Advances in Nuclear Physics: Volume 5 PDF
Preview Advances in Nuclear Physics: Volume 5
ADVANCES IN NUCLEAR PHYSICS VOLUME 5 Contributors to This Volume Akito Arima Department of Physics State University of New York Stony Brook, New York L. M. Delves Department of Computational and Statistical Science University of Liverpool Liverpool, England Hisashi Horiuchi Institute for Fundamental Physics University of Kyoto Kyoto, Japan Kuniharu Kubodera Service de Physique Theorique C.E.N. de Sac/ay Sac/ay, France Donald W. L. Sprung Physics Department McMaster University Hamilton, Ontario, Canada Noboru Takigawa Department of Physics University of Tokyo Tokyo, Japan ADVANCES IN NUCLEAR PHYSICS Edited by Michel Baranger Department of Physics Massachusetts Institute of Technology Cambridge, Massachusetts Erich Vogt Department of Physics University of British Columbia Vancouver, B.C., Canada VOLUME 5 ~PLENUM PRESS • NEW YORK-LONDON • 1972 Library of Congress Catalog Card Number 67-29001 ISBN-13: 978-1-4615-8233-5 e-ISBN-13: 978-1-4615-8231-1 DOL 10.1007/978-1-4615-8231-1 ©1972 Plenum Press, New York A Division of Plenum Publishing Corporation 227 West 17th Street, New York, N. Y. 10011 United Kingdom edition published by Plenum Press, London A Division of Plenum Publishing Company, Ltd. Davis House (4th Floor), 8 Scrubs Lane, Harlesden, London, NW10 6SE, England All rights reserved No part of this publication may be reproduced in any form without written permission from the publisher Volume 4 The Investigation of Hole States in Nuclei by Means of Knockout and Other Reactions Daphne F. Jackson High-Energy Scattering from Nuclei Wieslaw Czyi Nucleosynthesis by Charged-Particle Reactions C. A. Barnes Nucleosynthesis and Neutron-Capture Cross Sections B. J. Allen, J. H. Gibbons, and R. L. Macklin Nuclear Structure Studies in the Z = 50 Region Elizabeth Urey Baranger An s-d Shell-Model Study for A = 18-22 E. C. Halbert, J. B. McGrory, B. H. Wilden thai, and S. P. Pandya IN PREPARATION Volume 6 Nuclear Fission A. Michaudon The Microscopic Theory of Nuclear Effective Interactions and Operators Bruce R. Barrett and Michael W. Kirson Two-Neutron Transfer Reactions and the Pairing Model Ricardo Broglia, Ole Hansen, and Claus Riedel ARTICLES PLANNED FOR FUTURE VOLUMES Nucleon-Nucleus Collisions and Intermediate Structure Aram Mekjian Beta Strength Functions Gregers Hansen Gamma Ray Strength Functions G. A. Bartholemew, E. D. Earle, J. W. Knowles, and M. A. Lone Static Quadrupole Moments of Nuclear Excited States Jack S. Greenberg and Allan Kleinfeld ARTICLES PUBLISHED IN EARLIER VOLUMES Volume 1 The Reorientation Effect J. de Boer and J. Eichler The Nuclear SU ~ Model M.Harvey The Hartree-'Fock Theory of Deformed Light Nuclei G. Ripka The Statistical Theory of Nuclear Reactions E. Vogt Three-Particle Scattering - A Review of Recent Work on the Nonrelativistic Theory I. Duck Volume 2 The Giant Dipole Resonance B. M.Spicer Polarization Phenomena in Nuclear Reactions c. Glaslzausser and J. Thirion The Pairing-Plus-Quadrupole Model D. R. Bes and R. A. Sorensen The Nuclear Potential P. Signel/ Muonic Atoms S. Devons and I. Duerdotlz Volume 3 The Nuclear Three-Body Problem A.N.Mitra The Interaction of Pions with Nuclei D. S.Koltun Complex Spectroscopy I. B. French, E. C. Halbert, I. B. McGrory, and S. S. M. Wong Single Nucleon Transfer in Deformed Nuclei B. Elbek and P. O. Tj~m Isoscalar Transition Rates in Nuclei from the (0:, 0:') Reaction A. M. Bernstein PREFACE In both the present volume of Advances in Nuclear Physics and in the next volume, which will follow in a few months' time, we have stretched our normal pattern of reviews by including articles of more major proportions than any we have published before. As a result we have only three review articles in Volume 5. From the beginning of this series it has been our aim, as editors, to achieve variation in the scope, style, and length of individual articles sufficient to match the needs of the individual topic, rather than to restrain authors within rigid limits. It has not been our experience that this flexibility has led to unnecessary exuberance on the part of the authors. We feel that the major articles now entering the series are entirely justified. The article by Professor Delves on "Variational Techniques in the Nuclear Three-Body Problem" is an authoritative, definitive article on a subject which forms a cornerstone of nuclear physics. If we start with two body interactions, then the three-nucleon system is, perhaps, the only many nucleon system whose exact description may lie within the scope of human ingenuity. In recent years some new techniques of scattering theory, origi nating mostly in particle physics, have led to a great deal of new interest in the nuclear three-body problem. In this series we have had two articles (by Mitra and by Duck) on the new approaches. These approaches have had some formidable success but they have been restricted in the kind of two body interactions for which they are tractable. In parallel with them the more traditional approach by variational techniques has been greatly im proved also. A recent review article by Delves and Phillips [Rev. Mod. Phys. 41: 497 (1969)] describes the recent accomplishments along these lines. It seemed to us that these accomplishments existed in a framework of tech niques which were powerful, beautiful, and applicable to a wide range of problems in nuclear physics, particle physics, and atomic physics, and yet relatively unknown and inaccessible to most. We were therefore very pleased when Professor Delves agreed to write the present article. It is our belief that it will stand for years as the major reference for this topic. vii viii Preface The article by Sprung on "Infinite Nuclear Matter" concerns that other many-body system-in this case an abstraction, not a real system-whose description appears to be within the scope of human ingenuity. Much has been written about the subject, including a number of review articles in the past. The understanding of the subject has not converged with time as rapidly as expected in its infancy: indeed, the convergence has been asymp totic, with many setbacks intermingled among the forward steps. Now things seem to have settled down, however, to a reasonably consistent, if not complete, picture. As editors, we felt it would be timely to ask Sprung, who has been one of the principal contributors to the subject, to analyze the main achievements as well as the remaining difficulties. While the article by Delves may establish a tradition, the article by Arima et al. on "Clustering in Light Nuclei" may start a revolution. The systematic treatment of alpha-particle clusters by anything more than hand waving arguments has been largely forgotten in the forward rush on the nuclear shell model during the past decade. There is a wealth of physical ideas on this topic, largely in the recent work of the Japanese school from which this review has been coaxed. It is our hope that the wakening of interest inspired by the present article will soon make the subject sweep beyond the results reported here. ERICH VOGT MICHEL BARANGER May, 1972 CONTENTS Chapter 1 Variational Techniques in the Nuclear Three-Body Problem L. M. Delves 1. Introduction 2. Variation Principles for Bound States 10 2.1. The Rayleigh-Ritz Principle . . II 2.2. Bound States of Three Bosons . 15 2.3. Estimates of the Accuracy of a Variational Calculation 19 3. Numerical Techniques . . . . . . . . . . . . . . . . . 37 3.1. Choice of Trial Function . . . . . . . . . . . . . 37 3.2. The Calculation of the Normalization and Hamiltonian Integrals . . . . . . . . . . . . . . . . . . . . 66 3.3. Solution of the Finite Matrix Equation . . . . . . 80 3.4. Control and Analysis of Round-off and Integration. Errors . . . . . . . . . . . . . .. ..... 85 4. Bound State Calculations with Realistic Local Potentials 100 4.1. Reduction of the Equations of Motion . 101 4.2. Construction of the Trial Function . . . 102 4.3. Calculations in the Harmonic Oscillator Basis 109 4.4. Calculations Using Hard-Core Potentials 110 5. Variational Methods for Scattering States . . 126 5.1. Two-Body Single Channel Scattering 127 5.2. Many-Body Two-Particle Scattering States 135 5.3. Variation Principles for Many-Body Scattering States 143 5.4. Neutron-Deuteron Elastic Scattering. . . . . 152 5.5. Alternate Variational Principles for the Schrodinger. Equation . . . . . . . . . . . . . . . . . . . . 167 6. Variational Methods for the Faddeev Equations . . . . 174 6.1. Variational Principle for Inhomogeneous Equations. 175 ix x Contents 6.2. The Two-Body t Matrix (DA 72) 176 6.3. Variational Principle for the Faddeev Equations 187 6.4. Alternative Principles for the Breakup Reaction 193 7. Summing Up . 197 Acknowledgments . . . . . . . . . . 198 Appendix A ............ . Convergence of Variational Methods 198 References 219 Chapter 2 Nuclear Matter Calculations Donald W. L. Sprung 1. Introduction 225 1.1. Outline. 225· 1.2. Introduction to the Theory 226 2. Calculation of the G Matrix . 239 2.1. General Principles. . . . 239 2.2. Bethe-Goldstone Equation 240 2.3. Kallio-Day Method . . . 243 2.4. Reduction to Partial Waves 243 2.5. Day's Derivation of the Radial Wave Equation 246 2.6. Binding Energy . . . . . . . . . . . . 249 2.7. The Brueckner Self-Consistency Problem 252 3. Comparison of Methods for Constructing the G Matrix 254 3.1. Kallio-Day Method 254 3.2. Kohler's Method . . . 255 3.3. Related Methods ... 257 3.4. Tabakin-Haftel Method 258 3.5. Coester-Day-Vincent-Cohen Method 259 3.6. Brueckner-Gammel Method . . 260 3.7. Dahl-Ostgaard-Brandow Method 261 3.8. Moszkowski-Scott Method 261 3.9. Summary. . . . . . . . . . . 263 4. Results of Calculations for Realistic Forces 264 4. I. Potentials Considered . . . . . . . 264 4.2. Results for Group One Potentials 265 4.3. Importance of Various Partial Waves 267 4.4. Results for OBEP and Supersoft-Core Forces 272
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