The Physics of Compact Stars - LIPIA Dissertation submitted to the Department of Physics in partial...
Transcript of The Physics of Compact Stars - LIPIA Dissertation submitted to the Department of Physics in partial...
THE FLORIDA STATE UNIVERSITY
COLLEGE OF ARTS AND SCIENCES
PHYSICS OF COMPACT STARS
By
JUTRI TARUNA
A Dissertation submitted to theDepartment of Physics
in partial fulfillment of therequirements for the degree of
Doctor of Philosophy
Degree Awarded:Spring Semester, 2008
The members of the Committee approve the Dissertation of Jutri Taruna defended on
March 7, 2008.
Jorge PiekarewiczProfessor Directing Dissertation
Ettore AldrovandiOutside Committee Member
Simon CapstickCommittee Member
Paul EugenioCommittee Member
Laura ReinaCommittee Member
Approved:
Mark Riley, ChairDepartment of Department of Physics
Joseph Travis, Dean, College of Arts and Sciences
The Office of Graduate Studies has verified and approved the above named committee members.
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This thesis is dedicated to my parents: Ch. Taruna and Lisna Taruna.May their souls rest in peace in heaven.
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ACKNOWLEDGEMENTS
I would like to acknowledge everyone who has supported me throughout my studies.
My special thanks goes to my advisor Jorge Piekarewicz for being a patient advisor, whose
guidance and kindness meant so much for me. I would also like to thank Simon Capstick for
the support and advice he has given me all these years at FSU.
To Paul Eugenio, Laura Reina, and Ettore Aldrovandi for the time they spent as my
graduate committee. To the faculty members of the nuclear theory group at FSU for being
very supportive and helpful whenever I need advice and suggestions.
To my best friends Alvin Kiswandhi and Suharyo Sumowidagdo for being my personal
diary in the ups and downs of my graduate life. Special thanks to Haryo who despite being
far away in Fermilab, has managed to keep his presence close and be my living physics
dictionary. I couldn’t have done it without you.
To my friends at nuclear theory group Tony Sumaryada, Naureen Ahsan and Olga
Abramkina for the good time we share that has brightened my days at FSU. Thanks to
all my friends Harianto Tjong, Paula Sahanggamu, and others who have enlightened my life
in Tallahassee and has made it less lonely. I will miss you all.
Last but not least, I owe much gratitude to my sisters whose love and support made it
possible for me to achieve my dreams.
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TABLE OF CONTENTS
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x
1. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Stellar evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Neutron stars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Supernova neutrinos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.4 This study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2. PEDAGOGICAL INTRODUCTION TO PHYSICS OF COMPACT STARS . 112.1 Hydrostatic Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.2 Degenerate Free Fermi Gas . . . . . . . . . . . . . . . . . . . . . . . . . 132.3 White Dwarf Stars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.4 Neutron Stars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3. VIRTUES AND FLAWS OF THE PAULI POTENTIAL . . . . . . . . . . . . 283.1 Free Fermi Gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.2 Pauli Potential: A New Functional Form . . . . . . . . . . . . . . . . . . 323.3 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.4 Comparison to other approaches . . . . . . . . . . . . . . . . . . . . . . . 383.5 Finite-Size Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4. EQUATION OF STATE FOR NUCLEAR PASTA . . . . . . . . . . . . . . . 454.1 Modeling the Nuclear Pasta . . . . . . . . . . . . . . . . . . . . . . . . . 464.2 Nuclear Matter Equation of State . . . . . . . . . . . . . . . . . . . . . . 484.3 Semi Empirical Mass Formula . . . . . . . . . . . . . . . . . . . . . . . . 504.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
5. CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 635.1 Introduction to Physics of Compact Stars . . . . . . . . . . . . . . . . . 635.2 Virtues and Flaws of the Pauli Potential . . . . . . . . . . . . . . . . . . 645.3 Equation of State for Nuclear Pasta . . . . . . . . . . . . . . . . . . . . . 65
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A. Monte Carlo Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66A.1 The estimator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66A.2 The Metropolis Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 67A.3 Equilibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
B. Woods-Saxon Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
BIOGRAPHICAL SKETCH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
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LIST OF TABLES
1.1 Parameters for the sun and typical white dwarf, neutron star and black holeparameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
3.1 Strength (in MeV) and range parameters (dimensionless) for the variouscomponents of the Pauli potential. . . . . . . . . . . . . . . . . . . . . . . . 36
4.1 Models parameters for the spin-independent term. . . . . . . . . . . . . . . . 53
4.2 Binding energy per nucleon, charge radii and predictions for neutron skin . . 56
4.3 Semi empirical mass formula best fit results for both parameter sets. . . . . . 59
B.1 Table of nuclei and their Fermi momentum. . . . . . . . . . . . . . . . . . . 71
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LIST OF FIGURES
1.1 A schematic cross section of a neutron star. . . . . . . . . . . . . . . . . . . 4
1.2 The neutron EoS for 18 Skyrme parameter sets. . . . . . . . . . . . . . . . . 5
1.3 Theoretical predictions of the mass (in units of solar mass) versus radius ofneutron stars for several EoSs. . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.4 The Nuclear Landscape. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.1 A star in hydrostatic equilibrium. . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2 The interplay between various physical effects on the “toy model” problem ofa M=1M star. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.3 Mass-versus-radius relation for white dwarf stars with a degenerate electrongas EoS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.4 Mass-radius relations for neutron stars with a degenerate neutron gas EoS . 27
3.1 Average kinetic energy of a system of N = 1000 identical fermions at atemperature of τ=T/TF =0.05. . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.2 Momentum distribution of a system of N = 1000 identical fermions at atemperature of τ = T/TF = 0.05 for a variety of densities (expressed in unitsof ρ0 =0.037 fm−3). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.3 Two-body correlation function for a system of N=1000 identical fermions ata temperature of τ=T/TF =0.05 for a variety of densities (expressed in unitsof ρ0 =0.037 fm−3). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.4 Comparison between the Pauli potential introduced in this work Eq. (3.18)and earlier approaches based on Eq. (3.16). . . . . . . . . . . . . . . . . . . . 40
3.5 “Kinematical” velocity distribution of a system of N=1000 identical fermionsat a temperature of τ = T/TF = 0.05 for a variety of densities (expressed inunits of ρ0 =0.037 fm−3). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
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3.6 Finite-size effects on the canonical momentum distribution (left-hand panel)and the two-body correlation function (right-hand panel) . . . . . . . . . . . 43
4.1 Binding energy predicted by the Semi Empirical Mass formula for various nuclei. 51
4.2 Two-body correlation function g(r) as a function rpF . . . . . . . . . . . . . . 53
4.3 Energy per particle of symmetric nuclear matter as a function of densities ρ/ρ0. 54
4.4 Energy per neutron of pure neutron matter as a function of densities ρ/ρ0. . 55
4.5 Binding energy per particle versus number of particles A for various nuclei. . 56
4.6 Neutron skin versus total number of particles for various nuclei . . . . . . . . 57
4.7 Charge radii versus total number of particles . . . . . . . . . . . . . . . . . . 58
4.8 Monte Carlo snapshots of a configuration of N= 800 neutrons and Z= 200protons at density 0.025 fm−3 (left) and 0.01 fm−3 (right). . . . . . . . . . . 60
4.9 Neutron-neutron two-body correlation function . . . . . . . . . . . . . . . . . 61
4.10 Proton-proton two-body correlation function . . . . . . . . . . . . . . . . . . 61
4.11 Proton-neutron two-body correlation function . . . . . . . . . . . . . . . . . 62
A.1 Monte Carlo thermalization step vs energy per nucleon . . . . . . . . . . . . 68
B.1 Typical shapes for Woods-Saxon potential. . . . . . . . . . . . . . . . . . . . 70
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ABSTRACT
This thesis starts with a pedagogical introduction to the study of white dwarfs and
neutron stars. We will present a step-by-step study of compact stars in hydrostatic
equilibrium leading to the equations of stellar structure. Through the use of a simple finite-
difference algorithm, solutions to the equations for stellar structure both for white dwarfs
and neutron stars are presented. While doing so, we will also introduce the physics of the
equation of state and insights on dealing with units and rescaling the equations.
The next project consists of the development of a “semi-classical” model to describe
the equation of state of neutron-rich matter in the “Coulomb frustrated” phase known
as nuclear pasta. In recent simulations we have resorted to a classical model that, while
simple, captures the essential physics of the nuclear pasta, which consists of the interplay
between long range Coulomb repulsion and short range nuclear attraction. However, for
the nuclear pasta the de Broglie wavelength is comparable to the average inter-particle
separation. Therefore, fermionic correlations are expected to become important. In an effort
to address this challenge, a fictitious “Pauli potential” is introduced to mimic the fermionic
correlations. In this thesis we will examine two issues. First, we will address some of the
inherent difficulties in a widely used version of the Pauli potential. Second, we will refine
the potential in a manner consistent with the most basic properties of a degenerate free
Fermi gas, such as its momentum distribution and its two-body correlation function. With
the newly refined potential, we study various physical observables, such as the two-body
correlation function via Metropolis Monte-Carlo simulations.
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CHAPTER 1
INTRODUCTION
Compact objects –white dwarf stars, neutron stars, and black holes– are born when normal
stars “die”, that is when the stars can no longer generate thermonuclear fusion to counter
act gravity [1, 2]. At this point the stars will need to generate another internal mechanism to
prevent themselves from collapsing. This will be the focus of my dissertation. We will first
examine the simplest mechanism, in which there are no interactions between constituents of
the stars other than through the Pauli exclusion principle. Later in the dissertation we will
discuss a more realistic model incorporating all the essential interactions.
1.1 Stellar evolution
Thermonuclear fusion drives stars through many stages of combustion; the hot center of the
stars allows hydrogen to fuse into helium. Once the core has burned all available hydrogen
it will contract until another source of support becomes available. As the core contracts and
heats, transforming gravitational energy into kinetic (or thermal) energy, the burning of the
helium ashes begins. For stars to burn heavier elements, higher temperatures are necessary
to overcome the increasing Coulomb repulsion and allow fusion through quantum-mechanical
tunneling. Thermonuclear burning continues until the formation of an iron core. Once iron
–the most stable of nuclei– is reached, fusion becomes an endothermic process. As a result,
no further energy can be produced by nuclear fusion. When thermonuclear fusion can no
longer support the stars against gravitational collapse, either because they are not massive
enough or because they have developed an iron core, the stars die and compact objects are
ultimately formed.
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Table 1.1: Parameters for the sun and typical white dwarf, neutron star and black holeparameters.
Massa Radiusb Mean Density Surface PotentialObject (M) (R) (g cm−3) (GM/Rc2)
Sun M R 1 10−6
White dwarf ≤ M ∼ 10−2 R ≤ 106 ∼10−4
Neutron star ∼1–3 M ∼ 10−5 R ≤ 1015 ∼10−1
a M = 1.989 x 1033 gb R = 6.9599 x 1010 cm
The primary factor in determining what type of compact objects the stars will form, is
their initial mass. White dwarfs are believed to originate from light stars with masses M ≤4 solar mass M. A good example of this type of star is our Sun. The Sun is not massive
enough to produce fusion of heavier elements. Once all of the hydrogen and helium in the
core has been burned, it will die as a white dwarf star [3]. Towards the final stages of burning,
the star will expand and expel most of the outer matter to create a planetary nebula. At the
beginning, the core contracts and heats up through conversion of gravitational energy into
thermal kinetic energy. However, at some point the Fermi pressure of the degenerate electrons
begins to dominate, the contraction is slowed down, and the core becomes a compact object
known as a white dwarf, cooling steadily towards the ultimate cold, dark, static black dwarf
star.
Neutron stars, on the other hand, result from one of the most cataclysmic events in the
universe, the death of a star with an initial mass much bigger than the mass of our Sun.
During the collapse of the core, a supernova shock develops, ejecting most of the mass of
the star into the interstellar space and leaving behind an extremely dense core –the neutron
star. As the star collapses, it becomes energetically favorable for electrons to be captured by
protons, making neutrons and neutrinos. The neutrinos carry away 99% of the gravitational
binding energy of the compact object, leaving neutrons behind to support the star against
further collapse [4]. The pressure provided by the degenerate neutrons, like degenerate
electron pressure for white dwarf stars, has a limit on the mass it can bear. Beyond this
limiting mass, there is no source of pressure that can prevent gravitational contraction. If
such is the case, then the star will continue to collapse into a black hole.
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Relative to normal stars of comparable mass, compact stars have much smaller radii and
therefore, much stronger surface gravitational field as shown in Table 1.1 [5]. Hence, to
understand the physics of compact objects other than white dwarfs we need to go beyond
Newtonian mechanics and take into account the effect of general relativity [6, 7].
1.2 Neutron stars
Shortly after the discovery of the neutron in 1932 [8], Baade and Zwicky proposed the
idea of a neutron star; a gravitationally bound, dense compact object, as the remnant of a
supernova explosion [9]. In 1939 Oppenheimer and Volkoff [10], assuming that the star is
a gravitationally bound object supported by neutron degeneracy pressure, made their first
theoretical calculation of the properties of the star. They obtained a maximum neutron-star
mass of around 0.7 M, radii of about 10 km and a central density ρc of about 5x1015 gr/cm3
≈ 3 fm−3 which is about 20 times that of normal nuclei ρ0 '0.15 fm−3. Thus, the predicted
star was one of the densest form of matter in the universe. However, the star remained but
a theoretical conjecture for another twenty eight years. In 1967, a major breakthrough came
when Jocelyn Bell and her advisor Antony Hewish [11] discovered the first pulsar–a rapidly
rotating neutron star. Since then, observational discoveries and theoretical predictions have
made these objects powerful tools in understanding matter on Heaven and Earth.
Fig. 1.1 shows a schematic cross section of a typical neutron star [12]. In the outer
part of the star, due to Coulomb repulsion, nuclei form a body centered cubic (bcc) lattice
immersed in roughly uniform sea of electrons to make it charge neutral. This region is
referred to as the outer crust. As the density increases, weak interactions come into play
making the nuclei neutron rich via electron capture. At density around 4 x 1011 gr/cm3 nuclei
become so neutron rich that the last occupied neutron levels are no longer bound; neutrons
begin to drip out of nuclei. This region is known as the inner crust of the star. Below this
region, at the core of the star with density around 1014 gr/cm3, nuclei disappear and the
system dissolves into uniform neutron-rich matter. In the region between the inner crust and
outer core, at densities around 1013 – 1014 gr/cm3 , competition between nuclear attraction
and Coulomb repulsion leads to a very complex ground state that involve various shapes.
This region is referred to as the “nuclear pasta” phase. In this sub-nuclear density region,
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Figure 1.1: A schematic cross section of a neutron star. The possible hadronic componentsof the matter are listed, and the horizontal scale gives estimates of the radial dimension (notdrawn to scale)
matter can be described very well in terms of interacting protons, neutrons and electrons.
As we go deeper towards the center of the star and thus increasing density and pressure, the
strong interactions play an increasingly important role. Hence, we might have to include the
contributions from heavier baryons or even their quark and gluon constituents.
The extreme density range in a neutron star poses challenging problems, since all four
known fundamental interactions (strong and weak nuclear forces, electromagnetic, and
gravity) play significant roles. Hence a neutron star serves as a melting pot for nuclear,
particle, and astrophysics studies [13, 14, 15, 16]. One of the underlying problems in studying
neutron stars is our limited understanding of the equation of state (EoS) for nuclear matter.
So far, the available experimental data can only probe stable nucleonic matter around nuclear
matter saturation density. As a result, the EoS of nuclear matter at extreme densities (both
at sub-saturation and high densities) is not very well constrained, as shown in Fig. 1.2 [17].
This also implies that the symmetry energy, which is the difference in energy between pure
neutron matter and symmetric nuclear matter, is also largely uncertain. A consequence of
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Figure 1.2: The neutron EoS for 18 Skyrme parameter sets. The filled circles are theFriedman-Pandharipande (FP) variational calculation and the crosses are SkX. The neutrondensity is in units of neutron/fm3.
this uncertainty is that different models predict different masses and radii of neutron stars,
as shown in Fig. 1.3 [18]. It has been known that stars calculated with a stiffer symmetry
energy EoS will yield greater maximum masses and larger radii than stars derived from a
soft equation of state [19]. Thus constraining the EoS of neutron rich matter is essential and
remains a fundamental problem in nuclear physics and astrophysics [20, 21, 22, 23, 24].
A potential constraint on the EoS will come from the Lead Radius Experiment (“PREX”)
at the Jefferson Laboratory. This experiment will use the parity violating weak neutral
interaction to measure the RMS neutron radius Rn of 208Pb to 1% accuracy. Thus it will
provide the first accurate measurement of Rn, and hence of the neutron skin, which is defined
as the difference between the root-mean-square radii of the neutron relative to that of the
proton [25]. A large neutron skin implies a greater neutron matter pressure (∂E/∂ρ). The
same phenomenon also applies to neutron stars as both –the radii of neutron rich nuclei and
the radii of neutron stars– are governed by the same equation of state. This strong correlation
is understandable since it is the same neutron pressure that support a neutron star against
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Figure 1.3: Theoretical predictions on mass (in units of solar mass) versus radius of neutronstars for several EoSs. The left hand side is for stars containing nucleons and, in some cases,hyperons. The right hand side is for stars containing more exotic components, such as mixedphases with kaon condensates of strange quarks, or pure strange quark matter stars.
gravitational collapse. Hence, models that predict thicker neutron skins will most likely
predict neutron stars with larger radii [26]. Similar events are happening at other facilities
as well. The Relativistic Heavy Ion Collider (RHIC) at Brookhaven National Laboratory and
the Facility for Rare Isotope Beams (FRIB) will also probe large neutron-proton asymmetries,
thus providing constraints on the equation of state of asymmetric matter.
On the other hand, the operation of more powerful telescopes has also started to turn
neutron stars from just theoretical curiosities into powerful observational tools [27, 28]. Since
neutron stars contain highly asymmetric matter (N Z) these observations will also bring
a better understanding of nuclear matter beyond the valley of stability. A chart of nuclei
is displayed in Fig. 1.4. The black squares represent stable nuclei and nuclei with half-
lives comparable to or longer than the age of the Earth. Stable light nuclei lie close to the
N=Z line, but heavier nuclei require an excess number of neutrons to be stable. The stable
nuclei are surrounded by the yellow area representing the present limits of unstable nuclei for
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Figure 1.4: The Nuclear Landscape. The black squares represent stable nuclei and the yellowsquares indicate unstable nuclei that have been studied in the laboratory. The unstable nucleiyet to be explored are indicated in green (Terra incognita). The red lines show the magicnumbers.
which there is experimental information. The green area comprises the unknown, proton and
neutron rich regions. A better understanding of neutron stars will be beneficial in unveiling
the neutron rich terra incognita.
1.3 Supernova neutrinos
Neutrinos play an important role in supernova explosions due to their weak interaction
with matter. As explained in section 1.1, during the collapse of the star, electrons become
energetic enough to react with protons through inverse beta decay (electron capture) making
neutrons and neutrinos. The neutrinos (νe) escape and carry away most of the energy, while
the newly formed neutrons then undergo beta decay emitting away electron anti neutrinos
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(νe), i.e.
p+ e− → n+ νe (electron capture) , (1.1a)
n→ p+ e− + νe (beta decay) . (1.1b)
The above processes of absorption and reemission of free electrons are responsible for a
major energy loss in the star due to the emission of neutrinos and anti neutrinos [29]. The
pairs of reactions in Eq. (1.1a) and Eq. (1.1b) were proposed by George Gamow and Mario
Schoenberg in 1941 as the mechanism for stellar collapse and supernova explosions later
known as the direct Urca-process. As Gamow recounted “We called it the Urca process,
partially to commemorate the casino in which we first met, and partially because the Urca
process results in a rapid disappearance of thermal energy from the interior of a star, similar
to the rapid disappearance of money from the pocket of the gamblers on the Casino da Urca”
[30].
At this stage the star is essentially transparent to the neutrinos, and thus neutrinos can
escape effortlessly through the body of the star. However, this process does not last very
long, because the stellar core very rapidly becomes opaque to neutrinos. The transparency
of matter to neutrinos depends on the ratio of neutrino mean free path λ and the radius
R of the system. If λ R then matter is transparent, otherwise matter is opaque. As a
result of the opacity, neutrinos are no longer freely streaming out of the core. Instead, they
will start to interact with matter. Because neutrinos are the main transporters of energy in
stellar collapse, our understanding of neutrinos and their interaction with matter is essential
in determining the mechanism of supernova explosion and hence the type of compact objects
formed as the remnant of the collapse.
Up until 1973, neutrinos had only been observed to participate in charged-current weak
interactions via the exchange of W+ or W− boson. However, in the summer of 1973, the
predicted neutral-current interaction was experimentally confirmed. This new type of weak
interaction via the exchange of a neutral boson (Z0) opened a new chapter in the study of
neutrinos. Neutrinos could merely scatter from nucleons or electrons without changing the
charge of the particles. In 1975, Tubbs and Schramm found that neutral-current neutrino
scattering might boost the neutrino scattering cross section off nucleons through coherent
neutrino scattering [31].
8
Such is the case when the density of the core reaches ' 1012 gr/cm3, where the neutrino
nucleus elastic scattering is thought to temporarily trap the neutrinos. This trapping is
important for the electron-per-baryon fraction Ye of the supernova core. It hinders further
conversion of electrons to neutrinos and thus provides the lepton degeneracy pressure that
helps support the star from collapsing.
1.4 This study
In this study I focus on two main ideas. The first is a pedagogical introduction to the physics
of compact stars focusing on white dwarfs and neutron stars in hydrostatic equilibrium. As
will be shown later, the only missing ingredient in this formalism is the equation of state. I
will start with the simplest case using the free Fermi gas EoS. This formalism is based on
the assumption that there is no interaction between particles, and thus the pressure is solely
generated by the Pauli exclusion principle. Using the free Fermi gas EoS, I will study the
relations between mass and radius of white dwarfs and neutron stars.
While the free Fermi gas EoS works very well for white dwarfs, the same conclusion can
not be drawn for neutron stars. Therefore, in the second part of this dissertation I will
focus on constructing a more realistic model of the equation of state by incorporating all the
essential interactions. This work is an extension of previous studies [32, 33], where I will add
a spin-dependent term to the nuclear pasta Hamiltonian. The spin-dependent contribution
is introduced via a semi-classical “Pauli potential” that mimics fermionic correlations of a
quantum free Fermi gas. The reason for this modification is to generalize our model such
that it can include both the vector and axial-vector contribution in studying neutrino–pasta
scattering. In the process I need to re-adjust and optimize variable parameters based on the
constraints from free Fermi gas and nuclear matter properties.
The manuscript is organized as follows. In Ch. 2 I start with a pedagogical introduction
to the physics of white dwarfs and neutron stars. The discussion is based on the assumption
of a static and spherically symmetric star in hydrostatic equilibrium. I will present results
of the mass-radius relationships for white dwarfs and neutron stars using the free Fermi gas
equation of state. In Ch. 3 I discuss the formulation of semi-classical “Pauli potential”.
I will present the existing Pauli potential models and the problems encountered in these
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formalisms. Next, I introduce a modified model to overcome these problems. These two
chapters are essentially excerpts from papers I wrote with Piekarewicz and collaborators
[34, 35].
In Ch. 4 I introduce a “semi-classical” model for nuclear pasta. Nuclear pasta is a
frustated system, which is characterized by the inability to minimize all interactions (short-
range nuclear interactions and long-range Coulomb repulsions) simultaneously. I will present
the full Hamiltonian for the nuclear pasta and will discuss the optimization of parameters sets
based on the free Fermi gas two-body correlation function, and constraints from symmetric
nuclear matter and finite nuclei properties. Using this refined Hamiltonian, I will also discuss
the nuclear pasta two body correlation function via Monte Carlo simulations. Finally a
summary and conclusion are presented in Ch. 5. A brief discussion of the Monte Carlo
algorithm used in this work is included in Appendix A. Lastly, detailed calculations of the
Fermi momentum for finite nuclei are given in Appendix B.
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CHAPTER 2
PEDAGOGICAL INTRODUCTION TO PHYSICS
OF COMPACT STARS
2.1 Hydrostatic Equilibrium
In this section we will focus our attention on a static and spherically symmetric star, the so
called Schwarzschild star. We start by considering a small volume element located between
radii r and r + ∆r, of cross-sectional area ∆A, and volume ∆r ∆A (see Fig. 2.1). The
gravitational force (Fg) acting on the small volume element is
r∆
∆Α
r
∆P(r+
r)
P(r)F
Figure 2.1: A star in hydrostatic equilibrium. The radial force (both internal pressure andgravity) acting on a small mass element a distance r from the center of the star.
11
Fg = −GM(r)∆m
r2, (2.1)
where G represents the gravitational constant, ∆m is the mass of the small volume element,
and M(r) is the mass enclosed by a spherical shell of radius r which can be expressed in
differential form as
dM(r) = 4πr2ρ(r)dr , (2.2)
with ρ(r) denotes the mass density of the star. If the internal pressure (P ) acting on the
outer surface of the volume element is not equal to the pressure acting on the inner surface,
then the total radial force acting on the volume element is equal to
Fr = −GM(r)∆m
r2− P (r + ∆r)∆A+ P (r)∆A = ∆m
d2r
dt2, (2.3)
with the minus sign referring to the vector in −r direction. Expanding the above equation
to lowest order in ∆r we get
− GM(r)ρ(r)
r2− dP
dr= ρ(r)
d2r
dt2. (2.4)
Assuming the star is in hydrostatic equilibrium (r = r ≡ 0) we arrive at the fundamental
equations describing the structure of Newtonian stars. That is,
dP
dr= −GM(r)ρ(r)
r2, P (r=0) ≡ Pc ; (2.5a)
dM
dr= +4πr2ρ(r) , M(r=0) ≡ 0 . (2.5b)
It is simple to see that in hydrostatic equilibrium, the pressure of the star is a decreasing
(or at least not increasing) function of r; otherwise the star collapses. Further, the enclosed
mass is obviously an increasing (or non-decreasing) function of r. Note that the radius of
the star R is defined as the value of r at which the pressure vanishes [P (R) = 0] and the
mass of the star as M=M(R).
However for more massive stars, where general relativity plays an important role, we have
to modify Eq. (2.5a) to include corrections from general relativity. For a Schwarzschild star,
we are going to use the Tolmann-Oppenheimer-Volkoff (TOV) equations, that is
dP (r)
dr= −GM(r)E(r)
c2r2
[1 +
P (r)
E(r)
] [1 +
4πr3P (r)
c2M(r)
] [1− 2GM(r)
c2r
]−1
, (2.6)
where E(r) is the energy density.
12
The first term on the right hand side corresponds to the non-relativistic case, that is
when P/E 1 and 2GM/c2r 1. The second and third terms arise from the pressure
being a form of energy density, and the last term is the correction due to the curvature of
space in the strong gravitational field of the star. Eq. (2.6) together with
dM
dr=
4πr2E(r)
c2. (2.7)
determine the structure of a relativistic star. The above set of equations, together with their
associated boundary conditions, must be completed by an equation of state (EoS), namely
a relation P = P (ρ) between the density and pressure. For pedagogical purposes we will
start with the EoS of a zero-temperature Fermi gas [36].
Even though stars have high temperature, the zero-temperature Fermi gas assumption
is reasonable, based on the following reason. For a compact object such as a white dwarf
star, the number density is very high and so is the Fermi energy. Typical electron Fermi
energies are of the order of 1 MeV which correspond to a Fermi temperature of TF'1010 K.
As the temperature of the system is increased (say from T = 0 to T = 106 K) electrons try
to jump to a state higher in energy by an amount of the order of kBT , but fail, as most
of these transitions are Pauli blocked. Only those high-energy electrons that are within
kBT ' 100 eV from the Fermi surface can make the transition, but those represent a tiny
fraction T/TF' 10−4 of the electrons in the star. Hence, for the purpose of computing the
pressure of the system, it is extremely accurate—to 1 part in 104—to describe the electrons
as a Fermi gas at zero temperature.
The same assumption will be made for neutron stars. Indeed, all of the systems dealt with
in this paper will be treated as degenerate Fermi gas at zero temperature. Although core
temperatures in neutron stars may increase by as much as two orders of magnitude relative
to that in white dwarf stars, the density typically increases by 8-9 orders of magnitude.
Thus, it is safe to use a zero temperature Fermi gas of neutrons to model the pressure of the
system. However, the density in a neutron star is so large that interactions between nucleons
may no longer be ignored. This will be discussed in Chap. 4
2.2 Degenerate Free Fermi Gas
The main assumption behind the Fermi gas hypothesis is that no correlations (or interactions)
are relevant to the system other than those generated by the Pauli exclusion principle. To
13
start, the Fermi wavenumber kF is defined as the momentum of the fastest moving fermion
and is solely determined by the number density (n≡N/V ) of the system. That is,
N = 2∑k
Θ(kF − |k|) = 2
∫V
(2π~)3d3kΘ(kF − |k|) = V
k3F
3π2~3, (2.8)
or equivalently
kF =(3π2~3n
)1/3. (2.9)
In Eq. (2.8), Θ(x) represents the Heaviside (or step) function. Having defined the Fermi
wavenumber kF, the energy density of the system is obtained by simply measuring the
energy of a system in which all single-particle momentum states are progressively filled in
accordance with the Pauli exclusion principle. For a degenerate (spin-1/2) Fermi gas at
zero temperature, exactly two fermions occupy each single-particle state below the Fermi
momentum pF = ~kF; all remaining states above the Fermi momentum are empty. In this
manner one obtains the following expression for the energy density:
E ≡ E/V = 2
∫d3k
(2π~)3Θ(kF − |k|)ε(k) , (2.10)
where ε(k) is the single-particle energy of a fermion with momentum k and E is the energy.
In what follows, the most general free-particle dispersion (energy vs. momentum) relation is
assumed, namely, one consistent with the postulates of special relativity. That is,
ε(k) =√
(~kFc)2 + (mc2)2 = mc2√
1 + x2F,
(with xF ≡
~kFc
mc2
). (2.11)
Performing the integral in Eq. (2.10) we obtain
E = E0E(xF) , (2.12)
where E0 is a dimensionful constant that may be written using elementary dimensional
analysis
E0 ≡(mc2)4
(~c)3, (2.13)
and E(xF) is a dimensionless function of the single variable xF given by
E(xF) ≡ 1
π2
∫ xF
0
x2√
1 + x2 dx =1
8π2
[xF
(1 + 2x2
F
) √1 + x2
F− ln
(xF +
√1 + x2
F
)].
(2.14)
14
The pressure of the system may now be directly obtained from the energy density by using
the following thermodynamic relation, which is valid only at zero temperature:
P = −(∂E
∂V
)N,T≡0
= −(∂(V E)
∂V
)N,T≡0
= P0P . (2.15)
In analogy to the energy density, dimensionful and dimensionless quantities for the pressure
have been defined:
P0 = E0 =(mc2)4
(~c)3, (2.16a)
P (xF) ≡[xF
3E ′(xF)− E(xF)
]. (2.16b)
Here E ′ is the first derivative of the energy density. With an expression for the pressure in
hand, we are finally in a position to compute its derivative with respect to xF (a quantity
that is labeled as η). As we shall see in the next section, η — a function closely related to
the zero-temperature incompressibility — is the only property of the EoS that Newtonian
stars are sensitive to. We obtain
η ≡ dP
dxF
= P0
[xF
3E ′′(xF)− 2
3E ′(xF)
]=
P0
3π2
x4F√
1 + x2F
, (2.17)
where E ′′ is the second derivative of the energy density. The above expression has a
surprisingly simple form that depends on the energy density only through its derivatives.
Alternatively, we could have bypassed the above derivation in favor of the following general
relation valid for a zero-temperature Fermi gas:
dP
dxF
= ndεFdxF
, (2.18)
where the Fermi energy εF is defined as the energy of the fastest moving electron. In view of
Eq. (2.18), it seems unnecessary to go through the trouble of computing the energy density
and the corresponding pressure if all that is required is the dependence of the Fermi energy on
xF. However, this is a necessary step to be taken when we want to include general relativity.
While Newtonian stars indeed depend exclusively on η, the structure of Schwarzschild stars
(such as neutron stars) are highly sensitive to corrections from general relativity. These
corrections depend on both the energy density and the pressure.
15
2.3 White Dwarf Stars
2.3.1 Toy Model of White Dwarf Stars
Before returning to attempt a numerical solution to the equations of hydrostatic equilibrium,
we consider as a warm-up exercise a simple, yet highly illuminating, toy model of a white
dwarf star. Assume a white dwarf star with a uniform, spherically symmetric mass density
of the form
ρ(r) =
ρ0 = 3M/4πR3 , if r ≤ R ;
0 , if r > R ,(2.19)
where M and R are the mass and radius of the star, respectively. For such a spherically
symmetric star, the gravitational energy released during the process of “building” the star
is given by
EG = −4πG
∫ R
0
M(r)ρ(r)r dr , (2.20a)
M(r) =
∫ r
0
4πr′2ρ(r′) dr′ . (2.20b)
Further, if the white dwarf star has a uniform mass density as assumed in Eq. (2.19), it
becomes straightforward to perform the above two integrals. Hence, the gravitational energy
released by “building” such a star is given by
EG(M,R) = −3
5
GM2
R. (2.21)
From the above relation, we conclude here that without a source of support against
gravity, a star with a fixed mass M will minimize its energy by collapsing into an object
of zero radius, namely, into a black hole. We know, however, that white dwarf stars are
supported by the quantum-mechanical pressure from its degenerate electrons, which (at
temperatures of about 106 K) are fully ionized in the star (recall that 1 eV'104 K). In what
follows, we assume that electrons provide all the pressure support of the star but none of its
mass, while nuclei (e.g., 4He, 12C, . . . ) provide all the mass but none of the pressure. The
electronic contribution to the mass of the star is inconsequential, as the ratio of electron to
nucleon mass is approximately equal to 1/2000.
The energy of a degenerate electron gas was computed in the previous section. Using
Eqs. (2.9) and (2.12) one obtains,
EF(M,R) = 3π2Nmec2E(xF)
x3F
, (2.22)
16
where me is the rest mass of the electron. Naturally, the above expression depends on the
mass and the radius of the star, although this dependence is implicit in xF. While the toy-
model problem at hand is instructive of the simple, yet subtle, physics that is displayed in
compact stars, it also serves as a useful framework to illustrate how to scale the equations.
2.3.2 Scaling the Equations
One of the great challenges in astrophysics, and the physics of compact stars is certainly
no exception, is the enormous range of scales that one must simultaneously address. For
example, in the case of a white dwarf star it is the pressure generated by the degenerate
electrons (constituents with a mass of me = 9.110×10−31 kg) that must support stars with
masses comparable to that of the Sun (M=1.989×1030 kg). This represents a disparity in
masses of 60 orders of magnitude! Without properly scaling the equations, there is no hope
of dealing with this problem numerically.
We start by defining fF≡EF/Nmec2 from Eq. (2.22), a quantity that is both dimension-
less and intensive (i.e., independent of the size of the system). That is,
fF(xF) = 3π2E(xF)
x3F
, (2.23)
where the dimensionless Fermi momentum xF was introduced in Eq. (2.11) and a closed-form
expression for E(xF) has been displayed in Eq. (2.14). Note that the scaled Fermi momentum
xF quantifies the importance of relativistic effects. At low density the Fermi momentum is
small (xF 1), therefore the corrections from special relativity are negligible and electrons
behave as non relativistic Fermi gas. In the opposite high-density limit when the Fermi
momentum is large (xF1) the system becomes ultra-relativistic and the “small” (relative
to the Fermi momentum) electron mass may be neglected. We shall see that in the case of
white dwarf stars, the most interesting physics occurs in the xF∼1 regime.
The dynamics of the star consists of a tug-of-war between gravity, that favors the collapse
of the star, and electron-degeneracy pressure that opposes the collapse. To efficiently
compare these two contributions, the contribution from gravity to the energy must be scaled
accordingly. Thus, in analogy to Eq. (2.23), we form the corresponding dimensionless and
intensive quantity for the gravitational energy (fG≡EG/Nmec2) in Eq. (2.21). One obtains,
fG(M,R) = −3
5
(GM
Rc2
) (M
Nme
)= −3
5
(GM
Rc2
) (mN
Yeme
). (2.24)
17
Note that in the above equations we have assumed that the mass of the star, M = AmN ,
may be written exclusively in terms of its baryon number A and the nucleon mass mN
(the small difference between proton and neutron masses is neglected). This is an accurate
approximation, as both nuclear and gravitational binding energies per nucleon are small
relative to the nucleon mass. Further, Ye≡Z/A represents the electron-per-baryon fraction
of the star (e.g., Ye =1/2 for 4He and 12C, and Ye =26/56 in the case of 56Fe).
The final step in the scaling procedure is to introduce dimensionful mass M0 and radius
R0 quantities that, when chosen wisely, will embody the natural mass and length scales in
the problem. To this effect we define
M≡M/M0 and R≡R/R0 . (2.25)
In terms of these natural mass and length scales, the gravitational contribution to the energy
of the system in Eq. (2.24) takes the following form:
fG(M, r) = −[3
5
(GM0
R0c2
) (mN
Yeme
)]M
R. (2.26)
While the dependence of the above equation on M and R is already explicit, the Fermi gas
contribution to the energy depends implicitly on them through xF. To expose explicitly
the dependence of fF on M and R we perform the following simple manipulation aided by
relations derived earlier in Sec. 2.2. One obtains
x3F =
(~kFc
mec2
)3
=
[(9π
4Ye
) (M0
mN
) (~c/mec
2
R0
)3]M
R3 . (2.27)
While we have already referred earlier to M0 and R0 as the natural mass and length
scales in the problem, their values have yet to be determined. Thus, they are still at our
disposal. Their values will be fixed by adopting the following choice: let the “complicated”
expressions enclosed between square brackets in Eqs. (2.26 and 2.27) be set equal to one.[3
5
(GM0
R0c2
) (mn
Yeme
)]=
[(9π
4Ye
) (M0
mn
) (~c/mec
2
R0
)3]
= 1 , (2.28)
This choice implies the following values for white dwarf stars with an electron-to-baryon
ratio equal to Ye = 1/2:
M0 =5
6
√15πα
−3/2G mnY
2e = 10.599M Y
2e −→
Ye=1/2= 2.650M , (2.29a)
R0 =
√15π
2α−1/2G
(~cmec2
)Ye = 17 250 kmYe −→
Ye=1/2= 8 623 km . (2.29b)
18
Here the dimensionless strength of the gravitational coupling between two nucleons has
been introduced as
αG =Gm2
N
~c= 5.906× 10−39 . (2.30)
The aim of this toy-model exercise is to find the minimum value of the total (gravitational
plus Fermi gas) energy of the star as a function of its radius for a fixed value of its mass.
Before doing so, however, a few comments are in order. First, from merely scaling the
equations and with no recourse to any dynamical calculation we have established that white
dwarf stars have masses comparable to that of our Sun but typical radii of only 10 000 km
(recall that the radius of the Sun is R≈ 700 000 km). Further, we observe that while R0
scales with the inverse electron mass, the mass scale M0 is independent of it. This suggests
that neutron stars, where the neutrons provide all the pressure and all the mass, will also
have masses comparable to that of the Sun but typical radii of only about 10 km (i.e., 2000
times smaller than those found in white dwarf stars).
Now that the necessary “scaling” machinery has been developed, we return to our original
toy-model problem. Taking advantage of the scaling relations, the energy per electron in
units of the electron rest energy is given by the following remarkably simple expression:
f(M,xF) = fG(M,xF) + fF(xF) = −M2/3xF + 3π2E(xF)
x3F
. (2.31)
The mass-radius relation of the star may now be obtained by demanding hydrostatic
equilibrium: (∂f(M,xF)
∂xF
)M
= 0 . (2.32)
While a closed-form expression has already been derived for the energy density E(xF) in
Eq. (2.14), it is instructive to display explicit non-relativistic and ultra-relativistic limits,
both of which are very simple. These are given by
fF(xF) =
1 + 3
10x2
F , if xF 1 ;34xF , if xF 1 .
(2.33)
To conclude this section, Fig. 2.2 has been included to illustrate how simple, within the
present approximation, it is to compute the radius of an arbitrary mass star (see Fig. 2.2).
For the present example, a 1 M star has been used. First, scales for input quantities
19
Figure 2.2: The interplay between various physical effects on the “toy model” problem ofa M = 1M star. The negative of the derivative of the gravitational energy is displayedas the lower horizontal line. The derivative of the Fermi energy in the ultra-relativisticlimit is displayed as the upper horizontal line. The (blue) line displays the derivative ofthe Fermi gas energy in the non-relativistic limit. Finally, the exact Fermi gas expression,which interpolates between the non-relativistic and the relativistic result, is displayed withthe (red) curve.
such as the dimensionful mass (M0) and length (R0), are defined. Next, energies and their
derivatives are computed and a plot displaying the latter is generated. The derivative of the
gravitational energy (actually the negative of it) is constant and is displayed as the lower
horizontal line. Similarly, the derivative of the Fermi energy in the ultra-relativistic limit
(upper horizontal line) is also a constant equal to 3/4 independent of the mass of the star.
The (blue) line with a constant slope displays the derivative of the Fermi gas energy in the
non-relativistic limit. Finally, the exact Fermi gas expression, which interpolates between
the non-relativistic and the relativistic result, is displayed with the (red) curve.
The equilibrium density of the star is obtained from the intersection of the red and blue
lines with the gravitational line. In the non-relativistic case the solution may be computed
analytically to be xF0 =5M2/3/3. However, this non-relativistic prediction overestimates the
Fermi pressure and consequently also the radius of the star. The non-relativistic predictions
20
for the radius of a 1 solar-mass white dwarf star is RNR =7162 km. In contrast, the result with
the correct relativistic dispersion relation in Eq. (2.11) is considerably smallerRRel =4968 km.
Yet an even more dramatic discrepancy emerges among the two models. While the non-
relativistic result guarantees the existence of an equilibrium radius for any value of the star’s
mass (RNR = 3/5M1/3
), the correct dispersion relation predicts the existence of an upper
limit beyond which the pressure from the degenerate electrons can no longer support the
star against gravitational collapse. This upper mass limit, known as the Chandrasekhar
mass, is predicted in the simple toy model to be equal to:
MCh = (3/4)3/2M0 = 1.72M . (2.34)
As it will be shown later in Fig. 2.3, accurate numerical results yield (for Ye = 1/2) a
Chandrasekhar mass of MCh = 1.44 M. Thus, not only does the toy model predict the
existence of a maximum mass star, but it does so with an 80% accuracy. Note that it is
important not to confuse this mass with the original mass of the star forming a white dwarf
which is around ≤ 4 M, since the collapsing star will loose some of its initial mass to form
a white dwarf star.
2.3.3 The Real Model of White Dwarf Stars
We now return to the exact (numerical) treatment of white dwarf stars. While the toy-model
problem developed in Sec. 2.3.1 provides a particularly simple framework to understand
the interplay between gravity, quantum mechanics, and special relativity, a quantitative
description of the systems demands the numerical solution of the hydrostatic equations
[Eqs. (2.5a and 2.5b)]. For the present treatment, however, one continues to assume that
the equation of state is that of a simple degenerate Fermi gas as discussed in Sec. 2.2. In
this case the equation of state is known analytically and it is convenient to incorporate it
directly into the the differential equation. In this way Eq. (2.5a) becomes
dxF
dr= −GM(r)ρ(r)
r2η, (2.35)
where the equation of state enters only through a quantity directly related to the zero-
temperature incompressibility. This quantity, η = dP/dxF, was defined and evaluated in
Eq. (2.17). Moreover, the density of the system ρ(r) can be expressed in terms of xF. It is
21
given by
ρ =
(mec
2
~c
)3mN
3π2Ye
x3F. (2.36)
At this point all necessary relations have been derived and the equations of hydrostatic
equilibrium in Eqs. (2.5a and 2.5b) using the equation of state in Sec. 2.2 may now be
written in the following form:
dxF
dr= −
[(GM0
R0c2
) (mn
Yeme
)]M
r2
√1 + x2
F
xF
, xF(r=0) ≡ xFc ; (2.37a)
dM
dr= +
[(3π
4Ye
) (M0
mn
) (~c/mec
2
R0
)3]−1
r2x3F, M(r=0) ≡ 0 . (2.37b)
Here the dimensionless distance r defined in Eq. (2.25) and the central (scaled) Fermi
momentum xFc have been introduced. The structure of the above set of differential equations
indicates that our goal of turning Eqs. (2.5a and 2.5b) into a well-posed problem, by directly
incorporating the equation of state into the differential equations, has been accomplished.
But we have done better. By defining the natural mass and length scales of the system (M0
and R0) according to Eq. (2.28), the two long expressions in brackets in the Eqs. (2.37a and
2.37b) reduce to the simple numerical values of 5/3 and 1/3, respectively. Finally, then, the
equations of hydrostatic equilibrium describing the structure of white dwarf stars are given
by the following expressions:
dxF
dr= f(r;xF,M) , xF(r=0) ≡ xFc ; (2.38a)
dM
dr= g(r;xF,M) , M(r=0) ≡ 0 , (2.38b)
where the two functions on the right-hand side of the equations (f and g) are given by
f(r;xF,M) ≡ −5
3
M
r2
√1 + x2
F
xF
and g(r;xF,M) = +3 r2x3F. (2.39)
This coupled set of first-order differential equations may now be solved using standard
numerical techniques, such as the Runge Kutta algorithm [37]. As in the toy-model problem,
solutions is presented in Fig. 2.3 for the full relativistic dispersion relation as well as the non-
relativistic approximation, where, in the latter, case the square-root term appearing in the
function f(r;xF,M) is set to one.
22
Figure 2.3: Mass-versus-radius relation for white dwarf stars with a degenerate electron gasEoS. While the non-relativistic calculation guarantees the existence of an equilibrium radiusfor any value of the stars’ mass, the correct dispersion relation (for Ye = 1/2) predicts theexistence of an upper mass limit known as the Chandrasekhar mass of MCh =1.44 M.
2.4 Neutron Stars
The escape velocity from the surface of a neutron star is about 50% of the speed of light.
This fact alone already suggests the importance of general-relativistic corrections for this
group of stars, that also includes the hitherto undiscovered hybrid and quark stars. Still, the
calculations for these stars can be patterned after the white dwarf calculations in Sec. 2.3.3
by incorporating a few important differences. First, in a white dwarf star the pressure is
produced by the quantum pressure of the degenerate electrons, while in a neutron star it is
due to the degenerate neutrons. Neutrons become degenerate at a much larger density so
neutron stars have masses comparable to those of white dwarf stars, but significantly smaller
radii. Second, in a white dwarf star gravity is weak enough that it can be described equally
well by Newtonian or general-relativistic dynamics. In Sec. 2.3.3 Newtonian mechanics was
chosen for simplicity. However, the significantly smaller radii of neutron stars demand a
general-relativistic treatment. As we will soon show, Eq. (2.5b) remains basically unaltered:
23
the mass density gets replaced by the full energy density that includes rest mass, kinetic,
and binding energy effects. In contrast, Eq. (2.5a) picks up three correction factors with all
of them working in favor of gravity. Finally, the equation of state for spherically symmetric,
general relativistic stars in hydrostatic equilibrium is more difficult than just replacing
degenerate electron pressure with degenerate neutron pressure. At the extreme densities
encountered in neutron stars, the equation of state is strongly influenced by the strong
interactions among the constituents. However, for pedagogical purposes of this exercise, it
is sufficient to treat the pressure as though it came only from the degenerate neutrons. We
will discuss the full extent of the potential, taking into account the strong interactions in
Ch. 4.
Before delving into the formalism of Schwarzschild stars, let us derive the appropriate
mass and radius parameters for the case of neutron stars. Since in the present approximation
neutrons provide both the mass and the pressure, these dimensionful parameters may be
easily obtained through the following substitutions into Eqs. (2.29a and 2.29b): Ye→1 and
me→mn. This yields
M0 =5
6
√15πα
−3/2G mn = 10.6M , (2.40a)
R0 =
√15π
2α−1/2G
(~cmnc2
)= 9.39 km . (2.40b)
As advertised earlier, neutron stars have masses comparable to those of white dwarf stars but
considerably smaller radii. In the general-relativistic regime the stellar structure equations—
the Tolman-Oppenheimer-Volkoff (TOV) equations—are given by
dP (r)
dr= −GM(r)E(r)
c2r2
[1 +
P (r)
E(r)
] [1 +
4πr3P (r)
c2M(r)
] [1− 2GM(r)
c2r
]−1
, (2.41a)
dM(r)
dr= +
4πr2E(r)
c2. (2.41b)
Relative to the corresponding equations for Newtonian stellar structure [see Eqs. (2.5a
and 2.5b)], the TOV equations display three corrections, that have been enclosed in square
brackets. A more subtle correction involves the replacement of the rest-mass density of the
star ρ(r), by its corresponding energy density E(r). From general relativity we know that
gravity “couples” to the energy density. This implies, in the particular case of a simple Fermi
24
gas of neutrons, that the overall mass of the star receives contributions not only from the
rest mass of the constituent neutrons, but in addition, from their kinetic energy. To make
further contact with the Newtonian equations, as well as to introduce some simplification in
the notation, the TOV equations in Eqs. (2.41a and 2.41b) are rewritten as
dP (r)
dr= −GM(r)ρ(r)
r2Γ(r) , (2.42a)
dM(r)
dr= +4πr2ρ(r) γ(r) , (2.42b)
where the corrections from general relativity are embodied in the functions γ(r) and Γ(r),
with the latter one being a short-hand notation for the four general-relativistic corrections
Γ(r) ≡[γ(r)Γ1(r)Γ2(r)Γ3(r)
]. (2.43)
The individual corrections have been defined as follows:
γ(r) ≡ E(r)/c2ρ(r) −→ E(xF)/(x3F/3π2) , (2.44a)
Γ1(r) ≡[1 +
P (r)
E(r)
]−→
[1 +
P (xF)
E(xF)
], (2.44b)
Γ2(r) ≡[1 +
4πr3P (r)
c2M(r)
]−→
[1 + 9π2r3P (xF)
M(r)
], (2.44c)
Γ3(r) ≡[1− 2GM(r)
c2r
]−1
−→[1− 10
3
M(r)
r
]−1
. (2.44d)
Formally, Eq. (2.42a) may be regarded as a Newtonian equation for hydrostatic equilibrium
with a “slowly varying” gravitational constant of the form Geff(r)≡GΓ(r). What results
is particularly interesting in that all four relativistic corrections are greater than one, i.e.,
Geff(r)≥1 for all r. Thus, general relativity enhances the unrelenting pull from gravity. Note
that the arrows in the above equations are meant to represent the properly scaled form of
the relativistic corrections that must be incorporated into the Newtonian “source terms”
displayed in Eq. (2.39). Hence, the suitably scaled TOV equations, amenable to a numerical
treatment, are given in complete analogy to the Newtonian case as follows:
dxF
dr= F(r;xF,M) , xF(r=0) ≡ xFc ; (2.45a)
dM
dr= G(r;xF,M) , M(r=0) ≡ 0 , (2.45b)
25
where now the source terms are defined through the two functions (F and G), defined as
follows:
F(r;xF,M) = f(r;xF,M)Γ(r;xF,M) and G(r;xF,M) = g(r;xF,M)γ(xF) . (2.46)
Recall that the two functions f and g have been previously defined in Eq. (2.39), and that
Γ(r;xF,M) is given as the product of the four (scaled) corrections displayed in Eq. (2.44).
In Fig. 2.4 results are displayed for a neutron star supported exclusively by the pressure
from its degenerate neutrons. To set a baseline, the limiting (Chandrasekhar) mass
for a neutron star without general-relativistic corrections is simply equal to 4 times the
Chandrasekhar mass of a Ye =1/2 white dwarf star, or 5.76 M [recall that the mass of the
star scales as Y 2e ; see Eq. (2.29a)]. To understand the role of the various general-relativistic
corrections we incorporate them one at the time, in order of increasing importance. The
curve labeled Γ1 [as in Eq. (2.44b)] incorporates only one such correction. Although there
are quantitative changes relative to the structure of white dwarf stars, for example the
Chandrasekhar mass has been reduced from 5.76 M to approximately 3.7 M, they both
display qualitatively similar mass-radius relations. In particular, the limiting mass is attained
at zero radius. The shape of the curve with the “next” general-relativistic correction,
however, is dramatically different. The curve labeled Γ1 +Γ2 attains it maximum mass of
Mmax'1.9 M at a finite radius of Rmax'6.6 km. Note that beyond this limiting value the
mass-radius relation becomes a double (or even multiple) valued function. While the TOV
equations guarantee hydrostatic equilibrium, they do not guarantee that the equilibrium will
be stable. Beyond the limiting value the star is indeed in hydrostatic equilibrium, but the
equilibrium is unstable; the smallest of perturbations will convert the neutron star into a
black hole. As expected, adding more relativistic corrections pushes the limiting mass to
smaller and smaller values (recall that all corrections increase the effect from gravity), yet
the qualitative shape ceases to change and one ultimately recovers the Oppenheimer-Volkoff
result of 1939. That is, a neutron star supported exclusively by the pressure from its neutrons
attains its limiting mass of Mmax'0.7 M at a radius of Rmax'9.1 km.
26
Figure 2.4: Mass-radius relations for neutron stars with a degenerate neutron gas EoS. Thedifferent curves include general-relativistic corrections one at a time.
27
CHAPTER 3
VIRTUES AND FLAWS OF THE PAULI
POTENTIAL
Insights into the complex and fascinating dynamics of Coulomb frustrated systems across
a variety of disciplines are just starting to emerge (see, for example, [38] and references
therein). In the particular case of neutron stars, one is interested in describing the equation
of state of neutron-rich matter across an enormous density range using a single underlying
theoretical model. In recent simulations the author of Ref. [32, 33, 39] have resorted
to a “semi-classical” model that, while exceedingly simple, captures the essential physics
of Coulomb frustration and nuclear saturation. The model includes competing interactions
consisting of a short-range nuclear attraction (adjusted to reproduce nuclear saturation) plus
a long-range Coulomb repulsion. The charge-neutral system consists of electrons, protons,
and neutrons, with the electrons (which at these densities are no longer bound) modeled as
a degenerate free Fermi gas.
So far, the only quantum effect that has been incorporated into this “semi-classical”
model is the use of an effective temperature to simulate quantum zero-point motion. The
main justification behind the classical character of the simulations is the heavy nature of
the nuclear clusters. Indeed, at the low densities of the neutron-star crust, the de Broglie
wavelength of the heavy clusters is significantly smaller than their average separation.
However, this behavior ceases to be true in the transition region from the inner crust to the
outer core. At the higher densities of the outer core, the heavy clusters are expected to “melt”
into a collection of isolated nucleons with a de Broglie wavelength that becomes comparable
to their average separation. Thus, fermionic correlations are expected to become important
in the crust-to-core transition region. Unfortunately, in contrast to classical simulations that
routinely include thousands — and even millions — of particles, full quantum-mechanical
28
simulations of many-fermion systems suffer from innumerable challenges (see [40] and
references therein). One of the problems with doing the full quantum-mechanical simulations
is the need to antisymmetrized the wave functions. For a system of N particles, one needs
to deal with N components to the wave function even for a single slater determinant, which
can be computationally demanding. In an effort to “circumvent” — although not solve
— some of these formidable challenges, classical simulations of heavy-ion collisions and of
the neutron-star crust have resorted to a fictitious “Pauli potential”. Within the realm
of nuclear collisions, the first such simulations were those of Wilets and collaborators [41].
Other simulations with a more refined Pauli potential have followed [42, 43, 44, 45, 46],
but the spirit has remained the same: introduce a momentum dependent, two-body Pauli
potential that penalizes the system whenever two identical nucleons get too close to each
other in phase space.
A goal of the present contribution is to show that the demands imposed by such a Pauli
potential are too weak to reproduce some of the most basic properties of a zero-temperature
(or cold) Fermi gas. Thus, we aim at refining such a potential in a manner that reproduces
three fundamental properties of a cold Fermi gas. These are: (i) the kinetic energy (as others
have done before us), (ii) the momentum distribution, and (iii) the two-nucleon correlation
function.
3.1 Free Fermi Gas
The zero temperature Fermi gas is the simplest many-fermion system. Such a system displays
no correlations beyond those imposed by the Pauli exclusion principle and is described by
the following free Hamiltonian:
H =N∑
i=1
p2i
2m. (3.1)
Here m is the mass of the fermions and N denotes the (large) number of particles in the
system. As no interaction of any sort exists among the particles, the eigenstates of the system
are given by a product of (single-particle) momentum eigenstates, suitably antisymmetrized
to fulfill the constraints imposed by the Pauli principle. For simplicity, we assume that the
fermions reside in a very large box of volume V =L3 and that the momentum eigenstates
satisfy periodic boundary conditions. We will be interested in the thermodynamic limit of
N → ∞ and V → ∞, but with their ratio fixed at a specific value of the number density
29
ρ≡N/V .
The (“box”) normalized momentum eigenstates are simple plane waves. That is,
ϕp(r) =1√Veip·r . (3.2)
Given that the eigenvalue problem is solved in a finite box using periodic boundary
conditions, the resulting single-particle momenta are quantized as follows:
p(n) =2π
Ln ≡ 2π
L(nx, ny, nz) , with ni = 0,±1,±2 , andi = x, y, z . . . (3.3)
with the corresponding single-particle energies given by ε(p) = p2/2m.
Up to this point the spin/statistics of the particles has not come into play. We are now
interested in describing the ground state of a system of N non-interacting, identical fermions
and the resulting many-body correlations. Such a zero-temperature state is obtained by
placing all particles in the lowest available momentum state, consistent with the Pauli
exclusion principle. Using fermionic creation and annihilation operators satisfying the
following anti-commutation relations [47],Ap, A
†p′
= δp,p′ and
Ap, Ap′
=
A†p, A
†p′
= 0 , (3.4)
the ground state of the system may be written as follows:
|ΦFG〉 =
pF∏p=0
A†p|Φvac〉 , (3.5)
where |Φvac〉 represents the (non-interacting) vacuum state and the Fermi momentum pF
denotes the momentum of the last occupied single-particle state. Note that henceforth, no
intrinsic quantum number (such as spin and/or isospin) will be considered. In essence, one
assumes that all intrinsic degrees of freedom have been “frozen”, thereby concentrating on
a single fermionic species (such as neutrons with spin up). In what follows, we compute
expectation values of various quantities in the Fermi gas ground state (|Φvac〉).We start by computing the Fermi momentum pF in terms of the number density of the
system ρ=N/V . That is,
N =∑n
nFD(n) −→V→∞
V
∫d3p
(2π)3nFD(p) =
T=0Vp3
F
6π2, (3.6)
30
or equivalently,
pF =(6π2ρ
)1/3. (3.7)
Note that in, Eq. (3.6), nFD(p) denotes the Fermi-Dirac occupancy of the single-particle
state denoted by p (or n) and the thermodynamic limit has been assumed. As all ground-
state observables will be computed over a spherically-symmetric Fermi sphere, we define the
Fermi-Dirac momentum distribution f(q) as follows:
f(q) = 3q2nFD(q) , with
∫ ∞
0
f(q)dq = 1 , (3.8)
where the dimensionless quantity q≡ p/pF is the momentum of the particle in units of the
Fermi momentum.
All classical simulations performed and reported in the next sections must be carried out
by necessity at finite temperature. Thus, we now incorporate finite temperature corrections
to the various observables of interest. For temperatures T that are small relative to the
Fermi temperature TF (with TF ≡ εF), finite-temperature corrections may be implemented
by means of a Sommerfeld expansion [48]. For example, to lowest order in τ ≡ T/TF the
momentum distribution becomes
f(q, τ) =3q2
exp
[(q2 − 1 +
π2
12τ 2
) /τ
]+ 1
−→τ→0
3q2Θ(1− q) . (3.9)
Here Θ(x) is the “Heaviside step function” appropriate for a zero-temperature Fermi gas.
Similarly, the energy-per-particle of a “cold” Fermi gas may be readily computed. One
obtains [48]
E/N = εF
∫ ∞
0
q2f(q)dq =3
5εF
[1 +
5π2
12τ 2 +O(τ 4)
], (3.10)
where the Fermi energy is defined by εF =p2F/2m.
The last Fermi-gas observable that we focus on is the “two-body correlation function”
g(r). This observable measures the probability of finding two particles at a fixed distance
r from each other. Moreover, the two-body correlation function is a fundamental quantity
whose Fourier transform yields the static structure factor, an observable that may be directly
extracted from experiment. As such, the two-body correlation function is the natural meeting
place of theory, experiment, and computer simulations [49]. The two-body correlation
function may be derived from the density-density correlation function [47]. That is,
g(r) =1
ρ2ρ2(x,y) =
1
ρ2
⟨ψ†(x)ψ†(y)ψ(y)ψ(x)
⟩, (3.11)
31
where ψ(x) is a fermionic field operator and the Dirac brackets denote a thermal expectation
value. As the two-body correlation function for a non-interacting Fermi gas may be readily
evaluated at zero temperature [47], we only provide its extension to finite temperatures. To
lowest order in τ≡T/TF (and for a single fermionic species) one obtains
g(r) = 1−(
3j1(z)
z
)2 [1− π2
12z2τ 2
], (3.12)
where z≡pFr and j1(z) is the spherical Bessel function of order n=1, namely,
j1(z) =sin(z)
z2− cos(z)
z. (3.13)
Equations (3.9), (3.10), and (3.12) display the three fundamental observables of a cold
Fermi gas that we aim to reproduce in this work via a momentum-dependent, two-body
Pauli potential. Note that in most (if not all) earlier studies of this kind, only the kinetic
energy of the Fermi gas [Eq. (3.10)] was used to constrain the parameters of the Pauli
potential [41, 42, 43, 44, 45, 46]. We are unaware of any earlier effort at including more
sensitive Fermi-gas observables to constrain the parameters of the model. Clearly, it should
be possible to reproduce the kinetic energy even with an incorrect momentum distribution.
Thus, while we build on earlier approaches, we also highlight some of their shortcomings.
3.2 Pauli Potential: A New Functional Form
In the previous section the wave function of a zero-temperature Fermi gas was introduced
as follows:
|ΦFG〉 =
pF∏p=0
A†p|Φvac〉 . (3.14)
Essential to the dynamical behavior of the system are the anti-commutation relations
[Eq. (3.4)] that enforce the Pauli exclusion principle (i.e., (A†p)2 ≡ 0). As it is often done,
one may project the above “second-quantized” form of the many-fermion wave-function into
configuration space to obtain the well-known Slater determinant. That is,
ΦFG(p1, . . . ,pN ; r1, . . . , rN) =1√N !
∣∣∣∣∣∣∣∣∣ϕp1(r1) . . . ϕp1(rN)
. . . . .
. . . . .
. . . . .ϕpN
(r1) . . . ϕpN(rN)
∣∣∣∣∣∣∣∣∣ , (3.15)
32
where the single-particle wave-functions ϕp(r) are the (“box”) normalized plane waves
defined in Eq. (3.2). The Slater determinant embodies important correlations that were
discussed in the previous section and that we aim to incorporate into our classical simulations.
These are:
(a) As a consequence of the Pauli exclusion principle [(A†p)2≡0], the probability that two
fermions share the same identical momentum is equal to zero. Mathematically, this
result follows from the fact that the wave-function vanishes whenever two rows of the
Slater determinant are equal to each other.
(b) Similarly, the wave-function also vanishes whenever two columns of the Slater determi-
nant are identical. This fact precludes two fermions from occupying the same exact
location in space.
(c) At zero temperature, only momentum states having a magnitude |p| less than the Fermi
momentum pF are occupied; the rest are empty.
The first two properties are embedded in the momentum distribution of Eq. (3.9), namely,
a quadratic momentum distribution sharply peaked at the Fermi momentum pF (recall that
such a momentum distribution emerges after folding the Heaviside step function with the
phase space factor). The third property induces spatial correlations that are captured by the
two-body correlation function g(r) of Eq. (3.12). Indeed, the two-body correlation function
is related to the integral of the Slater determinant over all but two of the coordinates of the
particles (e.g., r1 and r2). Clearly, the Slater determinant vanishes whenever r1 = r2, and so
does the two-body correlation function at r= |r1−r2|≡0. It is the aim of this contribution to
build a Pauli potential that incorporates these three fundamental properties of a free Fermi
gas.
However, before doing so, we will briefly review the Pauli potential introduced by Wilets
and collaborators — and used by others with minor modifications — to simulate the collisions
of heavy ions and the properties of neutron rich matter at sub-saturation densities. Such a
Pauli potential is given by a sum of momentum-dependent, two-body terms of the following
form:
VPauli(p1, . . . ,pN ; r1, . . . , rN) =N∑
i<j=1
V0 exp(−s2ij/2) , (3.16)
33
where V0>0 and the dimensionless phase-space “distance” between points (pi, ri) and (pj, rj)
is given by
s2ij ≡
|pi − pj|2
p20
+|ri − rj|2
r20
. (3.17)
Here p0 and r0 are momentum and length scales related to the excluded phase-space volume
that is used to mimic fermionic correlations. That is, whenever the phase-space distance
between two particles is such that s2ij .1, then a penalty is levied on the system in an effort
to mimic the Pauli exclusion principle. Although the parameters of this Pauli potential
(V0, p0, and r0) can — and have — been adjusted to reproduce the kinetic energy of a free
Fermi gas, it fails (as we show later) in reproducing more sensitive Fermi-gas observables,
particularly, the momentum distribution f(p) and the two-body correlation function g(r).
Upon closer examination, the above flaws should not come as a surprise. In our previous
discussion of the Slater determinant it has been established that the probability of finding
two identical fermions in the same location in space “or” with the same momenta must
be identically equal to zero. Yet the Pauli potential of Eq. (3.16) fails to incorporate this
important dynamical behavior. Indeed, the above Pauli potential imposes a penalty on the
system only when both the location “and” momenta of the two particles are close to each
other (i.e., s2ij .1). In particular, no penalty is imposed whenever two fermions occupy the
same location in space, provided that their momenta are significantly different from each
other, i.e., |pi−pj|2p20. Thus, the Pauli potential of Eq. (3.16) will generate an incorrect
two-body correlation function, namely, one with g(r) 6= 0 as r tends to zero. By the same
token, an incorrect momentum distribution will be generated, although not necessarily its
second moment.
To remedy these deficiencies, a Pauli potential is now constructed so that the three
properties [(a), (b), and (c)] defined above are explicitly satisfied. To this end, we introduce
the following form for the Pauli potential:
VPauli(p1, . . . ,pN ; r1, . . . , rN) =N∑
i<j=1
[VA exp(−rij/r0) + VB exp(−pij/p0)
]+
N∑i=1
VC Θη(qi) , (3.18)
34
where rij = |ri − rj|, pij = |pi − pj|, qi = |pi|/pF, and Θη is a suitably smeared Heaviside-step
function of the following form:
Θη(q) ≡1
1 + exp[−η(q2 − 1)]−→η→∞
Θ(q) . (3.19)
The parameters of the model VA, VB, VC and r0, p0, η will be adjusted to reproduce both
the momentum distribution and two-body correlation function of a low-temperature Fermi
gas. Note that the phase-space dependence of the Pauli potential has been separated into
a “sum” of two-body pieces, with the first acting exclusively in configuration space and the
second one only in momentum space. The first term in the potential imposes a penalty as
the particles get too close (of the order of r0) to each other. Similarly, the second term in
the potential penalizes particles whenever their relative momenta becomes of the order of
p0. Finally, the third “one-body” term enforces the low-temperature behavior of the Fermi
gas, namely, that the probability of finding any particle with a momentum significantly
larger than the Fermi momentum is vanishingly small. Most of the parameters will depend
explicitly on the density of the system (see Sec. 3.3). This reflects the complex many-body
nature of the Pauli correlations and our inability to simulate them by means of a “simple”
(albeit momentum dependent) two-body potential.
3.3 Simulation Results
We start this section by listing in Table 3.1 the parameters of the Pauli potential. For the
strength parameters, the following simple density dependence is assumed:
Vi(ρ) = V 0i
(ρ/ρ0
)αi
, i = A,B,C , (3.20)
where ρ0 = ρsat/4 = 0.037 fm−3 is the density of a single fermionic species (for example,
neutrons with spin up) at nuclear-matter saturation density (ρsat = 0.148 fm−3). For the
range parameters (r0 and p0) the following scaling relation is adopted:
r0 = βA/pF , (3.21a)
p0 = βBpF . (3.21b)
Monte Carlo simulations for a system of N = 1000 identical fermions at the finite (but
small) temperature of τ = T/TF = 0.05 have been performed. Initially, the particles are
35
Table 3.1: Strength (in MeV) and range parameters (dimensionless) for the various compo-nents of the Pauli potential. See Eqs. (3.18), (3.20), and (3.21). These values have been usedto simulate a system of N=1000 identical fermions at a temperature of τ=T/TF =0.05.
V 0A V 0
B V 0C αA αB αC βA βB η
13.517 1.260 3.560 0.629 0.665 0.831 0.845 0.193 30
distributed randomly throughout the box with momenta that are uniformly distributed up to
a maximum momentum of the order of the Fermi momentum. After an initial thermalization
phase of typically 2000 sweeps (or 2 million Monte Carlo moves for both coordinates and
momenta) data is accumulated for an additional 2000 sweeps, with the data divided into 10
groups to avoid correlations among the data. It is from these 10 groups that averages and
errors are generated.
3.3.1 Kinetic Energy
The kinetic energy of a finite-temperature Fermi gas as a function of density in Eq. (3.10) is
displayed in Fig. 3.1. The (black) solid line represents the analytic behavior of a free Fermi
gas in Eq. (3.10) correct to second order in τ . The (red) line with circles is the result of
the Monte Carlo simulations with the Pauli potential defined in Eq. (3.18). The agreement
(to better than 5%) is as good as the one obtained with earlier parametrization of the Pauli
potential.
To our knowledge, reproducing the kinetic energy of a free Fermi gas is the sole constraint
that has been imposed on most of the Pauli potentials available to date. However, while a
host of Pauli potentials can reproduce such a behavior, it is unclear if these potentials can
also reproduce the full momentum distribution. Thus, we now show how the Pauli potential
defined in Eq. (3.18) is successful at reproducing two highly sensitive Fermi-gas observables,
namely, the momentum distribution and the two-body correlation function.
3.3.2 Momentum Distribution
The momentum distribution obtained from the Monte Carlo simulations is displayed in
Fig. 3.2 for a variety of densities. Note that the momentum distribution has been normalized
36
0.0 0.5 1.0 1.5 2.0ρ/ρ0
0
5
10
15
20
25
30
35
Eki
n/N
Monte CarloExact
N=1000τ=T/T
F=0.05
Figure 3.1: Average kinetic energy of a system of N = 1000 identical fermions at atemperature of τ = T/TF = 0.05. The line with circles is the result of the Monte Carlosimulations with the Pauli potential of Eq. (3.18). The solid line is the exact behavior of anon-relativistic Fermi gas, as given by Eq. (3.10).
to one and that the densities have been expressed in units of ρ0 =0.037 fm−3. As indicated
in Eq. (3.9), the momentum distribution of a cold Fermi gas depends solely on the two
dimensionless ratios q = p/pF and τ = T/TF. Thus, all curves must collapse into the exact
one — displayed by the (black) solid line — independent of density. It is gratifying to see
that this is indeed the case. In contrast, we show in the next section that the standard Pauli
potential of Eq. (3.16) fails to reproduce this behavior.
3.3.3 Two-Body Correlation Function
The two-body correlation function of a free Fermi gas is displayed in Fig. 3.3. The two-body
correlation function g(r) is related to the probability of finding two particles separated by
a distance r. For a free Fermi gas, the probability of finding two identical fermions at zero
separation is identically equal to zero [see Eq. (3.12)]. As this short-range (anti-)correlation
37
0.00 0.25 0.50 0.75 1.00 1.25 1.50q=p/p
F
0.0
0.5
1.0
1.5
2.0
2.5
3.0
f(q,
τ)ρ=0.2ρ=0.4ρ=0.6ρ=0.8ρ=1.0ρ=1.2ρ=1.4Exact
N=1000τ=T/T
F=0.05
Figure 3.2: Momentum distribution of a system of N = 1000 identical fermions at atemperature of τ = T/TF = 0.05 for a variety of densities (expressed in units of ρ0 =0.037 fm−3). The momentum distribution has been normalized to one [see Eq. (3.8)]. The(black) solid line with no symbols gives the exact behavior of a non-relativistic Fermi gas.
is the sole consequence of the Pauli exclusion principle, the “Pauli hole” disappears for
distances of the order of the inter particle separation (p−1F ). As in the case of the momentum
distribution, the correlation function depends only on two dimensionless variables (z= pFr
and τ). It is again gratifying that the simulation curves scale to the exact correlation
function, depicted here with a (black) solid line.
3.4 Comparison to other approaches
In this section we compare the Pauli potential introduced in Eq. (3.18) to earlier approaches
that are based on Eq. (3.16). Such approaches have been very successful in reproducing
the kinetic energy of a free Fermi gas for a wide range of densities. Indeed, the kinetic
energy displayed in Fig. 3.2 of Ref. [45] is as good — if not better — than the one obtained
here. However, a faithful reproduction of the kinetic energy does not guarantee that the
38
0 3 6 9 12 15z=p
Fr
0.00
0.20
0.40
0.60
0.80
1.00
g(z,
τ)ρ=0.2ρ=0.4ρ=0.6ρ=0.8ρ=1.0ρ=1.2ρ=1.4Exact
N=1000τ=T/T
F=0.05
Figure 3.3: Two-body correlation function for a system of N = 1000 identical fermionsat a temperature of τ = T/TF = 0.05 for a variety of densities (expressed in units ofρ0 = 0.037 fm−3). The (black) solid line with no symbols gives the exact behavior of anon-relativistic Fermi gas.
system displays the same phase-space correlations as that of a free Fermi gas. To illustrate
this point we compare in Fig. 3.4 the results from the two approaches for the momentum
distribution and two-body correlation function at a fixed density of ρ0 = 0.037 fm−3. The
left-hand panel in the figure displays the momentum distribution and indicates that while
earlier approaches (depicted with a blue line with circles) are accurate at reproducing the
second moment of the distribution (i.e., the kinetic energy) the momentum distribution
itself shows a behavior that differs significantly from that of a cold Fermi gas. The right-
hand panel shows deficiencies that are as — or even more — severe. Two points are worth
highlighting. First, the two-body correlation function g(r) generated with the standard form
of the Pauli potential does not vanish at r = 0. Second, for distances of the order of the
inter-particle separation and beyond, the two-body correlation function develops artificial
oscillations. Failure in reproducing the correct behavior of g(r) at r=0 is relatively simple
39
0 5 10 15 20z=p
Fr
0.0
0.2
0.4
0.6
0.8
1.0
1.2
g(z,
τ)
MaruyamaThis workExact
0.0 0.2 0.4 0.6 0.8 1.0 1.2q=p/p
F
0.0
1.0
2.0
3.0
4.0
5.0
6.0f(
q,τ)
N=1000ρ=ρ0
Figure 3.4: Comparison between the Pauli potential introduced in this work Eq. (3.18)and earlier approaches based on Eq. (3.16). The left-hand panel shows the momentumdistribution while the right-hand panel the two-body correlation function. The simulationshave been carried out for a system of N = 1000 identical fermions at a density ofρ0 = 0.037 fm−3. The (blue) line with circles is obtained from a Monte Carlo simulationusing Eq. (3.16) with the parameters of Ref. [45]. The (red) line with squares displaysthe results using the Pauli potential introduced in this work; the black line gives the exactbehavior of a non-relativistic Fermi gas.
to understand. Potentials based on Eq. (3.16) impose a penalty on the system only if both
the positions and momenta of the two fermions are close to each other. Yet the correlations
embodied in a free Fermi gas are significantly more stringent than that. Indeed, a Slater
determinant vanishes if either the positions or the momenta of the two fermions are equal
to each other. In contrast, the development of artificial structure in g(r) is a more subtle
effect that is intimately related to the momentum dependence of the potential and that we
now address.
The artificial oscillations present in g(r) are reminiscent of the structure of liquids and/or
crystals. The emergence of crystalline structure in a low-temperature/low-density system
would be expected if the energy of localization becomes small relative to the mutual repulsion
40
between the particles. Such, however, is not the behavior of a cold Fermi gas. While spatial
anti-correlations would favor the formation of a periodic structure, the large velocities of
the particles — resulting from the Pauli exclusion principle — would make their localization
extremely costly. What is not evident is the reason for the momentum distribution generated
with a standard Pauli potential to lead to crystallization, whereas not that of a real Fermi
gas (see left-hand panel in Fig. 3.4).
To our knowledge, the answer to this question was first provided by Neumann and Fai [50].
One must realize that any Hamiltonian that contains momentum-dependent interactions —
such as most (if not all) Pauli potentials — generates a “canonical” momentum distribution
that may (and does!) differ significantly from the corresponding “kinematical” momentum
distribution 〈π〉 ≡ 〈mr〉. Indeed, in a Hamiltonian formalism where the Hamiltonian
depends on the positions and canonical (not kinematical) momenta of all the particles,
namely, H = H(p1, . . . ,pN ; r1, . . . , rN) the kinematical velocities must be obtained from
Hamilton’s equations of motion, i.e., ri = ∂H/∂pi. For a Hamiltonian that contains
momentum-dependent interactions — as in the case of the Pauli potential — then the
kinematical momentum πi ≡ mri differs from the corresponding canonical momentum
pi. This suggests that while a given choice of Pauli potential may produce the correct
canonical momentum distribution, it may generate kinematical velocities that are too small
to prevent crystallization. Such a distinction between canonical and kinematical momenta is
an unwelcome, yet unavoidable, consequence of the approach. A cold Fermi gas — a system
subjected to no interactions — is the quintessential quantum system were such an artificial
distinction is not required.
It is worth noting that although some earlier choices yield an extremely soft kinematical
momentum distribution, the Pauli potential introduced in this work is not immune to such a
disease. Indeed, we believe that an unrealistically soft kinematical momentum distribution
is likely to be a general result of the Hamiltonian approach. Thus, we close this section
by displaying in Fig. 3.5 the kinematical velocity distribution obtained with the new choice
of Pauli potential introduced in Eq. (3.18). For comparison, the exact distribution of a
cold Fermi gas (solid black line) is also included. The dependence of the Pauli potential
on the canonical momentum distribution is responsible for generating such a soft velocity
distribution, with its peak around 1/10 of the Fermi velocity.
41
0.00 0.25 0.50 0.75 1.00 1.25u=v/v
F
0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
9.0
h(u,
τ)ρ=0.2ρ=0.4ρ=0.6ρ=0.8ρ=1.0ρ=1.2ρ=1.4Exact
N=1000τ=T/T
F=0.05
Figure 3.5: “Kinematical” velocity distribution of a system of N = 1000 identical fermionsat a temperature of τ = T/TF = 0.05 for a variety of densities (expressed in units ofρ0 = 0.037 fm−3). The velocity distribution has been normalized to one. The (black) solidline with no symbols gives the exact behavior of a non-relativistic Fermi gas.
And while we were able to avoid crystallization with the present set of parameters (see
Figs. 3.3 and 3.4), the risk of crystallization looms large (see next section).
3.5 Finite-Size Effects
We have observed that the Pauli potential introduced in Eq. (3.18), with its parameters
suitably adjusted, has been successful in reproducing a variety of Fermi-gas observables, such
as its kinetic energy, its (canonical) momentum distribution, and its two-body correlation
function. However, the first indication of a potential problem — and one that may be
generic to all approaches employing momentum-dependent interactions — is the emergence
of an unrealistically soft velocity distribution and with it, the possibility of artificial spatial
correlations (i.e., crystallization). Fortunately, with the choice of parameters adopted in this
work, the problem of crystallization was avoided (see Fig. 3.3). Yet, there is no guarantee
42
that crystallization will not become a problem as one examines the sensitivity of our results
to finite-size effects.
0 5 10 15z=p
Fr
0.0
0.2
0.4
0.6
0.8
1.0
g(z,
τ)
N=250N=500N=1000N=2000Exact
0.0 0.3 0.6 0.9 1.2 1.5q=p/p
F
0.0
1.0
2.0
3.0f(
q,τ)
τ=0.05ρ=ρ0
Figure 3.6: Finite-size effects on the canonical momentum distribution (left-hand panel) andthe two-body correlation function (right-hand panel) for a system of identical fermions ata temperature of τ = T/TF = 0.05 and a density of ρ = ρ0 = 0.037 fm−3. Simulations werecarried out for systems containing N =250, 500, 1000 and 2000 particles. The (black) solidline with no symbols gives the exact behavior of a non-relativistic Fermi gas.
In order to estimate the sensitivity of our results to finite-size effects, Monte Carlo
simulations were performed for a system containing N = 250, N = 500, N = 1000, and
N=2000, identical fermions (note that the results reported so far have been limited to 1000
particles). The conclusions from this study are mixed. First (and fortunately) no evidence of
crystallization or of significant finite-size effects were found. These findings are displayed in
Fig. 3.6 for both the canonical momentum distribution (left-hand panel) and the two-body
correlation function (right-hand) panel. Unfortunately, however, in order to preserve the
high quality of the results previously obtained with 1000 particles, a parameter of the Pauli
potential [V 0B in Eq. (3.18)] had to be fine tuned. Specifically, the following scaling with
particle number was used:
V 0B(N) = V 0
B(N=1000)
(1000
N
), (3.22)
43
with V 0B(N=1000)=1.26 MeV being the value listed in Table 3.1. This unpalatable fact may
be a reflection of the highly challenging task at hand: how to simulate fermionic many-body
correlations by means of a “simple” two-body Pauli potential.
44
CHAPTER 4
EQUATION OF STATE FOR NUCLEAR PASTA
As has been discussed earlier, the density range in neutron stars spans from essentially zero at
the edge of the star to about 0.8 baryons fm−3 at the center of the star. Due to this extreme
density range, modeling a realistic equation of state that works for the whole density region
is a difficult task. Therefore, in this chapter we will limit our study to constructing a realistic
equation of state for the pasta region in neutron stars.
The physics of nuclear pasta is essentially a competition between short range nuclear
attraction that tries to correlate nucleons into nuclei, and the long range Coulomb repulsion
that prevents protons from getting too close to each other. The first question that comes to
mind in dealing with nuclear pasta is whether it should be treated as a classical or a quantum
system. From statistical mechanics, we know that a system can be treated classically if the
mean inter particle distance (V/N)1/3 in the system is much larger than the mean thermal
wavelength λT . That is,
nλ3T ≡
nh3
(2πmkT )3/2 1 , (4.1)
where n is the baryon density, m is the mass of the particles in the system, k is the Boltzmann
constant, and h is the Planck constant. The mean thermal wavelength λT = h/(2πmkT )1/2.
For nuclear pasta the density is very low (around 10−3 nucleon/fm3) and the temperature
in the star is very high (of the order of 108 K ≈10−2 MeV). To illustrate this condition, let
us take a look at an iron nucleus in nuclear pasta. The value for nλ3T for a 56Fe nucleus in
this region is around 0.18. Therefore, in studying nuclear pasta we will use semi-classical
Monte Carlo simulations by incorporating the fermionic correlation via a semi-classical Pauli
potential.
45
4.1 Modeling the Nuclear Pasta
In this section we introduce a semi-classical model that while simple, captures the essential
physics of the nuclear pasta. In this model a neutral system consisting of electrons, protons
and neutrons is considered. The nucleons interact via a short range nuclear and screened
Coulomb potential. The electrons are assumed to form a degenerate free Fermi gas of
density ρe=ρp to ensure charge neutrality. The very slight polarization of electrons leads to
a screening length λ for the Coulomb interactions between protons. The full Hamiltonian is
of the form
H = K + VTot , (4.2a)
VTot =∑i<j
Vnuc(i, j) +∑i<j
Vc(i, j) , (4.2b)
where K is the kinetic energy and VTot represents the potential energy. In this model the
potential energy consists of nuclear potential and Coulomb potential. The two-body nuclear
potential (Vnuc) consists of spin-independent (VSI) and spin-dependent(VSD) terms of the
following form:
Vnuc(i, j) = VSI(i, j) + VSD(i, j) , (4.3a)
VSI(i, j) = ae−r2ij/Λ1 + be−r2
ij/Λ2 + cτz(i)τz(j)e−r2
ij/Λ3 , (4.3b)
VSD(i, j) = VA e(−rij/r0)δσiσjδτiτj
, (4.3c)
VA(ρ) = V 0A
(ρ/ρ0
)αA
, and r0 = βA/pF . (4.3d)
Here the distance between particles is denoted by rij = |ri − rj|, and pF is the Fermi
momentum of the nucleon as defined in Appendix B. The isospin of the ith particle is τz =
+1 for a proton and τz = -1 for a neutron, while σ denotes the z-component of the spin of
the particles.
The two-body spin-independent potential (VSI) in Eq. (4.3b) represents the characteristic
intermediate-range attraction and short-range repulsion of the nucleon-nucleon (NN) force.
The inclusion of the isospin dependence in the potential insures that while pure neutron
matter is unbound, the symmetric nuclear matter is bound appropriately. The spin-
independent potential contains free-parameters (a, b, c,Λ1,Λ2,Λ3) that will be adjusted to
46
reproduce as accurately as possible the following properties: a) the saturation density and
binding energy per nucleon of symmetric nuclear matter and b) the binding energy and
charge radii for several finite nuclei.
The two-body spin-dependent potential (VSD) in Eq. (4.3c) represents the fermionic
correlations between fermions. It is obtained from the Pauli-potential in Eq. (3.18) without
taking into account the momentum-dependent term of the potential. This is a necessary step
since the potential in Eq. (3.18) due to its momentum-dependence, presents problems in the
calibration process. That is, we could not find a best fit that satisfies the above mentioned
nuclear matter and finite nuclei properties. Also note that in Eq. (4.3c) we have generalized
the Pauli potential in Eq. (3.18) to include different fermionic species.
Finally, a screened Coulomb interaction of the following form is included :
Vc(i, j) =e2
rij
e−rij/λτp(i)τp(j) , (4.4)
where τp = (1 + τz)/2 is a proton projection operator and λ is the screening length that
results from the slight polarization of the electron gas. The screening length λ defined in a
Thomas-Fermi approximation is not significantly smaller than the length L of our simulation
box, unless for cases in which a very large number of particles is used. Therefore to control
finite-size effect we have arbitrarily adopted a screening length of λ ≡ 10 fm .
We perform simulations for a canonical ensemble with a fixed number of particles (A)
at a temperature T and a fixed baryon density (ρ). The simulation volume is then simply
V = A/ρ. To minimize finite size effects, periodic boundary conditions are adopted. The
distance between 2 particles (rij) is then calculated from the x, y, and z coordinates of the
ith and jth particles as follows:
rij =√
[xi − xj]2 + [yi − yj]2 + [zi − zj]2 , (4.5)
[li] = li − LR( liL
), i ≡ (x, y, z) . (4.6)
where [li] represents the minimum distance between two particles in the i-th axis. L is
the length of the simulation box which is represented by V 1/3 and R(
liL
)is a notation to
represent rounding off (li/L) to the next integer if (li/L) ≥ 0.5 and to the lower integer
otherwise.
47
The total energy (E) of the system is made of kinetic (K) and potential (VTot) energy
contributions. The average energy can be calculated as a thermal expectation value as
follows:
〈E〉 =1
Z(A, T, V )
∫d3Ar d3Ap H e−H/T , (4.7)
with the canonical partition function given by
Z(A, T, V ) =
∫d3Ar d3Ap e−H/T . (4.8)
However, since the potential energy defined in Eq. (4.2b) is independent of momentum,
the partition function for the system factorizes into a product of a partition function in
momentum space (ZP ) and a partition function in coordinate space (ZR), such that
Z(A, T, V ) = ZRZP , (4.9a)
ZR =
∫d3r1 d
3r2...d3rA e−VTot/T , (4.9b)
ZP =
∫d3p1 d
3p2...d3pA e−K/T . (4.9c)
Hence, Eq. (4.7) becomes
〈E〉 =1
ZR
∫d3r1 d
3r2...d3rA VTot e
−VTot/T +1
ZP
∫d3p1 d
3p2...d3pA K e(−K/T ) . (4.10)
The second term in Eq. (4.10), which is the expectation value of kinetic energy, reduces to
its classical value
〈K〉 =3
2AT, (4.11)
while the first term–the expectation value of the potential energy– will be evaluated using
Metropolis Monte Carlo simulations. For a detailed discussion on this algorithm please refer
to Appendix A.
4.2 Nuclear Matter Equation of State
Infinite nuclear matter is modeled as an infinite system of nucleons (protons and neutrons)
interacting via nuclear interactions but with the electromagnetic interaction turned off. De-
spite its idealization, understanding the properties of infinite nuclear matter is a prerequisite
for any consistent theory of nuclei. In this section we will discuss the equation of state (EoS)
of cold nuclear matter, that is, matter at zero temperature and in its lowest energy state
48
[51]. The EoS of cold nuclear matter describes the relationship between the binding energy
per nucleon versus density and proton fraction, which can be written in terms of isospin
asymmetry parameter δ ≡ (N − Z)/A as [24]
(E/A)(ρ, δ) = (E/A)(ρ, δ = 0) + (Esym/A)(ρ)δ2 +O(δ4) . (4.12)
The first term in Eq. (4.12) denotes the energy for symmetric nuclear matter (N = Z),
whereas the second term is the symmetry energy. To lowest order in δ, the symmetry energy is
the difference in energy between pure neutron matter and symmetric nuclear matter. Hence,
the symmetry energy may be viewed as the penalty imposed on the system by departing
from the symmetric (N=Z) limit.
Symmetric nuclear matter saturates. That is, there exists an equilibrium density ρ0 at
which the pressure vanishes. Expanding the energy of symmetric nuclear matter around
saturation density ρ0 yields
(E/A)(ρ, δ = 0) = (E/A)(ρ0, δ = 0) +1
18K
(ρ− ρ0
ρ0
)2
+ ... (4.13)
Here the first term represents the binding energy per nucleon for symmetric nuclear matter at
saturation density ρ0, and the second term is the compression modulus K. The compression
modulus defines the curvature of symmetric nuclear matter at ρ0. That is,
K = 9ρ20
∂2(E/A)(ρ, δ = 0)
∂ρ2
∣∣∣ρ=ρ0
. (4.14)
The larger the value of K, the stiffer the EoS, namely, the faster the energy increases with
density. A large value of K corresponds to a stiffer equation of state while a small value of
K represents a softer equation of state.
Experimental data on finite nuclei have placed constraints on the properties of symmetric
nuclear matter. The symmetric nuclear matter saturation density ρ0 is around 0.15 fm−3 and
has a binding-energy per nucleon E/A(ρ0, δ = 0) ' -16 MeV. The constant density of nuclear
matter and the proportionality of the binding energy to the mass number A represent an
important fact about nuclear forces. Nuclear forces show saturation [52]; that is, the forces
are attractive for a small number of nucleons but become repulsive for a larger number.
The saturation point corresponds to the equilibrium point (at zero temperature) of nuclear
matter and is characterized by the vanishing pressure of the system,
49
P = ρ2(∂(E/A)
∂ρ
)∣∣∣ρ=ρ0
= 0 . (4.15)
and positive curvature ∂P/∂ρ 0. This is all that is known about the EoS with a high
degree of experimental confidence. The next question is whether we will be able to probe the
vicinity of the saturation point to get a better information about the EoS. The first quantity
of interest in this respect is the compression modulus K which determines the shape of the
equation of state at zero temperature close to the saturation point. It should be noted that
while the linear term in Eq. (4.13) vanishes for symmetric nuclear matter (i.e., the pressure),
no such special saturation point exist in the case of the symmetry energy. Rather, it is the
symmetry energy at lower density of ρ0 ' 0.1 fm−3 that seems to be accurately constrained
(to within 1 MeV) by available ground-state observables [53]
4.3 Semi Empirical Mass Formula
In the previous section, we discussed the binding energy of infinite nuclear matter which
is an idealization of actual finite nuclei. We found that the binding energy per nucleon of
symmetric nuclear matter at saturation density is around -16 MeV. However, treating the
binding energy of actual nuclei requires further corrections due to the Coulomb repulsion
between protons and surface effects which have been neglected in the first step. This method
was first devised by von Weizsacker in 1935 and suggest a binding energy per nucleon of the
following form: [54, 55]
B
A= avol −
asurf
A1/3− acoul
Z2
A4/3− asym
(A− 2Z)2
A2. (4.16)
The leading term in the mass formula is the volume term, which indicates the constant
binding energy per nucleon of symmetric nuclear matter at saturation density. The second
term is the correction due to surface effects. The nucleons on the surface of the nucleus
are surrounded by less nucleons than those in the interior, and therefore give a smaller
contribution to the binding energy. For a spherical nucleus the surface area is related to the
volume as S ∝ V 2/3. The volume of the nucleus, under this assumption, will be proportional
to A and hence S ∝ A2/3 The third term describes the Coulomb energy, which indicates the
reduction in the binding energy due to the Coulomb repulsion between protons. The last
term describes the symmetry energy, showing the decrease in binding energy due to unequal
50
0 50 100 150 200 250A
0
2
4
6
8
10
12
14
16
Ebi
nd/A
(MeV
)
volume
surface
Coulomb
Total
symmetry
Figure 4.1: Binding energy predicted by the Semi Empirical Mass formula for variousnuclei. The solid line refers to contribution to the Binding energy from the volume term.Relative contributions to the binding energy after the inclusion of of the various other terms(asurf , acoul, asym) have been identified, leading to the predicted binding energy (Total)
number of protons and neutrons. A best fit of the various coefficients in Eq. (4.16) to many
nuclei in the mass table [56] gives the following values:
avol = 15.72 MeV (4.17)
asurf = 17.54 MeV (4.18)
acoul = 0.71 MeV (4.19)
asym = 23.38 MeV (4.20)
Fig. 4.1 displayed the relative importance of the various terms (avol, asurf , acoul, asym) to
the binding energy. The first parameter (avol) represents the binding energy per nucleon
for symmetric nuclear matter. After including corrections from various terms, the binding
energy per nucleon (except for the few lightest nuclei) is close to 8 MeV.
51
4.3.1 Spin-Dependent Potential
The spin-dependent potential as stated in Eqs. (4.3c and 4.3d) has three free parameters
(VA, αA, and βA). Using Eq. (3.20) expressed in terms of pF , the spin-dependent potential
becomes
VA(pF ) = 13.517(pF/pF0
)1.995
MeV , (4.21a)
r0 = 0.845/pF . (4.21b)
4.4 Simulation Results
We performed Metropolis Monte Carlo simulations for A = 250 identical particles at a fixed
temperature T = 1 MeV such that T/TF 1 using the parameters in Eq. (4.21). The
simulation results for the two- body correlation function are shown in Fig. 4.2. Also shown
in the figure is the analytic calculation of the free Fermi gas two-body correlation function
of Eq. (3.12). We find that our simulations give a reasonable results compare to the analytic
prediction.
4.4.1 Spin-Independent Potential
We now return to discuss the choice of spin-independent parameters (a, b, c,Λ1,Λ2,Λ3) that
must be used to reproduce various bulk properties of symmetric nuclear matter, such as
the saturation density, the binding energy per nucleon at saturation, and the compression
modulus. As mentioned earlier, symmetric matter saturates at a density of around 0.15
fm−3 and an energy per nucleon ' -16 MeV. With this information at hand, we are ready
to calibrate the model parameters. Results from the fit yields the two models listed in Table
(4.1).
Again, we ran Metropolis Monte Carlo simulations to reproduce the above properties
of cold nuclear matter. However, in semi-classical simulations it is impossible to perform
calculations at exactly zero temperature. Therefore, for simulation purposes the temperature
will be fixed arbitrarily at T = 1 MeV. This T = 1 MeV temperature should be regarded
as an additional model parameter. Table (4.1) displays the model parameters obtained from
the fits. The first three terms are in MeV while the last three terms are in fm2.
52
0 1 2 3 4 5 6 7 80
0.2
0.4
0.6
0.8
1
0 1 2 3 40
0.2
0.4
0.6
0.8
1
0 1 2 3 40
0.2
0.4
0.6
0.8
1ρ/ρ0=1 ρ/ρ0=0.17 ρ/ρ0=0.07
rpF
g(r)
Figure 4.2: Two-body correlation function g(r) as a function rpF for a system of A = 250identical particles at a temperature of T = 1 MeV for various densities. Only the spin-dependent part of the potential was used.
Table 4.1: Models parameters for the spin-independent term. Strength (in MeV) and rangeparameters (in fm2) for various components of the spin-independent potential.
Set a b c Λ1 Λ2 Λ3
I 108 -30 27 1.37 2.625 2.380II 102 -30 30 1.50 2.770 2.590
Fig. 4.3 displays the energy per particle versus density for symmetric nuclear matter
using both parameter sets for A=500 particles (N=Z=250). With these model parameters
nuclear saturation is achieved at a density of ≈ 0.15 fm−3 and has a binding energy/particle
≈ -18 MeV. For comparison, the energy per particle for the soft FSUGold model and the stiff
NL3 model [57, 58, 59] is also shown in Fig. 4.3. These two models based on the mean-field
approximation have been very successful in reproducing ground state properties (such as
binding energies, charge radii, separation energies, etc) for a variety of nuclei. In addition
to symmetric nuclear matter, we also performed simulations for pure neutron matter (N/A
53
0 0.25 0.5 0.75 1 1.25 1.5 1.75 2 2.25 2.5ρ/ρ0
-20
-10
0
10
20
30
40
50
E/A
(MeV
)
FSUNL3SM(set-1)SM(set-2)
Figure 4.3: Energy per particle of symmetric nuclear matter as a function of densities ρ/ρ0
for various parameter sets. The simulation was performed using A = 500 particles (Z/A =0.5) at a temperature of T = 1 MeV.
= 1) as shown in Fig. 4.4. Here we display the energy per neutron as a function of density
for A = N = 500 particles. Pure neutron matter is predicted to be unbound in all models.
Again for comparison, the energy per neutron for the soft FSUGold model and the stiff NL3
model is also shown. At high densities our results for symmetric nuclear matter show a close
resemblance to the soft FSUGold model, but for pure neutron matter the simulations tend
to resemble the stiff NL3 EoS. However, the behavior of nuclear matter at high densities is
largely unconstrained due to the limited amount of experimental data. Thus we will make
no attempt to discuss the properties of nuclear matter in the high-density region. Instead
we will limit our discussion to the region around saturation density and below.
We now proceed to analyzing the density dependence of the EoS. For symmetric nuclear
mater, the dynamics of the EoS around saturation density is controlled by the compression
modulus (K). Using Eq. (4.14) we evaluated the compression modulus for both parameter
sets. Our calculations yield K ≈ 249.3 MeV for the first parameter set and K ≈ 296.1 MeV
for the second parameter set. Hence the first parameter set yields a slightly softer EoS for
symmetric nuclear matter. For comparison the NL3 and FSUGold models give K = 271
54
MeV and K = 230 MeV respectively.
0 0.25 0.5 0.75 1 1.25 1.5 1.75 2ρ/ρ0
0
20
40
60
80
100
E/A
(MeV
)
FSUNL3NM(set-1)NM(set-2)
Figure 4.4: Energy per neutron of pure neutron matter (N/A=1) as a function of densityρ/ρ0 for various parameter sets. The simulation was conducted for A = 500 particles at atemperature of T = 1 MeV.
In the case of finite nuclei, the full Coulomb potential is included without the need for a
screening length. That is,
Vc(i, j) =e2
rij
τp(i)τp(j) . (4.22)
To evaluate the Fermi momentum for finite nuclei as required by Eq. (4.3d), we use a Woods-
Saxon density distribution , as explained in Appendix B. Again, Metropolis Monte Carlo
simulations (at T = 1 MeV) were used to compute the average potential energy for finite
nuclei. In Fig. 4.5 we plot binding energies for the ground state of several finite nuclei. Also
shown in the figure are the corresponding experimental values. Results from both models,
with the exception of a few finite nuclei, seem to reproduce the global trend of the binding
energy curve. For completeness, binding energies per nucleon for various parameter sets
have also been collected in Table (4.2). Also shown in the table are the predicted values
for the neutron skin and charge radii for several nuclei along with their experimental values.
55
0 25 50 75 100 125 150 175 200 225 250A
5
5.5
6
6.5
7
7.5
8
8.5
9
9.5
Ebi
nd/A
(MeV
)
ExpSet-1Set-2
Binding Energy per Nucleon for Finite Nuclei
12C
16O
20Ne
24Mg
40Ca
48Ca
62Ni90Zr 118Sn
150Sm
196Pt208Pb
232Th
238U
Figure 4.5: Binding energy per particle versus number of particles A for various nuclei.
Table 4.2: Binding energy per nucleon and charge radii for several nuclei from simulationsusing various parameter sets and the experimental data. In addition, predictions aredisplayed for the neutron skin of these nuclei.
Nucleus Observable Experiment NL3 FSUGold Set-1 Set-240Ca B/A(MeV) 8.55 8.54 8.54 8.10 8.43
Rch (fm) 3.45 3.46 3.42 3.48 3.58Rn-Rp(fm) – -0.05 -0.05 -0.07 -0.07
48Ca B/A(MeV) 8.67 8.64 8.58 8.21 8.57Rch (fm) 3.45 3.46 3.45 3.68 3.77
Rn-Rp(fm) – 0.23 0.20 -0.09 -0.0790Zr B/A(MeV) 8.71 8.69 8.68 8.61 8.87
Rch (fm) 4.26 4.26 4.25 4.34 4.53Rn-Rp(fm) – 0.11 0.09 0.18 0.20
208Pb B/A(MeV) 7.87 7.88 7.89 7.91 8.05Rch (fm) 5.50 5.51 5.52 5.67 5.98
Rn-Rp(fm) – 0.28 0.21 0.48 0.53
56
0 25 50 75 100 125 150 175 200 225A
0
0.1
0.2
0.3
0.4
S(fm
)
Set-1Set-2
Figure 4.6: Neutron skin versus total number of particles A for various nuclei.
The neutron skin thickness (S) of a nucleus is defined as the difference between the root-
mean-square radii of the neutron (Rn) relative to that of the proton (Rp) [60]. That is
S = Rn −Rp , (4.23a)
R2n =
1
N
N∑i=1
(ri −RCM)2 (4.23b)
R2p =
1
Z
Z∑i=1
(ri −RCM)2 , (4.23c)
where RCM is the center of mass coordinate.
The preliminary results for neutron-skin calculations are given in Fig. 4.6. The neutron
skin is strongly correlated to the slope of symmetry energy [61]. Increasing the slope of the
symmetry energy and thus the internal pressure of the system will result in a bigger neutron
skin. In our models, the second parameter set has a higher symmetry energy slope compared
to the first parameter set. Thus it explains why the neutron skin predicted with the second
parameter set is larger than with the first set. Next, we present a comparison between charge
radii obtained from the simulations to the experimental values in Fig. 4.7. The charge radii
57
0 25 50 75 100 125 150 175 200 225 250A
2
3
4
5
6
Rch
(fm
)
ExpSet-1Set-2
40Ca 48Ca
62Ni
90Zr
118Sn
150Sm
196Pt
208Pb232Th
238U
Figure 4.7: Charge radii versus total number of particles from simulations and the corre-sponding experimental data.
is defined as [62]
R2ch = (R2
p +R2P ) , (4.24)
where R2P is the mean square charge radius of an individual proton = 0.64 fm2 [63]. The
solid lines represent a best fit to the data. We find that our simulation results are in good
agreement with the experimental data.
We are also interested in using our simulations to obtain a best fit to the semi-
empirical mass formula. To implement the optimization, the amoeba subroutine from
Numerical Recipes [37] was used. The subroutine will find the best empirical parameters
(avol, asurf , aCoul, asym) by minimizing the Chi-squared function (χ2) such that
χ2 =1
A
∑ (Xi −Mi
Wi
)2
. (4.25)
Here Xi are the experimental values for binding energy, Mi are the numerically determined
values, Wi is the weight given to the experimental data, and A is the number of nuclei
used in the fit. Note that in this calculation the weight has been set equal to one. The
fitting results are shown in Table 4.3. The semi-empirical mass formula coefficients obtained
58
Table 4.3: Semi empirical mass formula best fit results for both parameter sets.
Set avol asurf acoul asym (MeV)I 16.80 22.54 0.72 25.67II 16.55 20.97 0.67 28.59
from both model parameters are in a good agreement with the ones obtained directly from
experimental data as given in Eqs. (4.17–4.20).
4.4.2 Two-body Correlation Function
In this section we present simulation results for nuclear pasta using the full Hamiltonian
described in Eqs. (4.2a and 4.2b) for both parameter sets given in Table (4.1). We perform
Metropolis Monte Carlo simulations at a fixed electron fraction Ye=0.2 and a temperature
of T=1 MeV. In core-collapse supernova the electron fraction starts at Ye around 0.5, but
drops with time, as the result of electron capture, to a smaller value of Ye of the order of 0.1.
Hence Ye = 0.2 is a good representation of a typical neutron-rich condition in the star. Since
the program is computationally intensive, in this preliminary study we run the simulations
for A = 1000 particles. At fixed Ye = 0.2, we run the simulations for 800 neutrons and 200
protons with equal number of spin-up and spin-down nucleons at a density of ρ = 0.025 fm−3,
which is typical for the pasta region. At this density the simulation volume has a length of
approximately L = 34.19 fm. As we have done before, we fix the electron screening length
arbitrarily to λ ' 10fm .
A sample configuration of A = 1000 particles at density ρ = 0.025 fm−3 is displayed in
Fig. 4.8. We found that all the protons and some of the neutrons are clustered into nuclei.
In addition there is a low density neutron gas between the clusters. At this density the
nuclei seem to cluster into cylindrical-like structures immersed in a dilute neutron gas. The
most interesting part of the simulations is that the decision of whether nucleons cluster in
nuclei or remain in the gas is being answered dynamically. As a comparison, if we lower the
density from ρ=0.025 fm−3 to ρ=0.01 fm−3, as is shown in the right hand side of Fig. 4.8, we
see that the system transforms into more conventional spherical shape neutron-rich nuclei
immersed in a dilute neutron gas.
59
Figure 4.8: Monte Carlo snapshots of a configuration of N= 800 neutrons and Z= 200 protonsat density 0.025 fm−3 (left) and 0.01 fm−3 (right). The following color code has been used:neutron spin up (tan); neutron spin down (white); proton spin up (blue),proton spin down(red)
The two-body neutron-neutron correlation function g(r) is shown in Fig. 4.9. The
neutron-neutron correlation function is measured by computing the relative distance between
neutron-neutron pairs. At short distances the correlation function is very small due to the
hard core in the NN potential. The two-body correlation function shows large broad peaks
between r = 2 fm and r = 6 fm. These peaks correspond to the other neutrons with the
same spin bound in the same and other clusters. Superimposed on this broad peak we find
three sharp peaks corresponding to the nearest, second-nearest, and third-nearest neighbors.
At larger distances between 8 and 15 fm, the correlation function shows a modest dip below
one, suggesting that the attractive NN interaction has shifted some neutrons from larger to
smaller distances to form the clusters.
The two body correlation function for proton-proton and neutron-proton are shown in
Fig. 4.10 and Fig. 4.11. Again the correlation function is measured by computing the relative
distance between proton-proton pairs and proton-neutron pairs respectively.
60
0 2 4 6 8 10 12 14 16r(fm)
0
0.5
1
1.5
2
2.5
3
g(r)
Set-1Set-2
Figure 4.9: Neutron-neutron two-body correlation function at a temperature of T = 1 MeV,an electron fraction Ye = 0.2, and density ρ = 0.025 fm−3.
0 2 4 6 8 10 12 14 16r(fm)
0
1
2
3
4
5
6
7
8
9
10
g(r)
Set-1Set-2
Figure 4.10: Proton-proton two-body correlation function at a temperature of T = 1 MeV,an electron fraction Ye = 0.2, and density ρ = 0.025 fm−3.
61
0 2 4 6 8 10 12 14 16r(fm)
0
1
2
3
4
5
6
7
8
9
g(r)
Set-1Set-2
Figure 4.11: Proton-neutron two-body correlation function at a temperature of T = 1 MeV,an electron fraction Ye = 0.2, and density ρ = 0.025 fm−3.
62
CHAPTER 5
CONCLUSIONS
5.1 Introduction to Physics of Compact Stars
Chapter 2 has been written in the spirit of giving students a pedagogical introduction to
the study of compact stars particularly white dwarfs and neutron stars. Students learned
several important lessons from this project. One of them relates to the usefulness of scaling
the equations. Without scaling, the problem would have been unsolvable. This is due to
the tremendous range of scales encountered in this problem; there are more than 60 orders
of magnitude between the minute electron mass and the immense solar mass. Another
important lesson learned is that, contrary to what seems to happen in the classroom, most
problems in physics have no analytic solution. Thus, numerical analysis is a necessary step
towards a solution.
In this project we relied on the free Fermi gas equation of state in calculating the
properties of white dwarfs and neutron stars. While the free Fermi gas EoS seems to be
sufficient for white dwarf stars leading to the Chandrasekhar upper mass limit, the same
conclusion can not be obtained for neutron stars. The structure of neutron stars poses
several additional challenges. First, Newtonian gravity must be replaced by general relativity.
This implies that the structure equations must be replaced by Tolman-Oppenheimer-Volkoff
equations. Second at higher densities encountered in the interior of neutron stars, the
equation of state receives important corrections from the interactions among the neutrons.
Thus, Pauli correlations are no longer sufficient to describe the equation of state. This is
evident in the upper mass limit for neutron stars which yield too small a mass at around 0.7
solar mass. In reality we have seen stars bigger than this limiting mass. In fact neutron stars
have been observed to have mass around 1–3 solar mass. Therefore, corrections from nuclear
interactions are crucial in the study of neutron stars. This is a topic of intense research
63
activity in our effort to understand the physics of neutron stars and the structure of exotic
compact objects known as hybrid and quark stars.
5.2 Virtues and Flaws of the Pauli Potential
In Chapter 3 we have conducted a systematic study of the standard version of the Pauli
potential. While simple and widely used, such a version fails to reproduce some of the most
basic properties of a free Fermi gas. We found that the constraints imposed by such a Pauli
potential, namely, the suppression of phase-space configurations for having two fermions with
both positions and momenta similar to each other, are too weak to faithfully reproduce some
basic properties of a free Fermi gas. By examining the well-known behavior of the Slater
determinant we suggest that phase-space configurations should be suppressed when either
the positions or the momenta of the fermions are close to each other. By incorporating these
features into a new form of the Pauli potential — and by carefully tuning the parameters
of the model — the momentum distribution and the two-body correlation function of a free
Fermi gas were accurately reproduced.
This new version — inspired by the properties of a Slater determinant — generated
accurate (canonical) momentum distribution and two-body correlation functions, while
avoiding crystallization. However, in the course of this study a pathology that is generic to
all momentum-dependent Pauli potential in conjunction with the Hamiltonian approach was
uncovered. The momentum distribution generated via Monte Carlo (or other) methods may
differ significantly from the resulting “kinematical” momentum distribution. This suggests
that while the kinetic energy of the free Fermi gas (computed from the canonical momenta)
may be accurately reproduced, the distribution of velocities may be grossly distorted. Indeed,
we found a distribution of velocities that significantly under-estimates — by a factor of 10
— that of a free Fermi gas. Such “sluggishness” among the particles could have disastrous
consequences by inducing artificial ordering in the system (e.g., “crystallization”). The
possible appearance of artificial long-range order in the system must be examined on a case
by case basis. For example, with the standard version of the Pauli potential [45] the system
displays an anomalous two-body correlation function suggestive of crystallization. On the
other hand, the Pauli potential introduced in this work faithfully reproduces the two-body
correlation function of a free Fermi gas. However, the possibility for generating artificial
64
correlations in the system (e.g., crystallization) remains large.
To avoid this problem, we will resort to using a momentum-independent Pauli Potential.
Consequently, the potential will not generate the correct momentum distribution and kinetic
energy of a free Fermi gas. However, we are still able to impose the correct two-body
correlation function with this new potential.
5.3 Equation of State for Nuclear Pasta
In the previous work by Piekarewicz and collaborators [32, 33], the spin-independent (vector
part) of the weak neutral-current neutrino pasta scattering has been studied via a spin-
independent semi-classical model. However, due to its spin-independent properties this
model is not sufficient in calculating the axial-vector (spin) response. Thus, a new model
that incorporate the spin-dependent is needed. The search for a new-modified model is the
main focus of this chapter.
In Chapter 4 we have employed a semi-classical model to simulate the dynamics of the
pasta phase of neutron-rich matter. Despite its simplicity, the model describes the crucial
physics of nuclear pasta that is the interplay between short-range nuclear attraction and long
range Coulomb repulsion. Apart from these features, the model has also incorporated the
fermionic correlations via a classical Pauli potential as discussed in Chapter 3. Using the
Metropolis Monte Carlo algorithm we obtained two best-fit parameter sets that satisfactorily
reproduced various bulk properties of symmetric nuclear matter: such as the saturation
density, the binding energy per nucleon at saturation, and the compression modulus, and
properties of several finite nuclei. The new parameter sets also captured the clustering of
symmetric nuclear matter below saturation density, a phenomena typical of nuclear pasta.
With the newly refined potential, we are interested in studying how the nuclear pasta
affects the neutrino transport which is a crucial factor in the core-collapse of a supernova. We
evaluated the two-body correlation function for nuclear pasta via Metropolis Monte Carlo
simulations. Although it is beyond the scope of the current dissertation, we are currently
working on the spin-dependent weak neutral current neutrino responses leading to the study
of neutrino transport properties in nuclear pasta.
65
APPENDIX A
Monte Carlo Simulation
A.1 The estimator
In our work we need to calculate the expectation value of energy, that is
〈E〉 =
∑µEµe
−βEµ∑µ e
−βEµ. (A.1)
However, the ideal route to calculate the expectation value by averaging over all states µ,
is impossible for a large system. Thus we resort to using Monte Carlo techniques. These
techniques work by choosing a subset of states µ –e.g. (µ1 ...µM)– at random based on some
probability distribution pµ which will be discussed later. The expectation energy 〈EM〉 for
this subset of states becomes
〈EM〉 =
∑Mi=1Eµip
−1µi e
−βEµi∑Mj=1 p
−1µj e
−βEµj
, (A.2)
where 〈EM〉 represents the estimator of 〈E〉, which in the limit M →∞, 〈EM〉=〈E〉.The next question is how to weight the states M such that EM will be an accurate
estimate of 〈E〉. This is related to choosing the pµ distributions that represents the correct
distribution of the system. For a classical simulation the best choice for pµ is the Boltzmann
probability distribution, that is
pµ = Z−1e−βEµ . (A.3)
Substituting Eq. (A.3) into Eq. (A.2) yields
〈EM〉 =1
M
M∑i=1
Eµi . (A.4)
The tricky part in Monte Carlo simulation is generating an appropriate random set of
states according to the Boltzmann probability distribution. All Monte Carlo simulations rely
66
on Markov processes to generate the random set of states. A Markov process is a mechanism
which, given a system in one state µ, generate a new state ν in a random fashion. We will
use a Markov process repeatedly to generate a Markov chain of states in such a way that
eventually the simulation will produce a succession of states with the desired Boltzmann
probability distribution. We call the process of reaching the Boltzmann distribution
A.2 The Metropolis Algorithm
One type of Markov chain Monte Carlo techniques is the Metropolis algorithm. To generate
Eµi in Eq. (A.4) using Metropolis algorithm we rely on the following procedures. Suppose
the simulation starts at an arbitrary point Xn. To generate Xn+1, we make a trial move to
a new point Xt. There is no strict rule on how to choose the new point. It can be chosen,
for example, uniformly at random within a multi dimensional cube of a small side δ about
Xn. The trial move will then be accepted or rejected according to the ratio
R =p(Xt)
p(Xn)= e−β(E(Xt)−E(Xn)) (A.5)
If R is larger than 1, then the move is always accepted and we let Xn+1 = Xt. If R is less
than 1, then the following rules apply. If R is bigger than η, an arbitrary random number
uniformly distributed in the interval [0,1], then the move is accepted. Otherwise the move
is rejected, and we have Xn+1=Xn. This procedure generates Xn+1, and we can generate
the next step by following the same process. Notice that by doing this selection, we always
accept transition to a new state which has lower or equal energy to the present state. If
the new state has a higher energy than the present state, then we may accept it with the
probability given above.
The next important question is how to efficiently choose the step size (δ). To answer this
question, suppose Xn is at the maximum of p, the most likely place to be. If δ is large, then
p(Xt) will be very much smaller than p(Xn) and most trial moves will be rejected. However,
if δ is small, most trial moves will be accepted, but the random walker will not move very
far. This will lead to a poor sampling of the distribution. Therefore, a good rule of thumb
is that the size of the trial step should be chosen so that about half of the trial steps are
accepted [64].
67
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
# of thermalization step
1
2
3
4
5
6
7
E/A
(MeV
)
<EP>/A<EK>/A<ET>/A
Figure A.1: Monte Carlo thermalization step vs energy per nucleon for symmetric nuclearmatter with A=500 particles at T=1 MeV
A.3 Equilibration
Once the selection process has taken place, the next question is when we should start
calculating our estimator EM . To answer this question we have to perform two computational
steps. The first step is the equilibration step. In the equilibration step, we run our simulation
for a suitably long period of time until the system comes to equilibrium. Then we measure
the quantity we are interested in –in our case the Energy– over another suitable long period
of time and average it, to evaluate the estimator in Eq. (A.4).
Fig. A.1 shows one of our equilibration time for symmetric matter simulations with
A=500 particles at density ρ 0.4 fm−3 and temperature T=1 MeV. By looking at the plot
we can guess that the system has reached equilibrium at around Monte Carlo step t ∼ 4000.
Up until this point the energy keeps changing, but after this point it just fluctuates around a
steady average value. However, it many cases it is possible that the system gets stuck in some
meta-stable state, and gives a roughly constant values for all quantities we are observing for
a while. We may mistake this local energy minimum–in which the system stays temporarily–
for the global energy minimum. Thus cautionary steps must be taken to avoid this trap[65].
68
APPENDIX B
Woods-Saxon Potential
A good representation of finite nuclei density is provided by the Woods-Saxon model, that
is
ρ(r) =ρ0
1 + exp[(r −R)/β], (B.1)
where ρ0 is the saturation density ≈ 0.16 baryon/fm−3, β is related to the width of the edge
region and is fixed at 0.55 fm. R is the mean nuclear radius and its value (R) is given by
normalizing the total number of baryons (A), such that
A =
∫4πρr2dr . (B.2)
Fig. B.1 displayed typical shapes for Wood-Saxons potential. It is evident from the figure
that even though larger nuclei have a larger mean diameter, the edge (β) region has a similar
width in all nuclei.
We are interested in evaluating the Fermi momentum (pF ) for finite nuclei as required
by Eqs. (4.21a and 4.21b). To do so, information on average density is needed. We use the
Woods-Saxon density distribution to calculate the average density, i.e.
ρavg =1
A
∫4πρ2r2dr . (B.3)
Once the value of average density is known, we can obtain the Fermi momentum by using
the following relation
pF =(3π2ρavg
2
)1/3
. (B.4)
Table (B.1) shows the finite nuclei and their corresponding Fermi momentum.
69
0 2 4 6 8 10 12 14 16 18 20r(fm)
0
0.05
0.1
0.15
0.2
ρ(fm
-3)
40Ca118Sn208Pb
ρ0
Figure B.1: Typical shapes for Woods-Saxon potential.
70
Table B.1: Table of nuclei and their Fermi momentum.
Nuclei Z N A pF (MeV)
12C 6 6 12 197.33016O 8 8 16 207.01320Ne 10 10 20 211.16224Mg 12 12 24 214.33340Ca 20 20 40 222.28648Ca 24 24 48 224.81362Ni 28 34 62 228.09990Zr 40 50 90 232.391118Sn 50 68 118 235.176150Sm 62 88 150 237.429194Pt 78 116 194 239.640196Pt 78 118 196 239.724208Pb 82 126 208 240.205232Th 90 142 232 241.062238U 92 146 238 241.257
71
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76
BIOGRAPHICAL SKETCH
Jutri Taruna
Education
• 2001-2008 Ph.D. in Theoretical Nuclear Physics, Department of Physics, Florida State
University.
• 1997-2000 M.Sc. in Theoretical Nuclear Physics, Department of Physics, University of
Indonesia.
• 1989-1995 B.Sc. in Physics Education, State University of Jakarta.
Experience
• 2003-2008 Research Assistant, Department of Physics, Florida State University.
• 2001-2007 Teaching Assistant, Department of Physics, Florida State University.
Publications
1. Taruna, J. and Piekarewicz, J. and Perez-Garcia, M. A., Virtues and Flaws of the Pauli
Potential, J. Phys. A: Math Theor 41 035308 (2008).
2. Jackson, Chris B. and Taruna, J. and Pauliot, S. L. and Ellison, B.W. and Lee, D.
D., and Piekarewicz, J., Compact Objects for Everyone: I. White dwarf stars, Eur. J.
Phys. 26 695 (2005).
77