Nghiên cứu phương trình trạng thái của chất hạt nhân cân bằng beta trong sao neutron và sao protoneutron (Equation of state of the betastable nuclear matter for neutron and protoneutron stars)

4 http://www.buzzle.com/articles/neutron-1.2 Density dependence of the energy per baryon of NM en-ergy given by the HF calculation using the CDM3Y3 and CDM3Y6 versions [8] of the M3Y-Par

VIETNAM ATOMIC ENERGY INSTITUTE

PhD THESIS

presented by

Ngo Hai Tan

EQUATION OF STATE OF THE BETA-STABLE NUCLEAR MATTER IN NEUTRON STARS AND

PROTO-NEUTRON STARS

Supervisor: Prof Dao Tien Khoa

This thesis presents the results of a consistent mean-field study for

the equation of state (EOS) of the β-stable baryonic matter containing

npeµν particles in the core of cold neutron star (NS) and hot proto-neutron

star (PNS) Within the non-relativistic Hartree-Fock formalism, differentchoices of the in-medium, density-dependent nucleon-nucleon (NN) inter-action have been used Although the considered density dependent NNinteractions have been well tested in numerous nuclear structure and/orreaction studies, they give rather different behaviors of the nuclear sym-metry energy at high baryonic densities which were discussed in the lit-

erature as the stiff and soft scenarios for the EOS of asymmetric NM A

strong impact of the nuclear symmetry energy to the mean-field prediction

of the cooling scenario for NS and thermodynamic properties of the PNSmatter has been found in our study In addition to the nuclear symmetryenergy, the nucleon effective mass in the high-density medium was found

also to affect the thermal properties of hot β-stable baryonic matter of

PNS significantly

Given the EOS of the crust of NS and PNS from the compressibleliquid drop model and relativistic mean-field approach, respectively, thedifferent EOS’s of the core of NS and PNS were used as input for theTolman-Oppenheimer-Volkov equations to obtain the structure of NS andPNS in the hydrostatic equilibrium, in terms of the gravitational mass,radius, central baryonic density, pressure and temperature For the PNSmatter, both the neutrino-free and neutrino-trapped baryonic matters in

β-equilibrium were investigated at different temperatures and entropy per

baryon S/A = 1, 2 and 4 The obtained results show consistently the

strong impact of the nuclear symmetry energy and nucleon effective mass

on the thermal properties and composition of hot PNS matter Maximal

ii

very massive PNS to black hole, based on the results of the hydrodynamic

simulation of a failed supernova of the 40 M ⊙ protoneutron progenitor.The effective, density dependent CDM3Yn interaction has been shown to

be quite reliable in the mean-field description of the EOS of both the coldand hot asymmetric NM

iii

First and foremost, I gratefully express my best thanks to my visor, Prof Dao Tien Khoa for his longtime tutorial supervision of myresearch study at the Institute for Nuclear Science and Technology (INST)

super-in Hanoi, ever ssuper-ince I graduated from Hanoi University of Pedagogy Prof.Khoa has really inspired me to pursuit research in nuclear physics by hisdeep knowledge in teaching and coaching his students and young collabo-rators, and his strict demand on every detail of the research work I wouldalso like to thank Dr J´erˆome Margueron from IPN Lyon for his collabo-ration work in the topic of my PhD Thesis and support of my short visit

to IPN Lyon as well as my attendance at some international meetings inEurope I have gained good skills of the nuclear physics research during

my short visits to IPN Orsay and IPN Lyon, and I am deeply grateful toProf Nguyen Van Giai from IPN Orsay for his help and encouragement

I would like to thank my fellow PhD student, Ms Doan Thi Loan,who gave very important contribution to our common research project onthe mean-field description of the equation of state of nuclear matter Wehave accomplished together many interesting tasks and share a lot of jointmemories during the years working at INST as PhD students I wish toexpress my thanks also to my colleagues in the nuclear physics center atINST, in particular, Dr Do Cong Cuong and Mr Nguyen Hoang Phucfor their useful discussions and kind friendship that made the workingatmosphere in our group very pleasant and lively The helpful discussions

on different physics problems with Dr Bui Minh Loc, a frequent visitor atINST from University of Pedagogy of Ho Chi Minh City, are also thankfullyacknowledged

The present research work has been supported, in part, by NationalFoundation for Science and Technology Development (NAFOSTED) of

iv

LIA collaboration in nuclear physics research between MOST of Vietnamand CNRS and CEA of France I am also grateful to INST and NuclearTraining Center of VINATOM for hosting my research stay at INST withinthe PhD program of VINATOM.

v

Abstract ii

Acknowledgements iv

Abbreviations vi

List of tables xi

List of figures xix

1 Introduction 1 2 Hartree-Fock formalism for the mean-field study of NM 9 2.1 Effective density-dependent NN interaction 13

2.1.1 CDM3Yn effective interaction 14

2.1.2 M3Y-Pn interactions 18

2.1.3 Gogny interaction 20

2.1.4 Skyrme interaction 22

2.2 Explicit Hartree-Fock expressions 23

2.2.1 The finite range interactions 23

2.2.2 Zero-range Skyrme interaction 26

2.3 HF results for the cold asymmetric nuclear matter 27

2.3.1 Saturation properties 27

2.3.2 Total energy of cold NM 31

2.3.3 Nuclear matter pressure 33

2.3.4 Symmetry energy 35

vii

3 HF study of the β-stable NS matter 40

3.1 β equilibrium constraint 41

3.2 EOS of the β-stable npeµ matter 43

3.2.1 Composition of the npeµ matter 43

3.2.2 The cooling of neutron star 47

3.2.3 Pressure of the β-stable npeµ matter 49

3.3 Cold neutron star in hydrodynamical equilibrium 51

3.3.1 Mass-radius relation 52

3.3.2 Total baryon mass 57

3.3.3 Surface red-shift 59

3.3.4 Binding energy 60

3.3.5 Causality condition 60

4 Hartree-Fock study of hot nuclear matter 63 4.1 Explicit HF expressions 66

4.1.1 The finite range interactions 66

4.1.2 Zero-range Skyrme interaction 69

4.2 HF results for the EOS of hot ANM 70

4.2.1 Helmholtz free energy 70

4.2.2 Free symmetry energy 75

4.2.3 Impact of nucleon effective mass on the thermaldy-namical properties of NM 79

4.2.4 Entropy 83

5 HF study of the β-stable PNS matter 89 5.1 β equilibrium constraint 90

5.2 EOS of PNS matter 93

5.2.1 Impact of the free symmetry energy 93

5.2.2 Impact of the in-medium nucleon effective mass 101

5.3 Proto-neutron star in the hydrodynamical equilibrium 103

viii

References 118

List of author’s publications in the present research topic 129

ix

List of Tables

2.1 Parameters of the central term V (C) (r12) in the original M3Y

Paris and M3Y-Pn (n=3,4,5) interactions [15] 15

2.2 Parameters of the density dependence (2.20) of CDM3Yn

interaction [8, 9] 17

2.3 Ranges and strengths of Yukawa functions used in the

ra-dial dependence of the M3Y-Paris, M3Y-P5, and M3Y-P7

interactions [15, 16] 18

2.4 Parameters of the density-dependent term v (DD) (n b , r12)[15,

16] 19

2.5 Ranges and strengths of Gaussian functions used in the

ra-dial dependence of the D1S and D1N interactions [10, 11] 21

2.6 HF results for the NM saturation properties using the

con-sidered effective NN interactions The nucleon effective mass

m ∗ /m is evaluated at δ = 0 and E0 = E(n0, δ = 0)/A Ksym

is the curvature parameter of the symmetry energy (2.6),

and K τ is the symmetry term of the nuclear

incompressibil-ity (2.57) determined at the saturation densincompressibil-ity n δ of

asym-metric NM 29

x

RG, and moment of inertia IG; maximum central densities

nc, ρc and pressure Pc; total baryon number A; surface

red-shift zsurf; binding energy Ebind 56

5.1 Properties of the ν-free and ν-trapped, β-stable PNS at

en-tropy per baryon S/A = 0, 1, 2 and 4, given by the

solu-tions of the TOV equasolu-tions using the EOS’s based on the

CDM3Y3, CDM3Y6 interactions [9] and their soft CDM3Y3s,

CDM3Y6s versions [6] Mmax and Rmax are the maximum

gravitational mass and radius; n c , ρ c , P c , and T c are the

baryon number density, mass density, total pressure, and

temperature in the center of PNS T s is temperature of the

outer core of PNS, at baryon density ρ s ≈ 0.63×1015 g/cm3

Results at S/A = 0 represent the stable configuration of cold

(ν-free) NS [6]. 1055.2 The same as Table 5.1 but obtained with the EOS’s based

on the SLy4 version [19] of Skyrme interaction, M3Y-P7

interaction parametrized by Nakada [16], and D1N version

[18] of Gogny interaction 107

xi

List of Figures

1.1 Structure of neutron star star-facts.html) 4

(http://www.buzzle.com/articles/neutron-1.2 Density dependence of the energy (per baryon) of NM

en-ergy given by the HF calculation using the CDM3Y3 and

CDM3Y6 versions [8] of the M3Y-Paris interaction, which

are associated with the nuclear incompressibility K = 218

and 252 MeV, respectively 6

2.1 Total NM energy per particle E/A at different

neutron-proton asymmetries δ given by the HF calculations using

the CDM3Y3 (lower panel) and CDM3Y6 (upper panel)

in-teractions The solid circles are the saturation densities of

the NM at the different neutron-proton asymmetries 30

2.2 Total NM energy per particle E/A for symmetric NM (upper

panel) and pure neutron matter (lower panel) given by the

HF calculation using different interactions The circles and

crosses are results of the ab-initio calculation by Akmal,

Pandharipande and Ravenhall (APR) [52] and microscopic

Monter Carlo (MMC) calculation by Gandolfi et al [53],

respectively 32

xii

using the effective NN interactions given in Table 2.6 The

shaded areas are the empirical constraints deduced from the

HI flow data [59] 34

2.4 HF results for the NM symmetry energies Esym(n b) given

by the density-dependent NN interactions under study The

shaded (magenta) region marks the empirical boundaries

de-duced from the analysis of the isospin diffusion data and

double ratio of neutron and proton spectra data of HI

col-lisions [60, 61] The square and triangle are the constraints

deduced from the consistent structure studies of the GDR

[62] and neutron skin [50], respectively The circles and

crosses are results of the ab-initio calculation by Akmal,

Pandharipande and Ravenhall (APR) [52] and microscopic

Monter Carlo (MMC) calculation by Gandolfi et al [53],

respectively 36

3.1 The fractions x j = n j /nb of constituent particles of the NS

matter obtained from the solutions of Eqs (3.4) and (3.6)

using the mean-field potentials given by the M3Y-P5 and

D1N interactions 44

3.2 The same as Fig 3.1 but using the mean-field potentials

given by the M3YP7 and Sly4 interactions The circles are

n j values calculated at the maximum central densities nc

given by the solution of the TOV equations 45

3.3 The same as Fig 3.2 but using the mean-field potentials

given by the CDM3Y6 and its soft version interactions 46

xiii

3.4 The proton fraction x p of the β-stable NS matter obtained

from the solutions of Eqs (3.4) and (3.6) using the

mean-field potentials given by the stiff-type CDM3Yn and Sly4

interactions The circles are np values calculated at the

max-imum central densities n c given by the TOV equations The

thin lines are the corresponding DU thresholds (3.11) 47

3.5 The pressure inside the NS matter obtained with the

in-medium NN interactions that give stiff (upper panel) and

soft (lower panel) behavior of E sym (n b), in comparison with

the empirical data points deduced from the astronomical

ob-servation of neutron stars [83] The shaded band shows the

uncertainties associated with the data determination The

circles are P values calculated at the corresponding

maxi-mum central densities given by the TOV equations 50

3.6 The NS gravitational mass versus its radius in comparison

with the empirical data (shaded contours) deduced by ¨Ozel

et al [83] from recent astronomical observations of neutron

stars The circles are values calculated at the maximum

cen-tral densities The thick solid (red) line is the limit allowed

by General Relativity [79] 53

3.7 The same as Fig 3.6, but in comparison with the empirical

data (shaded contours) deduced by Steiner et al [80] from

the observation of the X-ray burster 4U 1608-52 54

3.8 The gravitational mass M given by different EOS’s of the NS

matter plotted versus the corresponding total baryon mass

M b The shaded rectangle is the empirical value inferred

from observations of the double pulsar PSR J0737-3059 by

Podsiadlowski et al [81] . 58

xiv

soft-type (lower panel) in-medium NN interactions The

thick solid (red) lines are the subluminal limit (vs 6 c),

and the vertical arrows indicate the baryon densities above

which the NS matter predicted by the M3Y-P7 interaction

becomes superluminal (see details in the text) 61

4.1 Free energy per particle F/A of symmetric nuclear matter

(SNM) and pure neutron matter (PNM) at different

temper-atures given by the HF calculation using the CDM3Y3 (right

panel) and CDM3Y6 (left panel) interactions [9] (lines), in

comparison with the BHF results (symbols) by Burgio and

Schulze [27] 71

4.2 The same as Fig 4.1 but for the HF results obtained with the

M3Y-P5 (right panel) and M3Y-P7 (left panel) interactions

parametrized by Nakada [15, 16] 72

4.3 The same as Fig 4.1 but for the HF results obtained with

the D1N version [18] of Gogny interaction (left panel) and

SLy4 version [19] of Skyrme interaction (right panel) 73

4.4 Pressure of ANM (upper panel) and PNM (lower panel) at

T=0,20,40 MeV compare to the analysis of the collective

data measured in relativistic HI collision [59] 74

4.5 Free symmetry energy per particle Fsym/A of pure neutron

matter (δ = 1) at different temperatures given by the HF

calculations (4.29) using the CDM3Y3 and CDM3Y6

inter-actions [9] and their soft versions CDM3Y3s and CDM3Y6s

[6] (lines), in comparison with the BHF results (symbols) by

Burgio and Schulze [27] 75

xv

4.6 The same as Fig 4.4 but for the HF results given by the

M3Y-P5 and M3Y-P7 interactions parametrized by Nakada

[15, 16], and the D1N version [18] of Gogny interaction and

SLy4 version [19] of Skyrme interaction 76

4.7 Free symmetry energy (4.29) (lower panels) and internal

symmetry energy (4.31) (upper panels) at different

temper-atures, given by the HF calculations using the CDM3Y6

interaction [9] (Fsym/A)/δ2 curves must be very close if the

quadratic approximation (4.30) is valid 77

4.8 Density profile of the neutron- (upper panel) and proton

ef-fective mass (lower panel) at different neutron-proton

asym-metries δ given by the HF calculation using the CDM3Y6

and CDM3Y3 interactions [9], and the D1N version of Gogny

interaction [18], in comparison with the BHF results

(sym-bols) by Baldo et al [71]. 80

4.9 The same as Fig 4.8 but for the HF results obtained with

the M3Y-P7 and M3Y-P5 interactions [15, 16], and the SLy4

version [19] of Skyrme interaction 81

4.10 Density profile of temperature in the isentropic and

symmet-ric NM given by the HF calculation using different density

dependent NN interactions, in comparison with that given

by the approximation (4.34) for the fully degenerate Fermi

(DF) system at T ≪ T F 82

4.11 Density profile of entropy per particle S/A of symmetric

nuclear matter (SNM) and pure neutron matter (PNM) at

different temperatures, deduced from the HF results (lines)

obtained with the CDM3Y6 and CDM3Y3 interactions [9],

in comparison with the BHF results by Burgio and Schulze

(symbols) [27] 84

xvi

4.13 The same as Fig 4.6 but for the HF results obtained with

the D1N version [18] of Gogny interaction and SLy4 version

[19] of Skyrme interaction 86

4.14 Symmetry part of the entropy per particle (4.33) of pure

neutron matter at different temperatures given by the CDM3Y3and CDM3Y6 interactions [9] and their soft CDM3Y3s and

CDM3Y6s versions [6] Ssym/A is scaled by the

correspond-ing temperature to have the curves well distcorrespond-inguishable at

different T 87

5.1 Total pressure (5.10) of the isentropic ν-free (left panel) and

ν-trapped (right panel) β-stable PNS matter at different

baryon densities n b and entropy per baryon S/A = 1, 2 and

4 The EOS of the PNS crust is given by the RMF

calcu-lation by Shen et al [5], and the EOS of the uniform PNS

core is given by the HF calculation using the CDM3Y6

in-teraction [9] (upper panel) and its soft CDM3Y6s version

[6] (lower panel) The transition region matching the PNS

crust with the uniform core is shown as the dotted lines 94

5.2 Neutron-proton asymmetry δ of the ν-free (left panels) and

ν-trapped (right panels) β-stable PNS matter at different

baryon number densities n b and entropy per baryon S/A =

1, 2 and 4 The EOS of the homogeneous PNS core is given

by the HF calculation using the CDM3Y6 interaction [9] and

its soft CDM3Y6s version [6] 95

xvii

5.3 Entropy per baryon (upper panel) and temperature (lower

panel) as function of baryon number density n b of the

β-stable PNS matter given by the CDM3Y6 interaction [13, 9]

(thick lines) and its soft CDM3Y6s version [6] (thin lines)

in the ν-free (left panel) and ν-trapped (right panel) cases. 96

5.4 Particle fractions as function of baryon number density n b in

the ν-free and β-stable PNS matter at entropy per baryon

S/A = 1, 2 and 4, given by the CDM3Y6 interaction [9]

(upper panel) and its soft CDM3Y6s version [6] (lower panel) 97

5.5 The same as Fig 5.4 but for the ν-trapped, β-stable matter

of the PNS 98

5.6 The same as Fig 5.4, but given by the SLy4 version [19] of

Skyrme interaction (upper panel) and M3Y-P7 interaction

parametrized by Nakada [16] (lower panel) 99

5.7 The same as Fig 5.6 but for the ν-trapped, β-stable PNS

matter 100

5.8 Density profile of neutron and proton effective mass in the

ν-free and β-stable PNS matter at entropy per baryon S/A =

1, 2 and 4, given by the HF calculation using the CDM3Y6

[9] and M3Y-P7 [16] interactions (left panel), the D1N

ver-sion of Gogny interaction [18] (right panel) and SLy4 verver-sion

[19] of Skyrme interaction (right panel) 102

5.9 Density profile of temperature in the ν-free and β-stable

PNS matter at entropy per baryon S/A = 1, 2 and 4,

de-duced from the HF results obtained with the same density

dependent NN interactions as those considered in Fig 5.8 103

xviii

at entropy S/A = 1, 2 and 4 as function of the radius (in

km), based on the EOS of the homogeneous PNS core given

by the CDM3Y6 interaction [9] (upper panel) and its soft

CDM3Y6s version [6] (lower panel) The circle at the end

of each curve indicates the last stable configuration 1045.11 The same as Fig 5.10 but given by the CDM3Y3 interaction

[9] (upper panel) and its soft CDM3Y3s version [6] (lower

panel) 1065.12 The same as Fig 5.10 but given by the SLy4 version of

Skyrme interaction [19] (upper panel) and M3Y-P7

interac-tion parametrized by Nakada [16] (lower panel) 108

5.13 Delay time tBH from the onset of the collapse of a 40 M ⊙

progenitor until the black hole formation as function of the

enclosed gravitational mass MG (open squares) given by the

hydrodynamic simulation [76, 4], and Mmax values given by

the solution of the TOV equations using the same EOS for

the ν-free and β-stable PNS at S/A = 4 (open circles) The

Mmax values given by the present mean-field calculation of

the ν-free and β-stable PNS at S/A = 4 using different

den-sity dependent NN interactions are shown on the correlation

line interpolated from the results of simulation 111

xix

Chapter 1

Introduction

With the physics of unstable neutron-rich nuclei being at the front of modern nuclear physics, the determination of the equation of state(EOS) of asymmetric neutron-rich nuclear matter (NM) becomes also animportant research goal in many theoretical and experimental studies Al-though, in general concept, asymmetric NM is an idealized infinite uniformmatter composed of strongly interacting baryons and (almost free) leptons

fore-at different mass densities and neutron-proton asymmetries, it is in fact a

real physical condition existing in neutron stars which can be observed

from Earth through their radiation of X-rays or radio signals Up to now,about 2000 neutron stars have been detected (mostly as radio pulsars) inthe Milky Way and Large Magellanic Cloud, with the observed gravitationmass of the most massive neutron stars reaching around or slightly above

two solar masses (M G ≈ 2.01 ± 0.04 M ⊙) Above this value stars evolve

into black holes For different theoretical studies, such a large neutron starmass should be possible with the realistic EOS of neutron star matter

In the terrestrial laboratories, the interior of a heavy neutron-rich nucleuslike lead or uranium can be considered as a small fragment of asymmetric

NM, and some basic properties of asymmetric NM were deduced from thestructure studies of heavy nuclei with neutron excess Very important are

1

the saturation properties of NM, in particular, the internal energy pressure

of the symmetric NM (infinite nuclear matter with the same neutron and proton densities) around the saturation baryon number density n0 ≈ 0.17

fm−3 In terms of thermodynamics, EOS often means the dependence of

the pressure P on the mass density ρ and temperature T of NM, while in

the many-body studies of NM it is often discussed as the dependence ofthe internal NM energy on the baryon number density and temperature.From the nuclear astrophysics viewpoint, a realistic EOS of neutronstar matter is a vital input for the astrophysical studies for the structureand formation of cold neutron star (NS) as well as hot proto-neutron star

(PNS) Proto-neutron stars are compact and very hot and neutrino-rich

stellar objects which have the shortest stellar life time in the Universe (it

is around one minute between the birth of PNS following the gravitationalcollapse of a massive progenitor and the appearance of a black hole or

a neutron star) Nevertheless, many complex physics phenomena occurduring these seconds, with PNS contracting, cooling down and eventuallylosing all its neutrino content Very important for the hydrodynamicalmodeling of a compact PNS or NS are its gravitational mass and radius in

the hydrostatic equilibrium With a given EOS of the β-stable neutron rich matter, the mass-radius (M/R) relation of NS or PNS can be determined

from the solution of the Tolman-Oppenheimer-Volkov (TOV) equations[1], which were derived from the Einstein theory of the general relativityassuming the spherical symmetry of the stellar object

where G is the universal gravitational constant, P and ρ are the pressure and mass energy density of NS or PNS, r is the radial coordinate in the Schwarzschild metric, and m is the gravitational mass enclosed within the

sphere of radius r As discussed below in the present thesis, the TOV tions (1.1) are solved numerically using the realistic P (ρ) relation given by

equa-a chosen EOS of the mequa-atter inside NS or PNS As such, EOS meequa-ans not

only P (ρ) but also the composition of the stellar matter Nuclear

astro-physics is an interdisciplinary field that is actively developed and carriedout at different nuclear physics centers in recent years [2] The nuclearastrophysical modeling of the stellar object is based in many cases on theastrophysical observations and extrapolations of what we consider reliablephysical knowledge of the EOS of NM tested in terrestrial laboratories

In fact, the combined use of the astrophysical and nuclear physics data inthe astrophysical studies also offers a unique opportunity to test differenttheoretical nuclear models

In the hot environment of PNS, the entropy per baryon S/A is believed

to be of the order of 1 or 2 Boltzmann constant k B [3] (in the present work

we assume k B = 1) However, the recent astrophysical studies have

sug-gested that S/A might well exceed 4 in the hot core of PNS during a failed

supernova [4], when a very massive progenitor collapses directly to blackhole Such an environment is so extreme that neutrinos are first trappedwithin the PNS matter in the beginning of the core collapse Then, on atime scale of 10-20 s PNS is cooling down mainly through electron neutrinoemission, and after about 40-50 s the PNS matter becomes transparent toneutrinos For a newborn NS, the cooling via neutrino emission can takeplace for 100 to 105 years before the γ cooling period begins [1] Thus,

the knowledge about the EOS of the hot, asymmetric NM is vital for theastrophysics studies of both NS and PNS From the surface of NS andPNS inwards, the baryon matter first forms a low-density, inhomogeneous

crust and, with the increasing baryon density, the matter gradually forms

a uniform core (see Fig 1.1).

Given the EOS’s of the crust of NS and PNS given by the compressible

Figure 1.1: Structure of neutron star.

(http://www.buzzle.com/articles/neutron-star-facts.html)

liquid drop model and relativistic mean-field approach, respectively, ferent EOS’s of the uniform core of NS and PNS predicted in the presentmean-field approach have been used as the input for the TOV equations(1.1) to obtain the basic stellar characteristics in the hydrostatic equi-

dif-librium, like the M/R relation between the gravitational mass and radius,

central matter density and pressure These results were compared with theavailable empirical data to verify the reliability of the considered EOS’s ofasymmetric NM in the astrophysical modeling of NS and PNS

The key quantity to distinguish different nuclear EOS’s of NM is the

nuclear field potential that can be obtained from a consistent

mean-field study, like the relativistic mean-mean-field (RMF) approach [5] or tivistic Hartree-Fock (HF) formalism [6] based on the realistic choice of thenucleon-nucleon (NN) interaction in the high-density nuclear medium Todeduce an in-medium NN interaction starting from the free NN interaction

nonrela-to the form amenable for different nuclear structure and reaction lations still remains a challenge for the microscopic nuclear many-bodytheories Therefore, most of the nuclear reaction and structure studies stilluse different kinds of the effective (density-dependent) NN interaction asin-medium interaction between nucleons Microscopic many-body studieshave shown consistently the strong effect by the Pauli blocking as well

calcu-as higher-order NN correlations with the increcalcu-asing baryon density Suchmedium effects are normally considered as the physics origin of the em-pirical density dependence introduced into various versions of the effective

NN interaction used in the HF mean-field approaches For example, thedensity dependent CDM3Yn versions [8, 9] of the M3Y interaction, whichwas originally constructed to reproduce the G-matrix elements of the Reid[10] and Paris [11] NN potentials in an oscillator basis

In searching for a realistic choice of the effective NN interaction for theconsistent use in the mean-field studies of NM and finite nuclei as well as

in the nuclear reaction calculations, we have performed in the present work

a systematic HF study of asymmetric NM at both zero and finite tures using the CDM3Yn interactions, which have been used mainly in thefolding model studies of the nuclear scattering [8, 12, 13, 14], and the M3Y-

tempera-Pn interactions carefully parametrized by Nakada [15, 16] for use in the

HF studies of nuclear structure For comparison, the same HF study hasalso been done with the D1S and D1N versions of the Gogny interaction[17, 18] and Sly4 version of the Skyrme interaction [19] In the mean-fieldstudies, the EOS of NM is usually associated with density dependence of

the total energy of NM (per baryon) which is expressed as

where the baryon number density (n b = n n + n p) is the sum of the neutron

and proton number densities, δ = (n n − n p )/n b is the asymmetry

parame-ter, and E sym (n b) is the so-called symmetry energy A foremost requisite

to the realistic in-medium NN interaction is that it should give the properdescription of the saturation properties of symmetric NM, i.e., the bindingenergy of symmetric NM of around E

A (n0, δ = 0) ≈ −16 MeV at the

satu-ration density n0, like the HF results obtained with the density dependentCDM3Y3 and CDM3Y6 versions [8] of the M3Y-Paris interaction shown

in Fig 1.2

-20 0 20 40 60 80

Figure 1.2: Density dependence of the energy (per baryon) of NM energy given by the

HF calculation using the CDM3Y3 and CDM3Y6 versions [8] of the M3Y-Paris

inter-action, which are associated with the nuclear incompressibility K = 218 and 252 MeV,

respectively.

Very vital quantity for the determination of the EOS of asymmetric

NM is the nuclear symmetry energy, especially, the behavior of E sym (n b)

with the increasing baryon number density Because the symmetry energy

of NM is determined entirely by the isospin- and density dependence of thein-medium NN interaction, it is directly related to the realistic description

of the structure of finite nuclei with neutron excess Moreover, the edge of the symmetry energy of NM is essential for the determination ofthe chemical potentials of the constituent baryons and leptons that in turndetermine the particle abundances in the stellar objects prior to or afterthe supernova [21] The recent HF studies of NM [22] have shown that thedensity dependence of the symmetry energy of NM given by the effective

knowl-NN interactions considered in the present work is associated with two

dif-ferent (soft and stiff ) behaviors at high baryon densities As a result, these

two families predict very different behaviors of the proton-to-neutron ratio

in the β equilibrium that can imply two drastically different mechanisms

for the neutron star cooling (with or without the direct Urca process) thermore, the difference in the NM symmetry energy is showing up also inthe main properties of NS in the hydrostatic equilibrium [6] that are read-ily obtained from the solutions of the TOV equations (1.1) In this thesis,the strong impact of the nuclear symmetry energy to the basic properties

Fur-of the NS matter as well as PNS matter is illustrated in details based onthe results of a consistent HF study of the NS and PNS matter in the

β-equilibrium.

Another fundamental physics quantity that also impacts the ior of the high-density NM is the nucleon effective mass in the medium,which is determined by the density- and momentum- dependence of thesingle-nucleon or nucleon mean-field potential [9, 23] Microscopic studies

behav-of the single-nucleon potential in the high-density nuclear medium haveshown the important link of the nucleon effective mass to different nuclearphenomena such as the dynamics of heavy-ion collisions at intermediate en-ergies, the damping of low-lying nuclear excitations and giant resonances

as well as the thermodynamical properties of the collapsing stellar ter [23] Therefore, an interesting research topic discussed in this thesis ishow the nucleon effective mass given by different density dependent NNinteractions affects the basic properties of the asymmetric NM at zero andfinite temperatures In particular, the density profiles of the temperature

mat-and entropy of the hot β stable PNS matter The impact of symmetry

energy as well as impact of nucleon effective mass to the properties of NSand PNS matter have been discussed in the two recent publishes by author

of this thesis and collaborations [6, 7]

The structure of this thesis is as follows: the next chapter presents the

HF formalism for the mean-field study of the EOS of asymmetric NM The

EOS of the β-stable NS matter and its composition at zero temperature

are discussed in Chapter 3, with the emphasis on the impact of the clear symmetry energy The results of a consistent HF study of the hot

nu-asymmetric NM and β-stable PNS matter are presented in Chapters 4 and

5, where the strong impact of the nuclear symmetry energy and nucleoneffective mass on the thermal properties and composition of hot PNS mat-ter is investigated In particular, the maximal gravitation masses obtained

with different EOS’s for the neutrino-free β-stable PNS at the entropy per baryon S/A ≈ 4 were used to assess the time of the collapse of a very

massive, hot PNS to black hole, based on the results of the hydrodynamic

simulation of a failed supernova of the 40 M ⊙ protoneutron progenitor.The summary of the present research and the main conclusions are given

in the final chapter

Chapter 2

Hartree-Fock formalism for the field study of NM

mean-Since the pioneer study by Brueckner et al. [24] and many others

in the late 1960’s, there have been many theoretical studies of the EOS

of isospin asymmetric nuclear matter The many-body problem in suchmatter can be treated in the microscopic approach where using the bare NNinteractions obtained from fitting experimental NN scattering phase shiftand deuteron properties [21], in the effective-field theory approach [21], or

in an semi-microscopic approach base on effective interaction Lagrangians

or effective nuclear forces In the latter manner, the nuclear mean fieldpotential is usually obtained from a self-consistent mean field study such asthe relativistic mean-field (RMF) theory or relativistic and non-relativisticHartree-Fock (HF) approaches

In our study, the HF method is applied for an infinite homogeneousnuclear matter with saturated spin, over a wide range of the neutron and

proton densities (n n and n p ) or equivalently of the baryon density n b and

the neutron-proton asymmetry δ At a given temperature, the total energy density E(T, n b , δ) = Ekin(T, n b , δ) + Epot(T, n b , δ) of the NM is determined

9

The single-particle wave function |kστ⟩ is a plane wave, and the

summa-tion in Eqs (2.1) and (2.2) is done separately over the neutron (τ = n) and proton (τ = p) single particle indices In this part we consider the HF

formalism for cold nuclear matter (T=0), that the total energy density is

denoted as E(n b , δ) The nucleon momentum distribution at zero

tempera-ture n στ (k, T = 0) in the spin saturated NM is a step function determined

with the Fermi momentum k F (τ ) = (3π2n τ)1/3 as

The existence of direct (vD) and exchange (vEX) parts in the central

in-medium NN interaction v c comes from the requirement of the NN wave

function to be anti-symmetric Dividing E(n b , δ) over the total baryon

number density n b, we obtain the total NM energy per baryon that is

defined as E/A The pressure P (n b , δ) and incompressibility K(n b , δ) of

the cold NM are determined from

be negligible This is equivalent to the quadratic dependence of the total

energy upon the neutron-proton asymmetry parameter δ up to high NM densities Therefore Esym can be defined as the energy required per particle

to change symmetric NM into pure neutron matter:

where J = Esym(n0) is the value of Esym(n b) at the symmetric NM

satura-tion density n0 ≈ 0.17 fm −3 known as the symmetry energy coefficient, L

and K sym are the slope and curvature of the symmetry energy at saturationdensity

According to Landau theory of infinite Fermi systems [25], the single

particle (s/p) energy e τ (k) in the NM is determined as

e τ (k) = ∂E

∂n τ (k) = t τ (k) + U τ (k) =

~2k2

2m τ + U τ (k), (2.8)

which is the change of the NM energy caused by the removal or addition

of a nucleon with the momentum k The single-nucleon potential U τ (k)

consists of both the HF and rearrangement terms

When the nucleon momentum approaches the Fermi momentum (k →

k F (τ ) ), e τ (k F (τ )) determined from Eqs (2.8)-(2.11) is exactly the Fermi energygiven by the Hugenholtz - van Hove theorem [26] Using the transformation

∂

∂n τ (k)

k →k (τ ) F

where Ω is the total volume of the NM in momentum space, the

rearrange-ment term of the s/p potential U τ at the Fermi momentum can be obtainedas

∂vc

∂k (τ ) F

An important quantity associated with the momentum dependence of the

nucleon s/p potential is the nucleon effective mass m ∗ τ, defined within thenonrelativistic mean-field formalism as

k τ F

]−1

where m is free nucleon mass, U τ is the single nucleon potential obtainedfrom the HF and rearrangement terms The effective mass for nucleon at

2.1 Effective density-dependent NN interaction 13

saturation density, m ∗ n , is about 0.6 to 0.7m n This is usually considered

as a constrain on the EOS caused by the many-body connections

The EOS of NM is determined in HF calculation based on some ent effective interactions Whereas straightforward application of the bare

differ-NN interaction is limited only to light nuclei, the effective interactions areemployed in description the medium effects of a large number of nucleons

by introducing the density-dependence The different density-dependenteffective interactions considered in this HF study of the EOS of NM arecurrently being used in HF calculations of the nuclear structure and insome nuclear reaction studies

inter-action

Because the HF method is the first order of many-body calculations, it

is necessary to have an appropriate in-medium NN interaction The uating an in-medium NN interaction started from the free NN interactionstill remains a challenge for the nuclear many-body theory There are so farstill using different kinds of the effective NN interaction in the microscopicnuclear matter calculations to effectively account for the higher-order NNcorrelations as well as to treat the Pauli blocking effect This effect playsimportant role for the many-body system in which the baryons at largedensities are influenced by the filled Fermi sea of nucleons In the presentresearch, we use different density-dependent versions of the M3Y interac-tion have been used in the HF calculation of symmetric and asymmetric

eval-NM, in the mean field studies of nuclear structure [15, 16] as well as in merous folding model studies of the nucleon-nucleus and nucleus-nucleusscattering [12, 8, 13] We introduce the density dependent versions of theoriginal M3Y effective interactions: CDM3Yn have been used in many

nu-folding model studies and M3Y-Pn have been used in studies of nuclearstructure The HF calculations for NM of these interactions would be com-pared to the same HF study with phenomenological forces of Gogny andSkyrme with free parameters adjusted to reproduce the nuclear experimen-tal data The realistic D1S[17], D1N[18] versions of Gogny interaction andSly4[20] version of Skyrme are briefly presented In this thesis, the HFcalculations are done for isotropic uniform nuclear matter therefore onlythe central part of the density-dependence effective interactions is consid-ered, matrix elements of the spin-orbit and tensor parts between HF statesvanish.

2.1.1 CDM3Yn effective interaction

The M3Y interactions were designed by the MSU group to reproducethe G-matrix elements of the Reid [10] and Paris [11] free NN potentials

in an oscillator basis (further referred to as M3Y-Reid and M3Y-Parisinteraction, respectively) The central part of the original M3Y effective

interaction are introduced in terms of three Yukawas (i = 1, 2, 3) which

respectively represent short, medium and long range of the force [7]:

where parameter 1/µ i = m i c/ ~ is interaction range and m i is the mass

of exchanged meson in NN interaction Yukawa form factor had beenproposed by Yukawa in 1937 in the one-pion exchange potential (OPEP)

to describe the interaction of nucleons with each other by the exchange of

2.1 Effective density-dependent NN interaction 15

one or several mesons of which OPEP is the simplest form Corresponding

to the masses of mesons ω, K and π about 790, 490 and 140 MeV, values of interaction range (Compton wavelength) 1/µ i and strength t i of the originalM3Y-Paris are showed in Table 2.3 The projection operators project ontothe singlet even (SE), triplet even (TE), singlet odd (SO) and triplet odd(TO) parts of the nuclear two-body wave function

Interaction i 1/a c i t(SE)i t(TE)i t(SO)i t(TO)i

M3Y-Paris 1 0.25 11446.0 13967.0 -1418.0 11345.0

2 0.40 -3556.0 -4594.0 950.0 -1900.0

3 1.414 -10.463 -10.463 31.389 3.488M3Y-P5 1 0.25 8027.0 5576.0 -1418.0 11345.0

2 0.40 -2650.0 -417.0 2880.0 -1780.0

3 1.414 -10.463 -10.463 31.389 3.488

The density dependent effective interactions CDM3Yn are based onthe original M3Y-Paris Since the original M3Y interaction gives a wrongdescription of the saturation properties of normal NM, the realistic densitydependence are introduced into this interaction to reproduce the satura-

tion properties of cold, symmetric NM at n b ≃ 0.17 fm −3 In the HF

calculations, CDM3Yn terms are usually introduced in the Direct (D) andExchange (EX) components:

vD(EX)(r) = v00D(EX)(r)+v10D(EX)(r)σ1·σ2+v01D(EX)(r)τ1·τ2+v11D(EX)(r)(σ1·σ2)(τ1·τ2),

(2.18)For the spin-saturated NM, the contributions of spin-dependent terms in

Eq (2.18) to the NM energy are averaged out, therefore only the

spin-independent (v00 and v01) components of the central NN interaction areneeded in the HF calculation of NM The density dependence have beenintroduced in the CDM3Yn interaction in the form of coupling constants

this density dependence, the radial parts v00(01)D(EX)(r) are kept unchanged as

determined from the M3Y-Paris interaction [11] in terms of three Yukawas:

2.1 Effective density-dependent NN interaction 17

Table 2.2: Parameters of the density dependence (2.20) of CDM3Yn interaction [8, 9]

Table 2.3 lists the parameters of original M3Y in Y form.

The isocalar density dependence F0(n b) has been set to reproduce thesaturation properties of symmetric NM and gives the nuclear incompress-

ibility K = 218 and K = 252 MeV with the CDM3Y3 and CDM3Y6

versions [8], respectively These interactions, especially the CDM3Y6 sion, have been well tested in numerous folding model analyses of the elas-tic nucleon-nucleus [30], nucleus-nucleus scattering [32], and the charge-exchange scattering to the isobar analog states [31, 35, 9] The parameters

ver-of the isovector density dependence F1(n b) were determined to consistentlyreproduce the BHF results for the IV term of the nucleon optical poten-tial (OP) in asymmetric NM (JLM)[43] as well as the measured charge-

exchange (p, n) and (3He,t) data for the isobar analog states in the folding

Table 2.3: Ranges and strengths of Yukawa functions used in the radial dependence of

the M3Y-Paris, M3Y-P5, and M3Y-P7 interactions [15, 16].

Interaction i a i Y00D(i) Y01D(i) Y00EX(i) Y01EX(i)

The M3Y-Pn interactions have been parametrized by Nakada [15, 16]

to consistently reproduce the NM saturation properties and ground stateshell structure for doubly-closed shell nuclei as well as unstable nuclei close

to the neutron dripline The author had made some modifications to theparameters of central part of the original M3Y-Paris that are shown inTable 2.2 in terms of spin-isospin channels or in 2.3 in terms of direct-

2.1 Effective density-dependent NN interaction 19

Table 2.4: Parameters of the density-dependent term v (DD) (n b , r12 )[15, 16]

Interaction α(SE) t(SE) α(TE) t(TE)

of doubly magic nuclei including 100Sn and the even-odd mass differences

of the Z =50 and N =82 nuclei as in [16] The longest-range part (i = 3)

of Yukawa functions are kept identical to the central channels of the pion exchange potential (OPEP) as in the original interaction The densitydependence in M3Y-Pn is introduced by adding a zero-range contact force

one-v (DD) [7]

v(n b , r) = v(C)(r) + v(DD)(n b , r) (2.23)

v (DD) (n b , r12) = (t(SE)PSE[n b (r1)]α (SE) +t (T E) PTE[n b (r1)]α(SE))δ(r12), (2.24)

with the parameters shown in Table 2.4

From the form of M3Y-Pn effective interaction, one can obtain theirdirect and exchange parts within HF calculations [7]

v00(01)D(EX)(n b , r) = v00(01)D(EX)(r) + dD(EX)00(01)(n b , r), (2.25)

2.1.3 Gogny interaction

Phenomenological effective interactions use the direct parametrization

of effective NN interactions They might be less fundamental than thoseobtained by fitting the HF mean field results to the experimental data asCDM3Y and M3Y-Pn, however their simplicity of the calculations can give

a somewhat better physical insight with the simple relations connectingdifferent nuclei properties In these phenomenological effective interactions,the finite-range Gogny interaction and zero-range Skyrme interaction havebeen used in several nuclear structure studies

First introduced in 1975 [44], the Gogny force in HF formalism for NMconsists of two Gaussians corresponding to medium range and short rangecentral force and a zero range density dependent term:

2.1 Effective density-dependent NN interaction 21

properties of many nuclei in the first parametrization D1 [45] However,the surface energy obtained by this version was too high and this versionwas not able to reproduce fission barriers, which are corrected in a newparametrization D1S [17] However D1S is still unable to reproduce theneutron matter EOS then the new versions of the Gogny D1N [18] remedy

to this defect and can reproduce the neutron matter EOS better thanthe D1S version while still giving simultaneously good descriptions of thenuclear structure properties and nuclear mass data

Table 2.5: Ranges and strengths of Gaussian functions used in the radial dependence of

the D1S and D1N interactions [10, 11].

-From the form of Gogny interaction in Eq (2.27), the central part can

be expressed in terms of Direct and Exchange as:

2.1.1... still unable to reproduce theneutron matter EOS then the new versions of the Gogny D1N [18] remedy

to this defect and can reproduce the neutron matter EOS better thanthe D1S version while