Thermal E ects in Supernova Matter - Graduate Physics and...
Transcript of 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
Stony Brook University
The Graduate School
Constantinos Constantinou
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
Constantinos Constantinou
Doctor of Philosophy
in
Physics and Astronomy
Stony Brook University
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.
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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 Zero Temperature . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.4.1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.5 Finite Temperature . . . . . . . . . . . . . . . . . . . . . . . . 67
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 Zero Temperature . . . . . . . . . . . . . . . . . . . . . . . . . 100
4.3.1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
4.4 Finite Temperature . . . . . . . . . . . . . . . . . . . . . . . . 103
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
APR x=0.5APR x=0.3Ska x=0.5Ska x=0.3
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
APR x=0.5APR x=0.3Ska x=0.5Ska x=0.3
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
APR x=0.5APR x=0.3Ska x=0.5Ska x=0.3
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
APR x=0.5APR x=0.3Ska x=0.5Ska x=0.3
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
APR x=0.5APR x=0.3Ska x=0.5Ska x=0.3
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)
• chemical potentials
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
• thermal pressure
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)
• thermal chemical potentials
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].
3.4 Zero Temperature
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
SkMs x=0.5SkMs 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
SkMs x=0.5SkMs 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
SkMs x=0.5SkMs 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
SkMs x=0.5SkMs 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
SkMs x=0.5SkMs 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
3.5 Finite Temperature
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
e(αi−ψi) + 1dαi
=∑i
21/2
π2T 3x1/2
∫ ∞0
(α3/2i + α
1/2i x)(αi
2x+ 1)1/2
e(αi−ψi) + 1dαi
=∑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
e(αi−ψi) + 1dαi
=
∑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
e(E∗ki−νi)/T + 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
e(E∗ki−νi)/T + 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)
3.5.1 Numerical Notes
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
∣∣∣∣∣ψi,ψj
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
∣∣∣∣∣ψi,ψj
−1∂ns∂ψi
∣∣∣∣∣ψj ,t
(3.99)
⇒ ∂t
∂ψi
∣∣∣∣∣ψj
=
Tcσt2− ∂ns
∂t
∣∣∣∣∣ψi,ψj
−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)
• thermal pressure
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
• thermal chemical potentials
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)
• thermal pressure
Pth =∑i
Tni
[1 +
ni4
(π
M∗T
)3/2]− PFi
− δV (3.136)
• thermal chemical potentials
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.
• entropy per particle
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
∣∣∣∣∣nn,np
+∂Up∂T
∣∣∣∣∣nn,np
+∂M∗
∂T
∣∣∣∣∣nn,np
[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
∣∣∣∣∣nn,np
=∂M∗
∂T
∣∣∣∣∣νn,νp
+∂M∗
∂νn
∣∣∣∣∣T,νp
∂νn∂T
∣∣∣∣∣nn,np
+∂M∗
∂νp
∣∣∣∣∣T,νn
∂νp∂T
∣∣∣∣∣nn,np
(3.140)
Similarly, the internal energy density Ui is a function of T , νi, and M∗. Hence
∂Ui∂T
∣∣∣∣∣nn,np
=∂Ui∂T
∣∣∣∣∣νi,M∗
+∂Ui∂νi
∣∣∣∣∣T,M∗
∂νi∂T
∣∣∣∣∣nn,np
+∂Ui∂M∗
∣∣∣∣∣T,νi
∂M∗
∂T
∣∣∣∣∣nn,np
(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
∣∣∣∣∣νn,νp
= − g2σ
m2σ
∂ns∂T
∣∣∣∣∣νn,νp
+∂M∗
∂T
∣∣∣∣∣νn,νp
[κ(M −M∗) +
λ
2(M −M∗)2
]Solving for ∂M∗
∂T
∣∣∣νn,νp
, we obtain
∂M∗
∂T
∣∣∣∣∣νn,νp
= − 1
cσ
∂ns∂T
∣∣∣∣∣νn,νp
(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
∣∣∣∣∣νn,νp
=∂ns∂T
∣∣∣∣∣νn,νp,M∗
+∂ns∂M∗
∣∣∣∣∣T,νn,νp
∂M∗
∂T
∣∣∣∣∣νn,νp
and therefore
−cσ∂M∗
∂T
∣∣∣∣∣νn,νp
=∂ns∂T
∣∣∣∣∣νn,νp,M∗
+∂ns∂M∗
∣∣∣∣∣T,νn,νp
∂M∗
∂T
∣∣∣∣∣νn,νp
Solving for ∂M∗
∂T
∣∣∣νn,νp
gives
∂M∗
∂T
∣∣∣∣∣νn,νp
= −
cσ +∂ns∂M∗
∣∣∣∣∣T,νn,νp
−1∂ns∂T
∣∣∣∣∣νn,νp,M∗
(3.149)
90
In similar fashion,
∂M∗
∂νn
∣∣∣∣∣T,νp
= −
cσ +∂ns∂M∗
∣∣∣∣∣T,νn,νp
−1∂ns∂νn
∣∣∣∣∣T,νp,M∗
(3.150)
∂M∗
∂νp
∣∣∣∣∣νn,T
= −
cσ +∂ns∂M∗
∣∣∣∣∣T,νn,νp
−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∗
∣∣∣∣∣T,νn,νp
=∂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
∣∣∣∣∣ψp,ψn
(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
∣∣∣∣∣ψp,ψn
∂t
∂T
∣∣∣∣∣M∗
= −ψnT
∂ns∂ψn
∣∣∣∣∣ψp,t
− ψpT
∂ns∂ψp
∣∣∣∣∣ψn,t
+1
M∗∂ns∂t
∣∣∣∣∣ψp,ψn
(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
In terms of the above, the neutron and proton number densities are
n = n(T, ν,M(T, ν, π)) (3.156)
p = p(T, π,M(T, ν, π)). (3.157)
The corresponding full differentials are given by
dn = nT |νMdT + nν|TMdν + nM |Tν(MT |νπdT +Mν|Tπdν +Mπ|Tνdπ)
(3.158)
dp = nT |πMdT + nπ|TMdπ + nM |Tπ(MT |νπdT +Mν|Tπdν +Mπ|Tνdπ)
(3.159)
Being that we are interested in the derivatives of the chemical potentials of
the two nucleon species with respect to T while fixing their number densities
we set
dn = 0 (3.160)
dp = 0, (3.161)
and then solve the system for dν (or dπ):
dν = dT(pT |πMnM |TνMπ|Tν − nT |νMpπ|TM − nT |νMpM |TπMπ|Tν − nM |TνMT |νπpπ|TM )
(nν|TMpπ|TM + nν|TMpM |TπMπ|Tν + nM |Tνpπ|TMMν|Tπ)
Thus,
νT |np =(pT |πMnM |TνMπ|Tν − nT |νMpπ|TM − nT |νMpM |TπMπ|Tν − nM |TνMT |νπpπ|TM )
(nν|TMpπ|TM + nν|TMpM |TπMπ|Tν + nM |Tνpπ|TMMν|Tπ)(3.162)
and
πT |np =(nT |νMpM |TπMν|Tπ − pT |πMnν|TM − pT |πMnM |TνMν|Tπ − pM |TπMT |νπnν|TM )
(nν|TMpπ|TM + nν|TMpM |TπMπ|Tν + nM |Tνpπ|TMMν|Tπ)(3.163)
A word of caution is in order here regarding the derivatives of ni, Ui, and
ns with respect to M∗ in the JEL framework: The prefactors of M∗3
π2 and M∗4
π2
92
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.
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
. (3.167)
In the non-degenerate limit,
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).
The results are shown below.
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.
4.3 Zero Temperature
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
4.4 Finite Temperature
4.4.1 Numerical Notes
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)
• thermal pressure
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
• entropy per particle
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.
In the degenerate limit,
E
A=E
A(T = 0) + aT 2.
Therefore,
CV = 2aT =S
A=
2(E/A)thT
. (4.97)
In the non-degenerate limit,
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.
The results are shown below.
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
• chemical potentials
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∂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
∣∣∣∣∣fi,gi
= −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
∣∣∣∣∣fi,gi
=−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