attempted at astrophysical energies, also due to the scarcity of experimental data in the energy region of interest. The high Coloumb barrier makes a direct experimental approach very- difficult at low energies. Current astrophysical calculations mostly employ a phenomenological SaxonWoods potential based on extensive data 10 . Future improved a potentials have to take into account the mass- and energy-dependence of the potential. Extended investigations of a scattering data 11,12 have shown that the data can best be described with folding potentials. Few attempts 6 have been made to construct such an improved global potential. Nevertheless, the postulation of such an optical potential close to or below the Coulomb barrier remains
281
!
* &
f
20
40
60
80
100
120
140
Figure 2. Production factors in the ejecta of a 15 MQ star relative to solar abundance.
one of the major challenges. More experimental data is clearly needed. 3
Further Inputs
Further nuclear input were updated rates from experimental cross sections for light as well as heavy nuclei and updated beta-decay rates. New predictions of weak rates 13 were also included. In respect to earlier simulations 14 we also used updated neutrino loss rates and opacity tables (OPAL95), and consider mass loss due to stellar wind. For further details, refer to our other papers 1,2 . 4
Results and Summary
For the first time, we studied consistently the production of all isotopes up to Bi during the pre-supernova evolution and the type II supernova explosion of a massive star. Exemplary for our results, the production factors of a 15 M 0 star are shown in Fig. 2. The revised weak rates introduce an important change mainly during core silicon burning and thereafter leading to an increase of the central Ye and smaller Fe core masses at the onset of core collapse. In addition to the well-known strong dependence of the stellar structure on the 12 C(a,7) 1 6 0 rate, we also found the (a,n)-(a,j) branching on 22 Ne to be an
282
important candidate for further laboratory study. It sensitively determines the strength of the s-process in the SN models. Summarizing, the progress in predictions of nuclear reactions has made it possible to consistently study the nucleosynthesis in a type II supernova model over a wide range of nuclear masses. The new investigations also underline the importance of new experimental and theoretical studies of specific nuclear properties. Acknowledgments This research was supported, in part, by DOE (W-7405-ENG-48), NSF (AST 97-31569, INT-9726315), and the Alexander von Humboldt Foundation (FLF1065004). T. R. acknowledges support by a PROFIL professorship from the Swiss NSF (grant 2124-055832.98). References 1. A. Heger, R.D. Hoffman, T. Rauscher, S.E. Woosley, in Proc. X Workshop on Nuclear Astrophysics, eds. W. Hillebrandt, E. Miiller (MPA, Garching 2000), in press, (astro-ph/abs/0006350) 2. T. Rauscher, R.D. Hoffman, A. Heger, S.E. Woosley, Ap. J. , in prep. (2000). 3. T. Rauscher, F.-K. Thielemann, K.-L. Kratz, Phys. Rev. C 56, 1613 (1997). 4. T. Rauscher, F.-K. Thielemann, At. Data Nucl. Data Tables 75, 1 (2000). 5. T. Rauscher, F.-K. Thielemann, J. Gorres, M.C. Wiescher, Nucl. Phys. A675, 695 (2000). 6. T. Rauscher, in Proc. Symp. "Nuclei in the Cosmos V", eds. N. Prantzos, S. Harissopoulos (Editions Frontieres, Gif-sur-Yvette 1998), p. 484. 7. S.E. Woosley, W.A. Fowler, J.A. Holmes, B.A. Zimmerman, At. Data Nucl. Data Tables 18, 306 (1978). 8. J.J. Cowan, F.-K. Thielemann, J.W. Truran, Phys. Rep. 208, 267 (1991). 9. R.D. Hoffman et al, Ap. J. 521, 735 (1999). 10. L. McFadden, G.R. Satchler, Nucl. Phys. 84, 177 (1966). 11. P. Mohr et al, Phys. Rev. C 55, 1523 (1997). 12. P. Mohr, Phys. Rev. C 6 1 , 045802 (2000). 13. K. Langanke, G. Martmez-Pinedo, Nucl. Phys. A673, 481 (2000). 14. S.E. Woosley, T A . Weaver, Ap. J. Suppl. 101, 181 (1995). 15. A. Heger, S.E. Woosley, G. Martinez-Pinedo, K. Langanke, Ap. J. , in prep. (2000).
WEAK INTERACTION RATES OF NEUTRON-RICH NUCLEI AND THE R-PROCESS NUCLEOSYNTHESIS I.N. BORZOV1'2, S. GORIELY2 'State Scientific Centre - Institute of Physics & Power Engineering 2
Institut d'Astronomie et d'Astrophysique, Universit'e Libre de Bruxelles, Belgium e-mail: iborzov@astro. ulb.ac. be
The rapid neutron-capture process, or r-process, is of fundamental importance for explaining the origin of about half of the stable nuclides heavier than iron observed in nature. Weak interaction rates for very neutron-rich nuclides relevant to the r-process are mostly beyond the experimental reach at the present time. For theoretical predictions a coherent extrapolation of different nuclear properties away from the experimentally known regions is needed. The results of prediction of the (5-decay and ve-capture rates in the framework of the ETFSI+cQRPA in a comparison with the other approaches are presented.
1
Introduction
We are involved in a program aiming at studies of a wide variety of phenomena encountered at subnuclear and nuclear densities during and after the stellar collapse associated with the supernova event in terms of a single, universal, effective nuclear interaction. Because of the huge number of nuclei relevant to the r-process, an efficient procedure has been found to be the so-called Extended Tomas-Fermi approach plus Strutinski Integral correction (ETFSI). The main achievment so far has been the development, for the first time, of a mass formula based entirely on a microscopic force, (ETFSI-1, see [1] and refs. therein). In improved ETFSI-2 version, the Skyrme force parameters are subject to the constraint that the neutron matter does not collapse at nuclear and sub-nuclear densities [2]. Here we present the results of the large-scale calculations of (5-decay and vcapture rates. We used the continuum quasi-particle random phase approximation (cQRPA) based on the same ETFSI-2 description of the ground states with the force SkSC17 which allows to describe the known nuclear masses with an rms error of 731KeV [2]. Coherence in calculation of the masses and weak rates is the main advantage of the method. It is of importance for the r-process modeling, as the elemental abundances are known to be more sensitive to the nuclear masses than to the weak rates. We study the predictive power of the method by comparing the ETFSI+cQRPA results with recent experimental data, as well as with the predictions obtained by phenomenological "gross theory" [3], non-selfconsistent QRPA [4] and RPA [5] global calculations, as well as the HFB+cQRPA [6] and shell model [7] approaches. 283
284
2
P-decay rates near the closed shells
Crucial for the r-process analysis are the p-decay half-lives around the closed-shell nuclei at N=50, 82, 126 which play the role of bottlenecks, hindering the r-process material to be driven to higher Z values. Recently much experimental effort has been devoted to study the decay properties near the doubly magic 78Ni and 132Sn. It can be shown that QRPA calculations systematically overestimate (by a factor of 2-3) the experimental half-lives near the closed shells nuclei with Z=28 and Z<50 undergoing a high-energy P-decay [2,3,4,6,8,9]. For the Z>50 nuclides in the region of the doubly magic 132Sn, a switch to the low-energy GT P-decay regime is observed by all QRPA approaches which results in a large overestimate (by a factor of 10-100) of the experimental half-lives. It is common belief that QRPA may correctly describe the experimental halflives for nuclides with high-energy allowed P-decays (o>=Qp) because the final states occur to be near the daughter ground state where the spreading effects play no significant role. In the case of low-energy p-decays (co«Qp), even a weak additional strength located at higher transition energies (co) affects the half-life strongly due to the sharp energy dependence of the lepton phase space factor. To improve the description of the half-lives an additional transition strength within the Qp-window or/and a shift of the strength to higher transition energies is certainly needed. Three major additional sources could provide a half-life reduction. First, in the cases of favorable selection rules, an additional strength comes from the contribution of forbidden transitions. The experimental evidence of such high energy first forbidden transitions near the Z=50, N=82 shell has been widely discussed. Second, effects beyond the QRPA should also be considered. Third, a major modification of the half-lives is due to the onset of nuclear deformation which has not been included in our calculations. It is clear that the impact of the additional strength depends strongly on the specific shell sequence. In Fig. 1, the p-decay half-lives for the N=82 isotones are presented. Our results are rather close to the one obtained within the HFB+cQRPA [6] (note that a 30-40% reduction of the ETFSI+cQRPA half-lives should be included for direct comparison due to the larger strength parameter of attractive spin-isospin pp interaction g'^ used by [6]) Since different possible remedies exist to improve the agreement between the experimental and theoretical P-decay half-lives, the direct adjustment of the g\ to measured p"-decay data [6] has to be taken with some reservation, especially if extrapolated or applied to large-scale calculations. The present prediction gives half-lifes with a Z-dependence very similar to HFB+cQRPA [6] and DF3+cQRPA [8]. In comparison with shell-model results [7], we obtain longer half-lives in the N=82 region. This partially reflects the QRPA neglect of more complex configurations than lp-lh. In particular, it would be of great interest to compare QRPA and shell-model predictions for nuclei with Z>50 near l32Sn, where complex configurations, as discussed above, are expected to give a predominant contribution.
285
3
Electron neutrino capture cross sections
The currently discussed r-process sites, such as the neutrino-driven wind at the surface of nascent neutron star at the center of a Type II supernova, or neutron star merger, assume that the r-process matter is exposed to extreme neutrino fluxes of all three flavors. Consistent r-process modeling requires a reliable prediction of neutrino-nucleus reaction rates for neutron-rich nuclei. The supernova neutrino spectrum can be given by a zero-chemical-potential Fermi-Dirac distribution [10]. As typical supernova electron neutrino temperatures correspond to the quite low average energy (E=l 1 MeV), the allowed transition approximation can be used for the ve-capture cross section. The latter involves the integrals of Fermi and Gamow-Teller strength functions weighted with the lepton phase space. The GT strength functions are calculated within the ETFSI+cQRPA including the quasi-particle damping width. As for the IAS energies, we prefer to use the systematic of Coulomb energies. For stable nuclei, the contributions of all vinduced excitations (o<mec2) is included, while for unstable ones, the v-mediated de-excitations are also considered, with lepton final states corresponding to nuclear transitions to the discrete states with a positive Q-values (oi>mec2). The reduced total capture cross sections and their partial components, averaged with supernova neutrino spectrum for Tv=4 MeV, are shown (as a function of A) for the Ni-isotopic chain in Fig. 1. In stable nuclei, the IAS and v-induced excitation components of the total cross section increase with N-Z. This is due to the shift of the IAS and GTR down in transition energy and towards the maximum of the neutrino energy spectrum. The contribution of the low energy states below the GTR is responsible for odd-even effect in the total cross sections, as illustrated in Fig. 1. In unstable nuclei, at increasing neutron excesses, the IAS and GT-excitation contributions to the cross section increase linearly with N-Z. For relatively small Z, the de-excitation part of the total cross section grows faster with neutron excesses, as the QP-value (and available phase-space) becomes larger. As seen in Fig. 1, for Ni isotopes, it exceeds the IAS contribution well before the neutron drip-line is reached. The impact of the de-excitation is smaller for heavier isotopic chains Z>50. For the cases available for comparison, our results are in general agreement with those from non-self-consistent RPA [5]. Note that an adjustment of the proton single particle energies in order to reproduce the IAS energy systematic, as done in [5], re-normalizes the Landau-Migdal constant needed to ensure correct position of the GTR, and thus changes the contribution of low-lying GT transitions to the neutrino capture cross section. We have not observed non-regularities in A dependence of the approaching the neutron drip-line, as seen in [5]. If ve were converted to the heavier neutrino species vx due to the matter enhanced oscillations, the ve-spectrum would be harder (<E>=25 MeV). The account of forbidden transitions turns to be necessary in that case [5,11]. Corresponding calculations are underway, the results will be reported elsewhere.
286 ~~*~ ETFSl+cORPA
J,
-----ETFSMST2
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-»*••- FRDM+QRPA
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Fig. 1. Comparison between the ETFSI+cQRPA, ETFSI+GT2 [4], FRDM+QRPA [5], shell model [7] and HFB+cQRPA [6] predictions of the (3decay half-lives along the N=82 isotone chain
_
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'/
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.-*' 40
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Fig. 2. The ETFSI+cQRPA total and partial v e -capture cross sections for Ni-isotopic chain.
—•— ias —•—gt< A
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—T— t0t
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The ETFSI+cQRPA fi-decay half-lives for about 700 short-lived nuclides (T1/2
References
1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.
Y. Aboussir, J.M. Pearson, A.K. Dutta, F. Tondeur, ADNDT 61,127 (1995). I.N. Borzov, S. Goriely Phys.Rev. C62 (2000). T. Tachibana, M. Yamada, and N. Yoshida, Prog. Theor. Phys. 84, 641 (1990). P. Moller, J.R. Nix, K.-L. Kratz. ADNDT 66, 131 (1997). A.Hektor, E.Kolbe, K.Langanke, J.Tojvanen, Phys.Rev. C61, 055803 (2000). J. Engel, M. Bender, J. Dobaczewski, et al., Phys. Rev. 60,1432(1999). G. Martinez-Pinedo, and K. Langanke, Phys. Rev. Lett. 83, 4502 (1999). I.N. Borzov, S.A. Fayans, E. Kromer, D. Zawischa, Z. Phys. A335 127,1996. I.N. Borzov, S. Goriely, and J.M. Pearson, Nucl. Phys. A621, 307c (1997). D.S.Miller, J.R.Wilson, R.W.Mayle, Astrophys.J. 415, 278 (1993). G.M.Fuller, W.C.Haxton, G.C.McLaughlin, Phys.Rev.D58, 085005 (1999)
SYSTEMATICS OF LOW-LYING LEVEL DENSITIES AND RADIATIVE WIDTHS A.V.IGNATYUK Institute of Physics and Power Engineering, 249020 Obninsk, Russia E-mail: [email protected] Comprehensive analysis of available experimental data on the neutron resonance spacings and cumulative numbers of low-lying levels is performed. New systematics of the constant temperature model parameters and the total radiative widths are proposed. An application of new systematics for improvement of astrophysical nuclear data is briefly discussed.
For calculations of neutron cross sections within the statistical theory of nuclear reactions it is important to use the level density models fitted to the reliable experimental data. The average neutron resonance spacings and the cumulative numbers of low-lying nuclear levels are usually considered as the most accurate data available. The recent compilations of such data were compiled in the framework of the Recommended Input Parameter Library (RIPL) project coordinated by the IAEA [1]. These data can be obtained via the Internet Web-site or requested from the IAEA Nuclear Data Section on a compact disk. Three models are most frequently used in applied calculations of the nuclear level densities: i) the back-shifted Fermi-gas model [2, 3], ii) the Gilbert-Cameron approach [4] that combines the Fermi-gas model with the constant temperature model for low excitation energies, and iii) the generalized superfluid model in the microscopic or phenomenological versions [5]. A brief description of these models is included in the RIPL starter file [1] together with the recommended parameter sets for each model. Because these parameters were fitted to the same data on the neutron resonance density and the cumulative number of low-lying levels there are no essential contradictions between the level densities calculated with different models for excitation energies close to the neutron binding energies of stable nuclei (6-7 MeV). However for lower and/or higher energies the models give in many cases rather diverse values of level densities. It should be noted that the data on the cumulative numbers of low-lying levels are available for much larger amount of nuclei than the neutron resonance data. Consideration of these numerous data should be very important for the level density systematics of nuclei far off the stability valley. 1
Low-Lying Level Densities
It was noted many years ago that the observed dependence of the cumulative level number N(U) on the excitation energy is well reproduced by the constant 287
288
temperature model N(U) = exp[{U-E0)/T\
,
(1)
where T is the nuclear temperature and EQ is the excitation energy shift. The values of these parameters obtained from the simultaneous analysis of the neutron resonances and the low-lying levels into the framework of the Gilbert-Cameron approach are shown in Fig. 1. The nuclear temperature dependence on the mass number is smooth and some deviations from the monotone behavior arise for near magic nuclei only. The even-odd splitting of nuclear level densities and strong shell effects are displayed clearly in the excitation energy shifts. The shell effects are well known as regular decreases in the level density parameter a(Bn) , obtained from the analysis of neutron resonance densities for near magic nuclei. The strong correlation of the ratio a(Bn) /A and the shell corrections 8EQ in the nuclear mass formulas can be used to construct the phenomenological systematics of the level density parameters [6]. These systematics are based on the relation a(U. Z,A) =~a(A)ll + ^ L [l - exp(-y(7]|
,
(2)
which parameterizes in a simple form the damping of shell effects with an increase of excitation energy. The asymptotic values of the level density parameters for high excitation energies may be evaluated in the form 3(A) = ccA + ^Am, and from the analysis of neutron resonance densities were obtained the values of corresponding phenomenological parameters (inMeV 1 ): cc= .078, /}= .115, and y= .055 -^0.65. The matching conditions of the Fermi-gas model and the constant temperature approach can be used to analyze the shell variations of nuclear temperatures. In accordance with Eq. (2) such variations can be described by the relation T = 17.60/ AsmJl + y8E0
,
(3)
the numerical coefficients in which are determined by a fitting to the resonance data. In Fig. 2 the constant temperature model parameters are shown obtained from the analysis of experimental data on low-lying levels of odd nuclei without any matching to the neutron resonance data [7]. The fluctuations of both parameters T and Eo are very strong, and for many nuclei the values of nuclear temperature are so low that they cannot be conformed to any systematics of the nuclear level densities at excitations corresponding to the neutron binding energies. The similar fluctuations of the parameters are obtained also for even-even and odd-odd nuclei [7]. The examples considered demonstrate mat for most nuclei a reliable estimation of both parameters T and E0 cannot be based on the available data on cumula-
289
0
50
100
150
200
250
mass number Figure 1. Nuclear temperature 7 and the energy shift Eo obtained from the simultaneous analysis of the neutron resonance spacing and the cumulative number of low-lying levels.
100
150
200
250
mass number Figure 2. Nuclear temperature T and the energy shift Eo obtained from the analysis of the cumulative numbers of low-lying levels in Ref [7] (solid symbols) and with T-systematics (open symbols).
tive numbers of low-lying levels only. It is more reasonable to estimate the nuclear temperature for nuclei, which have no direct data on neutron resonance densities, by means of Eq. (3) and to use the data on low-lying levels to determine the parameter £o only. The results of such estimations of £b are shown in Fig. 2 and they well agree with the results of the combined analysis of both the neutron resonance and low-lying level densities, presented in Fig. 1. From the fitting of obtained parameters the following relations can be proposed for the the energy shift systematics: £,=11.174
-.520-.0795E n
for even-even nuclei,
E0 = - . 3 9 0 - . 000584 -.079<5E0
for odd nuclei,
£„ =-11.174" 6 4 +.285-.079<5E„
for odd-odd nuclei.
(4)
Some remarks should be made about applications of microscopic models for calculations of low-lying level densities. The microscopic approaches based on the shell model enable to consider most consistently the shell, collective and pairing effects in the level density and other statistical properties of nuclei [5, 8, 9].
290
However, practical applications of such models always require a accurate adjustment of model parameters to the corresponding experimental data. Without an adjustment the microscopic models cannot pretend on a higher accuracy of level density calculations than the phenomenological systematics fitted to experimental data. 2
Systematics of Total Radiative Widths
For many nuclei beside the neutron resonance spacing the total radiative widths are determined too [1, 10]. The total radiative widths connected with the level densities by the relation
iru^-x l k w ^ ) ^ p * r
(5)
where A^(e,) is the radiative strength function for the corresponding electric EX or magnetic MX gamma-transitions. As a rule, the dipol electric transitions with energies of 2-3 MeV dominate into the integrant of Eq. (5). Using the power approximations for the radiative strength functions [11] and the constant temperature model, Eq. (5) can be transformed to the very simple form Ty =.624^ L 6 0 r 5
,
(6)
where the numerical coefficient corresponds to the widths expressed in eV. Relations similar to (5) were analyzed by many authors [10, 12, 13]. The radiative widths calculated by means Eqs. (3) and (6) are compared with the available experimental data in Fig. 3. Eq. (6) correctly reproduces main irregularities of the total radiative widths including their increases for the magic nuclei. As it can be seen from the lower part of Fig. 3, the average uncertainty of Eq. (6) is about 50% and for many nuclei this uncertainty is comparable with the errors of experimental data. For nuclei close to the stability valley, in which the direct experimental data on the decay schemes of low-lying levels and the radiative strength functions are available, die calculations of the total radiative widdis could be performed more accurately on me basis of Eq. (5). Of course, an accuracy of such calculations is higher man achievable one for simple approximations similar to Eq. (6). However, for nuclei far off the stability valley the data on discrete levels are rather limited and uncertainties of calculations based on Eq. (5) will be comparable with uncertainties of simple approximating formulas, like Eq. (6), fitted to experimental data. For astrophysical applications it is very important to know the level density of nuclei far off the beta-stability valley. In particularly, the properties of nuclei close to the neutron drip-line are needed for description of the r-process nucleo-synthesis [14]. The total radiative widths estimated in accordance with Eq. (6) for such nuclei
291
' ' '>
• * • '
i
3
10
>
: 0 D qp •
102
4: +"4r
s
H 0.5
^
50
100
150
200
250
mass number Figure 3. Mass dependence of the tatal radiative widths of neutron resonances (upper part) and the ratio of the widths calculated from Eq. (6) to the experimen-tal ones (lower part).
X
0
" ^
a
1.0 " \ .£
>
10'
. V -"v.
-
50
A
^ -v
100
: .
vV , ™«A^ 150
200
250
mass number Figure 4. Mass dependence of the tatal radiative widths (upper part) and the nuclear temperatures (lower part) for nuclei along the r-process path.
are shown in Fig. 4. The shell effects for the corresponding near-magic nuclei are rather strong and they should be taken into consideration in calculations of the neutron capture cross sections used for the r-process analysis. Possible improvements of the neutron capture cross section evaluations for astrophysical applications were discussed recently [15]. It was noted big differences between the neutron cross sections included into the current astrophysical nuclear data networks [16-18] for mass numbers around 130 and 190. Elimination of nuclear data contradictions for near-magic nuclei close to the neutron drip-line will be important for both evaluations of neutron flux densities in supernova explosions and estimations of a direct neutron capture contribution to the r-process nucleosynthesis. To improve accuracy of theoretical evaluations of the total radiative widths the additional experimental data about the averaged intensities of gamma-ray transitions between the low-lying levels should be included into consideration. Nowadays a vast amount of data on individual gamma-transitions is compiled [19] and the systematic statistical analysis of such data is an important topical task.
292
3
Conclusion
New systematics of the constant temperature model parameters is developed on the basis of the combined analysis of the low-lying nuclear levels and neutron resonance densities. The systematics considered could be especially useful for improvement of nuclear data evaluations for nuclei far off the stability valley that have no experimental data on the low-lying levels. References 1. Handbook for Calculations of Nuclear Reaction Data - Reference Input Parameter Library, IAEA-TECDOC-1034, IAEA, Vienna, 1998. 2. Dilg W., Schantl W., Vonach H., Uhl M. Nucl. Phys., A217, 269 (1973). 3. Huang Zhongfu, He Ping, Su Zongdi, Zhou Chunmei, Chin. J. Nucl. Phys., 13, 147 (1991). 4. Gilbert A., Cameron A., Can. J. Phys., 43, 1446 (1965). 5. Ignatyuk A.V., Statistical Properties of Excited Atomic Nuclei (Russian). Moscow: Energoatomizdat, 1983; Translated by IAEA, Report INDC-233(L), Vienna, 1985. 6. Ignatyuk A.V., Smirenkin G.N., Tishin A.S., J. Sov. Nucl. Phys., 21, 255 (1975). 7. Belgya T., Molner G., Fazekas B., Ostor J., Report INDC(NDS)-367, 1997. 8. Ignatyuk A.V., in Nuclear Reaction Data and Nuclear Reactors (Trieste, 1996). Ed. A.Gandini, G.Reffo, World Sci., Singapore, 1998, v. 1, p. 206. 9. Goriely S., Nucl. Phys., A605, 28 (1996). 10. Mughabghab S.F., Divadeenam M., Holden N.E., Neutron Cross Sections, vol. 1, New York - London: Academic Press, 1981. 11. Axel, P., Phys. Rev., 126, 271 (1962). 12. Reffo G., in Nuclear Theory for Applications, IAEA-SMR-43, Trieste, 1980. p. 205. 13. Nedwedyuk K., Popov Yu.P., Acta. Phys. Polonica, B 13, 51 (1982). 14. Rolfs C.E., Rodney W.S., Cauldrons in the Cosmos. Chicago-London: University of Chicago Press, 1988, ch. 9. 15. Ignatyuk A.V., in Proc. 10th Symposium on Capture Gamma-Ray Spectroscopy (Santa Fe, 1999), to be publ. 16. Thielemann F.K., Arnold M., Truran J., in Advances in Nuclear Astrophysics, Ed. E.Vangioni-Flam, Gif sur Yvette: Editions Frontiere, 1987, p. 525. 17. Hoffman R.D., Woosley S.E., Astrophys. J., 395, 202 (1992); Stellar Nucleosynthesis Data, www-ie.lbl.gov/astro (1992). 18. Goriely S., Phys. Lett., B436, 10 (1998); Hauser-Feshbach Reaction Rates, www-astro.ulb.ac.be (1998) 19. Firestone R.B. ed. Table of Isotopes, 8th edition, John Wiley and Sons, Inc., New York, 1995.
COOLING OF N E U T R O N STARS REVISITED: APPLICATION OF LOW E N E R G Y T H E O R E M S A. E. L. DIEPERINK, E. N. E. van DALEN, A. KORCHIN, R. TIMMERMANS Kernfysisch Versneller Instituut Zernikelaan 25, 9747AA Groningen, The Netherlands E-mail: [email protected] Cooling of neutrons stars proceeds mainly via neutrino emission; As an example we study the modified neutrino bremsstrahlung process nn —> nnuw. The radiated energy is small compared to other scales in the system. Hence one can use low-energy theorems to compute the neutrino emissivity in terms of the non-radiative process, ie the on-shell T matrix. We find that the use of the X— matrix as compared to previous estimates based upon one pion exchange leads a substantial reduction of the predicted emissivity.
1
Introduction
The thermal evolution of neutron stars is dominated by the weak interaction and in particular neutrino interactions with the hadronic matter. The hadronic information is contained in the socalled current-current correlator U(q), where q = (q, w) is the four-momentum of the neutrinos. In general one can distinguish two diiferent regimes: • neutrino scattering (space-like, UJ < q) • neutrino-pair emission (time-like, u> > q). In the latter case the quasi-particle response vanishes, i.e., the one-body process n —> n+ v + 9 is forbidden by energy-momentum conservation, and the socalled URCA process • n —> p + e~ + v p + e~~ —> n + v, is forbidden unless the n:p and e~ fermi momenta satisfy the unequality P F + PeF > P F ' which is very unlikely. Therefore the bremsstrahlung process can only take place in the presence of spectator nucleon (referred to as modified processes), e.g.: • n+N—^n + N + u + u (N = n,p) • n + n—>n+p + e~ + 9(+ inverse) In the pioneering work of Friman and Maxwell 2 these processes were computed in the extreme soft neutrino limit in Born approximation with the NN interaction presentated by a Landau type interaction plus a one-pion exchange part. With respect to this approach several questions can be asked; how accurate is 293
294
q Pi
-
i r
p?
-
u
Pi
Pi
T
Pi' r
1
Pz
P?
, p2'
Figure 1: Leading order diagrams for soft bremsstrahlung
the use of Born approximation, what is the contribution of other mesons, and how important are relativistic effects? It is the aim of the present study to address these issues by computing the emissivity using a low-energy theorem on the basis of the observation that the neutrino radiation is very soft compared to other scales in the system (u> « T = 1 MeV); this allows one to use an empirical T—matrix, fully determined by phase shifts. 2
Soft electro-weak b r e m s s t r a h l u n g
First we consider the "Soft-photon" amplitude in free space The original soft-photon theorem states that the first two terms in an expansion of the electromagnetic bremsstrahlungs amplitude in photon four-momentum q are fully determined by the amplitudes of the non-radiative process, l M = A/w + B + 0(u>) This result can be generalized in several ways, e.g. to virtual bremsstrahlung, (q2 > 0) and also to the case of the weak axial vector current. Here we shall restrict ourselves to the leading order, the A term, in which case only radiation off external legs contributes (see Fig. 1), and the amplitude can de expressed as M„ = TS(PI - <7)rM + rvs(K + q)T + {i ~ 2} The weak vector and axial vector vertices Y are given by low-energy neutral current Hamiltonian,
with the hadron current given by B^ = J2i=n,P^i'yIJ'((^-",i ~ Is^A^^iin the non-relativistic limit r M (vector) « gvJn -> <5M,o
Hence
295 r^(axial) « 3^757M ~* 9A° (* = P. n i ^
= -»A)
The two Feynman propagators (corresponding to prior and post emission) are given by S(p±q)
=- — ± . « — = ^ j - « ±"(1+ -y(P±q) — m p.q uj—p.q/m u>
Ofrq/mu,)).
Hence in this order the amplitude M can be expressed as a commutator of T with the space component of the axial weak current operator Mi =
^[T,Si]
where S = (a\ + cr2)/2, and in this order there is no vector contribution. Which terms survive the commutator [J' i r w e a k ] ? Looking at the structure of the NN amplitude given in the appendix one sees that there will be nonvanishing contributions from tensor, spin-orbit and quadratic spin-orbit terms in T. Thus one may conclude that only nn spin correlations contribute to the emissivity. However, we note that in the past the effective nn interactions were restricted to purely local interactions and hence the spin-orbit interactions were not considered. As an illustation we compute the cross section for NN electroweak bremsstrahlung in free space as a function of the relative nn momentum (in the nn cm system). In fig. 2a we show the results for full T— matrix and its separate components. In fig. 2b the cross section for the full T—matrix is compared with the result for n, -n + p and also a meson exchange. We note that the OPE exchange (which forms the basis of most standard "cooling scenarios" overestimates the results obtained with the full T—matrix by about a factor 5 for relative momenta in the range appropriate for neutron matter at normal density (pF = 1.3fm_1) (a similar result has been found in 4 ). It is also seen that the contribution from the OPE tensor force is largely canceled by that from p exchange, and that the sum of OPE and ORE is much closer to the T -matrix result than OPE . 3
T h e n p case
The np case is slightly different, since the axial vector couplings of neutron and proton have opposite values g\ — -g\ = -1.25. Hence in this case the net result can be expressed as the sum of a commutator and an anticommutator, where the latter corresponds to the exchange diagrams Mi = £±([Tdir, Di] + {T^ch,
Dt})
296
1e-22 -
^
—
-
-
^
•
.
*
yS
—
—
.
~
\
1e-23 -t
\
/ /
...
- • • • " '
< S
»
\ 1e-24 -
1
,
,
100
\ ,
250
p (MeV/c)
1S0
200
p (MeV c"1)
Figure 2: Cross section for weak nn bremsstrahlung for w= lMeV as a function of the relative momentum p = (pi — P2)/2; top: full T-matrix (solid line), tensor+spin-spin (dashed), spinorbit (dotted) and quadratic spin-orbit (dashed-dotted line); bottom: O P E (solid), O P R E (dotted), OPRSE (long dashed), direct O P E (short dashed), direct O P R E (dashed-doubly dotted) and full T (dashed-dotted line).
297 where D = {o\ - v2)/2, and Tdir'exch are denned in the appendix. In this case also the socalled Landau terms in the T-matrix, of the form ga\.a2and g'a\.a2T\.T2 contribute to the anti-commutator. In practice the proton fraction in a neutron star is quite small and therefore we find that the relative contribution from np is quite unimportant. 4
Emissivity in medium
The emissivity in the medium with given density and temperature T can be computed from the amplitude in two ways: (i) via the (imaginary part) of the current-current correlator, or (ii) using the Fermi golden rule. The former approach is more general, but the latter is simpler to use and is sufficient in the present case. In lowest order in the virial expansion it amounts to convoluting the free space matrix elements \M2\ with the hadronic Fermi-Dirac functions 6 = n, / J
d
3
^l^( wi ui2
p i
W l
+u;2)54(Pl + p2 -
P 3
-
P 4
-
q i
-
q2)F\
< Mj > | 2 ,
where F = / i / 2 ( l — h){^ — IA)- In practice the integrals are simplified by taking the absolute values of the nucleon momenta equal to the fermi momenta. If we consider the ratio of the emissivity calculated with the full T matrix and the one from the O P E 2 we obtain about a factor 4-5 reduction. This is inline with the conclusion in 4 for pure neutron matter. Other medium that need to be considered in further work are 1) replacement of the free T- by a G-matrix, which takes into account the Pauli blocking and other medium effects, 2) medium renormalization of the axial coupling vertex, 3) inclusion of higher order medium effects such as dressing Green functions. In the latter case one has to take care to conserve the symmetries (CVC, PCAC) of the problem (which are conserved in the present case). Acknowledgments We thank A. Sedrakian for stimulating discussions. This work has been supported by the Stichting voor Fundamenteel Onderzoek der Materie with financial support from the Nederlandse Organisatie voor Wetenschappelijk Onderzoek. Appendix: Structure of the on-shell N N amplitude The on-shell T-matrix is fully determined by the nn phaseshifts. consider nn; in a covariant approach one has Ton = Ta(s,t,u)u(p2)nau(p2)
x u(pi)n a «(pi)
First we
298 with ftQ =
{1,CTMI1/,7M,75,7M75}
Non-relativistically T = TC + Taax.ai + TTS12 + TsoL.S + TQQ12 In case of np there are twice as many components which can be distinguished by isospin / = 0,1 , i.e. Tdir = (T° + Tl)/2 and Texeh = (T° - T 1 )/2. References 1. 2. 3. 4.
F.E. Low, Phys. Rev. 110,974(1958) B.L. Priman and O.V. Maxwell, ApJ. 232,541(1979) R. Timmermans et al, in preparation C. Hanhart, D. Phillips and S. Reddy, astro-ph/0003445
THE ROLE OF ELECTRON SCREENING DEFORMATIONS IN SOLAR N U C L E A R FUSION REACTIONS A N D THE SOLAR N E U T R I N O PUZZLE
1
T h e o d o r e E . Liolios 1 ' 2 ' 3 " European Center for Theoretical Studies in Nuclear Physics and Related Villa Tambosi, 1-38050 Villazzano (Trento), Italy 62 University of Thessaloniki, Department of Theoretical Physics Thessaloniki 54006, Greece 3 Hellenic War College, BST 903, Greece
Areas
Thermonuclear fusion reaction rates in the solar plasma are enhanced by the presence of the electron cloud t h a t screens fusing nuclei. The present work studies the influence of electron screening deformations on solar reaction rates in the framework of the Debye-Hiickel model. These electron-ion cloud deformations, assumed here to be static and axially symmetric, are shown to be able to considerably influence the solar neutrino fluxes of the pp and the CNO chains, with reasonable changes in the macroscopic parameters of the standard solar model (SSM).
In the stellar plasma gravitational compression and quantum mechanical tunneling combine in order to achieve the classically impossible fusion between light nuclei. The electron gas that surrounds the nuclei acts as a catalyst in the reaction, by lowering the repulsive Coulomb barrier which prevents atomic nuclei from approaching each other 1»2»3'4'5'6. In the framework of the Debye-Hiickel (DH) model each nuclei is assumed to polarize its neighborhood creating a spherically symmetric but inhomogeneously charged ionic cloud around it. In this model, the potential V (r) of a given nucleus of charge Z\ is: VD(r)=^exp(-^-) r
(1) \
rD J
with r
D
47re2 kT
Y^ Zfni + neee
(2)
where rr> is the Debye-Hiickel radius at temperature T, 0e is the electron degeneracy factor, and n the number densities of ions (n,-) with atomic number Zi, and electrons ( n e ) , respectively. a
[email protected] Correspondence address
299
300
Studies of heavy nuclei fusion reactions have shown that theoretical predictions of cross section can be greatly improved7 by assuming rotations and deformations of the fusing nuclei. It is therefore plausible to consider similar effects in the study of screened thermonuclear reactions where the electron cloud is assumed to be deformed. In fact this deformation can be parametrized in the framework of the liquid-drop model so that the Debye-Hiickel radius is considered a measure of the electron cloud. The deformed DH radius is now: rD (e) = r^
[I+/3Y2° (cose)]
(3)
where the angle 9 is measured from the axis of symmetry i.e. the z axis. For all orientations the weak screening (WES) assumption ZlZ2e2 < 1 rD (0) kT
(4)
-0.8
(5)
yields the inequality:
where we have disregarded the contribution of volume conservation which is always less than 5%. The deformation parameter can take all the above values without violating the WES regime , thus rendering the use of the deformed screening formalism legitimate. If we disregard non-linear screening corrections10 the screening deformation effects on pp reactions can be simply represented by Salpeter's1 WES formula fD(9;/3)=exp
\kTrD
{6)
(6)
where the DH radius ro (9) is now orientation dependent. Using Eq. (3) we obtain:
fD(6;P) = (f$ e M ff
-1
(9
(7)
where g (9; /?) is the ratio rr> (9; 0) /ro: -1/3
g{?;P)
1 + /?
— 16TT
(3w
V
2
1)
du
00 20 1 i+N^zi* 16TT * - )
(8)
301
In most solar evolution codes the pp screening factor is evaluated by means of Salpeter's formula which has been proved to be valid and accurate in standard conditions6. In the deformed case the quantity fo should be used, instead. We can obtain an approximation of the neutrino flux uncertainties introduced due to the presence of such deformations by using the proportionality formulae8'9 which relate neutrino fluxes to screening factors. In order to isolate the pp uncertainty, we will assume that except for the pp reaction all the other neutrino-producing reactions remain unaffected by the deformations, thus obtaining a minimum of the total associated uncertainty. For various solar fusion reactions the ratios of the deformed neutrino fluxes <&D to the ones obtained in the WES regime <$WES, are as follows: ^fre+v.)!!':
( - £ J J U\
B'ipe-
{Jj^j
D
,v.) H* :
N 14 -( = ( - f° ^ j )°
\
/
i
\
(9)
-0.08
= (j^sY
(1°)
pp
fle7(p,7)58(e+,,e)58*
^ ( ^ C - O ^ e V e ) ^
:
5
:
^
"
(12)
ȣ- = ( ^ - )
(13)
=
(
£
y
According to the above formulae, for a collision along the z axis with /? = —0.8 we obtain fo = 1.16 and the corresponding uncertainties are at least 23% for the N, O and B8 neutrinos, 9.2% for the Be7 neutrinos and less than 2% for the pp, hep ones. Adding the effects of the screening factors of the other reactions, which have been disregarded so far, the uncertainties can be dramatic. It seems therefore that the presence of screening deformations in the sun can tune the predicted neutrino fluxes in order to reduce the observed deficit. It is crucial to study what deviations from the SSM parameters such deformations can induce. As was previously noted conservation of luminosity implies a reduction in the central temperature Tc as a result of an increase in Spp, since LQ ~ SPPT8. This implies:
302 ID \fWES
J'WES ~
Therefore, for the above considered deformations t h e (WES) central temperature of the sun would have to decrease by 1.2%. O n the other hand, as the ratio p/T3 is approximately constant along the solar profile, the new central density would now be given by: P°c Pc
nWES
( SD fWES \Jo
~ \
that is roughly decreased by 3.7% with respect to the (WES) assumption. Both values represent reasonable deviations from the (WES) SSM considering that the (WES) assumption itself causes a 0.6% deviation from the central temperature of the unscreened solar plasma and another 1.8% deviation from the respective central density. Acknowledgments This work was financially supported by the Hellenic State Grants Foundation (IKY) under contract #135/2000. It was initiated at the Hellenic War College while its revised version was written at ECT* during a nuclear physics fellowship. The author would like to thank the director of ECT* Prof. R. Malfliet for his kind hospitality and support. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.
E.E.Salpeter, Aust.J.Phys.7,373,(1954) H.C.Graboske, H.E.DeWitt, A&A, 181, 457 (1973) H.E.Mitler, ApJ 212, 513(1977) N.J.Shaviv, G. Shaviv, ApJ 468, 433(1996) H.Dzitko, S.Turck-Chieze, P.Delbourgo-Salvador , C.Lagrange, Apj. 447, 428 (1995) A.V.Gruzinov, J.N.Bahcall, ApJ, 504 ,996 (1998) P.Moeller, A.Iwamoto, Nucl.Phys.A. 575, 381(1994) J.N.Bahcall, 1989, Neutrino Astrophysics, Cambridge University Press B.Ricci, S.Degl'Innocenti, G.Fiorentini, Phys.Rev.C. 52, 1095 (1995) T.E.Liolios , Phys.Rev.C, 61, 055802 (2000)
NUCLEAR MASSES AND HALFLIVES: STATISTICAL MODELING WITH NEURAL NETS E. MAVROMMATIS, S. ATHANASSOPOULOS AND A. DAKOS Physics Department, Division of Nuclear and Particle Physics, University of Athens, GR -157 71 Athens, Greece K. A. GERNOTH UMIST, P.O. Box 88, Manchester M60J0D, United Kingdom J.W. CLARK McDonnell Centerfor the Space Sciences and Department of Physics, Washington University, St. Louis, Missouri 63310, USA Global statistical models of nuclear masses and halflives are developed based on multilayer feedforward neural networks with performance comparable to the models based on quantum theory.
1
Introduction
During the last few years, a statistical approach to nuclear systematics based on multilayer feedforward neural networks has been under development1121. The networks are trained on backpropagation algorithm or on its modifications and are applied to generate a "predictive" statistical model of different nuclear properties.To date, global neural-network models have been successfully developed for the stability/instability dichotomy, for the atomic-mass table, for neutron separation energies, for spins and parities, for decay branching probabilities of nuclear ground states and for • " decay halflives[37]. In the present work we report new results for nuclidic masses and for the halflives of unstable nuclear ground states that decay 100% via the • " mode. 2
Masses
The problem of devising global models of nuclidic (atomic) masses has a long history (for recent reviews, see refs.[8,9]). The primary aims are i) a fundamental understanding of the physics of the mass surface and ii) the prediction of the masses of "new" nuclides far from stability. The predicted masses are of great current interest in connection with present and future experimental studies of nuclei far from stability, conducted at heavy-ion and 303
304
radioactive-ion beam facilities. Moreover, the results are useful in such astrophysical problems as nucleosynthesis and supernova models. The existing global models of the mass table lie on a spectrum extending from low to high theoretical input (and correspondingly, high to low numbers of fitting parameters). Neural network mass models, as currently developed, rely on minimal theoretical input. The current work is a continuation of the program established in refs. [3,4,6] with improvement of certain aspects of coding and training110,111. In particular, we mention the use of a modified version of the backpropagation algorithm which proves in most cases to be more efficient in avoiding local minima of the cost function that the learning algorithm is intended to minimize. We report results obtained with a special database (MN) consisting of 1323 "old" (O) experimental masses (used as training set) that the 1981 Moller-Nix model[121 was designed to reproduce, together with 351 "new" (N) mass data that lie mostly beyond the edges of the 1981 data set (used as validation or prediction set). Three different input coding schemes have been used along with an analog output unit scaled in different ways. As performance measures we have used the rms error CX^^, as well as the number of patterns for which the output value deviates from experimental target value by less than 5%. Some of our results for the mass excess A M are reported in Table 1. Table 1: Comparison of neural network models of mass excess data with other models based on nuclear theory. Use is made of data basis MN[1323(0) - 351(N)]. Net type of model (I-Hr...-HL-0){P\ (16-10-10-10-1) [401] Z & N in binary (18-10-10-10-1) [421] Z &N in binary and analog (4-10-10-10-1)* [281] Z & N in analog and parity
Learning mode • RMS (MeV) Recalled Patterns 1172/1323 0.393
Validation (v) & Prediction (p) mode • «Mj(MeV) Recalled Patterns 3.575 (v) 246/351
0.331
1187
2.199 (v)
272
0.491
1141
1.416 (v)
280
(4-10-10-10-1)** [273]a)Z & N in analog and parity (4-10-10-10-1)* [281] Z & N in analog and parity (18-10-10-10-1) [421]b) Z &N in binary, A & Z-N in analog (4-40-1) [245]c)
0.617
1095
1.209 (v)
284
0.453
1242
1.200 (v)
298
Moller et a l *
0.673
a) see ref.[10]
5.981 (p)
0.828
1.068
b) see ref. [6]
-
0.735 (p)
c) see ref.[16]
d) see ref. [8]
3.036 (p)
-
The fifth network model listed in the table, with gross architecture (4 - 10 - 10 - 10 - 1)+(281) is the best neural-network model of atomic systematics created to date, if one makes a judgement based on the accuracy of the calculations for the "N" set. Most interesting is the comparison with the results from the
305
macroscopic/microscopic theoretical treatment developed by Moller, Nix and collaborators which gives Gms =0.735 MeV for the "N" set[8]. For the NB1 set, which is another set of 158 nuclei that lie outside the "O" and "N" databases19131 and which provides a more legitimate test set since it was not referred to at all in the training process, the values of (JUMS given by the selected neural-network model and the FRDM model of ref.[13] are 1.462 MeV and 0.697 MeV respectively. Based on the above and other tests and examples, it is evident that the current generation of neural-network models of the mass table represents a significant step toward extrapability levels competitive with those reached by the best traditional global models rooted in quantum theory. 3
Halflives
Concerning halflives attention is restricted here to the problem of predicting the halflives T'w of nuclear ground states that decay 100% by the • " mode. Prediction 72
of the • -decay lifetimes of neutron-rich nuclei is of great current interest from the perspective of nuclear astrophysics, main-stream nuclear physics and nuclear technology1141. Models rooted in quantum theory and involving large-scale computations entail simplifications and approximations, with the consequence that the results for • " halflives sometimes deviate from experiment by a factor of 10. The neural network statistical approach offers a promising alternative. Successful multilayer neural network models of • " decay systematics have already been described in ref. [7]. Here we present results from the best network model obtained so far when an additional input is employed together with Z and N, namely the Q-value of the decay process. (A preliminary report on our results has been given in ref. [10]). The network in question has architecture (17 - 10 - 1). Standard backpropagation with a momentum term has been adopted as the learning algorithm.The data set has been formed from data available in early 1995 from the Brookhaven National Nuclear Data Center by including nuclei with halflives not greater than 106 sec. This data set consists of 692 nuclides, of which 518 are reserved for learning and 174 for prediction. Binary coding of Z, N and analog coding for Q have been used at the input layer and a single analog unit that generates a coded value of lnT,/ has been employed at the output layer. The /2
performance of our network in learning and prediction mode is illustrated in Table 2 for the odd - odd, odd - even and even - even nuclides of our base with halflives not greater than 103 sec. Performance is assessed in terms of the deviation (x)M(M10 = 1 0 ^ * ) and associated standard deviation <7M (Gm = 1 0 ^ ) introduced by Moller et al[13]. Our results are compared with those of the conventional global models of Homma et al.[15] and Moller et al.[131. The level of
306
performance displayed by the network model is similar to (and in some cases better than) that of the latter models. Table 2: Illustration of the power of the neural network model [ 1 7 - 1 0 - 1 ] to yield the values of • " decay halflives and comparison with the results of Moller et al. [13] and H o m m a et al. [15]. Smaller M 1 0 implies a better fit or prediction. Learning
T
K
(sec) <1
0-0
o-e e-e <10
0-0
o-e e-e <100
0-0
o-e e-e <1000
0-0
o-e e-e
Prediction
Moller etai. [13]
H o m m a et a].[15]
n
M1"
. l0
n
IT
. 10
n
M10
. '"
n
M10
10
32 52 19 67 101 43 92 156 59 110 204 73
1.15 1.07 1.61 1.17 1.04 1.19 1.18 1.05 1.19 1.19 0.98 1.14
2.27 2.03 1.71 2.25 1.91 2.09 2.18 1.93 1.97 2.13 1.99 1.95
10 20 5 18 30 12 39 58 25 52 91 31
2.05 1.08 1.79 2.26 1.19 1.31 1.76 1.12 0.98 2.22 1.22 0.93
2.31 2.38 2.71 5.42 2.44 2.30 5.19 3.15 2.67 6.25 5.50 4.78
29 35 10 59 85 34 88 133 54 115 194 71
0.59 0.59 3.84 0.76 0.78 2.50 2.33 1.11 2.61 3.50 2,77 6.86
2.91 2.64 3.08 8.83 4.81 4.13 49.19 9.45 4.75 72.02 71.50 58.48
28 31 10 66 81 34 85 127 52 93 157 63
1.75 0.60 1.15 1.89 0.92 1.01 3.15 1.07 1.13 3.02 1.10 1.39
4.96 2.24 2.36 4.60 3.84 2.93 10.51 4.29 3.58 10.25 5.55 6.10
Q Experimental —
—
Tl/2
0 Neural Net
(sec)
^f »Cu
1
80
Zn
8]
Ga
i~kni .n1 h. 1 U. .X. 83
Ga
130 c d 131 I n 133 I n HI S n M5 sb i38 T e
Fig 1: • decaying nuclides that are found on or near the r - process path
Two qualifications should accompany on judjement of the relative merits of neural network and traditional approaches. On the one hand, comparison is hindered by the absence of a clear distinction between the aspects of fitting and prediction in the conventional treatments and on the other, the neural network model has many more adjustable parameters than the traditional models. At any rate, the good performance of the (17 - 10 - 1) network model is demonstrated in Fig. 1, where the results given by this network for nuclides on or near the r-process path are
307
compared with experiment. The encouraging results of these and other computer studies provide a strong incentive for seeking further improvements of network performance and extending the approach to other decay modes. 4
Acknowledgement
This research was supported in part by the U.S. National Science Foundation under Grant No. PHY-9900713 and by the University of Athens under Grant No. 70/4/3309. References 1. J. W. Clark in Scientific Applications of Neural Nets (Springer - Verlag, Berlin, 1999), J. W. Clark, T. Lindenau, and M. L. Ristig, eds. (Springer-Verlag, Berlin, 1999), p. 1 2. K. A. Gernoth in Scientific Applications of Neural Nets, J. W. Clark, T. Lindenau, and M. L. Ristig, eds. (Springer-Verlag, Berlin, 1999), p. 139 3. S. Gazula, J. W. Clark and H. Bohr, Nucl. Phys. A540 (1992) 1 4. K. A. Gernoth, J. W. Clark, J. S. Prater and H. Bohr, Phys. Lett. B300 (1993) 1 5. K. A. Gernoth and J. W. Clark, Neural Networks 8 (1995) 291 6. K. A. Gernoth and J. W. Clark, Comp. Phys. Commun. 88 (1995) 1 7. E. Mavrommatis, A. Dakos, K. A. Gernoth and J. W. Clark, in Condensed Matter Theories, Vol. 13, ed. by J. da Providencia and F. B. Malik (Nova Science Publishers, Commack, NY 1998) 423 8. P. Moller and J. R. Nix, J. Phys. G 20 (1994) 1681 9. a) C. Borcea and G. Audi, Rom. J. Phys 38 (1993) 455; b) G. Audi, O. Bersillon, J. Blachot and A. H. Wapstra, Nucl. Phys. A624 (1997) 1 10. E. Mavrommatis, S. Athanassopoulos, A. Dakos, K. A. Gernoth, and J. W. Clark, Proceedings of the Ninth Panhellenic Symposium on Nuclear Physics, 1998 (in press) 11. S. Athanassopoulos, E. Mavrommatis, K. A. Gernoth, and J. W. Clark, to be published 12. P. Moller and J. R. Nix, At. Data Nucl. Data Tables 26 (1981) 165 13. P. Moller, J. R. Nix and K. L. Kratz, At Data Nucl. Data Tables 66 (1997) 131 14. H. V. Klapdor, Prog. Part. Nucl. Phys. 10 (1983) 131 15. H. Homma, E.Bender, M. EBrsch, K. Muto, H. V. Klapdor-Kleingrothaus and T. Oda, Phys. Rev. C54 (1996) 2972
QUASI-THERMAL P H O T O N BATH FROM BREMSSTAHLUNG P. M O H R , M. B A B I L O N , J. E N D E R S , T. H A R T M A N N , C. H U T T E R , K. V O G T , S. V O L Z , A N D A. Z I L G E S Institut fur Kernphysik, Schlossgartenstrafie E-mail:
Technische Universitat 9, D-64289 Darmstadt, [email protected]
Darmstadt, Germany
A superposition of bremsstrahlung spectra was used to simulate the thermal photon bath in stars at temperatures of 2-3 billion degrees which are typical for supernova explosions. In this astrophysical environment the neutron-deficient p-nuclei are synthesized by a series of photon-induced reactions. As first example we measured the (7,n) reaction rates of several platinum isotopes by the photoactivation technique.
1
Introduction
The bulk of the heavy nuclei beyond the abundance peak around iron has been synthesized by neutron capture in the s- and r-process. However, many of the neutron-deficient nuclei with A > 100 cannot be produced by neutron capture. The main production mechanism for these so-called p-nuclei is photodisintegration in the 7-process by (7,n), (7,p), and (7,a) reactions of heavier seed nuclei synthesized in the s- and r-process. Typical parameters for the 7-process are temperatures of 2 < Tg < 3 (T9 is the temperature in 109 K), densities of about p w 106 g/cm 3 , and time scales T in the order of seconds. Several astrophysical sites for the 7-process have been proposed, and the oxygen- and neon-rich layers of type II supernovae seem to be a good candidate. It has to be pointed out that no experimental data are available for the cross sections resp. for the reaction rates of the 7induced reactions at astrophysically relevant energies. All reaction rates have been derived theoretically using statistical model calculations. Details about the 7 process can be found in Refs. 1.2-3.4.5>6. 2
Reaction rates in a thermal photon bath
The reaction rate A of a photodisintegration reaction B(7,x)A is given by X(T)=
cn.,(E,T)ah,x)(E)dE Jo 308
(1)
309 with the speed of light c and the cross section of the 7-induced reaction <7(7)X)(.E). Obviously, A is also the production rate of the residual nucleus A. The photon density ny(E,T) is the number of 7-rays at energy E per volume and per energy interval:
^E-T»=(^(^)3exp(E/Ir)-l
•
<">
The integrand of Eq. (1) is given by the product of the photon density n 7 which decreases with the energy E and the photodisintegration cross section a which increases with the energy E. The integrand has a sharp maximum at energies of about kT/2 above the neutron threshold with a typical width of about 1 MeV. A measurement of the cross section in this narrow window is already sufficient to derive the astrophysical reaction rate. 3
Experimental simulation of a thermal photon bath
A quasi-thermal photon spectrum is obtained in a given energy range by a careful superposition of bremsstrahlung spectra at different endpoint energies: Cnj(E,T)
« $[f rems (T) = £ > ( T ) x d>brems(£o,;)
(3)
i
where the a,i(T) are strength coefficients which can be adjusted for any temperature T relevant for the 7 process. The superposition is shown in Fig. 1 for the temperature of Tg = 3.0, i.e. for T = 3 billion degrees. We have irradiated several platinum samples of natural isotopic composition with the above shown bremsstrahlung spectra, and we have measured the activation yields of the 19 °. 192 > 198 pt(7 )n ) reactions. The half-lives of several lines in the activation spectra were determined, and the results agreed well with the adopted half-lives 7 . The experiment was performed at the new set-up for real photon experiments 8 installed at the superconducting electron accelerator S-DALINAC at the Technische Universitat Darmstadt. The electrons with typical energies up to 10 MeV and beam currents of about 30 — 40 fiA were stopped completely in a massive copper radiator. The bremsstrahlung was collimated in a 95 cm copper collimator. The target was located about 150 cm behind the radiator and was sandwiched between two thin boron layers. The incoming photon flux was monitored by measuring the scattered photons in the n B ( 7 , 7 ' ) reaction. The experimental yield Yi per target nucleus is given by Yi=
[*biena(Eo,i)(rh,x)(E)dE
.
(4)
310 10T 9 = 3.0 E(Bremsstrahlung) Bremsstrahlung 10 (7,n)threshold _
s io '> r-™10"2
10"
10
5000 E 7 (keV)
10000
Figure 1. The superposition of several bremsstrahlung spectra 3>JJrem C^) (^ u " ' ' n e ) with different endpoint energies EQ is compared to the thermal Planck spectrum ^(EjT) (dashed line) at the temperature of Tg = 3.0. A good agreement is found from about 7 to 10 MeV with the superposition of only six endpoint energies. The six contributing bremsstrahlung spectra *brems(^o,i) are shown as dotted lines. Typical thresholds of (7,11) reactions are indicated by an arrow.
A comparison of Eq. (4) with Eqs. (1) and (3) relates the astrophysical decayrate A directly to the experimental yields F; by
A(r) = ^ a , ( r ) x y ,
(5)
The average deviation between the thermal Planck distribution [Eq. (3), l.h.s.] and the bremsstrahlung-approximated quasi-thermal distribution [Eq. (3), r.h.s] is about 10% in the relevant energy region around f?eff » £
Conclusions and outlook
In further experiments we plan to measure (7,n) cross sections for many nuclei. The aim is to obtain a data basis for the 7 process calculations which is
311 Table 1. Summary of the results of the platinum activation experiment: decay rates A for several platinum isotopes at the temperature of TQ = 2.5. Note the much larger reaction rate of 1 9 8 P t which is a consequence of the significantly lower threshold energy.
190
Pt 192p t 198p t
Ethr (keV) 8911.4 8676.3 7556.6
Afs-1) «0.4 0.37±0.07 62±9
based on experimental data. The new results should also be used to test the statistical model calculations for a broad range of nuclei. This might help to improve the model predictions for unstable nuclei where our photoactivation method is not applicable. Furthermore, we will try to improve our experimental setup to measure also (7,a) reactions. (7,a) reaction rates are an additional important ingredient for the understanding of the nucleosynthesis in the 7 process as has been pointed out e.g. in 9 , 1 ° . Acknowledgments We want to thank the S-DALINAC group around H.-D. Graf for the reliable beam during the photoactivation and A. Richter and U. Kneissl for valuable discussions. This work was supported by the Deutsche Forschungsgemeinschaft (contracts Zi510/2-1, Ri242/12-2, FOR272/2-1). References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.
D. L. Lambert, Astron. Astrophys. Rev. 3 (1992) 201. S. E. Woosley and W. M. Howard, Astrophys. J. Suppl. 36, 285 (1978). M. Rayet et al., Astron. Astrophys. 227, 271 (1990). B. S. Meyer, Ann. Rev. Astron. Astrophys. 32, 153 (1994). M. Rayet et al., Astron. Astrophys. 298, 517 (1995). V. Costa et al, astro-ph/0005513, 2000. P. Mohr et al., Europ. Phys. J. A 7, 45 (2000). P. Mohr et al, Nucl. Inst. Meth. Phys. Res. A 423, 480 (1999). T. Rauscher, Proc. Nuclei in the Cosmos V, Editions Frontieres, 1998. C. Grama and S. Goriely, Proc. Nuclei in the Cosmos V, Editions Frontieres, 1998.
NUCLEAR STRUCTURE NEAR THE NEUTRON DRIP-LINE AND RPROCESS CALCULATIONS
WILLIAM B. WALTERS Department
of Chemistry
and Biochemistry, University of Maryland, 20742 USA E-mail: [email protected]
College Park, MD
KARL-LUDWIG KRATZ AND BERND PFHFFER Institute f r Kemchemie, E-mail:
Universitt Mainz, D-55128 Mainz, [email protected]
Germany
Recent experiments continue to provide evidence for the reduction of shell strength and the associated neutron binding energies for heavy isotopes of elements 40 t Z t50 for nuclides with N f 82. Results of these experiments are presented and their implication is discussed.
1
Introduction
Nature has provided the abundances of the chemical elements and isotopes created in past astrophysical events, including supernovae, thereby challenging investigators with the dual task of developing models for nucleosynthesis and models for nuclear structure that would reproduce these abundances. [4] From the raw data, it is possible to determine the abundances of isotopes produced in explosive nucleosynthesis known as the r-process. This abundance curve exhibits five features, two large peaks (A - 130 and -190) , one small peak, one modest valley (A - 180), and several sections where the yields do not vary much, including a region between A = 112 and A = 124. The two large peaks have long been attributed to buildup of nuclei along the closed neutron shells at N = 82 and N = 126 as described by the waiting-point hypothesis. Moreover, the dip at A = 180 is also ascribed to the strong binding of neutrons just below the N = 126 shell closure that depresses the photodisintegration rate and lowers the equilibrium concentrations of these nuclides. Similar reasoning would suggest a similar dip should be expected for A - 120, but none is observed. 2
New data
In Figure 1 are shown the energies of the 2+ levels for Pd, Cd, and Te nuclides with neutron numbers that range from the closed shell at N= 50 and the closed shell at N = 82. This figure includes new data from the study of the decay of the heavy Ag nuclides isolated at ISOLDE by the use of the Resonance Ionization Laser Ion Source (RILIS) as well as new data for Pd-118 from fission spectroscopy. [1] [3] 312
313
Also shown are the ratios of the energies of the first 4+ levels to the energies of the first 2+ levels. Note that the points on the graph for the very heavy Pd
Figure 1. Energies for the first 2+ levels in the even-even Pd, Cd, and Te isotopes along with the ratio of the energy of the first 4+ level to that of the first 2+ level. Note that the values shown for the heavy Pd nuclides are calculated values.
2000
I calculated 2.5
1500 >
1.5
.§1000 E? CD C CD
+
1
+ CM
500
fcuTalxi 0.5 i?+ energies j(keV) |
Q
t i l , » 11*.i I M I I I I i li.liiil I. I t.l..i.liili.liill.Hiili.l .1 i«iii*nfl I M n f l Hil.Ai.it Hi,ifi li I ' I I ' ' ' '
45
50
55
60 65 70 75 Neutron number
80
Q
85
nuclides are calculated values. [2] These values were calculated by methods that have been able to provide an excellent fit to the structure of the lighter Pd nuclides. What can be noted in these values is the clear departure from a smooth rise in the energies of the 2+ level in the heavy Cd nuclides, contrary to the behavior shown by the Te nuclides approaching N = 82 and by the behavior shown for both lighter Cd and Pd nuclides.
314
3
Calculations
The effect shell-quenching is readily illustrated as shown by Kratz et al., [4] in their Figure 2. They show that without the inclusion of the shell quenching, the large expected dip in yields is calculated, comparable to that calculated for A - 180, whereas, with the shell quenching, a flat yield curve is obtained that is much closer to the experimental values.
4
Conclusions
The new experimental data provide only the first step toward an understanding of the nuclear structure of the r-process nuclides in the region 100 t A f 125. However, they do point out the strong role played by experimental data in providing constraints in astrophysical models for r-process nucleosynthesis. 5
Acknowledgements
The authors are grateful for the outstanding work performed by the members of the ISOLDE collaboration during the collection of the data presented here. Support from the U. S. Department of Energy and the German BMFT is also appreciated. References 1. M. Houry et al., Structure of neutron rich palladium isotopes produced in heavy ion inducedfission,Eur. Phys. J. A 6 (1999) 43. 2. T. Kautzsch et al., New states in heavy Cd isotopes and evidence for N = 82 shell-quenching, Eur. Phys. J. A in press (2000). 3. Kim, K.-H, et al., IBM-2 calculations of even-even Pd nuclei, Nucl. Phys. A 504 (1996) 163. 4. Kratz, K.-L., G rres, J., Pfeiffer, B., Wiescher, M., Nuclear structure near the particle drip-line and explosive nucleosynthesis processes, J. Radioanal. Nucl. Chem. TA2> (2000) 133.
ANALYSIS OF T H E N E U T R I N O P R O P A G A T I O N IN N E U T R O N STARS IN THE F R A M E W O R K OF RELAITVISTIC NUCLEAR MODELS
R. N I E M B R O , S. M A R C O S Departamento
de Fisica Moderna, Facultad de Ciencias, Universidad E-39005 Santander, Spain E-mail: [email protected] E-mail: [email protected]
de
Cantabria,
P. B E R N A R D O S Departamento E.T.S.I.I.T.,
de Matemdtica Aplicada y Ciencias de la Computacion, Universidad de Cantabria, E-39005 Santander, Spain E-mail: [email protected] M. L O P E Z - Q U E L L E
Departamento
de Fisica Aplicada, Facultad de Ciencias, Universidad E-39005 Santander, Spain E-mail: [email protected]
de
Cantabria,
Three relativistic nuclear models: a + u>,
1
Introduction
Neutrino transport properties in assymmetric nuclear matter play an important role in astrophysics, in particular, to describe the evolution of neutron stars (NS). This problem involves the description of the interacting NS matter and relativistic nuclear models should be suitable. The aim of this work is to study the neutrino opacity at a NS stage of evolution where the zero temperature approximation should be appropriate. This scenario is typical of an old NS where high densities are reached at the central core. Since a nuclear equation of state still awaits, the results obtained might be model dependent. Therefore, an interesting topic to deal with is to identify the model magnitudes to which neutrino diffusion is dependent and evaluate their relevance. 315
316
2
Neutral current neutrino cross sections
We make use of three relativistic nuclear models *, namely: a + u>, a + w + npv + P and + TTMIX + P because of the mesons that take part. The second model includes the pion with a pure pseudovector form whereas the last model considers the pion with a mixed pseudoscalar and pseudovector coupling. The pN coupling is considered with its vector and tensor components. They all proceed from effective lagrangians, where the a and u coupling constants are fitted to the nuclear matter saturation conditions: /?o=0.1484 fm~3 and E/A(p0)=-lb.7b MeV. We consider a multicomponent system made up of neutrons, protons, electrons and muons. The concentration of each component is determined once the charge neutrality and the chemical potential equilibrium conditions are fulfilled. The neutrino differential cross section per volume is the observable to calculate and it is expressed in the following way 2 :
1 rf»«r _ V dPSldE
GF E±Im{L^v)
(1)
32TT2 EV
where GF is the weak coupling constant, Ev and E'v are the initial and final neutrino energies, respectively, L^v is the neutrino tensor and I P " is the polarization tensor. It embodies the description of the nuclear system and is defined for each component as:
KM
= - • / -j^tr
{Gl(P)J,Gi(p + q)Jv) ,
(2)
i = n, p, e, p, G(p) is the particle propagator and (J^, Jv) are the currents which stem from the weak interaction and are replaced by {y^,fv), (TM75,7^75) and (7^75,71/) giving rise to the vector, axial-vector and mixed polarizations, respectively. This magnitude depends on q, the four momentum transfer of the neutrino and the NS. We perform the calculations at the Hartree (H) and Hartree-Fock (HF) approaches. To evaluate 1 1 ^ we get the expression:
KM
= -i / Jo
dp / J-i
- 7 - £ - F ; „ 0(|P4-?I-P5,)I?(PJ,-P)« (E; - s;+q + &p
qo)
&p+q
(3) where x = cosQ is the angle between the momentum transfer q and the initial target particle momentum p. F1^ groups together traces of Dirac matrices, Elp — E*1 — So(p) is the self-consistent single particle energy, with E*% =
317 Jp2 + M*'2 and M*1 = M2' + E s ( p ) being the target effective mass. E' s (p) and Sg(p) are the scalar and timelike self-energies ( ^ 0 for i = n , p ) . At a first stage, we do not consider the nucleon vector self-energy E y , since | E y |
Results and conclusions
It is a well known result of relativistic nuclear models t h a t the single particle energy spectrum is smoother in HF t h a n in H. Therefore, a larger density of states close to the Fermi surface is obtained in the former approach. This result is powered the higher is the baryon density. Thus, in principle, we would expect larger neutrino cross sections in HF t h a n in H.
Figure 1: Neutrino differential scattering cross section as a function of the ratio of energy to momentum transfer at zero temperature. Contributions from neutrons, protons, electrons and muons are summed up. The lines correspond to the models: dotted line (H (a + w))> dot-dashed line (HF (cr + u>)), dashed line (HF (a + w + wPV + p)) and full line (HF (cr -f- ui + TTMIX + />))• ( a ) The baryon density pg is po- (b) The baryon density pg is 2po-
In Fig. 1 the neutrino differential scattering cross section is shown for the three models chosen as well as the () Hartree model, in the sake of comparison. T h e m o m e n t u m transfer is |g|=50 MeV and the initial neutrino
318
energy is i? i/ =100 MeV. The baryon density pB considered in Fig. 1 (a) is pa. The cross section has increased when going from H to any of the HF models studied here. As we see in the figure, as the maximum increases, the range of actuation reduces. The responsible of this mechanism is the vneutron contribution, in particular, the momentum dependence of the neutron self-energies. From energy-momentum conservation (denoted by the argument of the <5 function in Eq. (3)) we easily inferred that q™ax{HF) < q^iH). It is due, mainly, to the negative contribution of £Q (p) — Sg(p + q) in HF. On the other side, the height of the //-neutron cross section has increased in the HF case versus H because of the slopes of the neutron self-energies around the Fermi surface. However, the //-proton cross section almost remains unchanged in HF because the proton self-energies are nearly constant. The fraction of protons in the equilibrium is around 5% at po • The role of the isovector mesons, with the pure pseudovector coupling of the pion, does not introduce significant changes, in comparison to the a + u HF results. However, the mixing model causes an increase in the neutrino cross section motivated by the steep self-energies for neutrons which brings about the pseudoscalar component of the pion. The same models are used in Fig. 1 (b) to calculate the neutrino cross section at 2po while keeping the same values for \q\ and Ev. To elucidate the differences from the previous results at p0, one should realize that the kinematical limits to produce particle-hole pairs excitations are extended now. That makes, in all cases, the cross sections become expanded in the
H Y P E R O N I C CRYSTALLIZATION IN H A D R O N I C M A T T E R M.A. PEREZ-GARCIA 1 E-mail: [email protected] J. DIAZ-ALONSO 1 ' 2 , L. MORNAS 1 , J.P. SUAREZ 1 (*) Dpto. de Fisica, Universidad de Oviedo. Avda. Calvo Sotelo 18, E-33007 Oviedo, Asturias, Spain (2) DARC, Observatoire de Paris - Meudon, F-92195 Meudon, France The possible formation of a spatially ordered phase in neutron star matter is investigated in a model where hyperonic impurities are localized on the nodes of a cubic lattice.
We propose a method for analyzing the formation of spatially ordered configurations in hadronic matter, related to the crystallization of the hyperonic sector in a lattice. The method allows the determination of the density ranges where such ordered configurations are energetically favoured with respect to the usual gaseous configurations. This leads to the determination of the parameters of a first order phase transition from the fluid to the crystallized state as well as the equation of state of the plasma in the ordered phase. We show some preliminary results obtained by the application of the method to a very simplified model which shows the efficiency of the mechanism of confinement of the hyperons in an ordered phase proposed here. The eventual existence of such a new phase should induce changes in the structure and cooling of dense stars. At densities about 1.5 times nuclear saturation and beyond, hyperons are present in hadronic matter. The equation of state taking into account the presence of nucleons and hyperons which interact through the exchange of several mesons has been extensively studied using phenornenological lagrangian models 1 . In the mean field approximation, the hyperonic component (as the other hadronic components in the ground state) is supposed to form a spatially uniformly distributed Fermi fluid undergoing the action of the mesonic mean field created by the whole baryonic distribution in the plasma. The hadronic interaction in our model is reduced to the exchange of scalar and vector meson fields. Such a description is the simplest one allowing for an acceptable fit of nuclear saturation 2 . We consider a unique species of neutral hyperons. The interaction Lagrangian for every baryon species B reads L
i=
Y*
9
B=p,n,h
319
+ guB^'full9
(1)
320
where the coupling constants to the meson fields are weaker for hyperons than for nucleons. The leptonic sector is reduced for simplicity to the electron. Using relativistic condensed matter techniques, we investigate the possibility of the existence of energetically favourable solid configurations, where pairs of antiparallel spin hyperons are confined on the nodes of a regular lattice in the ground state of the hadronic plasma. In this situation the hyperons should behave as impurities which induce a redistribution of the surrounding nucleons. Nevertheless, as a first approximation to the solution for the dynamics of the nucleon in the medium, we consider the lattice of neutral hyperons as a uniform background, in an analogous way to the "free electron model" in a Coulomb lattice 4 . In evaluating the energetic contribution of a crystal array of hyperons surrounded by the nucleon liquid component in the plasma, the hyperons are assumed to be localized on the nodes of a cubic lattice under the action of their mutual interaction and the mean fields created by the surrounding nucleons. In this way the total potential at every lattice site is approximated by a harmonic potential which is determined selfconsistently as the superposition of the potentials created by the gaussian clouds of two antiparallel spin hyperons on all the other nodes of the lattice, to which the interaction with the mean field of the nucleon sector is added. In a first approximation we neglect the central and spin-spin interaction between the hyperons in the same node. We shall consider such interaction in future work, but we have already verified that their effect improves the confining character of the whole potential and facilitates the crystallization. In calculating the lattice potential we have also introduced monopolar form factors accounting for the composite structure of the hyperon. In this analysis, vanishing temperature is assumed. For the nucleonic background in the crystal, a relativistic Hartree approximation is used to calculate the contribution to the total energy density of the system, as in the case of the liquid ground state to which it is compared. The relative abundances of protons, neutrons and hyperons can be now obtained by establishing the equations of /^-equilibrium between all these particles. The chemical potentials of protons, neutrons and electrons are the Fermi energies of these particles which are in Fermi gaseous phases. The chemical potential of the hyperons are calculated as the energy gained by the system when a new hyperon is added in the n = 1 level of the harmonic potential in a node. We need also the equations for the mean scalar and vector fields generated by the uniform background of nucleons. The abundance of hyperons resulting from this system is an ingredient in the calculation of the potential created by the lattice on every node. Consequently, for the periodic configuration the self-consistent set of equa-
321
tions for the beta equilibrium of chemical potentials in the plasma coupled to the mean field equations, and the self consistent equations for the confining fields in the nodes are neiectrons = nprotons [^neutrons — ^protons
(charge neutrality) ' /^electrons
[^neutrons — f^hyperons Tlbaryons 2
m ml
a
< a + aext
=
^neutrons
™a < aext m
u
<
u
i ^protons "r f^hyperons
< * 7 ° * > -guHnhyperons
> = gcrHH-hyperons
ext >
=
(3) (4)
> = gaN < * * > +gaHnhyperons
< Up + LJ°xt > = ~guN
(2)
-guHUhyperons
(5) (6) (7) (8) (9)
The crystal size cell a is related to the hyperonic density nhyverons of hyperons paired in a S-state. 2 l^hyperons —
o"
\*^)
a6 The solution of this system gives the magnitudes characterizing the configuration, such as crystal cell size, characteristic oscillation frequency and width of hyperon wave functions, the energy levels, etc as functions of the total baryonic density 5 . In Figure 1 we show the confining potential in a node of the hyperonic lattice for a set of values of the parameters of the model (a=2.2142 fm, cutoff for the scalar and vector fields, A^ = 1.4 GeV, A u = 2.5 GeV, value of the coupling constants gaH/g
322 E ^4.54-
3.53-
2.52-
1.5-
1 0.52
Figure 1. Self consistent confining potential
2.5
3
3.5 ,
4
Figure 2. Lattice size as a function of density
phases in the transition region is determined by looking for the cross point between the liquid and solid configuration lines in the diagram of partial hyperon pressure versus hyperonic chemical potential. This phase transition is of first order, leading to a change in the equation of state of the plasma with a modification of the mechanical properties which should have consequences on the hydrostatic equilibrium of neutron stars and modify the analysis of the cooling processes. References 1. M. Prakash, I. Bombaci, M. Prakash, J.M. Lattimer, P. Ellis, R. Knorren, Phys. Rep. 280, 1 (1997). N.Glendenning, Compact Stars (Springer-Verlag NY, 1997). 2. J.D. Walecka. Ann. Phys. 83, 491 (1974). 3. R. Machleidt, The Meson Theory of Nuclear Forces and Nuclear Structure Adv. in Nucl. Phys. 19, J.W. Negele and E.Vogt eds. (Plenum Press NY, 1989). 4. N.W. Ashcroft, N.D. Mermin, Solid State Physics (Saunders College Publishing, Fort Worth, 1976). 5. J. Diaz Alonso, L. Mornas, M.A. Perez-Garcia, J.P. Suarez, in preparation. 6. J. Diaz Alonso, A. Perez Canyellas, Nucl. Phys. A 526, 623 (1991).
RADIOACTIVE WITNESSES OF THE LAST EVENTS OF NUCLEOSYNTHESIS IN THE NEIGHBOURHOOD OF THE NASCENT SOLAR SYSTEM V.P. CHECHEV V.G. Khlopin Radium Institute, Saint Petersburg, Russia E-mail: [email protected] Based on the total set of radionuclidic and meteoritic data on the extinct natural Al, Mn, Fe, Pd, I, Sm and Pu, the parameters of the last nucleosyndiesis events in the neighbourhood of the nascent solar system have been evaluated.
1
Introduction
It is well known that the relatively short-lived radionuclides with half-lives of millions to hundreds of millions years, although now extinct, can provide information of the earliest stages of the solar system evolution /1-4/. In particular, it refers to such an important parameter as a time interval, 8, between the last events of nucleosynthesis in the neighbourhood of the nascent solar system and the formation of the first solid bodies of the solar system (meteorites). The extinct short-lived radionuclides were still alive at the time of the meteorites formation and their subsequent decay had to lead to isotopic anomalies on their daughter stable or long-lived decay products. A revelation of these anomalies is the direct evidence for the original presence of the short-lived radionuclides in the early solar system. Nowadays this presence has been well established for the following radionuclides: 107Pd /5/ - half-life, T,/2, of 6.5 Myr (million years), 1Z9I /6/ - T1/2 of 16 Myr, 146Sm 111 - T1/2 of 103 Myr, 244Pu /8/ - T1/2 of 80 Myr and the most shortlived 26A1 /9/ - T1/2 of 0.7 Myr, 53Mn /10/ - T1/2 of 3.7 Myr and 60Fe / l l / - T1/2 of 1.5 Myr. The information of these nuclides is given in Table 1. Trace of the latter radionuclide, 60Fe, was discovered not so long ago, in 1992 /11.12/. The different astrophysical processes could be responsible for contribution of the short-lived radionuclides to the early solar system matter. It could be nucleosynthesis during a supernova explosion or recurrent thermal pulses of burning He-shell on the asymptotic giant branch phase (AGB star). Recently Wasserburg et al. IM have shown that 26A1, ^Fe and 107Pd can be produced in sufficient quantities for contamination of the early solar system from a single AGB star or by continuous injection into me interstellar medium ISM from many stars and subsequent mixing.
323
324 Table 1. Short-lived radioactive nuclides used in the chronology of the early solar system and isotopic abundance ratios used as monitors of the nucleosynthesis burst prior to formation of the solar system
Radio- M e a n lifetime, x , nuclide Myr, and decay Monitor mode lu ,u 'Pd 'Pd/1U8Pd 9.4 (p~) 129T 129T;127T 22.6 (PI ,46 ,46 Sm/142Nd Sm 149 (a) 244 244pu Pu/238U 116 (a,SF) a 2K A1 Ai/ f l Ai 1.04 (P+e) 53 53 Mn/55Mn Mn 5.34 (e) 60 60 Fe Fe/56Fe 2.16 ( p ) 2
Relative abundance, Relative rate of n synthesis, a (meteoritic data) (model values) (1.5 -1.8) 10 b ~1 (0.8 - 2.3) 1 0 4 0.5 -1.5 0.0047 - 0.015 0.5 -1.5 0.005 - 0.035 0.1 - 1.0 2.10-7- 5 i 0 5 ~1 10 3 - 1 0 2 (1.29+0.07) 10 6 10" - 1 0 3 (3.9±0.6)10 9
Method of Evaluation
In this paper I am distracted from details of production of the short-lived radionuclides, and I use only the limits of relative rates of their production and relative meteorite abundances making up my mind to determine the value of 8, common for three short-lived radionuclides: 26A1, 53Mn and 60Fe, as well as the value of a relative contribution of the last nucleosynthesis events to production of these radionuclides, the S-value. There are different models of galactic nucleosynthesis. For definiteness one chooses a model of continuous galactic nucleosynthesis with one additional steep increase not long before die formation of the solar system. First this model was suggested by Fowler /13/. It could be a supernova or matter injection from nearby AGB star. In such a model for uniform continuous galactic nucleosynthesis with the burst of S intensity the concentration of radioactive nuclides at the time of (A+5) meteorites solidification, Nr, is described by the equation /3/: Nr = arA exp(-5/x){(T/A)[l-exp(-A/T)](l-S)+S} (1) Here ar is a production rate of the given nuclide in nucleosynthesis, A is a duration of the uniform galactic nucleosynthesis, 8 is the time interval between the isolation of the solar system from all nucleosynthesis sources and the formation of the meteorite solid substance which retains radionuclides, x is a mean lifetime of the radionuclide, S is the ratio of the number of nuclei formed during the flash (burst) of nucleosynthesis to the number of nuclei formed during the continuous galactic nucleosynthesis. For short-lived nuclides x « A and hence Nsh = ash Aexp(-8/x)[(x/A)(1-S)+S] (2) For stable nuclides Nst = ast A (3)
325
From here for relative values n = N^/N1" and a=ash/ast we have the correlation: n = a exp(-8/-c){S[l-(T/A)]+(T/A)} (4) Taking into account that x/A is small, we have finally the following estimation for S: S = (n/a)[exp(o/-t)]-(i:/A) (5) The last relation can be used for determination of the S and 8 permissible values range based on the data in Table 1. The pairs of nuclides shown in the third column of this Table serve as monitors of the last events of nucleosynthesis. In the fourth column their relative concentrations at the time of formation of the meteorite solid substance, n, are given from meteoritic data. In the last column their relative rates of production in nucleosynthesis obtained from theoretical models are presented. It is obvious from the equation (5) that the ratios of n/a and the mean radionuclide lifetimes x determine the range of permissible values of S and 5. 3
Results
For construction of the above range for the three radionuclides (see Figure) the minimum and maximum values of n/a obtained with account of the results scattering and uncertainties of meteorite measurements and theoretical calculations were used/1-4.11,12,14/. From Figure it is seen that the range of values of 5=5-10 Myr and S<0.6 % corresponds to allowable set of data for all three radionuclides: 26A1,53Mn and 60Fe. This field of S(8) is restricted by the curves of the least allowable values of n/a for 26 A1/27A1 and the greatest allowable values of n/a for 60Fe/56Fe. The data on 53 Mn/55Mn admit more wide range of S and 8 values. Analogous curves of dependence of S(8) obtained earlier in /3/ for the set of data on 107 Pd/™Pd, 129I/1271,146Sm/H2Nd and 244Pu/238U showed that they can conform to the nucleosynthesis burst with 8!=40-60 Myr and S^O.4 %. These curves (except 107 Pd/10gPd) do not overlap dependences for the three most short-lived radionuclides 26 ( A1,53Mn, 60Fe). Thus, for explanation of the available data set on the seven short-lived radioactive witnesses of the last nucleosynthesis events shown in Table 1 it is required, at least, the presence of two spike bursts in the neighbourhood of the nascent solar system. Each of them gives a contribution to general galactic nucleosynthesis less than 0.6 %. The first spike can be moved away by time from meteorite solidification on Si=40-60 Myr and the second spike - on 82=5-10 Myr
326
52> Myr Figure. Bounds to permissible S-values for 26A1 (thick solid lines), 53Mn (dashed lines) and 60Fe (thin solid lines) in dependence on the 8 value.
References 1.
2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14.
G.J.Wasserburg and D.A.Papanastassiou. In: Essays in Nuclear Astrophysics. Edited by C.A.Barnes, D.D.Clayton, and D.N.Schramm. Cambridge University Press, 1982.P.85. Ya. M.Kramarovsky and V.P.Chechev, Synthesis of Elements in the Universe. M., Nauka, 1987 (in Russian). V.P.Chechev and Ya.M.Kramarovsky, Bull. Acad.Sci. USSR, Phys.Ser. 53(1989) 138. G.J. Wasserburg, Astroph. J. 424 (1994) 412. W.R.Kelly and G.J.Wasserburg, Geophys. Res.Lett. 5(1978) 1079. J.H. Reynolds, Phys.Rev.Lett. 4 (1960) 8. G.W.Lugmair, T.Shimamura, R.S.Lewis and E.Anders, Science 222 (1983) 1015. M.W.Rowe and P.K.Kuroda, J.Geophys. Res. 70 (1965) 709. T.Lee, D.A.Papanastassiou, and G.J.Wasserburg, Geophys. Res. Lett. 3 (1976) 109. J.-L.Birck and C.J.Allegre, Geophys. Res. Lett. 12(1985) 745. A.Shukolyukov and G.W.Lugmair, Lunar Planet. Sci. XXIII (1992) 1295. A.Shukolyukov and G.W.Lugmair, Science 259 (1993) 1138. W.A. Fowler. In: Cosmology, Fusion and Other Matters. Edited by F. Reines, Colorado, Boulder, 1972. P. 67. A. Shukolyukov and G.W. Lugmair, Lunar Planet Sci. XXIII (1992) 823.
Section II. Experimental Aspects of Nuclear Astrophysics
BOUND STATE BETA-DECAY AND ITS ASTROPHYSICAL RELEVANCE P. KIENLE Physik Department
E 12, Technische
Universitat Miinchen, D-85748
Garching
Bound state P-decay of fully ionized 16 Dy and 187Re nuclei, circulating in a storage cooler ring has been observed from the growth of their hydrogenlike daughter nuclei using two methods. As neutral atom, 163Dy is stable, and ,87Re decays with T1/2 = 43Gyr. The half lifes of ls3Dy66+ and I87 Re75+ determined by bound state P-decays to their respective K-orbits have been determined to T1/2 = (48 ± 3) d and Ti/2 = 32.9 ± 2.0 yr, respectively. The astrophysical relevance of both observations will be presented. The observed bound state P-decay of ""Dy66* explains the high abundance of I64Er, due to a s-process sequence including bound state P-decay of 163Dy66+. From the observation of the bound state P-decay of 187Re75, the 187Re-187Os-cosmochronometer may be calibrated using chemical star evolution models, and including reastration effects. From recent abundance measurements of Re- and Os-isotopes in meteorites the age of the Galaxy was deduced as (15±2)Gyr, with our new calibration.
1
Introduction
In hot stellar plasmas the atoms become highly ionized, which can lead to drastic changes of weak decay properties of nuclei. Of particular interest is the bound (3decay (pb-decay) in which the decay electron is captured in a bound atomic state rather than being emitted into the continuum [1]. It is a two body weak decay process with a monoenergetic antineutrino carrying away the total decay energy, and it is the time reversed process to orbital electron capture. Its decay probability is therefore proportional to the electron density at the nucleus and thus especially large for decays to empty K-orbits of highly ionized atoms. For neutral atoms pb is only of minor importance. It might become a strong and in some cases the only decay channel for highly ionized atoms in stellar plasmas during nuclear synthesis [2]. The astrophysical relevance of (Jb in particular for the s-process and for cosmochronometry has been pointed out during the eighties [3,4,5]. For fully ionized atoms the Q-value of pVdecay into K-orbits is given by: Ql =Qp- ABf (Z +1, Z) + Bf (Z +1) with Qp denoting the Q-value of the continuum p-decay of a neutral atom with atomic number Z. The difference of the total electron binding energies of the neutral daughter (Z+l) and mother atom (Z) is denoted by AB'e°'(Z+l,Z), and Bf(Z+\) is the binding energy of a K-electron in the hydrogenlike daughter atom. From equation (1) one notes pb becomes energetically favored compared 328
329 with continuum (3-decay, because the K-orbit binding energy Bf ( Z + l ) is always larger than AB'f.
In some cases even nuclei which are energetically stable in
neutral atoms Q * <0, are expected to become unstable if completely ionized when g« - AB'f' +Bf (Z +1) becomes positive. An example for such a nucleus is 163
Dy, which is stable as a neutral atom with an abundance of 24.9%, but as fully ionized 163Dy66+ is expected to decay by pb-decay with Qf = (50.3+1) keV into 163
Ho66+ with the decay electron bound in the K-shell of 163Ho66+ with | B^o„6+ 1= 65.137 keV. Indeed the first observation of pb-decay has been reported by our group [6] by storing completely ionized 163Dy66+ in the Experimental Storage Ring (ESR) of GSI, Darmstadt, for periods of time up to 85 minutes. From the number of 163Ho66+ daughter ions, measured as function of the storage time a half life of 47 +4 d was derived. The pb-decay of 163Dy occurring in hot star plasmas can explain the unusual high abundance of 164Er, shielded against a r-formation process. It is formed by an s-process chain starting from 163Dy, which decays by Pbdecay to 1S3Ho. After the capture of a neutron, 164Ho is formed which produces 164 Er by p-decay. In the following experiment we focused on the observation of the pb-decay of fully ionized 187Re and its application for calibration of the 187Re-1870s cosmochronometer [6]. The abundances of most elements heavier than iron in the solar system are the result of prior generations of stellar nucleosynthesis via the sand r-neutron-capture process. The duration of the nucleosynthesis until the formation of the solar system can be estimated from the abundances of long lived radioisotopes, such as 232Th, 238U and 187Re using models for the effective nucleosynthesis rates. Compared with chronometers like 232Th and 238U, for which the relative r-process yield must be calculated, a rather model dependent procedure, the 187Re-1870s cosmochronometer introduced by Clayton [7] in 1964 has several advantages. One is the very long half life of 42.3+1.3Gyr [8], but the main advantages is that the long lived 187Re is only produced by the r-process, whereas its daughter 187 0s is produced mainly radiogenetically with some modifications due to s-process formation from 186Os and some destruction from a 9.75 keV excited state of 187Os. Corrections for these processe have been calculated [5]. However one large uncertainty in the calibration of the 187Re-1870s chronometer remained, as has been pointed out by Takahashi et al [3,4]. In the hot plasma of a star 187Re may become highly ionized during a reastration period, with the consequence of a fast pb-decay, which decreases its effective half life, thus leading to a too long duration of the nucleosynthesis time if no correction is made. In order to contribute to a calibration of the 187Re-187Os-chronometer we started a
330
program to measure the half life of completely ionized 187Re75+ in the ESR storage ring using similar techniques as for 163Dy [6]. The potential decay modes of neutral and fully ionized 187Re are shown in Fig. 1,
450-
400-
F 1 &
50-
ijp-2.66
0J...-T.
"Re"
key
l/r
Q
Os,++«n/
Figure 1. Decay schemes for neutral (bottom) and fully ionized (top) P-transitions indicated by arrows.
,7
Re with the energetically allowed
including the relevant energetic facts, such as electron binding energies and decay energies. For neutral 187Re° only a 2.66 keV unique, first forbidden groundstate transition to 187Os is energetically possible. The small matrix element and small Q -value of 2.663 [19] keV [9] lead to the long half life of 42 Gyr [8]. As the strongly bound orbits are occupied with electrons in neutral Re, pb-decay contributes less than 1 % [10]. For fully ionized 187Re75+, fS-decay to the continuum of 187Os76+ is energetically forbidden, instead bare 187Os76+ may decay back to 1S7 .75+ Re by capturing an electron in the plasma of a star [11]. Bare Re however is unstable against pYdecay with the electron capture in the K( g f t =72.97 keV) [12] or in the L-shell (g ft =9.07 keV). Yokoi, Takahashi and Arnould [4] realized that also the first excited state of 187Os at 9.75 keV can be fed by a non-unique first forbidden transition with a substantially larger matrix element and Qp = 63.22 keV thus dominating the 187Re75+ decay. They estimated the half life of bare 187Re to T1/2= 14y [5], which is more than a billion times shorter than that for neutral 187Re. Thus reastration can reduce the effective half life of cosmogenic 187Re appreciably, and a measurement of the fib-decay of bare 187 Re would put the calibration of the 187Re-1870s clock on safer grounds.
331
Measurements of the half life of 187Re75+ 187
Re50+ ions injected into the heavy ion synchrotron SIS were accelerated to an energy of 347 AMeV, extracted, stripped with a 100 mg/cm Cu-foil to bare 187 Re75+ with an efficiency of about 75 %, and finally injected into the storage ring ESR, shown in the sketch of Fig 2.
Schottkypick up
S
target
electron^" T—*- g> cooler
Figure 2. Sketch of the experimental storage ring (ESR) at GSI. The position of the internal gas jet target, the electron cooler, the Schottky pick up system and the particle detectors (PD) are indicated as well as the pathof ,87 Os 76+
Electron cooling was applied to achieve a small momentum spread (10 5 ) and a small emittance (0.1 n mm mrad) of the coasting ion beam with currents up to 2mA, corresponding to 108 bare 187Re75+ ions. The storage losses due to collisions with atoms of the residual gas (10"nmb), and atomic charge change reactions in the electron cooler section, lead to an effective storage half life of 4.5 hrs. With about 108 stored 187Re75+ ions, several hundred 187Os75+ ions were produced by pV decays of 187Re7S+ during storage times up to 5 hours. The 187Os75+ ions were circulating with nearly the same frequency (within 4ppm) as the main beam due to the small m/q difference. Their number Nos(ts) grows proportional to the storage time tj (ts « T1/2), according to the relation NosOs) = ( V Y )
N
Re (ts) t.
187
NRe(ts) denotes the number of circulating Re75+ ions at time U, Xpb the decay 187 75+ probability in the Re rest frame and y = E/mc2 the Lorentz factor, which was determined experimentally to y = 1.373 [2].
332
For separation of hydrogen - like 187Os75+ ions from the 187Re75+ mother nuclei, the bound p-decay electron was stripped by turning on a gas jet target which crossed the ion beam and produced 187Os76+. Two methods were used for the determination of the number of 187 0s 76+ ions. In the first one, the Schottky noise frequency spectroscopy, we measured the number of circulating ions as function of the revolution frequency, which allows a unique identification of 187 0s 76+ daughter nuclei. The circulating ions induce a noise signal in a pair of capacitive pick up plates. When Fourier transformed, the noise signal reveals the corresponding revolution frequency (or harmonics) of each species of stored ions. This frequency is a unique measure of the mass/charge ratio since the velocity of all ions is forced to be equal to that of the cooler electrons. Fig. 3 shows a Schottky noise frequence spectrum taken after storing 187Re75+ for 1.8 h and after stripping the 187 0s ions by the gas jet, which was turned on for 200 s containing 3 x 1012 argon atoms/cm2. All lines observed in this spectrum can be assigned to nuclei produced by reactions of 187Re with nuclei in the gas jet (mainly by loss of a few nucleons) except for the pVdecay daughter 176 0s 76+ . Only this line grows linearly with the storage time as demonstrated in the inset of Fig. 3, proving its origin from the pbdecay of 187Re75+.
3
i
I M 1
1.0005
1.001
1.0015
1.002
relative revolution frequency Figure 3. Schottky noise frequency spectrum after a storage time of 1.8 h and after the reaction of the coasting beam with the jet target. Besides a number of nuclides produced by nuclear reaction the pVdecay daughter 187Os76+ is seen. The inset demonstrates that the intensity of the 187Os76+ line increases expectedly when the storage time is increased from 1.8 h to 4.7 h.
A small 187Os contribution originating from a nuclear charge exchange reaction in 187Re collisions with argon atoms of the gas jet could be determined from the intensity for zero storage time. The absolute number of 187Os75+ions
333
produced by 1S7Re (3b-decay was determined from the area of the Schottky-noise signal of fully ionized 1870s7s+and corrected for the experimentally determined electron stripping efficiency of the gas jet. The area of the Schottky noise lines were calibrated absolutely in terms of particle numbers by measuring currents at particle numbers exceeding 105 and counting single ions for very small particle numbers. In an independent experiment we measured the position of ions that had interacted with the gas jet target and were deflected by the following dipole magnet stronger than the coasting beam. The dispersion of the magnet displaced 1870s76+ ions from 187Re75+ ions by 75 mm at the location of our detector. A gas microstrip counter measured the deflection position with a resolution of 0,4 mm. It was operated with 1 bar of an argon/isobutane (70:30) mixture. In front of this counter and in the same gas volume the energy loss was measured by an ionization chamber. In spite of the small energy loss of 60 MeV we achieved a resolution with respect to the nuclear charge of AZ = 1.5. By a condition on the pulse height in the ionization chamber other elements than osmium could be suppressed in the position spectra. Storage time = 9.7 min, NJtJ = 8x10', NJtJ = 14 (± 31)
Distance to beam [mm] Storage time = 4 h, N„(tJ = 6x10', N„,(tJ = 182 (^ 475)
55 60 65 70 75 80 85 90 95 100 Distance to beam [mm] Figure 4. shows position spectra of particles detected with a multistrip gas counter with breeding times of 9.7 min (upper) and 4 h (lower).
334
Fig. 4 shows position spectra taken with the multistrip gas counter with a pulse height condition from the ionization chamber set such that all elements except osmium being suppressed, taken after a short (9.7m) (upper) and a long (4h) (lower) breeding time. The spectrum with the short breeding time represents the background due to Rutherford scattering (continuously decreasing part) and reaction products (broad peak) from interactions with the Ar gas jet. The spectrum with the 4 h breeding time reveals a narrow line at the position for cooled 187Os76+ daughter nuclei superimposed on the Rutherford scattering and nuclear reaction background. With either method a dozen measurements were performed with storage time t ranging from U ~ 0 (determination of background from nuclear reactions) to U = 5 h. After calibrating the areas of the Schottky signals and of the line in the particle detector in absolute particle numbers and after determining the number of primary 187Re75+ ions (with a beam current transformer), taking into account the various losses, we obtain for the pVdecay probability ^pb = (6.29 ± 0.19 ± 0.40) ' lO^V1 from the Schottky-noise analysis and Xpb = (7.05 ± 0.28 ± 0.34) • 10"10 s"1 from the position spectra, where first the statistical errors are given and then the estimated systematical errors. Adding these two errors algebraically in each case and then taking the average of both results, we get (x^ \ = (6.7 ± 0.4) ' 10"10 s"1 and Tic = (32.9 ± 2.0) yr. The measured Apb is practically equal to the pYdecay probability into the K- shell of 187Os, because the decay into the L- shell is about 4 orders of magnitude less probable. From the measured Tm we deduce log ft = 7.87 ± 0.03. Note also, that the decay of bare 187Re is dominated by the nonunique transition to the first excited state of 187Os, since the decay to the ground state has a much smaller matrix element (log ft =11.0, from the decay of neutral 187Re). Re ions in a stellar plasma can have average numbers of bound electrons between 1 and more than 20, depending on temperature and density. With the measured ft value the decay rate of 187Re in any charge state, and hence at any temperature, can now be calculated. Our measurement is very close to logy? = 7.5, which was assumed by Yokoi et al. [4] in their study of the 187Re-187Os cosmochronometry. 3
Cosmochronometric Results
Recently Takahashi et al. [13] reviewed the progress made towards a reliable determination of the age of the Galaxy from the 187Re-187Os abundance ratios in meteorites with the aim to set a lower bound for the age of the Universe. This is of current interest, especially in view of the latest results from the observation of redshifts of type la supernovae, from which the cosmological parameters, i.e. the mass density £lM, the cosmological-constant energy density £2A, and what we are
335
mostly concerned, values for the Hubble time are deduced. The values quoted for the dynamical age of the Universe, respectively Hubble time, are 14.2±1.5 Gyr [14] and 14.9 ±{J Gyr [15] respectively. Our measurement of the pVdecay of 187Re75+ gives an experimental value for the key transition matrix element to the 9.75 keV excited state of 1870s, which allows now reliable evaluations for the 187Re P-decay and 1870s electron capture rates in stellar interiors with the use of the basic formalism and the thermodynamic conditions (temperature, density and composition) given by realistic stellar evolution models. It requires modeling of the evolution of various size stars which is needed in the first instance for the evaluation of the effects of astration in the stellar interiors, such as transmutation of 187Re and 187Os by enhanced pYdecays and ecaptures as well as destruction by neutron capture. Additionally it provides the input data for a quantitative modeling of the chemical evolution of the stars (stellar life times, remnant masses, yields of various elements). The most difficult problem is the modeling of the chemical evolution of the matter from which the solar system is formed. A simple one-zone evolution model was used with the allowance for "infall", and imposing as many observational constraints as possible. In an analysis of recent data on isotopic abundances of 187Re, 186Os, 1870s and 188 0s in various types of meteorites [16, 17, 18, 19, 20] Faestermann [21] derived from these data the input quantities for Re/Os chronometry with much better precision: ?i(187Re) = (1.666 ± 0.010) x 10"nyr \ the solar abundances at the time of formation of the first meteorites 4.56 Gyr ago namely for 1870s/1880s = 0.09536 with 0.2 % and for 187Re/188Os = 0.423 with 2.5 % precision. The 186Os/188Os ratio is well known as 0.12035 with an uncertainty below 0.1 %.
1.02
1 | I:
-a 0.98
0.96
10
12
14
16
18
20
T c (Gyr)
Figure 5. calculated 187Re abundances Xcai relative to the solar value X© for various evolution models as function of the age of the Galaxis TG. The probable TG intervals thus derived are displayed at the bottom.
336
Fig. 5. shows the computed 187Re abundances relative to the solar value for various star evolution models [see ref. 13 for details] as function of TG the age of the Galaxy. The lo error bars attached are theoretical and reflect the spreads in the parameter space. It is clear that the 187Re-187Os chronometry still leads to a considerable spread in the TG-values derived, particularly owing to the uncertainties in modeling of the chemical evolution. On the other hand it is comfortable to see that the most likely value for TG lies around TG = (15±2)Gyr reasonably close to the recent values of the Hubble times [14, 15]. Better chemical evolution models can be constructed, which take into account more observational data in near future. 4
Acknowledgements
This work was supported by the "Sonderforschungsbereich 375-95 fur Astroteilchenphysik der deutschen Forschungsgemeinschaft", by the Beschleuniger-Laboratorium der Miinchner Universitaten and by the GSI, Darmstadt. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.
R. Daudel, M. Jean and M. Lecoin, J.Phys.Radium 8Z 238, (1947) J.N. Bahcall, Phys.Rev. 124z495, (1961) K. Takahashi and K. Yokoi, Nucl.Phys, A404,578, (1983) K. Yokoi, K. Takahashi and M. Arnould, Astron. Astrophys. 117, 65, (1983) K. Takahashi et al., Phys.Rev. C36, 1522, (1987) M. Jung et al., Phys.Rev.Lett. 66, 2164, (1992); F. Bosch et al., Phys. Rev. Lett. 77, 5190, (1996) D.D. Clayton, Astrophys.J. 139, 637, (1964) M. Lindner et al, Geochim.Cosmochim. Acta 53, 1597, (1989) G. Audi and A.H. Wapstra, Nucl.Phys. A595, 409, (1995) Z. Chen, L. Rosenberg and L. Spruch, Phys. Rev. A35,1981, (1987) M. Arnould, Astron.Astrophys. 21, 401, (1972) W.R. Johnson and G. Soff, AtData Nucl. Data Tables, 33, 405, (1987) K. Takahashi et al., Proc. 9th Workshop on "Nuclear Astrophysics", Ringberg Castle, Tegernsee 1998, MPA/P10, June 1998, eds: W. Hillebrandt and E. Miiller, Max-Planck-Institut fur Astrophysik, Garching b. Miinchen A.G. Riess et al., Astronomical Journal 116, 1009, (1998) S. Perlmutter et.al., Astronomical Journal 517, 565, (1999) M.I. Smoliar, R.J. Walker, J.W. Morgan, Science 271, 1099, (1996)
337
17. J.J. Shen, D.A. Papanastassion, G.J. Wasserburg, Geochim. Cosmochim. Acta 60, 2887, (1996) 18. T. Meisel, R.J. Walker, J.W. Morgan, Nature 383, 517 (1996) 19. H. Becker, R.J. Walker et al., Proc. 29 Lunar and Planatary Science Conf., 1998 20. K.P. Jochum, Geochim. Cosmochim. Acta 60, 3353, (1996) 21. T. Faestermann, Proc. 9th Workshop on "Nuclear Astrophysics", Ringberg Castle, Tegernsee 1998, MPA/P10, June 1998, eds: W. Hillebrandt and E. Miiller, Max-Planck-Institut fur Astrophysik, Garching b. Miinchen
SEARCHING FOR SIGNALS FROM THE DARK UNIVERSE
R. B E R N A B E I , P. B E L L I , R. C E R U L L I , F . M O N T E C C H I A Dip. di Fisica, II Universitd di Roma and INFN Sez. Roma2, via delta Scientifica 1, 1-00133 Roma, Italy E-mail: [email protected]
Dip.
Ricerca
M. A M A T O , G. I G N E S T I , A. I N C I C C H I T T I , D . P R O S P E R I di Fisica, Universitd di Roma and INFN Sez. Roma, P.le A. Moro 2, 1-00185 Roma, Italy C. J. DAI, H. L. H E , H. H. K U A N G , J. M. M A IHEP, Chinese Academy, P.O. Box 918/3 Beijing 100039, China
DAMA experiments are running deep underground in the Gran Sasso National Laboratory of I.N.F.N.. Results on various rare event searches have been obtained. In particular, here the most recent results achieved in the investigation of the WIMP annual modulation signature are summarized.
1
Introduction
DAMA is dedicated to searches for rare events by developing and using low radioactive scintillators; its main aim is the search for relic particles embedded in the galactic halo mainly by investigating the annual modulation of the WIMP "wind" which would continuously hit the Earth. The success of this kind of search depends on the possibility to build large mass detectors with high intrinsic radiopurity; they are needed both to significantly search for candidates with extremely reduced rate (such as the neutralino) and to exploit peculiarities of the WIMP "wind" in order to unambiguously select a signal from the background. The relevance of performing experiments with a proper signature is clear. Here we only briefly recall that the DAMA WIMP searches are carried out by using: i) CaF2(Eu) detectors 1; ii) ~ 2 1 liquid Xenon (LXe) pure scintillator 2 ; iii) ~ 100 kg highly radiopure Nal(Tl) set-up 3 . A relevant part of the experimental DAMA activity regards the ~ 100 kg Nal(Tl) set-up 3 . Studies on possible future applications of large mass highly radiopure Nal(Tl) set-ups have also been carried out 4 . Moreover, searches for /?/?, charge-non-conserving, Pauli-exclusion-principle-violating processes, etc. 5 are performed as well as other approaches to Dark Matter search 5 . 338
339 In this conference we have presented the recent results on the investigation of the annual modulation signature during four annual cycles; some main points will be briefly summarized in the following. 2
Generalities
Due to the known intrinsic uncertainties in the comparison among the results obtained by different experiments - mainly when different target-nuclei and/or different techniques are used - it is necessary to realize experiments offering their own effective distinctive signature. For this purpose we investigate the socalled annual modulation signature, which is based on the annual modulation of the WIMP rate induced by the Earth's motion around the Sun 3 ' 6 . This signature is quite strong since it requires the satisfaction of all the following specifications: i) presence in the rate of a modulated part varying as a cosine function; ii) with proper period (1 year); iii) with proper phase (about 2 June); iv) only in a well-defined low energy region; v) for those events in which only one detector of many actually "fires", since the probability of a WIMP multiple scattering is negligible; vi) with modulation amplitude in the region of maximal sensitivity < 7%. To fake this signature possible systematics must also satisfy all these six requirements; therefore, for some other effect to mimic it is highly unlikely. The detailed description of the ~ 100 kg Nal(Tl) setup, of its radiopurity and of its performances has been given in Ref. 3 . Here we only recall that the detectors used in the annual modulation studies are nine 9.70 kg Nal(Tl) scintillators especially built for this purpose. The bare Nal(Tl) crystals are encapsulated in suitably radiopure Cu housings; 10 cm long Tetrasil-B light guides act as optical windows on the two end faces of the crystals and are coupled to EMI9265-B53/FL photomultipliers (PMT). The two PMTs work in coincidence and collect light at single photoelectron threshold, while the software energy threshold has been cautiously taken at 2 keV 3 . The detectors are inside a low radioactive sealed copper box installed in the center of a low radioactive Cu/Pb/Cd-foils/polyethylene/paraffin shield. The copper box is maintained in a high purity (HP) Nitrogen atmosphere in slightly overpressure with respect to the external environment. Furthermore, also the whole shield is sealed and maintained in the HP Nitrogen atmosphere. The installation is air conditioned. On the top of the shield a glove-box (also maintained in the HP Nitrogen atmosphere) is directly connected to the inner Cu box, housing the detectors, through Cu pipes. The pipes are filled with low radioactive Cu bars, which can be removed to allow the insertion of source holders for calibrating the detectors in the same running condition, without any contact with external
340 Table 1: Released data sets.
3
period
statistics (kgday)
DAMA/NAI-1
4549
DAMA/NaI-2
14962
DAMA/NaI-3
22455
DAMA/NaI-4
16020
Total statistics
57986
+ DAMA/NaI-0
limits on recoils by PSD
air. A hardware/software monitoring system is operating to careful control the running conditions; in particular, several probes are read out by the d a t a acquisition system and stored with the production d a t a , while self-controlled computer processes are operational to automatically control various related parameters and to manage alarms. The only d a t a treatment, which is performed on the raw d a t a , is to eliminate obvious noise events (sharply decreasing with the increase of the number of available photoelectrons) present below ~ 10 keV. T h e followed procedure is described in ref. 3 . Energy spectra in various energy regions have been published 3 , s . 3
R e s u l t s o n t h e i n v e s t i g a t i o n for t h e W I M P a n n u a l m o d u l a t i o n signature
Results on the d a t a collected during four annual cycles (see Table l a ) have been released so far 3 . An immediate evidence of the presence of annual modulation in the lowest energy experimental rate during the considered four annual cycles is given by the single hit events residual rate in the cumulative 2-6 keV energy interval as a function of the time (see fig. 1) 3 ; this is completely model independent. In fact, the x2 test on the d a t a of fig. 1 disfavours the hypothesis of unmodulated behaviour giving a probability of 4 • 1 0 - 4 (x2/d.o.f. = 48/20), while fitting these residuals with the function A- cos[ui • (t — ^o)] (integrated in each of the considered t i m e bin), one gets T = ^ = (1.00 ± 0.01) year, when fixing tQ "The DAMA/NaI-0 running period is also listed there. Its d a t a were analysed in terms of PSD, obtaining upper limit on recoils 3 , which has been accounted in the particular model dependent search for a candidate performed in ref.
341 0.1 0.05
<*• - 0 . 0 5
-0.1
-
-
500
1000
1500 time (days)
Figure 1: Model independent residual rate for single hit events, in the 2-6 keV cumulative energy interval, as a function of the time elapsed since January 1-st of the first year of d a t a taking. The expected behaviour of a WIMP signal is a cosine function with minimum roughly at the dashed vertical lines and with maximum roughly at the dotted ones 3 .
at 152.5 days and to = (144 ± 13) days, when fixing T at 1 year. Similar results, but with larger errors, are found in case all the three parameters are kept free. As it is evident the period and the phase fully agree with the ones expected for a WIMP induced effect. This model independent analysis gives evidence for the possible presence of a WIMP signal. This is further supported by the negative result of the search for possible systematics able to mimic such a modulation 3 . Within the analyses, which have been performed 3 , here we only summarize as an example arguments related to the particular cases of temperature, Radon and energy calibrations in the two last annual cycles. Full discussions are available in ref.3. Sizeable temperature variations could only induce a light output variation, which is negligible considering: i) that around our operating temperature, the average slope of the light output is < -0.2%/°C; ii) the energy resolution of these detectors in the keV range; iii) the role of the intrinsic and routine calibrations 3 . Moreover, a time correlation analysis of the temperature data gives a modulation amplitude (considering the same period and phase as for WIMPs) compatible with zero: (0.021 ± 0.046) °C and (0.064 ± 0.058) °C for DAMA/NaI-3 and DAMA/NaI-4 respectively. Therefore, a temperature effect can be excluded. As regards the Radon case, we examined the behaviour of external Radon level with time; the fitted cosine annual modulation amplitude with the WIMP
342 expected phase results (0.14 ± 0.25) B q / m 3 and (0.12 ± 0.20) B q / m 3 , for the two periods respectively; they are consistent with zero. Therefore, a Radon effect can be excluded firstly because our detectors are maintained in H P Nitrogen atmosphere, then because the external Radon itself does not show any modulation. Moreover, in every case, a modulation induced by Radon would fail some of the six requirements of the annual modulation signature, inducing e.g. modulation also in other energy regions t h a n the one of interest for W I M P induced recoils, which has not been observed in the d a t a . As regards the energy calibration, we recall t h a t - in long term running conditions - the knowledge of the energy scale is assured by periodical calibration with 2 4 1 A m source and by continuously monitoring within the same production d a t a (grouping t h e m every ~ 7 days) the position and resolution of the 2 1 0 P b peak (46.5 k e V ) 3 . As in refs. 3 , the distribution of the relative variations of the energy calibration factors, estimated from the position of the 210 P b peak for all the 9 detectors during b o t h D A M A / N a I - 3 and D A M A / N a I - 4 taken without any correction, has been investigated; it shows a gaussian behaviour with a — (0.95 ± 0.04)%. Since the results of the routine calibrations are obviously properly taken into account in the d a t a analysis, such a result allows to conclude t h a t the energy calibration factors for each detector are known with an uncertainty < 1%. Due to the relatively poor energy resolution of the detectors at low energy, this could give rise only to an additional relative energy spread < 10 ~ 4 in the lowest energy region and < 1 0 " 3 at 20 keV, which is totally negligible. T h e absence of modulation in energy regions not involved in the particle Dark Matter direct detection, has been verified (see ref. 3 ) . T h e complete investigations of all the known sources of possible systematic effects, which could affect the energy spectrum, credit a percentage systematic error 3 of order of < 10~ 3 . Moreover, the results on the analysis of the rate integrated above 90 keV as a function of the time has excluded the presence of a possible overall background modulation (excluding also significant contribution e.g. from possibly surviving neutrons from the environment 3 ). In addition, no known systematics able to satisfy all the same six requirements for a W I M P induced effect - as it is necessary to mimic the signature - has been found so far. The investigation of possible competing physical processes has led to the same conclusion; in fact, we found u p to now only the muon modulation reported in ref. T , but it would give in our set-up modulation amplitudes < < 10~ 4 c p d / k g / k e V , t h a t is much smaller t h a n we observe. Moreover, it will also fail some of the six requirements necessary to mimic the signature; so it can be safely ignored.
343
As a further step in order to investigate the nature of a possible candidate, a full time and energy correlation analysis of the data has to be performed and it is necessary to assume a particular model, which would require not only the choice e.g. of a particular coupling, of a particular velocity distribution, but also to fix every needed parameter on a certain value. According to ref. 3 , the events between 2 and 20 keV have been analysed by using the standard maximum likelihood method considering in particular a candidate with spinindependent interaction and mass above 30 GeV (for details see ref. 3 ) . In the minimization procedure the WIMP mass has been varied up to 10 TeV. Totally similar results are achieved when using different approaches as \ 2 method, Feldman and Cousins method, etc.. As usual in particle Dark Matter direct i
i
i
i
i
20
50
100
200
500
-
3 —
to - 5
a.
an 10" 6
10
1000
Mw (GeV) Figure 2: Regions allowed at 3a C.L. by the complete global analysis: i) for VQ = 220 k m / s (dotted contour); ii) when accounting for VQ uncertainty (170 k m / s < VQ < 270 k m / s ; continuous contour); iii) when considering also a possible bulk halo rotation as in Ref. 9 (dashed contour). Consistency with the upper limit on recoils 3 measured by applying the pulse shape discrimination on the statistics collected during the running period labelled DAMA/NaI-0 has been required here.
searches standard hypotheses have been used in the calculations of the distribution of astrophysical velocities , while a detailed investigation of the effects induced by their uncertainties 8 has been performed in ref. 9 . The model dependent results of every cycle independently are consistent; therefore, a global analysis has been carried out. In addition, also the consistency with the results of DAMA/NaI-0 3 has been required. The obtained 90% C.L. allowed region is shown in fig. 3. A dedicated discussion quantitatively comparing this particular model dependent result with the model independent one has been carried out in ref.
344 3
The theoretical implications of the observed effect in terms of neutralino with dominant spin-independet interaction and mass above 30 GeV has been discussed in ref. 10 . 11 . 12 ) the case for an heavy neutrino of the fourth family has instead been introduced in ref. 13 . The inclusion of present uncertainties on some nuclear and particle physics parameters would enlarge these regions; full estimates are in progress. As an example, let us notice that, as for the case of vo and possible bulk rotation, uncertainties can also arise from standard nuclear and particle physics assumptions used in the evaluation of the expected unmodulated term and of the modulation amplitude, Sm. We have also mentioned the case of the form factor, which depends on the nuclear radius and on the thickness parameter of the nuclear surface 14 . Varying e.g. by 20% the values of the latter parameters with respect to standard assumptions 14 , the locations of the minimum move toward slightly larger M\y and lower £crp, while the calculated Sm in the 2-6 keV energy interval increases by about 15%. 4
Conclusion
To further investigate the observed effect, the data of a fifth annual cycle, available at end of July 2000, will be analysed. A significant upgrading of the electronics and DAQ of the ~ 100 kg Nal(Tl) set-up is in progress. Moreover, in 1999 the construction of radiopure detectors to fulfil the DAMA Nal installation (final target-detector mass: ~ 250 kg) has been funded by INFN after a dedicated R&D and the work is in progress. Finally investigation on the model framework and on the role of the uncertainties associated to the parameters to be used in model dependent calculations are in progress. References 1. R. Bernabei et al., Astrop. Phys. 7, 73 (1997); P. Belli et al., Nucl. Phys. B 563, 97 (1999). 2. P. Belli et al., Nuovo Cimento A 103A, 767 (()1990); P. Belli et al., Nucl. lustrum. Methods A 336, 336 (1993); P. Belli et al., Nuovo Cimento C 19, 537 (1996); P. Belli et al., Phys. Lett. B 387, 222 (1996); R. Bernabei et al., Phys. Lett. B 436, 379 (1998); R. Bernabei et al., New Jurnal of Physics 2, 15.1 (2000). 3. R. Bernabei et al., Phys. Lett. B 389, 757 (1996); R. Bernabei et al., Phys. Lett. B 424, 195 (1998); R. Bernabei et al., Phys. Lett. B 450,
345
4.
5.
6. 7. 8.
9. 10.
11. 12. 13. 14.
448 (1999); R. Bernabei et al., Nuovo Cimento A 112, 545 (1999); P. Belli et al., in the volume "3K - Cosmology" (AIP pub. (1999) 65); R. Bernabei et al., Nuovo Cimento A 112, 1541 (1999); R. Bernabei et al., Phys. Lett. B 480, 23 (2000); R. Bernabei et al., ROM2F/2000-26. R. Bernabei et al., Astrop. Phys. 4, 45 (1995); I.R. Barabanov et al., Astrop. Phys. 8, 67 (1997); R. Bernabei et al., Proc. of IDM96 (World Sc. Pub., Singapore (1997) 574); I.R. Barabanov et al., Nucl. Phys. B 546, 19 (1999). P. Belli et al., Astrop. Phys. 5, 217 (1996); R. Bernabei et al., Nuovo Cimento A 110, 189 (1997); R. Bernabei et al., Phys. Lett. B 408, 439 (1997); P. Belli et al., Astrop. Phys. 10, 115 (1999); P. Belli et al., Phys. Lett. B 460, 235 (1999); R. Bernabei et al., Phys. Rev. Lett. 83, 4918 (1999); P. Belli et al., Phys. Rev. C 60, 065501 (1999); P. Belli et al., Phys. Lett. B 465, 315 (1999); P. Belli et al., Phys. Rev. D 61, 117301 (2000); R. Bernabei et al., ROM2F/2000/24, submitted for publication. K.A. Drukier et al., Phys. Rev. D 33, 3495 (1986); K. Freese et al., Phys. Rev. D 37, 3388 (1988). M. Ambrosioet al., Astrop. Phys. 7, 109 (1997). P.J.T. Leonard and S. Tremaine, Astrop. J. 353, 486 (1990); C.S. Kochanek, Astrop. J. 457, 228 (1996); K.M. Cudworth, Astron. J. 99, 590 (1990). P. Belli et al., Phys. Rev. D 61, 023512 (2000). A. Bottino et al., Phys. Lett. B 423, 109 (1998); Phys. Rev. D 59, 095004 (1999); Phys. Rev. D 59, 095003 (1999); Astrop. Phys. 10, 203 (1999); Astrop. Phys. 13, 215 (2000); hep-ph/0001309 to appear on Phys. Rev. D. R.W. Arnowitt and P. Nath, Phys. Rev. D 60, 044002 (1999). E. Gabrielliet al., hep-ph/0006266. D. Fargion et al., Pis'm.a Zh. Eksp. Teor. Fiz. 68, (1998); JETP Lett. 68, 685 (1998); Astrop. Phys. 12, 307 (2000). R.H. Helm, Phys. Rev. 104, 1466 (1956); A. Bottino et al., Astrop. Phys. 2, 77 (1994).
E X P E R I M E N T A L STUDIES RELATED TO s- A N D r- P R O C E S S ABUNDANCES K. WISSHAK, F.VOSS, F. KAPPELER Forschungszentrum Karlsruhe, Institut fur Kernphysik, Postfach 3640, D-76021 Karlsruhe, Germany E-mail: [email protected] The accurate determination of the neutron capture cross sections at stellar temperatures of kT=10—100 keV is an essential prerequisite for investigations of the nucleosynthesis of heavy elements. For this purpose a 4-7T Barium Fluoride Detector was constructed at Karlsruhe and implemented at the 3.75MV Van de Graaff accelerator. Over the last decade the cross sections of 57 isotopes have been measured in the mass range from Cd to Ta with typical uncertainties of 1—2%, an improvement up to factors 5—10 compared to other methods. These data were used in extensive s—process studies based on phenomenological and stellar models for deriving reliable sets of s— and r—process abundances.
1
Introduction
Neutron reactions are responsible for the formation of all elements heavier than iron. The corresponding scenarios relate to helium burning in Red Giant stars and to supernova explosions. In the first case, moderate neutron fluxes are produced in the slow neutron capture process (s process) by (a,n) reactions on 13 C and 22 Ne, which imply neutron capture times much longer than typical /3—decay times. Thus the reaction path follows the valley of beta stability and the respective neutron capture reactions are accessible to laboratory experiments. In the second case, neutron fluxes are more than ten orders of magnitude larger, giving rise to the rapid neutron capture process (r process). On average, the observed solar abundances of the heavy elements, N Q , are produced by both processes in equal parts. The reliable separation of the two components represents a key problem for the comparison with the abundance distributions predicted by nucleosynthesis studies. Since the s—process part can be determined for all isotopes 1 the r—process contribution follows simply from the difference to the solar values, N r = N© - N s . An essential requirement for the determination of the s-process abundances N s is the accurate determination of the respective stellar (n,7) cross sections. Therefore a new effort was started in Karlsruhe some 10 years ago to set up an experiment which allows to determine the required (n,7) cross sections at keV energies with uncertainties of ~ 1 % . These data were used in extensive s- process studies were reliable sets of s- and r- process abundances were derived. 346
347
2
Experiment
K
flight path 77cm
4n BaF2 detector
— H
neutron collimator
Figure 1: Schematic view of the experimental setup at the accelerator.
A schematic sketch of the experimental setup is shown in Fig.l. Continuous neutron spectra in the energy range from 3 to 225 keV were produced Yia the 7 Li(p,n) 7 Be reaction using the pulsed proton beam of the Karlsruhe Van de Graaff accelerator. Compared to other sources, neutron production by nuclear reactions leads to relatively low 7-ray backgrounds. Hence, a rather compact shielding of the neutron target is sufficient, allowing for short light paths and correspondingly favorable neutron fluxes. In the present experiment, the neutron energy is determined by the time-of-flight technique using a flight path of only 77 cm. The complete 7-cascade from neutron capture events is registered by the 4frBaF2 detector 2 , which consists of a spherical shell of barium fluoride with 20 cm inner diameter and 15 cm thickness subdivided into 42 hexagonal and pentagonal crystals. The 4w array is characterized by an energy resolution of 7.1% at 2.5 MeV, a time resolution of ^500 ps, and an efficiency > 90% in the 7-ray energy range up to 10 MeV. Up to 9 samples can be mounted on a sample changer, 3 positions being reserved for the gold reference sample, a graphite sample for measuring the background due to scattered neutrons, and an empty sample can. This means that six samples can be investigated in an experiment, corresponding to a complete isotope sequence along the s-process chain in practically all elements.
348
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3
Main features
With respect to accurate neutron capture cross section measurements the present setup is characterized by several relevant features: The main advantage results from the unique combination of good energy resolution and high 7-ray detection efficiency. Accordingly, most capture events fall in a sharp line at the neutron binding energy (see left part of Fig.2), and 98% of all capture events are observed above the detection threshold at 1.6 MeV. This implies that the efficiency is independent of the multiplicity of the capture cascades, thus avoiding the most crucial correction in other types of experiments. The neutron flux at the sample position is determined via the gold standard used in the same experiment. This eliminates many of the systematic uncertainties correlated with electronics, accelerator stability, and detector efficiency. To a large extent, backgrounds from natural radioactivity and from the neutron beam, which are concentrated at low energies, can be discriminated by the good energy resolution, whereas true capture events are found at higher 7-ray energies. A further background discrimination is provided by the multiplicity. This is illustrated in the right part of Fig.2 which shows that typically 90% of the true capture events are recorded with multiplicities >3, while background events are mainly restricted to multiplicities one or two. Backgrounds from isotopic impurities and scattered neutrons can be discriminated by 7-ray energy as can be seen at the example of 164 Dy in Fig.3. More than 60% of the capture events on the 163 Dy impurity are located above the 164 Dy peak, a region which is excluded from the evaluation of the cross
349 5.0-1 o 4 : 164
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Figure 3: Correction for isotopic impurities and capture of scattered neutrons.
section. The respective background is subtracted by a normalized spectrum measured with the 163 Dy sample (middle part in Fig.3). The remaining background due to capture of sample scattered neutrons in the barium isotopes of the scintillator is eliminated using a normalized spectrum recorded with a scattering sample. The final spectrum in the right part of Fig.3 shows good but not perfect compensation of both backgrounds. It is to be emphasized that these corrections can be calculated versus neutron energy and that the good resolution allows to check the result in each neutron energy interval. Thus reliable results are obtained even at low neutron energies where the correction for scattered neutrons is larger. An important feature of the present setup is the short primary flightpath. Consequently, events due to capture of scattered neutrons are significantly delayed in time compared to true capture events. Due to the small capture cross section of the barium isotopes, neutrons are scattered in the scintillator 20 times on average before they are captured. Therefore, most of the corresponding background is sufficiently delayed and does not interfere with the time window of true events. In the last years neutron capture cross sections of 57 isotopes have been measured. Most of the results can be found in the compilation of Bao et al? and have been used in a recent determination of the s- and r- process abundances 4 . References 1. 2. 3. 4.
F. Kappeler, H. Beer, K. Wisshak, Rep. Prog. Phys. 52 (1989) 945. K. Wisshak, et al., Nucl. Instr. Meth. A 292 (1990) 595. Z.Y. Bao, et al. Atomic Data and Nuclear Data Tables (in print). C. Arlandini, et al., Ap. J. 525 (1999) 886.
EXPERIMENTAL STUDY OF T H E ELECTRON SCREENING EFFECT IN T H E D(3HE,P)4HE FUSION REACTION
S. ZAVATARELLI (ON BEHALF O F THE LUNA COLLABORATION) Istituto Nazionale di Fisica Nucleare, Sezione di Genova, via Dodecaneso 16146, Genova, Italy E-mail: [email protected]
33
Due to the low background present at LNGS the d(3He,p)4He cross section has been measured at E = 4.2 to 13.8 keV using the LUNA underground accelerator facility. The experiment was performed to determine the magnitude of the atomic screening effect and to establish values for the energy loss used in data reduction. The observed stopping power of the 3He ions in the D2 target is in good agreement with the standard compilation. Using these stopping power values the data lead to an electron-screening potential energy Ue= 132±9 eV, which is significantly higher than the estimated value of 65 eV from an atomic-physics model
1
Introduction
In laboratory studies the electron screening effect give rise to an enhancement of the fusion cross section over that of bare nuclides which is, conversely, of astrophysical interest. In particular, the lower is the interacting particles energy, the higher is the expected effect. The resulting enhancement of the electron-screened cross section, o"s(E), over that for bare nuclei, Ob(E), is described by the expression [1]: os(E)/ob(E) = Ss(E)/Sb(E) = E/(E+Ue)*exp(7rriUe/E)
(1)
where Ue is assumed to be a constant electron screening potential energy. The effect has been observed in several fusion reactions [6-10], at energies from a few keV to a few tens of keV. However, the deduced screening potentials were significantly larger than could be accounted from the adiabatic limit, i.e. the difference in electron binding energies between the colliding atoms and the compound atom. In data analysis the computation of effective beam energy in the target involves energy-loss corrections, which are extracted from a standard compilation [2]. The compilation is based on experimental data down to energies around the Bragg peak, while at lower energies -relevant to nuclear astrophysics- the experimental data are extrapolated with theoretical guidance. However, new energy-loss measurements of low-energy protons and deuterons in a helium gas yielded [3] significantly lower values than tabulated [2]. Using these lower values, a reanalysis of the 3He(d,p)4He data led [4] to Ue = 134+8 eV, in fair 350
351
agreement with the adiabatic limit. It is not clear whether this solution is also applicable to other reaction studies, due to the lack of experimental energy-loss data at the relevant low energies. As part of an ongoing program on electron-screening effects, we have restudied at the LUNA underground accelerator facility, situated at the Laboratori Nazionali del Gran Sasso (LNGS), the D(3He,p)*He low-energy cross section including an estimate of the associated energy loss (stopping power). 2
The experimental apparatus and energy loss measurements
Technical details of the LUNA facility have been reported [5-7]. Briefly, the 50 kV accelerator facility consists of a duoplasmatron ion source, an extraction and acceleration system, a double-focusing 90° analysing magnet, a gas-target system, and a beam calorimeter. The 3He beam energy ranged from Eb = 11 to 35 keV with a spread less than 20 eV and 60 (xA maximum current. The beam entered the target chamber of the differentially pumped gas-target and was stopped in the calorimeter. The D2 gas pressure in the target chamber was varied from 0.05 to 0.30 mbar. The beam size at the calorimeter was investigated by means of a variable diameter shadowing plate and the complete beam collection was carefully checked. The detector setup consisted of eight, 1 mm thick Si detectors of 5x5 cm2 area (each) placed around the beam axis: they formed a 12 cm long parallelepiped in the target chamber and were shielded by a 27 (im thick Al foil in order to stop the 4 He ejectiles, the elastic scattering products and the light induced by the beam. At a given incident energy Eb, the reaction yield Y(Eb,p), was obtained as a function of the pressure. The yield is related to the cross section , from which one arrives - from a Taylor expansion of the cross section as a function of energy - to the expression [8]: <xY(Eb,p) = o(Eb,p)/a(Eb,p=0)=(l-e(Eb)*p0*zav*(jm-l)*p/(po*Eb) +...)
(2)
where a is a normalization constant, p0 and p0 are the density and pressure of the D2 gas at STP, respectively, and e(Eb) is the energy loss (stopping power) of 3He ions in the D2 gas. Equation 2 assumes a negligible energy dependence of the S(E) factor and of the energy loss e(E) over the energy range of the target thickness, which is well fulfilled. Using for o(Eb,p=0) the intercept a of the experimental o(Eb,p) data and for do/dp the slope b of these data, one arrives at the energy loss value e(Eb)=b/a*Eb*R*T/(M*zav*(jcn-D)
(3)
352
where R, T, and M are the gas constant, absolute gas temperature, and molecular weight of the D2 gas, respectively. The measured stopping powers came out in good agreement with the compilation. For example at Eb = 14, 19, and 28 keV we found energy loss values e(E„) of 0.89+0.10, 0.94+0.11, and 1.28±0.37 keV/|ig/cm2 in good agreement, respectively, with the values of 0.85, 0.99, and 1.19 keV/jig/cm2 from [2]. 3
The D( He,p) He astrophysical factor and the screening potential
The cross section values measured at E = 4.2 to 13.8 keV (o(E) = 6.7 pb to 4.9 Hb) are displayed in Fig. 1. The data include measurements at the same incident energy and different gas pressure. Previous data [6,9,10], normalized as described in the following, are also shown in Fig. 1. For the analysis of electron screening effects, one must extrapolate the bare cross section c b (E) at high energies (E > 30 keV) to low energies. Recent measurements at E = 36 to 385 keV included also a polarized deuteron beam [11] and led to a consistent Sb(E) energy dependence at low energies, which we have adopted: Sb(E) = 6.70 + 2.43xl0 -2 E + 2.06xl0 -4 E2 MeV-b (with E in keV). With this Sb(E) function (including a free normalisation factor N) and equation 1, the resulting fit of the data at E < 60 keV (Fig. 2) led to Ue = 132 eV±0.3 eV (statistical) ±9 eV (systematical), N = 0.93±0.01 (statistical) +0.07 (systematical) and a reduced %2= 2.07. In the fit, present data and those in [6,9] were considered, together with the subset in [10] with 30< E< 60 keV. Statistical errors only were taken into account in the fit. Each data set has been normalized with a proper constant determined by the %2 minimization procedure. In this way, the effects of systematical uncertainties on the four data sets on Ue are reduced. Normalization constants turned out to be within the systematical errors quoted by each author [6,9,10]. The corresponding S(E) factor curves for bare and shielded nuclei are shown in Fig. 1 as dashed and solid curves, respectively. The experimental Ue value is much larger than the expected value of 65 eV predicted with inclusion of the Coulomb explosion effect, but in agreement with the value calculated for the atomic case. Calculations reported in [12] lead to a small momentum transfer to the spectator nucleus, so that the decrease in screening energy due to the breaking of a molecular bond should be negligible. According to the same calculations, such a decrease for a molecular target with respect to the atomic case could derive from the angle dependence of the screening energy and the subsequent angle-averaging procedure. The latter effect has not been investigated experimentally in detail so far. A similar experiment of 3He(d,p)4He, where Coulomb explosions effects are not present, including energy-loss measurements, is in progress.
353
bare nuclides shielded nuclides
8
10
E
(keV)
30
50
Figure 1. S(E) factor data for the D(3He,p)4He reaction from previous works [6,9,10] (open points) normalized by a fitting procedure (see text) and present work (filled-in points). The dashed curve represents the S(E) factor for bare nuclei and the solid curve that for shielded nuclei with Ue=l 32 eV
References 1. 2.
HJ.Assenbaum, K.Langanke, C.Rolfs: Z.Phys. A327 (1987) 461 KAndersen, J.F.Ziegler: The Stopping and Ranges of Ions in Matter (Pergamon, New York, 1977) 3. R.Golser, D.Semrad: Phys.Rev.Lett. 66 (1991) 1831 4. K.Langanke, T.D.Shoppa, C.A.Barnes, C.Rolfs: Phys.Lett. B369 (1996) 211 5. U.Greife et al.: Z.Phys. A351 (1995) 107 6. P.Prati et al.: Z.Phys. A350 (1994) 171 7. R.Bonetti et al.: Phys.Rev.Lett. 82 (1999) 5205 8. H. Costantini et al.: Phys. Lett. B482 (2000) 43 9. S.Engstler et al.: Phys.Lett. B202 (1988) 179, Z.Phys. A342 (1992) 471 10. A.Krauss et al.: Nucl.Phys. A465 (1987) 150 11. W.H.Geist et al.: Phys.Rev. C60 (1999) 5403 12. T. D. Shoppa et al.: Nuc. Phys. A605 (1996) 387
T H E SOLAR N E U T R I N O PROBLEM: LOW E N E R G Y M E A S U R E M E N T S OF T H E 7 B e ( p , 7 ) 8 B CROSS SECTION F. HAMMACHE", G. BOGAERT6, A. COC,M. JACOTIN, J. KIENER, A. LEFEBVRE, V. TATISCHEPF, J. P. THIBAUD CSNSM, IN2P3-CNRS, 91405 Orsay et Universite de Paris-Sud, Prance P. AGUER, J. F. CHEMIN, G. CLAVERIE, J. N. SCHEURER, E. VIRASSAMYNAIKEN CENBG, IN2PS-CNRS, et Universite de Bordeaux I, 33175 Gradignan, France L. BRILLARD, M. HUSSONOIS, C. LE NAOUR IPN,IN2P3-GNRS et Universite de Paris-Sud, 91406 Orsay, France S. BARHOUMI, S. OUICHAOUI Institut de Physique, USTHB, B.P. 32, El-Alia, Bab Ezzouar, Algiers, Algeria C. ANGULO Centre de Recherche du Cyclotron, UGL, 1348 Louvain La Neuve, Belgium We have measured 7Be(p,7)8B cross sections for Ec.m. = 185.8 keV, 134.7 keV and 111.7 keV (the lowest energy ever reached at this day) using a radioactive 7 Be target (132 mCi). /?+ and a particles from 8 B and 8Be* decay, respectively, were detected in coincidence using a large solid angle spectrometer. The zero energy S factor inferred from our data is in agreement with the most recent direct measurements 1
The 8 B solar neutrinos
Solar neutrino oscillations have been invoked to explain the discrepancy between the measured neutrino flux and the predicted one by solar models. However, the oscillation parameters, square neutrino mass differences and mixing angles, depend on these models and cross sections of nuclear reactions operating in the sun core. In this respect, the most important nuclear physics parameter is the S factor of the 7 Be(p,7) 8 B reaction which gives rise to the crucial 8 B neutrinos 1 . The 7 Be(p,7) 8 B cross section is much too low for being known by measurements at solar energy ( 20 keV). Thus, cross sections are measured at higher energies and then extrapolated down to solar energy by using theoretical energy dependence 2 . "Present address : GSI mbH, Planckstr. 1, D-64291 Darmstadt, Germany, E-mail: [email protected] ''Permanent address : LPNHE, Ecole polytechnique, 91128 Palaiseau, Prance 354
355
The previous direct measurements of the 7 Be(p/y) 8 B cross section 3,4,5,6 are in agreement with regard to the energy dependence but divide into two groups with S(0) values differing by 30%, making this quantity the most uncertain input to solar models. Therefore, it appeared highly desirable to perform new measurements of the 7 Be(p,7) 8 B cross section. In a previous work 7 , we have measured cross sections for the 7 Be(p,7) 8 B reaction for E c . m = 0.35 - 1.4 MeV using radioactive 7 Be targets. The resulting S(0) factor was found close to 19 eV b in agreement with the results of Filippone et al. 6 and Vaughn et al. 5 . Here, we report on new direct measurements of the 7 Be(p,7) 8 B cross section performed at the PAPAP electrostatic accelerator 8 at energies below 200 keV, where extrapolation to solar energies is expected to be almost free of theoretical uncertainties 2 . 2
Experimental methods
We used a highly radioactive 7 Be target (131.7 mCi) in which a small amount ( 3.1016 atoms) of 9 Be was also introduced 9 . To reduce the background, the 7 Be(p,7) 8 B cross section was measured by detecting in coincidence the delayed alpha and beta + particles from 8 B(/3+) 8 Be*(2a) decays at backward and forward direction respectively, with respect to the beam direction. The target and the a — /? detection system were positionned in a solenoidal superconducting magnet SOLENO 1 0 which thanks to its focusing properties, allowed us to obtain a very high detection efficiency while moving away the detectors from the high 7-ray flux of the 7 Be decay. The a and /? detectors were placed on each side of the target which was located perpendicular to the symmetry axis of the field and near the solenoid center. /3+ particles (E TOax = 14 MeV) were detected in a set of 6 successive cylindrical plastic scintillators centered on the field axis and 22 cm away from the target. The overall /?+ efficiency as given by GEANT was 25% of 4n. a particles (from 8 Be* decay) with energies lower than 3.5 MeV were deflected towards the field axis and detected in a array of 6 x 4 Si detectors (22 mm x 45 mm x 0.1 mm). The detectors were mounted in a cylindrical geometry aligned on the solenoid axis. The overall a efficiency was found to be 11.5% (see below), very close to that deduced from realistic simulations using GEANT code. Both singles and coincidence events between a and /?+ particles were recorded, the latter providing extremely clean spectra even at lower bombarding energies. For the delayed detection purpose, the beam passed through an electro-
356
static deflector which was alternately switched on and off by time periods of 1.5 s. When the beam was on the target, we detected the prompt events from 9 Be(p,a) 6 Li and 9 Be(p,d) 8 Be for normalization of the cross section and energy calibration of a detectors and when the beam was away from the target, we detected the delayed events from 7 Be(p,7) 8 B. 3
Results
Cross section measurements were performed at three proton energies, 217 keV, 160 keV and 130 keV. The absolute value of the cross section at the proton energy of 217 keV was obtained using singles delayed a counting in the range from 1 MeV to 3.36 MeV. The corresponding singles delayed a particle spectrum after background substraction, is shown in figure l.a. :. ~ ':~T~,
a)
:
u
LA 11
b)
1
Ep«217keV
i
F._=217keV
kn
\
/HfSLft*..,*. L«. EJMeV)
^-rirrli.... i , I F R t A L ^ Ea (MeV)
Figure 1: a). Energy spectrum of singles delayed a particles, b). Energy spectrum of delayed a particles detected in coincidence with delayed /? particles.
For comparison, the coincidence delayed a particle spectrum measured in the same runs is shown in figure l.b. In particular, it can be seen in figure l.b that the low energy (< lMeV) component in the singles spectrum corresponding to pileup events due to photoelectrons created by the 478 keV 7 rays has completely vanished in the coincidence spectrum. The solid curve in figure l.b is obtained from a least squares fit to this background free coincidence spectrum. We can see in figure l.a, that the same curve (after normalization to counting) gives also a perfect fit to singles data as expected from an unbiased background substraction process. The a detection efficiency in the 1-3.36 MeV energy range was determined with the same experimental setup (except an enriched 7 Li target) from an analysis of the reaction 7 Li(p,7i) 8 Be* at E p = 160 keV, by detecting in coincidence the e + - e _ pairs created by the 14.8 MeV 71 line and the a particles coming from the decay of 8 Be* (the same a's as in the channel 8 B(/?+) 8 Be*). The a detection efficiency was deduced from the ratio of the coincidence silicon-plastic
357
counts and of the single plastic counts due to 71 only. Finally, the detection efficiency was ea = 0.115 ± 0.008 in the 1.00-3.36 MeV energy range, in fair agreement with GEANT simulations. The 7 Be total activity at the beginning and at the end of the run at 217 keV, the target area (0.47 ± 0.02 cm 2 ) and the 7 Be activity profile were accurately determined with the same instruments and methods like in the experiment described in ref. 7 . The value of 16.7 ± 2.1 nb at the incident proton energy of 217 keV was deduced. This value takes into account a 1% correction due to the 8 B diffusion on platinum atoms and escape out of the target 1 3 , c .This correction was calculated using a TRIM 1 4 simulation with target thickness and composition determined from consistent RBS, (d,p) and PIXE analysis measurements performed during the course of the experiment. The measured target thickness of 9.6 ± 1.0 keV for protons of 217 keV leads to an effective energy of 212.4 keV (ECTO = 185.8 keV) and an S factor value of 17.2 ± 2.1 eV b. The cross sections at E p = 130 keV and 160 keV were determined using a-/3 coincidence measurements only, due to the high background in the singles spectra. On the contrary, the corresponding coincidence a energy spectra which are shown in figure 2 together with time difference spectra between a and 0+ particles were very clean. The 3 peaks in the time spectra correspond to 3 different kind of trajectories of the a particles in the magnetic field of SOLENO. S factors at E p = 160 keV and E p = 130 keV, relative to the one measured at E p = 217 keV, were obtained by normalization to the a yield from the reaction 9 Be(p,a) 6 Li through the relation:
S7(£7roi)
RW,*
S9(E»mJ
(l)
where the subscripts 1, 2 and 3 label the three runs at E p = 217 keV, 160 keV and 130 keV respectively and the superscripts 7 and 9 label the reactions 7 Be(p,7) 8 B and 9 Be(p,a) 6 Li respectively. R(Ei) 7 ' 9 is the coincidence yield normalized to the a yields from 9 Be(p,a) 6 Li, S9 is the astrophysical S factor of the reaction 9 Be(p,a) 6 Li at the corresponding c m . effective energy. K is a c
Backscattered 8 B nuclei which decay outside the target are lost for delayed detection inducing smaller measured cross sections. For heavy target backing, the yield of the 8 B loss depends strongly on the beam energy and on the target composition and thickness. In this experiment at low energy, the target deposit was thick enough to stop most of backscattered 8 B nuclei resulting in small 8 B loss.
358
constant accounting for the changes in dead times, in effective time parameters (see parameter 0 in formula 4 of ref. 6 ) and in angular distributions of alphas in 9 Be(p,a) 6 Li with the bombarding energies. All the last corrections were found very small (less than a few percents). The exponential term accounts for 7 Be target activity decrease. s
is
§ ',% j12 IO 8 6 ^
2 O
J "
Ep=I60
O
%
E JV...
Ep=160
40
10 0
% *°
krV
-
1
.
Ep=130furV
30
L
20 JO
i ihJIWLA I
ll
20
Ep=130k*V
O
0
30
L
to 5
k*V
2
1. 3
4 5 £a>
O
J_
1/ loo
t-A.
. _ . . , . .
200
300 Af^p (rial
Figure 2: a). Energy spectra obtained at 160 keV and 130 keV for delayed a particles detected in coincidence with delayed /? particles, b). Corresponding spectra for time difference between a and /3+ particles obtained using the first of the six plastic scintillators. A null time of flight difference is found arbitrarily at 200 ns due to delays in the electronics.
The interest of this normalization comes from the fact that the amount of Be over 7 Be remains constant on the whole surface. This was checked by the comparison of r Be 7 ray scan with a 9 Be scan using (d,p) reaction analysis. Such normalization allowed us then to be free from target non uniformity and target degradation problems. For the calculations of the proton energy losses at 160 and 130 keV, we took into account the loss of target material thanks to the monitoring of the 9 Be content through the 9 Be(p,a) 6 Li reaction. Taking into account the 8 B escape effect13 which did not exceed 1%, we deduced the astrophysical S factors values of S(134.7 keV) = 19.5 ± 3.1 eV b and S(111.7 keV) = 15.8 ± 2.7 eV b. Results in the form of astrophysical S factors are shown in figure 3 together with those obtained previously at higher energies 7 . Extrapolation from the three S factor measured values to zero energy gives S(0) = 18.5 ± 2.4 eV b (the calculation of Descouvemont et al. 15 was used for extrapolation purpose). This value is in perfect agreement with the value S(0) = 19.1 ± 1.2 eV b deduced from our previous measurements performed at higher energies when the experimental values are corrected from the 8 B escape effect13'9 and using the calculation of Descouvemont et al. (S(0) = 18.5 ± 1.0 eV b was deduced from uncorrected data in reference 7 ). The averaged S(0) values corresponding to the three experiments, in the 9
359
Figure 3: Measured S factors from present work and from reference [7] after backscattering correction. Error bars represent relative uncertainties. The curve through t h e d a t a is a fit t o the three sets of data using the calculation of Descouvemont et al. [15] for the non resonant capture.
energy range up to 1.4 MeV and up to 0.43 MeV are respectively, 19.1 ± 1.1 eV b and 19.2 ± 1.2 eV b. The quoted error bars are experimental only and neglect any uncertainty arising from extrapolation process. Note that this low S-value was also inferred recently by Hass et al. 16 and also by indirect measurements of Kikuchi et a l . 1 7 , Iwasa et a l . 1 8 and Azhari et a l . 1 9 . References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19.
H. Schlattl et al., Phys. Rev. D 60, 113002 (1999). E. Adelberger et al. , Rev. Mod. Phys, (1998). P.D. Parker , Phys. Rev. 150, 851 (1966). R.W. Kavanagh et al., 1969, Bull. Am. Phys. Soc. 14, 1209 F.J. Vaughn et al., Phys. Rev. C 2, 1657 (1970). B.W. Filippone et al., Phys. Rev. Lett. 50, 412 (1983). F. Hammache et al., Phys. Rev. Lett. 80, 928 (1998). G. Bogaert et al., Nucl. Instrum. Methods B 89, 8 (1994). F. Hammache, Thesis, (1999), http://www-csnsm.in2p3.fr/ J.P. Shapira et al., Nucl. Instrum. Methods 224, 337 (1984). D. Zahnow et al., Z. Phys. A 351, 229 (1995). D. Zahnow et al., Z. Phys. A 359, 211 (1997). L. Weissman et al., Nucl. Phys. A 630, 678 (1998). J.F. Ziegler, TRIM, http://www.research.ibm.com/ionbeams/ P. Descouvemont and D. Baye, Nucl. Phys. A 567, 341 (1994). M. Hass et al., Phys. Lett. B 462, 237 (1999). T. Kikuchi et al., European. Phys. J. A 3, 213 (1998). N. Iwasa et al., Phys. Rev. Lett. 83, 2910 (1999). A. Azhari et al., Phys. Rev. Lett. 82, 3960 (1999).
D E T E R M I N A T I O N O F T H E A S T R O P H Y S I C A L S - F A C T O R S S17 A N D S18 F R O M 7 B E ( d , n ) 8 B A N D 8 B ( d , n ) 9 C C R O S S - S E C T I O N S D. BEAUMEL, S. FORTIER, H. LAURENT, J.-M. MAISON, S. PITA Institut de Physique Nucleaire, IN2P3-CNRS, 91406 Orsay cedex, France T. KUBO, T. TERANISHI, H. SAKURAI, T. NAKAMURA, N. AOI, N. FUKUDA, M. HIRAI, N. IMAI, H. IWASAKI, H KUMAGAI, S.M. LUKYANOV, K. YONEDA, M. ISHIHARA RIKEN (The Institute of Physical and Chemical Research), 2-1 Hirosawa, Wako, Saitama 351-01, Japan T. MOTOBAYASHI Department of Physics, Rikkyo University, Toshima-ku, Tokyo 171, Japan H. OHNUMA Department of Physics, Chiba Institute of Technology, Chiba 275, Japan The astrophysical S-factor of the 7 Be(p,7) 8 B and 8 B(p,7) 9 C capt ure reactions were determined from 7 Be(d,n) 8 B and 8 B(d,n) 9 C transfer cross-sections at 12 and 14.4 MeV/u, respectively. Results extracted for 5i7(0) are close to the adopted value. The value obtained for Sis is lower than those predicted by existing calculations.
Radiative c a p t u r e reactions such as (p,7) play an i m p o r t a n t p a r t in basic astrophysical processes such as hydrogen burning. In general, the relevant incident energies for such reactions are well below the coulomb barrier, making direct measurements of their cross-sections very difficult. T h e situation becomes even worse when the capturing nucleus is radioactive. In such cases, the use of indirect m e t h o d s can b e appropriate. A few years ago, such an indirect approach based on measurements of peripheral one-proton transfer cross-sections on 7 B e was p r o p o s e d 1 ' 2 as an alternative way to determine 5i7(0), t h e astrophysical S-factor of the long-studied 7 B e ( p , 7 ) 8 B reaction at solar energies 3 . It consists in extracting nuclear quantities called Asymptotic Normalization Coefficients (ANC) for the virtual decay 8 B —> 7 B e + p , t h r o u g h a Distorted Wave Born Approximation (DWBA) analysis of peripheral transfer process. Knowing these quantities, the S-factor of the c a p t u r e reaction can then b e reliably calculated. T h e condition of peripherality in the transfer reaction requires incident energies of typically 10 MeV per nucleon or less. We report here on an experimental study of the 7 B e ( d , n ) 8 B and 8 B ( d , n ) 9 C proton transfer reaction at 12 and 14.4 M e V / u incident energy, respectively. 360
361 From these cross-sections, t h e S-factor of the 7 B e ( p , 7 ) 8 B and the 8 B ( p , 7 ) 9 C c a p t u r e reactions can b e derived using the ANC m e t h o d . T h e 7 B e ( p , 7 ) 8 B c a p t u r e reaction at solar energies (near 20 keV) is well-known to b e closelylinked to the so-called solar neutrino puzzle but its S-factor still suffers from large u n c e r t a i n t i e s 3 . This reaction ends the pp-III chain in the sun. In other stars such as the (still hypothetic) supermassive stars, t e m p e r a t u r e a n d densities can be such t h a t the 8 B ( p , 7 ) 9 C capture reaction may compete with the (3 decay of 8 B and become a possible alternative p a t h to the synthesis of C N O elements (the so called hot pp-chain). A recent calculation of the S-factor for this reaction was performed 4 and the result was found to be in disagreement with a previous evaluation 5 . On the experimental side, only an a t t e m p t was m a d e by measuring the Coulomb dissociation of 9 C 6 . T h e experiment was performed at the R I K E N Accelerator Research Facility where the 7 B e ( d , n ) 8 B and 8 B ( d , n ) 9 C cross-sections were measured in inverse kinematics. T h e low energy 7 B e and 8 B beams were produced by fragm e n t a t i o n of a 70 A-MeV 1 2 C primary b e a m at the R I P S 7 fragment separator using a combination of thick 9 B e target and degrader. Incident particles were identified unambiguously event-by-event, by measuring the time of flight between the two last focal planes of R I P S . T h e ejectile detection system was composed of three thin plastic scintillators, placed at 38cm downstream of the target. T h e first two detectors, 0.25mm thick, were used as A E - E telescope for ejectiles identification and the last detector (1mm thick) served as a veto detector to reject b e a m particles. Recoiling neutrons were detected in coincidence with ejectiles by BC408 plastic detectors placed in the backward hemisphere. 7
T h e experimental cross-section for the peripheral transfer reactions B e ( d , n ) 8 B and 8 B ( d , n ) 9 C can b e written as: „(()\-(r
^1,3/2(0) ,r ,2^1,1/2(0) (ll,3/2) ,2 + (Ci,i/2)-72 "l,3/2
m
I1)
1,1/2
where aij are the calculated D W B A cross-sections, and b 2 • are given by the ratio (v.ij{r)/W+(r))2 at large radius , uij(r) being the single-particle wavefunctions used in the D W B A calculation as form factors, and W+ (r) the Whittaker function. C i i 3 / 2 a n d Ci^/2 are the two ANC's of interest. In our case, the ratio (Tij/bj • is almost independent of j within 1% accuracy (for 0CM < 15°), due to the peripherality of the reaction studied. Consequently: CTW
T h e sum {Cij/2)2
= ((C 1 ,3 / 2 ) 2 + ( C
+ (Ci^/2)2
1
,
1 / 2
)
2
)^^
(2)
just determines the m a g n i t u d e of the c a p t u r e
362 cross-section at astrophysical energies 8 , a n d can be extracted by normalizing "•1,3/2 t ° t h e transfer data. D W B A cross sections aij were calculated using t h e zero-range D W B A code D W U C K 4 1 0 . We have used optical potentials for deuterons u ' 1 2 a n d neutrons 13 ' 14 > 15 derived from global formulae. As stressed in ref. 16 , t h e determination of t h e optical potential in the entrance channel is i m p o r t a n t for t h e accuracy on t h e final value of t h e S-factors. This point is crucial in t h e case of S17 for which more precision is aimed. Consequently, we have measured t h e 7 B e ( d , d ) 7 B e elastic scattering cross-section at t h e Orsay T a n d e m Accelerator using a radioactive 7 B e target. T h e S-factors for t h e c a p t u r e reactions were calculated from t h e A N C ' s using a Direct Radiative C a p t u r e Model with a two-body description for t h e 8 B ground state. In such approach, t h e capture cross-section at astrophysical energies is proportional to t h e s u m {Ci^/2)2 + ( C ^ i / 2 ) 2 which defines completely its overall normalization. Figure 1 shows preliminary values of t h e astrophysical S-factor Si7(0) of t h e 7 B e ( p , 7 ) 8 B c a p t u r e reaction deduced from ANC's extracted by using different combinations of the optical potentials mentioned above. For t h e three upper points, t h e optical potential deduced from our elastic scattering measurement was used. T h e fluctuations with respect to the average over all t h e plotted values are relatively small, of t h e order of ± 6 % . T h e three values deduced using t h e optical potential parameters coming from the elastic scattering measurement are thought to be t h e most reliable. T h e y are compatible with results using transfer with heavier targets 8 ' 9 a n d stand very close to t h e presently adopted value, bringing another evidence t h a t t h e two former direct measurements leading to high values (25-27 eV.b) of S17 are incorrect. O u r results thus plead in favor of a reduction of the adopted u p p e r error b a r on S17, still large enough to remain compatible with these larger direct measurements. Concerning t h e S-factor Sis for the 8 B ( p , 7 ) 9 C , we extracted a preliminary value Sis = 45 ± 13 eV.b at 100 keV. T h e error was estimated taking into account t h e statistical error (much larger t h a n for t h e former reaction) a n d t h e error due t o t h e choice of the optical potentials. This result stands by nearly a factor of two below t h e calculated value reported in ref. 4 , where a microscopiccluster description of t h e 9 C structure was used. Such tendency of microscopic models to overestimate absolute cross-sections was already observed in t h e case of 5*17. In ref. 5 , the mean value of S i s , averaged over t h e energy range Ep < 0.8MeV was found to be 5 i s ~ 210eV.6, much higher t h a n our results. T h e origin of such a high value probably comes from t h e value of 2.5 used for the spectroscopic factor, which is incompatible with our present transfer d a t a from which a spectroscopic factor of nearly 0.75 can b e deduced.
363
d(7Be,8B)n 12 MsV/u i
20 -
I »
15 10 5 0
i
i
i
esm-WATS esm-WILM esra-BECH
•
DAE-WATS DAE-W1LM DAE-BECH PER-WATS PER-WILM PER-BECH
• * ! + |
i
• •
_
•j
-
4
l
l
10
15
r,
20
i
25
30
S 17 (0) Figure 1: Values of Si7(0) obtained by using different combinations of optical potentials in the entrance and exit channels (see text). The label "esm" refers to the optical potential deduced from our elastic scattering measurement. The dashed line shows the currently adopted value while the solid lines indicate the adopted error bars.
References 1. H.M. Xu et al., Phys. Rev. Lett. 73, 2027 (1994). 2. A.M. Mukhamedzhanov et al, Phys. Rev. C 56, 1302 (1995). 3. E.G. Adelbergeret al., Rev. Mod. Phys. 70, 1265 (1998) and references therein. 4. P. Descouvemont, Nucl. Phys. A 646, 261 (1999). 5. M. Wiescher et al, Astrophys. J 343, 352 (1989). 6. I. Hisanaga et al., to be submitted 7. T. Kubo et al, Nucl. Instrum. Methods B 70, 309 (1992). 8. A. Azhari et al. Phys. Rev. Lett. 82, 3960 (1999). 9. A. Azhari et al. Phys. Rev. C 60, 055803 (1999). 10. P.D. Kunz, code DWUCK4 University of Colorado (unpublished) 11. C M . Perey and F.G. Perey, Phys. Rev. 132, 755 (1963). 12. W.W. Daehnick, J.D. Childs and Z. Vrcelj, Phys. Rev. C 21, 2253 (1980). 13. F.D. Becchetti and G.W. Greenlees, Phys. Rev. 182, 1190 (1969). 14. D. Wilmore and P.E. Hodgson, Nucl. Phys. 55, 673 (1964). 15. B.A. Watson, P.P. Singh, R.E. Segel, Phys. Rev. 182, 977 (1969). 16. J.C. Fernandes, R. Crespo, F.M. Nunes and I.J. Thompson, Phys. Rev. C 59, 2865 (1999).
A MEASUREMENT OF THE 1 X ( a , a ) DIFFERENTIAL CROSS SECTION AND ITS APPLICATION ON THE 13C(a,n) REACTION. M. HEIL1, A. COUTURE2, J. DALY2, R. DETWILER2, J. GORRES2, G. HALE3, F. KAPPELER1, R. REIFARTH1, U. GIESSEN2, E. STECH2, P. TISCHHAUSER2, C. UGALDE2, M. WIESCHER2 1
Forschungszentrum
Karlsruhe
2
University of Notre Dame, IN
3
Los Alamos National Laboratory,
NM
13
The C(ce,n) reaction is considered as the main neutron source for the s process in AGB-stars of 1-3 solar masses. The reaction takes place at temperatures of about 10s K which corresponds to a Gamov peak at 190 keV. At this energy a direct measurement of the 13C(a,n) cross section is extremely difficult and therefore one has to rely on an extrapolation of the measured cross section to low energies. Since the cross section is suspected to be influenced by a subthreshold resonance and a resonance just above the threshold, R-matrix analyses are especially suited for the extrapolation. An exact measurement of the excitation function and the angular distribution of the ,3 C(a,a) reaction, which represents another decay channel of the 1 7 0 compound nucleus, can provide additional constraints to improve the quality and reliability of the R-matrix extrapolation. New results for the 13C(oc,a) differential cross section in the energy range 2.6 - 6.2 MeV are presented. The experiment was performed with the 10 MV tandem accelerator at the University of Notre Dame. Angular distributions for 29 angles between 44° and 165° could be obtained for 375 energies. Preliminary results for the reaction rate at kT = 8 keV are presented.
1
Introduction
Advanced stellar models [1] suggest that low mass (M ~ 3M 0 ) TP-AGB (thermally pulsing asymptotic giant branch) stars play the major role in producing i-process elements. These stars show alternate H and He burning episodes in two shells separated in mass by a thin He-rich intershell. During each thermal pulse the temperature becomes high enough to activate the 22Ne source resulting in high neutron densities (about 1010 n/cm3) while during the interpulse phase at temperatures of kT ~ 8 keV only the 13C reaction is active leading to lower neutron densities of about 107 n/cm3. A lower 13C(a,n) cross section than previously reported [2,3] would suggest that 13C is not completely burned during the interpulse phase leading to much higher neutron densities during the successive thermal pulse. Also the production mechanism of the 13C pocket is still not quite understood. One plausible scenario is the mixing of protons into the He shell region during the third dredge-up and the production of 13C via the reaction chain 12C(p/y)13N(p+v)13C. 364
365
Knowledge of the 13C(a,n) cross section at the relevant energies would eliminate an essential uncertainty with respect to the overall neutron balance, thus allowing for an improved test of the 13C production, and hence of the mixing mechanism predicted by stellar models. 2
Experiment
The experiment was performed at the University of Notre Dame using a 10 MV tandem accelerator which provided a beam of doubly charged alpha particles. The beam was focused on 13C targets with 8 mm diameter (99.9% enrichment). The beam spot on the target was limited to 4 mm diameter by two slit pairs. Build-up of 12 C on the sample was minimized by a cold trap located at the entrance of the chamber. The scattering chamber contained 29 silicon detectors at laboratory angles of 43.9, 48.9, 54.0, 58.9, 63.9, 68.9, 74.0, 75.8, 79.0, 80.8, 84.0, 85.8, 89.0, 90.8, 94.0, 95.8, 99.0, 100.8, 103.9, 105.8, 110.8, 115.8, 120.8, 125.8, 130.8, 140.8, 150.8, 160.8, and 165.8°. The detector solid angles were defined by collimators of 5.5 mm diameter. The distance from the center of the sample to the collimators was 58.7 cm. Figure 1 shows a sketch of the experimental setup.
Figure 1: Schematic sketch of the experimental setup.
The 13C targets with 9.6 to 12.1 ng/cm2 thickness were mounted on a sample ladder at an angle of 45° relative to the beam axis. The beam current was measured with a Faraday cup at the exit of the chamber. The example of Fig. 2 illustrates mat
366
the experimental resolution allowed clear separation of the 13C signal from background due to 12C build-up.
glOO
u 940 960 Channel number
1000
Figure 2: Spectra of scattered a-particles in a minimum of the 13C cross section. In general, the 12C peak is as small as the 1 6 0 feature in the right part of the spectrum.
3
R-matrix analyses
Extrapolation of the 13C(oc,n) cross section to the relevant energies was performed by means of a multilevel R-matrix fit. All open reaction channels (160(n,tot), 160(n,n)160, lsO(n,a)13C, 13C(a,n)160, 13C(a,a)130) were included in the fit and channel spins up to 1 = 5 were considered. In total, the data sets contained 22,458 data points as indicated in Table I. Reaction 16
0(n,tot) 0(n,n) 16 0 16 0(n,a) 13 C 13 C(a,n) 16 0 I6
I3
C(a,a) 13 0
Energy range (MeV) Ea= 0 -14.9 En= 0.2 - 6.2 En= 3.6 - 8.8 E a = 0.4 - 5.2
Number of data points 533 12959 292 246
E a = 2.6-6.2
8428
Author [4], [5] [6], [7] [8], [9], [10], [11] [12], [13], [14], [15], [16], [17] This work
Table I: Summary of data sets used in R-matrix fit.
Figure 3 shows the present status of the R-matrix analyses for the S-factor of the 13 C(a,n) reaction. For further improvement of this analysis it is intended to continue the scattering cross section measurement down to an a-energy of 1 MeV.
367
i jj
• Experimental data R-matrix fit
E a in MeV Figure 3: Preliminary result of the R-matrix fit based the present scattering data.
4
First results
The present status of our analysis yields a reaction rate of 1.40 10"14 cm3/mole/s at T9=0.09 (kT = 8 keV), somewhere in between the values given by Caughlan and Fowler [2] (0.89 10"14 cm3/mole/s) and by the NACRE compilation [3] (2.22 10"14 cm3/mole/s). References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17.
R. Gallino et al., Ap. J. 497, 388 (1998). G.R. Caughlin and W.A. Fowler, At. Data Nucl. Data Tables 40, 283 (1988). C. Angulo et al., Nucl. Phys. A656, 3 (1999). D. G. Foster and D.W. Glasgow, Phys. Rev. C 3, 576 (1971). W. Ohkubo, private communication. I. Shouky and S. Cierjacks, report KFK-2379, FZK (1976). R. O. Lane et al., J. Appl. Phys. 12,135 (1961). E.A. Davis et al., Nucl. Phys. 48,169 (1963). D. Dandy, J.L. Wankling, C.J. Parnell, AWRE-060/68, Aldermaston (1968). J. Seitz and P. Huber, Helv.,Phys. Acta 28, 227 (1955). R.B. Walton, J.D. Clement, andF. Borelli, Phys. Rev. 107,1065 (1957). S.E. Kellogg, R.B. Vogelaar, R.W. Kavanagh, Bull. Am. Phys. Soc. 34, 1192 (1989). K.K. Sekharan, A.S. Divatia, M.K. Netha, Phys. Rev. 156, 1187 (1967). C.N. Davids, Nucl. Phys. A110, 619 (1968). H.O. Cohn, J.K. Bair, and H.B. Willard, Phys. Rev. 122, 534 (1961). C.R. Brune, I. Licot, andR.W. Kavanagh, Phys. Rev. C 48, 3119 (1993). H.W. Drotleff et al., Ap. J. 414, 735 (1993).
NEUTRON CROSS SECTIONS MEASUREMENTS FOR LIGHT ELEMENTS AT ORELA AND THEIR APPLICATION IN NUCLEAR CRITICALITY AND ASTROPHYSICS
K.H. GUBER, L.C. LEAL, R.O. SAYER, R.R. SPENCER, P.E. KOEHLER, T.E. VALENTINE, H. DERRffiN, J. ANDRZEJEWSKI, Y.M. GLEDENOV, AND J A . HARVEY Oak Ridge National Laboratory,
P.O. Box 2008, Oak Ridge, TN USA E-mail: guberkh @ ornl.gov
37921-6354
The Oak Ridge Electron Linear Accelerator (ORELA) was used to measure neutron total and capture cross sections of aluminum and natural silicon in the energy range from 100 eV to -600 keV. These measurements were carried out in order to support the Nuclear Criticality Safety Program. More accurate nuclear data are not only needed for these calculations but also serve as input parameters for s-process stellar models. Most recently developed models require more precise cross section data in the neutron energy range characteristic of the site of the s-process.
1
Introduction
To support the Nuclear Criticality Safety Program, neutron cross section measurements have been initiated at the Oak Ridge Electron Linear Accelerator (ORELA). This will not only help to resolve inconsistencies between different data sets but will also improve the representation of the cross sections since most of the available evaluated data rely only on old measurements. Usually these were done with poor experimental resolution or only over a very limited energy range which is insufficient for current applications. To clarify inconsistencies in the criticality calculations for systems including Al and 235U, the total and capture cross sections of Al in the energy range from about 100 eV to several hundred keV have been measured. To understand systems including sand the total cross section of natural silicon has been measured with much higher resolution than has been done previously. More accurate neutron cross section data are not only needed for these calculations but also serve as input parameters for s-process stellar models. Due to their large abundance, light elements could be neutron poisons for the s-process. The silicon isotopes in the mainstream presolar SiC grains found in meteorites originated most 368
369 probably from the envelope of an asymptotic giant branch (AGB) star [1]. These grains show a Si isotopic ratio which cannot be explained by AGB stellar models. And, in addition, up-to-date stellar models require Maxwellian averaged cross section at lower temperatures (kT= 8 keV) than previous models in order to calculate the s-process nucleosythesis abundance. Table 1
Energy kT (keV) 5 8 10 15 20 25 30 35 40 50 100
2
ORELA (mb) 9.43±0.91 6.80+0.54 5.94+0.49 4.78+0.36 4.10+0.31 3.63+0.26 3.30+0.24 3.04±0.22 2.83+0.20 2.50+0.18 1.59+0.12
Maxwellian averaged capture cross section of Al
Ref. [8] (mb) 11.2 7.60 6.8 5.4 4.6 4.1 3.74+0.3 3.3 3.0 2.3
Ref. [9] (mb)
Ref. [10] (mb)
Ref. [7] (mb)
7.51 6.72 5.39 4.64 4.14 3.80+0.3 3.55 3.34
4.75 4.24 3.89+0.27 3.63 3.42
3.9
Experimental setup
ORELA is the only high power white neutron source with excellent time resolution still operating in the USA and is ideally suited for these experiments. We used two extremely high purity (0.01520 a/b and 0.04573 a/b) rectangular aluminum samples for the neutron capture measurements. For the silicon capture measurements, we used a high purity natural silicon metal sample with a thickness of 0.07831 a/b. The samples and the C6D6 detectors were located at a distance of 40 meters from the neutron target. This capture system [2] has been re-engineered to minimize the amount of structural material surrounding the sample and detectors in order to reduce the prompt neutron sensitivity. A 0.5-mm thick 6Liglass scintillator served as the neutron flux monitor. Pulse-height weighting was employed with the C6D6 detectors; normalization of the capture efficiency was carried out in a separate measurement using the "black resonance" technique by means of the 4.9-eV resonance from a gold sample [3]. For the Al transmission measurements, the two extremely high purity (0.0189a/b and 0.1513 a/b) samples were mounted in the sample changer positioned at about 10 meters from the neutron target in the beam of ORELA. A pre-sample collimation limited the beam size to about 2.54-cm on the samples and allowed
370
only neutrons from the water moderator part of the neutron source to be used. The neutron detector was an 11.1-cm diameter, 1.25-cm thick 6Li-glass scintillator viewed on edge by two 12.7-cm diameter photomultipliers and positioned in the beam at 79.815 meters from the neutron source. Additional measurements were made in both experiments for the open beam, and measurements with a thick polyethylene sample were used to determine the gamma-ray background from the neutron source. 3
Results
The capture and transmission data sets were analyzed with the R-matrix code SAMMY [4]. Using the newly determined r n values for Al, SAMMY calculated the corrections for self-shielding and multiple scattering to the capture data. From these resonance parameters, we calculated the Maxwellian average capture cross section, which are compiled in Table 1, together with previous experiments and evaluations. We find an overall reduction of about 10% in the Maxwellian average cross sections. This is a result of the underestimated neutron sensitivity in the older measurements and an improved calculation of the weighting function. Table 2 Maxwellian averaged capture cross section of Si isotopes 29
^Si Energy kT (keV) 5 8 10 15 20 25 30 40 50 70 100
ORELA (mb) 0.41±0.02 0.42+0.02 0.50±0.02 0.83+0.02 1.12+0.07 1.29+0.08 1.37+0.09 1.39+0.09 1.33+0.08 1.16+0.07 0.95+0.06
Ref. [8] (mb) 0.29 0.86 1.9 2.5 2.8 2.9+0.3 2.8 2.7 1.9
ORELA (mb) 6.69+0.65 9.09+0.8 9.52+0.78 9.06+0.69 7.87+0.60 6.69+0.51 5.69+0.44 4.17+0.32 3.26+0.24 2.24+0.15 1.53+0.09
30
Si Ref. [8] (mb) 10.3 14.4 13.3 11.3 9.5 7.9+0.9 5.8 4.4 1.7
Si Ref [8] ORELA (mb) (mb) 124 18.29+1.78 11.01+1.01 43 8.25+0.74 22 4.61+0.39 13 2.97+0.24 8.8 2.13+0.17 6.5+0.6 1.68+0.12 3.8 1.31+0.08 2.6 1.22+0.08 1.19+0.08 0.96 1.10+0.08
For analysis of the silicon capture data, we used the Tn values from the latest ENDF/B-VI evaluation of the cross section for the different isotopes [5]. From the resonance parameters resulting from our SAMMY analysis, we calculated the Maxwellian average cross sections for the different isotopes. The results are compiled the Table 2 and compared with the most recent evaluation [8]. We find significant changes in the cross sections compared to Ref. [8], which was based
371
only on one experiment [6] and re-evaluations of that experiment. The measurement was performed with the previous capture setup at ORELA that had underestimated neutron sensitivity. For the astrophysical interpretation, only the isotopic ratios of 29Si/28Si and 29 Si/28Si are of interest. Since the cross-sections for 28Si and 29Si have changed by the same percentage the AGB stellar model is calculating no change in the isotopic abundance ratio. The 30Si abundance is dominated in the stellar model by the 33 S(n,a)30Si reaction, which has a cross section orders of magnitude higher than the neutron capture cross section. Therefore, the new much lower neutron capture cross-section for the 30Si isotope has no impact. The same holds for the Al cross section, which is much lower than the previous measurements. It is most likely not as good a candidate for a neutron poison. 4
Acknowledgements
This research is sponsored by the Office of Environmental Management, U.S. Department of Energy, under contract No. DE-AC05-00OR22725 with UTBattelle, LLC. References 1. M. Lugaro, E. Zinner, R. Gallino, and S. Amari, ApJ, 527, 369 (1999). 2. R. L. Macklin and B. J. Allen, Nucl. Instrum. Methods, 91, 565 (1971). 3. R. L. Macklin, J. Halperin, and R. R. Winters, Nucl. Instrum. Methods, 164, 213 (1979). 4. N.M. Larson, Oak Ridge National Laboratory technical report No. ORNL/TM9179/R4,1998 5. D.M. Hetrick, D.C. Larson, N.M. Larson, L.C. Leal, and S.J. Epperson, Oak Ridge National Laboratory technical report No. ORNL/TM-11825,1997 6. J.W. Boldeman, B.J. Allen, A.R. de L. Musgrove, and R.L. Macklin, Nucl. Phys. A252 (1975) 62. 7. B.J. Allen et al., "Neutron Capture Mechanism in Light and Closed Shell Nuclide" in Nuclear Cross Sections and Technology, edited by R.A. Schrack and C D . Bowman, Washington DC, 1975, pp 360-362. 8. Bao et al., Atomic Data Nuc. Data, 75 (2000) 1. 9. H. Beer, F. Voss, and R.R. Winters, Astropys. J. Suppl. 80, 403 (1992). 10. K. Wisshak, F. Kappeler, and G. Reffo, Nucl. Sci. and Eng. 88, 594 (1984).
2 0 8
T H E STELLAR N E U T R O N C A P T U R E RATE OF
PB
H. B E E R Forschungszentrum
Karlsruhe, Institut fur Kernphysik, D-76021 Karlsruhe, Germany E-mail: [email protected]
P. O. Box
3640,
Institut, Universitat Tubingen, Auf der Morgenstelle D-72076 Tubingen, Germany
14,
W. R O C H O W Physikalisches
P. M U T T I , F . C O R V I CEC, JRC,
Institute
for Reference B-2440
Materials and Measurements, Geel, Belgium
Retieseweg,
K.-L. K R A T Z , B. P F E I F F E R Institut
fur Kernchemie, Universitat Mainz, Fritz-StrassmannD-55099 Mainz, Germany
Weg 2,
The Maxwellian average neutron capture (MAC) cross section of 2 0 8 P b has been reinvestigated. With the activation technique the MAC cross section was measured at a thermonuclear energy of feX=52 keV. The termination of the s-process has been analyzed. Independent calculations of s - and r-process yield a consistent decomposition for the the P b and Bi isotopic s- and r-contributions.
The neutron capture cross section of the nucleus 2 0 8 Pb is in the keV energy region the smallest capture cross section of the heavy stable isotopes (A>56). Theoretical calculations suggest that direct capture (p-wave capture) 1 is of the same order of magnitude than resonance capture. 2 0 8 Pb is an extreme bottleneck in the s-process termination at 209 Bi and plays an important role in the decomposition of the isotopic solar abundances of Pb and Bi into the s—, r—, and radiogenic contributions from U and Th 2 ' 3 . Resonance capture of 207 > 208 pb and of 209 Bi have been measured by timeof-flight at the electron linear accelerator GELINA using 4 C 6 D 6 liquid scintillation detectors 2 ' 4 . Direct and resonance capture of 2 0 8 Pb can be investigated by the activation technique. An activation measurement at A;T=25keV has been carried out previously at the Karlsruhe 3.75 MV Van de Graaff accelerator 5 . In the present work a new activation measurement was performed at the Tubingen Van de Graaff accelerator to determine the MAC cross section at JfcT=52keV. The neutron spectra were generated using proton energies close above the T(p,n) reaction threshold at EPilab=1091 keV. The 209 Pb(3.253h) (3 372
373 T
•
1
•
1
•
1
•
1
•
1
'
1
•
r
Figure 1. The MAG cross section of 2 0 8 P b . The data reported by Beer et al. 2 (resonance, direct and total capture, i.e., the sum of resonance + direct capture) are shown as solid lines. Estimated uncertainties are indicated by shaded areas for resonance capture (lower curve) and total capture (upper curve). The direct capture contribution theoretically calculated by Rauscher et al. 1 is plotted without estimated uncertainties. The open square symbol represents an activation measurement at feT=25keV 5 . The present measurement at 52keV is indicated by a solid square. In these measurements at 25 and 52 keV, respectively, total capture has been determined directly.
activity produced in the irradiations of samples with natural composition was counted with a 4TT/3 Si (Li) spectrometer. In Fig. 1 the results are summarized. Previous data shown are, the value at fcT=25 keV reported by Ratzel 5 and the resonance capture, the direct and the total capture (i.e., the sum of direct + resonance capture) from Beer et al. 2 . Our MAC cross section at JfcT=52keV was determined from six individual activation runs. The runs were carried out relative to the well-known 197 Au standard cross section. The uncertainty is dominated by the statistical error in counting the 2 0 9 Pb activity. The present MAC cross section at A;T=52keV is lower than the value predicted from the combined data (experimental resonace + theoretical direct capture) of Beer et al. 2 , but still in agreement with the quoted uncertainties. With the new compilation of Bao et al. 6 a new analysis of the solar abundances was carried out using the double pulse s-process model 2 (figure2).
374
D s-dominant isotopes • s-only isotopes
I 1
10 faW'oodtoj! o 10
u °
™ 10 1
io- 3
I
I I I I I I' I I I • r-residual isotopes • r-only isotopes
sJ V ^ V B * !a'** Vd
|
I I;
4
j*>
5f 80
100
120
140
160
180
200
MASS NUMBER Figure 2. (Top) A new s-process analysis on the basis of a new compilation of Bao et al. 6 and (Bottom) calculated r-process abundances N r = N e - N s . The peak at A=206-208 in the N r distribution is due to accumulated r-contributions from transbismuth nuclei.
An essential improvement of the s-process calculations compared to previous results 2 was obtained especially for the s-only 142 Nd because of an improved 142 Nd capture cross section. However, the N r distribution which is obtained by subtracting the s-process abundances N s from the solar abundances N© remained practically unchanged. At the Pb and Bi isotopes the r-process residuals comprise also the accumulated N r contributions from short lived transbithmuth nuclei. This leads to a maximum in the empirical r-process distribution (figure2). The new decomposition of the Pb and Bi isotopes is shown in detail in figure 3. Our s-process results and the results of the r process calculations from Cowan et al. 3 add to a consistent analysis of the solar Pb and Bi isotopic abundances. This analysis is, therefore, also a check of the r-process calculations from Bi to the termination of the r-process. In conclusion, the new MAC cross section result confirms the very small 2 0 8 Pb capture cross section which makes 2 0 8 Pb an extreme bottleneck so that the sprocess is already terminated to a large extent at 2 0 8 Pb with the consequence that the last stable isotope 209 Bi within s-process reach has to be mainly an
375 1
i
10° r ^ Nr : — o — Nrt 5 • + N s+rt W O
<
Q
Z
10 1
i
i
i
/
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"a
>
204
'
205
i
i
i
i
206
207
208
209
210
MASS NUMBER Figure 3. Consistent picture of the sotopic P b and Bi abundances. Solar P b is renormalized at 2 0 4 P b . Note the large s-process contributions of 206,207,208 p ^ D e c a u s e 0 f the small MAC cross sections. For 2 0 9 Bi N s is small in spite of a small MAC cross section. This is caused by the bottleneck function of 2 0 8 P b . Calculated direct ( N r ) and total (NTt) r-process contributions are from Cowan et al.3.
r-process nucleus (figure 3). Support was provided by the Volkswagen-Stiftung (Az: 1/72286). References 1. T. Rauscher, R. Bieber, H. Oberhummer, K.-L. Kratz, J. Dobaczewski, P. Moller, and M. M. Sharma, Phys. Rev. C 57, 2031 (1998). 2. H. Beer, F. Corvi, and P. Mutti, Ap. J. 474, 843 (1997). 3. J.J. Cowan, B. Pfeiffer, K.-L- Kratz, F.-K. Thielemann, C. Sneden, S. Buries, D. Tyler, and T.C. Beers, Ap. J. 521, 194 (1999). 4. P. Mutti, F. Corvi, K. Athanassopulos, H. Beer, and P. Krupchitsky in Nucl. Data for Sci. and Technology, eds. G. Reffo, A. Ventura, C. Grandi (Italian Physical Society, Bologna, 1997) p. 1584. 5. U. Ratzel, Diplomarbeit 1988, unpublished. 6. Z.Y. Bao, H. Beer, F. Kappeler, F. Voss, K. Wisshak, and T. Rauscher, Atomic Data Nucl. Data Tables, in press
T H E R-PROCESS AS T H E T H E M I R R O R IMAGE OF T H E S-PROCESS: H O W DOES IT W O R K ?
Dipartimento
Fisica
Osservatorio
Forschungszentrum
Lunatic
Asylum,
R. G A L L I N O Generate, Universitd di Torino, 1-1015 Torino, Italy E-mail: [email protected]
Via P. Giuria
M. B U S S O Astronomico di Torino, Via Osservatorio 1-10025 Pino Torinese, Italy E-mail: [email protected] F. K A P P E L E R Institut fur Kernphysik, Karlsruhe, Germany E-mail: [email protected]
Karlsruhe,
Postfach
1,
28,
3640,
G. J. W A S S E R B U R G Division of Geological and Planetary Sciences, California of Technology, Pasadena, California 91125, USA E-mail: isotopes@gps. caltech. edu
D-76021
Institute
We present a brief summary of the researches that in recent years led to a crisis in the traditional ideas on neutron capture processes for the formation of the heavy elements beyond Fe. A strong revision is underway, both for the rapid neutron captures occurring in supernovae and for the slow neutron captures occurring in red giants. After a brief discussion of the astrophysical implications of the detection of short-lived r-process isotopes 1 2 9 I and 1 8 2 Hf in early solar system material, which require diverse supernova sources for the two isotopes, we concentrate on the researches on low mass stars in their final evolutionary stages, which finally led to a profound modification in our knowledge of the astrophysical mechanisms that guarantee the production of heavy s-elements.
1
Introduction
Two major neutron capture mechanisms need to be invoked for the build-up of almost all heavy nuclei starting from Fe seeds, the r-process and the s-process (Burbidge et al. 2 ). During the r-process (r is for "rapid neutron capture"), very intense neutron fluxes of short duration have to be produced, possibly during a supernova explosion, with an extremely high peak neutron density (n n > 10 20 n/cm 3 ). Under these conditions, all stable nuclei are shifted far away from the valley of /3-stability, preferentially feeding the highly unstable neutron-rich isotopes with magic neutron numbers. Immediately after the 376
377
neutron irradiation ceases, all nuclei decay back towards the stability valley, in particular feeding the r-abundance peaks at Te-I-Xe, at Os-Ir-Pt-Au, and all the actinides. Until recently, the idea was that of a "unique" r-process, i.e. that any star suffering such an explosive phenomenon would produce roughly the same distribution of r-process isotopes as in the solar system. The existence in early solar system condensates of a number of short-lived radioactive nuclei led to the apparent impossibility of reconciling in a unique scenario the production of the r-process isotopes 129 I (Jeffrey & Reynolds 1 0 ), 182 Hf (Lee & Halliday 15 ) and of the actinides (Chen & Wasserburg 5 ), with the consequence that the r-process is at least bimodal (Wasserburg et al. 2 2 ) . At least two different types of supernovae must contribute to the r-process: the first, tentatively identified as type II supernovae, produce all the r-nuclei above A ~ 140, together with 182Hf and the actinides, at a typical rate of 1 event every 10 Myr in the solar neighborhood. The second type, occurring at a lower rate of 1 event every 100 Myr, produces 1 2 9 I, and probably r-process nuclei lighter than A = 140. This suggestion is receiving continuous confirmations on independent grounds, based on observations of very low metallicity stars that appear to be enhanced in the heavy r-process nuclei, ascribed by Wasserburg et at 22 to the shorter living supernovae (see e.g. Sneden et al. 18>19). The general result is that the traditional picture of a unique r-process is now put in question. One has to say that, from a theoretical point of view, the astrophysical origin of the r-process nuclei is still a debated problem. Often, the r-process isotopic fractions are estimated as r residuals, after subtracting the s-fractions from solar abundances. This tool however assumes that a unique mechanism provided the s-process distribution in the solar system, at least in the atomic mass range 88 < A < 208 (see below).
2
The s-process: general background
The s-process (s is for "slow neutron capture") requires a relatively low neutron density, nn < 108 n/cm 3 , so that any unstable nucleus built by neutron capture would preferentially decay before capturing another neutron. Consequently, the s-process path moves along the valley of /3-stability, in particular feeding the abundance peaks at Sr-Y-Zr, at Ba-La-Ce-Pr-Nd, and at Pb. The classical analysis of the s-process is based on simple rules that are related to knowledge of the neutron capture cross sections and of the /?-decay lives of unstable isotopes. Of the highest importance is the consideration of the existence of neutron magic nuclei N = 50, 82, 126, with low neutron capture cross sections. This means that these nuclei act as bottlenecks during the neutron irradiation, so that their production is largely favored with respect to that of
378
the non neutron-magic nuclei in between, where, in turn, a local equilibrium is established:
fN56
p(T) =
x e
r
°
To
where / Nse is the total number of iron seeds and To is called mean neutron exposure. For a modern account of the phenomenological approach see Kappeler et al.11, Kappeler 14 , Arlandini et al.1, Busso et al.3. 3
Recent AGB models: crisis of the classical s-process approach
The main component is ascribed to the s-processing occurring in Asymptotic Giant Branch (AGB) stars of low mass (M = 1.5 to 4 M©). In AGB stars, H and He burn alternatively in two thin shells, while the star suffers from strong mass loss. Eventually, the remaining degenerate C-O core will become a white dwarf. Partial mixing mechanisms are expected to operate at the H/He discontinuity left behind by the penetration of the convective envelope into the top layers of the He intershell, favoring the formation of proton-rich layers (Hollowell & Iben 9 , Herwig et al.8; Singh et al.17). As a matter of fact, in most recent calculations the extension of the proton-enriched layers and/or the ensuing 13 C concentration in them have been assumed as free parameters. New stellar model calculations for the AGB phases by Straniero et al.20,21 showed that, whichever the abundance of 13 C produced in the intershell is, it burns locally in the radiative layers of the He intershell, before a new convective pulse develops. The thermal conditions are rather different from those established inside the pulses, in particular the temperature is lower (0.8 — 0.9
379
T
1
1
1
1
1
1
1
1
1
1
1
1
i
1
r
[Fe/H]
Figure 1: The enhancement factors relative to solar abundances of some representative elements in the s-processed material cumulatively mixed to the surface of 2 MQ AGB models, for the standard choice of the 13 C-pocket used by Gallino et al.7, as function of the initial metallicity.
x 10 8 K ) , a n d t h e average neutron density does not exceed 1 x 10 7 n / c m 3 for AGB stars of solar metallicity (Gallino et al.7). One of the main consequences of the new models, which produce s-elements directly in the thin 1 3 C pocket at very high efficiency, is t h a t the distribution of neutron exposures is now built in a very complex way, mostly in radiative (from 1 3 C ) , and partly in convective (from 2 2 Ne) conditions. A verification of the above results soon came from the strong improvements in cross section measurements obtained in recent years. At t h e high precision level of up-to-date measurements (a few percent in uncertainty) the classical t r e a t m e n t of branchings has failed to reproduce the solar aNs curve in the regions of tin, barium and neodymium (Kappeler 1 3 ' 1 4 ) . T h e analysis by Arlandini et al.l shows in particular t h a t the new Nd cross sections are incompatible with the results of the phenomenological approach, but well inside stellar model predictions. Other major problems with the classical analysis are related t o important overproductions predicted for several isotopes, for
380 example of
86
Kr,
87
R b , and i34,i36,i38 Ba
A major consequence of considering a "primary" 1 3 C neutron source (i.e. deriving from nuclear reactions starting on H and He, the concentration of 13 C nuclei in the pocket deriving from proton captures on the newly synthesized 12 C) is the fact t h a t the ensuing s-process distribution is extremely dependent on t h e initial abundance of Fe group seeds, i.e. on stellar metallicity. Indeed, the neutron exposure r is roughly proportional to the number of available 13 C nuclei per Fe seed. Then, starting from an A G B star of initial solar composition, the neutron exposure increases with decreasing metallicity. This has a dramatic effect on t h e s-process abundance distribution, as shown in Fig. 1 as a function of [Fe/H] for some representative elements. Here [Fe/H] = log(Fe/H) — log(Fe/H)0 is the usual notation in stellar spectroscopy for the iron content (metallicity). W i t h the ST choice of the 1 3 C-pocket used by Gallino et al. 7 , at solar metallicity essentially s-elements belonging t o t h e Sr-Y-Zr peak at neutron magic TV = 50 are produced; decreasing Fe one leaves more neutrons per Fe seed, thus bypassing the bottleneck at TV = 50 and progressively feeding elements at the second neutron magic peak at Ba-La-Ce-Pr-Nd, with a maximum production yield at [Fe/H] = —0.6; for even lower metallicity there are enough neutrons per Fe seed to feed 2 0 8 P b at the magic neutron number TV = 126. P b production peaks at [Fe/H] = —1.3. Eventually, for Galactic Halo AGB stars with even lower metallicities, all t h e initial Fe is efficiently converted directly to P b , but the shortage of iron seeds becomes dominant, so t h a t even the production of P b decreases with metallicity. In summary, t h e s-process in A G B stars of different metallicity driven by the primary 1 3 C neutron source gives rise to a wide spectrum of different abundance distributions. Quite similar trends versus metallicity are obtained with other choices of the 1 3 C pocket. On the whole, the solar system distribution of s-elements is clearly not a 'unique' process, and results from the integrated chemical evolution of the Galaxy, which mixes in the interstellar medium the o u t p u t s of many different stars, with yields changing with initial metallicity, stellar mass and other physical properties. Although the classical analysis may no more be considered a perfectly predictive tool, it remains however of useful help t o infer detailed properties of branchings on the s-path, the general distribution of abundances, and t h e r residuals in the solar system. A c k n o w l e d g m e n t s We t h a n k C. Arlandini, A. Chieffi, M. Limongi, M. Lugaro, O. Straniero, C. Travaglio, S. Cristallo, A. Dalmazzo, for continous help in a long-standing effort. Work supported in part by t h e Italian M U R S T project Cofm98.
381
References 1. Arlandini, C , Kappeler, F., Wisshak, K., Gallino, R., Lugaro, M., Busso, M. and Straniero, O., Astrophys. J. 525, 886 (1999). 2. Burbidge, E.M., Burbidge, G.R., Fowler, W.A. and Hoyle, F., 1957. Rev. Mod. Phys. 29, 547 (1957). 3. Busso, M., Gallino, R., and Wasserburg, G.J., Ann. Rev. Astron. Astrophys. 37, 239 (1999). 4. Cameron, A.G.W, Thielemann, F.-K. and Cowan, J.J., Phys. Rep. 227', 283 (1993). 5. Chen, J.H. and Wasserburg, G.J., Analyt. Chem. 53, 2060 (1981). 6. Clayton, D.D. and Rassbach, M.E., Astrophys. J. 148, 69 (1967). 7. Gallino, R., Arlandini, C , Busso, M., Lugaro, M. and Travaglio, C , Astrophys. J. 497, 388 (1998). 8. Herwig, F., Blocker, T. and Schonberner, D., Astron. Astrophys. 324, L81 (1997). 9. Hollowell, D., and Iben, I. Jr, Astrophys. J. 333, L25 (1988). 10. Jeffrey, P.M. and Reynolds. J.H., J. Geophys. Res. 66, 3582 (1961). 11. Kappeler, F., Beer, H. and Wisshak, K., 1989. Rep. Progr. Phys. 52, 945 (1989). 12. Kappeler, F., Gallino, R., Busso, M., Picchio, G. and Raiteri, CM., Astrophys. J. 354, 630 (1990). 13. Kappeler, F., Nucl. Phys. A 621, 221c (1997). 14. Kappeler, F. in Nuclei in the Cosmos V, ed. N. Prantzos and S. Harissopoulos (Editions Frontieres, Gyf-sur-Yvette 1999), p. 174. 15. Lee, D.-C. and Halliday, A.N., Nature 378, 771 (1995). 16. Seeger, P.A., Fowler, W.A. and Clayton, D.D., Astrophys. J. Suppl. 11, 121 (1965). 17. Singh, H.P., Roxburgh, I.W. and Chan, K.L., Astron. Astrophys. 340, 178 (1998). 18. Sneden, C , McWilliam, A., Preston, G.W., Cowan, J.J., Burris, D.L. and Armoski, B.J., Astrophys. J. 467, 819 (1996). 19. Sneden, C , Cowan, J.J, Burris, D.L. and Truran, J.W., Astrophys. J. 496, 235 (1998). 20. Straniero, O., Astrophys. J. 440, L85 (1995). 21. Straniero, O., ChiefS, A., Limongi, M., Busso, M., Gallino, R. and Arlandini, C , Astrophys. J. 478, 332 (1997). 22. Wasserburg, G.J., Busso, M. and Gallino, R., Astrophys. J. 466, L109 (1996).
T H E N E U T R O N C A P T U R E CROSS SECTION OF STELLAR E N E R G I E S
147
P m AT
C. ARLANDINI, M. HEIL, R. REIFARTH, F. KAPPELER Forschungszentrum Karlsruhe, Institut fur Kernphysik, Postfach 3640, D-76021 Karlsruhe, Germany E-mail: reifarthQikS.fzk.de P.V. SEDYSHEV Frank Laboratory of Neutron Physics, JINR, 14980 Dubna, Moscow Region, Russia E-mail: [email protected] The stellar (11,7) cross section of the unstable isotope 1 4 7 Pm (t 1 / 2 =2.62 yr) has been measured at a thermal energy of kT=25 keV. The experiment was carried out at the Karlsruhe 3.7 MV Van de Graaff accelerator by means of the activation method. In spite of a comparably small sample mass of only 28 ng and considerable backgrounds from various impurities, both partial cross sections feeding the 5.37 d ground state and the 41.3 d isomer in 148 Pm could be derived with an overall uncertainty of 10%.
1
Introduction
The three s-process branchings in the Nd-Pm-Sm region (Fig. 1) are well defined by the relative abundances of two s-only isotopes 148 Sm and 150 Sm. While 148 Sm is partially bypassed by the reaction flow, 150 Sm is not affected by the branchings and can be used for normalizing the resulting abundance pattern. The observed 1 4 8 Sm/ 1 5 0 Sm ratio is completely determined by the stellar neutron density since the /3-decay rates of all three branching points are practically independent of temperature and electron density*. The remaining nuclear physics uncertainties are due to the cross sections of the unstable branch point isotopes, the existing calculated values having uncertainties of 20 to 50%. While measurements on the short-lived branch points 147 Nd and 1 4 8 Pm are not feasible at present, the activation technique has been used for determining the (n, 7) cross section of 1 4 7 Pm. 2
Measurements and Results
The applied activation method 2 ' 3 is based on the irradiation of a sample in a quasi-stellar neutron spectrum which can be obtained by bombarding a thick metallic Li target with protons of 1912 keV, slightly above the reaction threshold at 1881 keV. The 7 Li(p, n) 7 Be reaction then yields a continuous energy distribution with a high-energy cutoff at i? n =106 keV. The resulting neutron 382
383 p process
147
Sm
149
'«Sm
Sm
k k k lw
Pm
1M
Sm
'«Pm
"»Pm
147
" 8 Nd
\ 146
Nd
Ndi—
k v._._
160 N ( j
r procesB
Figure 1: The s-process branchings between Nd and Sm are characterized by the relative abundances of 1 4 8 S m and 1 5 0 Sm. These two s-only isotopes are shielded against r-process /3-decays by their Nd isobars.
spectrum closely resembles a Maxwellian distribution for &T=25 keV, thus exhibiting almost exactly the shape required to determine directly the stellar cross section. The sample was produced by adding minute amount of 1 4 7 Pm to enriched 147 Sm (98.27%) which served as a carrier. The corresponding solution was converted to oxide, mixed with 10 mg of fine graphite powder and pressed into a pellet of 6 mm diameter and 0.4 mm thickness. The 1 4 7 Pm mass was determined by comparing the intensity of the 121 keV 7-activity with a calibrated 57 Co source. The resulting activity of 982 ± 41 kBq corresponds to (1.17±0.05)xl0 14 atoms or 28.7±1.2 ng. During irradiation, the sample was sandwiched between two thin gold foils with 6 mm diameter and placed on top of the neutron producing target 3 . Two activations were carried over periods of 60 hours and 12 days. After activation, the integral neutron flux was determined via the induced 198 Au activity 2 ' 3 . The much weaker 1 4 8 Pm activity was measured with an arrangement of two HPGe Clover detectors facing each other. The sample was centered in the 3 mm gap between the detectors. The 7-ray efficiency of the system was determined in the range from 50 to 1200 keV using a set of calibrated sources complemented with a Monte Carlo analysis made with the GEANT code. Typical uncertainties of the measured efficiencies are 2%. Even with the high detection efficiencies achievable in add-back mode, single lines from the induced 1 4 8 Pm activities could not be identified because of interfering backgrounds from 1 4 6 Pm and Nd impurities in the sample and from
384
natural radioactivity. This background could only be reduced by looking at the 7-ray cascades in the 1 4 8 Pm decay. Thanks to the granularity and high efficiency of the Clover detectors, the respective coincident events were identified off-line with sufficient statistics. At the example of the 914.9/550.3-keVcascade from the 1 4 8 g Pm decay Fig. 2 illustrates the unambiguous background separation provided by this technique. i48gpm decay 4
I—
™
± 1 I ID
-i
LU -3 -4
"5-5
-4
- 3 - 2 - 1 0 1 2 3 4 5 ENERGY SHIFT CkeV)
Figure 2: Coincident events due to 915/550-keV-cascades from the decay of 1 4 8 s p m are clearly identified in the center, well separated from Compton scattered background due to the 1461 keV mK line, which appears as a band in the lower left comer.
Preliminary values for the partial (n/y) cross sections to ground state and isomer in 1 4 8 Pm add to a stellar 1 4 7 Pm cross section = 685 keV with an uncertainty of ~10%. This result is about two times smaller than theoretical predictions 4 , thus indicating that systematic trends near mafpc neutron number N=82 were not adequately considered in the calculations. Therefore, the present result is important not only for direct use in the branching analysis but also to test and to adjust the parameter systematica for statistical model calculations, in this particular region. 1. K. Takahashi and K. Yokoi, Atomic Data Nucl. Data Tables 36, 375 (1987). 2. H. Beer and F. Kappeler, Phys. Rev. C 21, 534 (1980). 3. K. Toukan et aL, Phys. Rev. C 51, 1540 (1995). 4. Z. Bao et «!., Atomic Data Nucl. Data Tables (2000), in print.
Applications of Nuclear Physics
Section I. Fission, Spallation and Transmutation
THE ENEA ADS PROJECT: Accelerator Driven System Prototype R&D and Industrial Program GIUSEPPE GHERARDI ADS Project Leader, ENEA, Bologna ITALY E-mail: Gherardi @ bologna.enea.it Hybrid reactors (Accelerator Driven Sub-critical Systems, ADS), coupling an accelerator with a target and a sub-critical reactor, could simultaneously burn minor actinides and transmute long-lived fission products, while producing a consistent amount of electrical energy. A group of Italian research and development (R&D) organizations and industries have set up a team, which is studying the design issues related to the construction of an 80 MWth Experimental Facility. The planned activities and the (tentative) time schedule of the Italian program are presented.
1
Introduction
Starting from 1995 [1], a growing interest on the Accelerator Driven Systems concepts has taken place in Italy and has given origin to several basic R&D activities and to an industrial program involving ENEA (the Italian National Agency for New Technologies, Energy and the Environment), INFN (the Italian National Institute for Nuclear Physics) and various industrial partners. This interest is confirmed by the national program which was sponsored by MURST (the Ministry of the University, Scientific and Technological Research) in 1998, under the leadership of INFN for the accelerator and of ENEA for the subcritical system. This program is considered of particular relevance for the creation of a well mixed group of competencies and it will provide results of relevant importance in support of any related industrial program. In parallel to the basic activities above, the aforementioned industrial program is being proposed in two main steps (the first has been approved and funded): a) on-going short term activities in the Italian context to issue preliminary design of the ADS demonstration prototype; a reference configuration has been proposed and submitted to the European partners [2] as a contribution to the discussion (Road Map for developing an ADS demonstration facility) to converge upon the prototype objectives and upon a configuration on which the detailed engineering design will be based; main supporting R&D needs will also be assessed; b) medium term activities in an European context with the aim to perform the detailed engineering design, the realization phase and the commissioning of the demonstration prototype along with all the supporting parallel R&D deemed necessary.
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388
The present memo focalizes the attention on the on-going short term activities aiming at the preparation of the proposed preliminary design, leaving the deal to define the details of the subsequent medium term activities to the expected common program in the European context. The funding sources for the industrial program and for the supporting R&D needs are assumed to be provided both at the European participating nations government level (mainly for the industrial activities) and at the EU level (mainly for R&D activities).
2
Background and Considerations
The industrial program presently ongoing in Italy aiming at the design, realization and commissioning of a demonstration prototype of an Accelerator Driven Subcritical System for nuclear waste transmutation is based on the following considerations and background: (a) me program objectives are the design, realization, commissioning and operation in the medium term (10-12 years) of a "low power" demonstration prototype (order of 100 Mwth); (b) the prototype is considered the key (and the only) choice to assess the demonstration of the ADS process and its industrial feasibility; it specifically has the objective to demonstrate in subsequent steps: • the accelerator and sub-critical system coupling; Q innovative fuels qualification and operation; Q Plutonium, Minor Actinides and Long Lived Fission Products transmutation efficiency; (c) supporting parallel R&D activities would be originated from the prototype design needs; this is seen as the proper and best efficient way to match design and experimental activities with funding to be made available; (d) starting from the conceptual configurations developed in the last years there is the need to finalize a demonstration prototype design by means of a sound engineered approach; (e) the technical options and alternatives must converge in the definition of an agreed upon reference configuration to be taken as the basis for the design development; (f) the design of the demonstration prototype must be based as much as possible on proven technological and engineering solutions making reference to the large experience available in nuclear power plants and accelerators designers community; (g) the design, realization and commissioning of the demonstration prototype must be developed in an European context making also reference, where needed, to similar on-going projects in the USA, in Japan, in South Korea and in China; (h) a strong industrial interest exists for the ADS technology in general and for the realization of a prototype in particular; such interest has been witnessed in the
389 common signature by Ansaldo, Framatome, NNC and Siemens of the document "An European Nuclear Industry Interest for the Accelerator Driven Systems Technology to Assess an European Reference System Configuration", which has been transmitted to the European Union; (i) basic activities on ADS technology and key features are, or are being, launched in several European countries involving Research Bodies, Universities and Industries; (j) long lasting cooperation experience and formal links already exist between Italian R&D bodies and nuclear industry and the equivalent European countries organizations (in particular with France); this makes easier, at least in principle, the definition of agreements for a common program sponsored by related governments as well as the European Union; (k) the Fifth European Community R&D Program, started in 1999, is a source of funding for ADS specific R&D activities within the specific program "Nuclear Fission Safety"; (1) to reach the objective of the realization and commissioning in a decade, the European program for detailed design development should be launched within the next EU Framework Program (2002-2006).
3
The R&D and the Industrial Program
The purpose of the short term activities is to make the main technical key choices leading to the definition of an engineered reference configuration of the demonstration prototype. The result of this preliminary engineering effort will be documented with the issue of the description of the prototype plant that will be made available for the evaluation of the European partners. Starting from the conceptual configurations presently available in literature and more specifically from the Energy Amplifier Project conceived and developed at CERN by Prof. Carlo Rubbia and his staff, the purpose of the on-going work is to define/confirm the main technical features applicable to an ADS demonstration prototype, hi particular the main technical issues which are being investigated are: Q plant safety and functional requirements (top tiers), Q accelerator configuration (including proton beam transport line to the subcritical reactor), • target/window configuration, Q fuel element type, composition and thermal hydraulic evaluation, • core and supporting structures configuration, • core reference cycle, a primary coolant fluid and circulation, Q structural materials, • plant thermal cycle and heat removal system (normal and emergency) configuration,
390
Q P a Q • a •
fuel handling system and storage configuration, containment system configuration, main components preliminary design, primary system auxiliaries configuration, instrumentation and control architecture, plant building preliminary layout (plot plan and general arrangements), civil structures preliminary design. The design activities for the definition of the ADS Reference Configuration are split among accelerator, core/fuel and sub-critical system. An adequate integration and coordination between working teams assures interface control and overall design consistency.
4
Experimental Plant Main Technical Key Features
The main key technical features presently taken as reference for the configuration of experimental plant are the following [3]: Q subcritical reactor power : 80 Mwth; • reactor pool type; Q design life : 20 equivalent full power years; P proton accelerator : two stages cyclotron scheme based on the PSI configuration [one source/injector LINAC (3-6 MeV, 2-6 mA), one intermediate separate sectors cyclotron (100 MeV, 2-6 mA), one booster separate sectors cyclotron (600 MeV, 2-6 mA]; another promising solution for the experimental plant or for the long term is the super-conductive LINAC studied by INFN; • target : alternative hot window and windowless solutions, spallation medium separated from primary coolant fluid; Q spallation fluid: liquid Lead-Bismuth eutectic (45% Pb, 55% Bi); Q primary coolant: liquid Lead-Bismuth eutectic (45% Pb, 55% Bi); for the longterm solution: liquid lead; Q fuel composition : U02+Pu02 (19%) [proven SPX fuel]; for the long-term solution: innovative fuels; Q average power density: 250 w/cm3 P fuel element type : hexagonal; P k e f f :0,95; P primary coolant circulation : natural enhanced by gas bubbles injection above core; Q normal heat removal : intermediate secondary system with release to environment through air cooled heat exchangers; P decay heat removal: direct release to environment through air cooling;
391
5
Partners Involvement
The ongoing short term activities of the industrial program are performed by the following organizations: 1) Ansaldo Nuclear Division, which also provides the program overall coordination, is the technical coordinator of the subcritical system design activities and is involved in all the main working tasks in order to assure integration and consistency; 2) ENEA, which is the technical coordinator of core and fuel design activities, provides also analytical support to target and sub-critical system design, and has in charge the experimental activities on the liquid metal systems and fuel reprocessing systems; 3) INFN, which is the technical coordinator of accelerator configuration and provides interface data at core/sub-critical system boundary; 4) Centro di Ricerca, Sviluppo e Studi Superiori in Sardegna (CRS4), which provides support to core/fuel and sub-critical system design activities; 5) Ansaldo Energia (Magnets Unit), which provides support to accelerator configuration.
6
Planning
An industrial and R&D program overall planning is under definition, to be presented by the Italian representatives in the Technical Working Group (TWG) subgroup for the Road Map. The proposed planning is consistent with the objective of prototype commissioning and operation within 12 years from present. The most significant milestones/activities are the following: a) completion of the preliminary design; [Italian context]; b) discussion and convergence upon a European Road Map for developing the ADS DEMO (fall 2000); c) preliminary engineering design (starting 2001); [including confirmation tests of some critical components and first interaction with Licensing Bodies]; d) choice for the accelerator architecture: "stand alone" or "multipurpose" machine; e) final detailed design (2004 - 2006) [including the safe operation demonstration of the spallation module: MEGAPIE experiment and MYRRHA Project]; f) the French government decision (30/12/91 law) on the closure of the fuel cycle (2006);
392
g) prototype construction (2006 - 2012); h) prototype commissioning (2010 - 2012) [the commissioning of the accelerator and of the spallation module will start well in advance to the fuel loading and plant completion]; i) start of prototype operations (2012); 7
Conclusions
The results obtained till now, though preliminary and not exhaustive, allow us to outline a consistent DF configuration. The experimental work performed by ENEA on the liquid metal technologies and experiments in physics and thermal hydraulics performed in many European institutes demonstrate that there is no technological obstacle to the realization of the experimental facility. References
1. Rubbia C. et al., Conceptual design of a fast neutron operated high power energy amplifier, report CERN/AT/95-44(ET) (1995) 2. Cinotti L. et al., Issues related to the design of an ADS cooled by the Pb-Bi eutectic, in : Heavy Liquid Metal Coolants in Nuclear Technology, (Obninsk, October 1998) 3. CRS4 web page on ADS: www.crs4.it/Areas/ea/
Heat Deposit Calculation in Spallation Unit
F . I . K a r m a n o v , A.A. Travleev Institute
of Nuclear Power Engineering, 249020 Obninsk, Russia E-mail: [email protected] L.N. Latysheva
Institute for Nuclear Research, 117312 Moscow, Russia E-mail: [email protected]
ENEA, E-mail:
M. Vecchi 40129 Bologna, Italy [email protected]
The present study concerns the calculation of the heat deposition in one of the EAP-80 basic units - the spallation module including the beam window, leadbismuth spallation target and primary liquid metal cooling system. It is assumed that the model of sub-critical reactor under investigation is based on ANSALDOINFN-ENEA-CRS4 reference configuration 1 . The calculation have been done by means of a couple computer codes: INCC 2 and GEANT3.21 3 . These codes have been preliminary tested on the experimental d a t a obtained in for the case of interaction of proton beam and lead-bismuth targets at the energy Ep= 800MeV which is close to energy range relevant for ADS configuration.
1
Benchmark calculation
In order to check the solution behaviour and to select some environmental parameters of the above mentioned codes a set of runs have been done for the variants of the targets discussed in Ref. 4 . T h e calculation have been done for the case of gaussian profile beam with the proton energy of Ep= 800 MeV. It was assumed t h a t targets are shaped of the cylinder with radii of R = 5cm or 10cm and the length of 60cm. Target materials which have been chosen for calculation are lead and bismuth. T h e results of our calculations show t h a t the discrepancy between calculational and experimental d a t a are in the range of 10% — 15%. Some additional INCC runs have been done to estimate the photon contribution to the heat released from the reaction TC° H->• 7 7 and the contribution of the photons arising from the de-excitation of the nuclei after inelastic interactions. As one can expect it is possible to improve the agreement of the I N C C results and the experimental d a t a taking into account the estimation of photon contribution to the total heat deposit. 393
394 Some noticeable difference in results produced by the INCC and GEANT can be seen in the range of Bragg pick only. Unfortunately, experimental data are absent in this region. So, one can hope that both of these codes are able to reproduce the heat deposit distribution with the accuracy required.
Figure 1: Distribution of heat deposition in the target window and lead-bismuth volume. Smooth curves on upper figures - analytical approximation. Solid and dashed histograms GEANT and INCC results respectively. Total heat release is equal to 4.29 and 7.76 k W / m A for the steel and tungsten window. Left bottom figure is the R, Z-distribution, where R is the radius of the cylindrical volume inside the spallation module, and right bottom figure is the R, ^-distribution, where R is the radius of spherical layer with centre located at the point Z = 0. The point Z = 0 in figures corresponds to the centre of the sphere and point Z = 10 cm is the beam entry point.
2
Heat deposit calculation in beam window
The lay-out of the ADS spallation module is presented in Ref.[l]. The geometry of the target window suggested for the Energy Amplifier Prototype is close to
395 the hemispherical layer with the wall thickness smoothly changing from 1.5 at the beam axis to 3.0 mm at the junction of the window with cylindrical stainless steel beam transport structure with outer radius of 10 cm. The material of the wall currently examined is either W-Re alloy, or stainless steel. The proton beam is defocused to be spread over the target window surface according to the parabolic distribution function to make thermal and radiation loads over the window surface more uniform. The beam current may vary from 2 to 6 mA. The proton energy Ep= 600MeV. It is evident that the profile of distribution of energy deposition over the thin target window will be determined by the spread of the defocused primary proton beam over the target window. The latter is described with the following distribution function:
'M =71-rg §(l-(f)r 2)> 0
where 1$ is the beam current, ro = 7.5 cm is the beam radius. So, the energy deposition in the unit volume can be now described with the relation: q(9)[kW/mA/cm3]
= (2/ 0 /jrrg)(l -
{r/r0)2){dE/d£)[MeV/cm},
where dEjdl is the electronic stopping power. Application of the above derived formula to calculate heat deposition gives the value of q0 = 0.15SkW/mA/cm3 for 600-MeV protons incident in the stainless steel target window near the beam axis (r close to zero, maximum proton flux) and (dE/dl) ~ UMeV/cm. Comparison of heat deposition in the target window calculated using the above approach (smooth curves) with those obtained using the GEANT code 3 (solid histograms) is given in Fig.l for two candidate structural materials tungsten and stainless steel. It is evident from the figure that the results obtained using the approximate approach are underestimated as compared to the Monte-Carlo simulation. The Q value obtained integrating over the histogram for stainless steel Q = 4.29kW/mA (GEANT), exceed the above cited value of Q = 2.86kW/mA by 50 % . The difference is explained by the contribution of ionization losses of energy by secondary particles and heat release due to nuclear interactions in the total heat release. Thus, the contribution of the photons to the total heat deposition can be seen from the comparison of the solid and dot histograms in Fig.l (both of them are the GEANT runs). 3
Heat deposit calculation in lead-bismuth volume
Three-dimensional distributions of heat deposition in the eutectic volume of the spallation module calculated using the GEANT code are also presented
396 in F i g . l . Such type of distributions is used for thermal-hydraulic analysis of the spallation module. Two peaks are observed in F i g . l . T h e first peak is produced by the particle b e a m incident in the target window (both primary and secondary particles) while the second one is the Bragg peak. According to our calculations the total energy released in the volume of spallation module is equal to 398kW/mA (GEANT) and 430kW/mA (INCC). An additional calculation has been done t o estimate the contribution of low energy neutron source (En < 20MeV) to the total heat deposition in the volume of spallation module taking into account the influence of the core also. The spatial-energy distribution of this neutron source have been taken from the results of C A S C A D E / I N P E r u n 6 . Our M C N P 5 calculations in the frame of homogeneous model of the core show t h a t the above mentioned contribution of low energy neutron source is of about 45kW/mA or of about 10% of the total energy deposition. T h e run analogous t o the previous one has been done for the case of the absence of any fissile elements in the volume of the core. It's result gives the value of energy deposition of about 30kW/rnA. Similar calculations aimed to evaluate the contribution of the photon source created as a results of C A S C A D E / I N P E run and including the photons arising from high energy (n, 7)—reactions give the value of heat deposition of about lOkW/mA. Therefore, one can conclude t h a t the influence of the core (keff ~ 0.92) 6 on the heat deposition in spallation module is not so essential and in this context the core and spallation module can be treated as the independent elements of facility. References 1. Energy Emplifier Demonstration Facility, Reference Configuration. ANSALDO, Divisione Nucleare, EA B0.00 1 199 - Rev.O, 22.12.98 2. A.S.Iljinov, et al., Intermediate Energy Nuclear Physics.CKS Press,1993 Ye.S.Golubevaet al., Nucl. P h y s . A 5 3 7 ( 1 9 9 2 ) , p.393; A 5 6 2 ( 1 9 9 3 ) , p.389 N.M.Sobolevsky, Preprint Bl-2-5458, Dubna, 1970. 3. "GEANT. Detector Description and Simulation Tool", C E R N P r o g r a m Library Long Writeup W5013, Geneva, 1994. 4. V.I.Belyakov-Bodin et al.,Calorimetric Measurements and Monte Carlo Analysis of Medium-Energy Protons Bombarding Lead and Bismuth Targets, N u c l . I n s t r . M e t h . , A 2 9 5 , p.l40(1990). 5. Ju.F.Briesmeister, MCNP - A General Monte Carlo N-Particle Transport Code. Version 4B. Los Alamos, March 1997. 6. A.Yu.Konobeev et al., Study of Accelerator-Driven Reactor Systems, Kerntechnik, 64,N5-6, p.458(1999);
NUCLIDE COMPOSITION OF PB-BI HEAT TRANSFER IRRADIATED IN 80 MW SUB-CRITICAL REACTOR A. YU. KONOBEYEV Institute of Nuclear Power Engineering (INPE), 249020 Obninsk, Russia E-mail: [email protected] M. VECCHI Enteper le Nuove Tecnologie, I'Energia e lAmbiente (ENEA), 40129 Bologna, Italy E-mail: [email protected] Nuclide composition of Pb-Bi coolant of 80 MW sub-critical reactor has been investigated. Particle spectra for different parts of the reactor including spallation (target) module, the core and other units have been calculated. Special data files for nucleon induced reaction crosssections including data for spallation reactions, fission porduct and heavy cluster yields at the energies of primary particles up to 1 GeV have been prepared. The special attention has been paid to the uncertainty of the results obtained, the influence of the choice of nuclear model parameters used for nuclear reaction cross-section calculations and to the comparison of the results obtained using different nuclear data libraries applied for nuclide composition calculations.The reactor configuration with the "window" for the proton beam and the "windowless" design were investigated.
1
Introduction
The concept of sub-critical accelerator driven-reactors is extensively examined in European countries nowadays. In comparison with coventional fission reactors such reactors are believed to possess special safety features, flexibility in operation, possibility of transmutation of long-lived radioactive wastes, which make the concept of such reactor attractive for nuclear power engineering in the future. Here the nuclide composition of such sub-critical reactor has been investigated. The principal layout of the reactor with the "window" is shown in Fig.l. The typical fuel composition is U-Pu MOX fuel [1]. The detail description of the reactor configuration with window is given in Ref.[l], "windowless" design is described in Ref.[2]. The characteristic dimensions of the reactor vessel, the core geometry and the spallation (target) module were taken from the Ref.[l]. Accelerator current is adopted equal to 3,02 mA. Primary proton beam energy is equal to 600 MeV. Source criticality factor It, was adopted equal to 0,966 to provide 80 MW power of the facility.
397
398
Figure 1. The layout of EAP-80 facility designed with the "window" for the proton beam with the designation of the different parts of the unit. "C", "P", "Bj", "Hi" mean core, plenum, "boxes" and pure coolant regions, correspondingly.
2
Method of nuclide composition calculation
The method of nuclide concentration calculation is described in details in Ref.[3]. The SNT code designed in INPE [4] was used for the calculation of nuclear reaction rates and the solving of the system of kinetic equations. Result of the test calculations and the comparison with recently obtained experimental data for the lead target irradiated by high energy protons is described in Ref. [3]. Computer code package CASCADE/INPE [5] including new version of high energy transport code CASCADE modified in INPE, MCNP/4B code [6] and other routines has been used for the particle spectra calculation for the sub-critical reactor. The cross-sections of nuclear reactions were taken from i) FENDL/A-2.0 data library for neutrons of the energies below 20 MeV, ii) MENDL-2 [7,8] library for protons and neutrons in the energy range from 20 to 100 MeV (this library was supplemented by fission product yields for Pb, Bi isotopes), iii) special file prepared for neutron and proton induced reactions including fission and heavy cluster (7Be, C etc.) emission at the energies 0.1-1 GeV, iv) file with tritium production crosssections for nucleon induced reactions at the energies up to 1,6 GeV [9]. 3
The results of calculations
The calculations were performed for the reactor designed with and without "window". "One loop" and "two loops" configurations were investigated. The time of the operation from 1 month up to 10 years has been considered. The results of the
399
calculation for the reactor with "window" are given in Ref.[3,10]. Some results for the "windowless" design are shown below. Example of the partice spectra calculation for the spallation (target) module for the reactor configurations designed with and without "window" are shown in Fig.2.
10"'
10''
10*
10*
10*
10*
10*
10''
10°
10'
10'
Energy (MeV)
Figure 2. Neutron and proton spectra calculated for spallation (target) module for the reactor with and without the "window". For neutrons 8000 energy group division for spectra is used.
Table 2 presents the amount of different nuclei produced in the target module (Pb-Bi inventory is 3.73 tons) and in the primary loop (Pb-Bi inventory is 2346 tons) for the reactor designed without "window" after one year of the operation.
Table 2. Mass of nuclides (grams) produced after 1 year irradiation of Pb-Bi coolant in target loop (spallation module) and primary loop for the reactor designed without the "window". Data corresponds to the end of irradiation. The nuclei are selected which contribution in the total activity after shut down is more than 1 per cent of the total induced activity.
Nuclide
84 210 84 209 84 208 83 210 83 210m 83 208 83 207 83 206 83 205 82 205 82 203 82 202 82 200 81 204
T1/2 (day)
1.384E+02 3.726E+04 1.058E+03 5.013E+00 1.096E+09 1.343E+08 1.387E+04 6.243E+00 1.531E+01 5.219E+09 2.169E+00 1.936E+07 8.958E-01 1.380E+03
Target loop
3.786E+01 5.435E-01 1.354E+00 1.644E+00 3.272E+01 6.576E+01 3.547E+01 4.935E-01 1.078E+00 6.997E+01 1.996E-01 1.891E+01 4.282E-02 3.024E+00
Primary loop
2.337E+03 1.485E-04 1.965E-05 1.016E+02 2.029E+03 9.872E+01 1.789E+01 2.215E-01 4.474E-01 1.762E+03 8.019E-02 6.247E+00 9.201E-03 1.141E+00
Target loop in relation to the total yield (%) 1.59 99.97 100.00 1.59 1.59 39.98 66.47 69.02 70.67 3.82 71.34 75.17 82.31 72.61
400
Table 2 continued 81201 81 200 80 197 80 194 79 195 78 193 78 191 78 188 77 192m2 77 189 77 188 73 179 41 94 38 90 1 3
3.046E+00 1.087E+00 2.671E+00 1.899E+05 1.861E+02 1.826E+04 2.900E+00 1.020E+01 8.797E+04 1.320E+01 1.729E+00 6.649E+02 7.415E+06 1.064E+04 4.500E+03
2.008E-01 6.233E-02 1.087E-01 6.971E+00 4.241E+00 6.315E+00 5.890E-02 1.314E-01 4.323E-01 2.005E-01 2.245E-02 5.241E-01 1.513E-01 1.181E-01 1.922E-01
5.091E-02 1.371E-02 1.062E-02 3.641E-01 2.782E-01 2.686E-01 1.618E-03 2.047E-03 5.248E-03 3.723E-03 3.519E-04 2.550E-03 3.071E-03 9.602E-03 1.955E-02
79.77 81.97 91.10 95.04 93.84 95.92 97.33 98.47 98.80 98.18 98.46 99.52 98.01 92.48 90.77
References 1. Energy Amplifier Demonstration Facility Reference Configuration, Ansaldo, EA B0.00 1 199 Rev.0, 22.12.1998, Rev.0, January 1999 2. ANSALDO, ADS Demonstration Facility, ADS 4 Tnix 0025, 27/01/2000, 1 EA 4 Tnlx0004, 15/9/99, ADS 5 Dmmx 0052 010, ADS 5 Dmmx 0104 030. 3. Konobeyev A.Yu. and Vecchi M., Nuclide composition of Pb-Bi heat transfer, Proc. Workshop on spallation module, ENEA, Bologna, 18 October 1999, ENEA GRX-TM-00001, pp.70-129 4. Korovin Yu.A, Konobeyev A.Yu. and Pereslavtsev P.E., The code for calculation of nuclide production and induced activity of irradiated materials, Voprosy Atomnoi Nauki i Techniki. Ser. Yadernije Konstanti, 3-4 (1992) pp.117-121. 5. Konobeyev A.Yu., Korovin Yu.A., Sosnin V.N and Vecchi M., Study of accelerator-driven reactor systems Kerntechnik, 64 (1999) pp.284-293. 6. Briesmeister Ju.F., Report LA-12625-M, March 1997. 7. Konobeyev A.Yu., Korovin, Yu.A., Lunev V.P., Masterov V.S. and Shubin Yu.N., Nuclear data library to study transmutation and activation of materials irradiated by intermediate energy neutrons, Voprosy Atomnoi Nauki i Techniki. Ser. Yadernije Konstanti, 3-4 (1992) pp.55-58. 8. Shubin Yu.N., Lunev V.P., Konobeyev A.Yu. and Dityuk A.I., Report IAEA, INDC(CCP)-385, May 1995. 9. Konobeyev A.Yu. and Korovin Yu.A., Tritium production in materials from C to Bi irradiated with nucleons of intermediate and high energies Nucl. Instr. Meth., B82 (1993) pp. 103-115. 10. Konobeyev A.Yu. and Vecchi M., Nuclide production in lead-bismuth coolant of EAP-80 and other problems, ENEA, July 2000.
RADIOLOGICAL ASPECTS OF HEAVY METAL LIQUID TARGETS FOR ACCELERATOR-DRIVEN SYSTEMS AS INTENSE NEUTRON SOURCES
E.V. GAI, A.V. IGNATYUK, V.P. LUNEV, YU.N. SHUBIN Institute of Physics and Power Engineering,
249020
Obninsk,
RUSSIA E-mail:
[email protected]
General problems arising in development of intense neutron sources as a part of accelerator-driven systems and first experience accumulated in IPPE during last several years are briefly discussed. The calculation and analysis of nuclear-physical properties of the targets, such as the accumulation of spallation reaction products, activity and heat release for various versions of heavy liquid metal targets were performed in IPPE. The sensitivity of the results of calculations to the various sets of nuclear data was considered. The main radiology characteristics of the lead-bismuth target, which is now under construction in the frame of ISTC Project # 559, are briefly described. The production of shortlived nuclides was estimated, the total activity and volatile nuclide accumulation, residual heat release, the energies of various decay modes were analysed.
1 Introduction Various properties of hybrid electronuclear systems, which can be used for scientific investigations or energy production, to utilize plutonium or to transmute long-lived radioactive waste, in a large degree depend on target characteristics: safety, lifetime and productivity as neutron source. First concepts of ADS supposed the use of accelerator having proton energy about 1-2 GeV and current 100-300 mA to have necessary intensity of neutrons. To provide the effective heat removal it was proposed that in the targets liquid heavy metal should be used. Reliable estimations of radiological properties of targets are necessary to design electronuclear facility, which secures all safety requirements. Below we analyze the status of the problem and estimate the target radiology characteristics for some examples of Pb, Pb-Bi and Hg targets. 2 The Analysis of Long-Lived Radioactivity The results of calculations of the total activity and activities of some most important nuclides for the lead target are shown in Figl. One can see that various isotopes of Pt, Au, Hg, Tl, Pb and Bi make more significant contribution to the long-lived (Ti/2>100 d) activity than 210Po isotope for both targets [1,2]. The nuclides making main contributions to the total activity of the targets after different cooling times were identified. The isotope 195Au (half-life Ti/2=186 d) provides main contribution in the cooling time range 10 days 1 year 401
402
Figure 1. Total and partial activities of lead target as a function of cooling time.
ACTIVITY OF TARGET
10 "* 10 "* 10 "° 10 "' 1
10 10 ' 10 * 10 * 10 " 10 * 10 * 10 " 10 * 10 " TIME OP COOLING (DAY)
Figure 2: Total and partial activities of mercury target as a function of cooling time.
together with the isotope 204T1 (Ti/2=3.78 years, begins to dominate after 3 years). The 193Pt contribution becomes significant later. For lead target the 210Po contribution is by a few orders of magnitude lower than the contribution of those isotopes. Alternative isotopes dominate also in lead-bismuth target activity for the same range of cooling times. The 210Po activity is only 2.5 times lower than the
403
activity of Au. After three years the total activity is determined by Bi nuclide (Ti/2=32.2 years). The comparison of the activities of lead and lead-bismuth targets irradiated by 800 MeV protons for one year demonstrates that the total activities of the targets begin to differ significantly only after one year. This is due to the formation of longlived bismuth isotopes in the (n,2n) and (n,3n) reactions in lead-bismuth target (207Bi and, for longer times, 208Bi). The calculations and analysis of the mercury target activity for the same irradiation conditions were carried out (Fig. 2). One can see that there is some difference in the list of the main contributing nuclides to the long-lived activity in comparison with lead or lead-bismuth targets. The other feature of mercury target is the determining contribution of fission products to the long-lived activity at very large cooling times. To evaluate correctly the possible uncertainties of the calculations and the effect of cross section data errors on the results it is necessary to analyse the spectral contributions of neutrons and protons to the accumulating activities. The components of the long-lived radioactivity in lead target, accumulated due to the (p,xn) and (n,xn) reactions, are presented in Figure 3. The activities induced by the soft part of neutron spectra with En<20 MeV are also presented. These results indicate that the dominating long-lived activities determined by platinum, gold, mercury and thallium isotopes are formed by hard components of proton and neutron spectra with the energies above 20 MeV. The component of neutron spectrum with the energies below 20 MeV corresponding to 96.6 % of the total neutron flux makes some 103-104 times lower contribution to the total long-lived activity (T1/2> 1000 d) than protons and neutrons from the high energy part of the spectra comprising less than 4 % of the total flux of the particles. It must be pointed out however that the accumulation of long-lived isotopes 207 Bi (T1/2= 1.39104 d), 208Bi (T1/2= 1.34 10s d) and 210Po (T1/2=138 d) is due to the (n,2n), (n,3n) and (n,y) reaction on soft neutrons. The total activity of those isotopes is more than 1000 times lower than that of gold, mercury and thallium for lead and lead-bismuth targets for cooling times longer than 1 year. Our analysis demonstrates that the uncertainties of the results of calculations of the long-lived activity of the targets are determined by the errors of the cross sections of the threshold reactions at intermediate energies of protons and neutrons. The results obtained for the lead-bismuth target under construction in IPPE now in the frame of ISTC Project No. 559 are shown in Fig.4. 3 Activity of volatile nuclides. The activity of volatile radionuclides at the moment of the accelerator shutdown was calculated. In contrast to the task of the accumulation of the total activity in this case the considerable contribution give short-lived radionuclides. Their quantities in
404
the target are determined by the relation between the production rate and decay rate. For this purpose the decay chains have been analyzed for all isotopes, the number of the decay chains differs from three (for Br) to nine (for cesium, xenon and iodine). One can see that the contribution of mercury to the activity of volatile nuclides comprises at the moment more than 75 %. Pb t a r g e t ( 5 0 x 1 0 0 ) P r o t o n e n e r g y 0.8 GeV I = 30 m i One y e a r i r r a d i a t i o n
10 • .10 'lO10 10 10
N e u t r o n flux 1.34:Xl0la n / c m s P r o t o n flux 1.94x10 p / c m s Contribution Total activity | ooooo Neutron spectrum below I I I I I I Neutron spectrum
below
>o©eoc Neutron spectrum below 100 MeV'{99.2«) 10 ' ^:***** Neutron spectrum below 800 MeV (iOO.Ox) Proton spectrum below 100 MeV (8.0 %) 10
io -* io ~* io -' io
10 10 * 10 a 10 4 10 " 10 " 10 ' Time of cooling (day)
10 * 10 * 10 '
Figure 3. Activity of lead target due to various spectral components.
Target Pb-Bi (M0x60) cm E p =800MeV, l p =1.25mA 91 days irradiation
10*
10 3
102
101
10°
10'
10z
103
10*
105
10 7
10"
T i m e of c o o l i n g ( d a y )
Figure 4. Total and partial activities of lead-bismuth target after three month irradiation
405
Activities of the target due to various components of neutron spectrum are shown in Fig. 5. 103 102
Target Pb-Bi (ii0x60) cm E =800 MeV, 1=1.25 mA
101
91 days irradiation ADL-93, MEMX-2, CASCADE
p
P
2 >
~
H„-2
t5 10
10
10 4
NedronapecautncottiiMlian total Neutron spMnm brio* LlfcV MeUn»mmliiirnbe)ow20. MeV - ^ - Neutron spectrum betowl 00. MeV
10 3
10 2
10"1
10°
101
102
103
104
105
10S
107
108
109
Time of cooling (day)
Fig. 5. Activities of lead-bismuth target after three month irradiation with 0.8 GeV, 1.25 mA proton beam due to various neutron spectral components.
The radiology characteristics have been estimated for all main components of the system [3]. The system is a complicated engineering construction, that includes such a components as target active part; electromagnetic pump; heat exchanger, pipelines; volume compensator; drainage tank; electric heating systems; control and instrumentation system; support metallic structure; system of radiation shielding, etc. Thus, the radiation danger of the target circuit is determined by radiological properties of all those components at all regimes. The main contributors into radiation hazard are the following components: radioactive Pb-Bi coolant, protective gas system, an air in the container, and cooling water. The preliminary calculation carried out in order to validate the radiation safety showed, that the total specific activity of the coolant at the end of the irradiation (6 months) is about -500 Ci/kg. The total activity of isotopes Po-208, Po-209, and Po-210 equals ~1 Ci/kg , activity of mercury isotopes -35 Ci/kg, and activity of the volatile nuclides (Cs, Xe, I, Rb, Kr, Br) is -3 Ci/kg. The total activity accumulated on the surfaces of the system has the following structure: polonium nuclides - 0.005 Ci, Hg -3850 Ci, Cs ~ 0.8 Ci, I -2.8 Ci, Br - 7 Ci, Rb -11 Ci. As for total activity of the gas system, it is determined by xenon and krypton nuclides and is about -240 Ci.
406
4
Conclusions
The obtained results showed that long-lived radioactivity accumulates mainly due to primary nuclear reactions. Secondary reactions are responsible for the production of small number of long-lived isotopes Bi-207, Po-210 and some others, being generated by radiative capture of low energy neutrons. It is possible to make a conclusion that neutrons in the energy range 20 - 800 MeV and protons with energy above 100 MeV give main contribution to the total activity generation although these parts of spectra inside the target give comparatively small contribution to the total flux. The correct consideration of short-lived nuclides contribution is the main problem in the analysis of the target behaviour in the case of short accelerator shutdowns. They make the determining contribution to the both activity and the heat release at the first moments after the accelerator shutdown, creating the intermediate links and additional channels for the long-lived nuclides accumulation chains. The strong dependence of calculated concentrations of short-lived nuclides on the choice of the cross section data library for the determining of the reaction rates was noted, particularly for volatile nuclides. The most dangerous are gaseous and volatile radionuclides, which are produced due to thermal diffusion and evaporation from the lead-bismuth mirror of volume compensator into protective gas system (helium). Among them there are noble gases krypton and xenon, radionuclides of polonium, mercury, cesium, iodine, bromine, and rubidium. However, even in the case of target window braking the direct danger for the personal is excluded, because of hermetic container under biological shielding. References 1.
Shubin Yu. N., Ignatyuk A. V., Konobeyev A. Yu., Lunev V. P.and Kulikov E.V. Analysis of energy release, beam attenuation, radiation damage, gas production and accumulation of long-lived activity in pb and pb-bi targets.In Proc. 2d Int. Conf. On Accelerator-Driven Technologies and Applications, June 3-7, 1996, Kalmar, Sweden, Ed. by H. Conde(Gotab, Stockholm, 1997), pp.953-959. 2. Gai E. V, Ignatyuk A. V., Lunev V. P. and Shubin Yu. N., Safety Aspects of Targets for ADS. In Proc. 3d Int. Conf. ADTT'99, 3-7 June 1999, Praha, Chekhia (CD- edition). 3. Efimov E. I., Ignatiev S. V., Levanov V. I., Pankratov D. V. and Shubin Yu. N., Radiological Properties of Heavy Liquid Metal Targets of Accelerator Driven Systems. Ibid.
INTERMEDIATE-ENERGY NUCLEAR DATA FOR RADIOACTIVE ION BEAMS AND ACCELERATOR-DRIVEN SYSTEMS M. V. RICCIARDI, P. ARMBRUSTER, T. ENQVIST, F. REJMUND, K. -H. SCHMIDT, J. TAIEB GSI, Planckstrasse 1, D- 64291 Darmstadt, Germany E-mail:
[email protected]
J. BENLLIURE, E. CASAREJOS Universidad de Santiago de Compostela, E-15706 Santiago de Compostela,
Spain
M. BERNAS, B. MUSTAPHA, L. TASSAN-GOT IPN Orsay - IN2P3, ¥-91406 Orsay, France A. BOUDARD, R. LEGRAIN, S. LERAY, C. STEPHAN, C. VOLANT, W. WLAZLO CEA Saclay, F-91191 Gifsur Yvette, France S. CZAJKOWSKI, J. P. DUFOUR, M. PRAVIKOFF CEN Bordeaux-Gradignan,
F-33175, Gradignan,
France
Formation cross sections of isotopes produced in inverse kinematics from the spallation/fragmentation and fission of 197Au (at 800 AMeV), 208Pb (at 1 A-GeV) and 238U (at 1 A-GeV) are presented. These data are extremely important for the design of acceleratordriven systems and new radioactive-ion-beam facilities. The data have been measured at GSI with the FRagment Separator, which allows precise measurements of the cross sections of the fragments and of their velocities. The knowledge of the velocity enables to deduce the reaction mechanisms and leads to a better understanding of the physics of intermediateenergy nuclear reactions. Thanks to this, nuclear models capable to predict the isotopic formation cross sections have been implemented and benchmarked with the measured data. Results are presented.
1
Introduction
The design of accelerator-driven systems and radioactive-ion-beam facilities requires the knowledge of formation cross sections of isotopes produced in spallation and fission reactions at intermediate energies. At the moment, the existing experimental data can by no means provide the information needed. Measuring all the reactions of interest would be a long and expensive task, therefore computational programs seem to be a more practical tool. However, the predictive power of theoretical models is often far from the performance required for the technical application, mostly due to the lack of knowledge on the mechanisms involved in these nuclear reactions. Therefore, selected nuclear reactions must be investigated 407
408
experimentally in order to throw light upon the physics involved in such reactions and supply data for benchmark tests. For this purpose, an experimental program has been started 4 years ago at GSI to measure formation cross sections of spallation and fission residues produced in the interaction of protons and deuterons with some selected nuclei (197Au, 208Pb, 238 TJ, 56Fe) at relativistic energies (500-1000 MeV) in inverse kinematics. In addition, Monte Carlo codes that can predict the isotopic formation cross sections have been implemented. In this paper, some results on proton- and deuteron-induced spallation and fission reactions are presented. Comparisons between Monte Carlo predictions and experimental data are discussed too. 2
Experimental results
The experiments have been performed at GSI [9] in inverse kinematics, i.e. the relativistic beams of 197Au, 208Pb, 238U were accelerated and directed onto a H2 or D2 target. The spallation or fission fragment, originating from that part of the projectile that survives, was selected and identified in-flight with the fragment separator (FRS) [10]. The in-flight separation allows to measure the yields of the residues prior to their p-decay and to obtain the whole isotopic distribution for every produced element. Indeed, the measurements can cover the full (N,Z) map of produced residues in every projectile-target reaction. As a test experiment, we firstly investigated part (due to lack of time) of the residues produced in the reaction 238U (1 A-GeV) + 208Pb [6]. At the moment, the formation cross sections of fission and spallation isotopes produced in the ' 7Au (800 A-MeV) + p, in the 208Pb (1 A-GeV) + p and in the 208Pb (1 A-GeV) + d reactions are also available [1, 7, 12, 13] or will be available soon. Some data from the 232U (1 A-GeV) + p, d reactions are partially analysed whilst those from the 208Pb (500 A-MeV) + p reaction still have to be analysed. Due to the limited time for the experiments, the cross-sections have been measured down to 0.1 mbarn. The accuracy of the measurements is about 10%. The measured formation cross sections for the isotopes produced in the reaction U (1 A-GeV) + Pb are qualitatively compared with those measured in Pb (1 A-GeV) + p (figure 1). The results can be interpreted on the basis of the reaction mechanisms. The data are grouped in two zones: a fragmentation (or spallation) corridor and a fission area. The corridor is filled with those residual nuclei that are formed at the end of an evaporation chain. Depending on the impact parameter, a certain amount of excitation energy is acquired by the surviving projectile nucleus, which can de-excite by evaporating nucleons and light particles. The length of the fragmentation corridor is given by the maximum energy deposited in the nucleus, which in turn is limited by the total beam energy in the centre of mass. That is why
409
the spallation corridor is shorter in the case of Pb->p reaction (Ec m ~ 1 GeV) than in the 238U-»Pb reaction (Ecm. - I l l GeV). In competition to the evaporation of particles fission can occur, providing that the excitation energy of the compound nucleus is enough to overcome the fission barrier. The latter is very low for "*U, and the energy transferred in the electromagnetic interaction with the target can be high enough to induce fission. The fission fragments will form a double-humped yield distribution, as it is expected for the low-energy (asymmetric) fission of uranium. At the same time, the nuclear interaction can transfer a higher amount of energy, and a large group of actinides can undergo high-energy (symmetric) fission, producing fragments that, after neutron evaporation, can lie even in the area of the (3-stability. In the case of Pb, due to the higher barrier, high-energy (symmetric) fission after nuclear interaction is the only possible fission channel. Since the fissility increases with Z2/A, the proton-rich Pb isotopes, produced after the evaporation of some neutrons, are the best candidates for fission. So the fission fragments will be located near the valley of stability.
Figure 1. Measured formation cross sections plotted on the chart of the nuclides for the isotopes produced in the reactions 238U (1 A-GeV) + Pb (left) and 208Pb (1 A-GeV) + p (right).
3
The models
One of the most important advantages of the FRS is that the in-flight separation allows precise measurements of the momenta of the fragments. Thanks to the measured velocity distributions of the produced isotopes, the reaction mechanisms can be deduced and fission products distinguished from spallation ones. The new experimental information has been exploited to develop nuclear models and to implement Monte Carlo codes to predict the isotopic formation cross sections. The codes are based on a first fast stage (intra-nuclear cascade from Cugnon [4] for
410
nucleon-nucleus reactions or an abrasion model [5] for nucleus-nucleus reactions) and on a successive slow de-excitation, in which two competitive mechanisms can occur (evaporation or fission). Both the evaporation and the fission model have been developed at GSI [2]. A statistical model is used to describe the evaporation of particles. The emission probability of a certain particle is given by the ratio between its decay width and the sum of all the decay widths. In the decay widths the level densities of the initial and final states take into account the influence of shell and pairing effects, as reported in Ref. [11]. A physical quantity that plays an important role for the description of the isotopic distributions of the produced elements is the Coulomb barrier of charged particle since it affects the competition between neutrons and charged particles evaporation. In figure 2 the isotopic distribution of the isotopes of rhenium from the reaction 197Au (800 A-MeV) + p are reproduced by the GSI Monte Carlo code. The results of two other simulations, in which the proton evaporation barrier was arbitrarily increased or decreased by 2 MeV, are also shown. The picture shows clearly how the proton evaporation affects the production of neutron-deficient isotopes.
• O A O
95
100
105
Exp. data Model Model, B„-2MeV Model, Bp*2MeV
110
Neutron number
Figure 2. Influence of proton-evaporation barrier: distribution of the isotopes of rhenium (Z=75) produced in the spallation of 197Au (800 A-MeV) + p.
The evaporation chain goes on until the excitation energy of the pre-fragment falls below the lowest particle threshold or until fission occurs. The fission decay width is calculated by the transition-state method of Bohr and Wheeler [3]. Nuclear friction is included according to Ref. [8], with the following value of the viscosity coefficient: (i = 210"21 s"1. The mass distribution of the fissionfragments is based on a semi-empirical description of the fission process [2] and depends mostly on the description of the level density above the conditional fission barrier. The mean value of the N/Z-ratio of fission fragments and its fluctuations depend on the energy of the nucleus at the scission point. Excited fission fragments can evaporate particles according to the previously described model. Some experimental data for the spallation and fission fragments of 197Au (800 A-MeV) + p and 208Pb (1 A-GeV) + p are compared to the prediction of the GSI
411
code in figures 3 and 4. The error bars of the experimental data correspond to the statistical error, while no error bars are given for the computational predictions.
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i i i i i i i-1
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170 IBB M) 100 BO
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Figure 3. Measured formation cross sections for some isotopes produced in the spallation of 197Au (800 A-MeV) + p (left) and 208Pb (1 A-GeV) + p (right). The experimental data (dots) are compared to the predictions of the GSI code (lines).
1
„HU
h t\
A
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Figure 4. Measured formation cross sections for some isotopes produced in the fission of 197Au (800 A-MeV) + p (left) and 208Pb (1 A-GeV) + p (right). The experimental data (dots) are compared to the predictions of the GSI code (lines). References 1. J. Benlliure, P. Armbruster, M. Bernas, A. Boudard, J. P. Dufour, T. Enqvist, R. Legrain, S. Leray, B. Mustapha, F. Rejmund, K.-H. Schmidt, C. Stephan, L.
412
2.
3. 4. 5. 6.
7.
8.
9. 10. 11.
12.
13.
Tassan-Got, C. Volant, Isotopic production cross sections of fission residues in 197Au-on proton collisions at 800 A MeV, submitted to Nucl. Phys. A (2000). J. Benlliure, A. Grewe, M. de Jong, K.-H. Schmidt, S. Zhdanov, Calculated nuclide production yields in relativistic collisions of fissile nuclei, Nucl. Phys. A628(1998)pp.458-478. N. Bohr, J. A. Wheeler, The mechanism of nuclear fission, Phys. Rev. 56 (1939) pp. 426-450. J. Cugnon, Monte Carlo calculation of high-energy heavy-ion interactions, Phys. Rev. C 22 (1980) pp. 1885-1895. Y. Eisenberg, Interaction of heavy primary cosmic rays in lead, Phys. Rev. 96 (1954) 1378. T. Enqvist, J. Benlliure, F. Farget, K.-H. Schmidt, P. Armbruster, M. Bernas, L. Tassan-Got, A. Boudard, R. Legrain, C. Volant, C. Bockstiegel, M. de Jong, J. P. Dufour, Systematic experimental survey on projectile fragmentation and fission induced in collisions of 238U at 1 a GeV with lead, Nucl. Phys. A 658 (1999) pp. 47-66. T. Enqist, W. Wlazlo, P. Armbruster, J. Benlliure, M. Bernas, A. Boudard, S. Czajkowski, R. Legrain, S. Leray, B. Mustapha, M. Pravikoff, F. Rejmund, K.H. Schmidt, J. Taieb, L. Tassan-Got, C. Volant, Isotopic yields and kinetic energies of primary residues in 1 A GeV 208Pb+p reactions, submitted to Nucl. Phys. A (2000). A. Heinz, B. Jurado, J. Benlliure, C. Bockstiegel, H.-G. Clerc, A. Grewe, M. de Jong, A. R. Junghans, J. Miiller, K.-H. Schmidt, S. Steinhauser, Signature of dissipation deduced from fission after peripheral heavy-ion collision, GSI Scientific Report 1999, p. 30. Http://www.gsi.de. Http://www-wnt. gsi.de/frs A. R. Junghans, M. de Jong, H.-G. Clerc, A. V. Ignatyuk, G. A Kudyaev, K.-H. Schmidt, Projectile-fragment yields as a probe for the collective enhancement in the nuclear level density, Nucl. Phys. A 629 (1998) pp. 635-655. F. Rejmund, B. Mustapha, P. Armbruster, J. Benlliure, M. Bernas, A.Boudard, J.P. Duffour, T. Enqvist, R.Legrain, S. Leray, K.-H. Schmidt, C.Stephan, J. Taieb, L. Tassan-got, C.Volant, Measurement of isotopic cross sections of spallation residues in 800 A MeV 197Au+p collisions, submitted to Nucl. Phys. A (2000). W. Wlazlo, T. Enqvist, P. Armbruster, J. Benlliure, M. Bernas, A. Boudard, S. Czajkowski, R. Legrain, S. Leray, B. Mustapha, M. Pravikoff, F. Rejmund, K.H. Schmidt, C. Stephan, J. Taieb, L. Tassan-Got, C. Volant, Cross sections of spallation residues produced in 1 A GeV 208Pb on proton reactions, Phys. Rev. Lett. 84 (2000) 5736.
A C T I N I D E N U C L E O N - I N D U C E D FISSION R E A C T I O N S U P TO 150 M E V V.M. MASLOV Radiation Physics and Chemistry Problems 220109, Minsk-Sosny, Belarus
Institute,
A. HASEGAWA Japan Atomic Energy Research Institute, Tokai-mura, Naka-gun, Ibaraki-ken, Japan, 319-1195 Fission/evaporation mechanism for description of nucleon-induced fission of actinide nuclei up to ~150 MeV excitation energy is claimed to be valid. Damping of collective modes contribution to the level density as dependent on deformation is investigated, n— and p— induced first-chance fission cross sections up to 150 MeV are found to be much different. Possible differences of measured p— and n— induced fission cross sections are attributed to the influence of the isovector term of the coupled channel optical potential.
1
Introduction
Statistical model analyses have been performed for n— and p—induced fission of U, Np and Pu targets l'2<3 up to En(p) ~m 150 MeV. Modified version of statistical model code STAPRE 4 was used. Calculated a(n, f) is a complex function of reaction cross section, pre-equilibrium emission of secondary particles and fission probabilities. These factors were disentangled below 20 MeV for 238U{n,f) cross section. 2
Statistical model
We obtained neutron optical model potential (OMP) for 238U m with 0+-2+4+-6 + -8 + coupling scheme within rigid rotator model, fitting total data 5 up to 150 MeV 6 ' 7 . Adopting weaker volume absorption 8 we avoid severe problems with fitting (n,/)and (n,xn) data at En > 10 MeV. To predict p +238 U cross section using n + 2 3 8 U OMPs one needs the isovector 9 and Coulomb terms. Real Vjl and imaginary surface Wfi terms both have isovector terms, which depend on the symmetry parameter 7 = (N — Z)/A 9 .These isovector terms are introduced into V^ and Wp' with opposite signs, i.e. V£ = V£ - 2aj, Wp — W^ - 2(3j. We assumed roughly linear decrease of a and j3 values from 16 and 8 at Ep ~ 40 MeV up to 5 and 3 at 150 MeV, respectively 10 . 413
414
Total nuclear level density is represented in adiabatic approximation as the factorized contribution of quasiparticle and collective states n , quasiparticle level densities pqp(U,J,Tr) 12 take into account the shell, pairing and collective effects: p(U, J, TT) = Krot(U, J)Kvib(U)Pqp(U,
J, TT).
(1)
The shell effects are modelled with the energy dependence of a—parameter12. We assume that a-values for equilibrium and fission saddle deformations are equal, i.e. a,f/an ratio is solely dependent upon respective shell correction values. The collective rotation enhancement factor Krot(U,J) depends on deformation order of symmetry 13 . At excitation energies U > Ur, damping of rotational modes was anticipated 14 , it is usually described with a Fermi function F(U) = (1 + exp(C/ - Ur)/dT)~l as K™t(U) = (Fxt-l)F(U)
+ l,
(2)
where F± is the momentum of inertia (perpendicular to the symmetry axis), t is thermodynamic temperature. For triaxially asymmetric nuclides (inner saddle deformations) the rotational enhancement factor is K^(U)
= K^(U)((2y/2^F\t
- \)F{U) + 1).
(3)
Here, F^ is the momentum of inertia (parallel to the symmetry axis). It was assumed in 14 that Ur ~ e 2 , e being quadrupole deformation. That means for actinides at excitations up to ~ 150 MeV there would be no damping at saddle deformations, that is at odds with new finding 15 , predicting Ur — 40 MeV and dr — 10 MeV, independent of quadrupole deformation e. We assumed Ur — 20 MeV and dr = 5 MeV for axially symmetric nuclei and Ur = 15 MeV 15 and dr = 5 MeV for triaxial nuclear shape. 3
Results
The main contribution to a(n, f) of 238U comes from ~ 9 fissioning nuclei (see Fig 1). Calculated cr(n, / ) is somewhat discrepant with measured data at U ~ 35-70 MeV as compared with other Pu,Np and U targets. Assuming no damping at saddles, (n,nf) reaction makes major contribution to 2 3 8 C/(n,/) cross section at En > 70 MeV. When damping is assumed, major contribution comes from (n, 3n/) reaction. To decrease calculated a(n, / ) when U ~ 100 MeV is rather difficult: the reduction of contribution of predecessors (lower chance fission) leads to immediate increase of that of successors (higher chance fission).
415 U fission cross section
Pu fission cross section o • 3
-
o
Staples etal.,1998 Shcherbakovetal., 1999 (n,F) . no dumping at saddles
A
_
(n,f)
-
Lisowski et a]., 1997 Shcherbakovetal., 1999 - (n,F) . no dumping at saddles
(n,f) 2 ~^—" (n,nf)
—
&j&§ig6£J?>i: — - (n,2nf) • (n,3nt) JB*^ T^ISSS t ~ - (nMf)Jp -
. .1..
10 neutron energy, MeV Fig. 1
100
U(p,f) cross section
10 100 neutron energy, MeV Fig. 2 First chance fission contribution to total fission cross section
'
10
100 proton energy, MeV Fig. 3
1000 0
50 100 excitation energy, MeV Fig. 4
ii
150
Fission data 2 trends for 239Pu and 2i0Pu and for lower fissility targets 242 Puand 2UPu could not be fitted simultaneously 16 . The most challenging feature of the Pu data 2 is the dependence of observed a(n, f) at high excitations on the fissility of the target, which was questioned recently 3 (see Fig. 2). Data trend for 239Pu(n, / ) 3 is consistent with data 2 for 240,242,244Pu_ T h a t means at U ~ 150 MeV there is almost no dependence of Pu measured a{n, / )
416
on the target fissility. Contributions of i th-chance (n, xnf) reactions depend on collective effects damping either at saddle and equilibrium deformations or energy dissipation during stretching over fission path, while a(n, / ) is almost insensitive to these assumptions. In case of no damping of collective modes at saddle deformations, the main contribution to the 239Pu(n, / ) above En <~ 60 MeV, would come from 239Pu(n,nf) reaction. When damping is independent of e, (n, nf) reaction contribution decreases above En ~ 30 MeV, while those of (n,2nf) and (n,3n/) will increase appreciably. Fission suppression leads only to redistribution of fission strength over the decay chain. Another controversial point is the reason of difference of 238U(p,f) and 238 U{n, / ) cross sections at En^p) ~ 40-150 MeV 17 (see Fig. 3). Description of 238U(p, n) and 238 £/(n,3n) 236s ./Vp cross sections is a validation of the consistency of the first chance fission cross section estimate 18 . Contributions of first chance fission to the measured a(n,f) of 238U and 238Np are compared on Fig. 4. Analysis of a(n, f) of 239Np nuclei, formed in p + 238JJ interaction is complemented with the 237Np(n, / ) data 3 analysis, fission probability of 239 Np is defined in 19 . It could be concluded that in case of (p, f) reaction contribution of non-emissive fission to a(p, f) is much higher than in case of (n, / ) reaction. To describe the lower a(p,f) one needs lower absorption cross section, hence very low isovector term of potential term Wp. Adopted estimates of a and 0 give 238U(p,f) cross section which is higher than that of 238U(n,f) reaction above En{p) ~ 30 MeV (Fig. 4). Summarizing, we argue the validity of fission/evaporation mechanism for nucleon-induced fission of U, Np and Pu nuclei up to U ~ 150 MeV. Fission dynamics effects are not evidenced in observed a(n,f). Collective modes damping is independent on the deformation. Dependence of observed a(n, / ) on the projectile at 2?n(p) > 40 MeV is interpreted as being due to optical reaction cross sections. We observe the dependence of the first chance fission contribution to the measured fission cross section on the projectile. 3.1
Acknowledgments
This research was supported by International Science and Technology Center under Project Agreement B-404 and International Atomic Energy Agency (Vienna, Austria) under Research Contract 9837.
417
References 1. Lisowski P. et al. Proc. Specialists' Meeting on Neutron Cross Section Standards for the Energy Region above 20 MeV, Uppsala, Sweden, May 21-23, 1991, p. 177, OECD, Paris, 1991. 2. Staples P., Morley K., Nucl. Sci. Eng. 129, 149 (1998). 3. Shcherbakov O.A. et al., Proc. of 7th Int. Seminar on Neutron Interactions with Nuclei, Dubna, Russia, May 25-28, 1999, p.357 4. Uhl M., Strohmaier B., IRK-76/01, IRK, Vienna (1976). 5. Shutt R.L., Shamu R.E., Lisowski P.W. et al., Phys. Lett. B, 203, 22 (1988). 6. Hasegawa A., Maslov V.M., Porodzinskij Yu.V., Shibata K., In: [3], p.89. 7. Maslov V.M. et al. Proc. of 8th Int. Seminar on Neutron Interactions with Nuclei, Dubna, Russia, May 17-20, 2000. 8. Young P.G., IAEA-TECDOC-1034, p.131, 1988, Vienna. 9. Madland D. Proc. of the Spec. Meeting on Nucleon-Nucleus Optical Model up to 200 MeV, 13-15 Nov., 1996, Bruyeres-le-Chatel, France, p. 129. 10. Delaroche J.P. et al.Proc. Int. Conf. Nucl. Data for Sci. and Techn. Trieste, Italy, 1997, p. 206. 11. Bohr A. and Mottelson B., Nuclear Structure, vol. 2, (Benj.,NY, 1975). 12. Ignatjuk A.V.,Istekov K.K.,Smirenkin G.N. Sov.J.Nucl.Phys. 29,450(1979). 13. Howard W.M., Moller P. At. Data and Nucl. Data Tables,25,219(1980). 14. Hansen G., Jensen A.S., Nucl. Phys. A406, 236, (1983). 15. Junghans A.R., et al. Nucl. Phys. A629, 635 (1998). 16. Maslov V.M. In: [3], p.249. 17. Conde H. et al. Proc. of the Second Int. Conf. on ADTT, Kalmar, Sweden, June 3-7, 1996, p.599. 18. Maslov V.M. Atomnaya Energia, 69, 252 (1990). 19. Maslov V.M., Porodzinskij et al., INDC(BLR)-011, IAEA, Vienna,1998.
T H E A U S T R O N SPALLATION S O U R C E P R O J E C T G. BADUREK AND E. JERICHA Nuclear Physics Institute, University of Technology Stadionallee 2, A-1020 Vienna, Austria E-rnail:badurek
AUSTRON
Vienna,
H. WEBER Project Group, Schellinggasse 1/9, A-1010 Vienna, E-mail:helmut. weberOnetway.at
E. GRIESMAYER FH Wiener Neustadt, Johannes Gutenbergstrasse 3, A-2700 Wiener E-mail: [email protected]
Austria
Neustadt
The organization scheme as well as the accelerator, target and instrumentation concepts of the planned 0.5 MW/10 Hz neutron spallation source AUSTRON are presented. Its relation to existing and planned pulsed sources is discussed.
1
Introduction
The project of an interdisciplinary International Research Centre is proposed for the fine analysis of matter by neutron scattering in the basic and applied fields of materials science, biology, engineering, biotechnology, life sciences, chemistry and physics. This Centre will serve the purpose of supporting the growth of the overall Central European research infrastructure (universities, industrial and academic laboratories), through an internationally competitive access and use, also inducing a quality benchmarking in the existing research programs. The fine analysis of the structure of matter is essential for the development of new materials, products and processes, and the scientific support for products ranging from microelectronics to new materials and new pharmaceutical compounds. It has, therefore, contributed to the most extensive and continuous growth of new industries in this century. The Centre emerges from the AUSTRON project developed by a large international team on the basis of indications from all interested users from the Central European countries. The AUSTRON project is based on a pulsed high-flux neutron spallation source providing one of the world-best research-service facilities for neutron scattering experiments for the next two decades. On an international scale the project AUSTRON was incorporated in the global plans to counteract the occurring "neutron gap" by the OECD-Megascience Forum. The major
418
419
40 60 Energy per Pulse (kJ)
ao
Figure 1. Planned pulsed neutron sources (updated 09/99).
projects of neutron sources of the next generation are the ESS in Europe, the SNS in USA and J J P in Japan (Fig. 1). New sources in Europe are the projects of the reactor FRM II (under construction) in Munich, a second target station at ISIS, UK and AUSTRON. AUSTRON is proposed as a regional neutron source to catalyze science in Central Europe across the borders of present and future EU countries. It can be built with existing techniques and will provide features not available at existing sources. The 10 Hz repetition rate favours high resolution condensed matter spectroscopy and new instruments will broaden the scope of neutron spectroscopy in general. 2
Accelerator and Target Concept
The AUSTRON project is based on a well-founded accelerator concept using available state-of-the-art technologies to allow for a relatively short construction period and to create a favourable ratio of cost to scientific and technological potential 1 . Several successive accelerator stages of different types will be used to generate a proton beam with 1.6 GeV energy per particle, an average beam current of 0.311 mA and a total beam power of 500 kW. The accelerator chain comprises an H~-ion source, a radio frequency quadrupole and a drift tube LINAC, providing a final ion energy of 130 MeV, from which the ions enter a rapid cycling synchrotron (RCS) passing a stripper foil which removes their electrons to enable the acceleration of a high-intensity proton beam to 1.6 GeV final energy. Using a dual frequency magnetic cycle, losses should be kept at about 0.5% occurring at lowest energies during trapping only. This is of particular importance with respect to the question of induced activity
420
which can be answered positively under these circumstances. The operation frequency of this acceleration process has been determined with 50 Hz. However, since there is strong demand for cold neutrons, a preference for a lower operation frequency of 10 Hz without reduction of beam power was expressed in order to avoid frame overlap problems of successive neutron pulses in the long wavelength regime. This can be achieved by adding an additional storage ring to the accelerator complex which works as a bunch accumulator for the proton bunches leaving the RCS (see Fig. 2). Employing such an installation, the stacking of up to 4 proton bunches is feasible. Extracting these bunches together with the bunch which has just reached the final energy in the RCS, a 10 Hz source is realized with 1.6 GeV protons (~ 2 x 1014 protons in total), which deposit 50 kJ per pulse on the spallation target. The average thermal neutron flux is expected to be about 7x 10 12 neutrons/cm 2 s with a peak flux of ~ 3.7 x 10 16 neutrons/cm 2 s. This configuration will make AUSTRON a truly unique facility among present neutron sources. The effective flux for certain classes of neutron instruments will be increased by a factor of 15 - 20 compared to the present standard. With more than an order of magnitude higher performance, the exploration of completely new fields of research can be envisaged. Furthermore, the 10 Hz option takes the increasing demand for cold neutron scattering into account and no flux penalty will be experienced by those instruments which would usually be operated at higher frequencies. Concerning AUSTRON's relation to the European Spallation Source (ESS) in this respect, it should be noted that these facilities will belong to two different generations of neutron sources which will be separated by a decade in time and an order of magnitude in beam power. In the present concept of the target design a flat target geometry is proposed. The target material under consideration is solid W-5%Re according to its excellent thermal and mechanical properties. Dimensions of a target block are 10x30x60 cm 3 (height x width x length) where, due to the edge-cooling concept, cooling channels are only installed within 2 cm from the top and bottom surface. Calculations of the temperature distribution in the target, based on a 0.5 MW version of AUSTRON running at 50 Hz, yield a temperature maximum of 1200-1300 °C. Edge-cooling of a target is possible under these conditions and an advanced cooling system has been designed. Material properties of W-5%Re like ductility, thermal conductivity or self-healing after irradiation damage look favourable for this temperature range and reveal that such a target is indeed feasible. The operation of AUSTRON with 10 Hz/0.5 MW leads to a marginal temperature increase of less than 10 °C only. It is also suggested to operate the AUSTRON target at even higher temperatures, above 2000 °C, and to cool by radiation cooling alone, which would help to avoid thermally induced stress inside the target
421
Figure 2. RCS and storage ring of the 10 Hz version of AUSTRON.
block. The final decision on the target design will be made immediately after approval of the AUSTRON project. 3
Instrumentation
The key part of the AUSTRON facility is naturally represented by the neutron instrumentation which is illustrated in Fig. 3. A set of 21 instruments has been proposed by an international working group. When taking up service 6-8 of them will be ready for operation 2 . For optimal instrument performance 4 moderators will be required, one at ambient or intermediate temperature and three cold moderators. • Instruments at the ambient/intermediate moderator: high resolution powder diffractometer (covering a very large detector solid angle), diflractometer for liquids and amorphous materials (emphasis on low- and small-angle scattering), direct chopper time-of-flight spectrometer (magnetic excitations and vibrational spectroscopy), crystal analyzer spectrometer for molecular excitations, radiography and tomography facility (providing the option of timegated energy selection), engineering research beam line. • Instruments at the high resolution cold moderator: general purpose powder diffractometer (following the recommendations in the Autrans report for new developments in this field), two single crystal diffractometers (one en-
422
abling protein crystallography, the other dedicated for the investigaton of samples with polarized nuclei), phase refiectometer (allowing a model-independent and unique reconstruction of the investigated surface profiles), high resolution crystal analyzer spectrometer (with several diffraction options), neutron resonance spin-echo spectrometer optimized for a pulsed neutron source and two development beam lines for general and for neutron optics developments, respectively. • Proposed instruments at coupled cold moderators: general purpose refiectometer, instrument for combined small and wide angle scattering, high resolution SANS instrument with neutron spin echo option, SANS project based on spin echo technique, multi-chopper TOF spectrometer with variable energy resolution, TOF spectrometer based on phase space transformation for high-resolution spectroscopy studies, neutron optics research station. This set of instruments seems to be well balanced with respect to the variety of instrumental and scientific possibilities. There is an adequate mixture of established and partly absolutely novel concepts and techniques. Most instruments are particularly suited to be installed at a pulsed neutron source and a majority of them will profit considerably from the 10 Hz operation of the source. Detailed consideration was given to an optimized sample environment. The proposed installation of a clean-room area (including temperature stabilization) combined with vibration isolation conditions represents a novel concept of sample environment for neutron sources. It will eventually contain about one quarter of the AUSTRON instruments such as reflectometers and single crystal diffractometers, which will gain from these possibilities for the investigation and development of advanced materials. The clean room area will also offer high stability conditions for neutron optics experiments of extreme sensitivity. Another special environment facility is the proposed engineering research area which allows heavy or large industrial samples to be delivered and investigated under full operating conditions on dedicated instrumentation. 4
Summary
As a summary AUSTRON is planned timely to counteract the foreseeable neutron gap. The impact on education and the possibility to train students will also contribute to avert the existing brain-drain in the region. The investment cost of AUSTRON was carefully estimated to be 340 Mill € over a construction period of seven years, the operative cost will be some 37 Mill € . The creation of this international research centre will need a workforce of about 280 newly employed. It's multidisciplinary character, ranging from
423 Neutron optics research station Clean room area
General purpose reflectometer
Diffractometer for polarised nuclei Single crystal 1 _ diffractometer
Radiography & Tomography facility
High resolution spectrometer Single-chopper TOF spectrometer
Phase reflectometer
General purpose diffractometer
Diffractometer for liquids and amorphous materials
Electromagnetic focusing beam line
Multi-chopper TOF spectrometer 0
10
20
• 50 m
High resolution diffractometer
Figure 3. Layout of the proposed AUSTRON instrument set.
materials and solid state physics to biology, polymers, chemistry, liquids, magnetism and superconductivity, will stimulate research in new fields bridging natural and life science. References 1. Ph.- Bryant, E. Griesmayer, E. Jericha, H. Rauch, M. Regler, H. Schonauer, Proc. 1999 IEEE Particle Accelerator Conf., Vol. 4, 2957 (1999). 2. H. Rauch, M. Regler, H. Weber, Physica 276-278, 33(2000).
Section II. Other Applications of Nuclear Physics
R E C E N T MODEL D E V E L O P M E N T S FOR N U C L E O N I N D U C E D REACTIONS U P TO 200 M E V E. B A U G E , J . P . D E L A R O C H E , M. G I R O D , S. H I L A I R E , J. L I B E R T ' * ) , B . M O R I L L O N , A N D P. R O M A I N Commissariat
a VEnergie
Atomique, Service Bruyeres-le-Chatel,
de Physique France.
Nucleaire,
BP 12,
91680
Improvements in the modeling of nucleon-nucleus interactions up to medium energies are presented. The dispersive optical model potential (OMP) is extended from spherical to permanently deformed nuclei in the actinide region, and a semimicroscopic OMP that is Lane-consistent is established. In support of our program for improving various reaction models, dedicated microscopic nuclear structure calculations based on the Gogny force are performed. These deal with level densities, spectroscopy of super-deformed states in actinide nuclei and fission barriers at finite temperature.
1
Introduction
The contemporary concepts designed for accelerator driven systems (ADS), have stimulated the revival of nuclear reaction models. Progress has recently been accomplished in making the Intra-Nuclear Cascade (INC) models amenable to improved predictions of proton and neutron emission spectra in the GeV region. However, it is not expected that the INC models are physically sound for physics studies down to thermal neutron energies. At incident nucleon energies typically lower than 200 MeV, simulation of ADS should rely upon solid nuclear data. Evaluated nuclear data files exist for many nuclei up to 20 MeV incident neutron energies, but are rarely available beyond this energy for both protons and neutrons. It is the task of nuclear models to fill the gap between 20 and 200 MeV, and assist evaluators in producing extended data libraries for stable or unstable nuclei present as target materials or spallation products in ADS. For these reasons the various reaction models (OMP, statistical/preequilibrium models, fission) should be set on better grounds than previously achieved to increase their predictive power. 2
Dispersive and semi-microscopic OMP
The optical model plays a key role in many facets of nuclear reactions. Unfortunately, only a modest effort has been put so far on building a smooth and physically sound close form for its energy dependence over a broad energy range (i.e. 1 keV < E < 200 MeV) in phenomenological OMP studies. The 425
426
same situation prevails in microscopic OMP studies.
Figure 1. Neutron total cross sections between 1 keV and 200 MeV for 1 8 1 Ta, 2 3 8 U , 2 3 9 P u and 2 4 2 P u . Comparisons between measurements and present coupled-channel, dispersive OM predictions.
Our strategy adopted to cure this problem is twofold. First, develop optimum and smooth E-dependent parameterizations of real and imaginary components of the central and spin-orbit phenomenological OMPs whenever many scattering and reaction observables have been measured for an individual target nucleus. Second, build a semi-microscopic OMP inspired from calculations in nuclear matter. These goals are now achieved as discussed below. In Fig. 1 are shown (solid lines) total neutron cross sections (
Figure 2. Elastic scattering of protons and neutrons on 2 0 8 P b , and quasi-elastic (p,n) scattering to IAS. Comparison between measurements and present Lane-consistent, semimicroscopic OM predictions.
427
lated in the coupled channel (CC) approach using dispersive OMPs built in the dispersion relation framework explained in 1 . This sophisticated model description yields excellent agreements between measured (dots) and calculated (solid curves) a? values. The overall quality of these (and other) data representations sets on solid grounds our predicted reaction cross section (and transmission coefficients) that is used below for describing the 2 3 9 Pu fission cross section in an evaporation/preequilibrium reaction model. In Fig. 2 is displayed for 2 0 8 Pb a comparison between calculated and measured (p,p) and (n, n) scattering cross sections as well as (p, n) cross section to the isobaric analog state (IAS). The predictions which are in good agreement with the measurements are obtained from our recent Lane-consistent, semi-microscopic OMP built from the model described in 2 . The quality of (p,p), (n,n) and (p, n) observable descriptions shown in this figure is representative of the overall quality achieved by our model to represent the elastic and quasi-elastic scattering measurements available for many 48 < A < 208 nuclei at incident energies between 20 and 200 MeV. This OMP should provide reliable scattering and reaction cross section predictions for stable and unstable A > 30 nuclei near and away from the /^-stability line provided that the asymmetry is restricted to |(iV — Z)/A\ < 0.35. This statement is based on challenging our model predictions with proton elastic and inelastic scattering cross sections measured recently for 3 °- 4 2 S in inverse kinematics in the 20 — 60 MeV energy range 3 , and for other proton and neutron measurements performed earlier up to 200 MeV for rare-earth and actinide nuclei 4 . The sole inputs to these OMP calculations are radial densities obtained from self-consistent, microscopic calculations based on D1S Gogny force 5 . 3
Level densities
In recent years, much efforts have been devoted to predicting level densities as functions of mass, excitation energy, spin and parity. In our Laboratory, we have adopted a combinatorial method which does not rely upon the so-called saddle-point approximation to the partition function. The combinatorial analysis is based on realistic single-particle, neutron and proton levels deduced from microscopic bound state calculations again made using the Gogny force. Preliminary report on this approach focusing on the excitation energy range E* > Bn is found in 6 . This work has been extended to energies lower than Bn, as illustrated in Fig. 3 for cumulated level histograms at low excitation energy in 2 3 8 Pu and 1 8 4 W. As can be seen good agreements are obtained between predictions and measurements below 1 MeV and 1.6 MeV for 2 3 8 Pu and 184 W, respectively.
428
„: X.
a: z
/
•
lC :
: :
/
•
/
;
/
:I :
/
*•
>'
i
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/
"*.
1
_ ;
x i 2
3
4
5
6
Excitation energy (MeV)
7
'"o
I
^
3
J)
5
E
?
Exdtalionoiergy(MeV)
Figure 3. Cumulated level histograms of 1 8 4 W and 2 3 8 P u . Comparison between measurements and present microscopic model predictions (upper solid curves).
4
Fission m e c h a n i s m : coupling t o class-II s t a t e s
As usual, the fission process of heavy nuclei is described using the Bj0rnholm and Lynn theory 7 implemented in evaporation/preequilibrium reaction models. Double-humped, phenomenological fission barriers in one dimension are assumed. Since states may take place in the secondary potential at large elongation (i.e. shape isomers and super-deformed levels often identified as class-II states in fission studies), an enhancement in the fission barrier transmission is expected. This resonant behaviour takes place whenever the excitation energy
Incident neutron energy (MeV)
Figure 4. Neutron-induced fission of 2 3 9 P u as calculated with including (upper solid curve) or ignoring (dashed curve) the coupling to class-II states.
of the compound nucleus gets close to t h a t of a class-II state. T h e impact
429 of such enhancements in the predicted ra+239Pu fission cross section (07) is shown in Fig. 4 at incident energies in the vicinity of the second-chance fission. As can be seen, the sharp increase observed above ~ 5.5 MeV in the measurements is well described by these calculations (higher solid curve). Ignoring the fission transmission enhancements inhibits the second chance fission process (dashed curve). Good fit to the 07 data beyond 5.5 MeV may indeed be restored by playing with symmetry-breaking level density enhancements at saddles, which then turn out to be at odd with solid theoretical predictions. 5
Class-II states and fission barriers at finite temperature in a microscopic model
The fission cross section calculated with the above mentioned model improvements is in good agreement with measurements up to 30 MeV. Prior to exploring higher incident energies, it has been judged pertinent to evaluate the reliability of the empirical model inputs: class-II states and fission barriers. A systematic study of class-II states in Th, U, Pu and Cm isotopes is underway through calculations based on a configuration mixing (CM) method. The reliability of this CM method has recently been tested for A ~ 190 superdeformed nuclei 8 . The CM predictions compare rather well (i.e. to within a few hundreds keV) with measurements 9 available for the yrast and /?vibrationnal SD bands of 238 U.
Figure 5. Potential energy of 2 4 0 P u as a function of axial deformation. Predictions at various temperatures for the pairing field (top panel) and binding energy (bottom panel).
To investigate how the fission barriers are modified with increasing exci-
430
tation energy, axially symmetric potential energy curves have been calculated as functions of temperature. The results obtained for 2 4 0 Pu are shown in Fig. 5. The top panel displays pairing energy. This energy gradually decreases with increasing T, and vanishes for T ~ 1 MeV. For T = 1 MeV, the potential shape (bottom panel) displays a secondary, SD minimum deeper than for T = 0. This is because the pairing energy at SD shape decreases more rapidly than at the top of inner barrier when T increases. As T grows beyond 1 MeV, the shell structure of 2 4 0 Pu gradually disappears, and by T ~ 4 MeV this nucleus gets a spherical shape. At this high excitation energy, the fission barrier of 2 4 0 Pu looks like that of a charged liquid drop. 6
Outlook
Significant progress has recently been achieved in optical model and level density studies for stable and unstable nuclei. These are based on i) self-consistent microscopic calculations for finite nuclei, and ii) mean-field predictions in nuclear matter. The reaction models should benefit from improvements in our understanding of the fission process that are gained through the revival of dedicated microscopic approaches in our laboratory. All the model developments too briefly described in this report are gradually inserted into the new reaction code TALYS developed in collaboration with A.J. Koning (Laboratory NRG, Petten, Netherlands). (*' Present address: Institut de Physique Nucleaire, Orsay, France. References 1. P. Romain and J.P. Delaroche, in Nucleon-Nucleus Optical Model up to 200 MeK(OECD Report, Paris, 1997), p.167. 2. E. Bauge et al, Phys. Rev. C 58, 1118 (1998). 3. F. Marechal et al, Phys. Rev. C 60, 034615 (1999). 4. E. Bauge et al, Nucl. Phys. A 654, 829c (1999). E. Bauge et al, Phys. Rev. C 6 1 , 034306 (2000). 5. J. Decharge and D. Gogny, Phys. Rev. C 2 1 , 1568 (1980). J.F Berger et al, Comput. Phys. Commun. 63, 365 (1990). 6. S. Hilaire et al, in Nuclear Data For Science and Technology (Conference Proceedings, Vol.59, Italian Physical Society, Trieste 1997), p.694. 7. S. Bj0rnholm and J.E. Lynn , Rev. Mod. Phys. 52, 725 (1980). 8. J. Libert et al, Phys. Rev. C 60, 054301 (1999). 9. U. Goerlach et al, Phys. Rev. Lett. 48, 1160 (1982).
MULTISTEP DESCRIPTION OF N U C L E O N P R O D U C T I O N SPECTRA IN NUCLEON-INDUCED REACTIONS AT INTERMEDIATE ENERGY
E. RAMSTROM Uppsala University, Department of Radiation Sciences, S-611 8% Nykoping, Sweden E-mail: Elisabet. Ramstrom©'studsvik. uu. se
Studsvik
H. LENSKE University of Giessen, Department of Theoretical Physics D-35392 Giessen, Germany E-mail: Horst.Lenske&theo.physik.uni-giessen.de H.H. WOLTER University of Munich, Department of Physics D-85748 Garching, Germany E-mail: Hermann. [email protected] The Tamura-Udagawa-Lenske (TUL) formulation of the multistep direct (MSD) reaction approach to the pre-equilibrium processes is reviewed. It takes into account consistently the coherent and the statistical aspects of such processes. We also show calculations of neutron-induced charge exchange reactions at 100 MeV, an energy region of great interest in applications to accelerator-driven systems.
1
Introduction
T h e increased interest in recent years in nuclear reaction models in t h e medium-energy region is due to fundamental nuclear physics questions, b u t also caused by applications such as accelerator-driven techniques and neutron cancer therapy. Most of t h e proposed accelerator-driven systems can be expected t o have hard neutron as well as proton spectra with substantial components of spallation neutrons and protons. T h u s feasibility calculations of such systems, which include items such as neutron and energy balance, t h e radiotoxicity of spallation products, damage and activation, rely critically on well-tested nuclear d a t a in the whole energy region from thermal energy u p t o several hundreds of MeV. Nuclear reaction d a t a for actinides, target materials, structure materials as well as for fission products constitute an i m p o r t a n t p a r t of t h e d a t a needed in these modelling calculations. However, it has been stated t h a t a global calculation scheme for accelerator driven systems should consist of a combination of evaluated d a t a libraries u p t o 150 MeV and in-
431
432 tranuclear cascade codes at higher energies *. Well-benchmarked d a t a files t h a t cover a great p a r t of t h e nuclear reactions u p t o 20 MeV have been constructed earlier in connection with fission and fusion reactor studies. T h e only practical way t o meet t h e need of the hugh amount of still missing intermediate energy nuclear d a t a in t h e energy interval above 20 MeV u p t o 150 MeV is t o rely on theoretical model calculations for most of t h e cross sections. T h u s relevant nuclear reaction model codes have t o b e developed and t o be firmly tested on t h e relatively few existing experimental data. This paper deals in particular with calculations of double-differential cross sections for (n,p) charge-exchange reactions at an incident neutron energy of 100 MeV. 2 2.1
Spectrum calculations Models for pre-equilibrium
reactions
T h e pre-equilibrium reaction mechanism can be looked upon as t h e conceptual link between t h e two extremes of nuclear reaction models, i.e. t h e compound nucleus and t h e direct reaction models. Then, pre-equilibrium models are t h e most appropriate approach t o use for t h e description of nucleon induced reactions at intermediate excitation energies t o t h e continuous p a r t of t h e spectrum. D u e t o t h e huge number of complicated eigenstates involved in continuum transitions, certain statistical assumptions axe incorporated into such models. Q u a n t u m mechanical calculations of multistep processes for nucleon induced pre-equilibrium reactions have primarily been carried out using two different approaches: t h e Feshbach, Kerrnan, and Koonin (FKK) 2 and t h e Tamura, Udagawa, and Lenske (TUL) 3 ' 4 theories. A third approach by Nishioka, Weidenmuller, and Yoshida (NWY) 5 relaxes some of t h e statistical assumptions in F K K and T U L in a way, which is still much debated, and it is certainly more difficult t o implement. T h e main differences and similarities between t h e F K K model, which is by far t h e one most commonly used in applications, and t h e T U L model will b e discussed below. T h e subject of q u a n t u m mechanical pre-equilibrium models was discussed at a workshop in Trento, a s u m m a r y of which appeared in ref. 6 . In t h e F K K model as well as in the T U L model t h e double-differential cross sections are expressed as products of quantities dependent on t h e nuclear s t r u c t u r e and on t h e dynamics of t h e reaction. For t h e first step t h e doubledifferential cross section has t h e same form in b o t h models d2o-W ^ do-CD dEdQ,
^
xy
' '
dil
K
' ''
y
'
433 as a sum over multipolarities A of products of a structure factor S\ (E,0) and a D W B A cross sections da^{E; A)/dQ. In F K K t h e s t r u c t u r e factor is given by p\ (Iplh, E), t h e ph level density (nuclear structure dependent) of multipolarity A a t excitation energy E. In the T U L model S\{Efi) is t h e ground s t a t e response function, describing t h e energy averaged transition strength function t o states of multipolarity A t o final states a t average excitation energy E. In recent applications of T U L ' this has been calculated in an energy averaged Q R P A m e t h o d which has t h e advantage of including collective and non-collective components simultaneously. Collective features strongly affect the response functions u p t o t h e giant resonance region and are thus i m p o r t a n t in pre-equilibrium reactions. T h e D W B A cross section is generally given as
^ r ( E ; \ ) = \<xJ\Fx(r)\Xi>\\
(2)
where \ij a r e distorted waves and F\(r) gives t h e average radial behaviour of t h e transition operator of multipolarity A, which, in principle, also depends on t h e average excitation energy. This form factor is also specified differently in t h e two models. In F K K it is customarily taken as V0exp{—JMT) / r, i.e. as a Yukawa of \x — lfm range with an adjustable strength parameter V0, which has been investigated empirically as a function of target mass and of incident energy 9 ' 1 0 . In T U L two approaches have been used. I n t h e macroscopic approach t h e derivative of t h e real p a r t of t h e optical potential is used, which agrees with t h e phenomenology in low energy collective transitions. In a microscopic approach, which has been applied in t h e present work, t h e m e a n formfactor is specified as
Fx(r) = ST 1 / 2 J dr'TNN
(r - r ' ) < 6xQRPA(r')
>,
(3)
where TJVJV (r ~r') is t h e effective nucleon-nucleon interaction a n d < 6^ (r1) > is t h e microscopic, energy averaged Q R P A transition form factor. For t h e second order MSD cross sections, which are not used here, larger differences appear between t h e two models. In T U L this is given as a folding over strength functions for t h e first and second step and by an averaged twostep D W B A cross section. In F K K t h e last quantity is calculated by a folding of one-step cross sections, which raises fundamental questions of t h e reaction formalism e .
434
56Fe,un.par.states J=1, L=0
Ex=0.5MeV Ex=15.5M8V Ex=30.5MeV
Q
2
4
6
8
4
Radlus(fm)
6
8
Radius(fm)
Figure 1. Calculated transition densities for a couple of natural and unnatural parity states at three different excitation energies.
2.2
Calculations
with the TUL
model
In t h e present work the microscopic version of t h e T U L model for only t h e first step has been applied t o describe experimental d a t a from studies of t h e 56 Fe(n,p) and 9 0 Zr(n,p) reactions at an incident neutron energy of 100 MeV performed at t h e T h e Svedberg Laboratory in Uppsala, Sweden 11 > 12 . T h e d a t a cover angular and excitation energy ranges of 0° — 30° and 0 — 40 MeV, respectively. T h e T U L model should be particularly appropriate t o describe t h e present experimental d a t a , a characteristic property of which is t h a t collective phenomena like giant resonances of different multipolarities are superimposed on an almost structureless background. This indicates t h a t collective and statistical features are competing. Then a statistical approach should retain characteristic structural aspects of t h e nuclei, as is t h e case in t h e T U L model with Q R P A response functions. Also, t h e predictive power of a MSD model is very i m p o r t a n t in applications. Since t h e observables in continuum reactions are heavily averaged, the main information is in t h e magnitude and t h e energy dependence of t h e cross sections. In t h e microscopic version of T U L these are strictly related t o effective interactions in a consistent way using Q R P A and Green function techniques. T h e F K K approach, in contrast, contains adjustable effective parameters, which makes predictions in connection
435
Theta (deg)
Excitation energy (MeV)
Figure 2. Experimental double-differential cross sections of the B6 Fe(n,p) reaction at 100 MeV (filled circles) The solid and dashed lines represent the theoretical cross sections calculated with the microscopic TUL model including and excluding the G T contributions, respectively.
with different applications more uncertain. In Fig. (1) representative microscopic transition densities, i.e. the quantity < 6\® (r') > of Eq. (3), for J = 1,3 and for natural and unnatural parity for several averaged excitation energies are shown. It is to be noted that the radial behaviour of these form factors does not depend strongly on the excitation energy. This was the assumption behind using energy independent phenomenological form factors in previous calculations. It is seen that this assumption is justified reasonably well. These transition densities for Ltransfers up to 6 have been used in the calculations of the cross sections for the 56 Fe(n,p) reaction which are shown as solid lines in Fig. (2). One obtains a reasonable fit to the data for excitation energies up to about 10 MeV except at very low excitation energies and forward angles, where the steep increase can be traced back to the strong contribution of the Gamow-Teller giant resonance, which is seen also in the J = l , L=0, E x =0.5 MeV transition density
436
in Fig. (1). It has been observed in other calculations t h a t t h e Love-Franey effective interaction 1 3 , which has been used here, overestimates this strength due t o a very strong tensor force . T h u s as a first estimate we have eliminated t h e G T contribution and obtain t h e dotted curve in Fig. (2), which improves t h e fit t o t h e d a t a substantially. T h e same trend has also been observed for t h e 9 0 Zr(n,p) d a t a . As shown in t h e figure t h e microscopic T U L model, after t h e elimination of t h e G T contribution, describes rather well t h e low excitation energy p a r t of t h e spectra u p t o about 10 MeV, b u t underpredicts t h e d a t a at higher excitation energies drastically. T h u s there is a need t o better understand additional contributing reaction mechanisms, in particular t h e knockout process. Work is in progress in this field. 3
Conclusions
There is a need for a statistical multistep description of nucleon induced reactions in t h e energy region 100-200 MeV for different applications. Such a model should b e free of any adjustable parameters and strictly related t o known physical parameters and quantities. In this work it has been demonstrated t h a t t h e unified microscopic T U L approach appears promising as such a formalism. References 1. A.J. Koning, J.-P. Delaroche and O. Bersillon, Nucl. Phys. A 4 1 4 , 49 (1998). 2. H. Feshbach, A. K e r m a n and S. Koonin, Ann. Phys. 125, 429 (1980). 3. T. Tamura, T. Udagawa and H. Lenske, Phys. Rev. C 26, 379 (1982). 4. H. Lenske and H.H. Wolter, Nucl. Phys. A 5 3 8 , 483c (1992). 5. H. Nishioka, H.A. Weidenmuller and S. Yoshida, Ann. Phys. 1 8 3 , 166 (1988). 6. M. Chadwick et al, Acta Physica Slovaca 4 9 , 365 (1999). 7. H.Lenske et al, Proc. Int. Conf. on Nuclear D a t a for Science and Technology, Trieste, Italy, May 19-24, 1997, ed. G.Reffo et al, Societa Italiana di Fisica, Atti di Conferenze, Vol.59, 231 (1997). 8. E. R a m s t r o m et al, ibid. 241. 9. W.A. Richter et al, Phys. Rev. C 4 6 , 1030 (1992). 10. Y. W a t a n a b e et al, Phys. Rev. C 5 1 , 1891 (1995). 11. T. Ronnqvist et al, Nucl. Phys. A 5 6 3 , 225 (1993). 12. H. Conde et al, Nucl. Phys. A 5 4 5 , 785 (1992). 13. W . G . Love and M.A. Franey, Phys. Rev. C 3 1 , 488 (1985).
HADRON CANCER THERAPY: ROLE OF N U C L E A R R E A C T I O N S
M. B . C H A D W I C K University of California, Theoretical Division, E-mail:
Los Alamos National Laboratory, Los Alamos, NM 87545, USA [email protected]
Recently it has become feasible to calculate energy deposition and particle transport in the body by proton and neutron radiotherapy beams, using Monte Carlo transport methods. A number of advances have made this possible, including dramatic increases in computer speeds, a better understanding of the microscopic nuclear reaction cross sections, and the development of methods to model the characteristics of the radiation emerging from the accelerator treatment unit. This paper describes the nuclear reaction mechanisms involved, and how the cross sections have been evaluated from theory and experiment, for use in computer simulations of radiation therapy. The simulations will allow the dose delivered to a tumor to be optimized, whilst minimizing the dose given to nearby organs at risk.
1
Introduction
A number of research programs have been initiated whose aim is to accurately simulate the nuclear collisions and radiation transport involved in hadron therapy. The nucleon energy range below a few-hundred MeV is crucial for these studies. Proton therapy is typically performed with energies in the 60250 MeV range, and fast neutron therapy utilizes energies up to about 70 MeV. The evaluation of proton and neutron interaction cross sections in this energy region requires particular care - the energies are too low for intranuclear cascade model assumptions to hold, and instead, nuclear reaction models that include more details of the nuclear structure properties should be applied. A further difficulty is the paucity of experimental data to test and validate the calculations. The present paper focuses on a description of models for direct, preequilibrium, and Hauser-Feshbach nuclear reaction mechanisms, and their use in producing cross section databases for radiation transport simulations. The need for accurate nuclear reaction cross sections is greatest for fast neutron therapy. This is because neutrons interact with matter only through the nuclear force, and the energy deposition and transport depend sensitively on nuclear cross sections and emission spectra. Nuclear reactions are important to a lesser extent for protons as these also have electromagnetic interactions, and perhaps their greatest impact is due to neutron production processes which can influence absorbed dose distributions, and which need to 437
438
be understood for shielding. To respond to these needs, a project to determine nuclear reaction cross sections up to 150 MeV for neutrons, and 250 MeV for protons, making use of advanced model calculations and measurements, has been underway at Los Alamos 1>2. The cross sections are represented in the ENDF format in evaluated nuclear data files for H, Li, C, N, 0 , Al, Si, P, Ca, Cr, Fe, Ni, Cu, Nb, W, Hg, Pb, and Bi, the suite of evaluations being known as the "LA150 Library". This work has been documented in an International Commission on Radiation Units and Measurements (ICRU) report that has been recently issued 3 . Various laboratories have begun to develop radiation transport codes that can utilize accurate nuclear cross sections in evaluated data libraries that extend up to 150 - 250 MeV. At the Lawrence Livermore National Laboratory, the Peregrine code 4 is being developed specifically for cancer radiotherapy applications. Its main focus is on conventional photon therapy, but preliminary capabilities have also been developed for neutron and proton therapy. At the Los Alamos National Laboratory, the MCNPX transport code 5 can be used for a variety of transport applications. 2
Nuclear Model Calculations for Medical Applications
Nuclear reaction calculations have played an important role in determining reaction cross sections for hadron radiotherapy. A variety of codes, implementing different physics models, have been used in the past (e.g. intranuclear cascade calculations using Brenner and PraePs code 6 , and the preequilibrium and Hauser-Feshbach calculations undertaken by the present author using the GNASH code 7 ) . Below, an overview of the different reaction mechanisms involved is given. The total, elastic, total nonelastic, and inelastic scattering cross sections to low-lying nuclear states were determined through optical model analyses, which are also needed for generating transmission coefficients and wavefunctions in the equilibrium and preequilibrium calculations. Elastic scattering processes are important because elastic scattering frequently constitutes a significant fraction of the scattering, and the scattered particle's energy and angular distribution must be known to describe the transport through matter. In addition, the recoil energy of the target nucleus contributes to the kerma (and absorbed dose). The Feshbach-Kerman-Koonin quantum mechanical theory 8 , and the semiclassical exciton model, were the basis of calculations of preequilibrium nucleon emission in which the interaction of a projectile nucleon with a target nucleus is modeled as taking place through a number of stages of increasing
439
complexity 9 . Initially, the projectile interacts with a nucleon within the nucleus, exciting a particle-hole pair. The excited nucleons may then undergo further interactions until all the energy brought in by the projectile is shared amongst the target nucleons in an equilibrated state. Particles may also be emitted in the early stages of the reaction. These preequilibrium secondary particles typically have high energy and a forward-peaked angular distribution. After the preequilibrium phase of the reaction the residual nucleus, which is usually left in an excited state, decays by sequential equilibrium particle or gamma-ray emission, calculated with the Hauser-Feshbach theory. 2.1
Neutrons
In figure 1 (left) an illustrative example is provided, for the angle-integrated emission spectrum of protons following 60 MeV neutron bombardment on oxygen. The calculation, shown as a solid line, is compared with measurements by Subramanian et al. 10 and by Benck et al. n . The dashed line shows the intranuclear cascade results from Brenner and Prael 6 . The calculated solid line contains contributions from a number of different emission mechanisms: the increase at low emission energies is due to compound nucleus equilibrium decay processes; and the higher energy contribution to the spectrum, extending from about 10 MeV to 50 MeV, is due to preequilibrium reactions. The reader is referred to Refs. 1,s for numerous additional comparisons. Kerma, an acronym for "kinetic energy released in matter", is an important concept in neutron dosimetry. Since the kerma coefficient can be calculated from the product of the charged-particle production cross sections and their average energies, it represents the interface between microscopic nuclear reaction cross sections, and macroscopic calculations of energy deposition. Recommended total kerma coefficients for various biologically-important elements, as well as elements present in accelerator collimeter structures, are compared against measurements extensively in Refs. 2 ' 3 , and the agreement was found to be good. Figure 1 (right) shows on a logarithmic scale the total kerma coefficient for ICRU-muscle up to 150 MeV, and contributions from individual elements comprising ICRU-muscle. The hydrogen kerma coefficient is seen to play a crucial role, with the contribution from oxygen becoming dominant at the highest energies. 2.2
Protons
In proton therapy, nuclear reactions result in protons being removed from the primary beam. Reaction products include secondary protons, neutrons, photons, and heavier recoils, some of which deposit energy outside the path
440
60MeV
Elemental kerma contributions in ICRU-muscle
6
0(n,xp)
Equilibrium emission S \ A
v« $**iiil*
Preequibrium i^~-~-"' emission
1 A
10 20 30 40 50 Proton emission energy, E /MeV
Neutron energy, En / MeV
Figure 1. (a.)left: The angle-integrated emission spectrum of protons from 60 MeV neutrons incident upon oxygen. The full circles indicate the data of Subramanian et al. 1 0 , and the crosses indicate these same measured values but based upon a more accurate angleintegration procedure 1 2 . The triangles are the data of Benck et al. n . The solid curve represents GNASH calculations 3 ' 1 2 , and the dashed line shows calculations by Brenner and Prael 6 ';(b)right: The total kerma coefficient for ICRU-muscle up to 150 MeV, together with contributions from individual elements comprising ICRU-muscle 2 | 3 .
of primary photons. Neutrons are particularly troublesome as they penetrate large distances and produce secondary heavy charged particles with enhanced biological effect, thereby complicating dosimetric and clinical results. Even more problematic are secondary neutrons generated by primary protons striking beam modification devices upstream of the patient. These neutrons pose a significant shielding problem and illuminate large portions of the patient outside the treatment volume. One of the most important quantities is the proton total nonelastic cross section, since this governs the rate at which protons are removed from the primary therapy beam. Figure 2 (left) shows the evaluated proton nonelastic cross sections, based upon optical model calculations, for oxygen up to 300 MeV. This result is seen to be in good agreement with measured data. Figure 2 (right) shows an illustrative example of our calculated 200 MeV C(p,xp) proton emission spectra compared with measurements recently taken at the National Accelerator Center 13 , for data at various angles. There is qualitative agreement between the measured preequilibrium data and the calculations, though significant quantitative discrepancies are evident, especially at the backward angles. This probably reflects the difficulties inherent in applying statistical preequilibrium and compound models for such light nuclei. A particularly interesting application of nuclear reaction physics is the
441
200 MeV
C(p,xp)
a Measurement — ONASH calculation
«
4
"S 10
100 200 incident proton energy, El MeV
50 100 150 Proton emission energy, E (MeV)
Figure 2. (a.)left: Evaluated proton total non-elastic cross sections compared with data 14 ;(b)right: Calculated C(p,xp) emission spectra compared with NAC data 1 3
proposed use of Positron-Electron Tomography (PET) to trace the location of the Bragg-peak in real time, to ensure that the proton therapy beam is depositing its maximum energy at the intended treatment volume 15 ' 16 . Radionuclides that are beta-emitters (created in proton-nucleus collisions) produce positrons that quickly fall into an orbit with an electron, producing a positronium state that subsequently annihilates to produce two back-to-back gamma-rays. The detection of these gamma-rays in coincidence allows the location of the Bragg-peak to be inferred. This is because the excitation function for the production of /3 + emitters peaks at relatively low proton energies (e.g. the 1 6 0(p, a) 1 3 N cross section peaks in the 8-15 MeV region), near the range of the primary protons. Further details, and a comparison between the calculated and measured p+O excitation functions for the production of radionuclides, are given in the ICRU report 3 . References 1. M. B. Chadwick, P. G. Young, S. Chiba, S. Frankle, G. M. Hale, H. G. Hughes, A. J. Koning, R. C. Little, R. E. MacFarlane, R. E. Prael, and L. S. Waters. Cross section evaluations to 150 MeV for accelerator-driven systems and implementation in MCNPX. Nucl. Sci. Eng., 131:293-328, 1999. 2. M. B. Chadwick, H. H. Barschall, R. S. Caswell, P. M. DeLuca, G. M. Hale, D. T. L. Jones, R. E. MacFarlane, J. P. Meulders, H. Schuhmacher, U. J. Schrewe, A. Wambersie, and P. G. Young. A consistent set of
442
3.
4.
5.
6. 7. 8.
9. 10. 11.
12. 13.
14. 15.
16.
neutron kerma coefficients from thermal to 150 MeV for biologically important materials. Med. Phys., 26:974-991, 1999. ICRU Report 63. Nuclear Data for Neutron and Proton Radiotherapy and for Radiation Protection. International Commission on Radiation Units and Measurements, Bethesda, MD, 2000. C Hartmann Siantar et al.. In Proc. of the International Conference on Mathematics and Computations, Reactor Physics, and Environmental Analyses, pages 857-865. Portland, Oregon, April 30 - May 4, 1995, American Nuclear Society. H. G. Hughes et al.. In J. M. Aragones, editor, Proc. of the Mathematics and Computation, Reactor Physics and Environmental Analysis in Nuclear Applications, page 939. Madrid, Spain, September 27-30, 1999, Senda Editorial, S. A., Madrid. D. J. Brenner and R. E. Prael. Atomic Data and Nuclear Data Tables, 41:71-130, 1989. P. G. Young, E. D. Arthur, and M. B. Chadwick. Technical Report LA-12343-MS, Los Alamos National Laboratory, Los Alamos, NM, 1992. H. Feshbach, A. Kerman, and S. Koonin. The statistical theory of multistep compound and direct reactions. Ann. Phys. (N. Y.), 125:429-476, 1980. E. Gadioli and P. E. Hodgson. Pre-Equilibrium Nuclear Reactions. Oxford University Press, Oxford, UK, 1992. T. S. Subramanian et al. Phys. Rev. C, 34:1580-1587, 1986. S. Benck, I. Slypen, J. P. Meulders, and V. Corcalciuc. Experimental double-differential cross sections and derived kerma factors for oxygen at incident neutron energies from reaction thresholds to 65 MeV. Phys. Med. Biol., 43:3427-3447, 1998. M. B. Chadwick and P. G. Young. Nucl. Sci. Eng., 123:1-16, 1996. M. B. Chadwick, D. T. L. Jones, G. J. Arendse, A. A. Cowley, W. A. Richter, J. J. Lawrie, R. T. Newman, J. V. Pilcher, F. D. Smit, G. F. Steyn, J. W. Koen, and J. A. Stander. Nuclear interaction cross sections for proton therapy. Nucl. Phys. A, 654:1051c-1057c, 1999. R. F. Carlson. Atomic and Nucl. Data Tables, 63:93-116, 1996. S. M. Qaim. Radioactivity in medicine: Achievements, perspectives, and role of nuclear data. In G. Reffo, editor, Proc. of the International Conference on Nuclear Data for Science and Technology, pages 31-37. Trieste, Italy, May 18-24, 1997, Societa Italiana di Fisica, Bologna, Italy. D. W. Litzenberg. Online monitoring and PET imaging of the positronemitting activity created in tissue by proton radiotherapy beams. Medical Physics, 25:254-254, 1998.
ACCELERATOR-BASED SOURCES OF EPITHERMAL NEUTRONS FOR BNCT
E. BISCEGLE, P. COLANGELO, N. COLONNA, V. VARIALE htituto Nazionale Fisica Nucleare, V. Amendola 173, 70126 Bari, Italy E-mail: [email protected] P. SANTORELLI Dip. Fisica, Universita Federico II, 80125 Napoli,
Italy
The treatment of deep-seated tumors with BNCT requires high-intensity neutron beams in the epithermal energy region (1 eV<E„<10 keV). Therapeutic beams with optimal spectral features could be produced with high-current accelerators, through a suitable neutron-producing reaction. We report here on the results of a systematic investigation of different low-energy (p,n) and (d,n) reactions that could reveal useful for the setup of accelerator-based neutron sources for BNCT.
1
Introduction
Boron Neutron Capture Therapy (BNCT), is a promising modality for the treatment of some malignant tumors, such as Glioblastoma Multiforme, that are currently difficult to treat with ordinary methods (1). This new modality relies on the cellular damage induced by high LET a-particles produced in the neutroncapture reaction by 10B. To obtain a therapeutic effect, a suitable level of 10B has to be selectively accumulated in the tumor tissues, and a neutron beam of adequate energy and intensity has to be used for irradiation. While clinical trials are currently being performed at nuclear reactors, accelerator-based neutron sources could be more conveniently used in the future to produce epithermal neutron beams with energy and intensity needed for the therapy. Such sources would present fewer safety problems, lower construction and maintenance costs relative to reactors, and could be installed in many medical centers, located in metropolitan areas. In this paper, we report on a study of different neutron-producing reactions that could be used to produce epithermal neutron beams for BNCT, in conjunction with high-current accelerators. The main features of the reactions, and the need on the primary particle beam, are presented and discussed, in particular with regard to the potential use of such sources in hospital-based BNCT facilities. 2
Energy and intensity of therapeutic neutron beams
The spectral features and the intensity of the neutron beams needed for the treatment of deep-seated tumors, in particular Glioblastoma Multiforme, can be 443
444
determined by means of Monte Carlo simulations of the dose produced in the therapy. To this aim, simulations have been performed for a realistic head model using the package Geant/Micap for neutron transport (2). A tumor region located at 5 cm depth in the brain was assumed in the simulations. The standard ICRU composition of the brain was used, with an addition of 10B in concentrations of 10 and 43 parts-per-million in healthy and tumor tissues respectively. Monoenergetic neutron beams were generated and followed through the head model. To determine the optimal neutron energy we have analyzed the therapeutic gain, defined as the ratio between the average dose to the tumor region and the maximum dose to the healthy tissues. As evident in fig. la), the therapeutic gain as a function of neutron energy displays a pronounced peak around few keV neutron energy. Lower energy neutrons do not penetrate deep enough towards the tumor, while high energy neutrons (E„>20 keV) are responsible for a large dose to the head surface, due to recoiling protons. An optimal neutron beam would therefore be a monoenergetic one, with energy centered around the few keV region. Panel b) in the figure shows the depth-dose profile for neutrons in this energy region. It is particularly important, as evident from the behavior of the therapeutic gain, to minimize the high-energy component, which is responsible for degradation of the beam quality. 2.5 2.25 2 c 'o 1.75 O 1.5 "3 1.25 » a. 1
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Depth in brain (cm)
Figure 1. Panel a): Therapeutic gain as a function of neutron energy for a tumor region located at 5 cm. Panel b): Depth-dose distribution for a monoenergetic neutron beam of 4 keV energy. The curves show the contribution of the different neutron-induced reactions in the tissues.
3
Neutron-producing reactions
Neutron beams with suitable spectral features and intensity can be produced with accelerators, through a suitable proton- or deuteron-induced
445
reaction. The primary beam current needed is, in most cases, of the order of several milliamps. To meet medical requirements and to minimize the demands on the accelerator, in particular the beam current, one has to choose reactions characterized by large yields of low-energy neutrons (En3 MeV, are being considered for use in the first generation of accelerator-based neutron sources for BNCT (3). Both reactions produce neutrons with adequate energy and yield, but their use is complicated by the low melting point of the Li target and the production of the radioactive 7Be residue for the Li(p,n) reaction, and the higher proton energy and poorer neutron beam quality for the Be(p,n). An approach potentially useful for accelerator-based neutron sources, is represented by (d,n) reactions at low energy. As most of these reactions present a positive Q-value, deuteron beams of energy as low as 1 MeV could be used, which would require a relatively simple and inexpensive accelerator. To investigate this possibility, we have performed a systematic study of the yield, energy and angular distribution of neutrons emitted in several low-energy deuteron-induced reactions (4). The measurements were performed at the 88" Cyclotron of Lawrence Berkeley National Laboratory, USA, in collaboration with groups of the Nuclear Science and Life Science Divisions of LBNL. Neutrons were detected with an array of 5 liquid scintillator cells covering between 15 and 150 degrees. The neutron energy was reconstructed by their time-of-flight relative to the cyclotron Radio-Frequency. A threshold of 100 keV on the neutron energy was achieved in the measurements. Among the different targets investigated at deuteron energies Ed<2 MeV, only 9 Be and 13C present a yield sufficiently high to be of interest for the production of epithermal neutrons for BNCT, with the Be target producing almost a factor of 2 more neutrons than the 13C one. Figure 2 shows the energy distribution (and TOF spectrum) for both targets. Because of the large contamination of high energy neutrons, the 9Be(d,n) reaction does not appear appropriate to produce epithermal neutron beams with the necessary spectral purity for BNCT. On the contrary, the relatively large yield and low contamination of high energy neutrons make the 13 C(d,n)14N reaction at Ed=1.5 MeV potentially interesting for an hospital-based facility, thanks also to the good thermal and mechanical properties of the Carbon target and the low-energy of the primary beam. Simulations performed with a beam shaping assembly made of a moderator of LiF 25 cm thick, a reflector of A1203, a delimiter of lithiated polyethilene and a 6Li layer for thermal neutron filtering, indicate that the current required for a treatment time of 15 minutes for this reaction is around 100 mA, a value within reach of the high-intensity accelerator technology currently being developed. Finally, a class of suitable neutron sources is represented by the nearthreshold reactions. In this case, the low neutron yield is compensated by the very small energy of the emitted neutrons, that can be easily and efficiently moderated to the therapeutic requirements. The use of the 7Li(p,n) reaction at 1.95 MeV, with
446
a beam shaping assembly made of a moderator of BeO 8 cm thick and a Pb reflector, would require an estimated proton beam current of 5 mA. Such an intense beam could become feasible with commercial accelerators or with the highcurrent RFQ's like the one presently being developed at the INFN Laboratori Nazionali Legnaro within the TRASCO project. ,„8 1 \J
"C(d,n)"N,E,= 1.5MeV C
1
1
1
1
1
FT"'
1
TT-T
M
10'
rc
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T—i—i—|—i—i t I
1
2
4 E„ (MeV)
6
|—rj-j
0
L
0
|
1(X>
ToF (nsec)
0
2
4
6
E. (MeV)
Figure 2. Efficiency-corrected energy spectrum and Time-of-Flight distributions (inside panel) of neutron emitted in the 13C(d,n)14N (panel a) and 9Be(d,n)10B reactions (panel b), at incident energy of 1.5 MeV.
In conclusion, among the different reactions studied, the 13C(d,n) and the near-threshold Li(p,n) reactions represent, in our opinion, the most interesting solutions for an hospital-based BNCT facilities, because of the relatively low cost and small size of the accelerators needed for the therapy. References 1. Barth R.F., Soloway A.H. and Brugger R.M., Boron Neutron Capture Therapy of brain tumors -Past History, current status and future potential, Cancer Investigation 14 (1996) pp. 534-550. 2. Bisceglie E., Colangelo P., Colonna N., Santorelli P., and Variale V., On the optimal energy of epithermal neutron beams for BNCT, Phys. Med. Biol., 45 (2000) 49-58. 3. Wheeler F.J. et al., Boron Neutron Capture Therapy (BNCT) - Implications of neutron beams and boron compound characteristics, Med. Phys. 26 (1999) 1237-1244. 4. Colonna N., Beaulieu L., Phair L., Wozniak G.J., Moretto L.G., Chu W.T. and Ludewigt B.A., Measurements of low-energy (d,n) reactions for BNCT, Med. Phys., 26 (1999), 793-798.
STUDY OF THE LIGHT ION BEAM FRAGMENTATION IN THICK TISSUE - LIKE MATTER USING TISSUE-LIKE TRACK DETECTOR S.P.TRETYAKOVA AND A.N. GOLOVCHENKO Joint Institute for Nuclear Research, 141980 Dubna, Russia E-mail: [email protected] R. HJC AND J. SKVARC
Josef Stefan Institute, Jamova 39, 1001 Ljubljana, Slovenia E-mail: [email protected]
1
Introduction
Experimental studies of interaction of light accelerated ions (C, N, O, Ne) with elements of human tissue are of particular interest for therapeutic and diagnostic medicine. Besides the evident advantages of light ions (i.e. a perfect depth-dose distribution, enhanced Relative Biological Efficiency and minimal angular scattering in matter) they exhibit the unfavourable effect of fragmentation. Fragmentation reactions have been studied theoretically and experimentally for many years. However there is still a lack of experimental data, especially for light systems and for energy ranges below ~ 100 MeV/n, and their disagreement with theoretical models is sometimes quite large. This work presents some results of investigation of a possibility obtaining information on the nuclear fragmentation with Z > 3 of heavy ion beams (C, O, Ne) in tissue-like materials (water, plexiglas, paraffin) by a CR-39 nuclear track detectors also having a tissue like composition (C12H1807). 2
Experimental method
The experimental procedure of measuring characteristics of the ion beam is based on the automatic counting of the nuclear tracks and measurements of their geometrical parameters (track diameters and length) and is given in [1-5] in details. Here this procedure can be briefly discussed as follows. In the case that the velocity of the fragments is approximately the same as that of the projectiles there is the smooth dependence of the track diameter on the ion charge. Therefore, the resulting track distribution can be easily converted to that of charges with typical resolution
447
448
oz - 0.2 e. Then, on the basis of the dependence of the track diameter on LET (calibration curve) in CR-39, the LET distribution caused by the projectiles and the fragments can be derived at any penetration point of the beam of interest. From the analysis of the elemental fragment distributions as function of depth in tissue-like absorber we can obtain total and partial charge-changing reaction cross sections. The angular spreads of the initial beam, and probably fragments, can be obtained by measuring the incident angles in each the CR-39 foil along the stack with o e N 0.01°. Typical the experimental set up to be irradiated with ions consisted of a stack of tissue equivalent absorber (water, Plexiglas, paraffin) and CR-39 track detectors (400 - 700 |iun thick). After exposures all detector foils were etched in 6-7 N NaOH solution at 70 - 80°C for different times, depending on experimental requirements. 3
Experimental result Typical spectrum of track diameters obtained in experiment with 77MeV/u Ne is given in Fig.l. Similar track distributions can be measured behind each
20
30
40
50
Track diameter (|jm) Figure 1. Distribution of nuclear track in a CR-39 foil situated just behind a 9.5 mm thick water absorber in which a 77 MeV / u 20Ne beam was stopped. Only secondary particles are seen [1].
tissue equivalent absorber located along the beam path. Fig.2 show an example of a LET spectrum of a fragmented 292 MeV / u 19F beam at the exit side of a 2.5 cm thick Plexiglas target. The penetration distance of the beam can then be found by summing up all the thicknesses of absorbers and the CR-39 foils untill primary beam tracks disappear. The longitudinal range straggling can be determined by measuring the lengths of fully etched tracks if all the ions are stopped in one CR-39 foils, or by counting the number of tracks in the last CR-39 foils where the beam is stopped. On the basis of the calibration curve and track diameter distributions measured along the beam path, the complete dose profile have been obtained. In Fig.3 shows an example for the dose profile for a 70 MeV/u lsO beam in the region of the Bragg peak [3]. As example of behaviour of the total charge changing crosssection vs energy, Fig. 4 compiles the experimental results from different sources.
449
10
15
20
25
30
35
LET^in Plexiglas (keV / pm) Figure 2. Experimental LET_ - in Plexiglas distribution. Number denotes the charges of ions [2].
and above mentioned models for the system l2 C + CH2 in the energy interval from - 30 to ~ 1000 MeV/u.
™
22'
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—• experimental points - - • - - - calculated for one 1 s O ion (no fragmentation, no range straggling)
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P e n e t r a t i o n d e p t h in C R - 3 9 (mm) Figure. 3. Relative dose versus penetration depth of the oxygen beam at 67.7 MeV/u in CR-39, which corresponds to the region of Bragg [3].
The obtained results show that the each track technique provides a nearly complete description of the properties of beam intended for radiotherapy and permit to obtain the total picture of and detailed information about the heavy ion interaction with tissue along the range of the primary ion beam and its fragmentations.
450 2250 C + CH, 2000 • -this work • - Schallet ai (1996) A -Fukumura (1999) • - Webberstal(1990) solid line - Sihver el al (1993) dotted line - Tripathi et al (1996)
. o 1750
1500
1250
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S
750
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carbon beam energy (MeV/u) Figure 4. Computation of the experimental and theoretical results for system 12C + CH2 [4,5].
References 1.
Golovchenko A. N., Tretyakova S. P., Tostain C, G, Bimbot R., Clapier F., Kubica B. and Borcea C, CR-39 Charge Resolution and Relative Yields Of Nuclear Fragments produced in Interaction of 77.1 Mev/u. In: Proc.of the II Intern.Workshop "Solid State Nuclear Track Detectors and their Applications Dubna, 24-26 March 1992, Dubna (1993) pp.57-61.
2.
Golovchenko A. N., Skvarc J, Tretyakova S. P., Ilic R., Bimbot R., Clapier F. and R. Freeman. Characterization of a 292 MeV/u 19F beam after passage through a thick Plexiglas target. Nucl. histrum. Meth. B 114 (1996) pp. 221.
3.
Golovchenko A. N., Skvarc J., Slavic S., Ilic R., Freeman R., Pauwels N, Tretyakova S .P. and Bimbot R., Dosimetry of the 40 and 70 MeV/u 160 beams. Radiat. Meas. 28 (1997) pp.455-462.
4.
Golovchenko A. N., Skvarc J., Ilic R., Sihver L., Tretyakova S. P., Schardt D., Tripathi R.K. and J.W. Wilson. Fragmentation of 200 and 244 MeV/u carbon beams in thick tissue-like absorbers. NIM B 159 (1999) pp. 233 - 240.
5.
Golovchenko A. N., Skvarc J.,Yasuda N., Ilic R., Tretyakova S. P., Ogura K. and Mukarami T., Total charge-changing and partial cross section of 110 MeV/u 12C with parafin, Radiat. Meas., (2001) in press.
ANISOTROPY FUNCTIONS FOR PALLADIUM MODEL 200 INTERSTITIAL BRACHYTHERAPY SOURCE ROBERTO CAPOTE 1 , ERNESTO MAINEGRA*, ERNESTO LOPEZ Centro de Estudios Aplicados al Desarrollo Nuclear, Calle 30#502 e/5ta y 7ma,Habana 11400, Cuba Anisotropy function for low energy 103Pd seed model 200 source is examined. Absolute dose rates have been estimated by mean of the EGS4 Monte Carlo Simulation System. DLC136/PHOTX cross section library, water molecular form factors, bound Compton scattering and Doppler broadening of the Compton-scattered photon energy were considered in the calculations. Binding corrections and phantom material selection have been found to have no influence on the anisotropy function. The accuracy of the geometrical source models used for the Monte Carlo calculations was validated against experimental measurements of in-air relative fluence at 100 cm from the source. More detailed knowledge about geometrical design of 103 Pd seed model 200 is needed in order to improve the agreement with experimentally measured in-air fluence. Variations due to differences in the design of Pd seeds' end welds are responsible for large discrepancies between calculation and experiment under 40°. Our results have estimated statistical uncertainties of 0.5%-3.0% at l a level within clinically relevant regions, but could contain systematic uncertainties related to the assumed geometrical details. These systematic uncertainties could be as large as 10% in some cases.
1
Introduction
Anisotropy effects of interstitial brachytherapy sources can not be neglected when small number of sources regularly arranged are used, i.e., in temporary brain implants and ophthalmic plaque applications. Single seeds, especially those with average emission energy below 80 keV, present a marked anisotropy in dose distribution around the longitudinal axis. In this study we present anisotropy function for palladium model 200 seed, comparing them with published experimental and theoretical values. 2
Methods and Materials
2.1
Dose calculation formalism, brachytherapy sources and phantoms
The AAPM Radiation Therapy Committee Task Group No.43 [1] recommends formalism developed by the Interstitial Brachytherapy Collaborative [email protected]; [email protected] [email protected]
451
452
ing Group (ICWG 1990) [2] to predict the two-dimensional dose distribution around cylindrically symmetric sources. Pd brachytherapy seed under study have been extensively described in previously published works [1,3-5]. In this study a cylindrical phantom was used and a brachytherapy source was located in the center of the phantom with its long axis coincident with the phantom central axis. Phantom materials included water and solid water. 2.2
Monte Carlo calculations
Monte Carlo calculations were performed using the EGS4 code system [6]. The photon cross section compilation, DLC-136/PHOTX cross section library [7] contributed by the National Institute of Standards and Technology and implemented for EGS4 use by Sakamoto [8] was employed in the calculations. Bound Compton scattering and Doppler broadening of the Compton-scattered photon energy were considered in the calculations by including the Low-Energy Photon-Scattering expansion for the EGS4 Code [9]. Molecular form factors [10] for coherent scattering in water were also included. Primary energy spectra of source photons are taken from the NUDAT database [11]. The theoretical approach was explained in detail in an earlier published work [5]
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Distance from source center r [cm] Figure 1. Anisotropy function for a 103Pd seed model 200 as function of distance.
2.3
Results
With its lower average photon energy, 103Pd-source exhibits stronger anisotropy effects and a faster dose falloff with distance than 125I sources. Therefore, it is of interest to determine the anisotropy function at very close distances from the source. Anisotropy function values were calculated from the Monte Carlo simulations in liquid and solid water phantoms and plotted in figure 1 as a function of distance for different angles ranging from 0° to 80° and from 0.3 cm up to 5 cm. As
453
can be observed from the similar results for both media, there is practically no influence of phantom material on the anisotropy function for 103Pd photon energies. Although not graphically shown, calculations with and without binding corrections have been also performed finding not influence at all on the anisotropy function. A strong anisotropy and a marked dependence with distance on the longitudinal axis can be observed. This dependence with distance decreases considerably already at 10°. Anisotropy function uncertainties due to statistical fluctuations in computed dose distributions are under 1.0% at angles over 30° and under 2% elsewhere excluding the longitudinal axis. Along this axis uncertainties are somewhat higher, being under 1% below 1cm, under 3% up to 2cm and increasingly higher over this distance. Values for the anisotropy function obtained by us are available from the authors (we do not include them because of limited space). We compared our results with anisotropy values obtained from the experimental results of ChiuTsao and Anderson [3] and the AAPM TG 43 experimental reference data of Nath [1,4]. These two measurements were performed in solid water using LiF thermoluminescent dosimeters (TLD). In general there is poor agreement of our results with the experimental values. However, the data of Chiu-Tsao and Anderson are in better agreement with our results at larger distances from the source ( r > 3 cm ) and less scattered, and they reproduce the physical fact that the dose on the longitudinal axis near the source is higher than that on the transverse axis. Recently, Weaver [12] reported anisotropy functions for 103Pd-source model 200. Data were generated through a two-step process, determining first the source intrinsic radiation emission pattern from in-air measurements at 100 cm from source center and then using these data as input to Monte Carlo calculations of the fluence distribution in water. 20
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454
Data from this work were also included in figure 1 for comparison. The largest discrepancy with our values is found along the longitudinal source axis where Weaver's values are lower than ours and do not reproduce the physical fact that the dose on the longitudinal axis near the source is higher than that on the transverse axis. At 10° agreement is excellent between both data sets except near the source and then at higher angles Weaver's values become systematically higher than ours. In order to understand the observed discrepancies with the results of Weaver we modeled his experimental setup [12]. We found necessary to model the 103Pd electroplated on graphite cylinders as a 0.3 \im layer. The layer thickness was deduced by fitting the Weaver experimental results. The left panel in figure 2 shows a comparison of the Monte Carlo simulation with values from the best representation curve of the experimental fluence. Although the minimum in both data sets is located almost at the same position, our values are somewhat lower than Weaver's. The maximum in Weaver's data is observed at 80° and in our calculation at 87 °. Irregularities in source design that were reported by Weaver [12] and the lack of an accurate source description are likely to be the reasons for remaining discrepancies and set the limits for the prediction capability of Monte Carlo generated data. Acknowledgments We are deeply indebted to Drs. M.C. Schell and D.W.O. Rogers for useful comments and considerable support in accessing key information for this work. References 1. Nath R., Anderson L.L., Luxton G., Weaver K.A., Williamson J.F., Meigooni A.S., Med. Phys. 22(1995) 209. 2. "Interstitial Brachy therapy: Physical, Biological, and Clinical Considerations", ed. L.L. Anderson, R. Nath and K.A. Weaver, Raven, NY, 1990, p.21 3. Chiu-Tsao S.T. and Anderson L.L., Med. Phys. 18(1991) 44. 4. Nath R., Meigooni A.S., Muench P. and Melillo A., Med. Phys. 20(1993) 1465 5. Mainegra E., Capote R., Lopez E , Phys.Med.Biol. 43(1998) 1557. 6. Nelson W.R., Hirayama H. and Rogers D.W.O., "The EGS4 code system, Version 4", Stanford Linear Accelerator Center Report SLAC-265, USA, 1985 7. RSIC Data Package DLC-136/PHOTOX, "Photon Interaction Cross Section Library" contributed by National Institute of Standards and Technology, 1993. 8. Sakamoto Y. 1993, Proc. Third EGS4 User's Meeting in Japan, KEK Proceedings 93-15, National Institute of High Energy Physics, Japan 9. Namito Y„ Ban S. and Hirayama H., Nucl.Instr.& Meth. A349(1994), 489; "LSCAT: Low-energy photon scattering expansion for the EGS4 code", KEK Internal Report 95-10, National Institute for High Energy Physics, Japan 10. Morin L.R.M., J. Phys. Chem. Ref Data. 11(1982) 1091. 11. Brookhaven National Laboratory, Upton, N.Y., USA 1996, NUDAT database 12. Weaver K.A., Med. Phys. 25(1998) 2271.
PRODUCTION OF RADIOPHARMACEUTICALS BASED ON THE ™STL AND 2 n AT FOR MYOCARDIUM DIAGNOSTIC AND CANCER THERAPY O.V.FOTINA, D.O.EREMENKO, V.O.KORDYUKEVICH, S.YU.PLATONOV, E.I.SIROTIN1N, A.V.TULTAEV, O.A.YUMTNOV Institute of Nuclear Physics, Moscow State University, 119899 Moscow, Russia E-mail: yuminov(d}p5-lnr.npi. msu.su
It is coasidered the possibilities of use of radiopharmaceuticals based on the 199T1 and aemitting 2ll At isotopes for the perfusion scintigraphy of myocardium and cancer therapy respectively.
The specific interest to the 199T1 and 2ll At isotopes is connected with medical use of radiopharmaceuticals based on the Tl and At nuclides for the perfusion scintigraphy of the myocardium and cancer therapy. The yields of the 200' '"• 198m *n and 2,1 At isotopes were measured for the 197 Au+a and 209Bi+a reactions respectevly on the thick gold and bismuth targets in the beam energy range from 20 to 30 MeV with the U-120 cyclotron of Moscow State University. The results of experiments and our theoretical estimations performed for the thick targets are presented on Fig. 1 The analysis of the isotope yields was performed in the frames of the statistical theory of nuclear reactions. The result of the experimental date analisis demonstrates that the 30 MeV otparticles energy is an optimal for production of the 199T1 and 211At isotopes with minimal yield of other longlived radioactive isotopes. At this energy the yields of the 199T1 and 2ll At isotopes are approximately 1 mKu/(hr • mkA). The estimations of radiation dose after intravenous injection of the Tl and At radiopharmaceuticals were obtained using a human body phantom by the M1RD method [1]. Our own data on the Residence Times for the radiopharmaceuticals with the 201 Tl, 199T1,99mTc isotopes were used for the estimations of Radiation Dose for the reference adult. Comparative analysis of the radiation effects on different organs and whole human body is presented on Fig 2. As it follows from date of Fig 2, after injection of 74 MBq "Thallium chloride, ,99 T1" the dose exposure is minimal compared to other radiopharmaceutical preparations of a similar action.
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457
The data on the pharmacokinetics of 211At and 123131J in the physiological solutions was obtained using 120 rats (160-180 grams of weight). Preliminary results demonstrate that more than 30% of injected activity of2" At accumulated eventually in the thyroid gland as in the case of J. The radiation dose predictions for the reference adult are presented on Fig. 3. They were obtained using the biokinetic test data by the method discribed in [2]. io -
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458
We can emphasise too different dose exposures for male and female which is due to by the differences in the pharmacokinetics for male and female. Thereby this data indicates the possibility to use At therapy to treat some kinds of the thyroid gland cancer. In conclusion, it seems that the At radiopharmaceuticals are more preferable for the thyroid cancer therapy of female, while the J radiopharmaceuticals are more convinient for the male cancer therapy. In addition, it should be pointed out that the same doze exposure on the thyroid might be achivied by injection of the At physiological solution with the activity of a few order less than in the cases of the J physiological solution. This work was supported in part by the Russian Foundation of Basic Research (grant N°. 98-04-48840-a). References 1. Cristy M., Eckerman K.F., Specific absorbed fractions of energy at various ages from internal photon sources. Oak Ridge National Laboratory(ORNL), ORNL/TM-8381, 1987, VI-V7. 2. DO. Eremenko et al., Experimental study of '"Tl-thallium chloride radiopharmaceutical for perfusion Scintigraphy of the myocardium. Med. Radiology and Radiation Protection, 44 (1999) pp.41-46.
HORIZONTAL COMPILATIONS OF N U C L E A R DATA. Ill
Z.N.SOROKO, S.I.SUKHORUCHKIN, D.S.SUKHORUCHKIN Petersburg Nuclear Physics Institute, Gatchina, 188300, RUSSIA E-mail: [email protected] Compilations of resonance parameters for all nuclei (horizontal-type compilations) derived from study of reactions with neutrons and charge particles are described. Systematic trends in experimental d a t a due to the differences in energy scales of spectrometers were found after comparison of d a t a for many nuclei. The combination of d a t a from horizontal-type compilations with d a t a from the well-known file ENSDF expands the usefulness of these d a t a files.
Resonance parameters seen in reactions with neutrons, protons, deuterons, tritons, 3 He and a-particles correspond to properties of highly excited states of compound nuclei. The most of them are not included in Evaluated Nuclear Structure Data File (ENSDF) [1]. Data contained in resonance parameters files are important for evaluation of characteristics of statistical models, for study of door-way structure effects in excitations, in estimation of astrophysical reaction rates and evaluation of parameters for reactor design calculations. Study of the collected at PNPI neutron resonance file NRF [2,3] and proton resonance file PRF [3] allowed to show the usefulness of a combined study of data from NRF, PRF and ENSDF [4,5]. The combination of data from NRF, PRF and ENSDF was used for a study of spacing distributions between all excited states of a given nucleus [3,6]. For example, in files ENSDF+NRF for 41 Ca and ENSDF+PRF for 4 0 Ca the presence of stable intervals D=3353 keV and D=3355 keV equal to 0+ excitation in 4 0 Ca (E*=3353 keV) was observed by sharp maxima (widths about 3-7 keV) in spacing distributions [3]. In this paper we continue [3,4] description of horizontal compilations and discuss the files containing data measured with high energy resolution. To preserve the quality of measured data during the production of the combined files ENSDF+NRF and ENSDF+PRF the determination of absolute energies of excited states should be performed with the accuracy comparable with resolution achieved. Comparison of neutron resonance data for several isotopes [5,7] has shown systematic differences in positions of the same resonances measured with different time-of-flight spectrometers (see Fig.l). To avoid the double-counting of resonances [2] their positions should be corrected. The method of correction uses the fact that the positions of resonances of Cr, Fe and Ni measured at long flight paths of ORELA and GELINA spectrometers coincide within 3 x l 0 - 4 and could be taken as energy standards [2]. Positions of resonances measured with other spectrometers could be used to 459
460
derive the slope of the shift needed to correct resonance positions in earlier data measured with worse accuracy in energy scales. Differences of E0 in data for 52 Cr and 104 Pd obtained in 70-ties at 40m flight path of ORELA and in corresponding data from GELINA are shown in Fig.l. Proximity of the estimated linear slopes A E 0 / E 0 in these data - 1.94xl0 - 3 and 1.96xl0~ 3 - could be used to perform a linear correction for positions of all resonances obtained in the measurements with 40m flight path of ORELA. >
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I
,
i . . . i . . . t . . , i . . .
I
200 250
10
E0 (keV)
(keV)
F i g . l . Differences in resonance positions o b t a i n e d a t O R E L A (40m flight p a t h ) a n d a t G E L I N A for 5 2 C r (left) a n d 1 0 4 P d (right).
The updating of NRF by inclusion of recently published resonance parameters and resonance capture 7-ray spectra is now in progress and a part of new data (numbers of data lines) is presented in Table 1. Table 1: N u m b e r s of recently m e a s u r e d n e u t r o n r e s o n a n c e s ( G E L - G E L I N A , O R X - O R E L A , T I T - T o k y o Inst, of Technology, R P I - R e n s s e l a e r P o l y t e c h . I n s t . ) . A
Z
A
n
14N
Z
ORL
16
O 20 19 F 5 23 Na ORL 25 Mg 27 A1 ORL 28 Si 32 S 9 37 Ar 4
50
A
1
93
n
A
7,
113
n
N b 31 GEL C d 2 3 Cr 234 GEL Cr 192 ORL " T c 659 GEL 1 1 5 I n 36 26 116 Fe TIT GEL 1 0 3 R h 160 Sn 211 ORL 104 58 N i 9 P d 275 GEL 1 2 0 Sn 99 ORL 82 Kr GEL 1 0 6 P d 281 GEL 1 3 7 B a 143 ORL 84 Kr GEL 1 0 8 P d 220 GEL 1 3 3 Cs 50 86 Kr GEL 1 1 0 P d 2 1 3 G E L 1 3 9 La 88 142 Sr 126 ORL 1 0 7 Ag 53 N d 4 4 ORL 91 144 Z r 7 R P I 1 0 9 Ag 78 N d 8 9 ORL 52
TIT GEL ORL GEL ORL TIT
n
A
Z
165
Ho Er 167 Er 169 Tm 166
n
A
8 1 7 5
208p b
182 W
4
183 W
1 4
184 W
2
186 W
2
206p b
Z
n
209gj 232
T h 16 U 282 235 u 4 g 23 «U 37 237 N p 28 238 Np 8 239 P u 128 233
In proton resonance spectroscopy a very high resolution (better than 1 keV) was achieved by the stabilization of beam energy and the use of thin targets. The calibration of beam energy becomes important to include the results of measurements with good resolution into the total spectrum of excited states of
461
the compound nucleus. In Table 2 the comparison of experimental results and accepted values in PRF and ENSDF for 55 Co are given to show the unwanted effect of the shift in the resonance position (E 0 ) which should be avoided when E* are combined into the single file. Adopted E0 in PRF (left) and ENSDF (right) are derived from parameters obtained in independent measurements of 7-ray yield [8,9] and proton scattering [10,11]. Positions of strong resonances, clearly seen in figures of the original papers (boxed), were used to identify resonances and to estimate differences in E 0 (left). In ENSDF a part of these resonances has been doubled because they were considered as different ones (marked by asterisks). The doubling of adopted E* in ENSDF could be seen in the central part of Table 2 where the data from 7-yield [8] and proton scattering [10] are shifted by 6 keV. These shifts are taken into account in PRF where corresponding E* are also calculated. It is shown in the bottom part of Table 2 that for E0 >3465 keV the energy shift in the data unexpectedly disappears. The shift due to different energy scales in measurements of proton scattering [10,11] (shown in the central part of Table 2 [10,11]) is seen also in the left part of Fig.2 where E 0 from data for 57 Co from [12,13] are compared (shift about 7 keV). We performed such comparison of resonance positions and parameters in large sets of data for proton reactions for 15 different target isotopes (from 36 Ar up to 62 Ni) and in many cases differences in E<, of the order of 5-7 keV were observed. In Fig.2 it is shown for 57 Co: systematic shift is the result of the different energy scales in the two scattering measurements [12,13]. 8
%
&
m
Z
Co
A
nil
^S
z
4- =_
••-*
2 z O z z — 2 z~ -1 1 1 1 1 1 — 43.1 3 . 2
.
•* •
*
%..
«"*
•
* r »•. »
1 1 I 1 1 • • • 1 1 • • 1 1 1 1 1 1 1 1 1 1 1 1 1
3.3
3.4-
3.5
3.6
3.7
• •
1 1 1 1 1 1 1 1
3.8
y* -
... 1
3.9 4E0 ( M e V ) Fig.3. Differences in proton resonance positions in S7Co from scattering experiment in [12,13] (circles) and in [11,12] (triangles, left). Independent absolute calibration of excitation energies E* could be performed by studying 7-transitions from individual resonances but for most nuclei with Z>20 such energy calibration was not done. Performance of these measurements will greatly extend known excitation spectra of many nuclei.
462 Table 2: C o m p a r i s o n of m e a s u r e d and a d o p t e d proton r e s o n a n c e s in [10] PRF [8] [9] E o t keV E 0 , keV 2J E 0 , keV E 0 , keV 2J
2305.5 1 -
25
2309.6
5.1
2342.7
4.0
2338.7 2342.7 2349.9 2353.4
3.5
2353.4 2358.4 1 -
2359.3
3277.0 3 "
3277.9 3 ~ 3283.2 3287.6 3294.9
3281.5 (1) 3285.1 3289.0 1,3 3293.5 3 _ 3365.2 3 ~
3370.7 3386.5
3386.5 3
-
3277.9* 40* 3281.5* 1400*
20
3287.6 3 ~ 3294.9 1 -
1400 300
3370.7 t=\
65
5.5
260
6.7
450
5.5 5.7
3392.0 450 3408.4* 3414.1 50
3386.5 5+
3408.4 (3) 3414.1 1 - 3 -
3414.1
40
2304.5* 2309.6 2338.7* 2342.7 2349.9* 2353.4 2359.3* 2362.1
3283.2 3 ~
3392.0 3 ~
3392.0
60
3.4 6.4 3281.4 3 ~ 1400 6.2
2362.7
2362.1
Co.
diff. ENSDF [11] r , eV E 0 , keV 2J r , eV keV E 0 , keV T, eV
2304.5 2309.6
55
3288.3 1 -
500
6.6
" 50
3283.2
20
3287.6 3294.9 3365.2* 3369.5 3386.5
1400 300 65 260
3464.7
3465.0 9+
3466.6
1.7
3465.0
78
3644.3
3644.7 5+
3644.3 5+
1.2
3644.7
195
195
References 1. ENSDF, Nucl. Data Sheets (Acad. Press, N.Y., current issues). 2. S.I.Sukhoruchkin, Z.N.Soroko, V.V.Deriglazov, Landolt Bornstein New Series, v. I/16B, ed. H.Schopper (Springer, 1998. ISBN 3-540-63277-8). 3. Z.N.Soroko et al., Proc. ISINN-8, JINR E3-2000-192, p.420. 4. S.I.Sukhoruchkin et al., Proc. CGS10, Santa Fe, 1999. AIP 529, p. 307. 5. S.I.Sukhoruchkin, Proc. ISINN-6, JINR E3-98-202, pp.343, 141. 6. Z.N.Soroko et al: Nucl. Phys. A 680, (2001) (in press). 7. Z.N.Soroko et al, Proc. ISINN-5, Dubna, 1997, JINR E3-97-213, p.435. 8. G.U.Din, J.A.Cameron, Phys. Rev. C 40, 577 (1989). 9. R.Hanninen, G.U. Din, Phys. Scr. 22, 439 (1980). 10. D.S.Flynn, E.G.Bilpuch, G.E.Mitchell, Nucl. Phys. A 288, 301 (1977). 11. D.P.Lindstrom et al., Nucl. Phys. A 168, 37 (1971). 12. W.A.Watson III, PhD thesis (Duke University, 1980, unpublished). 13. E.Arai, T.Takahashi, J.Kato: Nucl. Phys. A 324, 63 (1979).
T U N I N G EFFECT IN N U C L E A R DATA
Petersburg
Nuclear
S.I.SUKHORUCHKIN Physics Institute, Gatchina,
188300,
RUSSIA
The presence of stable mass/energy intervals in nuclear excitations and nucleon separation energies with common (for different shells) values is considered as a tuning effect due to the influence of nucleon structure. Several parameters of nuclear tuning effects turned to be proximate to the electromagnetic mass differences of leptons, nucleon and pion and this fact was connected with the discussed in literature correlations in particle masses which include nucleon and its A-excitation as well as lepton and pion (tuning effect in particle masses).
Suggested by S.Devons [1] influence of nucleon structure on fine nuclear effects (tuning effect) was connected in [2-4] with a recent state of Standard Model which will explain all interactions and particle masses and which, according to Y.Nambu [5], at its recent stage of development, could be much benefited by the observed semiempirical correlations in data. Analysis of three blocks of nuclear data containing excitations and binding energies is presented. In the first block which includes Evaluated Nuclear Structure Data File (ENSDF) integer relations (numbers k) in excitations (E*) of near-magic nuclei with Z=27-29 and Z=51 were found (Table 1). A period 85 keV is seen as ground state splitting E*=84.5 keV of near-magic 68 Cu and as stable intervals D in 55 Co and 65 Cu [6,7]. A maximum at E*=e o =1022(2) keV (k=12) in E*distribution of all nuclei with A<150 [2] and stable intervals D=1021(2) keV (k=12) in near-magic 38 Ar [10], D=342 keV in 4 3 Ca (k=4), D=682 keV in 43 Sc, 47 V, 62 Ni (k=8) [8] as well as 511 keV=m e (electron rest mass) at k=6 [2-4] belong to this system. Grouping of E* was noticed at D 0 =1293 keV (mass difference of nucleons m„-m p =1293 keV), at 1/2 D 0 and 3/2 D 0 [6,7]. Stable character of intervals D=4090 keV and D=3355 keV in 4 1 Ca and 40 Ca (close to 4£ o =4088 keV and E*(0+)=3353 keV of 40 Ca) [3] is similar to the stable character of interval D=3576 keV=3|e 0 in several light nuclei (close to E?(0 + )=3576 keV in 18 Ne while E^(0+)=4590 keV is close to 4±e 0 =4599 keV). We should mention the proximity of E*(0+,T=2) = 27595(2) keV in 12 C to 27e 0 =27594 keV and observed in [10] integer relations with a period 85 keV/2= =42.5 keV (and hence with £,,=12x85 keV) in the energies of strong 7-ray transitions in nine nondeformed nuclei (from 46 Sc up to 1 9 8 Au). Data from the second block which contains neutron and charge particles resonance parameters (not included in ENSDF) were used to study fine structure effects in spacing distributions in near-magic nuclei. In Table 2 [7,8] integer relations between intervals seen in neutron resonances are given. Inter463
464
Table 1: Comparison of E* in nuclei with Z=27-29,51 and period 85 keV=e 0 /12.
Nucleus
57Ni
57
E*,keV 768 kx85keV 766
1/2" 1113 1107
Cu 3/2" 5/2" 1028 1022
k
9
13
12
Nucleus
130
3/2" 5/2"
TTT
J
0
Sb 8-5+
E*, keV 0 diff., keV kx85keV
k
54
1/21106 1107
13
Co 0+ 1+ 937 937 11
58
T=l 2+ 1447 1448 17
131
?-
?-
84.7 84.7
341 341 341 4
85 1
Sb 3/2+ 1142 84.4
85 1
59
Ni
Cu
0+ 1+ 848 851 10
T=l 3/2" 2+ (5/2") 1449 339 1448 341 17 4
132
Sb
11/2+ 1226 255.1 255 3
7/2+ 1481 339 341 4
4+ 0
3+
2+ 85.5 426 85.5 340 426 341 426 85 1 4 5
vals of fine-structure are expressed as parts (m) or integers (n) of parameter 9.5 keV=8e'=<5' where 4.75 keV=4e' ( m = l / 2 ) is the directly observed stable spacings and 18.8 keV=16e'=8' (m=2) is the observed grouping of neutron resonance positions [8]. A period e'=1.2 keV ( m = l / 8 ) was estimated in [11] as (a/2w)e0 with a/2ir - dimensionless Scaling Factor close to Electrodynamics Radiative Correction (SFERC) which connects 4e'=4.75 keV with hyperfine structure parameter 5.5 eV=4e". A period 85 keV=£ 0 /12 corresponds to n=9 while intervals 161 keV and D 0 =1293 keV=8x 161 keV correspond to n=17. Introduction of SFERC=a/27T was motivated by small deviations of muon mass and mass splitting of pions from integer numbers of me [3,11]. It was noticed in [11] that two nearly equivalent empirical ratios 1/27x32= = 1 . 1 5 7 x l 0 - 3 and a/27r=1.16xl0- 3 exist between doubled value of large interval 441 MeV in particle masses introduced by R.Sternheimer [4] (close to 3/2 of A-excitation of nucleoli) and parameters (periods) e0, s' and e" in nuclear data [3,4,11]. The same SFERC-factor was determined later as a ratio between the mass of muon and Z-boson 1.1587(l)xl0 - 3 (close to SFERC=1.1596xlO- 3 ) [3]. For this reason numbers x=0,l,2,3 were assigned to powers of the scaling factor SFERC by presentation of different mass/energy intervals in Table 2. Interconnection between intervals of fine and hyperfine structure corresponding to n=13-17 in both structures, namely between 0+ excitations in some heavy nuclei (E*(0+)=kx246 keV, E*=D 0 ) and stable intervals of hyperfine structure (D=kx286 eV, D=8 x 187 eV=1500 eV) was noticed [4,12], but to check this effect more resonance data for nuclei with Z=48-59 are needed.
465
Table 2: Parameters of tuning effects in particle masses and nuclear d a t a (E*, D, A E B , resonance positions EQ) are expressed as nxmx(8e 0 =16ra e ) X (a/2n)x.
9 4 n quant. X m 61Ni D 4.8=4e' 2 1/8 140 61 Ce, Ni 1/4 E c 9.3-9.5* 21.5* 61 19 Ni,many D,E„ 1/2 83-87 39 1 51 Cr, 61 Ni D,E* 51 D Cr 2 (76) 170=e o / 6 43 341 152 D,E* 4 Ca nucleus A Z
many 3 1/8 1 76 As, Nd 124 Sb 2 88 4 Sr,many many 1 1-2 many 6 1 1-2 part .mass 6 0 1 part.mass
E 0 5.5=4£" 44 D,E„ D,E 0 (88) D,E 0 176 AE B 32.8=4(5 AEB
m. Ami
13
17
Units Comment keV Ohkubo*
122 (246) (492)
161
[8] [8] [8]
D 0 /2
&np
eV Ideno* 187 372 [8] 394* Ohkubo* 748* MeV 147 140 441 6 = 8s0 409 nip + me mw — me MeV AMA 6 — 16me i Wg=44]L 409 GeV Mz-91.2 143* (286) 572*
The third block of data used in the analysis consists of A E B - differences of binding energies EB and second differences of Eg, known as parameters of residual nucleon interaction, for example e p n = A S p ( A N = l ) etc. Grouping of resonance positions [9] allows to suggest that the same structures as in E* could be seen in EB- For valence nucleons above shells with Z=28,50,82 and N=82 parameters £ p n =682 keV and 341 keV were derived from linear dependencies of S p or Sn upon N or Z [7]. In case of 51-st proton e p n =340 keV and intervals D=340 keV=e 0 /3 are boxed in Tables 1-2. Discreteness in S2P and S4P with period e0 was found in [2,6]. A E B connected with AZ=2, AN=4 were found to be rational with a period A=4|e c ,=4.6 MeV close to mass splitting of pion [9,11]. In nuclei differing by four a-particles A E B were found to be close to m^ = 140 MeV and to the parameter of nucleon A-excitation A M A = 147 MeV=(ra A -m N )/2=32A (see Table 3). In Fig.la,b) groupings of A E B = 1 4 7 MeV in nuclei differing by four a-particles (Z<26) and in nuclei differing by AZ=8,AN=14 are shown. Groupings were observed even at larger intervals AE B =441 MeV=3xl47 MeV (Fig.lc) and 409 MeV [9].
466
w
I 147.1 MeV
30 25 20 T 15
Jl \k w
~
10 7 5
ft*jW 140
,
i , ,
150
160
440
450
AE8(MeV)
Fig.l. AEB-distributions in nuclei differing by AZ=AN=8 (a), AZ=8, AN=14 (b) and in all odd-odd nuclei (c). EB are from AME-95 by G.Audi and H.Wapstra. Proximity of A E B = 1 4 7 MeV to AM A =32A and baryon constituent quark mass M q to 3 A M A (Table 3) means that in several hadronic effects common tuning phenomena (connected with m e ) exist. Relation between vector boson mass M z , lepton ratio L=23x9=13xl6-1 and 3 A M A and proximity of kx A M A to the radial excitation of charmonium (cc) and to intervals in masses of vector mesons (with AJ=2) support the distinguished character of A M ^ . Table 3: Relations in particle masses ( P a r t . D a t a Group, 1998) with AM A [3,4].
Parameter A M A raA»•F -m„ Mi; MeV 293.5 kxl47MeV 294 k (2)
Mq Mq M ^ / 3 mH- / 3 1/2+ 1/2+ 440.3 450 441 441 3 3
Mq Md 1/2+ 436 441 3
AmCc A n ^ ArriK' AS=1 AJ=2 A J = 2 11-3- 1-3589 588 4
885(4) 884(7) 882 882 6 6
Mz/L estim. 1440.5 441 3
References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.
S.Devons, Proc. Rutherford Jub. Conf., ed. J.Birks, 1961, p.611. S.I.Sukhoruchkin, Proc. Conf. Exp. Phys.. AIP 495, p.482 (1999). S.I.Sukhoruchkin, Symm. in Subatomic Phys., AIP 539, 2000, p.142. S.I.Sukhoruchkin, Symm. Meth. Phys., JINR E2-94-347, pp.528,536. Y.Nambu, Progr. Theor. Phys., 7, 595 (1952). Z.N.Soroko, S.I.Sukhoruchkin et al., Nucl. Phys.k 680, (2001) (in press). S.I.Sukhoruchkin, Phys. At. Nucl. 6 1 , 1855 (1998) and ref. therein. S.I.Sukhoruchkin etal., Proc. ISINN-8, JINR E3-2000-192, pp.428,420. S.I.Sukhoruchkin, J. Phys.G: Nucl. Part. Phys. 25, 921 (1999). O.I.Sumbajev, A.I.Smirnov, L.N.Kondurova, JETF 6 1 , 1276 (1971) . S.I.Sukhoruchkin, Stat. Prop. Nucl.,ed. J.Garg. PL Press, 1972, p.215. S.I.Sukhoruchkin, Proc. ISINN-8, JINR E3-95-307, 1995, pp. 182,330.
List of Participants
468 Abrosimov Valery [email protected] Afanasjev Anatoli [email protected] Alhassid Yoram [email protected] Allal Nassima-Hosni nallal@caramail. com Andreyev Andrei andrei. [email protected]. be Antinori Federico federico. antinori@cern. ch Apagyi B a r n a b a s [email protected] Aslanyan Petros [email protected]. ru Azaiez Faisal [email protected] Badurek Gerald badurek@ati. ac. at B a k t a s h Cyrus baktashc Qornl. gov B a r a n Virgil [email protected] Basile Maurizio [email protected]
Institute for Nuclear Research Prospect Nauki 47, Kiev UKRAINE Technical University, Muenchen Garching D-5747 GERMANY Yale University, Physics Dept P . O . Box 298124, New Haven, USA Institut de Physique U S T H B B P 3 2 , El-Alia ALGERIA I.K.S. Celestijnenlaan, 200D, Leuven BELGIUM Dip.to di Fisica, Univ. Padova Via Marzolo 8, Padova ITALY Technical University of Budapest Budapest Budafoki U.8 H - l l l l HUNGARY Joint Institute for Nuclear Research Dubna, Moscow Region, P.O. 141980 RUSSIA Institut de Physique Nucleaire 15 Rue Georges Clemenceau, Orsay FRANCE Nuclear Physics Institute Stadionallee 2, Vienna AUSTRIA Oak Ridge N a t . L a b . Physics Div. B L D G 6000, MS 6371 Oak Ridge USA Laboratori Nazionali del SUD Via S. Sofia 44, C a t a n i a ITALY Dip.to di Fisica, Univ. Bologna Via Irnerio 46, Bologna ITALY
469
Universite Blaise Pascal Aubiere CEDEX 63177 FRANCE C.S.N.S.M. Bauchet Armand Orsay Campus bauchet@csnsm. in2p3.fr FRANCE Institut de Physique Nucleaire Beaumel Didier 15, Rue Georges Clemenceau, Orsay [email protected] FRANCE Forschungszentrum Karlsruhe Beer Hermann P. O. Box 3640 Karlsruhe [email protected] GERMANY Beghini Silvio I.N.F.N. Via Marzolo 8, Padova beghini@pd. infn. it ITALY Benczer-Koller Noemie Dept. of Physics and Astronomy Rutgers Univ., New Brunswick nkoller@physics. rutgers.edu USA Dipartimento di Fisica, Roma "Tor Vergata" Bemabei Rita Via della Ricerca Scientifica 1, Roma Bernabei@roma2. infn.it ITALY Bernardos Pilar Universidad de Cantabria Avda de los Castros s/n Santander SPAIN Dipartimento di Fisica Univ. Bologna Bertin Antonio [email protected] Via Irnerio 46, Bologna ITALY ICN-UNAM Bijker Roelof [email protected] AP 70-543 MEXICO Blecher Marvin Virginia Polytechnic Institute blecherm @alphamb2.phys.vt. edu Blacksburg VA 24061 USA Laboratori Nazionali del Sud Bonasera Aldo Via S. Sofia 44, Catania [email protected] ITALY Dipartimento di Fisica, Univ. Bologna Bonsignori Giovanni Carlo Via Irnerio 46, Bologna [email protected] ITALY Bastid Nicole bastid@clermont. in2p3.fr
470
Borderie Bernard borderie @ipno. in2p3.fr Bortignon Pier Francesco pierfrancesco. bortignon @mi. infn.it Borzov Ivan borzovQippe. rssi. ru B o t t a Elena [email protected] Botvina Alexandre botvina Qganac4-in2p3.fr Bracco Angela Bracco @mi. infn. it Brandolini Franco brandolin i @pd. infn. it Brant Slobodan [email protected] Bressani Tullio [email protected] Broglia Ricardo broglia @mi. infn. it Bruno M a u r o bruno @bo. infn. it Bulgac Aurel [email protected] Burgio Fiorella fiorella. burgio @ct. infn. it
Institut de Physique Nucleaire 15, R u e Georges Clemenceau, Orsay FRANCE Dipartimento di Fisica, Univ. Milano Via Celoria 16, Milano ITALY Institute of Physics a n d Power Engineering Bondarenko S q . l , Obninsk RUSSIA Dip.to di Fisica Sperimentale, Univ. Torino Via P. Giuria 1, Torino ITALY G.A.N.I.L B P 5027, Caen Cedex 5 FRANCE Dipartimento di Fisica, Univ. Milano Via Giovanni Celoria 16, Milano ITALY Dipartimento di Fisica, Univ. Padova Via Marzolo 8, Padova ITALY D e p a r t m e n t of Physics , Zagreb University Bijenicka 32, Zagreb CROATIA Dip.to di Fisica Sperimentale, Univ. Torino Via P. Giuria 1, Torino ITALY Dipartimento di Fisica, Univ. Milano Via Giovanni Celoria 16, Milano ITALY Dipartimento di Fisica Univ. Bologna Via Irnerio 46, Bologna ITALY Max-Planck-Institut fuer Kernphysik Heidelberg GERMANY I.N.F.N. Corso Italia 57, C a t a n i a ITALY
471
Dipartimento di Fisica, Univ. Milano Via Giovanni Celoria 16, Milano ITALY Dipartimento di Fisica, Univ. Trieste Camerini Paolo Via Valerio 2, Trieste cameriniSts. infn. it ITALY ENEA Canetta Elisabetta Via Don G.Fiammelli 2, Bologna sirio68Qyahoo.com ITALY I.N.F.N. Cannata Francesco [email protected]. it Via Irnerio 46, Bologna ITALY Center of Applied Studies for Nuclear Dev. Capote Noy Roberto Calle 11-702 esq.A Habana capote ©bologna, enea. it CUBA Cappuzzello Francesco Laboratori Nazionali del Sud cappuzzello @lns.infn. it Via S. Sofia 44, Catania ITALY I.N.F.N. Cardella Giuseppe [email protected] Corso Italia 57, Catania ITALY Jefferson Laboratory Cardman Laurence S. cardman @jlab. org 12000 Jefferson Ave, Virgina VA 23606 USA Casten Richard Yale University, Physics Department [email protected] P.O. Box 208124 New Haven USA Catara Francesco Dipartimento di Fisica, Univ. Catania [email protected] Corso Italia 57, Catania ITALY Chadwick Mark Los Alamos National Lab., Theoretical Division [email protected] Los Alamos NM 87545 USA V. G. Khlopin Radium Institute Chechev Valery chechevQatom. ntu. ru 28 Second Murinsky Avenue, St. Petersburg RUSSIA Cheon Il-Tong Yonsei University, Department of Physics itcheon @phya. yonsei. ac.kr Seoul 120-749 REPUBLIC OF KOREA
Camera Franco camera@mi. infn. it
472
Chiavassa Emilio [email protected] Chomaz Philippe [email protected] Ciofi Degli Atti Claudio [email protected] Cole Brian [email protected] Colo Gianluca [email protected] Colonna Nicola [email protected] Coniglione Rosa [email protected] Coraggio Luigi [email protected] Corradi Lorenzo [email protected] Cotanch Stephen [email protected] D'Agostino Michela [email protected] De Angelis Giacomo [email protected] De Souza Romualdo [email protected]
I.N.F.N. Via P. Giuria 1, Torino ITALY G.A.N.I.L. Boulevard Henri Becquerel FRANCE Dipartimento di Fisica, Univ. Perugia Via A. Pascoli, Perugia ITALY Columbia University 2960 Broadway, New York USA Dipartimento di Fisica, Univ. Milano Via Celoria 16, Milano ITALY Dipartimento di Fisica, Univ. Bari Via Amendola 173, Bari ITALY Laboratori Nazionali del Sud Via S. Sofia 44, Catania ITALY Sezione I.N.F.N. Via Cintia, Napoli ITALY Laboratori Nazionali di Legnaro Via Romea 4, Legnaro (PD) ITALY North Carolina State University, Dept. of Physics Raleigh, North Carolina USA Dipartimento di Fisica, Univ. Bologna Via Irnerio 46, Bologna ITALY Laboratori Nazionali di Legnaro Via Romea 4, Legnaro (PD) ITALY G.A.N.I.L. Boulevard Henri Becquerel FRANCE
473
Oak Ridge National Laboratory Bldg. 6003, M.S. 6373 USA Centre d'Etudes de Bruyeres-le-Chatel Delaroche Jean-Paul jean-paul. delaroche Qbruyeres. cea.fr Bruyeres-le-Chatel FRANCE Universite Libre de Bruxelles Descouvemont Pierre [email protected] Brussels BELGIUM Institut de Physique Nucleaire Desesquelles Pierre 15 av. Clemenceau, Orsay [email protected] FRANCE Kernfysisch Versneller Instituut Dieperink Alex Zernikelaan 25, Groningen [email protected] The Netherlands RIKEN Rl-beam Factory Project Dinh Dang Nguyen [email protected] Wako City Saitama JAPAN Dobaczewski Jacek Institute of Theoretical Physics Hoza 69, Warsaw [email protected] POLAND Doering Joachim G.S.I Planckstrasse 1, Darmstadt j . doering@gsi. de GERMANY School of Physics and Astronomy Donadille Laurent Edgbaston, Birmingham I. donadille @bham. ac. uk UNITED KINGDOM Departamento de Fisica Dorso Claudio Pab.l Ciudad Universitaria, Buenos Aires codorso @df. uba. ar ARGENTINA Draayer Jerry Louisiana State University Baton Rouge, Louisiana [email protected] USA Drago Alessandro Dipartimento di Fisica, Univ. Ferrara [email protected] Via Paradiso 12, Ferrara ITALY Egido Jean Luis Universidad Autonoma de Madrid egido Qeldorado.ft. uam. es Madrid Dean David [email protected]
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TRIUMF Theory Group Vancouver CANADA Universidad de Valencia Farnea Enrico [email protected] Apartado de correos 2085, Valencia SPAIN Fedorov Dimitri I.F.A. University of Aarhus Ny Munkegade Aarhus C fedorov@ifa. an. dk DENMARK I.N.F.N. Feliciello Alessandro Via P. Giuria 1, Torino feliciello @to. infn. it ITALY Fernandez Francisco Facultad de Ciencias, Grupo de Fisica Nuclear fernandez [email protected] Universidad de Salamanca SPAIN I.N.F.N. Filippi Alessandra Via P. Giuria 1, Torino filippi @to. infn. it ITALY Dip. di Fisica, Univ. Bologna Finelli Paolo Via Irnerio 46, Bologna [email protected] ITALY Finger Miroslav Faculty of Mathematics and Physics finger@mbox. troja. mff. cuni. cz V Holesovickach 2, Praha 8 CZECH REPUBLIC Finger Michael Joint Institute for Nuclear Research Dubna, Moscow [email protected] RUSSIA Laboratori Nazionali del SUD Finocchiaro Paolo finocchiaro @lns. infn. it Via S.Sofia 44, Catania ITALY Centre d'Etudes de Saclay DSM/DAPNIA/SPHN Fioni Gabriele Orme des Merisiers, Gif-sur-Yvette CEDEX [email protected] FRANCE Dip. di Fisica, Univ. Catania Foti Antonino Corso Italia 57, Catania [email protected] ITALY I.N.F.N Fragiacomo Enrico enrico.fragiacomo @trieste. infn. it Via Valerio 2, Trieste ITALY Escher Jutta es cher&triumf. ca
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Physik-Dept., Technische Univ. James Franck Str. 1 Garching GERMANY Inst, of Nuclear Research and Energy Gaidarov Mitko Blvd. Tzarigradsko chaussee 72, Sofia gaidarov @inrne. bas. bg BULGARIA Universitaet Muenchen Gaitanos Theodoras Theo. Gaitanos@Physik. Uni-Muenchen.DE Am Coulombwall 1, Garching GERMANY Dip.to di Fisica Generale, Univ. Torino Gallino Roberto Via P. Giuria 1, Torino gallino @ph02xd.ph. unito. it ITALY I.I.T. Department of Physics Gambhir Yogendra Mumbai [email protected]. ernet.in INDIA Massachusetts Inst, of Technology Gao Haiyan 77 Mass. Ave Cambridge [email protected]. edu USA I.N.F.N. Sezione di Napoli Gargano Angela Via Cintia, Monte S. Angelo gargano @na. infn. it ITALY Instituto de Estructura de la Materia Garrido Eduardo Serrano 123 Madrid imteg57@pinarS. csic.es SPAIN Hahn Meitner Institut Gebauer Burckhard Glienicker Str. 100, Berlin gebauer [email protected] GERMANY NSCL - MSU Michigan State Univ. Gelbke C.Konrad East Lansing gelbke @nscl. msu. edu USA Laboratori Nazionali del Sud Geraci Elena Via Santa Sofia 44, Catania geraci@lns. infn. it ITALY ENEA Gherardi Giuseppe V. M. di Monte Sole 4, Bologna [email protected] ITALY Ghetti Roberta Lund University roberta. ghetti @kosufy. lu.se Box 118, Lund SWEDEN
Friese Juergen [email protected]
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Giacomelli Giorgio [email protected]
Dipartimento di Fisica, Univ. Bologna Via Irnerio 46, Bologna ITALY Giannatiempo Angela Dipartimento di Fisica, Univ. Firenze giannatiempo @fi. infn.it Largo E. Fermi 2, Firenze ITALY Giannini Mauro Dipartimento di Fisica, Univ. Genova giannini @ge. infn. it via Dodecaneso 33, Genova ITALY Ginocchio Joseph Los Alamos National Laboratory MS gino @t5. lanl. gov Los Alamos, New Mexico USA Giorgini Miriam Dipartimento di Fisica, Univ. Bologna giorginim @bo.infn. it Via Irnerio 46, Bologna ITALY Giusti Paolo Dipartimento di Fisica, Univ. Bologna Via Irnerio 46, Bologna [email protected] ITALY Russian Research Center, Kurchatov Institute Gloukhov Iouri Kurchatov sq. 4 glukhov @dni.polyn.kiae. su RUSSIA Institut de Physique Nucleaire de Lyon Gobet Franck Boulevard du 11 Novembre 43,Villeurbanne [email protected] FRANCE Department of Eng.Physics Gonul Bulent gonulQalpha. bim.gantep. edu. tr Gaziantep TURKEY Laboratori Nazionali di Legnaro Gramegna Fabiana Via Romea 4, Legnaro (Padova) ITALY Physik Dept. der Technischen Univ. Munchen Griesshammer Harald hgrie @physik. tu-muenchen. de Garching GERMANY Hahn-Meitner Institut Gross Dieter Glienickerstr.100 Berlin [email protected] GERMANY Phys. Dept. Aristotle Univ. Grypeos Michael Thessaloniki grypeos @ccf. auth.gr GREECE
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Phys. Dept. Aristotle Univ. Thessaloniki GREECE Max-Planck-Institut fur Kernphysik Gu Jianzhong Postfach 103980 Heidelberg gu @daniel. mpi- hd. mpg. de GERMANY Dipartimento di Fisica, Univ. Milano Guazzoni Paolo Via Celoria 16, Milano [email protected] ITALY Oak Ridge National Laboratory Guber Klaus Oak Ridge [email protected] USA Guttormsen Magne Dept. of Physics, Oslo University P.O.Box 1048 Blindern, Oslo magne.guttormsen @fys. uio.no NORWAY Fakultat fur Physik, Freiburg Univ. Haberland Helmut Hermann Herderstr. 3, Freiburg [email protected] GERMANY Hagemann Gudrun Niels Bohr Institute hagemann @nbi. dk Blegdamsvej 17 Copenhagen DENMARK Hammache Fairouz G.S.I. /. hammache @gsi. de Planckstrasse 1, Darmstadt GERMANY Heil Michael Forschungszentrum Karlsruhe [email protected] Hermann-von-Helmholtz Platz 1 GERMANY Heyde Kris Dept. of Subatomic and Rad. Physics kris.heyde@rug. ac. be Proeftuinstraat 86, Gent BELGIUM Hofmann Prank Giessen University [email protected] Heinrich-Buff Ring 16, Giessen GERMANY Horowitz Charles J. Nuclear Theory Center charlie @iucf. indiana. edu Indiana University, Bloomington USA Iachello Francesco Sloane Physics Lab., Yale Univ. francesco. iachello @yale. edu 217 Prospect Street, New Haven USA Grypeos Agnes grypeos @ccf. auth. gr
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I.N.F.N. Via P. Giuria 1, Torino ITALY Ignatyuk Anatoly Institute of Physics and Power Engineering Bondarenko Squ. 1, Obninsk ignatyuk@ippe. rssi. ru RUSSIA Iori Ileana Dipartimento di Fisica, Univ. Milano Via Celoria 16, Milano [email protected] ITALY Itaco Nunzio Dip.to di Scienze Fisiche, Univ. Napoli [email protected] via Cintia, Monte S. Angelo, Napoli ITALY Jaqaman Henry Dept. of Physics, Bethlehem Univ. Bethlehem hjaqaman@bethlehem. edu PALESTINE Jastrzebski Jerzy Heavy Ion Laboratory, Warsaw University Pasteura 5a, Warszawa jastj@nov. slcj. uw. edu.pl POLAND Niels Bohr Institute Jensen Dennis R. Blegdamsvej 17 Copenhagen ringkbng @alf. nbi. dk DENMARK Jungclaus Andrea Phys. Institut, Univ. Goettingen II jungclaus @physik2. uni-goettingen. de Bunsenstrasse 7-9, Goettingen GERMANY Kaubler Lutz Institute fur Kern-und-Hadronenphysik Dresden kaeubler@fz-rossendorf. de GERMANY Institute of Nuclear Power Engineering Karmanov Fedor Studgorodok 1, Obninsk [email protected] RUSSIA Kellerbauer Alban C.E.R.N. Division E P Geneve 23 a. kellerbauer@cern. ch SWITZERLAND Institut de Physique Nucleaire Khan Elias khan @ip no. in2p3.fr 15, rue Georges Clemenceau, Orsay FRANCE TU-Munchen Physik Department Kienle Paul Paul. Kienle @Physik. Tu-Muenchen. de James Franck-Strasse, Garching GERMANY Iazzi Felice [email protected]
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Kirson Michael [email protected] Klein Abraham [email protected] Kolybasov Victor kolybasvQsci. lebedev. ru Konobeyev Alexandre [email protected] Korichi Amel [email protected] Korten Wolfram [email protected] Koshelkin Andrey koshelkn@gpd. mephi. msk. su Kowalski Seweryn [email protected] Krasznahorkay Attila [email protected] Kroell Thorsten thorsten.kroell@pd. infn. it Kukulin Vladimir [email protected] Kurepin Alexei kourepin @vxcern. cern. ch Kuss Michael [email protected]
The Weizmann Institute of Science Rehovot ISRAEL Department of Physics Pennsylvania University, Philadelphia USA Lebedev Physical Institute Leninsky prospect 53, Moscow RUSSIA Institute of Nuclear Power Engineering Studgorodok 1, Obninsk RUSSIA CSNSM Orsay Orsay FRANCE CEA Saclay DAPNIA/SPhN l'Orme des Merisiers, Gif-sur-Yvette FRANCE Institute for Physics and Engineering Kashirskoye sh. 31, Moscow RUSSIA Physics Inst., Silesia Univ. ul. Uniwersytecka 4, Katowice POLAND ATOMKI P.O. Box 51, Debrecen HUNGARY Dipartimento di Fisica, Univ. Padova Via Marzolo 8, Padova ITALY Nuclear Physics Inst., Moscow Univ. Moscow RUSSIA Institute for Nuclear Research, Moscow 60-th October Anniversary prospect 7 A RUSSIA Jefferson Laboratory, Physics Division 12000 Jefferson Avenue, Newport News USA
480 Kuyucak Serdar sekl [email protected]. edu. au Lalazissis Georgios glalazis Qphysik. tu- muenchen. de Lanchais Ariane lanchais @bo. infn. it L a t o r a Vito [email protected] Lavagno Andrea
Laville Jean-Louis [email protected] Le Neindre Nicolas [email protected] Leeb Helmut leeb @kph. tuwien. ac.at Lenske Horst [email protected] Lenzi Silvia M. silvia.lenziQpd. infn. it Leonardi Renzo leonardi @ science.unitn. it Levai Geza [email protected] Leviatan A m i r a m [email protected]. ac.il
Dept. of Theoretical Physics Australian National Univ., C a n b e r r a AUSTRALIA Aristotle University Thessaloniki GREECE Dipartimento di Fisica, Univ. Bologna Via Irnerio 46, Bologna ITALY Dipartimento di Fisica, Univ. C a t a n i a Corso Italia 57, C a t a n i a ITALY Politecnico di Torino Corso D u c a Degli Abruzzi, 24, Torino ITALY G.A.N.I.L. Boulevard Henry Bequerel, Caen FRANCE I.N.F.N. Via Irnerio 46, Bologna ITALY Institut fur Kernphysik der T U Wien Wiedner Hauptstrasse 8-10/142, Wien AUSTRIA Institut fur Theoretische Physik Heinrich-Buff-Ring 16, Giessen GERMANY Dipartimento di Fisica, Univ. Padova Via F . Marzolo 8, Padova ITALY Dipartimento di Fisica, Univ. Trento Povo, Trento ITALY ATOMKI P.O. Box 51,Debrecen HUNGARY R a c a h Institute of Physics Jerusalem ISRAEL
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Dept. of Physics, Arkansas State Univ. 2009 Cedar Heights, Jonesboro USA Aristotle University Liolios Theodore Thessaloniki theolioW.au1h.gr GREECE Argonne National Laboratory Physics Div. 203 Lister Christopher J. 9700 S. Cass. Ave, Argonne [email protected]. anl.gov USA Dip.to di Matematica e Fisica Univ. Camerino Lo Bianco Giovanni lobianco @camserv. unicam. it Via Madonna delle Carceri, Camerino (Mc) ITALY DSM/DAPNIA/SPHN CEA Saclay Lobo Georges Gif-sur-Yvette Cedex [email protected] FRANCE Laboratori Nazionali del SUD Lu Jun Via S. Sofia 44, Catania [email protected] ITALY G.S.I. Lukasik Jerzy Planckstrasse 1, Darmstadt J.Lukasik@gsi. de GERMANY Dipartimento di Fisica, Univ. Padova Lunardi Santo Via Marzolo 8, Padova lunardi @pd. infn. it ITALY Dipartimento di Fisica, Univ. Padova Maglione Enrico Via Marzolo 8, Padova maglione @pd. infn. it ITALY ENEA Maino Giuseppe Via Don Fiammelli 2, Bologna maino Qbologna. enea. it ITALY Dipartimento di Fisica, Univ. Bologna Malaguti Franco Via Irnerio 46, Bologna malaguti @bo. infn. it ITALY I.N.F.N. Marcello Simonetta Via Pietro Giuria 1, Torino [email protected] ITALY Universidad de Cantabria Marcos Saturnino Marcoss @besaya. unican. es Avda. de los Castros S/N, Santander SPAIN
Li Bao-An [email protected]. astate. edu
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Martino Jacques jmartino Qcea.fr
Service de Physique Nucleaire CEN Saclay Gif-sur-Yvette Cedex FRANCE Maslov Vladimir Radiation Phys. and Chem. Problems Inst. maslov@sosny. baa-net. by Minsk-Sosny BELARUS Massa Ignazio Dipartimento di Fisica, Univ. Bologna massa [email protected] Via Irnerio 46, Bologna ITALY Massen Stelios Aristotle University massen@physics. auth. gr Thessaloniki GREECE Laboratori Nazionali di Legnaro Mastinu PierPrancesco mastinu @lnl. infn. it Via Romea 4, Legnaro (PD) ITALY Dipartimento di Fisica, Univ. Firenze Matera Francesco Largo Enrico Fermi 2, Firenze matera @fi. infn. it ITALY Physics Dept., Athens University Mavrommatis Eirene Athens GREECE ENEA Mengoni Alberto Via Don Fiammelli 2, Bologna [email protected] ITALY Laboratori Nazionali del SUD Migneco Emilio Via Santa Sofia 44, Catania migneco Olns. infn. it ITALY Dipartimento di Fisica Univ. Trieste Milazzo Paolo Via A. Valerio 2, Trieste milazzo @trieste. infn. it ITALY Belgrade Institute of Physics Milosevic Jovan Pregrevica 118, Belgrade jmilos@phy. bg. ac. yu YOUGOSLAVIA Institute for Nuclear Research and Energy Minkov Nikolay Tzarigrad Road 72, Sofia nminkov @inrne. bas. bg BULGARIA Mishustin Igor Goethe University mishits @th.physics. uni-frankfurt. deRobert-Meyer Str. 8-10, Frankfurt am Main GERMANY
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Institut fur Kerniphysik der TU Darmstadt Schlossgarstenstrasse 9, Darmstadt GERMANY Sapporo Gakuin University Morita Hiko Ebetsu-shi Hokkaido hiko @earth. sgu. ac.jp JAPAN Dipartimento di Fisica, Univ. Firenze Mosconi Bruno Largo E. Fermi 2, Firenze mosconi @fi. infn. it ITALY Michigan State University, N.S.C.L. Mueller Wilhelm S. Shaw Lane, East Lansing mueller@nscl. msu. edu USA G.S.I. Munzerberg Gottfried Planckstrasse 1, Darmstadt [email protected] GERMANY Dept. of Physics, Chiba University Nakada Hitoshi Yayoi-cho 1-33 Inage, Chiba nakada @c. chiba-u. ac.jp JAPAN Michigan State University, N.S.C.L. Nakamura Takashi [email protected]. ac.jp S. Shaw Lane, East Lansing USA I.N.F.N. Nannini Adriana nannini @fi. infn. it Largo E.Fermi 2, Firenze ITALY Brookhaven National Laboratory Nara Yasushi Upton, New York [email protected]; USA Nazarewicz Witold Dept. of Physics and Astronomy witek-nazarewicz@utk. edu Knoxville (Tennessee) USA Neergard Greges Niels Bohr Institute [email protected] Blegdamsvej 17, Copenhagen DENMARK School of Physics and Astronomy Nicoli Marie-Paule Edgbaston, Birmingham [email protected] UNITED KINGDOM Odegard Stein Dept. of Physics, Oslo University stein @lynx. uio.no PB 1048, Oslo NORWAY Mohr Peter mohr@ikp. tu- darmstad. de
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Oglobin Alexey [email protected]
Russian Research Center, Kurchatov Institute 1, Kurchatov Square, Moscow RUSSIA Orlandini Giuseppina Dipartimento di Fisica, Univ. Trento orlandinQscience, unitn. it Povo (TR) ITALY Otsuka Takaharu Dept. of Physics, Tokyo University [email protected] Hongo, Bunkyo-ku Tokyo JAPAN Outenkov Vladimir Joint Institute for Nuclear Research [email protected]. dubna.su Dubna, Moscow RUSSIA I.N.F.N. Pagano Angelo Pagano @ct. infn. it Corso Italia 57, Catania ITALY Panitkin Sergei Centre of Nuclear Research panitkin @cher. star, bnl.gov Kent State University USA Paar Nils Physik-Department der Technischen Univ. NilsPaar@Physik. TU-Muenchen.DB Garching, Muenchen GERMANY Parreno Assumpta Institute for Nuclear Theory parrenoQphys. Washington, edu Washington University, Seattle USA Pawlovski Piotr Institut de Physique Nucleaire [email protected] 15, Rue Georges Clemenceau, Orsay FRANCE Pearson Michael J. Universite de Montreal pearson@LPS. UMontreal.CA Montreal (Qc) CANADA Institut fur Kernphysik Peitzmann Thomas peitzmann @ikp. uni- muenster. de Wilhelm-Klemm Str.9, Muenster GERMANY Departamento de Fisica, Univ. de Oviedo Perez-Garcia M. Angeles Campus de Llamaquique Oviedo, Asturias aperez@pinon. ecu. unto vt. es SPAIN Dip.to di Matematica e Fisica, Univ. Camerino Petrache Costel via Madonna delle Carceri, Camerino [email protected] ITALY
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Pezer Robert [email protected]. hr Piantelli Silvia [email protected] Piccinini Maurizio [email protected] Pierella Fabrizio [email protected] Pignanelli Marcello Marcello.Pignanelli QMUNFN.IT Pirrone Sara [email protected] Platonov Sergey [email protected] Plujko Vladimir plujko @kinr. kiev. ua Podolyak Zsolt Z. Podolyak@ surrey, ac.uk Poggi Giacomo [email protected] Politi Giuseppe [email protected] Popa Gabriela [email protected] Porquet Marie-Genevieve [email protected]
Dept. of Physics, Zagreb University Bijenicka cesta 32, Zagreb CROATIA Dipartimento di Fisica, Univ. Firenze Largo E.Fermi 2, Firenze ITALY I.N.F.N. Via Irnerio 46, Bologna ITALY Dipartimento di Fisica, Univ. Bologna Via Irnerio 46, Bologna ITALY Dipartimento di Fisica, Univ. Milano Via Celoria 16, Milano ITALY I.N.F.N. Corso Italia 57, Catania ITALY Nuclear Physics Inst., Moscow State Univ. Moscow
RUSSIA Institute for Nuclear Research Kiev Prospect Nauki, 47 Kiev ITALY Dept. of Physics, Surrey University Guildford, Surrey UNITED KINGDOM Dipartimento di Fisica, Univ. Firenze Largo E.Fermi 2, Firenze ITALY Dipartimento di Fisica, Univ. Catania Corso Italia 57, Catania ITALY Louisiana State University Baton Rouge, Louisiana USA C.S.N.S.M. Orsay Campus FRANCE
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Porto Francesco porto @lns. infn. it
Laboratori Nazionali del SUD Via Santa Sofia 44, Catania ITALY Pshenichnov Igor Institute for Nuclear Research 60th October Anniversary Str. 7a Moscow pshenichnov@allO. inr. troitsk. ru RUSSIA Quaglia Maria Rosa Dipartimento di Fisica Teorica, Univ. Torino [email protected] Via P. Giuria 1, Torino ITALY Quesada Jose Manuel Universidad de Sevilla Apartado 1065, Sevilla [email protected] SPAIN Laboratori Nazionali del SUD Raciti Giovanni [email protected] Via Santa Sofia 44, Catania ITALY Nat. Inst, of Physics and Nucl. Engineering Raduta Alexandra Horia hraduta @ifin. nipne. ro Magurele, Bucharest ROMANIA Nat. Inst, of Physics and Nucl. Engineering Raduta Adriana Rodica Magurele, Bucharest araduta @ifin. nipne. ro ROMANIA Ramstrom Elisabet Department of Radiation Sciences Elisabet. Ramstrom@studsvik. uu.se Uppsala University SWEDEN LBNL Randrup Jorgen Berkeley, California [email protected] USA Rauscher Thomas Institut fur Theoretische Physik Klingelberg Stabe 82, Basel Thomas. Rauscher@unibas. ch SWITZERLAND Institut fur Kernphysik, FZ Karlsruhe Reifarth Rene Hermann Von Helmholtz Platz 1, Karlsruhe reifarth @ik3.fzk. de GERMANY I.N.F.N. Riccati Lodovico via Pietro Giuria 1,Torino [email protected] ITALY Laboratori Nazionali di Legnaro Ricci Renato Angelo Via Romea 4, Legnaro (Padova) raricci @lnl. infn. it ITALY
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I.N.F.N. Largo E. Fermi 2, Firenze ITALY G.S.I. Ricciardi Maria Valentina KP2 Planckstrasse 1, Darmstadt m. v. ricciardi @gsi. de GERMANY Dipartimento di Fisica, Univ. Genova Ricco Giovanni Via Dodecaneso 33, Genova [email protected] ITALY Laboratoire de Physique Theorique, Strasbourg Richert Jean 3, rue de l'Universite, Strasbourg [email protected] FRANCE Physics Department, Technical Univ. Munich Ring Peter Franck James Str.l, Garching [email protected] GERMANY Institut de Physique Nucleaire Rivet Marie-France 15, Rue Georges Clemenceau, Orsay rivet@ipno. in2p3.fr FRANCE Inst, fur Physik mit Ionenstrahlen Rolfs Claus rolfs @ep3. ruhr-uni- bochum. de Bochum GERMANY Universite de Laval, Departement de Physique Roy Rene Cite universitaire Sainte-Foy, Quebec roy @phy. ulaval. ca CANADA R.I.K.E.N. Sakurai Hiroyoshi 2-1 Hirosawa, Wako-shi Saitaura [email protected] JAPAN Universita di Camerino Saltarelli Alessandro Madonna delle Ceneri, Camerino [email protected] ITALY I.N.F.N. Sambataro Michelangelo [email protected] Corso Italia 57, Catania ITALY Laboratori Nazionali del Sud Sapienza Piera Via S.Sofia 44, Catania sapienza @lns. infn. it ITALY Sarriguren Pedro C.S.I.C., Instituto Estructura de la Materia imtpsSl @pinar2. csic. es Serrano 123 Madrid SPAIN
Ricci Paolo [email protected]
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Sartorelli Gabriella sartorelli@bo. infn. it
Dipartimento di Fisica, Univ. Bologna Via Irnerio 46, Bologna ITALY ENEA Saruis Anna Maria Via Don Fiammelli 2, Bologna ITALY Satoh Eiji Kanto Gakuin University 1162-2 Ogikubo, Odawara-shi, Kanagawa [email protected]. ac.jp JAPAN Satz Helmut Universitat Bielefeld Pf 100131 Bielefeld [email protected] GERMANY Dipartimento di Fisica, Univ. Bologna Savoia Mirko savoia@bo. infn. it Via Irnerio 46, Bologna ITALY Savushin Lev State University for Telecommunications nab.r.Moiki 61, St.Petersburg savush @ph. turn, de RUSSIA Jefferson Laboratory Schiavilla Rocco 12000 Jefferson Ave., Newport [email protected] USA G.S.I. Schmidt Karsten Planckstrasse 1, Darmstadt K. SchmidtSgsi. de GERMANY G.S.I. Schwarz Carsten Planckstrasse 1, Darmstadt G. Schwarz@gsi. de GERMANY Forschungszentrum Rossendorf Schwengner Ronald [email protected] Dresden GERMANY Dipartimento di Fisica, Univ. Bologna Semprini Cesari Nicola Via Irnerio 46, Bologna [email protected] ITALY Physics Department, Technical Univ. Munich Serra Milena [email protected] Franck-James-Str. 1, Garching GERMANY Institute of Physics and Power Engineering Shubin Yuri Bondarenko Sq.l, Obninsk shubin@ippe. rssi. ru RUSSIA
489 Signorini Cosimo [email protected]
Dipartimento di Fisica, Univ. Padova Via Marzolo 8, Padova ITALY Sinha Shrabani Indian Inst, of Technology, Bombay Dept. of Physics [email protected] Powai - Mumbai INDIA Siwek Wilczynska Krystina Institute of Experimental Physics [email protected] Hoza 69, Warszawa POLAND Soramel Francesca Dipartimento di Fisica Univ. Udine [email protected] Via delle Scienze 208, Udine ITALY Soroko Zoya Petersburg Nuclear Physics Institute [email protected] Gatchina, Leningrad District RUSSIA Stander Anton Department of Physics, Stellenbosch University [email protected] Matieland SOUTH AFRICA Stefanini Alberto M. Laboratori Nazionali di Legnaro [email protected] via Romea 4, Legnaro ITALY Stoitcheva Guergana Louisiana State University [email protected] Baton Rouge, Louisiana USA Stroebele Herbert Universitat Frankfurt [email protected] August Eulerstr.6, Frankfurt GERMANY Sugarawa-Tanabe Kazuko Otsuma Women's University [email protected] Karakida, Tama, Tokio JAPAN Sukhoruchkin Sergey Petersburg Nuclear Physics Institute [email protected] Gatchina, Leningrad district RUSSIA Suleymanov Mais Joint Institute for Nuclear Research [email protected] Dubna, Moscow RUSSIA Suzuki Toshio College of Humanities and Sciences, Nihon Univ. [email protected] Sakurajosui 3-25-40 Setagaya-ku, Tokio JAPAN
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Sviratcheva Kristina [email protected]
Department of Physics and Astronomy Baton Rouge, Louisiana USA Talmi Igal The Weizmann Institute of Science igal. talmi@weizmann. ac. il Rehovot ISRAEL Department of Physics,Tohoku University Tamura Hirokazu [email protected] Tohoku Aramaki, Aoba-ku Sendai JAPAN Dept. of Physics, Saitama University Tanabe Kosai [email protected] Kurawa Saitama JAPAN Thielemann Friederich Basel University Klingelberg Stabe 82, Basel [email protected]. unibas.ch SWITZERLAND Thoenessen Michael Michigan State University, NSCL East Lansing, Michigan [email protected] USA Hahn Meitner Institut Thummerer Severin Glienicker Str. 100, Berlin thummerer@hmi. de GERMANY Dip.to di Fisica Teorica, Univ. Trieste Treleani Daniele S. Costiera 11, Miramare, Grignano, Trieste [email protected] ITALY Joint Institute of Nuclear Research Tretyakova Svetlana Dubna, Moscow [email protected]. ru Russia Laboratori Nazionali di Legnaro Trotta Monica Via Romea 4, Legnaro [email protected] ITALY I.N.F.N. Ur Calin A. Via F. Marzolo 8, Padova [email protected] ITALY Dipartimento di Fisica, Univ. Bologna Vagnoni Vincenzo Via Irnerio 46, Bologna vagnoni@bo. infn. it ITALY G.A.N.I.L. Van Isacker Pieter B.P. 5027 Caen 5 [email protected] FRANCE
491 Vannini Gianni [email protected] Ventura Alberto ventura ©bologna, enea. it Venturelli Luca [email protected] Venturi Giovanni armitage @bo. infn. it Villa M a u r o [email protected] Viola Vic vicv@iucf. indiana. edu Vitturi Andrea vitturi @pd. infn. it Volpe Cristina volpe @ipno.in2p3.fr Von B r e n t a n o Peter [email protected] Von Neumann-Cosel Peter vnc @ikp. tu- darmstadt. de Von Oertzen Wolfram oertzen @hmi. de Voukelatou K o n s t a n t i n a nadia @bologna. enea. it Vretenar Dario [email protected]
Dipartimento di Fisica, Univ. Bologna Via Irnerio 46, Bologna ITALY ENEA Via Martiri di Monte Sole 4, Bologna ITALY Dipartimento di Fisica, Univ. Brescia Via Vallotti 9, Brescia ITALY Dipartimento di Fisica, Univ. Bologna Via Irnerio 46, Bologna ITALY Dipartimento di Fisica, Univ. Bologna Via Irnerio 46, Bologna ITALY Indiana University, I U C F Bloomington, IN 47408 USA Dipartimento di Fisica, Univ. Padova Via Marzolo 8, Padova ITALY Institut de Physique Nucleaire 15, Rue Georges Clemenceau, Orsay FRANCE Institut fur Kernphysik der Uni. Koeln Ziilpicher Strasse 17, Koeln GERMANY Technische Universitat D a r m s t a d t Schlossgartenstr. 9, D a r m s t a d t GERMANY Han-Meitner Institut Glienickerstr.1000, Berlin GERMANY ENEA Via Martiri di Monte Sole 4, Bologna ITALY Dept. of Physics, Zagreb University Bijenicka cesta 32, Zagreb CROATIA
492
Vyvey Katrien Katrien. VyveyQfys.ktileuven.ac.be
Instituut voor Kern- en Stralingsfysica Celestijnenlaan 200 D, Leuven BELGIUM Walters William B. Dept. of Chemistry and Biochemistry ww3@umail. umd. edu College Park, Maryland USA Austron Project Office Weber Helmut helmut.weber@netway. at Schellinggasse 1, Wien AUSTRIA Weise Wolfram Technische Universitat Munchen weise @physik. tu-muenchen. de James-Franck-Strasse, Garching GERMANY N.S.C.L., Michigan State University Westfall Gary East Lansing, Michigan westfalWnscl.msu.edu USA Physicalishes Institut der Heidelberg Univ. Wienold Thomas Heidelberg [email protected] GERMANY Institute For Nuclear Studies Wilczynski Janusz Swierk-Otwock januszw @iriss. ipj.gov.pl POLAND Institut fur Kernphysik Wirzba Andreas Forscungszentrum Julie, Julich a. wirzba @fz-juelich. de GERMANY Institut fur Kernphysik, Fz Karlsruhe Wisshak Klaus Karlsruhe [email protected]. de GERMANY Universitat Munchen, Sektion Physik Wolter Hermann hermann. wolter@physik. uni-muenchen. de An Coulombwall 1, Garching GERMANY Dept. of Physics and Astronomy Woods Philip Mayfield Road -Edinburgh [email protected]. ed.ac. uk UNITED KINGDOM School of Physics, Tel Aviv Univ. Yavin Avivi 14 Oppenheimerstreet, Tel Aviv [email protected] ISRAEL
493 Yennello Sherry yennello @comp. tamu. edu
Texas A and M Univ., Cyclotron Institute
USA Kansai University Yoshida Nobuaki [email protected] Miyada-cho Takatsuki-shi JAPAN Nuclear Physics Inst., Moscow State Univ. Yuminov Oleg Moscow [email protected] RUSSIA WNSL, Physics Department, Yale University Zamfir N. Victor [email protected]. edu 272 Whitney Ave., New Haven USA I.N.F.N. Zavatarelli Sandra Via Dodecanneso 33, Genova zavatare@ge. infn. it ITALY The Racah Institute of Physics Zeldes Nissan Jerusalem zeldes Qvms.huji. ac.il ISRAEL N.S.C.L., Michigan State University Zelevinski Vladimir zelevinsky @theol 2. nscl. msu. edu East Lansing, Michigan USA Dipartimento di Fisica Univ. Milano Zetta Luisa Via Celoria 16, Milano luis a. zetta @mi. infn. it ITALY Dipartimento di Fisica Univ. Bologna Zichichi Antonino Via Irnerio 46, Bologna ITALY Zilges Andreas Technische Universitat Darmstadt Schlossgartenstr. 9, Darmstadt zilges @ikp. tu- darmstadt. de GERMANY Dipartimento di Fisica Univ. Bologna Zoccoli Antonio Via Irnerio 46, Bologna [email protected] ITALY Zuffi Lina Dipartimento di Fisica Univ. Milano Via Celoria 16, Milano [email protected] ITALY
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AUTHOR INDEX
Boudard A. 407 Bressani T. 73, 146 BrilliardL. 354 Bulgac A. 261 Burgio G. F. 257 Busso M. 376
Aguer P. 354 Amato M. 338 Andrzejewski J. 368 Angulo C. 354 Aoi N. 360 Apagyi B. 140 Arlandini C. 382 Armbruster P. 407 Aslanian P. Z. 114 Athanassopoulos S. 303
Calvo D. 146 Camerini P. 134 Capote R. 56, 451 Casarejos E. 407 Cerulli R. 338 Chadwick M. B. 437 CHAOS Coll. 134, 162 Chechev V. P. 323 Chemin J. F. 354 Cheon I. T. 86 Ciofi degli Atti C. 224 Clark J. W. 303 Claverie G. 354 Coc A. 354 Colangelo P. 443 Colonna N. 443 Corvi F. 372 Cotanch S. R. 209 Couture A. 364 Czajkowski S. 407
Babilon M. 308 Badurek G. 418 Baldo M. 257 Barhoumi S. 354 Bathory B. 140 Bauge E. 425 Beaumel D. 360 Beer H. 372 Belli P. 338 Benlliure J. 407 Bernabei R. 338 Bernardos P. 44, 315 Bernas M. 407 Bijker R. 193 Bisceglie E. 443 Blanco L. A. 215 Blecher M. 205 BNL E930 Coll. 106 Bogaert G. 354 Borzov I. N. 283 Botta E. 158
Dai C. J. 338 Daly J. 364 Dakos A. 303 Delaroche J. P. 495
425
496 Derrien H. 368 Descouvemont P. 267 Detwiler R. 364 Di'az-Alonso J. 319 Dieperink A. E. L. 293 Doornbos J. 146 Dufour J. P. 407 E95-001 Coll. 181 Egido J. L. 28 Enami K. 48, 52 Enders J. 308 Enqvist T. 407 Eremenko D. O. 455 Farine M. 273 Fedorov D. V. 68 Feliciello A. 154 Fernandez F. 215 Filippi A. 150 FINUDA Coll. 100 Fortier S. 360 Fotina O. V. 455 Fragiacomo E. 162 Friese J. 175 Fukuda N. 360 Gorres J. 364 Gai E. V. 401 Gallino R. 376 Gao H. 181 Gargano A. 38 Gernoth K. A. 303 Gherardi G. 387 Giannini M. M. 187 Giessen U. 364 Girod M. 425 Gledenov Y. M. 368 Golovchenko A. N. 447
Goriely S. 273, 283 Gorres F. 364 Griesmayer E. 418 Griesshammer H. W. Grypeos M. 60 Guber K. H. 368
219
HADES Coll. 175 Hale G. 364 Hammache F. 354 Harman Z. 140 Hartmann T. 308 Harvey J. A. 368 Hasegawa A. 413 He H. L. 338 Heger A. 277 Heil M. 364, 382 Higashiyama K. 52 Hilaire S. 425 Hirai M. 360 Hoffman R. D. 277 Hussonois M. 354 Hutter C. 308 Iazzi F. 146 Ignatyuk A. V. 287, 401 Ignesti G. 338 Ilic R. 447 Imai N. 360 Incicchitti A. 338 Ishihara M. 360 Iwasaki H. 360 Jacotin M. 354 Jensen A. S. 68 JEFF. Lab A Coll. Jeong M. T. 86 Jericha E. 418
230
497
Kappeler F. 346, 364, 376, 382 Karmanov F. I. 393 KEK E419 Coll. 106 Kiener J. 354 Kienle P. 328 Kimura M. 90 Koehler P. E. 368 Kolybasov V. M. 110, 170 Konobeyev A. Y. 397 Korchin A. 293 Kordyukevich V. O. 455 Koutroulos C. 60 Kratz K. -L. 312, 372 Kuang H. H. 338 Kubo T. 360 Kukulin V. I. 166 Kumagai H. 360 Kuss M. 230 Latysheva L. N. 393 Laurent H. 360 Le Naour C. 354 Leal L. C. 368 Lefebvre A. 354 Legrain R. 407 LEGS SPIN Coll. 205 Lenske H. 431 Leray S. 407 Leviatan A. 193 Libert J. 425 Liolios T. E. 299 Llanes-Estrada F. J. 209 Lombard R. 44 Lopez E. 451 Lopez-Quelle M. 44, 315 Lukyanov S. M. 360 LUNA Coll. 350 Lunev V. P. 401
Ma J. M. 338 Magierski P. 261 Mainegra E. 56, 451 Maison J. -M. 360 Marcello S. 236 Marcos S. 44, 315 Maslov V. M. 413 Massen S. E. 64 Mavrommatis E. 303 Mohr P. 308 Montecchia F. 338 Morillon B. 425 Morita H. 224 Mornas L. 319 Mosconi B. 240 Motobayashi T. 360 Moustakidis C. C. 64 Mustapha B. 407 Mutti P. 372 Nakamura T. 360 Niembro R. 44, 315 OBELIX Coll. 150, 158 Ohnuma H. 360 Onsi M. 273 Orlandini G. 2 Ouichaoui S. 354 Parreno A. 96 Pearson J. M. 273 Perez-Garci'a A. 319 Pfeiffer B. 312, 372 Pita S. 360 Platonov S. Yu. 455 Pravikoff M. 407 Prosperi D. 338
498 Ramstrom E. 431 Rauscher T. 277 Reifarth R. 364, 382 Rejmund F. 407 Ricci P. 240 Ricciardi M. V. 407 Ring P. 34 Robledo L. M. 28 Rodriguez-Guzman R. R. Rochow W. 372 Romain P. 425 Rummel A. 34
28
Sakurai H. 360 Santopinto E. 187 Santorelli P. 443 Satoh E. 90 Sayer R. O. 368 Scheurer J. N. 354 Schiavilla R. 10 Schmidt K. -H. 407 Schulze H. -J. 257 Sedyshev P. V. 382 Serra M. 34 Shahbazian B. A. 114 Shebeko A. 60 Shubin Yu. N. 401 Sirotinin E. I. 455 Skvarc J. 447 Soroko Z. N. 459 Spencer R. R. 368 Stech E. 364 Stephan C. 407 Suarez J. P. 319 Sukhoruchkin D. S. 459 Sukhoruchkin S. I. 459, 463 Taieb J. 407 Tamura H. 106
Tanabe K. 48, 52 Tassan-Got L. 407 Tatischeff V. 354 Teranishi T. 360 Thibaud J. P. 354 Thielemann F. K. 246 Timmermans R. 293 Tischhauser P. 364 Tondeur F. 273 Traini M. 199 Travleev A. A. 393 Treleani D. 224 Tretyakova S. P. 447 Tultaev A. V. 455 Ugalde C.
364
Valcarce A. 215 Valentine T. E. 368 Van Dalen E. N. E. 293 Variale V. 443 Vecchi M. 393, 397 Venturelli L. 100 Villa M. 127 Virassamynaiken E. 354 Vogt K. 308 Volant C. 407 Volz S. 308 Voss F. 346 Walters W. B. 312 Wasserburg G. J. 376 Weber H. 418 Weise W. 119 Wiescher M. 364 Wisshak K. 346 Wlazlo W. 407 Wolter H. H. 431 Woosley S. E. 277
499
Yoneda K. 360 Yoshinaga N. 48, 52 Ypsilantis K. 60 Yuminov O. A. 455
Zavatarelli S. 350 Zelevinsky V. 20 Zilges A. 308
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