Table Of ContentNuclear Matter Theory
Nuclear Matter Theory
Omar Benhar
Stefano Fantoni
CRC Press
Taylor & Francis Group
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Library of Congress Cataloging-in-Publication Data
Names: Benhar, Omar, author. | Fantoni, S. (Stefano), author.
Title: Nuclear matter theory / Omar Benhar, Stefano Fantoni.
Description: Boca Raton : CRC Press, 2020. | Includes bibliographical
references and index.
Identifiers: LCCN 2019048245 | ISBN 9780815386667 (hardback) | ISBN
9781351175340 (ebook)
Subjects: LCSH: Nuclear matter.
Classification: LCC QC793.3.N8 B46 2020 | DDC 539.7--dc23
LC record available at https://lccn.loc.gov/2019048245
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Contents
Preface ix
hapter
C 1(cid:4) INTRODUCTION 1
1.1 NUCLEARMATTERINATOMICNUCLEI 1
1.2 NUCLEARMATTERINNEUTRONSTARS 5
hapter
C 2(cid:4) NUCLEAR DYNAMICS 7
2.1 THEPARADIGMOFMANY-BODYTHEORY 7
2.2 EMPIRICALFACTSONNUCLEARFORCES 8
2.3 PHENOMENOLOGICALPOTENTIALS 10
2.3.1 The nucleon-nucleon potential 10
2.3.2 Three-nucleon forces 14
2.4 BOSON-EXCHANGEPOTENTIALS 17
2.5 POTENTIALSBASEDONCHIRALLAGRANGIANS 17
hapter
C 3(cid:4) NUCLEAR MATTER PROPERTIES 21
3.1 THEFERMIGASMODEL 21
3.1.1 Energy-density and pressure of the degenerate Fermi gas 23
3.1.2 Transition to the relativistic regime 23
3.1.3 Extension to non-zero temperature 24
3.2 THEEQUATIONOFSTATE 25
3.2.1 Equation of state of cold nuclear matter 26
3.2.2 Symmetry energy 27
3.2.3 Pressure 28
3.3 SINGLE–NUCLEONPROPERTIES 29
3.3.1 Green’s function and spectral function 30
hapter
C 4(cid:4) NUCLEAR MATTER THEORY 33
4.1 THEMEAN-FIELDAPPROXIMATION 33
4.1.1 Limits of the mean-field approximation 35
4.2 RENORMALISATIONOFTHENNINTERACTION 36
4.3 G-MATRIXPERTURBATIONTHEORY 37
4.4 THEJASTROWVARIATIONALAPPROACH 43
v
vi (cid:4) Contents
4.4.1 Cluster expansion 46
4.4.2 Kinetic energy 55
4.4.3 Low-order variational calculation of nuclear matter energy 57
4.5 ADVANCEDPERTURBATIVEMETHODS 60
4.5.1 Coupled cluster method 60
4.5.2 Self-consistent Green’s function method 62
4.6 MONTECARLOMETHODS 65
4.6.1 Variational Monte Carlo 66
4.6.2 Auxiliary field diffusion Monte Carlo 67
4.7 RELATIVITY 71
4.7.1 Boost corrections to the nucleon-nucleon potential 71
4.7.2 Dirac-Brueckner formalism 73
4.7.3 Relativistic mean-field approximation 74
hapter
C 5(cid:4) ADVANCED VARIATIONAL METHODS 81
5.1 CORRELATEDBASISFUNCTIONSTHEORY 81
5.2 HYPER-NETTED-CHAINSUMMATIONSCHEME 83
5.2.1 Fermi Hyper-Netted Chain 83
5.2.2 RFHNC equations 86
5.3 EXTENSIONTOSPIN-ISOSPINDEPENDENTCORRELATIONS 91
5.3.1 Diagrammatic rules 93
5.3.2 RFHNC/SOC approximation 96
5.3.3 Determination of the correlation functions 98
5.3.4 Applications to the study of nuclear matter properties 99
5.4 CBFEFFECTIVEINTERACTION 102
hapter
C 6(cid:4) NEUTRON STARS 109
6.1 NEUTRONSTARFORMATION 109
6.2 NEUTRONSTARSTRUCTURE 111
6.2.1 Crust region 111
6.2.2 Core region 113
6.3 EQUATIONOFSTATEOFNEUTRONSTARMATTER 114
6.4 HYDROSTATICEQUILIBRIUM 116
6.4.1 The equations of Tolman, Oppenheimer and Volkoff 117
hapter
C 7(cid:4) CONSTRAINTS FROM ASTROPHYSICAL DATA 121
7.1 MEASUREMENTSOFMASSANDRADIUS 121
7.2 NEUTRINOEMISSIONANDCOOLING 124
7.3 GRAVITATIONAL-WAVEOBSERVATIONS 126
7.3.1 Neutron star merger 127
7.3.2 Quasi-normal Modes 129
Contents (cid:4) vii
Outlook 135
ppendix
A A(cid:4) Two- and Three-Body Cluster Contributions 137
Bibliography 142
Index 155
Preface
Nuclear matter can be thought of as a giant nucleus, consisting of an infinite number of
protons and neutrons subject to strong interactions only. Theoretical studies of such a
system, which greatly benefit from the simplifications granted by translation invariance,
are a necessary intermediate step towards the description of atomic nuclei, and provide the
basis for the development of accurate models of matter in the interior of compact stars.
While being a very lively research field, and the subject of a large number of original
papers every year, nuclear matter theory—which lies at the interface of Nuclear Physics
and the Physics of Quantum Fluids—has been seldom discussed in books, and never in
a systematic and comprehensive fashion. In Nuclear Physics textbooks, nuclear matter is
typicallyconfinedtoonechapteratmost,whilemonographsonQuantumFluidsfailtogive
proper emphasis to the complexity of nuclear dynamics.
Thesystematicsofthenuclearcharge-densitydistributionsclearlyindicatesthatinterac-
tions between protons and neutrons are strongly repulsive at short distance, and cannot be
treated in perturbation theory using the basis of eigenstates of the non interacting system.
Moreover, they exhibit a strong dependence on the total spin and isospin of the interacting
particles,S andT,whichentailsacomplexoperatorstructureofthenuclearwavefunction.
The very fact that a two-nucleon bound state is only observed with total spin and ispospin
S = 1 and T = 0—the nucleus of 2H, or deuteron—signals a significant spin-isospin
dependence of the interaction.
Inspiteofthefactthatthedescriptionofnuclearmatterisfundamentalinmanyareasof
nuclearphysicsandastrophysics,doctoralstudents,aswellasyoungresearchersandsenior
scholars approaching this subject, have to resort largely to technical papers, or depend
on the help of more learned colleagues, a problem that was made all the more severe by
the groundbreaking progress of the past two decades. This book, providing a concise but
exhaustiveaccountofnuclearmattertheory,fromtheearlyapproachestothemostadvanced
developments, is meant to fill an empty spot in the existing literature.
Inanefforttokeepthebookasself-containedaspossible,wehaveincludedanintroduc-
tory discussion of the models of nuclear dynamics and of the basic concepts of many-body
theory.Thedifferenttheoreticalapproachestothenuclearmany-bodyproblemareanalysed
followingtheirhistoricaldevelopment,withanemphasisonthemodelsthathavebeenmore
widely applied to study the properties of nuclear matter.
Special attention is given to the recent applications of nuclear matter theory to the
description of neutron star properties. A prominent role, in this context, is played by the
studiesofgravitational-waveemissionfromneutronstars,whoseresultswillbeofparamount
importance in the dawning age of gravitational-wave astronomy. The development of novel
approaches—capable to provide a consistent description of a variety of equilibrium and
non-equilibrium properties, and based on dynamical models applicable over the whole rel-
evant density range—will be needed to fully exploit the potential of future detections of
gravitational wave signals.
This book has greatly benefited from countless discussions with our colleagues, collabo-
rators, and students, whose advice and constructive criticisms we have deeply appreciated.
ParticularmentionisowedtoIngoSick,authorofthepictureappearingonthebookcover,
ix
