thermal e ects in supernova matter - graduate physics and...

Thermal Effects in Supernova Matter

A Dissertation Presented

by

Constantinos Constantinou

to

The Graduate School

in Partial Fulfillment of the Requirements

for the Degree of

Doctor of Philosophy

in

Physics and Astronomy

Stony Brook University

January 2013


The Graduate School


We, the dissertation committee for the above candidate for the Doctor ofPhilosophy degree, hereby recommend acceptance of this dissertation.

Jim Lattimer - AdvisorProfessor, Department of Physics and Astronomy

Madappa Prakash - AdvisorProfessor, Department of Physics and Astronomy, Ohio University

Thomas Kuo - Committee ChairProfessor Emeritus, Department of Physics and Astronomy

Harold MetcalfDistinguished Teaching Professor, Department of Physics and Astronomy

Derek TeaneyAssistant Professor, Department of Physics and Astronomy

Roy LaceyProfessor, Department of Chemistry

Nuggehalli Narayan AjitanandSenior Research Scientist, Department of Chemistry

This dissertation is accepted by the Graduate School

Charles TaberInterim Dean of the Graduate School

ii

Abstract of the Dissertation

Thermal Effects in Supernova Matter

by


Doctor of Philosophy

in

Physics and Astronomy


2013

A crucial ingredient in simulations of core collapse supernova (SN)

explosions is the equation of state (EOS) of nucleonic matter for

densities extending from 10−7 fm−3 to 1 fm−3, temperatures up

to 50 MeV, and proton-to-baryon fraction in the range 0 to 1/2.

SN explosions release 99% of the progenitor star’s gravitational

potential energy in the form of neutrinos and, additionally, they are

responsible for populating the universe with elements heavier than56Fe. Therefore, the importance of understanding this phenomenon

cannot be overstated as it could shed light onto the underlying

nuclear and neutrino physics.

A realistic EOS of SN matter must incorporate the nucleon-nucleon

interaction in a many-body environment. We treat this prob-

lem with a non-relativistic potential model as well as relativistic

mean-field theoretical one. In the former approach, we employ the

Skyrme-like Hamiltonian density constructed by Akmal, Pandhari-

pande, and Ravenhall which takes into account the long scatter-

iii

ing lengths of nucleons that determine the low density characteris-

tics. In the latter, we use a Walecka-like Lagrangian density sup-

plemented by non-linear interactions involving scalar, vector, and

isovector meson exchanges, calibrated so that known properties of

nuclear matter are reproduced. We focus on the bulk homogeneous

phase and calculate its thermodynamic properties as functions of

baryon density, temperature, and proton-to-baryon ratio. The ex-

act numerical results are then compared to those in the degenerate

and non-degenerate limits for which analytical formulae have been

derived. We find that the two models bahave similarly for densities

up to nuclear saturation but exhibit differences at higher densities

most notably in the isospin susceptibilities, the chemical potentials,

and the pressure.

The importance of the correct momentum dependence in the sin-

gle particle potential that fits optical potentials of nucleon-nucleus

scattering was highlighted in the context of intermediate energy

heavy-ion collisions. To explore the effect momentum-dependent

interactions have on the thermal properties of dense matter we

study a schematic model constructed by Welke et al. in which the

appropriate momentum dependence that fits optical potential data

is built through finite-range exchange forces of the Yukawa type.

We look into the finite-temperature properties of this model in the

context of infinite, isospin-symmetric nucleonic matter. The exact

numerical results are compared to analytical ones in the quantum

regime where we rely on Landau’s Fermi-Liquid Theory, and in

the classical regime where the state variables are obtained through

a steepest descent calculation. Detailed comparisons with simi-

larly calibrated Skyrme models are also performed. We find that

the high-density behavior of the thermal pressure is once again a

differentiating feature. We attribute this to the temperature de-

pendence of the energy spectrum of the finite-range and the meson-

exchange models which leads to a higher specific heat and thus a

lower pressure.

iv

Contents

List of Figures ix

List of Tables xi

Acknowledgements xii

1 Introduction 1

1.1 Stellar Evolution . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Core-Collapse Supernovae . . . . . . . . . . . . . . . . . . . . 2

1.3 Equation of State . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3.1 Supernova EOS . . . . . . . . . . . . . . . . . . . . . . 4

1.3.2 Experimental Constraints of the EOS . . . . . . . . . . 6

1.3.2.1 Neutron Star Observations . . . . . . . . . . . 6

1.3.2.2 Laboratory Constraints . . . . . . . . . . . . 7

1.4 Scope and Organization . . . . . . . . . . . . . . . . . . . . . 11

2 Non-Relativistic Potential 13

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2 Single-Particle Energy Spectrum . . . . . . . . . . . . . . . . . 15

2.3 Zero Temperature . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.3.1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.4 Nuclear Matter at Finite Isospin

Asymmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.5 Finite Temperature . . . . . . . . . . . . . . . . . . . . . . . . 29

2.5.1 Thermal Effects . . . . . . . . . . . . . . . . . . . . . . 31

2.5.2 Numerical Notes . . . . . . . . . . . . . . . . . . . . . 33

2.5.2.1 Results . . . . . . . . . . . . . . . . . . . . . 35

vi

2.5.3 Limiting Cases . . . . . . . . . . . . . . . . . . . . . . 39

2.5.3.1 Degenerate Limit . . . . . . . . . . . . . . . . 39

2.5.3.2 Non-degenerate Limit . . . . . . . . . . . . . 44

2.5.3.3 Results . . . . . . . . . . . . . . . . . . . . . 46

2.5.4 Specific Heat . . . . . . . . . . . . . . . . . . . . . . . 49

3 Mean Field Theory 53

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.2 Mean Field Approximation . . . . . . . . . . . . . . . . . . . . 55

3.3 Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . 55


3.4.1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 63


3.5.1 Numerical Notes . . . . . . . . . . . . . . . . . . . . . 69

3.5.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

3.5.3 Susceptibilities . . . . . . . . . . . . . . . . . . . . . . 73

3.5.3.1 Results . . . . . . . . . . . . . . . . . . . . . 78

3.5.4 Limiting Cases . . . . . . . . . . . . . . . . . . . . . . 79

3.5.4.1 Degenerate Limit . . . . . . . . . . . . . . . . 79

3.5.4.2 Non-Degenerate Limit . . . . . . . . . . . . . 80

3.5.4.3 Results . . . . . . . . . . . . . . . . . . . . . 85

3.5.5 Specific Heat . . . . . . . . . . . . . . . . . . . . . . . 88

4 Finite Range Interactions 95

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

4.2 Model Hamiltonian and Calibration . . . . . . . . . . . . . . . 99


4.3.1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 101


4.4.1 Numerical Notes . . . . . . . . . . . . . . . . . . . . . 103

4.4.2 Degenerate Limit . . . . . . . . . . . . . . . . . . . . . 103

4.4.3 Non-Degenerate Limit . . . . . . . . . . . . . . . . . . 104

4.4.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

4.4.5 Specific Heat . . . . . . . . . . . . . . . . . . . . . . . 115

vii

5 Conclusions 119

5.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

5.2 Advances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

5.3 Findings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

5.4 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

Bibliography 125

A APR State Variables 126

B JEL Notes 133

C MDYI Non-Degenerate CV 137

C.1 Number Density . . . . . . . . . . . . . . . . . . . . . . . . . . 137

C.2 Kinetic Energy Density . . . . . . . . . . . . . . . . . . . . . . 138

C.3 Exchange Potential . . . . . . . . . . . . . . . . . . . . . . . . 139

viii

List of Figures

2.1 T=0 E/A: APR vs. Ska . . . . . . . . . . . . . . . . . . . . . 21

2.2 T=0 Pressure: APR vs. Ska . . . . . . . . . . . . . . . . . . . 21

2.3 T=0 Neutron chemical potential : APR vs. Ska . . . . . . . . 22

2.4 T=0 Proton chemical potential : APR vs. Ska . . . . . . . . . 22

2.5 T=0 Neutron-Neutron Susceptibility : APR vs. Ska . . . . . . 23

2.6 T=0 Neutron-Proton Susceptibility : APR vs. Ska . . . . . . 23

2.7 Approximations to E/A of pure neutron matter . . . . . . . . 25

2.8 Tracing the minima of E/A. . . . . . . . . . . . . . . . . . . . 28

2.9 T=20 MeV Energy per particle : APR vs. Ska . . . . . . . . . 35

2.10 T=20 MeV Pressure : APR vs. Ska . . . . . . . . . . . . . . . 36

2.11 T=20 MeV Chemical potential : APR vs. Ska . . . . . . . . . 36

2.12 T=20 MeV Entropy per particle : APR vs. Ska . . . . . . . . 37

2.13 T=20 MeV Isentropes : APR vs. Ska . . . . . . . . . . . . . . 37

2.14 T=20 MeV Neutron-neutron susceptibility : APR vs. Ska . . 38

2.15 T=20 MeV Neutron-proton susceptibility : APR vs. Ska . . . 38

2.16 APR thermal energy with limits at 20 MeV . . . . . . . . . . 46

2.17 APR thermal pressure with limits at 20 MeV . . . . . . . . . . 47

2.18 APR chemical potential with limits at 20 MeV . . . . . . . . . 47

2.19 APR entropy with limits at 20 MeV . . . . . . . . . . . . . . . 48

2.20 APR neutron-neutron susceptibility with limits at 20 MeV . . 48

2.21 APR neutron-proton susceptibility with limits at 20 MeV . . . 49

2.22 APR specific heat with limits at 20 MeV . . . . . . . . . . . . 52

2.23 T=20 MeV specific heat : APR vs. Ska . . . . . . . . . . . . . 52

3.1 T=0 E/A: MFT vs. SkM* . . . . . . . . . . . . . . . . . . . . 64

3.2 T=0 Pressure: MFT vs. SkM* . . . . . . . . . . . . . . . . . . 64

3.3 T=0 Neutron chemical potential : MFT vs. SkM* . . . . . . . 65

ix

3.4 T=0 Proton chemical potential : MFT vs. SkM* . . . . . . . 65

3.5 T=0 Neutron-Neutron Susceptibility : MFT vs. SkM* . . . . 66

3.6 T=0 Neutron-Proton Susceptibility : MFT vs. SkM* . . . . . 66

3.7 T=20 MeV Energy per particle : MFT vs. SkM* . . . . . . . 71

3.8 T=20 MeV Pressure : MFT vs. SkM* . . . . . . . . . . . . . 72

3.9 T=20 MeV Chemical potential : MFT vs. SkM* . . . . . . . . 72

3.10 T=20 MeV Entropy per particle : MFT vs. SkM* . . . . . . . 73

3.11 T=20 MeV Neutron-neutron susceptibility : MFT vs. SkM* . 78

3.12 T=20 MeV Neutron-proton susceptibility : MFT vs. SkM* . . 78

3.13 MFT thermal energy with limits at 20 MeV . . . . . . . . . . 85

3.14 MFT thermal pressure with limits at 20 MeV . . . . . . . . . 86

3.15 MFT chemical potential with limits at 20 MeV . . . . . . . . 86

3.16 MFT entropy with limits at 20 MeV . . . . . . . . . . . . . . 87

3.17 MFT neutron-neutron susceptibility with limits at 20 MeV . . 87

3.18 MFT neutron-proton susceptibility with limits at 20 MeV . . . 88

3.19 MFT specific heat with limits at 20 MeV . . . . . . . . . . . . 94

3.20 Specific heat : MFT vs. SkM* . . . . . . . . . . . . . . . . . . 94

4.1 Optical Potential: Microscopic Calculations vs. Fits to Data . 97

4.2 Optical Potential: Microscopic Calculation vs MDYI . . . . . 98

4.3 MDYI energy per particle at T = 0. . . . . . . . . . . . . . . . 101

4.4 MDYI pressure at T = 0. . . . . . . . . . . . . . . . . . . . . . 102

4.5 MDYI chemical potential at T = 0. . . . . . . . . . . . . . . . 102

4.6 MDYI E/A with limits at 20 MeV . . . . . . . . . . . . . . . 111

4.7 MDYI pressure with limits at 20 MeV . . . . . . . . . . . . . 112

4.8 MDYI chemical potential with limits at 20 MeV . . . . . . . . 112

4.9 MDYI S/A with limits at 20 MeV . . . . . . . . . . . . . . . . 113

4.10 T=20 MeV Thermal Energy : MDYI vs. SkM* . . . . . . . . 113

4.11 T=20 MeV Thermal Pressure : MDYI vs. SkM* . . . . . . . . 114

4.12 T=20 MeV Thermal Chemical Potential : MDYI vs. SkM* . . 114

4.13 T=20 MeV Entropy : MDYI vs. SkM* . . . . . . . . . . . . . 115

4.14 MDYI specific heat with limits at 20 MeV . . . . . . . . . . . 118

4.15 Specific heat : MDYI vs. SkM* . . . . . . . . . . . . . . . . . 118

x

List of Tables

2.1 Parameter values for HAPR . . . . . . . . . . . . . . . . . . . . 15

2.2 Parameter values for HSka. . . . . . . . . . . . . . . . . . . . . 20

2.3 Saturation properties of symmetric nuclear matter. . . . . . . 20

2.4 Asymmetry Coefficients. . . . . . . . . . . . . . . . . . . . . . 29

2.5 Non-relativistic JEL coefficients. . . . . . . . . . . . . . . . . . 34

3.1 MFT Calibration . . . . . . . . . . . . . . . . . . . . . . . . . 62

3.2 Parameter values for HSkM∗ . . . . . . . . . . . . . . . . . . . 62

3.3 Saturation properties of symmetric nuclear matter:MFT vs. SkM* 62

3.4 Asymmetry Coefficients. . . . . . . . . . . . . . . . . . . . . . 63

3.5 JEL Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . 70

4.1 MDYI Calibration . . . . . . . . . . . . . . . . . . . . . . . . 100

xi

Acknowledgements

I would like to thank my advisors Gerry Brown, Madappa Prakash, and Jim

Lattimer for their teaching, support, and patience with me over the course of

my graduate studies; with the clarification that any and all mistakes herein,

are my own.

Thanks are also due to my collaborators Ken Moore and Brian Muccioli

both of whose help has been invaluable for the completion of this dissertation.

Chapter 1

Introduction

1.1 Stellar Evolution

A star begins its life when a relatively large (M ∼ 104M), cool (T ∼ 100 K)

interstellar cloud collapses to a high density. The onset of the collapse takes

place when the Jean criterion [1]:

GM2

R>

3

2kBT

M

m, (1.1)

is met; here M , R, and T , are the mass, radius, and temperature of the cloud

respectively, and m is the mean molecular weight of its components. The Jean

criterion says that collapse ensues when the cloud’s gravitational potential

energy (GPE) exceeds its thermal energy.

For a bound object to form, the virial theorem must be satisfied: Half of

the GPE released in the collapse must be stored in the system and the other

half must be radiated away. The collapse halts when the temperature at the

center of the protostar is large enough (∼ 107 K) so that hydrogen fusion

begins, thus producing the energy needed to comply with the virial theorem

without the need for further contraction.

A star remains in a steady-state until the hydrogen in its core is exhausted.

This sets the stage to ignite hydrogen in a layer surrounding the core which

heats the core sufficiently for helium burning to start. At this point, small stars

(M? < 8 M) will shed mass in excess of the Chandrasekhar limit (1.4 M)

and eventually become white dwarfs. Massive stars (M? > 8 M) however, will

1

proceed to fuse 42He into 12C and so on up to 56Fe. This procedure terminates

at iron because this is the most tightly bound nucleus and hence beyond iron,

fusion becomes endothermic.

1.2 Core-Collapse Supernovae

The iron core of an evolved massive star is at a temperature T ∼ 109 K and a

density n ∼ 1010 gcm−3. Under these conditions, the Fermi energy of electrons

is high enough that electron capture becomes favorable. This process leads to

the neutronization of matter and the production of neutrinos:

p+ + e− → n+ νe

At this stage, the total entropy per baryon is [2]

S = XHSFe + Sexc

56+XnSn +XeSe = 0.93 (1.2)

where

SFe =5

2+ ln

[(56mT

2πh2

)3/2 1

nFe

]' 17 (1.3)

is the entropy due to the translational motion per iron nucleus,

Sn =5

2+ ln

[(mT

2πh2

)3/2 1

nn

]' 12.9 (1.4)

is the entropy of translational motion per free neutron,

Sexc = 56π2

2

(T

TF

)' 4.8 (1.5)

is the entropy per excited state of iron nuclei, and

Se =π2T

µe' 1.1 (1.6)

is the entropy per electron. Above XH , Xn, Xe are the nucleons-in-nuclei

fraction, the free neutron fraction, and the electron fraction, respectively.

While the number of neutrinos produced in electron capture is small, they

2

escape the system. Thus e− capture is out of equilibrium and as such it

contributes to the entropy 0.43 units. The neutrinos carry away 0.28 units.

However, as the number of neutrinos increases, their Fermi energy becomes

large enough, such that the neutrino mean free path

λν ∝1

ε2ν

gets very small and the neutrinos become trapped. This means that β-

equilibrium is achieved and no further increase in entropy takes place (the

trapped neutrinos heat up the system somewhat, but this only contributes

0.04 units of entropy).

Being that dripped neutrons require S ' 8, it follows that nuclei persist

until the core contracts to a density around nuclear saturation no (' 0.16

fm−3), at which point homogeneous nucleonic matter emerges. This is further

compressed to ∼ 3no where the Fermi energy of nucleons dominates their

attraction hence inhibiting contraction causing the core to rebound and create

an outward moving shock wave that ejects the envelope of the star and releases

its GPE (∼ 1053 ergs) primarily in the form of neutrinos (for which the star has

now become transparent). At later times, as ionized matter becomes dilute

and cools, it recombines; opacity is reduced and photons can also escape.

The fact that the collapse is almost isentropic allows us to draw rough

conclusions about how the temperature and the density are related to each

other by simply examining entropy adiabats.

1.3 Equation of State

In general, an equation of state (EOS) provides the link between the micro-

scopic interactions of the elementary constituents of the system under consid-

eration and its macroscopic thermodynamic properties.

For simplicity, consider a classical, monoatomic gas with two-body, position-

dependent interactions. Its energy has the form

E(p, q) =N∑a=1

p2a

2m+ U(q). (1.7)

3

The corresponding partition function is given by

Z =∫e−E(p,q)/TdΓ (1.8)

=∫e−

p2

2m1

Td3p1 . . . d

3pN

∫e−

UT d3r1 . . . d

3rN , (1.9)

which leads to the free energy

F = −T lnZ = Fideal − T ln(

1

V NZ). (1.10)

Assuming that the gas is rarefied enough such that only one pair of atoms can

collide at any time, and that N 1 gives

F = Fideal +N2T

VB(T ) (1.11)

where

B(T ) =1

2

∫ (1− e−

U12T

)dV (1.12)

In eq. (1.12), U12 is the interaction energy of a pair of atoms and it depends

only on the coordinates of the two atoms. From (1.11), we find the pressure

P = −∂F∂V

=NT

V

(1 +

NB(T )

V

). (1.13)

The second term is the contribution of the interaction to the gas pressure.

The above considerations illustrate how interactions affect the thermal

properties of state variables. In subsequent chapters, we will see how interac-

tions play a crucial role in dense matter (i.e. non-dilute situations).

1.3.1 Supernova EOS

For supernova as well as neutron star matter, nucleons are the fundamental

degrees of freedom. Therefore any realistic EOS must incorporate the nucleon-

nucleon interaction in a many-body environment.

There are two common ways by which this problem is treated: potential

models and field theoretical models. In the former case, a non-relativistic

Hamiltonian density is traditionally constructed by approximating non-local

exchange forces by contact interactions. In the latter, a relativistic Lagrangian

4

is often used where the nucleon-nucleon interaction is mediated by the ex-

change of mesons. Both kinds of models are calibrated to properties of nuclear

matter at saturation. Modern potential models are fit to the Nijmegen scatter-

ing database [3, 4] so as to reproduce observables such as the mass difference

between the charged pions and the neutral one, the phase shifts between np

and pp in the 1S0 channel and the nn scattering length and effective range.

These may also be supplemented by 3-body potentials gauged on the bind-

ing energies of light nuclei. The ground state (effectively, the energy density)

is obtained through minimization procedures (the variational approach is a

classic example) which take into account many-body correlations.

The thermodynamics corresponding to a specific model are then deter-

mined by using the temperature T , the density n = nn + np, and the proton

fraction x = np/n as inputs. The outputs of the EOS are the energy per

baryon E/A, the pressure P , the entropy per baryon S/A, the chemical po-

tentials µi as well as their derivatives with respect to n, T , and x. These

quantities are, in turn, used as inputs in calculations of neutron star structure

and in hydrodynamic simulations of core-collapse supernovae.

In the context of hydrodynamics, a moving fluid is described by its velocity

distribution ~v = ~v(~x, t) and any two state variables pertaining to the fluid (e.g.

pressure P (~x, t) and density n(~x, t)). Therefore the complete specification of

fluid dynamics requires five equations [5]. For the simple case of an ideal,

Newtonian fluid these are:

• Euler’s equations∂~v

∂t+ (~v · ∇)~v = −∇P

n+ ~f (1.14)

where ~f is an external force.

• Continuity equation∂n

∂t+∇ · (n~v) = 0 (1.15)

• Adiabatic equation∂S

∂t+ ~v · ∇S = 0 (1.16)

For a real fluid one has to consider shears, viscocities, heat conductivity

etc. Nevertheless, the important point is that knowledge of ~v, P , n, and the

5

EOS completely determines the thermodynamic state of a moving fluid such

as the nucleonic matter encountered in the supernova problem. In the case of

supernovae, general relativistic modifications to the above equations are also

necessary [6, 7].

1.3.2 Experimental Constraints of the EOS

1.3.2.1 Neutron Star Observations

The structure of a neutron star is governed by the Tolman-Oppenheimer-

Volkoff (TOV) equations [8]

m(r) = 4π∫ r

0ρ(r′)r′2dr′ (1.17)

dp

dr= −(ε+ p)[Gm(r) + 4πGr3p]

r[r − 2Gm(r)](1.18)

and the EOS :

p = p(ε). (1.19)

Here, ρ(r′) is the mass density at a radius r′, ε and p are the energy density and

the pressure of the system respectively, and G is the gravitational constant.

The TOV equations are derived from the Einstein equation

Gµν = 8πGTµν

when a perfect fluid described by the energy-momentum tensor

Tµν = (ε+ p)UµUν + pgµν ,

is placed in a general, static, spherically symmetric metric

ds2 = −e2α(r)dt2 + e2β(r)dr2 + r2dΩ2.

In eq. (1.17), m(r) is the mass inside a radius r, and equation (1.18) describes

hydrostatic equilibrium. The solution of this system of coupled equations for a

given EOS predicts the mass-radius diagram of a neutron star and accordingly

specific values for the maximum mass Mmax, the radius Rmax of the maximum

mass configuration and the maximum frequency Ωmax that a neutron star can

6

have.

Currently, observational evidence has placed a lower limit on the maximum

mass [9]:

Mmax ≥ 1.97 M.

This measurement was achieved by exploiting the Shapiro effect (gravitational

time delay) in binary pulsars where the pulse arrival times become longer when

the pulsar is behind its companion. This time delay is given by

∆t = −2∫

Φds (1.20)

where

Φ =GMcomp

r(1.21)

and the integration is over the flat spacetime path connecting the source and

the observer.

Thus the companion mass is determined and afterwards it is used in Kepler’s

Third Law

P 2 =4π2

G(Mcomp +Mpulsar)a3 (1.22)

to extract the pulsar mass.

Direct measurements of pulsar periods give [10]

Ωmax ≥ 114 rad s−1

The radius has been indirectly confined in a range [11]

10 km ≤ R ≤ 12.5 km

by analyzing photospheric emission data and the thermal spectra of neutron

stars.

1.3.2.2 Laboratory Constraints

In addition to neutron star observations, the EOS can be constrained from

laboratory experiments. In particular,

• High energy electron scattering from heavy nuclei produces a differential

7

cross-section that exhibits oscillations as a function of the scattering

angle θ. Successive maxima are separated by

qR ' π (1.23)

where q is the momentum transfer at a given scattering angle:

q = 2Ee sin(θ/2) (1.24)

and R is the liquid drop radius

R = roA1/3 ; A = N + Z. (1.25)

The nuclear saturation density is obtained from

no =(

4

3πr3

o

)−1

= 0.16± 0.01 fm−3. (1.26)

• Using the measured masses of free nucleons and of atomic nuclei in their

ground state, one can determine the nuclear binding energy:

B(N,Z) = M(N,Z)− (Nmn + Zmp) (1.27)

This is then fit to the liquid drop mass formula

B(N,Z) = EoA− bsurfA2/3 − S2(N − Z)2

A− bCoulZ2A−1/3 (1.28)

which gives

– Energy per particle

Eo =E

A= −16± 1 MeV

– Symmetry Energy

S2 = 30± 5 MeV

• Inelastic collisions produce collective excitations in nuclei known as giant

resonances. The peak energy of the giant monopole resonance for a given

8

nucleus is connected to its incompressibility KA via

EGMR =(

KA

m < r2 >

)1/2

(1.29)

where m is the nucleon mass and < r2 > is the mean square mass radius

of the ground state.

The liquid drop model parametrization for KA is

KA = K +KsurfA−1/3 +Kτ

(N − Z)2

A2+KCoul

Z2

A4/3(1.30)

By fitting the above equation to GMR data we get

– Compression Modulus

K = 230± 30 MeV

– Asymmetry Incompressibility

Kτ = −550± 100 MeV

• The energy distribution of neutrons scattered inelastically from heavy

nuclei is given by the statistical relation

N(En) ∝ ρ(U)Enσc(En, U), (1.31)

where ρ(U) is the level density of the final nucleus at the excitation

energy U , and σ(En, U) is the cross-section for the formation of an in-

termediate nucleus when the final one is at U .

The Fermi gas expression for the level density is

ρ(U) ∝ 1

U2e2(aU)1/2 . (1.32)

Therefore, neutron evaporation spectra can be used to extract the level

density parameter a and hence the effective mass m∗, because

a =π2m∗

2k2F

. (1.33)

9

This procedure leads to

M∗

M= 0.8± 0.1.

• The parameter

L = 3no∂S2

∂n

∣∣∣∣∣n=no

(1.34)

has been constrained in the range

40 MeV < L < 60 MeV

as the intersection of allowed regions in a number of different experi-

ments.

All these quantities (Eo, S2, Ko, Kτ ,m∗, L) are obtained at the nuclear satura-

tion density no.

• In heavy-ion experiments, the longitudinal and transverse momenta of

particles produced in the collisions of heavy nuclei are measured. These

momenta can be calculated theoretically as

< p⊥,‖ >=∫p⊥,‖f(~r, ~p, t)d3rd3p (1.35)

where the phase-space distribution function f evolves according to the

Boltzmann equation

df

dt=∂f

∂t+ ~v · ∇f + ~F · ∂f

∂~p(1.36)

where

~F = −∇U = −∇(∂H∂n

). (1.37)

Furthermore, f(~r, ~p, t = 0) is related to the Slater determinant states

making up the colliding nuclei. These states are also constructed from

the hamiltonian density H through a Thomas-Fermi or Hartree-Fock

procedure.

Thus, flow of matter, momentum, and energy in heavy-ion experiments can be

used as consistency checks for any H from which the EOS is to be calculated.

10

1.4 Scope and Organization

From the previous section, it should be clear that there is a profound lack

of empirical knowledge of the behavior of nuclear matter in the conditions

that are present in supernovae as laboratory experiments can only probe den-

sities up to nuclear saturation (and perhaps up to ∼ 3no in heavy-ion colli-

sions) involving known nuclei with relatively small proton-neutron asymme-

tries (Z/A ∼ 0.5 − 0.39). In view of this limited guidance from experiment,

the approach has thus far been to use low density results for the calibration

of models and extrapolate them into the high-density, high-isospin asymmetry

regimes.

One of the goals of the present work is to study the extent to which predic-

tions from potential and field-theoretical models differ, especially when models

in each class are calibrated to similar values of nuclear matter properties at

saturation. Particular attention is paid to the thermodynamic variables at su-

pernuclear densities as we attempt to answer the question whether or not this

similar calibration translates to comparable behavior over all temperatures,

densities, and isospin asymmetries of interest. The subnuclear regime where a

mixed phase of nucleons, nuclei, and, presumably, other structures (”pasta”)

exists [12] is not explored in this thesis and will be taken up elsewhere.

In chapter 2, we investigate the potential model sector, for which purpose

we employ the model of Akmal, Pandharipande, and Ravenhall [13]. This

choice permits the investigation of the effects of long scattering lengths, which

have not been examined until now. We focus on the bulk homogeneous phase

and calculate, for the first time, the finite temperature properties of this EOS

for all proton fractions. The numerical results are compared to approximate

ones in the classical and the quantum regimes for which analytical expressions

have been developed. Furthermore, detailed comparisons between this model

and the more traditional Skyrme model Ska [14]are performed to isolate

regions of agreement and/or disagreement.

In Chapter 3, we use a Walecka-type Lagrangian [15] in which the nucleon-

nucleon interaction is mediated by scalar, vector, and isovector meson fields

and augmented with non-linear self-interactions of the scalar field. Its finite

temperature and finite isospin asymmetry properties in the homogeneous phase

are studied numerically for all regions of degeneracy, and analytically, in the

11

degenerate and non-degenerate limits and compared to the SkM∗ [16]Skyrme

model.

Another objective in this work is to explore the possibility of calibrating

the EOS not only at saturation but also using results from relativistic heavy

ion experiments in which the high density (∼ 3no) regime is accessible. To this

end, in Chapter 4, we employ a finite- range model with explicitly momentum-

dependent interactions due to Welke et al. [17] which is fit to nucleon-nucleus

scattering data so as to reproduce the correct optical potential behavior. We

study its finite temperature properties for densities up to 2.5 no but only at

zero isospin asymmetry. Extensions to include arbitrary proton fractions are

underway.

Finally, Chapter 5 summarizes the objectives and results of this dissertation

ans the advances made in order to achieve them. We conclude the chapter with

new questions that arise and tasks to be performed as a consequence of the

present work.

12

Chapter 2

Non-Relativistic Potential

The present chapter is devoted to the study of a non-relativistic potential

model due to Akmal, Pandharipande, and Ravenhall which is tuned so that it

reproduces two-body scattering data and the properties of light nuclei. We be-

gin by briefly summarizing the basic ingredients involved in the construction of

the model and derive its energy spectrum using a variational procedure. Then

we address its state variables at zero temperature and explore their depen-

dence on isospin asymmetry. Next, we present finite-temperature results for

the numerical computation of which the JEL technology is employed. Finally,

we explicitly work out analytical expressions which permit independent inves-

tigation of the state variables in the degenerate and non-degenerate limits.

2.1 Introduction

The Hamiltonian density of Akmal, Pandharipande, and Ravenhall (APR) [13]

is a parametric fit to the microscopic calculations of Akmal and Pandharipande

[18]in which the nucleon-nucleon interaction is modeled by the Argonne v18 2-

body potential [19], the Urbana UIX 3-body potential [20], and a relativistic

boost potential δv [21] which is a kinematic correction when the interaction

is observed in a frame other than the rest-frame of the nucleons. Explicitly,

the APR Hamiltonian density is given by

13

HAPR =

[h2

2m+ (p3 + (1− x)p5)ne−p4n

]τn

+

[h2

2m+ (p3 + xp5)ne−p4n

]τp

+g1(n)[1− (1− 2x)2)] + g2(n)(1− 2x)2, (2.1)

where x = npnn+np

= npn

is the proton fraction and

ni =1

π2

∫dki

k2i

1 + e(εki−µi)/T(2.2)

τi =1

π2

∫dki

k4i

1 + e(εki−µi)/T(2.3)

are the number densities and kinetic energy densities of nucleon species i =

n, p, respectively.

In the low density phase (LDP)

g1L = −n2(p1 + p2n+ p6n2 + (p10 + p11n)e−p

29n

2

) (2.4)

g2L = −n2(p12

n+ p7 + p8n+ p13e

−p29n2

), (2.5)

whereas, in the high density phase (HDP)

g1H = g1L − n2[p17(n− p19) + p21(n− p19)2ep18(n−p19)

](2.6)

g2H = g2L − n2[p15(n− p20) + p14(n− p20)2ep16(n−p20)

]. (2.7)

The critical trajectory in the n− x plane along which the transition from

the LDP to the HDP occurs is obtained by solving

g1L[1− (1− 2x)2] + g2L(1− 2x)2 = g1H [1− (1− 2x)2] + g2H(1− 2x)2.

This gives a critical density nc = 0.32 fm−3 for symmetric nuclear matter

(x = 1/2) and nc = 0.192 fm−3 for pure neutron matter (x = 0).

The values of the parameters p1 − p21 as well as their dimensions so that

HAPR is in units of MeV fm−3 are summarized in Table 2.1.

14

Common LDP HDP

p3 = 89.8 MeVfm5 p1 = 337.2 MeVfm3 p13 = 0p4 = 0.457 fm3 p2 = −382.0 MeVfm6 p14 = 0p5 = −59.0 MeVfm5 p6 = −19.1 MeVfm9 p15 = 287.0 MeVfm6

p7 = 214.6 MeVfm3 p16 = 1.54 fm3

p8 = −384.0 MeVfm6 p17 = 175.0 MeVfm6

p9 = 6.4 fm6 p18 = −1.45 MeVfm4

p10 = 69.0 MeVfm3 p19 = 0.32 fm−3

p11 = −33.0 MeVfm6 p20 = 0.192 fm−3

p12 = 0.35 MeV p21 = 0

Table 2.1: Parameter values for the Hamiltonian density of Akmal, Pandhari-pande, and Ravenhall. The leftmost column lists parameters in Eq.(2.1) thatare common to both the low density phase (LDP) and the high density phase(HDP). Parameters distinct to the LDP and the HDP are listed in the secondand third columns, respectively.

2.2 Single-Particle Energy Spectrum

The single-particle energy spectrum that εki , (i = n, p) that appears in the

Fermi-Dirac (FD) distribution nki = 1

1+e(εki−µi)/T is obtained from the func-

tional derivatives of the Hamiltonian density :

εki = k2i

∂H∂τi

+∂Hni. (2.8)

This is a direct consequence of the fact that the expectation value of the

Hamiltonian is stationary with respect to variations of its eigenstates [22, 23]:

δ

δφk

(E −

∑k

εk

∫|φk(~r)|2d3r

)= 0, (2.9)

where εk is the eigenvalue corresponding to the eigenstate φk, E = 〈H〉 and k

is the set of all relevant quantum numbers.

For a many-body Hamiltonian, φk are the single particle states making up

the Slater determinant, and therefore the set of all εk is the single-particle

energy spectrum of the Hamiltonian.

15

Consider now a nucleonic Hamiltonian density H = H(τi, ni), where

τi(~r) =∑k,s

|∇φk(~r, s, i)|2 (2.10)

ni(~r) =∑k,s

|φk(~r, s, i)|2 (2.11)

are the kinetic energy density and number density respectively, of the nucleon

species with isospin i.

The variation of the number density with respect to φ is

δni =∑k,s

[δφ∗(~r, s, i)φ(~r, s, i) + φ∗(~r, s, i)δφ(~r, s, i)]. (2.12)

Imposing time-translational invariance leads to

φ(~r, s, i) = −2sφ∗(~r,−s, i) (2.13)

and δφ(~r, s, i) = −2sδφ∗(~r,−s, i). (2.14)

Therefore,

δni =∑k,s

[δφ∗φ+φ(−s)

2s× (−2s)δφ∗(−s)]

=∑k,s

[δφ∗φ+ δφ∗(−s)φ(−s)]

= 2∑k,s

δφ∗φ, (2.15)

as the sum is over all spins.Similarly,

δτi = 2∑k,s

∇δφ∗k · ∇φk. (2.16)

Furthermore,

E =∑i

∫d3r H(τi, ni). (2.17)

16

Combining this with (2.12) and (2.14) implies

δE =∑i

∫d3r

[∂H∂τi

δτi +∂Hniδni

]

=∫d3r

∑i

∂H∂τi

(2∑k,s

∇φ∗k · ∇φk) +∂Hni

(2∑k,s

δφ∗kφk)

=

∫d3r

∑k,s

[2δφ∗k

∑i

(−∇∂H

∂τi∇+

∂Hni

)φk

]. (2.18)

The minus sign is a consequence of the anti-hermiticity of the ∇ operator:

〈∇φ| = 〈φ|∇† = 〈φ|(−∇).

Finally, by inserting (2.18) into (2.9) we get

0 =∫d3r

∑k,s

2δφ∗k

[∑i

(−∇∂H

∂τi∇+

∂H∂ni

)]φk −

∫d3r

∑k,s

2δφ∗kεkφk

=∫d3r

∑k,s

2δφ∗k

[∑i

(−∇∂H

∂τi∇+

∂H∂ni

)− εk

]φk

⇒∑i

(−∇∂H

∂τi∇+

∂H∂ni

)− εk = 0

⇒ −∇∂H∂τi∇+

∂H∂ni− εki = 0. (2.19)

Thus in momentum space,

k2i

∂H∂τi

+∂H∂ni

= εki

.

2.3 Zero Temperature

At T=0, the nucleons are restricted to the lowest available quantum states.

Therefore, the Fermi-Dirac distribution that appears in the integrals of the

number density and the kinetic energy density, becomes a step-function:

nki = θ(εki − εFi), (2.20)

17

where εFi is the energy at the Fermi surface. Equivalently,

ni =1

π2

∫ kFi

0k2i dki =

k3Fi

3π2(2.21)

τi =1

π2

∫ kFi

0k4i dki =

k5Fi

5π2. (2.22)

Thus, the kinetic energy densities can be written as simple functions of the

number density n and the proton fraction x :

τp =1

5π2(3π2np)

5/3 =1

5π2(3π2nx)5/3 (2.23)

τn =1

5π2(3π2nn)5/3 =

1

5π2(3π2n(1− x))5/3. (2.24)

We can therefore write H(np, nn, τp, τn;T = 0) = H(n, x). Then, we can apply

standard thermodynamics relations to get the various quantities of interest

such as:

• energy per particleE

A=Hn

(2.25)

• pressure

P = n2∂(E/A)

∂n(2.26)

• chemical potentials

µp =∂H∂np

∣∣∣∣∣nn

=E

A+ n

∂(E/A)

∂n

∣∣∣∣∣x

+ (1− x)∂(E/A)

∂x

∣∣∣∣∣n

(2.27)

µn =∂H∂nn

∣∣∣∣∣np

=E

A+ n

∂(E/A)

∂n

∣∣∣∣∣x

− x ∂(E/A)

∂x

∣∣∣∣∣n

(2.28)

• isospin susceptiblilities

χij =

(∂µi∂nj

)−1

(2.29)

• incompressibility

K = 9dP

dn(2.30)

18

• symmetry energy

S2 =1

8

∂2(E/A)

∂x2(2.31)

• Landau effective masses

m∗i =

(∂εki∂ki

)−1

(2.32)

Of particular importance are the values of E/A, K, S2, and m∗ at the satu-

ration point (P = 0) of symmetric nuclear matter, as these are accessible to

experiments.

The skewness [24]

S = k3F

d3(E/A)

dk3F

∣∣∣∣∣x=1/2,no

= K

[−3 +

27n2o

K

∂3H∂n3

]x=1/2,no

, (2.33)

and the derivative of the symmetry energy at saturation density, characterized

by the parameter

L = 3nodS2

dn

∣∣∣∣∣no

(2.34)

are also measurable [25]. These values are summarized in Table 2.3 along

with the corresponding values resulting from the Ska model whose Hamiltonian

density is given by :

HSka =h2

2mn

τn +h2

2mp

τp + n(τn + τp)[t14

(1 +

x1

2

)+t24

(1 +

x2

2

)]+(τnnn + τpnp)

[t24

(1

2+ x2

)− t1

4

(1

2+ x1

)]+to2

(1 +

xo2

)n2 − to

2

(1

2+ xo

)(n2

n + n2p)[

t312

(1 +

x3

2

)n2 − t3

12

(1

2+ x3

)(n2

n + n2p)]nε (2.35)

The parameters to through t3, xo through x3, and ε are listed in Table 2.2.

The quantities S and L are related to the symmetry term Kτ of the liquid

drop formula for the isospin asymmetric incompressibility [26] via

Kτ = KS2 −LS

K, (2.36)

19

where KS2 = 9n2o

d2S2

dn2

∣∣∣∣∣no

. (2.37)

i ti xi ε

0 −1602.78 MeVfm6 0.02 1/31 570.88 fm3 02 −67.7 fm3 03 8000.0 MeVfm7 -0.286

Table 2.2: Parameter values for the Ska Hamiltonian density.The dimensionsare such that the Hamiltonian density is in MeV fm−3.

no E/A Ko S2 m∗/m S LAPR 0.160 -16.00 266.0 32.59 0.70 541.82 58.46Ska 0.155 -15.99 263.2 32.91 0.61 1278.91 74.62

Table 2.3: Saturation properties of symmetric nuclear matter. With the ex-ception of no which is in fm−3 and m∗/m which is unitless, all other quantitiesare in MeV.

2.3.1 Results

The results presented here are for zero temperature, bulk homogeneous nuclear

matter. The plots show comparisons of the APR and the Ska models at proton

fractions of 0.3 and 0.5.

The energy per particle of APR is higher than that of Ska for densities

less than 0.1 fm−3 and lower for densities above 0.2 fm−3. This is due to the

realistic treatment of the nuclear force in APR which makes it more repulsive

in the long range and more attractive in the short. At smaller proton fractions,

the models are less bound and their saturation points occur at lower densities.

The pressure curves exhibit the same degree of stiffness for both models.

This is related to the fact that the compression moduli of the two models are

nearly equal. The chemical potentials behave similarly to the pressure.

The isospin susceptibilities reveal how the chemical potentials vary with

nucleonic compositions. These exhibit large differences throughout the density

range.

20

-20

0

20

40

60

80

100

120

140

160

180

200

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

E/A

[M

eV

]

nB [fm-3

]

T = 0 MeV

APR x=0.5APR x=0.3Ska x=0.5Ska x=0.3

Figure 2.1: Zero temperature comparison of the energy per particle versusdensity of the EOS of APR (blue) and Ska (red) at proton fractions of 0.3(dotted lines) and 0.5 (solid lines).

-50

0

50

100

150

200

250

300

350

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

P [

MeV

/fm

3]

nB [fm-3

]

T = 0 MeV


Figure 2.2: Zero temperature comparison of the pressure versus density of theEOS of APR (blue) and Ska (red) at proton fractions of 0.3 (dotted lines) and0.5 (solid lines).

21

-100

0

100

200

300

400

500

600

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

µn [

MeV

]

nB [fm-3

]

T = 0 MeV


Figure 2.3: Zero temperature comparison of the neutron chemical potentialversus density of the EOS of APR (blue) and Ska (red) at proton fractions of0.3 (dotted lines) and 0.5 (solid lines).

-100

0

100

200

300

400

500

600

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

µp [

MeV

]

nB [fm-3

]

T = 0 MeV


Figure 2.4: Zero temperature comparison of the proton chemical potentialversus density of the EOS of APR (blue) and Ska (red) at proton fractions of0.3 (dotted lines) and 0.5 (solid lines).

22

0

200

400

600

800

1000

1200

1400

1600

1800

2000

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

dµ

n /

dn

n [M

eV

fm

3]

nB [fm-3

]

T = 0 MeV


Figure 2.5: Zero temperature comparison of the neutron-neutron susceptibilityversus density of the EOS of APR (blue) and Ska (red) at proton fractions of0.3 (dotted lines) and 0.5 (solid lines).

-500

0

500

1000

1500

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

dµ

n / d

np [M

eV

fm

3]

nB [fm-3

]

T = 0 MeV


Figure 2.6: Zero temperature comparison of the neutron-proton susceptibilityversus density of the EOS of APR (blue) and Ska (red) at proton fractions of0.3 (dotted lines) and 0.5 (solid lines).

23

2.4 Nuclear Matter at Finite Isospin

Asymmetry

The properties of asymmetric nuclear matter are relevant to the physics of su-

pernova explosions as well as to the physics of neutron stars and of heavy nuclei

as in all three cases, neutron excess is significant [27, 28]. In what follows,

the asymmetry (α = 1 − 2x = nn−npn

) dependence of the incompressibility at

fixed density, the density at fixed pressure, and the isobaric incompressibility

are explored.

Consider the expansion of the energy per particle E/A ≡ E in powers of

the asymmetry α :

Eα(n) = Eo(n) + S2(n)α2 + S4(n)α4 + . . . (2.38)

where Eo(n) is the energy per particle of symmetric matter, S2(n) is (as usual)

the symmetry energy and

S4(n) =1

4!

1

24

∂4(E/A)

∂x4=

1

4!

∂4(E/A)

∂α4. (2.39)

Then, pure neutron matter (α = 1) can be written in terms of symmetric

nuclear matter (α = 0) as

Eα=1(n) = Eo(n) + S2(n) + S4(n) + . . . (2.40)

This is a rapidly converging series as Fig.2.7 shows.

Due to the isospin invariance of the nucleon-nucleon interaction, potential

terms are only carried up to O(α2). Therefore, S4 receives contributions only

from the kinetic energy densities. Explicitly,

S4(n) =1

81εF

where

εF =k2F

2m∗=

1

2m∗

(3π2n

2

)2/3

.

24

0.0 0.2 0.4 0.6 0.8

0

50

100

150

200

250

n H fm-3 L

EA

HMeV

L

Figure 2.7: Approximations to the energy per particle of pure neutron matter(black line). Isospin symmetric matter is in red. The blue line representsE/A+ S2 and the orange one is E/A+ S2 + S4.

Only even powers of α survive in this series because the two nucleon

species are treated symmetrically in the Hamiltonian. Therefore for every

nγn =(

1+α2

)nγ there exists a nγp =

(1−α

2

)nγ such that

nγn + nγp =(n

2

)γ [(1 + γα +

γ(γ − 1)

2α2 + . . .

)+

(1− γα +

γ(γ − 1)

2α2 − . . .

)]

=(n

2

)γ (1 + γ(γ − 1)α2 + . . .

).

To lowest order in α2, the pressure corresponding to (2.38) is

P = n2∂E∂n

= n2

[∂Eo∂n

+∂S2

∂nα2

](2.41)

and the incompressibility

Kα(n) = 9

[2n

(∂Eo∂n

+∂S2

∂nα2

)+ n2

(∂2Eo∂n2

+∂2S2

∂n2α2

)]

= 9

[2n∂Eo∂n

+ n2∂2Eo∂n2

]+ 9

[2n∂S2

∂n+ n2∂

2S2

∂n2

]α2

= Ko(n)(1 + Aα2) (2.42)

25

where

Ko(n) = 9[2nE ′o(n) + n2E ′′o (n)] (2.43)

A(n) =9

Ko(n)[2nS ′2(n) + n2S ′′2 (n)]. (2.44)

The primes above denote differentiation with respect to the density n.

At the equilibrium density noα of isospin asymmetric matter,

E ′α(noα) = E ′o(noα) + S ′2(noα)α2 + . . . = 0. (2.45)

The next step involves the expansion of E ′o(noα) and of S ′2(noα) about the

equilibrium density no of symmetric matter:

E ′o(noα) = E ′o(no) + E ′′o (no)(noα − no)

= 0 +Ko

9n2o

(noα − no) ≡Ko

9n2o

δnα (2.46)

S ′2(noα) = S ′2(no) + S ′′2 (no)(noα − no)

= S ′2(no) + S ′′2 (no)δnα (2.47)

By inserting (2.46) and (2.47) into (2.45), we get

Ko

9n2o

δnα + α2[S ′2(no) + S ′′2 (no)δnα] = 0

⇒ δnα = −[Ko

9n2o

+ α2S ′′2 (no)

]−1

α2S ′2(no).

Thus, to lowest order in α2

δnα = −9n2o

Ko

S ′2(no)α2 (2.48)

By rearranging terms this can be written as an equation for the dimensionless

ratio noαno

:noαno

= 1− 9noKo

S ′2(no)α2 ≡ 1− Cα2 (2.49)

The solution of (2.49) for α utilized into (2.38) traces the locus of the minima

of the energy per particle for changing asymmetries. This scheme remains

26

accurate to within 2% for α ≤ 0.5 (⇒ x ≥ 0.25).

The rapid convergence of the series in (2.38) encourages one to try and

extend the range of α for which the above method is valid, by including terms

to O(α4). Doing so leads to

δnα =−9n2

oα2

Ko + 9n2oα

2S ′′2 (no)

[S ′2(no) + S ′4(no)α

2],

which implies that

noαno

= 1− −9noα2

Ko

1

1 +9n2oα

2S′′2 (no)

Ko

[S ′2(no) + S ′4(no)α

2].

Being that9n2oα

2S′′2 (no)

Ko 1, we use (1 + x)−1 ' 1− x+ . . . to write the above

expression as

noαno

= 1− 9noKo

S ′2(no)α2 − 9no

Ko

[S ′4(no)−

9n2oS′′2 (no)S

′2(no)

Ko

]α4. (2.50)

This, however, turns out to be a fruitless pursuit. The O(α4) term in (2.50)

underestimates the saturation point for α ≥ 0.5 whereas (2.49) overestimated

it (see Fig.2.8). Thus, to properly improve the scheme, one should expand

E ′o(noα) and S2(noα) to order δn2α:

E ′o(noα =Ko

9n2o

δnα +E ′′′o (no)

2δn2

α (2.51)

S ′2(noα) = S ′2(no) + S ′′2 (no)δnα +S ′′′2 (no)

2δn2

α (2.52)

and solve for α, which is then to be used in (2.38).

Terms of O[α4] are far less significant than δn2α terms because

δnα = no(noαno− 1

), and noα

nois of the same order of magniture as α (∼ 0.5).

27

æ

æ

æ

æ

æ

0.00 0.05 0.10 0.15

- 15

- 10

- 5

0

5

n H fm- 3 L

EA

HMeV

L

Figure 2.8: Tracing the locus of the minima of the APR energy per particle.The red curve represents expansions to α2 and δnα, the green curve correspondsto α2 and δn2

α, and the blue curve shows expansions to α4 and δnα. Thegreen curve is closest to the exact minima, shown in black dots, insofar as thesystem is bound. The thin lines correspond to E/A at proton fractions of 0.40(orange), 0.28 (purple), 0.20 (pink), 0.15 (light blue), and 0.10 (brown).

28

Returning to O(α2, δnα), the isobaric incompressibility at finite asymmetry

is obtained by first expanding Ko(n) about no :

Ko(n) = Ko(no) +K ′o(no)(n− no)

= Ko +K ′o(no)δnα

= Ko −K ′o(no)9n2

o

Ko

S ′2(no)α2 = Ko(1 +Bα2), (2.53)

where

B ≡ −K ′o(no)9n2

o

Ko

S ′2(no). (2.54)

Then from (2.40) and (2.48) we get

Kα(n) = Ko(1 +Bα2)(1 + Aα2)

' Ko[1 + (A+B)α2] ≡ Ko(1 + Aα2) (2.55)

to lowest order in α2.

The results for the asymmetry coefficients A, B, C, A evaluated at the

saturation density of symmetric matter for APR and Ska are displayed in

table 2.4.

Model A B C AAPR 0.933 -1.766 0.659 -0.833Ska 1.403 -3.079 0.851 -1.676

Table 2.4: Results for the coefficients that describe the isospin asymmetrydependence. For each model these are given at the equilibrium density ofsymmetric matter.

One observes that even though HAPR and HSka are calibrated to very sim-

ilar values of the symmetry energy and the compression modulus, the asym-

metry coefficients vary quite dramatically.

2.5 Finite Temperature

At finite-T, HAPR is a function of four independent variables; namely the

number densities ni and the kinetic energy densities τi of the two nucleon

29

species. These are, in turn, proportional to the F1/2 and F3/2 Fermi-Dirac

(FD) integrals respectively:

ni =1

2π2

(2m∗iT

h2

)3/2

F1/2i (2.56)

τi =1

2π2

(2m∗iT

h2

)5/2

F3/2i (2.57)

where Fαi =∫ ∞

0

xαie−ψiexi + 1

dxi (2.58)

xi =1

T

(k2i

∂H∂τi

)=

1

T

h2k2i

2m∗i≡ εki

T(2.59)

ψi =1

T

(µi −

∂H∂ni

)=µi − ViT

≡ νiT. (2.60)

The quantity ψi is known as the degeneracy parameter. Another important

quantity is the fugacity which is defined as zi = eψi .

Equation (2.65) can be written as

F1/2i = 2π2ni

(h2

2m∗iT

)3/2

, (2.61)

where one must keep in mind that m∗i is a function of the number densities of

both nucleon species. Thus,

∂F1/2i

∂ni= 2π2

(h2

2m∗iT

)3/2

− 3

2

1

m∗i

∂m∗i∂ni

2π2ni

(h2

2m∗iT

)3/2

=F1/2i

ni

(1− 3

2

nim∗i

∂m∗i∂ni

)(2.62)

and∂F1/2i

∂nj= −3

2

1

m∗i

∂m∗i∂nj

F1/2i. (2.63)

FD integrals of different order are connected through their derivatives:

∂Fαi∂ψi

= αF(α−1)i (2.64)

Therefore∂Fαi∂ni

=∂Fαi∂F1/2i

∂F1/2i

∂ni

=∂Fαi∂ψi

(∂F1/2i

∂ψi

)−1∂F1/2i

∂ni

30

= 2αF(α−1)i

F−1/2i

∂F1/2i

∂ni. (2.65)

Similarly,∂Fαi∂nj

= 2αF(α−1)i

F−1/2i

∂F1/2i

∂nj. (2.66)

Finally, beacause

∂

∂n=

∂

∂nn

∂nn∂n

∣∣∣∣∣x

+∂

∂np

∂np∂n

∣∣∣∣∣x

= (1− x)∂

∂nn+ x

∂

∂np

∂

∂x=

∂

∂nn

∂nn∂x

∣∣∣∣∣n

+∂

∂np

∂np∂x

∣∣∣∣∣n

= −n ∂

∂nn+ n

∂

∂np,

the derivatives of Fαi with respect to n and x are obtained as

∂Fαi∂n

= 2αF(α−1)i

F−1/2i

n

[(1− x)

∂Fαi∂nn

+ x∂Fαi∂np

](2.67)

∂Fαi∂x

= 2αF(α−1)i

F−1/2i

n

[∂Fαi∂np

− ∂Fαi∂nn

]. (2.68)

Using equations (2.65)-(2.68) we arrive to the following expressions for the

density derivatives of the kinetic energy density:

∂τi∂ni

=τini

[3F 2

1/2i

F3/2iF−1/2i

+5

2

nim∗i

∂m∗i∂ni

(1− 9

5

F 21/2i

F3/2iF−1/2i

)](2.69)

∂τi∂nj

=5

2

τim∗i

∂m∗i∂nj

(1− 9

5

F 21/2i

F3/2iF−1/2i

)(2.70)

∂τi∂n

= τi

[5

2

1

m∗i

∂m∗i∂n

+3F1/2i

F3/2iF−1/2i

((1− x)

∂F1/2i

∂nn+ x

∂F1/2i

∂np

)](2.71)

∂τi∂x

= τi

[5

2

1

m∗i

∂m∗i∂x

+3F1/2i

F3/2iF−1/2i

n

(∂F1/2i

∂np−∂F1/2i

∂nn

)]. (2.72)

These are necessary for the subsequent discussion of the finite-temperature

susceptibilities.

2.5.1 Thermal Effects

To infer the effects of finite temperature we focus on the thermal part of the

various state variables; that is, the difference between the T = 0 and the

31

finite-T expressions for a given thermodynamic function:

Xth = X(n, x, T )−X(n, x, 0) (2.73)

This subtraction scheme essentially discards terms that do not depend on the

kinetic energy density. The results are as follows:

• energy per particle

EthA

=E(T )

A− E(0)

A

=1

n

∑i

[h2

2m∗iτi −

h2k2Fi

2m∗i

3

5ni

]

≡ 1

n

∑i

[h2

2m∗iτi −

3

5TFini

]. (2.74)

• entropy per particle

S

A=

1

nT

∑i

[5

3

h2

2m∗iτi + ni(Vi − µi)

]

=1

n

∑i

ni

[5

3

F3/2i

F1/2i

− lnzi

]. (2.75)

• pressure

Pth = P (T )− P (0)

=2

3

∑i

Qi

[h2

2m∗iτi −

3

5TFini

], (2.76)

where Qi = 1− 3

2

n

m∗i

∂m∗i∂n

. (2.77)

Clearly, Qi are the consequence of the momentum-dependent interactions

in the Hamiltonian which lead to the Landau effective mass. For a free

gas, Qi = 1 and Pth = 2n3EthA

as usual.

• free energy density

Fth = F(T )−H(0)

= H(T )− nT SA−H(0)

32

=∑i

[h2

2m∗iτi −

3

5TFini − Tni

(5

3

F3/2i

F1/2i

− lnzi

)]. (2.78)


µith = µi(T )− µi(0) =∂Fth∂ni

∣∣∣∣∣nj

. (2.79)

where µi(T ) = Tψi + Vi (2.80)

• susceptibilities

χij,th = χij(T )− χij(0) =

(∂µith∂nj

)−1

(2.81)

where χii(T ) =

(∂µi∂ni

)−1

=

(T∂ψi∂ni

+∂Vi∂ni

)−1

=

T (∂F1/2i

∂ψi

)−1∂F1/2i

∂ni+∂Vi∂ni

−1

=

[2T

ni

F1/2i

F−1/2i

(1− 3

2

nim∗i

∂m∗i∂ni

)+∂Vi∂ni

]−1

, (2.82)

and χij(T ) =

[−3T

nj

F1/2i

F−1/2i

njm∗i

∂m∗i∂nj

+∂Vi∂ni

]−1

; i 6= j. (2.83)

2.5.2 Numerical Notes

For the purposes of numerical computation we use a scheme due to Johns,

Ellis, and Lattimer (JEL) [29, 30] whereby (in the non-relativistic case) the

FD integrals are written as algebraic functions of the degeneracy parameter:

ψi = 2(1 + fi/a)1/2 + ln

[(1 + fi/a)1/2 − 1

(1 + fi/a)1/2 + 1

](2.84)

F3/2i =3fi(1 + fi)

1/4−M

2√

2

M∑m=0

pmfmi (2.85)

F1/2i =fi(1 + fi)

1/4−M√2(1 + fi/a)

M∑m=0

pmfmi

[1 +m−

(M − 1

4

)fi

1 + fi

](2.86)

33

F−1/2i = − fia(1 + fi/a)3/2

F1/2i +

√2fi(1 + fi)

1/4−M

1 + fi/a

×M∑m=0

pmfmi

[(1 +m)2 −

(M − 1

4

)

× fi1 + fi

(3 + 2m−

[M +

3

4

]fi

1 + fi

)]. (2.87)

The coefficients M, a, pm that appear in (2.84)-(2.87) are supplied in

Table 2.5 [31].

Coefficient ValueM 3a 0.433

poe2

a

(π32

)1/2)

= 5.34689

p1 16.8441

p2a−1/4

3

(π2 − 8 + 88

5a

)= 17.4708

p332a−5/4

15= 6.07364

Table 2.5: Non-relativistic JEL coefficients.

One first specifies nn and np (or equivalently n and x) and then solves the

system

F1/2p(fp) = 2π2np

(h2

2m∗pT

)3/2

F1/2n(fn) = 2π2nn

(h2

2m∗nT

)3/2

for fn and fp. These are, in turn, used as inputs in (2.75)-(2.78) that determine

ψi (and thus µi) and the other Fαi on which the state variable depend.

The JEL method is, by construction, thermodynamically consistent and

furthermore it is more efficient than the standard methods of evaluating inte-

grals.

34

2.5.2.1 Results

In what follows we present comparative results for APR and SKa at 20 MeV, in

the case of isospin symmetric bulk homogeneous matter. The observed trends

at zero temperature persist at finite temperature as well.

Of particular interest is Fig. 2.13 which shows lines of constant entropy

versus temperature and density. The APR model is more disordered as can

also be verified from Fig. 2.12.

-20

0

20

40

60

80

100

120

140

160

180

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

E/A

[M

eV

]

nB [fm-3

]

T = 20 MeVx = 0.5

APRSka

Figure 2.9: Comparison of the energy per particle versus density of the EOSof APR (blue) and Ska (red) at T = 20 MeV.

35

0

50

100

150

200

250

300

350

400

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

P [M

eV

/fm

3]

nB [fm-3

]

T = 20 MeV

x = 0.5

APRSka

Figure 2.10: Comparison of the pressure versus density of the EOS of APR(blue) and Ska (red) at T = 20 MeV.

-100

0

100

200

300

400

500

600

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

µ [M

eV

]

nB [fm-3

]

T = 20 MeV

x = 0.5

APRSka

Figure 2.11: Comparison of the chemical potential versus density of the EOSof APR (blue) and Ska (red) at T = 20 MeV.

36

0

1

2

3

4

5

6

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

s/A

nB [fm-3

]

T = 20 MeVx = 0.5

APRSka

Figure 2.12: Comparison of the entropy per particle versus density of the EOSof APR (blue) and Ska (red) at T = 20 MeV.

0

20

40

60

80

100

120

0 0.2 0.4 0.6 0.8 1

T (

MeV

)

nB (fm-3

)

STotal = 1

2

3

Yp = 0.5

APR

Ska

Figure 2.13: Comparison of isentropes of the EOS of APR (solid lines) andSka (dotted lines) at T = 20 MeV.

37

0

500

1000

1500

2000

2500

3000

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

dµ

n /

dn

n [M

eV

fm

3]

nB [fm-3

]

T = 20 MeV

x = 0.5

APRSka

Figure 2.14: Comparison of the neutron-neutron susceptibility versus densityof the EOS of APR (blue) and Ska (red) at T = 20 MeV.

-1000

-500

0

500

1000

1500

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

dµ

n / d

np [M

eV

fm

3]

nB [fm-3

]

T = 20 MeVx = 0.5

APRSka

Figure 2.15: Comparison of the neutron-proton susceptibility versus densityof the EOS of APR (blue) and Ska (red) at T = 20 MeV.

38

2.5.3 Limiting Cases

To ensure the validity of our exact numerical results, we subject them to two

consistency checks. In particular, numerical results for the (thermal) state

variables are graphically compared with their analytical counterparts in the

degenerate (high n, low T ) and non-degenerate (ND) limit (high T , low n).

2.5.3.1 Degenerate Limit

For the degenerate regime (the results of which are useful in their own right

for neutron star applications), we make use of Landau’s Fermi Liquid Theory

(FLT) [32, 33]. The main idea in FLT is that the occupation numbers nk

of weakly excited states remain approximately valid even in the presence of

strong interactions. This means that excitations close to the Fermi surface of

an interacting gas can be constructed from the low-lying states of an ideal gas

by adiabatically switching-on the interaction. ”Adiabatic” refers to a timescale

that is longer than the resolution time of the quasiparticles (tres ∼ 1vF |k−kF |

),

but shorter than their lifetime (τ ∼ 1v2F |k−kF |2

).

This procedure establishes a one-to-one correspondence between the par-

ticles of the free system and the quasiparticles of the interacting one and thus

the entropy density s, and the number density n maintain their free forms.

For a single-component gas,

s = − 1

V

∑k,s

[nkslnnks + (1− nks)ln(1− nks)] (2.88)

n =1

V

∑k,s

nks, (2.89)

where nks =1

e(εks−µ)/T + 1(2.90)

and k and s stand for wave number and spin respectively. We also introduce

the quasiparticle density of states at T = 0, at the Fermi surface:

N(0) =1

V

∑k,s

δ(εoks − εF ), (2.91)

where εoks is the energy spectrum at T = 0.

39

In the continuum limit,

N(0) = 2∫ kF

0δ(εk − εF )d3k

= 24π

(2π)3

∫ εF

0δ(ε− εF )m∗(2m∗ε)1/2dε

=m∗kFπ2

=k2F

π2vF, (2.92)

where vF =∂εoks∂k

∣∣∣∣∣k=kF

=kFm∗

(2.93)

Above, vF denotes the velocity at the Fermi surface.

For the study of low-temperature physics, one begins by calculating the

variation of the entropy density as the temperature is varied:

δs = − 1

V

∑k,s

[δnkslnnks + δnks − δnksln(1− nks)− δnks]

= − 1

V

∑k,s

δnks [lnnks − ln(1− nks)]

= − 1

V

∑k,s

δnksln(

nks1− nks

)

Using (2.90) for nks

δs = − 1

V

∑k,s

δnks(εks − µ) (2.94)

and δnks = −[e(εks−µ)/T + 1

]−2[(εks − µT 2

)(−δT ) +

δεksT− δµ

T

]

= − 1

T

[e(εks−µ)/T + 1

]−2[−(εks − µT

)δT + δεks − δµ

]=

∂nks∂εks

[−(εks − µT

)δT + δεks − δµ

].

Sommerfeld expansions suggest that εks − µ ∝ T 2lnT + O(T 3). Thus, to

lowest order in temperature

δnks = −∂nks∂εks

(εks − µT

)δT (2.95)

⇒ δs = − 1

V

∑k,s

∂nks∂εks

(εks − µT

)2

δT. (2.96)

40

In the continuum limit

δs = −δTT 2

∫ ∞0

∂nks∂εks

(εks − µ)2 d3k. (2.97)

For small T (T εF ), the FD distribution differs only slightly from a

step-function. Therefore the derivative ∂nks∂εks

is non-zero only in a small neigh-

borhood of εF (everywhere else nks is flat and thus ∂nks∂εks

= 0). This restricts

momenta in a narrow region around kF :

δs = −δT24π

(2π)3

∫ ∞0

k2dk

dεdε∂n

∂ε

(ε− µT

)2

' −δT 1

π2

∫ +δε

−δεm∗kFdε

∂n

∂ε

(ε− µT

)2

; δε = |εF − µ|

= −δTN(0)∫ +δε

−δεdε∂n

∂ε

(ε− µT

)2

= −δTN(0)∫ +∞

−∞

∂

∂x

(1

ex + 1

)x2dx ;

ε− µT≡ x

= −δTN(0)∫ +∞

−∞

(x

ex + 1

)2

exdx.

The value of the integral is π2

3, which means that the low-T entropy density

is given by

s =π2

3N(0)T = 2anT, (2.98)

where the level density parameter a is defined as

a =π2

2kFvF=π2N(0)

6n. (2.99)

The generalization to a multi-component gas is straight-forward. The sums

in (2.88) and (2.89) would also go over particle type so that the end result for

the entropy density would read

s =π2

3T∑i

Ni(0) = 2T∑i

aini (2.100)

where ai =π2

2kFivFi=π2Ni(0)

6ni=π2

2

m∗ik2Fi

=π2

2

m∗i(3π2ni)2/3

(2.101)

The rest of the thermodynamics is obtained via Maxwell relations [34].

41

Their derivation proceeds along the following lines: The relevant thermody-

namic potential is the free energy F being that the energy, the entropy, and

the temperature of the system are allowed to vary. Explicitly,

F (E, T, S) = E − TS. (2.102)

Its differential is

dF = dE − TdS − SdT (2.103)

The energy functional is

E(S, V,N) = TS − pV +∑i

µiNi (2.104)

⇒ dE = TdS − pdV +∑i

µidNi. (2.105)

Combining the two previous equations gives

dF = −SdT − pdV +∑i

µidNi (2.106)

⇒ −dFdT

= S, − dF

dV= p,

dF

dNi

= µi. (2.107)

Thus

dS

dV= − d

dV

dF

dT=

d

dT

dF

dV=dp

dT(2.108)

⇔ dp

dT= s− nds

dn(2.109)

Similarly,

− dS

dNi

=dµidT⇔ − ds

dni=dµidT

(2.110)

anddp

dNi

=dµidV

(2.111)

Finally, (2.105) impliesdE

dS= T (2.112)

Thus we have:

42

• thermal energy

∫dE =

∫TdS

=2

n

∑i

aini

∫TdT

⇒ Eth =T 2

n

∑i

aini (2.113)

• thermal pressure

∫dp =

∫ T

0

(s− nds

dn

)dT

= 2∫ T

0

∑i

[aini − n

d(aini)

dn

]TdT

=∑i

[aini − n

d(aini)

dn

]T 2.

Using ai = π2

2

m∗i(3π2ni)2/3

we get

daidn

=−2ai3n

(1− 3

2

n

m∗i

dm∗idn

)

⇒ nd(aini)

dn= aini −

2ain

3

(1− 3

2

n

m∗i

dm∗idn

)

⇒ pth =2nT 2

3

∑i

ai

(1− 3

2

n

m∗i

dm∗idn

)

=2nT 2

3

∑i

aiQi, (2.114)

where Qi = 1− 3

2

n

m∗i

dm∗idn

. (2.115)

• thermal chemical potentials

∫dµi = −

∫ ds

dnidT

= −2∫ d

dni

∑j

ajnj

TdT

43

= − d

dni

∑j

ajnj

T 2

⇒ µith = −T 2

ai3

+∑j

njajm∗j

dm∗jdni

. (2.116)

2.5.3.2 Non-degenerate Limit

In the ND limit, the degeneracy (and hence the fugacity) is small, so that the

FD functions can be expanded in Taylor series about z = 0:

Fαi ' Γ(α + 1)

(zi −

z2i

2α+1+ . . .

)(2.117)

Then the F1/2 series is perturbatively inverted to get the fugacity in terms of

the number density and the temperature

zi =niλ

3i

γ+

1

23/2

(niλ

3i

γ

)2

, (2.118)

where λi =

(2πh2

m∗iT

)1/2

(2.119)

and γ = 2 (the spin orientations).

Subsequently, these are used in the other integrals so that they, too, are ex-

pressed as explicit functions of the number density and the temperature:

F3/2i =3π1/2

4

niλ3i

γ

[1 +

1

25/2

niλ3i

γ

](2.120)

F1/2i =π1/2

2

niλ3i

γ(2.121)

F−1/2i = π1/2niλ3i

γ

[1− 1

23/2

niλ3i

γ

](2.122)

Finally, we insert these into equations (2.74)-(2.83) from which we get:

• thermal energy

Eth =1

n

∑i

3

2Tni

1 +ni4

(πh2

m∗iT

)3/2− 3

5TFini

(2.123)

44


pth =∑i

TQini

1 +ni4

(πh2

m∗iT

)3/2− 2

5TFini

(2.124)

• entropy

S =1

n

∑i

ni

5

2− ln

(2πh2

m∗iT

)3/2ni2

+ni8

(πh2

m∗iT

)3/2 (2.125)


µith = −T

ln

(2πh2

m∗iT

)3/2ni2

− ni2

(πh2

m∗iT

)3/2

+3

2

nim∗i

dm∗idni

1 +ni4

(πh2

m∗iT

)3/2

+3

2

njm∗j

dm∗idnj

1 +nj4

(πh2

m∗jT

)3/2

− TFi[1− 3

5

nim∗i

dm∗idni

]+

3

5

njm∗j

dm∗jdniTFj

(2.126)

45

2.5.3.3 Results

This section hosts results pertaining to the APR EOS at T = 20 MeV, for

isospin symmetric matter. The agreement of the exact results with the de-

generate and non-degenerate limits in the corresponding regions of validity is

excellent for all state variables.

0

5

10

15

20

25

30

35

40

45

50

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

ET

herm

al [

MeV

]

nB [fm-3

]

T = 20 MeVx = 0.5

ExactDegenerate

Non-Degenerate

Figure 2.16: Comparison of the degenerate (red line) and the non-degenerate(blue line) approximations with the exact (black line) thermal energy of APRat T = 20 MeV.

46

0

0.5

1

1.5

2

2.5

3

3.5

4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

PT

he

rma

l [M

eV

/fm

3]

nB [fm-3

]

T = 20 MeV

x = 0.5

ExactDegenerate

Non-Degenerate

Figure 2.17: Comparison of the degenerate (red line) and the non-degenerate(blue line) approximations with the exact (black line) thermal pressure of APRat T = 20 MeV.

-200

0

200

400

600

800

1000

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

µ [M

eV

]

nB [fm-3

]

T = 20 MeVx = 0.5

ExactDegenerate

Non-Degenerate

Figure 2.18: Comparison of the degenerate (red line) and the non-degenerate(blue line) approximations with the exact (black line) chemical potential ofAPR at T = 20 MeV.

47

0

0.5

1

1.5

2

2.5

3

3.5

4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

s/A

nB [fm-3

]

T = 20 MeV

x = 0.5

ExactDegenerate

Non-Degenerate

Figure 2.19: Comparison of the degenerate (red line) and the non-degenerate(blue line) approximations with the exact (black line) entropy of APR at T =20 MeV.

0

500

1000

1500

2000

2500

3000

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

dµ

n / d

nn [M

eV

fm

3]

nB [fm-3

]

T = 20 MeVx = 0.5

ExactDegenerate

Non-Degenerate

Figure 2.20: Comparison of the degenerate (red line) and the non-degenerate(blue line) approximations with the exact (black line) neutron-neutron suscep-tibility of APR at T = 20 MeV.

48

-1000

-500

0

500

1000

1500

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

dµ

n / d

np [M

eV

fm

3]

nB [fm-3

]

T = 20 MeV

x = 0.5

ExactDegenerate

Non-Degenerate

Figure 2.21: Comparison of the degenerate (red line) and the non-degenerate(blue line) approximations with the exact (black line) neutron-proton suscep-tibility of APR at T = 20 MeV.

2.5.4 Specific Heat

The specific heat is a quantity that describes how a system accumulates energy

as its temperature is changed. It is a function of the dynamical structure of the

system; specifically, it depends on the number of degrees of freedom available to

the system in its current thermodynamic state. Furthermore, a discontinuity

in the specific heat corresponds to the occurence of a phase transition.

In the context of supernovae, the specific heat has a two-fold role:

1. It controls the time-scale over which the core of a collapsing star reaches

nuclear statistical equilibrium and begins to expand [35] and,

2. It controls the density at which the core rebounds.

For example, if an EOS has a lower specific heat relative to another, a

higher temperature will be necessary before the Fermi energy of neu-

trons becomes large enough to overcome their attraction. Being that the

energy per particle above nuclear saturation is an increasing function of

49

both the temperature and the density (cf. figures 2.1 and 2.9), then that

particular energy will be realized at a lower density. A lower specific

heat also means a higher pressure; this can be seen most clearly in the

case of an ideal gas for which

p ∝ n1+R/CV

where R is a constant.

For the calculation of the APR specific heat we begin by writing the energy

per particle asE

A=

1

n

∑i

h2

2m∗iτi + n-dependent terms

Then

CV =∂(E/A)

∂T

∣∣∣∣∣n

(2.127)

=1

n

∑i

h2

2m∗i

∂τi∂T

∣∣∣∣∣ni

The condition that ni are constant implies

dnidT

= 0 =∂ni∂T

∣∣∣∣∣F1/2i

+∂ni∂F1/2i

∣∣∣∣∣T

∂F1/2i

∂T

∣∣∣∣∣ni

⇒ ∂ni∂T

∣∣∣∣∣F1/2i

= − ∂ni∂F1/2i

∣∣∣∣∣T

∂F1/2i

∂T

∣∣∣∣∣ni

(2.128)

But

∂F1/2i

∂T

∣∣∣∣∣ni

=∂ψi∂T

∣∣∣∣∣ni

∂F1/2i

∂ψi

=1

2F−1/2i

∂ψi∂T

∣∣∣∣∣ni

(2.129)

where eq. (2.64) was used in going from the first to the second equality.

50

Solving for ∂ψi∂T

∣∣∣ni

gives

∂ψi∂T

∣∣∣∣∣ni

= − ∂ni∂T

∣∣∣∣∣F1/2i

(∂ni∂F1/2i

∣∣∣∣∣T

1

2F−1/2i

)−1

Using eq. (2.56) for the derivatives of ni with respect to T and F1/2i we get

∂ψi∂T

∣∣∣∣∣ni

= − 3

T

F1/2i

F−1/2i

(2.130)

The T -derivative of eq. (2.57) is

∂τi∂T

∣∣∣∣∣ni

= τi

5

2T+

1

F3/2i

∂F3/2i

∂T

∣∣∣∣∣ni

= τi

5

2T+

1

F3/2i

∂ψi∂T

∣∣∣∣∣ni

∂F3/2i

∂ψi

= τi

(5

2T+

9

2T

F 21/2i

F3/2iF−1/2i

)(2.131)

where equations (2.64) and (2.130) have been exploited for the last line. Thus

CV =5

2nT

∑i

h2τi2m∗i

(1− 9

5

F 21/2i

F3/2iF−1/2i

)(2.132)

In the degenerate limit,

E

A=E

A(T = 0) +

T 2

n

∑i

aini.

Therefore,

CV =2T

n

∑i

aini =S

A=

2(E/A)thT

. (2.133)

In the non-degenerate limit,

E

A=

1

n

∑i

3

2Tni

1 +ni4

(πh2

m∗iT

)3/2+ n-dependent terms

⇒ CV =1

n

∑i

3

2ni

1− ni8

(πh2

m∗iT

)3/2 . (2.134)

51

The results are shown below.

-1

-0.5

0

0.5

1

1.5

2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Cv

nB [fm-3

]

T = 20 MeVx = 0.5

ExactDegenerate

Non-Degenerate

Figure 2.22: Comparison of the degenerate (red line) and the non-degenerate(blue line) approximations with the exact (black line) specific heat of APR atT = 20 MeV.

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Cv

nB [fm-3

]

T = 20 MeV

x = 0.5

APRSka

Figure 2.23: Comparison of the specific heat versus density of the EOS of APR(blue) and Ska (red) at T = 20 MeV.

52

Chapter 3

Mean Field Theory

In this chapter, we consider an alternate approach to the nuclear many-body

problem which is based on the Lagrangian formulation of mechanics and in

which mesons are taken to be the quanta of the nuclear force. This approach

has the advantage of being manifestly Lorentz covariant which, among other

things, is useful in hydrodynamic calculations. After a short description of

the model and the justification of the assumptions on which it is founded, we

derive its equations of motion and its zero-temperature state variables. Then,

we turn our attention to its finite-temperature properties using the relativistic

generalization of the JEL method as well as analytical approximations for the

classical and the quantum regimes. Special emphasis is placed on the details

of the calculations of the isospin susceptibilities and the specific heat in the

JEL framework.

3.1 Introduction

As a basis for Mean Field Theory (MFT), we have chosen a Walecka-type

Lagrangian [15] in which the nucleon-nucleon interaction is mediated by the

exchange of σ, ω, and ρ mesons (scalar, vector, and isovector, respectively). In

addition to the free meson and meson-nucleon terms, non-linear self-couplings

of the scalar field have been included. Explicitly, the Lagrangian density is

53

L = Ψ[γµ

(i∂µ − gωωµ −

gρ2~ρµ.~τ

)− (M − gσσ)

]Ψ

+1

2

[∂µσ∂

µσ −m2σσ

2 − κ

3(gσσ)3 − λ

12(gσσ)4

]

+1

2

[−1

2fµνf

µν +m2ωω

µωµ

]+

1

2

[−1

2~Bµν

~Bµν +m2ρ~ρµ~ρµ

], (3.1)

where fµν = ∂µων − ∂νωµ (3.2)

~Bµν = ∂µ~ρν − ∂ν~ρµ (3.3)

are the field strength tensors of the ω and ρ fields, respectively, and ~τ are the

SU(2) isospin matrices.

This is a renormalizable Lagrangian. Renormalizability is important be-

cause it ensures that the extension of the theory to densities higher than

nuclear saturation density does not require additional parameters. However,

it is not essential; this model is, afterall, a low-energy effective theory which

must break down at the GeV scale (as suggested by Bjorken scaling the onset

of which signifies the resolution of substructure in nucleons).

The preservation of renormalizability prohibits the inclusion of vector-

vector couplings in the Lagrangian. This can be understood in terms of naive

power-counting arguments: The canonical dimension for the scalar field is

D = 1 which means that the corresponding couplings (up to σ4) will have

D ≥ 0. The vector fields have D = 2; so couplings of more than two such

fields will have D < 0 and thus the Lagrangian will be non-renormalizable.

Even D = 0 can be a problem. For the neutral field (ω) this is resolved because

it is coupled to a conserved current, namely the baryon density. The charged

field ρ enters through spontaneous SU(2) isospin symmetry breaking and thus

it, too, is renormalizable.

Additionally, local symmetry breaking generates a Higgs field which is,

however, very massive and since the Higgs mean field, to leading order, goes

as 1m2H

it is ignored. The pion field is also excluded. This is a pseudoscalar

and as such it couples states of opposite parity whereas the nuclear ground

state, in which we are interested, is assumed to be an eigenstate of parity.

54

Nevertheless, one can argue that pions are implicitly included via the σ field

which may be thought of as a parametrization of the 2π exchange.

3.2 Mean Field Approximation

The mean field approximation consists of the following assumptions:

1. The fluctuations in the meson fields are negligible and thus the meson

field operators can be replaced by their expectation values:

σ → 〈σ〉 ≡ σo (3.4)

ω → 〈ωµ〉 ≡ δµoωo (3.5)

ρjµ → 〈ρjµ〉 ≡ δµoδj3ρo. (3.6)

The nucleons then, move independently in these classical fields. This

assumption becomes more valid as the nucleon density increases, since

the nucleons are the sources of the meson fields.

2. The system under consideration is uniform and static. This means that

the meson fields are not functions of the spacetime coordinate xµ (i.e.

∂µφ = 0) and the Lagrangian becomes

LMFT = Ψ[iγµ∂

µ − γogωωo − γogρ2ρoτ3 − (M − gσσo)

]Ψ

−1

2

[m2σσ

2o +

κ

3(gσσo)

3 +λ

12(gσσo)

4

]

+1

2m2ωω

2o +

1

2m2ρρ

2o. (3.7)

3.3 Equations of Motion

With the LMFT of (3.7), the Euler-Lagrange (EL) equations for the meson

fields simplify to ∂L∂φ

= 0 and thus their equations of motion are:

• scalar field

∂L∂σo

= gσ〈ΨΨ〉 −m2σσo −

κ

2g3σσ

2o −

λ

6g4σσ

3o = 0

55

⇒ gσ〈ΨΨ〉 = gσns = m2σσo +

κ

2g3σσ

2o +

λ

6g4σσ

3o (3.8)

where 〈ΨΨ〉 = ns is the scalar density which is dynamical and is deter-

mined by the solution of the baryon field equation.

• vector field

∂L∂ωo

= −gω〈ΨγoΨ〉+m2ωωo = −gω〈Ψ†Ψ〉+m2

ωωo = 0

⇒ ωo =gωm2ω

〈Ψ†Ψ〉 =gωm2ω

n (3.9)

where n (the source of ωo) is the baryon density.

• isovector field

∂L∂ρo

= −gρ2〈Ψγoτ3Ψ〉+m2

ρρo = −gρ2〈Ψ†τ3Ψ〉+m2

ρρo = 0

⇒ ρo =gρ

2m2ρ

〈Ψ†τ3Ψ〉 =gρ

2m2ρ

(np − nn) (3.10)

since τ3 =

1 0

0 −1

and Ψp ∼

1

0

, Ψn ∼

0

1

.

Equations (3.8)-(3.10) are to be solved self-consistently for the corresponding

field.

The EL equation for the nucleon field is

∂µ∂L

∂(∂µΨ)− ∂L∂Ψ

= 0 (3.11)

Since LMFT has no ∂µΨ term, (3.11) reduces to ∂L∂Ψ

= 0 just like for mesons.

Thus, we have

∂L∂Ψ

=[iγµ∂

µ − γogωωo − γogρ2ρoτ3 − (M − gσσo)

]Ψ = 0

⇒[iγµ∂

µ − γo(gωωo +gρ2ρoτ3)−M∗

]Ψ = 0 (3.12)

where M∗ = M − gσσo (3.13)

56

is the Dirac effective mass. For the solution of (3.12), we make the ansatz

Ψ = ψ(~k, s, τ)ei~k·~x−iεt (3.14)

and use γµ = (β, β~α) (3.15)

∂µ = (∂t,∇~x) (3.16)

to write it as

(βε− β~α · ~k − βgωωo − βgρ2ρoτ3 −M∗)ψ(~k, s, τ) = 0

Multiplication on the left by −β leads to

(−ε+ ~α · ~k + gωωo +gρ2ρoτ3 + βM∗)ψ(~k, s, τ) = 0

⇒ (~α · ~k + βM∗)ψ(~k, s, τ) = (ε− gωωo −gρ2ρoτ3)ψ(~k, s, τ)

Finally, we multiply both sides by the Hermitian conjugate:

k2 +M∗2 = (ε− gωωo −gρ2ρoτ3)2

⇒ ε = ±(k2 +M∗2)1/2 + gωωo +gρ2ρoτ3 (3.17)

⇒ εki± = ±E∗ki +g2ω

m2ω

n+g2ρ

4m2ρ

(ni − nj), (3.18)

where the subscripts i, j refer to the nucleon species, the positive sign to the

particles, and the negative sign to the antiparticles.

Equation (3.18) gives the single-particle energy spectrum of the MFT we have

adopted [36].


The examination of the T = 0 thermodynamics of the system described by

LMFT requires the construction of its energy-momentum tensor:

Tµν =∂L

∂(∂µφ)∂νφ− gµνL

57

= −gµν

Ψ[iγµ∂

µ − γo(gωωo +gρ2ρoτ3)− (M − gσσo)

]Ψ

− 1

2

[m2σσ

2o +

κ

3(gσσo)

3 +λ

12(gσσo)

4

]

+1

2m2ωω

2o +

1

2m2ρρ

2o

+ ∂νΨ(iΨγµ)

The Ψ−Ψ bracket is the equation of motion of the nucleon field which is

equal to 0 and thus:

Tµν = iΨγµ∂νΨ +gµν2

[m2σσ

2o +

κ

3(gσσo)

3 +λ

12(gσσo)

4 −m2ωω

2o −m2

ρρ2o

](3.19)

For an isotropic system observed in its rest-frame, the energy density and

the pressure are given by Tµν ’s diagonal elements:

ε = 〈Too〉

= iΨγo∂tΨ +1

2

[m2σσ

2o +

κ

3(gσσo)

3 +λ

12(gσσo)

4 −m2ωω

2o −m2

ρρ2o

]

= iΨ†(−iε)Ψ +1

2

[m2σσ

2o +

κ

3(gσσo)

3 +λ

12(gσσo)

4 −m2ωω

2o −m2

ρρ2o

].

The ground state of a uniform, infinite system is obtained by filling all

available stages up to kF :

ε =∫ kF i

0Ψ†[(k2i +M∗2)1/2 + gωωo +

gρ2ρoτ3

]Ψd3ki(2π)3

+1

2

[m2σσ

2o +

κ

3(gσσo)

3 +λ

12(gσσo)

4 − g2ω

m2ω

n2 −g2ρ

4m2ρ

(np − nn)2

],

(3.20)

where equations (3.9), (3.10), and (3.17) have been used for ωo, ρo, and ε

respectively. The integral in (3.20) gives

∫ kF i

0Ψ†(k2

i +M∗2)1/2Ψd3ki(2π)3

+ gωωo

∫ kF i

0Ψ†Ψ

d3ki(2π)3

+gρ2ρo

∫ kF i

0Ψ†τ3Ψ

d3ki(2π)3

= 2∑i

∫ kF i

0(k2i +M∗2)1/2 d

3ki(2π)3

+ gωωon+gρ2ρo(np − nn)

58

= 2∑i

∫ kF i

0(k2i +M∗2)1/2 d

3ki(2π)3

+g2ω

m2ω

n2 +g2ρ

4m2ρ

(np − nn)2. (3.21)

Therefore the energy density is

ε = 2∑i

∫ kF i

0(k2i +M∗2)1/2 d

3ki(2π)3

+g2ω

2m2ω

n2 +g2ρ

8m2ρ

(np − nn)2

+1

2

[m2σσ

2o +

κ

3(gσσo)

3 +λ

12(gσσo)

4

]. (3.22)

The calculation of pressure proceeds along similar lines:

P =1

3〈Tii〉

=1

3

iΨβ~α · ∇Ψ− 3

2

[m2σσ

2o +

κ

3(gσσo)

3 +λ

12(gσσo)

4 −m2ωω

2o −m2

ρρ2o

].

(3.23)

In momentum space

iΨβ~α · ∇Ψ→ Ψγo~α · ~kΨ (3.24)

Furthermore, the nucleon equation of motion implies

Ψγo~α · ~kΨ = Ψ†(ε− gωωo −gρ2ρoτ3 − βM∗)Ψ (3.25)

Using our stationary state ansatz for Ψ and a spinor normalization such

that ψψ = M∗

E∗ψ†ψ we get

Ψ†βM∗Ψ = ψ†(k, s, i)βM∗ψ(k, s, i)

= M∗ψψ

=M∗2

E∗ψ†ψ. (3.26)

The insertion of (3.25) and (3.26) into (3.23) together with the meson

equations of motion lead to the final result:

59

P =1

3× 2

∑i

∫ kF i

0

k2i

(k2i +M∗2)1/2

d3ki(2π)3

+g2ω

2m2ω

n2 +g2ρ

8m2ρ

(np − nn)2

− 1

2

[m2σσ

2o +

κ

3(gσσo)

3 +λ

12(gσσo)

4

]. (3.27)

The Dirac effective mass M∗ of the nucleons is obtained by minimizing the

energy density with respect to M∗ or, equivalently, with respect to σo (since

M∗ = M − gσσo) [37]:

∂ε

∂σo=

∂

∂σo

2∑i

∫ kF i

0(k2i + (M − gσσo)2)1/2 d

3ki(2π)3

+g2ω

2m2ω

n2 +g2ρ

8m2ρ

(np − nn)2

+1

2

[m2σσ

2o +

κ

3(gσσo)

3 +λ

12(gσσo)

4

]

= −2gσ∑i

∫ kF i

0

M∗

E∗ki

d3ki(2π)3

+m2σ

(M −M∗

gσ

)

+κ

2g3σ

(M −M∗

gσ

)2

+λ

6g4σ

(M −M∗

gσ

)3

= −gσns +m2σ

gσ(M −M∗) +

κ

2gσ(M −M∗)2 +

λ

6gσ(M −M∗)3 = 0 (3.28)

ns ≡ 2∑i

∫ kF i

0

M∗

E∗ki

d3ki(2π)3

(scalar density) (3.29)

Expression (3.28) can be rearranged as a self-consistent equation for M∗:

M∗ = M − g2σ

m2σ

[ns −

κ

2(M −M∗)2 − λ

6(M −M∗)3

](3.30)

The (T = 0) chemical potentials are derived by evaluating the single-

particle spectrum at the Fermi surface:

µi = E∗Fi +g2ω

m2ω

n+g2ρ

4m2ρ

(ni − nj) (3.31)

Alternatively, one can arrive to (3.31) by differentiating the energy density

with respect to the individual nucleon number densities while keeping in mind

60

that

dε

dni

∣∣∣∣∣nj

=∂ε

∂ni

∣∣∣∣∣nj ,M∗

+∂ε

∂M∗

∣∣∣∣∣ni,nj

∂M∗

∂ni

∣∣∣∣∣nj

=∂ε

∂ni

∣∣∣∣∣nj ,M∗

(since∂ε

∂M∗ = 0) (3.32)

This consideration is also relevant for the incompressibility:

K = 9nd2ε

dn2= 9n

d

dn

(dε

dn

)(3.33)

When calculating dεdn

one again ignores dM∗

dn, dM∗

dx. The mass derivatives, how-

ever, become relevant in the second derivative. The final result is

K = 3

[(1− x)k2

Fn

E∗Fn+xk2

Fp

E∗Fp

]+ 9

g2ω

m2ω

n+9g2

ρ

4m2ρ

n(1− 2x)2

+9nM∗dM∗

dn

[(1− x)

E∗Fn+

x

E∗Fp

](3.34)

The symmetry energy is

S2 =1

8

d2(ε/n)

dx2

∣∣∣∣∣x=1/2

=k2F

6E∗F+ n

g2ρ

8m2ρ

(3.35)

and the susceptibilities

χ−1ii =

dµidni

=k2Fi

3niE∗Fi+M∗

E∗Fi

dM∗

dni+

g2ω

m2ω

+g2ρ

4m2ρ

(3.36)

χ−1ij =

dµidnj

=M∗

E∗Fi

dM∗

dnj+

g2ω

m2ω

−g2ρ

4m2ρ

(3.37)

Now the theory must be calibrated. This means that equations (3.22),

(3.27), (3.30), (3.34), (3.35) have to be solved for the couplings gσ, gω, gρ, κ,

and λ given the properties of nuclear matter at saturation, and the values of

the free particle masses. Input and results are summarized in Table 3.1.

61

Saturation Values Masses (MeV) Couplings

no = 0.155 fm−3 M = 939.0 gσ = 9.061E/A = −16 MeV mσ = 511.2 gω = 10.55M∗/M = 0.7 mω = 783.0 gρ = 7.475Ko = 225 MeV mρ = 770.0 κ = 9.194 MeVS2 = 30 MeV λ = −3.280× 10−2

Table 3.1: Equilibrium properties, masses, and couplings for the Lagrangianin Equation (3.1). The coupling strengths were calculated from a set of fixedmasses and input equilibrium parameters (n = no, T = 0, and x = 1/2).

Finally, we compare the asymmetry coefficients of our MFT with those of the

similarly calibrated Skyrme model SkM*. The parameters of this model are

given in Table 3.2, and its nuclear saturation properties in Table 3.3. The

comparison of the asymmetry coefficients at nuclear saturation is performed

in Table 3.4.

i ti xi ε

0 −2645.0 MeVfm6 0.09 1/61 410.0 fm3 02 −135.0 fm3 03 15595.0 MeVfm7 0

Table 3.2: Parameter values for the SkM* Hamiltonian density. The dimen-sions are such that the Hamiltonian density is in MeV fm−3.

no E/A Ko S2 m∗/m S LMFT 0.155 -16.00 225.0 30.00 0.70 -164.2 87.0SkM* 0.160 -15.80 216.6 30.03 0.79 913.7 45.8

Table 3.3: Saturation properties of symmetric nuclear matter. With the ex-ception of no which is in fm−3 and m∗/m which is unitless, all other quantitiesare in MeV.

62

Model A B C AMFT 1.738 -1.674 0.952 -0.064SkM* 0.548 -2.159 0.634 -1.611

Table 3.4: Results for the coefficients that describe the isospin asymmetrydependence. For each model these are given at the equilibrium density ofsymmetric matter.

Just as in the APR vs. Ska comparison, similar values of the symmetry en-

ergy at saturation density and the compression modulus do not imply similar

asymmetry properties at all densities.

3.4.1 Results

The results presented here are for zero temperature, bulk homogeneous nuclear

matter. The plots show comparisons of the MFT and the SkM* models at

proton fractions of 0.3 and 0.5.

The energy per particle of MFT is significantly higher than that of SkM*

for densities above 0.2 fm−3. This is related to the L parameter (see Eq. 2.34)

of MFT which is much higher than that of SKM*. At smaller proton fractions,

the models are less bound and their saturation points occur at lower densities.

The pressure curve of MFT is much stiffer than that of SkM* for densities

above saturation despite the fact that the compression moduli of the two

models are nearly equal. The chemical potentials behave similarly to the

pressure.

The isospin susceptibilities exhibit large differences throughout the density

range.

63

-50

0

50

100

150

200

250

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

E/A

[MeV

]

nB [fm-3]

T = 0 MeV

MFT x=0.5MFT x=0.3

SkMs x=0.5SkMs x=0.3

Figure 3.1: Zero temperature comparison of the energy per particle versusdensity of the EOS of MFT (blue) and SkM* (red) at proton fractions of 0.3(dotted lines) and 0.5 (solid lines).

-50

0

50

100

150

200

250

300

350

400

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

P [M

eV/fm

3 ]

nB [fm-3]

T = 0 MeV

MFT x=0.5MFT x=0.3


Figure 3.2: Zero temperature comparison of the pressure versus density of theEOS of MFT (blue) and SkM* (red) at proton fractions of 0.3 (dotted lines)and 0.5 (solid lines).

64

-100

0

100

200

300

400

500

600

700

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

µ n [M

eV]

nB [fm-3]

T = 0 MeV

MFT x=0.5MFT x=0.3


Figure 3.3: Zero temperature comparison of the neutron chemical potentialversus density of the EOS of MFT (blue) and SkM* (red) at proton fractionsof 0.3 (dotted lines) and 0.5 (solid lines).

-100

0

100

200

300

400

500

600

700

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

µ p [M

eV]

nB [fm-3]

T = 0 MeV

MFT x=0.5MFT x=0.3


Figure 3.4: Zero temperature comparison of the proton chemical potentialversus density of the EOS of MFT (blue) and SkM* (red) at proton fractionsof 0.3 (dotted lines) and 0.5 (solid lines).

65

0

200

400

600

800

1000

1200

1400

1600

1800

2000

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

dµn

/ dn n

[MeV

fm3 ]

nB [fm-3]

T = 0 MeV

MFT x=0.5MFT x=0.3


Figure 3.5: Zero temperature comparison of the neutron-neutron susceptibilityversus density of the EOS of MFT (blue) and SkM* (red) at proton fractionsof 0.3 (dotted lines) and 0.5 (solid lines).

-1000

-500

0

500

1000

1500

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

dµn

/ dn p

[MeV

fm3 ]

nB [fm-3]

T = 0 MeV

MFT x=0.5MFT x=0.3


Figure 3.6: Zero temperature comparison of the neutron-proton susceptibilityversus density of the EOS of MFT (blue) and SkM* (red) at proton fractionsof 0.3 (dotted lines) and 0.5 (solid lines).

66


At finite temperature the occupation number operator nki is

nki = ψ†kiψki =1

e(εki−µ)/T + 1(3.38)

(instead of θ(ki − kFi) ).

Therefore the number density is given by

n =∑s,i

∫ ∞0

d3ki(2π)3

1

e(εki−µi)/T + 1(3.39)

=∑i

1

π2

∫ ∞0

dkik2i

e(E∗ki−νi)/T + 1

, (3.40)

where νi = µi −g2ω

m2ω

n−g2ρ

4m2ρ

(ni − nj). (3.41)

Antiparticles are ignored in our calculations due to the following considera-

tions:

Baryon number conservation requires that the chemical potentials of particles

and antiparticles are equal and opposite

µp = −µa ≡ µ > 0.

Furthermore, for the low temperatures (T ≤ 50 MeV) in which we are inter-

ested µ ' εF , and states with ε > εF are mostly unoccupied. Thus,

e(εp−µ)/T 1

e(εa+µ)/T 1.

So, the contributions of antiparticles are negligible in supernova matter.

By defining the variables

αi ≡E∗kiT, x ≡ M∗

T, φi ≡

νiT

(3.42)

which imply

ki = T 2(α2i − x2)1/2 (3.43)

67

dki =Tαi

(α2i − x2)1/2

dαi (3.44)

equation (3.40) can be written as

n =∑i

T 3

π2

∫ ∞0

αi(α2i − x2)1/2

e(αi+φi) + 1dαi. (3.45)

Shifting the integration variables

αi → αi + x, φi → φi − x ≡ ψi =νi −M∗

T(3.46)

results in

n =∑i

T 3

π2

∫ ∞0

(αi + x)(α2i + 2αix)1/2

e(αi−ψi) + 1dαi

=∑i

T 3

π2

∫ ∞0

(αi + x)(2αix)1/2(αi2x

+ 1)1/2


=∑i

21/2

π2T 3x1/2

∫ ∞0

(α3/2i + α

1/2i x)(αi

2x+ 1)1/2


=∑i

21/2

π2T 5/2M∗1/2

∫ ∞0

α3/2i (αi

2x+ 1)1/2

e(αi−ψi) + 1dαi + x

∫ ∞0

α1/2i (αi

2x+ 1)1/2


=

∑i

21/2

π2T 5/2M∗1/2

[F3/2i(ψi, x) + xF1/2i(ψi, x)

], (3.47)

where Fγi =∫ ∞

0

αγi (αi2x

+ 1)1/2

e(αi−ψi) + 1dαi (3.48)

is the (dimensionless) Relativistic Fermi-Dirac (RFD) integral of order γ.

The RFD integrals satisfy the recursion relation

∂Fγ∂ψ

= γFγ−1 − x∂Fγ−1

∂x. (3.49)

Similar considerations lead to the following expressions for the kinetic en-

ergy density τ , the kinetic pressure pk, and the scalar density ns:

τ =∑s,i

∫ ∞0

d3ki(2π)3

E∗kie(E∗

ki−νi)/T + 1

(3.50)

68

=∑i

21/2

π2T 7/2M∗1/2

[F5/2i + 2xF3/2i + x2F1/2i

](3.51)

pk =∑s,i

1

3

∫ ∞0

d3ki(2π)3

k2i

E∗ki

1


(3.52)

=∑i

21/2

3π2T 7/2M∗1/2

[F5/2i + 2xF3/2i

](3.53)

ns =∑s,i

∫ ∞0

d3ki(2π)3

M∗

E∗ki

1


(3.54)

=∑i

21/2

π2T 3/2M∗3/2F1/2i. (3.55)

Equations (3.47), (3.51), and (3.53) can be solved as a system for F1/2i,

F3/2i, and F5/2i :

F1/2i =π2

21/2

1

T 3/2M∗5/2 (τi − 3pki) (3.56)

F3/2i =π2

21/2

1

T 5/2M∗3/2 (M∗ni − τi + 3pki) (3.57)

F5/2i =π2

21/2

1

T 7/2M∗1/2 (2τi − 3pki − 2M∗ni) (3.58)

Hence

ns =∑i

1

M∗ (τi − 3pki). (3.59)


For the numerical evaluation of the thermodynamic integrals we employ the

relativistic version of the JEL method [30] whereby the number density, the

kinetic energy density, and the kinetic pressure are expressed algebraically in

terms of the effective mass, the temperature, and the chemical potentials:

ni =M∗3

π2

fig3/2i (1 + gi)

3/2

(1 + fi)M+1/2(1 + gi)N(1 + fi/a)1/2

×M∑m=0

N∑n=0

pmnfmi g

ni

[1 +m+

(1

4+n

2−M

)fi

1 + fi

+(

3

4− N

2

)figi

(1 + fi)(1 + gi)

](3.60)

Ui = τi −M∗ni

69

=M∗4

π2

fig5/2i (1 + gi)

3/2

(1 + fi)M+1(1 + gi)N

M∑m=0

N∑n=0

pmnfmi g

ni

×[

3

2+ n+

(3

2−N

)gi

1 + gi

](3.61)

pki =M∗4

π2

fig5/2i (1 + gi)

3/2

(1 + fi)M+1(1 + gi)N

M∑m=0

N∑n=0

pmnfmi g

ni , (3.62)

where fi is given by the solution of

ψi =νi −M∗

T= 2(1 + fi/a)1/2 ln

[(1 + fi/a)1/2 − 1

(1 + fi/a)1/2 + 1

](3.63)

and gi = TM∗

(1 + fi)1/2 ≡ t(1 + fi)

1/2.

The derivatives of fi and gi are :

∂fi∂ψi

=

(∂ψi∂fi

)−1

=fi

1 + fi/a(3.64)

∂gi∂fi

=t

2(1 + fi)2=

t2

2gi(3.65)

∂gi∂t

= (1 + fi)1/2 =

git

(3.66)

∂fi∂t

= 0 (3.67)

The coefficients pmn for M = N = 3 and a = 0.433 are displayed in Table 3.5.

pmn n = 0 n = 1 n = 2 n = 3m = 0 5.34689 18.0517 21.3422 8.53240m = 1 16.8441 55.7051 63.6901 24.6213m = 2 17.4708 56.3902 62.1319 23.2602m = 3 6.07364 18.9992 20.02285 7.11153

Table 3.5: JEL coefficients pmn for M = N = 3 and a = 0.433

The entropy and the free energy follow from standard thermodynamic re-

lations:

s =1

T(ε+ p−

∑i

µini) (3.68)

F = ε− Ts (3.69)

70

3.5.2 Results

In what follows we present comparative results for MFT and SKM* at 20 MeV,

in the case of isospin symmetric bulk homogeneous matter. The observed

trends at zero temperature persist at finite temperature as well. However,

unlike the other state variables, the entropies of the two models are very nearly

the same for all densities.

-20

0

20

40

60

80

100

120

140

160

180

200

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

E/A

[MeV

]

nB [fm-3]

T = 20 MeV

x = 0.5

MFTSkMs

Figure 3.7: Comparison of the energy per particle versus density of the EOSof MFT (blue) and SkM* (red) at T = 20 MeV.

71

0

50

100

150

200

250

300

350

400

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

P [M

eV/fm

3 ]

nB [fm-3]

T = 20 MeVx = 0.5

MFTSkMs

Figure 3.8: Comparison of the pressure versus density of the EOS of MFT(blue) and SkM* (red) at T = 20 MeV.

-100

0

100

200

300

400

500

600

700

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

µ [M

eV]

nB [fm-3]

T = 20 MeVx = 0.5

MFTSkMs

Figure 3.9: Comparison of the chemical potential versus density of the EOSof MFT (blue) and SkM* (red) at T = 20 MeV.

72

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

s/A

nB [fm-3]

T = 20 MeVx = 0.5

MFTSkMs

Figure 3.10: Comparison of the entropy per particle versus density of the EOSof MFT (blue) and SkM* (red) at T = 20 MeV.

3.5.3 Susceptibilities

For the ∂νi∂nj

part of the susceptibilities, we exploit properties of partial deriva-

tives and perform the necessary variable changes so that they may be expressed

explicitly in terms of JEL-differentiable functions.

We begin by writing the differential of ni in terms of nj and νk, where the

subscript k can be either i or j:

dni =∂ni∂nj

∣∣∣∣∣νk

dnj +∂ni∂νk

∣∣∣∣∣nj

dνk. (3.70)

Then dnj and dνk are expressed in terms of ψi and t:

dnj =∂nj∂ψi

∣∣∣∣∣t

dψi +∂nj∂t

∣∣∣∣∣ψi

dt (3.71)

dνk =∂νk∂ψi

∣∣∣∣∣t

dψi +∂νk∂t

∣∣∣∣∣ψi

dt. (3.72)

73

and subsequently plugged back into (3.70):

dni =∂ni∂nj

∣∣∣∣∣νk

∂nj∂ψi

∣∣∣∣∣t

dψi +∂nj∂t

∣∣∣∣∣ψi

dt

+∂ni∂νk

∣∣∣∣∣nj

∂νk∂ψi

∣∣∣∣∣t

dψi +∂νk∂t

∣∣∣∣∣ψi

dt

.Collecting terms proportionalto dψi and dt we get:

dni =

∂ni∂nj

∣∣∣∣∣νk

∂nj∂ψi

∣∣∣∣∣t

+∂ni∂νk

∣∣∣∣∣nj

∂νk∂ψi

∣∣∣∣∣t

dψi+

∂ni∂nj

∣∣∣∣∣νk

∂nj∂t

∣∣∣∣∣ψi

+∂ni∂νk

∣∣∣∣∣nj

∂νk∂t

∣∣∣∣∣ψi

dt. (3.73)

Thus

∂ni∂ψi

∣∣∣∣∣t

=∂ni∂nj

∣∣∣∣∣νk

∂nj∂ψi

∣∣∣∣∣t

+∂ni∂νk

∣∣∣∣∣nj

∂νk∂ψi

∣∣∣∣∣t

(3.74)

∂ni∂t

∣∣∣∣∣ψi

=∂ni∂nj

∣∣∣∣∣νk

∂nj∂t

∣∣∣∣∣ψi

+∂ni∂νk

∣∣∣∣∣nj

∂νk∂t

∣∣∣∣∣ψi

. (3.75)

Solving (3.74) for ∂ni∂nj

∣∣∣νk

and substituting the result into (3.75)

∂ni∂t

∣∣∣∣∣ψi

=

∂ni∂ψi

∣∣∣∣∣t

− ∂ni∂νk

∣∣∣∣∣nj

∂νk∂ψi

∣∣∣∣∣t

( ∂nj∂ψi

∣∣∣∣∣t

)−1∂nj∂t

∣∣∣∣∣ψi

+∂ni∂νk

∣∣∣∣∣nj

∂νk∂t

∣∣∣∣∣ψi

. (3.76)

We then solve (3.76) for ∂ni∂νk

∣∣∣nj

:

∂ni∂νk

∣∣∣∣∣nj

=

∂ni∂t

∣∣∣ψi

∂nj∂ψi

∣∣∣t− ∂ni

∂ψi

∣∣∣t

∂nj∂t

∣∣∣ψi

∂νk∂t

∣∣∣ψi

∂nj∂ψi

∣∣∣t− ∂νk

∂ψi

∣∣∣t

∂nj∂t

∣∣∣ψi

. (3.77)

By inverting (3.77) we arrive at

∂νk∂ni

∣∣∣∣∣nj

=

∂νk∂t

∣∣∣ψi

∂nj∂ψi

∣∣∣t− ∂νk

∂ψi

∣∣∣t

∂nj∂t

∣∣∣ψi

∂ni∂t

∣∣∣ψi

∂nj∂ψi

∣∣∣t− ∂ni

∂ψi

∣∣∣t

∂nj∂t

∣∣∣ψi

. (3.78)

74

In the next step the goal is to express ∂nj∂ψi

∣∣∣t

and ∂nj∂t

∣∣∣ψi

in terms of ∂nj∂ψj

∣∣∣t

and

∂nj∂t

∣∣∣ψj

. Initially, dnj is written in terms of ψi and t:

dnj =∂nj∂ψi

∣∣∣∣∣t

dψi +∂nj∂t

∣∣∣∣∣ψi

dt, (3.79)

and afterwards, dψi is cast in terms of ψj and t

dψi =∂ψi∂ψj

∣∣∣∣∣t

dψj +∂ψi∂t

∣∣∣∣∣ψj

dt. (3.80)

Combining the two expressions we get

dnj =∂nj∂ψi

∣∣∣∣∣t

∂ψi∂ψj

∣∣∣∣∣t

dψj +∂ψi∂t

∣∣∣∣∣ψj

dt

+∂nj∂t

∣∣∣∣∣ψi

dt

=∂nj∂ψi

∣∣∣∣∣t

∂ψi∂ψj

∣∣∣∣∣t

dψj +

∂nj∂ψi

∣∣∣∣∣t

∂ψi∂t

∣∣∣∣∣ψj

+∂nj∂t

∣∣∣∣∣ψi

dt (3.81)

from which we deduce that

∂nj∂ψj

∣∣∣∣∣t

=∂nj∂ψi

∣∣∣∣∣t

∂ψi∂ψj

∣∣∣∣∣t

⇒ ∂nj∂ψi

∣∣∣∣∣t

=∂nj∂ψj

∣∣∣∣∣t

∂ψj∂ψi

∣∣∣∣∣t

. (3.82)

The cyclic property of partial derivatives says that

∂ψj∂ψi

∣∣∣∣∣t

= − ∂t

∂ψi

∣∣∣∣∣ψj

∂ψj∂t

∣∣∣∣∣ψi

(3.83)

With this, (3.82) becomes

∂nj∂ψi

∣∣∣∣∣t

= − ∂t

∂ψi

∣∣∣∣∣ψj

∂ψj∂t

∣∣∣∣∣ψi

∂nj∂ψj

∣∣∣∣∣t

. (3.84)

From (3.81), we also get

∂nj∂t

∣∣∣∣∣ψj

=∂nj∂t

∣∣∣∣∣ψi

+∂nj∂ψi

∣∣∣∣∣t

∂ψi∂t

∣∣∣∣∣ψj

75

=∂nj∂t

∣∣∣∣∣ψi

− ∂ψj∂t

∣∣∣∣∣ψi

∂nj∂ψj

∣∣∣∣∣t

, (3.85)

where (3.84) was used in going from the first line of (3.85) to the second.

Rearranging (3.85) leads to

∂nj∂t

∣∣∣∣∣ψi

=∂nj∂t

∣∣∣∣∣ψj

+∂ψj∂t

∣∣∣∣∣ψi

∂nj∂ψj

∣∣∣∣∣t

. (3.86)

Finally, we insert (3.84) and (3.86) into (3.78):

∂νk∂ni

∣∣∣∣∣nj

=

∂νk∂t

∣∣∣ψi

∂t∂ψi

∣∣∣ψj

∂ψj∂t

∣∣∣ψi

∂nj∂ψj

∣∣∣t+ ∂νk

∂ψi

∣∣∣t

(∂nj∂t

∣∣∣ψj

+ ∂ψj∂t

∣∣∣ψi

∂nj∂ψj

∣∣∣t

)∂ni∂t

∣∣∣ψi

∂t∂ψi

∣∣∣ψj

∂ψj∂t

∣∣∣ψi

∂nj∂ψj

∣∣∣t+ ∂ni

∂ψi

∣∣∣t

(∂nj∂t

∣∣∣ψj

+ ∂ψj∂t

∣∣∣ψi

∂nj∂ψj

∣∣∣t

) . (3.87)

The last ingredient one needs to consider in order for (3.87) to be truly JEL-

differentiable are the derivatives of ν with respect to ψ and t. Since

νi = Tψi +M∗ = Tψi + T/t (3.88)

we have

∂νi∂ψi

∣∣∣∣∣t

= T (3.89)

∂νi∂t

∣∣∣∣∣ψi

= −Tt2. (3.90)

Also,∂νi∂ψj

∣∣∣∣∣t

= T∂ψi∂ψi

∣∣∣∣∣t

= −T ∂t

∂ψj

∣∣∣∣∣ψi

∂ψi∂t

∣∣∣∣∣ψj

, (3.91)

where the cyclic property has been used, and

∂νi∂t

∣∣∣∣∣ψj

= T∂ψi∂t

∣∣∣∣∣ψj

− T

t2= −T

t2

1− t2 ∂ψi∂t

∣∣∣∣∣ψj

. (3.92)

In contrast with all previous results pertaining to the susceptibilities, the cal-

culation of ∂t∂ψi

∣∣∣ψj

is model-dependent.

76

We begin by writing equation (3.13) for the Dirac mass as

σo =M − T/t

gσ(3.93)

and insert it into the equation of motion of the scalar field (3.8):

T

t= M − g2

σ

m2σ

[ns −

κ

2(M − T/t)2 − λ

6(M − T/t)3

]. (3.94)

Then we act with ∂∂ψi

∣∣∣ψj

on both sides and solve for ∂t∂ψi

∣∣∣ψj

:

∂t

∂ψi

∣∣∣∣∣ψj

=t2

T

[m2σ

g2σ

+ κ(M − T/t) +λ

2(M − T/t)2

]−1∂ns∂ψi

∣∣∣∣∣ψj

(3.95)

≡ t2

T

1

cσ

∂ns∂ψi

∣∣∣∣∣ψj

. (3.96)

But∂ns∂ψi

∣∣∣∣∣ψj

=∂ns∂ψi

∣∣∣∣∣ψj ,t

+∂ns∂t

∣∣∣∣∣ψi,ψj

∂t

∂ψi

∣∣∣∣∣ψj

. (3.97)

Using (3.96), this becomes

∂ns∂ψi

∣∣∣∣∣ψj

=∂ns∂ψi

∣∣∣∣∣ψj ,t

+∂ns∂t


t2

T

1

cσ

∂ns∂ψi

∣∣∣∣∣ψj

. (3.98)

Collecting ∂ns∂ψi

∣∣∣ψj

on one side, gives

∂ns∂ψi

∣∣∣∣∣ψj

=

1− t2

T

1

cσ

∂ns∂t


−1∂ns∂ψi

∣∣∣∣∣ψj ,t

(3.99)

⇒ ∂t

∂ψi

∣∣∣∣∣ψj

=

Tcσt2− ∂ns

∂t


−1∂ns∂ψi

∣∣∣∣∣ψj ,t

(3.100)

with ns given by (3.59) using the JEL expressions for τi and pki.

One should be mindful of the fact that the M∗ terms, by which the JEL

functions are multiplied, are also acted upon by derivatives with respect to ψ

(f) and t. The same is true for the variable g.

77

3.5.3.1 Results

500

1000

1500

2000

2500

3000

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

dµn

/ dn n

[MeV

fm3 ]

nB [fm-3]

T = 20 MeVx = 0.5

MFTSkMs

Figure 3.11: Comparison of the neutron-neutron susceptibility versus densityof the EOS of MFT (blue) and SkM* (red) at T = 20 MeV.

-1000

-500

0

500

1000

1500

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

dµn

/ dn p

[MeV

fm3 ]

nB [fm-3]

T = 20 MeVx = 0.5

MFTSkMs

Figure 3.12: Comparison of the neutron-proton susceptibility versus densityof the EOS of MFT (blue) and SkM* (red) at T = 20 MeV.

78

3.5.4 Limiting Cases

Just as in the case of the non-relativistic potential, we check the validity of

our numerical results by comparing them with analytical expressions in the

degenerate and non-degenerate limits. Once again we are interested in the

thermal parts of the various state variables.

3.5.4.1 Degenerate Limit

In the low-temperature domain, FLT is employed [38]. By combining (3.18) for

the MFT energy spectrum with (2.90), we get for the level density parameter

ai =π2

2

E∗Fik2Fi

. (3.101)

Its derivatives with respect to density are

daidn

= − ai3n

1 +

(M∗

E∗Fi

)2 (1− 3n

M∗dM∗

dn

) (3.102)

daidni

= − ai3ni

1 +

(M∗

E∗Fi

)2 (1− 3ni

M∗dM∗

dni

) (3.103)

daidnj

= aiM∗

E∗Fi

dM∗

dnj. (3.104)

Using these along with S = 2Tn

∑i aini for the entropy in the appropriate

Maxwell’s relations we have:

• thermal energy

Eth =∫TdS =

T 2

n

∑i

aini (3.105)


pth =∑i

[aini − n

d(aini)

dn

]T 2

=T 2

3

∑i

aini1 +

(M∗

E∗Fi

)2 (1− 3n

M∗dM∗

dn

) (3.106)

79


µith =d

dni

∑j

ajnj

T 2

= −T2

3ai

2−(M∗

E∗Fi

)2

+∑j

ajai

(M∗

E∗Fj

)23njM∗

dM∗

dni

.(3.107)

In the preceding expressions, M∗ is the Dirac effective mass at T = 0.

3.5.4.2 Non-Degenerate Limit

In the non-degenerate limit, analytic expressions for the thermodynamic inte-

grals are obtained [39] by exploiting the smallness of the fugacity (z = eνT 1).

Consider, for example, the number density of a single nucleon species i :

ni =1

π2

∫ ∞0

dkik2i

1 + e(E∗ki−νi)/T

=1

π2

∫ ∞0

dkik2i

1 + exp[

(ki2+M∗2)1/2

T− νi

T

] . (3.108)

With the definitions

x ≡ M∗

T, yi ≡

kiM∗ (3.109)

the number density becomes

ni =1

π2

∫ ∞0

M∗dyi(M∗yi)

2

1 + z−1i exp [x(1 + y2

i )1/2]

=M∗3

π2

∫ ∞0

dyizi exp[−x(1 + y2

i )1/2]y2

i

1 + zi exp[−x(1 + y2i )

1/2]. (3.110)

The term w = zi exp[−x(1 + y2i )

1/2] is smaller than unity since zi 1 and

x, yi ≥ 0 (which means that the exponential is less than or equal to 1). Thus

we can write the denominator in (3.110) as a geometric series in w:

1

1 + w=

∞∑m=0

(−1)mwm. (3.111)

80

The number density, then, becomes

ni =M∗3

π2

∞∑m=0

(−1)mzmi

∫ ∞0

dyiexp[−mx(1 + y2i )

1/2]zi exp[−x(1 + y2i )

1/2]y2i

=M∗3

π2

∞∑m=1

(−1)m+1zmi

∫ ∞0

dyiexp[−mx(1 + y2i )

1/2]y2i . (3.112)

The substitution

yi = sinh ti (3.113)

⇒ (1 + y2i )

1/2 = cosh ti; dyi = cos tidti

leads to

ni =M∗3

4π2

∞∑m=1

(−1)m+1zmi

∫ ∞0

dti exp (−mx cosh ti)(cosh 3ti − cosh ti) (3.114)

The above expression can be cast in terms of the modified Bessel function

Kα(mx) =∫ ∞

0exp (−mx cosh t) coshαt dt, (3.115)

which obeys the recursion relation

Kα+1(mx) = Kα−1(mx) +2α

mxKα(mx). (3.116)

Therefore,

ni =M∗3

4π2

∞∑m=1

(−1)m+1zmi [K3(mx)−K1(mx)]

=M∗3

π2

∞∑m=1

(−1)m+1zmiK3(mx)

mx. (3.117)

Since zi is small, we only keep the first two terms in the series:

ni 'M∗3

π2

1

x

[z2iK2(x)− z2

i

K2(2x)

2

]. (3.118)

Similar manipulations lead to the corresponding expressions for the kinetic

81

energy density and the kinetic pressure:

τi =M∗4

π2

∞∑m=1

(−1)m+1zmi

[K1(mx)

mx+

3K2(mx)

m2x2

]

' M∗4

π2

zi

[K1(x)

x+

3K2(x)

x2

]− z2

i

[K1(2x)

2x+

3K2(2x)

4x2

](3.119)

pki =M∗4

π2

∞∑m=1

(−1)m+1zmiK2(mx)

m2x2

' M∗4

π2x2

[ziK2(x)− z2

i

K2(2x)

4

]. (3.120)

Next, (3.118) is perturbatively inverted for zi. We begin by writing it as

zi = ηi +1

2

K2(2x)

K2(x)z2i − . . . , (3.121)

where ηi ≡π2x

M∗3K2(x)ni. (3.122)

The first approximation is zi = ηi. Substituting this into (3.121), we get the

second order approximation

zi = ηi +1

2

K2(2x)

K2(x)η2i

=π2x

M∗3K2(x)ni

[1 +

1

2

π2x

M∗3K2(2x)

K22(x)

ni

]. (3.123)

Then, equation (3.123) is used in (3.119) and (3.120):

τi = M∗ni

[1 +

π2x

2M∗3K2(2x)

K22(x)

ni

] [K1(x)

K2(x)+

3

x

]

− π2x

2M∗2n2i

[K1(2x)

K22(x)

+3

2x

K2(2x)

K22(x)

](3.124)

pki =M∗nix

[1 +

π2x

2M∗3K2(2x)

K22(x)

ni

]− π2

4M∗2K2(2x)

K22(x)

n2i . (3.125)

In the context of supernova explosions we are interested in temperatures

at best up to 100 MeV, which means that x = M∗

Tis large and therefore

the exact Bessel functions can be substituted by their large-x expansions the

82

general form of which is given by

Kα(w) ∼(π

2w

)1/2

e−w[1 +

γ − 1

8w+

(γ − 1)(γ − 9)

2!(8w)2+ . . .

], (3.126)

where γ = 4α2. After some algebra, we arrive at

τi = M∗ni +3Tni

2

[1 +

ni4

(π

M∗T

)3/2

+5T

4M∗

](3.127)

pki = Tni

[1 +

ni4

(π

M∗T

)3/2]

(3.128)

From these, we must subtract their T = 0 counterparts in order to get the

thermal kinetic energy density and pressure:

τith = τi − TFini; pith = pki − PFi, (3.129)

where

TFi =1

8π2ni

[kFiE

∗Fi(E

∗2Fi + k2

Fi)−M∗4ln

(kFi + E∗Fi

M∗

)](3.130)

PFi =1

24π2

[kFiE

∗Fi(2k

2Fi − 3M∗2) + 3M∗4ln

(kFi + E∗Fi

M∗

)](3.131)

For the expressions of the total thermal energy and total thermal pressure

we must also calculate the difference

δV = V (σo(T ))− V (σo(0)), (3.132)

where V (σo) =1

2m2σσ

2o +

κ

6(gσσo)

3 +λ

24(gσσo)

4. (3.133)

Using the definition of the Dirac effective mass,

δV =1

2

m2σ

g2σ

[(M −M∗)2 − (M −M∗

o )2]

+κ

6

[(M −M∗)3 − (M −M∗

o )3]

+λ

24

[(M −M∗)4 − (M −M∗

o )4],

where M∗o is the T = 0 effective mass and M∗ the finite-T one.

83

Then, we define δM∗ ≡M∗ −M∗o so that

δV =1

2

m2σ

g2σ

[(M −M∗

o − δM∗)2 − (M −M∗o )2]

+κ

6

[(M −M∗

o − δM∗)3 − (M −M∗o )3]

+λ

24

[(M −M∗

o − δM∗)4 − (M −M∗o )4].

The scalar field varies slowly with temperature and thus so does M∗. This

implies that δM∗ is small relative to M∗. We are therefore justified in keeping

terms only to first order in δM∗ :

δV =1

2

m2σ

g2σ

[−2(M −M∗o )δM∗] +

κ

6[−3(M −M∗

o )δM∗)]

+λ

24[−4(M −M∗

o )δM∗)]

= −δM∗[m2σ

g2σ

(M −M∗o ) +

κ

2(M −M∗

o )2 +λ

6(M −M∗

o )3

](3.134)

Putting everything together, we have

• thermal energy

Eth = M∗ +1

n

∑i

3Tni

2

[1 +

ni4

(π

M∗T

)3/2

+5T

4M∗

]− TFini

+δV

n(3.135)


Pth =∑i

Tni

[1 +

ni4

(π

M∗T

)3/2]− PFi

− δV (3.136)


µith = µi(T )− µi(0)

= νi(T )− νi(0) = T lnzi − E∗Fi

= M∗ − E∗Fi + T

ln

[ni2

(2π

M∗T

)3/2]

+ni2

(π

M∗T

)3/2

− 15T

8M∗

(3.137)

84

For T lnzi, we use (3.123) together with (3.126) and the expansion

ln(1 + w) ' w − w2

2+ w3

3− . . . ; |w| < 1.


S =1

T

(Eth +

Pthn− 1

n

∑i

µini

)

=1

n

∑i

ni

5

2− ln

[ni2

(2π

M∗T

)3/2]

+ni8

(π

M∗T

)3/2

− 15T

4M∗

(3.138)

3.5.4.3 Results

This section hosts results pertaining to MFT at T = 20 MeV, for isospin

symmetric matter. The agreement of the exact results with the degenerate

and non-degenerate limits in the corresponding regions of validity is excellent

for all state variables.

0

5

10

15

20

25

30

35

40

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

ET

herm

al [M

eV]

nB [fm-3]

T = 20 MeV

x = 0.5

ExactDegenerate

Non-Degenerate

Figure 3.13: Comparison of the degenerate (red line) and the non-degenerate(blue line) approximations with the exact (black line) thermal energy of MFTat T = 20 MeV.

85

0

0.5

1

1.5

2

2.5

3

3.5

4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

PT

herm

al [M

eV/fm

3 ]

nB [fm-3]

T = 20 MeVx = 0.5

ExactDegenerate

Non-Degenerate

Figure 3.14: Comparison of the degenerate (red line) and the non-degenerate(blue line) approximations with the exact (black line) thermal pressure of MFTat T = 20 MeV.

-100

0

100

200

300

400

500

600

700

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

µ [M

eV]

nB [fm-3]

T = 20 MeVx = 0.5

ExactDegenerate

Non-Degenerate

Figure 3.15: Comparison of the degenerate (red line) and the non-degenerate(blue line) approximations with the exact (black line) chemical potential ofMFT at T = 20 MeV.

86

0

0.5

1

1.5

2

2.5

3

3.5

4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

s/A

nB [fm-3]

T = 20 MeVx = 0.5

ExactDegenerate

Non-Degenerate

Figure 3.16: Comparison of the degenerate (red line) and the non-degenerate(blue line) approximations with the exact (black line) entropy of MFT atT = 20 MeV.

500

1000

1500

2000

2500

3000

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

dµn

/ dn n

[MeV

fm3 ]

nB [fm-3]

T = 20 MeVx = 0.5

ExactDegenerate

Non-Degenerate

Figure 3.17: Comparison of the degenerate (red line) and the non-degenerate(blue line) approximations with the exact (black line) neutron-neutron suscep-tibility of MFT at T = 20 MeV.

87

-1000

-500

0

500

1000

1500

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

dµn

/ dn p

[MeV

fm3 ]

nB [fm-3]

T = 20 MeV

x = 0.5

ExactDegenerate

Non-Degenerate

Figure 3.18: Comparison of the degenerate (red line) and the non-degenerate(blue line) approximations with the exact (black line) neutron-proton suscep-tibility of MFT at T = 20 MeV.

3.5.5 Specific Heat

For the calculation of the MFT specific heat we begin by writing the energy

density as

ε = τn+τp+m2σ

2g2σ

(M−M∗)2+κ

6(M−M∗)3+

λ

24(M−M∗)4+n-dependent terms

Using τi = Ui +M∗ni, we recast it as

ε = Un+Up+M∗n+

m2σ

2g2σ

(M−M∗)2+κ

6(M−M∗)3+

λ

24(M−M∗)4+n-dep. terms

Then

CV =1

n

∂ε

∂T

∣∣∣∣∣nn,np

=1

n

∂Un∂T


+∂Up∂T


+∂M∗

∂T


[n− m2

σ

g2σ

(M −M∗)− κ

2(M −M∗)2 − λ

6(M −M∗)3

]88

(3.139)

The effective mass M∗ is a function of T , νn, and νp. Thus

∂M∗

∂T


=∂M∗

∂T

∣∣∣∣∣νn,νp

+∂M∗

∂νn

∣∣∣∣∣T,νp

∂νn∂T


+∂M∗

∂νp

∣∣∣∣∣T,νn

∂νp∂T


(3.140)

Similarly, the internal energy density Ui is a function of T , νi, and M∗. Hence

∂Ui∂T


=∂Ui∂T

∣∣∣∣∣νi,M∗

+∂Ui∂νi

∣∣∣∣∣T,M∗

∂νi∂T


+∂Ui∂M∗

∣∣∣∣∣T,νi

∂M∗

∂T


(3.141)

In order to make the connection with the JEL functions we take

Ui = Ui(ψi(T, νi,M∗), t(T,M∗))

where ψi = νi−M∗T

and t = TM∗

. Therefore

∂Ui∂T

∣∣∣∣∣νi,M∗

=∂Ui∂ψi

∣∣∣∣∣t

∂ψi∂T

∣∣∣∣∣νi,M∗

+∂Ui∂t

∣∣∣∣∣ψi

∂t

∂T

∣∣∣∣∣M∗

= −ψiT

∂Ui∂ψi

∣∣∣∣∣t

+1

M∗∂Ui∂t

∣∣∣∣∣ψi

(3.142)

∂Ui∂νi

∣∣∣∣∣T,M∗

=∂Ui∂ψi

∣∣∣∣∣t

∂ψi∂νi

∣∣∣∣∣T,M∗

=1

T

∂Ui∂ψi

∣∣∣∣∣t

(3.143)

∂Ui∂M∗

∣∣∣∣∣νi,T

=∂Ui∂ψi

∣∣∣∣∣t

∂ψi∂M∗

∣∣∣∣∣νi,T

+∂Ui∂t

∣∣∣∣∣ψi

∂t

∂M∗

∣∣∣∣∣T

= − 1

T

∂Ui∂ψi

∣∣∣∣∣t

− T

M∗2∂Ui∂t

∣∣∣∣∣ψi

(3.144)

These express the internal energy derivatives in the JEL language. The cor-

responding calculation of the effective mass derivatives is more laborious. We

begin with the definition of M∗ :

M∗ = M − g2σ

m2σ

[ns −

κ

2(M −M∗)2 − λ

6(M −M∗)3

]

where

ns =∑i=n,p

ni +Ui − 3piM∗ .

89

Next, we take the derivative of M∗ with respect to T at fixed νn and νp :

∂M∗

∂T


= − g2σ

m2σ

∂ns∂T


+∂M∗

∂T


[κ(M −M∗) +

λ

2(M −M∗)2

]Solving for ∂M∗

∂T

∣∣∣νn,νp

, we obtain

∂M∗

∂T


= − 1

cσ

∂ns∂T


(3.145)

cσ ≡m2σ

g2σ

+ κ(M −M∗) +λ

2(M −M∗)2 (3.146)

Similarly,

∂M∗

∂νn

∣∣∣∣∣T,νp

= − 1

cσ

∂ns∂νn

∣∣∣∣∣T,νp

(3.147)

∂M∗

∂νp

∣∣∣∣∣νn,T

= − 1

cσ

∂ns∂νp

∣∣∣∣∣νn,T

(3.148)

Now, consider the scalar density with the dependencies

ns = ns(T, νn, νp,M∗(T, νn, νp)).

This means that

∂ns∂T


=∂ns∂T

∣∣∣∣∣νn,νp,M∗

+∂ns∂M∗

∣∣∣∣∣T,νn,νp

∂M∗

∂T


and therefore

−cσ∂M∗

∂T


=∂ns∂T

∣∣∣∣∣νn,νp,M∗

+∂ns∂M∗


∂M∗

∂T


Solving for ∂M∗

∂T

∣∣∣νn,νp

gives

∂M∗

∂T


= −

cσ +∂ns∂M∗


−1∂ns∂T

∣∣∣∣∣νn,νp,M∗

(3.149)

90

In similar fashion,

∂M∗

∂νn

∣∣∣∣∣T,νp

= −

cσ +∂ns∂M∗


−1∂ns∂νn

∣∣∣∣∣T,νp,M∗

(3.150)

∂M∗

∂νp

∣∣∣∣∣νn,T

= −

cσ +∂ns∂M∗


−1∂ns∂νp

∣∣∣∣∣νn,T,M∗

(3.151)

In the JEL formalism, the scalar density is a function of ψn, ψp, and t. These,

however, carry implicit dependencies on νn, νp, T , and M∗:

ns = ns(ψn(T, νn,M∗), ψp(T, νp,M

∗), t(T,M∗))

Consequently,

∂ns∂M∗


=∂ns∂ψn

∣∣∣∣∣ψp,t

∂ψn∂M∗

∣∣∣∣∣νn,T

+∂ns∂ψp

∣∣∣∣∣ψn,t

∂ψp∂M∗

∣∣∣∣∣νp,T

+∂ns∂t

∣∣∣∣∣ψp,ψn

∂t

∂M∗

∣∣∣∣∣T

= − 1

T

∂ns∂ψn

∣∣∣∣∣ψp,t

− 1

T

∂ns∂ψp

∣∣∣∣∣ψn,t

− T

M∗2∂ns∂t


(3.152)

∂ns∂T

∣∣∣∣∣M∗,νn,νp

=∂ns∂ψn

∣∣∣∣∣ψp,t

∂ψn∂T

∣∣∣∣∣νn,M∗

+∂ns∂ψp

∣∣∣∣∣ψn,t

∂ψp∂T

∣∣∣∣∣νp,M∗

+∂ns∂t


∂t

∂T

∣∣∣∣∣M∗

= −ψnT

∂ns∂ψn

∣∣∣∣∣ψp,t

− ψpT

∂ns∂ψp

∣∣∣∣∣ψn,t

+1

M∗∂ns∂t


(3.153)

∂ns∂νn

∣∣∣∣∣T,M∗,νp

=∂ns∂ψn

∣∣∣∣∣ψp,t

∂ψn∂νn

∣∣∣∣∣M∗,T

=1

T

∂ns∂ψn

∣∣∣∣∣ψp,t

(3.154)

∂ns∂νp

∣∣∣∣∣T,M∗,νn

=∂ns∂ψp

∣∣∣∣∣ψn,t

∂ψp∂νp

∣∣∣∣∣M∗,T

=1

T

∂ns∂ψp

∣∣∣∣∣ψn,t

(3.155)

The last two quantities that need to be calculated are ∂νn∂T

∣∣∣nn,np

and ∂νp∂T

∣∣∣nn,np

.

First we switch to a more convenient notation:

nn → n, np → p

M∗ →M, T → T

νn → ν, νp → π

∂A

∂B

∣∣∣∣∣C,D,...

→ AB|CD...

91

must be accounted for. Explicitly,

∂ni∂M∗

∣∣∣∣∣νi,T

=3niM∗ −

1

T

∂ni∂ψi

∣∣∣∣∣t

+T

M∗2∂ni∂t

∣∣∣∣∣ψi

(3.164)

∂Ui∂M∗

∣∣∣∣∣νi,T

=4UiM∗ −

1

T

∂Ui∂ψi

∣∣∣∣∣t

+T

M∗2∂Ui∂t

∣∣∣∣∣ψi

(3.165)

∂ns∂M∗

∣∣∣∣∣νn,νp,T

=3nsM∗ −

∑i

1

T

∂ns∂ψi

∣∣∣∣∣t

+T

M∗2∂ns∂t

∣∣∣∣∣ψi

(3.166)

This is not the case for derivatives with respect to νi and T because these are

taken at fixed M∗.

With the calculation of the exact CV now concluded, we turn our attention

to the derivation of analytical expressions valid in the degenerate and non-

degenerate regimes.


E

A=E

A(T = 0) +

T 2

n

∑i

aini.

Therefore,

CV =2T

n

∑i

aini =S

A=

2(E/A)thT

. (3.167)


E

A=

1

nδV +M∗ +

1

n

∑i

3Tni

2

[1 +

ni4

(π

M∗T

)3/2

+5T

4M∗

]+n-dependent terms (3.168)

⇒ CV =1

n

d(δV )

dT+dM∗

dT

+1

n

∑i

3ni2

1− ni

8

(π

M∗T

)3/2

+5T

2M∗

− T

M∗dM∗

dT

[3ni8

(π

M∗T

)3/2

− 5T

4M∗

](3.169)

d(δV )

dT=

δV

δM∗dM∗

dT(3.170)

dM∗

dT=

(3T

2M∗− 1

)m2σ

g2σ+ 3Tn

2M∗2+ κ(M −M∗) + λ

2(M −M∗)2

(3.171)

93

where δV and δM∗ are given in equations (3.132) and (3.134).


0

0.5

1

1.5

2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Cv

nB [fm-3]

T = 20 MeVx = 0.5

ExactDegenerate

Non-Degenerate

Figure 3.19: Comparison of the degenerate (red line) and the non-degenerate(blue line) approximations with the exact (black line) specific heat of MFT atT = 20 MeV.

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Cv

nB [fm-3]

T = 20 MeV

T = 60 MeVT = 80 MeV

x = 0.5

T = 40 MeV

SkM*MFT

Figure 3.20: Comparison of the specific heat versus density of the EOS ofMFT (solid) and SkM* (dotted) at T = 20 MeV (black), T = 40 MeV (green),T = 60 MeV (blue), and T = 80 MeV (red).

94

Chapter 4

Finite Range Interactions

In this chapter, we explore the thermal properties of dense matter using the

schematic model constructed by Welke et al. [17] in which the appropriate

momentum dependence that fits optical potential data is built through finite-

range exchange forces of the Yukawa type. We refer to this interaction as

a momentum-dependent Yukawa interaction (MDYI). Here, the model’s finite

temperature properties are studied in the context of infinite, isospin-symmetric

nucleonic matter.

4.1 Introduction

The independent particle model is based on the assumption that the many-

body Hamiltonian can be replaced by one in which all interparticle interactions

are described by an average potential felt by a single particle. In the context

of scattering this is called the optical potential. The concept of the optical

potential is important for our purposes because it provides additional means for

calibrating the EOS. Specifically, it can be fit to heavy ion data on transverse

momentum flow [40, 41] and as such it affords us a handle on the high density

behavior of the EOS.

The optical potentials of APR and MFT can be derived from their single-

particle energy spectra. For APR we have:

(h2k2

i

2m+ Vop

)ψi = εkiψi (4.1)

95

=

(h2k2

i

2m∗i+∂H∂ni

)ψi. (4.2)

Defining βi(n, x) such that

m∗im

=1

1 + βi(n, x)(4.3)

leads to

h2k2i

2m∗i=

h2k2i

2m+h2k2

i

2mβi(n, x)

⇒ Vop =h2k2

i

2mβi(n, x) +

∂H∂ni

. (4.4)

In the case of MFT, (4.1) becomes

(h2k2

i

2M∗ + Vop

)ψi = (E∗ki + Vi(n, x))ψi, (4.5)

where Vi(n, x) =g2ω

m2ω

n+g2ρ

4m2ρ

(ni − nj). (4.6)

But

E∗ki = (k2i +M∗2)1/2

= M∗

1 +k2i

2M∗2 +1

8

(k2i

M∗2

)2

+ . . .

(4.7)

≡ k2i

2M∗ + αi(ki, n, x) (4.8)

⇒ Vop = αi(ki, n, x) + Vi(n, x). (4.9)

One observes that for both models the optical potential increases monoton-

ically with momentum. This is in direct contrast with experimental evidence

on nucleon-nucleus scattering which suggests that the momentum dependence

of the real part of the optical potential is such that it causes it to be at-

tractive for low energies whereas it becomes repulsive and saturates at high

energies [42, 43, 44, 45]. Results of microscopic nuclear matter calculations

are consistent with this picture (cf. Figures 4.1 and 4.2).

96

Figure 4.1: Nuclear optical potential as a function of energy at different den-sities. The blue and green lines correspond to variational calculations withthe UV14+TNI and UV14+UVII interactions respectively. The solid blackline represents a fit to data with an effective mass of m∗ = 0.7m whereas the red dotted and dashed lines are different fits with the same effectivemass of m∗ = 0.65m. Saturation at high energies is evident. Taken from P.Danielewicz, Nucl. Phys. A 673:375-410, 2000.

97

Figure 4.2: Comparison of the single particle potential of MDYI (solid lines)with variational calculations using the UV14+UVII interaction [46] thatdemonstrates reasonable quantitative agreement between the two. Taken fromC. Gale et al. Phys. Rev. C, 41:1545, 1990.

98

4.2 Model Hamiltonian and Calibration

The Hamiltonian density for MDYI is:

HMDY I =h2

2mτ + V (n) = ε (4.10)

V (n) =A

2

n2

no+

B

σ + 1

nσ+1

nσo+C

no

∫ d3k d3k′

(2π)6

nknk′

1 +(~k−~k′

Λ

)2 (4.11)

τ = 4∫ d3k

(2π)3k2nk (4.12)

n = 4∫ d3k

(2π)3nk (4.13)

nk =[1 + exp

(εk − µT

)]−1

. (4.14)

The single-particle energy spectrum εk is given by

εk =h2k2

2m+ U(n, k) (4.15)

U(n, k) = R(n, k) + An

no+B

(n

no

)σ(4.16)

R(n, k) =2C

no

∫ d3k′

(2π)3

nk′

1 +(~k−~k′

Λ

)2 . (4.17)

We calculate the entropy density from

s = 4∫ d3k

(2π)3[nklnnk + (1− nk)ln(1− nk)] (4.18)

and the pressure from the thermodynamic identity

P = −ε+ Ts+ µn. (4.19)

The chemical potential µ is the solution of (4.13) for given n and T .

The constants A, B, C, σ, and Λ and the nuclear matter properties to which

they correspond are summarized in Table 4.1.

99

Values at Saturation Constants

no = 0.16 fm−3 A = −110.44 MeVE/A = −16 MeV B = 140.9 MeVK = 215 MeV C = −64.95 MeVU(no, k = 0) = −75 MeV σ = 1.24

U(no,h2k2

2m= 300 MeV) = 0 Λ = 1.58koF = 415.6 MeV

Table 4.1: The right column lists the parameters in Equations (4.10) and (4.11)as determined by fits to properties of the optical potential and of equilibriumnuclear matter (left column).

These yield an effective mass m∗ = 0.67m at the Fermi surface.


At T = 0, nk = θ(k − kF ). Thus, for isospin symmetric matter

n =2k3

F

3π2⇒ kF =

(3π2n

2

)1/3

(4.20)

τ =2k5

F

5π2=

2

5π2

(3π2n

2

)5/3

. (4.21)

Also,

∫ kF

0

∫ kF

0

d3k d3k′

(2π)6

1

1 +(~k−~k′

Λ

)2

=1

6π4kFΛ2

[3

8− Λ

2kFtan−1

(2kFΛ

)

− Λ2

16k2F

+

(3Λ2

16k2F

+1

64

Λ4

k4F

)ln

(1 +

4k2f

Λ2

)](4.22)

Ro(n, k) =2C

no

∫ kF

0

d3k′

(2π)3

1

1 +(~k−~k′

Λ

)2

=1

4π2

CΛ3

no

k2F + Λ2 − k2

2kΛln

[(k + kF )2 + Λ2

(k − kF )2 + Λ2

]

+2kFΛ− 2

[tan−1

(k + kF

Λ

)− tan−1

(k − kF

Λ

)].(4.23)

100

Using these expressions we get the energy density at zero temperature εo as

a function of the number density n, and from it, the pressure P o and the

chemical potential µo :

P o = µon− εo (4.24)

µo =dεo

dn(4.25)

4.3.1 Results

The zero temperature results for the energy per particle, the pressure, and

the chemical potential of MDYI are displayed below. The equilibrium point of

nuclear matter is reproduced quite well as the minimum of E/A vs. n shows.

The minimum of the pressure and of the chemical potential at about 0.1 fm−3

signifies a spinodal instability which is related to the transition from the pure

nucleonic phase to the mixed phase.

-16

-14

-12

-10

-8

-6

-4

-2

0

2

0 0.05 0.1 0.15 0.2 0.25 0.3

E/A

[M

eV

]

nB [fm-3

]

T=0 MeVx=0.5

Figure 4.3: MDYI energy per particle at T = 0.

101

-1

0

1

2

3

4

5

6

7

8

0 0.05 0.1 0.15 0.2 0.25 0.3

P [M

eV

/fm

3]

nB [fm-3

]

T=0 MeVx=0.5

Figure 4.4: MDYI pressure at T = 0.

-25

-20

-15

-10

-5

0

5

10

15

20

0 0.05 0.1 0.15 0.2 0.25 0.3

µ [M

eV

]

nB [fm-3

]

T=0 MeV

x=0.5

Figure 4.5: MDYI chemical potential at T = 0.

102



At finite temperature, the numerical computation is complicated by the fact

that the calculation of R(n, k) requires knowledge, at all k′, of R(n, k′) which

appears in the energy spectrum in the Fermi-Dirac distribution nk′ . This self-

consistency problem is solved by an iterative scheme:

1. Assume, initially, that R(n, k) is equal to its T = 0 counterpart Ro(n, k).

2. Use Ro(n, k) in equation (4.15) to get an initial guess for the energy

spectrum ε(o).

3. Utilize this energy spectrum in equation (4.13) and solve for µ(o) for the

desired density and temperature.

4. With ε(o) and µ(o) as inputs, get the next approximation for R(n, k),

using equation (4.17).

5. The cycle is repeated until sufficient convergence is achieved.

4.4.2 Degenerate Limit

The quantum regime is handled, per our usual practice, by FLT. For a single

nucleon species (or equivalently for proton fraction x = 1/2), we have:

• thermal energy

Eth = aT 2 (4.26)


Pth =2nT 2

3aQ (4.27)

Q = 1− 3

2

n

m∗dm∗

dn(4.28)

• thermal chemical potential

µth = −aT2

3

(1 + 3

n

m∗dm∗

dn

)(4.29)

103


S = 2aT (4.30)

The effective mass m∗ is given by

m∗ = kF

∂εok∂k

∣∣∣∣∣k=kF

−1

(4.31)

= kF

kFm

+∂Ro

∂k

∣∣∣∣∣kF

−1

(4.32)

= kF

kFm

+CΛ2

2π2no

[1− 1

2

(1 +

Λ2

2k2F

)ln

(1 +

4k2F

Λ2

)]−1

(4.33)

and the level density parameter a by

a =π2

2

m∗

k2F

(4.34)

4.4.3 Non-Degenerate Limit

In the classical regime, the state variables are obtained through a steepest

descent calculation. Here, we begin by replacing the FD distribution with the

Maxwell-Boltzmann distribution

nk = exp[−(εk − µT

)](4.35)

which is equivalent to expanding the FD distribution as a Taylor series in

z 1 and keeping only the lowest order term. This allows us to write the

various thermodynamic integrals in the form

I =∫ ∞

0g(x)e−f(x)dx (4.36)

Under the assumptions that :

1. e−f(x) is sharply peaked about the extremum xo of f(x) (i.e. f ′(xo) = 0)

such that the range of integration can be expanded to x ∈ (−∞,+∞)

without incurring significant error, and

2. g(x) is a relatively flat (slowly varying) function of x,

104

the method of steepest descent can be used to evaluate I. In its simplest form,

I =∫ +∞

−∞g(x)e−f(x)dx =

√2πgoe

−fo√f ′′o

(4.37)

where the subscript o indicates the value of the function at xo. This expression

proved inadequate for our purposes.

However, the accuracy of the approximation can be substantially improved

in the following manner. First, we expand f(x) about xo :

f(x) ' fo +1

2f ′′o (x− xo)2 +

1

6f (3)o (x− xo)3 +

1

24f (4)o (x− xo)4 + . . . (4.38)

By defining

η ≡ f ′′1/2o (x− xo) (4.39)

⇒ x− xo =η

f′′1/2o

, dx =dη

f′′1/2o

(4.40)

(4.38) becomes

f(x) = fo +1

2η2 +

1

6

f (3)o

f′′3/2o

η3 +1

24

f (4)o

f ′′2oη4 + . . . (4.41)

Similarly,

g(x) ≡ go + g′o(x− xo) +1

2g′′o (x− xo)2 + . . .

= go

[1 +

g′o

gof′′1/2o

η +g′′o

2gof ′′oη2 + . . .

](4.42)

Thus,

I =goe−fo

f′′1/2o

∫ +∞

−∞

[1 +

g′o

gof′′1/2o

η +g′′o

2gof ′′oη2

]

× e−η2

2 exp

[−1

6

f (3)o

f′′3/2o

η3 − 1

24

f (4)o

f ′′2oη4

]dη (4.43)

105

Then, we expand exp [. . .] to O(η6) :

exp [. . .] = 1− 1

6

f (3)o

f′′3/2o

η3 − 1

24

f (4)o

f ′′2oη4 +

1

72

f (3)2o

f ′′3oη6 + . . . (4.44)

and insert it back into (4.43) to get:

I =goe−fo

f′′1/2o

∫ +∞

−∞dηe−

η2

2

×[1 +

g′′o2gof ′′o

η2 − 1

6

g′of(3)o

gof ′′2oη4 − 1

24

f (4)o

f ′′2oη4 +

1

72

f (3)2o

f ′′3oη6

],(4.45)

where terms with odd powers of η have been discarded since∫+∞−∞ ηαe−η

2dη = 0

for odd α. Using

∫ +∞

−∞ηαe−

η2

2 dη =Γ[(α + 1)/2]

2α−12

; α = 0, 2, 4, . . . , (4.46)

we arrive at

I =

√2πgoe

−fo√f ′′o

[1 +

5

24

f (3)2o

f ′′3o− 1

8

f (4)o

f ′′2o+

g′′o2gof ′′o

− g′of(3)o

2gof ′′2o

]. (4.47)

The leading term gives equation (4.37) which says that the value of the expo-

nential integral is determined by its maximum. The second and third terms

are corrections due to the exponent f(x), whereas the fourth and the fifth

terms are due to the factor g(x) in the integrand.

With equation (4.47) in place, we are ready to derive analytical expressions

for the state variables in the non-degenerate limit. We begin with R(n, k) for

the calculation of which we take the energy spectrum to be

εk′ =h2k

′2

2m+ A

n

no+B

(n

no

)σ. (4.48)

Neglecting R(n, k′) in εk′ is a valid approximation for finite-range interactions

in dilute matter insofar as the interparticle distance r ∼ n−1/3 is larger than

the range of the interactions (∼ 1Λ

) : r ≥ 0.5 fm.

106

Thus,

R(n, k) =2C

no

∫ d3k′

(2π)3

1

e(εk′−µ)/T + 1

1

1 +(~k−~k′

Λ

)2 (4.49)

' 2C

no

∫ dk′ k′2

(2π)3ze−

k′22mT

∫ 2π

0dφ∫ π

0dθ

sin θ

1 + k2+k′2−2kk′ cos θΛ2

(4.50)

where z = exp

1

T

[µ− A n

no−B

(n

no

)σ](4.51)

The angular part is analytically integrable and leads to

R(n, k) =2C

no

1

8π2

Λ2z

k

∫dk′ k′ ln

[Λ2 + (k + k′)2

Λ2 + (k − k′)2

]e−

k′22mT (4.52)

Then the factor of k′ is moved into the exponential :

R(n, k) =2C

no

1

8π2

Λ2z

k

∫dk′ ln

[Λ2 + (k + k′)2

Λ2 + (k − k′)2

]e−

k′22mT

+lnk′ (4.53)

From this expression we identify f(k′) and g(k′) as

f(k′) =k′2

2mT− lnk′ (4.54)

g(k′) = ln

[Λ2 + (k + k′)2

Λ2 + (k − k′)2

](4.55)

The stationary point is koR = (mT )1/2. Applying equation (4.47), we get

R(n, k) = αβ(k)z(n)ln

[Λ2 + (k +

√mT )2

Λ2 + (k −√mT )2

](4.56)

α ≡ 2C

no

Λ2

8π2

√π

e(mT ) (4.57)

β(k) ≡ 1

k

11

12

+k√mT (Λ2 + k2 +mT )[Λ4 + (k2 −mT )2 + 2Λ2(k2 − 3mT )]

ln[

Λ2+(k+√mT )2

Λ2+(k−√mT )2

][Λ4 + (k2 −mT )2 + 2Λ2(k2 +mT )]2

.(4.58)

107

Equation (4.56) gives the R(n,K) that is to be used as part of the energy

spectrum in the subsequent calculations of the number density n, the kinetic

energy density τ , and the exchange term of the potential Vk :

n =2

π2

∫ ∞0

dk k2 1

z−1ek2

2mT+RT + 1

=2

π2z∫ ∞

0dk k2e−

k2

2mT−RT

=2

π2z∫ ∞

0dk

[Λ2 + (k +

√mT )2

Λ2 + (k −√mT )2

]αβ(k)zT

e−k2

2mT+2lnk. (4.59)

Therefore,

f(k′) =k′2

2mT− 2 lnk′ (4.60)

g(k′) =

[Λ2 + (k +

√mT )2

Λ2 + (k −√mT )2

]αβ(k)zT

(4.61)

kon =√

2mT. (4.62)

For this calculation, the terms of equation (4.47) involving derivatives of g(k)

are neglected as they introduce higher orders of z (which is small in this limit).

The final result is

n =2

π2

23

12

√π

e(mT )3/2zg(kon; z) (4.63)

= c1zc−c3z2 , (4.64)

where c1 ≡2

π2

23

12

√π

e(mT )3/2 (4.65)

c2 =Λ2 +mT (

√2 + 1)2

Λ2 + (√mT − 1)2

(4.66)

c3 =αβ(kon)

T(4.67)

Equation (4.64) can be solved for z in terms of the so-called Lambert W -

function [47]:

z =−W [(−c3

c1lnc2)n]

c3lnc2

(4.68)

108

This determines the chemical potential :

µ = T lnz + An

no+B

(n

no

)σ(4.69)

The calculation of the kinetic energy density proceeds along the same lines:

τ =2

π2

∫ ∞0

dk k4 1

z−1ek2

2mT+RT + 1

=2

π2z∫ ∞

0dk

[Λ2 + (k +

√mT )2

Λ2 + (k −√mT )2

]αβ(k)zT

e−k2

2mT+4lnk (4.70)

Hence,

f(k′) =k′2

2mT− 4 lnk′ (4.71)

g(k′) =

[Λ2 + (k +

√mT )2

Λ2 + (k −√mT )2

]αβ(k)zT

(4.72)

koτ =√

4mT. (4.73)

The end-product is

τ =2

π2

(4

e

)2√π(mT )5/2 47

48zg(koτ ; z). (4.74)

For the exchange potential we have

Vk =C

no

∫ d3k d3k′

(2π)6

nknk′

1 +(~k−~k′

Λ

)2 (4.75)

=C

no

(2

π2

)2 Λ2

4z2∫ ∞

0dk ke−

k2

2mT−RT

×∫ ∞

0dk′ k′e−

k′22mT−R′T ln

[Λ2 + (k + k′)2

Λ2 + (k − k′)2

]. (4.76)

The steepest descent machinery yields for the k′ integral:

f(k′) =k′2

2mT− lnk′ (4.77)

g(k′) = ln

[Λ2 + (k + k′)2

Λ2 + (k − k′)2

] [Λ2 + (k′ +

√mT )2

Λ2 + (k′ −√mT )2

]αβ(k′)zT

109

= ln

[Λ2 + (k + k′)2

Λ2 + (k − k′)2

]× g(k′) (4.78)

k′oV =√mT (4.79)

Thus,

Ik′ =

√π

e(mT )

11

12g(√mT )ln

[Λ2 + (k +

√mT )2

Λ2 + (k −√mT )2

](4.80)

The unprimed integral is similar. So, finally

Vk =C

no

(2

π2

)2 Λ2

4z2[√

π

e(mT )

11

12g(√mT )

]2

ln

[Λ2 + (

√mT +

√mT )2

Λ2 + (√mT −

√mT )2

]

=C

no

(2

π2

)2 Λ2

4z2π

e(mT )2

(11

12

)2 [g(√mT )

]2ln

(Λ2 + 4mT

Λ2

). (4.81)

With complete expressions for n, τ , and Vk the energy per particle is ob-

tained from the Hamiltonian (4.10) as

E

A=Hn

=ε

n(4.82)

The Maxwell-Boltzmann expression for the entropy density is

s = − 2

π2

∫ ∞0

dk k2(nklnnk − nk), (4.83)

where nk = e(µ−εk)/T (⇒ lnnk =µ− εkT

). (4.84)

Thus,

s = − 2

π2

[1

T

∫ ∞0

(µ− εk)k2nkdk −∫ ∞

0k2nkdk

]= − 2

π2

[(µ

T− 1

) ∫ ∞0

k2nkdk −1

T

∫ ∞0

εkk2nkdk

]=

(µ

T− 1

)n+

2

π2

1

T

∫ ∞0

[k2

2m+ U(n, k)

]k2nkdk

=(µ

T− 1

)n+

τ

2mT+n

T

[An

no+B

(n

no

)σ]+

2VkT

=τ

2mT+

2VkT− n lnz + n. (4.85)

110

The pressure is given by

P = −ε+ Ts+ µn. (4.86)

using (4.69), (4.82), and (4.85) for µ, ε, and s respectively.

4.4.4 Results

The extent to which the degenerate and non-degenerate approximations are

able to reproduce the exact state variables are shown in Figures 4.6 through

4.9.

-20

-10

0

10

20

30

40

50

60

0 0.05 0.1 0.15 0.2 0.25 0.3

E/A

[M

eV

]

nB [fm-3

]

T=0 MeV

x=0.5

Non-DegenerateExact

Degenerate

Figure 4.6: Comparison of the degenerate (red line) and the non-degenerate(blue line) approximations with the exact (black line) energy per particle ofMDYI at T = 20 MeV.

111

0

1

2

3

4

5

6

7

8

9

10

0 0.05 0.1 0.15 0.2 0.25 0.3

P [M

eV

/fm

3]

nB [fm-3

]

T=20 MeVx=0.5

Non-DegenerateExact

Degenerate

Figure 4.7: Comparison of the degenerate (red line) and the non-degenerate(blue line) approximations with the exact (black line) pressure of MDYI atT = 20 MeV.

-60

-40

-20

0

20

40

0 0.05 0.1 0.15 0.2 0.25 0.3

µT

he

rma

l [M

eV

]

nB [fm-3

]

T=20 MeV

x=0.5

Non-DegenerateExact

Degenerate

Figure 4.8: Comparison of the degenerate (red line) and the non-degenerate(blue line) approximations with the exact (black line) chemical potential ofMDYI at T = 20 MeV.

112

-4

-2

0

2

4

6

8

10

0 0.05 0.1 0.15 0.2 0.25 0.3

s/A

nB [fm-3

]

T=20 MeV

x=0.5

Non-DegenerateExact

Degenerate

Figure 4.9: Comparison of the degenerate (red line) and the non-degenerate(blue line) approximations with the exact (black line) entropy per particle ofMDYI at T = 20 MeV.

Comparisons of the state variables of MDYI with those of the similarly

calibrated SkM* are presented in Figures 4.10 through 4.13.

10

12

14

16

18

20

22

24

26

28

30

0 0.05 0.1 0.15 0.2 0.25 0.3

E/A

[MeV

]

nB [fm-3]

T = 20 MeVx = 0.5

MDYISkM*

Figure 4.10: Comparison of the thermal energy versus density of the EOS ofMDYI (blue) and SkM* (red) at T = 20 MeV.

113

0

0.5

1

1.5

2

2.5

3

3.5

0 0.05 0.1 0.15 0.2 0.25 0.3

P [M

eV/fm

3 ]

nB [fm-3]

T = 20 MeVx = 0.5

MDYISkM*

Figure 4.11: Comparison of the thermal pressure versus density of the EOS ofMDYI (blue) and SkM* (red) at T = 20 MeV.

-60

-50

-40

-30

-20

-10

0

10

20

30

0 0.05 0.1 0.15 0.2 0.25 0.3

µ [M

eV]

nB [fm-3]

T = 20 MeVx = 0.5

MDYISkM*

Figure 4.12: Comparison of the thermal chemical potential versus density ofthe EOS of MDYI (blue) and SkM* (red) at T = 20 MeV.

114

1

2

3

4

5

6

7

0 0.05 0.1 0.15 0.2 0.25 0.3

s/A

nB [fm-3]

T = 20 MeVx = 0.5

MDYISkM*

Figure 4.13: Comparison of the entropy versus density of the EOS of MDYI(blue) and SkM* (red) at T = 20 MeV.

4.4.5 Specific Heat

For the calculation of the MDYI specific heat we begin by writing the energy

density as

ε =h2

2mτ +

C

noIexchange + n-dependent terms

where

τ = 4∫ d3k

(2π)3k2nk

Iexchange = 16∫ ∫ d3k d3k′

(2π)6

nknk′

1 +(~k−~k′

Λ

)2

Then

CV =1

n

∂ε

∂T

∣∣∣∣∣n

=1

n

(h2

2m

∂τ

∂T

∣∣∣∣∣n

+C

no

∂Iexchange∂T

∣∣∣∣∣n

)(4.87)

Next we define

ek ≡ εk −R (4.88)

ν ≡ µ−R (4.89)

115

so that all of the implicit temperature dependence of nk is absorbed in ν :

nk =[1 + e(

εk−µT )

]−1

nk =[1 + e(

ek−νT )

]−1

(4.90)

Here, εk and R are given by equations (4.15) and (4.17) respectively.

Thus

∂τ

∂T

∣∣∣∣∣n

= 4∫ d3k

(2π)3k2

(∂nk∂T

∣∣∣∣∣ν

+∂nk∂ν

∣∣∣∣∣T

∂ν

∂T

∣∣∣∣∣n

)(4.91)

∂Iexchange∂T

∣∣∣∣∣n

= 16∫ d3k

(2π)3k2

(∂nk∂T

∣∣∣∣∣ν

+∂nk∂ν

∣∣∣∣∣T

∂ν

∂T

∣∣∣∣∣n

)∫ d3k′

(2π)3

nk′

1 +(~k−~k′

Λ

)2

+16∫ d3k′

(2π)3k′2(∂nk′

∂T

∣∣∣∣∣ν

+∂nk′

∂ν

∣∣∣∣∣T

∂ν

∂T

∣∣∣∣∣n

)∫ d3k

(2π)3

nk

1 +(~k−~k′

Λ

)2

= 32∫ d3k

(2π)3k2

(∂nk∂T

∣∣∣∣∣ν

+∂nk∂ν

∣∣∣∣∣T

∂ν

∂T

∣∣∣∣∣n

)∫ d3k′

(2π)3

nk′

1 +(~k−~k′

Λ

)2

(4.92)

In the last line of eq (4.92) we gain a factor of 2 because the two terms appear-

ing after the first equality of∂Iexchange

∂T

∣∣∣n

are symmetric under the exchange of

primed and unprimed quantities.

The condition that n is constant implies

dn

dT= 0 =

∂n

∂T

∣∣∣∣∣ν

+∂n

∂ν

∣∣∣∣∣T

∂ν

∂T

∣∣∣∣∣n

⇒ ∂ν

∂T

∣∣∣∣∣n

=−∫d3k ∂nk

∂T

∣∣∣ν∫

d3k ∂nk∂ν

∣∣∣T

. (4.93)

Furthermore,

∂nk∂T

∣∣∣∣∣ν

=(ek − ν)

T 2exp

(ek − νT

)n2k

=(εk − µ)

T 2(1− nk)nk (4.94)

116

∂nk∂ν

∣∣∣∣∣T

=1

Texp

(ek − νT

)n2k

=1

T(1− nk)nk (4.95)

⇒ ∂ν

∂T

∣∣∣∣∣n

= − 1

T

∫d3k(εk − µ)(1− nk)nk∫

d3k(1− nk)nk(4.96)

This concludes the derivation of the exact MDYI specific heat.


E

A=E

A(T = 0) + aT 2.

Therefore,

CV = 2aT =S

A=

2(E/A)thT

. (4.97)


E

A=

1

n(τ + Vk) + n-dependent terms

⇒ CV =1

n

(∂τ

∂T

∣∣∣∣∣n

+∂Vk∂T

∣∣∣∣∣n

)(4.98)

∂τ

∂T

∣∣∣∣∣n

=∂τ

∂z

∣∣∣∣∣T

∂z

∂T

∣∣∣∣∣n

+∂τ

∂T

∣∣∣∣∣z

(4.99)

∂Vk∂T

∣∣∣∣∣n

=∂Vk∂z

∣∣∣∣∣T

∂z

∂T

∣∣∣∣∣n

+∂Vk∂T

∣∣∣∣∣z

(4.100)

where n, τ , and Vk are given by (4.64), (4.74), and (4.81) respectively.

The derivative of z with regards to T is a consequence of the constancy of n

when the temperature is changed:

∂z

∂T

∣∣∣∣∣n

=

∂n∂z

∣∣∣∣∣c1,c2,c3

−1 ∂n

∂c1

∣∣∣∣∣z,c2,c3

∂c1

∂T+

∂n

∂c2

∣∣∣∣∣z,c1,c3

∂c2

∂T+

∂n

∂c3

∣∣∣∣∣z,c1,c2

∂c3

∂T

(4.101)

The final equations are truly ”wonderful” but the page is too small to contain

them. However, dear reader, you may find them in Appendix C.


117

0

0.5

1

1.5

2

2.5

0 0.1 0.2 0.3 0.4 0.5 0.6

Cv

nB [fm-3]

T = 20 MeVExact

DegenerateNon-Degenerate

Figure 4.14: Comparison of the degenerate (red line) and the non-degenerate(blue line) approximations with the exact (black line) specific heat of MDYIat T = 20 MeV.

0

0.5

1

1.5

2

2.5

0 0.1 0.2 0.3 0.4 0.5 0.6

Cv

nB [fm-3]

T = 20 MeVT = 50 MeVT = 80 MeV

MDYISkM*

Figure 4.15: Comparison of the specific heat versus density of the MDYI EOS(solid) and SkM* (dotted) at T = 20 MeV (black), T = 50 MeV (red), andT = 80 MeV (blue). Notice that the MDYI specific heat exceeds 3/2 forT = 50 MeV and T = 80 MeV and is consistently higher than that of SkM*in a manner that inversely reflects the thermal pressure behavior of the twomodels (Fig 4.9).

118

Chapter 5

Conclusions

Core-collapse supernovae form an immensely complicated problem whose so-

lution requires the synergy of several fields of physics ranging from astronomy

and astrophysics to magnetohydrodynamics, nuclear and neutrino physics.

The overall purpose of this work is to demonstrate how the microscopic strong

force of nucleons affects the macroscopic equation of state. To that end, we

use modern models of the nucleon-nucleon interaction in a many body envi-

ronment and study their thermal effects which are crucial in understanding

core-collapse supernovae.

5.1 Objectives

This work was based on three objectives:

• To perform comparisons between different classes of equations of state

(EOS) in bulk homogeneous matter and establish whether finite-temperature

and finite-asymmetry effects depend on the model building approach.

• To determine the complete EOS in homogeneous matter for the non-

relativistic potential of Akmal, Pandharipande, and Ravenhall and pro-

duce tables of its thermodynamic variables for use in hydrodynamic su-

pernovae simulations.

• To investigate the possibility of using heavy ion data to further restrict

the parameter space for the EOS.

119

• To understand the effects of finite range interactions via the inclusion of

finite-range exchange terms in the Hamiltonian.

5.2 Advances

For the accomplishment of the afforementioned goals several advances were

made:

• We treated the nucleon-nucleon interaction with non-relativistic poten-

tial models as well as with a relativistic mean-field theoretical one. We

employed for the first time the Skyrme-like Hamiltonian density con-

structed by Akmal, Pandharipande, and Ravenhall which takes into ac-

count the long scattering lengths of nucleons that determine the low

density characteristics. As a basis for mean field theory (MFT) we used

a Walecka-like Lagrangian density supplemented by non-linear interac-

tions involving σ, ω, and ρ meson exchanges [3], calibrated so that known

properties of nuclear matter (e.g. equilibrium density, compression mod-

ulus, symmetry energy, etc) are reproduced. A numerical code was con-

structed for densities extending from 10−7 fm−3 to 1 fm−3, temperatures

up to 60 MeV, and proton-to-baryon fraction in the range 0 to 1/2.

• We focused on the bulk homogeneous phase and calculated its thermody-

namic properties (such as pressure, free energy, entropy, chemical poten-

tials, isospin susceptibilities, and effective masses) as functions of baryon

density, temperature, and proton-to-baryon ratio. For these, schemes

valid for all regimes of degeneracy and for general energy density func-

tionals other than Skyrme-like potentials and Walecka-like Lagrangians

have been devised. The exact results were compared to approximate

ones in the degenerate and non-degenerate limits for which analytical

formulae have been derived. Analytical expressions for the dependence

of the incompressibility on isospin asymmetry have also been developed.

• The importance of the correct momentum dependence in the single par-

ticle potential that fits optical potentials of nucleon-nucleus scattering

was highlighted in the context of intermediate energy heavy-ion colli-

sions. We explored the thermal properties of dense matter using the

120

schematic model constructed by Welke et al. in which the appropriate

momentum dependence that fits optical potential data is built through

finite-range exchange forces of the Yukawa type. We have studied the

finite-temperature properties of such a model in the context of infinite,

isospin-symmetric nucleonic matter. The exact numerical results were

compared to analytical ones in the quantum regime where we rely on

Landau’s Fermi-Liquid Theory, and in the classical regime where the

state variables are obtained through a steepest descent calculation.

5.3 Findings

The key results of the work are:

• The main distinguishing features as they result from the comparisons

between the various models are:

1. The density dependencies of the symmetry energies and the effective

masses.

2. The proton fraction dependence of the incompressibility

These heterogeneities are most prominent in the isospin susceptibilities

and in the chemical potentials. They also manifest themselves in the

high density behavior of the thermal pressure. This is a consequence of

the temperature dependence of the energy spectrum of the finite-range

and the meson-exchange models (through the exchange interaction and

the Dirac effective mass respectively), which leads to a higher specific

heat and, correspondingly, to a lower pressure.

• The low density behavior of the state variables, with the exception of

the susceptibilities, is similar for all models.

• The exploitation of heavy ion data in the calibration of models is feasible

and leads to comparable predictions with models that are adjusted to

nuclear data at the saturation density.

121

5.4 Future Work

There are several questions, not addressed here, that we plan to answer in the

near future:

• We will extend our calculations of the state variables to the subnuclear

region where nuclei are present. For the potential model, the equations

of motion arise through a variational procedure in which the Hamilto-

nian density is minimized with respect to baryon and isospin asymmetry

densities under the constraints of baryon number and charge conserva-

tion, respectively. The resulting Hartree-Fock equations will then be

solved both at zero and finite temperatures. The field-theoretical model

will be treated approximately at the Hartree level whereby the nuclei

are considered static and spherically symmetric. As a result of these cal-

culations we will obtain values for the parameters in the liquid droplet

nuclear energy formula (e.g. surface energy, neutron skin thickness, etc)

which are to be compared with experimental data.

• We will attempt to construct a single-particle energy spectrum which

conforms with the appropriate optical potential behavior by turning to

the original microscopic calculations of Akmal and Pandharipande in-

stead of the Hamiltonian density to which these calculations led, and

study the extent to which this new spectrum will affect the thermody-

namics. For the case of relativistic field theories we will try to achieve

the correct behavior by going beyond the mean field (Hartree) level via

the inclusion of exchange (Fock) terms.

• The finite range model MDYI will be extended to two nucleon species

so that its properties at finite isospin asymmetry can be probed.

• The analytical and computational tools developed in the study of MFT

models will be adapted for use in related chiral effective-field theoretical

(χEFT) calculations.

122

Bibliography

[1] C. E. Rolfs and W. S. Rodney. Cauldrons in the Cosmos. Chicago,

Chicago, 1988.

[2] H. A. Bethe, G. E. Brown, J. Applegate, and J. M. Lattimer. Nucl. Phys.

A, 324:487–533, 1979.

[3] J. R. Bergervoet et al. Phys. Rev. C, 41:1435, 1990.

[4] V. G. J. Stoks et al. Phys. Rev. C, 48:792, 1993.

[5] L. D. Landau and E. M. Lifshitz. Fluid Mechanics. Butterworth Heine-

mann, Oxford, second edition, 1987.

[6] F. D. Swesty, J. M. Lattimer, and E. S. Myra. Astrophys. Jl., 425:195–204,

1994.

[7] E. S. Myra et al. Astrophys. Jl., 318:744–759, 1987.

[8] S. Carroll. An Introduction to General Relativity. Addison Wesley, San

Francisco, 2004.

[9] P. B. Demorest et al. Nature, 467:1081, 2010.

[10] J. Hessels et al. Science, 311:1901–1904, 2006.

[11] A. W. Steiner, J. M. Lattimer, and E. F. Brown. Astrophys. Jl., 722:33–

54, 2010.

[12] D. G. Ravenhall, Pethick C. J., and J. R. Wilson. Phys. Rev. Lett.,

50:2066, 1983.

123

[13] A. Akmal, V. R. Pandharipande, and D. G. Ravenhall. Phys. Rev. C,

58:1804, 1998.

[14] H. S. Kohler. Nucl. Phys. A, 258:301–316, 1976.

[15] B. D. Serot and J. D. Walecka. The Relativistic Nuclear Many-Body

Problem. Plenum, New York, 1986.

[16] H. Krivine, J. Treiner, and O. Bohigas. Nucl. Phys. A, 336:155–184, 1980.

[17] G.M. Welke et al. Phys. Rev. C, 38:2101, 1988.

[18] A. Akmal and V. R. Pandharipande. Phys. Rev. C, 56:2261, 1997.

[19] R. B. Wiringa, V. G. J. Stoks, and R. Schiavilla. Phys. Rev. C, 51:38,

1995.

[20] B. S. Pudliner et al. Phys. Rev. Lett., 74:4396, 1995.

[21] J. L. Forest, V. R. Pandharipande, and J. L. Friar. Phys. Rev. C, 52:568,

1995.

[22] H. Friedrich. Theoretical Atomic Physics. Springer, Heidelberg, third

edition, 2006.

[23] D. Vautherin and D. M. Brink. Phys. Rv. C, 5:626, 1972.

[24] S. Sudaz et al. Phys. Lett., 285:183–186, 1992.

[25] J. M. Lattimer and Y. Lim. Constraining the symmetry parameters of

the nuclear interaction. arXiv, 1203:4286v1, 2012.

[26] H. Sagawa et al. Phys. Rev. C, 76:34327–1, 2007.

[27] J. P. Blaizot and P. Haensel. Acta Physica Polonica, B12:1157, 1981.

[28] M. Prakash and K. S. Bedell. Phys. Rev. C, 32:1118, 1985.

[29] P. P. Eggleton, J. Faulkner, and B. P. Flannery. Astron. & Astrophys.,

23:325–330, 1973.

[30] S. M. Johns, P. J. Ellis, and J. M. Lattimer. Astrophys. Jl., 473:1020–

1028, 1996.

124

[31] J. M. Lattimer. private communication.

[32] E. M. Lifshitz and L. P. Pitaevskii. Statistical Physics Part 2. Butterworth

Heinemann, Oxford, 1980.

[33] G. Baym and C. Pethick. Landau Fermi-Liquid Theory. Wiley Inter-

science, New York, 1991.

[34] L. D. Landau and E. M. Lifshitz. Statistical Physics Part 1. Butterworth

Heinemann, Oxford, third edition, 1980.

[35] R. Kippenhahn and A. Weigert. Stellar Structure and Evolution. Springer-

Verlag, Berlin, 1991.

[36] T. L. Ainsworth et al. Nucl. Phys. A, 464:740–768, 1987.

[37] M. Prakash et al. Windsurfing the Fermi Sea Volume II. Elsevier Science,

Amsderdam, 1987.

[38] G. Baym and S. A. Chin. Nucl. Phys. A, 262:527–538, 1976.

[39] S. Chandrasekhar. An Introduction to the Study of Stellar Structure.

Dover, 1967.

[40] C. Gale et al. Phys. Rev. C, 41:1545, 1990.

[41] M. Prakash, T. T. S. Kuo, and S. Das Gupta. Phys. Rev. C, 37:2253,

1988.

[42] S. Hama et al. Phys. Rev. C, 41:2737, 1990.

[43] E.D. Cooper et al. Phys. Rev. C, 47:297, 1993.

[44] P. Danielewicz. Nucl. Phys. A, 673:375–410, 2000.

[45] P. Danielewicz. Acta Physica Polonica, B33:45, 2002.

[46] R. B. Wiringa. Phys. Rev. C, 38:2967, 1988.

[47] D. Veberic. Having fun with lambert w(x) function. arXiv, 1003:1628v1,

2010.

125

Appendix A

APR State Variables

In this appendix we summarize results pertaining to the zero temperature state

variables of APR. Combining these, with the appropriate thermal expressions

from section 2.5.1 yields the corresponding expressions at finite temperature.

In the finite-T case, attention should be paid to the fact that τi is given by

(2.57) and not by (2.23)-(2.24) which are suitable for T = 0.

It is convenient to write HAPR as the sum of a kinetic part Hk, a part con-

sisting of the momentum-dependent interactions Hm, and a density-dependent

interactions part Hd :

HAPR = Hk +Hm +Hd (A.1)

where Hk =h2

2m(τn + τp) (A.2)

Hm = (p3 + (1− x)p5)ne−p4nτn + (p3 + xp5)ne−p4nτp (A.3)

Hd = g1(n)[1− (1− 2x)2)] + g2(n)(1− 2x)2 (A.4)

Furthermore, the following quantities are necessary:

f1L =dg1L

dn− 2g1L

n

= −n2[p2 + 2p6n+ (p11 − 2p2

9p10n− 2p29p11n

2)e−p29n

2]

(A.5)

f1H = f1L − n2 p17 + p21 [p19(−2 + p18p19)

+(2− 2p18p19)n+ p18n2]ep18(n−p19)

(A.6)

126

h1L =df1L

dn− 2f1L

n

= −n2[2p6n− 2p2

9(p10 + 3p11n− 2p29p10n

2 − 2p29p11n

3)e−p29n

2](A.7)

h1H = h1L − n2[2− 4p18p19 + p2

18p219

+(2p18 + 2p2

18 − 2p318p19

)n+ p3

18n2]p21e

p18(n−p19) (A.8)

w1L =dh1L

dn− 2h1L

n

= −n2(−3p11 + 6p2

9p10n+ 12p29p11n

2

−4p49p10n

3 − 4p49p11n

4)

2p29e−p29n

2

(A.9)

w1H = w1L − n2[4 + 2p18(1− 2p18p19 − 2p19 + p18p

219)

+2p18(1 + 2p18 − p218p19)n+ p3

18n2]p18p21e

p18(n−p19) (A.10)

f2L =dg2L

dn− 2g2L

n

= −n2(−p12

n2+ p8 − 2p2

9p13ne−p29n

2)

(A.11)

f2H = f2L − n2 p15 + p14 [p20(−2 + p16p20)

+(2− 2p16p20)n+ p16n2]ep16(n−p20)

(A.12)

h2L =dh2L

dn− 2h2L

n

= −n2[2p12

n3− 2p2

19p13(1− 2p29n)e−p

29n

2]

(A.13)

h2H = h2H − n2[2− 4p16p20 + p2

16p220 + (2p16 + 2p2

16 − 2p316p20)n

+p316n

2]p14e

p16(n−p20) (A.14)

w2L =dw2L

dn− 2w2L

n

= −n2[−6p12

n4+ 4p4

9p13(1 + n− 2p29n

2)e−p29n

2]

(A.15)

w2H = w2L − n2[4 + 2p16(1− 2p16p20 − 2p20 + p16p

220)

+2p16(1 + 2p16 − p216p20)n+ p3

16n2]p16p14e

p16(n−p20) (A.16)

The subscripts L and H imply the low density and the high density phase

respectively.

127

Then the state variables are:

• energy per particle

E

A=

EkA

+EmA

+EdA

(A.17)

EkA

=(3π2)5/3

5π2

h2

2mn2/3[(1− x)5/3 + x5/3] (A.18)

EmA

=(3π2)5/3

5π2

p3[(1− x)5/3 + x5/3] + p5[(1− x)8/3 + x8/3]

n5/3e−p4n

(A.19)

EdA

=1

n

g1[1− (1− 2x)2)] + g2(1− 2x)2

(A.20)

• pressure

P = Pk + Pm + Pd (A.21)

Pk =2

3nEkA

(A.22)

Pm =(

5

3− p4n

)nEmA

(A.23)

Pd = nEdA

+ f1[1− (1− 2x)2] + f2(1− 2x)2

(A.24)

• incompressibility

K = Kk +Km +Kd (A.25)

Kk = 10EkA

(A.26)

Km = (40− 48p4n+ 9p24n

2)EmA

(A.27)

Kd = 18EdA

+ 9

(4f1 + nh1)[1− (1− 2x)2]

+(4f2 + nh2)(1− 2x)2

(A.28)

• second derivative of pressure with respect to density

d2P

dn2=

d2Pkdn2

+d2Pmdn2

+d2Pddn2

(A.29)

d2Pkdn2

=20

27

1

n

EkA

(A.30)

128

d2Pmdn2

=(

200

27− 56

3p4n+ p2

9n2 − p3

4n3)

1

n

EmA

(A.31)

d2Pddn2

=2

n

EdA

+

(10f1

n+ 7h1 + nw1

)[1− (1− 2x)2]

+

(10f2

n+ 7h2 + nw2

)(1− 2x)2 (A.32)

• symmetry energy

S2 = S2k + S2m + S2d (A.33)

S2k =10

9

1

25/3

(3π2)5/3

5π2

h2

2mn2/3 (A.34)

S2m =10

9

1

25/3

(3π2)5/3

5π2

h2

2mn5/3e−p4n(p3 + 2p5) (A.35)

S2d =1

n(−g1 + g2) (A.36)

• first derivative of symmetry energy with respect to density

dS2

dn=

dS2k

dn+dS2m

dn+dS2d

dn(A.37)

dS2k

dn=

2

3

S2k

n(A.38)

dS2m

dn=

S2m

n

(5

3− p4n

)(A.39)

dS2d

dn=

S2d

n+

1

n(−f1 + f2) (A.40)

• second derivative of symmetry energy with respect to density

d2S2

dn2=

d2S2k

dn2+d2S2m

dn2+d2S2d

dn2(A.41)

d2S2k

dn2= −2

9

S2k

n2(A.42)

d2S2m

dn2=

S2m

n2

(10

9− 10

3p4n+ p2

4n2)

(A.43)

d2S2d

dn2=

1

n2(−2f1 + 2f2 − nh1 + nh2) (A.44)

129


µi = µik + µim + µid (A.45)

µik =5

3

(3π2)5/3

5π2

h2

2mn

2/3i (A.46)

µim =(3π2)5/3

5π2e−p4n

p5

[8

3n

5/3i − p4

(n

8/3i + n

8/3j

)]+p3

[8

3n

5/3i +

5

3n

2/3i nj + n

5/3j

−p4

(n

8/3i + n

5/3i nj + nin

5/3j + n

8/3j

)](A.47)

µid =1

n2

[4njg1 + 4ninjf1 + 2(ni − nj)g2 + (ni − nj)2f2

](A.48)

• susceptibilities

χii = χiik + χiim + χiid (A.49)

χiik =2

3

µikni

(A.50)

χiim = −p4µim +(3π2)5/3

5π2e−p4n

p5

[40

9n

2/3i −

8

3p4n

5/3i

]+p3

[40

9n

2/3i +

10

9n−1/3i nj

−p4

(8

3n

5/3i +

5

3n

2/3i nj + n

5/3j

)](A.51)

χiid =1

n2

[8njf1 + 4ninjh1 + 4(ni − nj)f2 + (ni − nj)2h2

](A.52)

χij = χijk + χijm + χijd (A.53)

χijk = 0 (A.54)

χijm = −p4µim +(3π2)5/3

5π2e−p4n

−8

3p4p5n

5/3j

+p3

[5

3n

2/3i +

5

3n

2/3j

−p4

(n

5/3i +

5

3n

2/3i nj +

8

3n

5/3j

)](A.55)

χijd =1

n2

[4g1 + 4nf1 + 4ninjh1 − 2g2 + (ni − nj)2h2

](A.56)

• Landau effective mass

m∗i =[

1

m+

2

h2 (np3 + nip5) e−p4n]−1

(A.57)

130

• derivatives of m∗i with respect to n, x, ni, and nj

dm∗idn

= −m∗i

n

(1− m∗i

m

)(1− np4) (A.58)

dm∗idx

=2

h2p5m∗2i ne

−p4n (A.59)

dm∗idni

= − 2

h2m∗2i [p3(1− np4) + p5(1− nip4)] e−p4n (A.60)

dm∗idnj

= − 2

h2m∗2i [p3(1− np4)− nip4p5)] e−p4n (A.61)

• single-particle energy spectrum

εki = k2i Ti + Vi (A.62)

Ti =∂H∂τi

=h2

2m∗i(A.63)

Vi =∂H∂ni

=∂Hm

∂ni+∂Hd

∂ni(A.64)

∂Hm

∂ni= [p3 + p5 − p4(np3 + nip5)] τi

+ [p3 − p4(np3 + njp5)] τj e−p4n (A.65)

∂Hd

∂ni= 4nj

g1

n2+ 4ninj

f1

n2+ 2(ni − nj)

g2

n2+ (ni − nj)2 f2

n2(A.66)

• derivatives of Vi with respect to ni and nj (for use in the finite-T sus-

ceptibilities)

∂Vim∂ni

=

[p3 + p5 − p4(np3 + nip5)]

(∂τi∂ni− p4τi

)− p4(p3 + p5)τi

[p3 − p4(np3 + njp5)]

(∂τj∂ni− p4τj

)− p4p3τj

e−p4n (A.67)

∂Vid∂ni

= 8njf1

n2+ 4ninj

h1

n2+

2g2

n2

+4(ni − nj)f2

n2+ (ni − nj)2h2

n2(A.68)

131

∂Vim∂nj

=

[p3 + p5 − p4(np3 + nip5)]

(∂τi∂nj− p4τi

)− p4p3τi

[p3 − p4(np3 + njp5)]

(∂τj∂nj− p4τj

)− p4(p3 + p5)τj

e−p4n

(A.69)

∂Vid∂nj

= 8njf1

n2+ 4ninj

h1

n2− 2g2

n2+ 4

g1

n2+ (ni − nj)2h2

n2(A.70)

132

Appendix B

JEL Notes

In this appendix we collect the JEL functions and their derivatives with respect

to the variables f , g, and t that are necessary for the numerical evaluation of

the isospin susceptibilities.

The JEL functions are:

• number density

ni =T 3

π2

1

t3fig

3/2i (1 + gi)

3/2

(1 + fi)M+1/2(1 + gi)N(1 + fi/a)1/2

×M∑m=0

N∑n=0

pmnfmi g

ni

[1 +m+

(1

4+n

2−M

)fi

1 + fi

+(

3

4− N

2

)figi

(1 + fi)(1 + gi)

](B.1)

• internal energy density

Ui = τi −M∗ni

=T 4

π2

1

t4fig

5/2i (1 + gi)

3/2

(1 + fi)M+1(1 + gi)N

M∑m=0

N∑n=0

pmnfmi g

ni

×[

3

2+ n+

(3

2−N

)gi

1 + gi

](B.2)

133

• pressure

pi =T 4

π2

1

t4fig

5/2i (1 + gi)

3/2

(1 + fi)M+1(1 + gi)N

M∑m=0

N∑n=0

pmnfmi g

ni (B.3)

• degeneracy parameter

ψi =νi −M∗

T= 2(1 + fi/a)1/2 ln

[(1 + fi/a)1/2 − 1

(1 + fi/a)1/2 + 1

](B.4)

The variables f , g, and t are connected through

gi =T

M∗ (1 + fi)1/2 = t(1 + fi)

1/2. (B.5)

Furthermore, we define

Fi ≡ Ui − 3pi (B.6)

=T 4

π2

1

t4fig

5/2i (1 + gi)

3/2

(1 + fi)M+1(1 + gi)N

M∑m=0

N∑n=0

pmnfmi g

ni

×[−3

2+ n+

(3

2−N

)gi

1 + gi

](B.7)

such that

nsi =τi − 3piM∗ = ni +

1

TtFi (B.8)

For the susceptibilities we need

∂ni∂t

∣∣∣∣∣ψi

=∂ni∂t

∣∣∣∣∣fi,gi

+∂ni∂gi

∣∣∣∣∣fi,t

∂gi∂t

∣∣∣∣∣fi

(B.9)

∂ni∂ψi

∣∣∣∣∣t

=

∂ni∂fi

∣∣∣∣∣gi,t

+∂ni∂gi

∣∣∣∣∣fi,t

∂gi∂fi

∣∣∣∣∣t

∂fi∂ψi

(B.10)

∂nsi∂t

∣∣∣∣∣ψi

=∂ni∂t

∣∣∣∣∣ψi

+1

T

∂(tFi)

∂t

∣∣∣∣∣ψi

=∂ni∂t

∣∣∣∣∣ψi

+1

T

Fi +∂Fi∂t


+∂Fi∂gi

∣∣∣∣∣fi,t

∂gi∂t

∣∣∣∣∣fi

(B.11)

∂nsi∂ψi

∣∣∣∣∣t

=∂ni∂ψi

∣∣∣∣∣t

+t

T

∂Fi∂fi

∣∣∣∣∣gi,t

+∂Fi∂gi

∣∣∣∣∣fi,t

∂gi∂fi

∣∣∣∣∣t

∂fi∂ψi

(B.12)

134

Thus, the most elementary ingredients are

∂fi∂ψi

=fi

1 + fi/a(B.13)

∂gi∂fi

∣∣∣∣∣t

=t

2(1 + fi)2=

t2

2gi(B.14)

∂gi∂t

∣∣∣∣∣fi

= (1 + fi)1/2 =

git

(B.15)

∂ni∂fi

∣∣∣∣∣gi,t

=T 3

π2

1

t3fig

3/2i (1 + gi)

3/2

(1 + fi)M+1/2(1 + gi)N(1 + fi/a)1/2

M∑m=0

N∑n=0

pmnfmi g

ni [

1 +m+(

1

4+n

2−M

)fi

1 + fi+(

3

4− N

2

)figi

(1 + fi)(1 + gi)

]

×[

1 +m

fi− 1

2a(1 + fi/a)− M + 1/2

1 + fi

]

+1

(1 + fi)2

[1

4+n

2−M +

(3

4− N

2

)gi

1 + gi

](B.16)

∂ni∂gi

∣∣∣∣∣fi,t

=T 3

π2

1

t3fig

3/2i (1 + gi)

3/2

(1 + fi)M+1/2(1 + gi)N(1 + fi/a)1/2×

M∑m=0

N∑n=0

pmnfmi g

ni [

1 +m+(

1

4+n

2−M

)fi

1 + fi+(

3

4− N

2

)figi

(1 + fi)(1 + gi)

]

×[(

3

2+ n

)1

gi+(

3

2−N

)1

1 + gi

]+

fi(1 + fi)(1 + gi)2

(3

4− N

2

)(B.17)

∂ni∂t


= −3nit

(B.18)

∂Fi∂fi

∣∣∣∣∣gi,t

=

[1−Mfifi(1 + fi)

]Fi

+T 4

π2

1

t4fig

5/2i (1 + gi)

3/2

(1 + fi)M+1(1 + gi)N

M∑m=0

N∑n=0

pmnfmi g

ni

m

fi

×[−3

2+ n+

(3

2−N

)gi

1 + gi

](B.19)

(B.20)

135

∂Fi∂gi

∣∣∣∣∣fi,t

=

[5/2 + (4−N)gi

gi(1 + gi)

]Fi +

(3/2−N)

(1 + gi)2pi

+T 4

π2

1

t4fig

5/2i (1 + gi)

3/2

(1 + fi)M+1(1 + gi)N

M∑m=0

N∑n=0

pmnfmi g

ni

n

gi

×[−3

2+ n+

(3

2−N

)gi

1 + gi

](B.21)

∂Fi∂t


=−4Fit

(B.22)

136

Appendix C

MDYI Non-Degenerate CV

This appendix contains the necessary ingredients for the calculation of the

analytical approximation to the MDYI specific heat in the non-degenerate

limit.

C.1 Number Density

The number density is given by:

n = c1zc−c3z2 (C.1)

c1 =16π

h3

23

12

√π

e(mT )3/2 (C.2)

c2 =Λ2 +mT (

√2 + 1)2

Λ2 +mT (√

2− 1)2(C.3)

c3 =α

(2mT )1/2

[11

12+

√2mT (Λ2 + 3mT )(Λ2 −mT )2

(Λ4 + 6Λ2mT +m2T 2)2 lnc2

](C.4)

α =2C

ρo

4

h3

√π

emπΛ2 (C.5)

Its partial derivatives with respect to c1, c2, c3, and z are:

∂n

∂c1

=1

c1

n (C.6)

∂n

∂c2

= −c3z

c2

n (C.7)

∂n

∂c3

= −z(ln c2)n (C.8)

137

∂n

∂z= (

1

z− c3 ln c2)n (C.9)

The derivatives of c1, c2, c3, and z with respect to T are :

∂c1

∂T=

3

2Tc1 (C.10)

∂c2

∂T=

4√

2Λ2m

[Λ2 +mT (√

2− 1)2]2(C.11)

∂c3

∂T= − 1

2Tc3 +

(T

αc3 −

11

12

1√2mT

)1

T+

3m

Λ2 + 3mT− 2m

Λ2 −mT

− 4√

2Λ2m(ln c2)−1

[Λ2 +mT (√

2− 1)2][Λ2 +mT (√

2 + 1)2]− 4m(3Λ2 +mT )

Λ4 + 6Λ2mT +m2T 2

(C.12)

∂z

∂T=

(∂n

∂z

)−1 (∂n

∂c1

∂c1

∂T+∂n

∂c2

∂c2

∂T+∂n

∂c3

∂c3

∂T

)(C.13)

C.2 Kinetic Energy Density

The kinetic energy density is given by:

τ =16π

h3

z

2m

(4

e

)2√π(mT )5/2 47

48c−c2τ z

1τ (C.14)

c1τ =Λ2 + 9mT

Λ2 +mT(C.15)

c2τ =α

(4mT )1/2

[11

12+

2mT (Λ2 + 5mT )(Λ4 + 2Λ2mT + 9m2T 2)

(Λ4 + 10Λ2mT + 9m2T 2)2 lnc1τ

](C.16)

Its partial derivatives with respect to c1τ , c2τ , T , and z are:

∂τ

∂c1τ

= −c2τz

c1τ

τ (C.17)

∂τ

∂c2τ

= −z(ln c1τ )τ (C.18)

∂τ

∂T=

5

2Tτ (C.19)

∂τ

∂z= (

1

z− c2τ ln c1τ )τ (C.20)

138

The derivatives of c1τ and c2τ , with respect to T are :

∂c1τ

∂T=

8Λ2m

(Λ2 +mT )2(C.21)

∂c2τ

∂T= − 1

2Tc2τ

+

(T

αc2τ −

11

12

1√4mT

)1

T+

5m

Λ2 + 5mT+

2m(Λ2 + 9mT )

Λ4 − 2Λ2mT + 9m2T 2

− 8Λ2m(ln c1τ )−1

(Λ2 +mT )(Λ2 + 9mT )− 4m(5Λ2 + 9mT )

Λ4 + 10Λ2mT + 9m2T 2

(C.22)

Finally, the partial derivative of τ with respect to T at fixed z is:

∂τ

∂T

∣∣∣∣∣z

=∂τ

∂T+

∂τ

∂c1τ

∂c1τ

∂T+

∂τ

∂c2τ

∂c2τ

∂T(C.23)

C.3 Exchange Potential

The excahnge potential is given by:

Vp =C

ρo

(16π

h3

)2 Λ2

4z2(π

e

)(11

12

)2

(mT )2lnc1v

(c−c2v z

1v

)2(C.24)

c1v =Λ2 + 4mT

Λ2(C.25)

c2v =α

(mT )1/2

[11

12+mT (Λ2 + 2mT )(Λ4 − 4Λ2mT )

(Λ4 + 4Λ2mT )2 lnc1v

](C.26)

Its partial derivatives with respect to c1v, c2v, T , and z are:

∂Vp∂c1v

= (1

ln c1v

− 2c2vz)Vpc1v

(C.27)

∂Vp∂c2v

= −2z(ln c1v)Vp (C.28)

∂Vp∂T

=2

TVp (C.29)

∂Vp∂z

= (1

z− c2v ln c1v)2Vp (C.30)

The derivatives of c1v and c2v, with respect to T are :

∂c1v

∂T=

4m

Λ2(C.31)

139

∂c2v

∂T= − 1

2Tc2v +

(T

αc2v −

11

12

1√mT

)1

T+

2m

Λ2 + 2mT− 4m

Λ2 − 4mT

− 4m

(ln c1v)(Λ2 + 4mT )− 8m

Λ2 + 4mT

(C.32)

Finally, the partial derivative of Vp with respect to T at fixed z is:

∂Vp∂T

∣∣∣∣∣z

=∂Vp∂T

+∂Vp∂c1v

∂c1v

∂T+∂Vp∂c2v

∂c2v

∂T(C.33)

140

thermal e ects in supernova matter - graduate physics and...

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