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IMPROVING MICROSCOPIC MODELS OF THE NEUTRON STAR CRUST A Thesis by SHUXI WANG Submitted to the Office of Graduate Studies of Texas A&M UniversityCommerce in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE August 2015 IMPROVING MICROSCOPIC MODELS OF THE NEUTRON STAR CRUST A Thesis by SHUXI WANG Approved by: Advisor: William Newton Committee: Kent Montgomery Kurtis Williams Head of Department: Matt A. Wood Dean of the College: Brent Donham Dean of Graduate Studies: Arlene Horne Copyright © 2015 Shuxi Wang iv ABSTRACT IMPROVING MICROSCOPIC MODELS OF THE NEUTRON STAR CRUST Shuxi Wang, MS Texas A&M UniversityCommerce, 2015 Advisor: William Newton, PhD The task of understanding the properties of neutron stars has attracted the attention of researchers in different fields, such as astronomy and nuclear and condensed matter physics. In this thesis, we focus on a region of matter called the nuclear “pasta phases,” exotic nuclear geometries which occur at the transition from the bottom of the inner crust to the superdense core of the neutron star. There are two simulation models applied in this work; one is the Three Dimensional Hartree Fock (3DHF) model, and the other is the Compressible Liquid Drop Model (CLDM). The Hartree Fock method is the most accurate way to simulate nuclear matter in recent times but is timeconsuming and contains spurious numerical effects. On the other hand, the Liquid Drop Model is efficient but oversimplifies the physics. The CLDM method treats nuclear matter as a “droplet” of nuclear matter, which loses sight of the behavior of each nucleon and the interaction between v them. The 3DHF model provides a more microscopic description of crustal matter in the neutron star. In this thesis we aimed to combine the advantages of both methods. We aimed to (i) devise a way of subtracting the spurious numerical effects from the results of 3DHF calculations, and then (ii) fit the results from the CLDM model to those of the 3DHF model by varying the parameters that determine the surface tension of nuclei in the model. This allowed the CLDM to accurately predict the composition of the inner crust and the 3DHF model to do precise calculations of crustal properties given that composition. vi ACKNOWLEDGEMENTS For the thesis to be completed, I gained the continued support from my advisor: Dr. William Newton. His guidance helped me in all the time of research and writing thesis. I would like to express my sincere thanks to him. Besides this, my sincere appreciation goes to the rest of my committee: Dr. Kent Montgomery and Dr. Kurtis Williams, for their patience and suggestions they offered me. I also take this opportunity to express gratitude to all of the department faculty and staff for their help and support. In addition, I would like to thank my parents and my friends; their support has kept me on the way to pursue the goals in my life. vii TABLE OF CONTENTS LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . x CHAPTER 1. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . 1 The CrustCore Transition and Nuclear Pasta . . . . . . . 2 Models of The Inner Crust and Nuclear Pasta . . . . . . . 5 Model Inputs . . . . . . . . . . . . . . . . . . . . 6 Model 1: The Compressible Liquid Drop Model (CLDM) . . . . . . . . . . . . . . . . . . . . . . . 8 Model 2: ThreeDimensional HartreeFock (3DHF) Method . . . . . . . . . . . . . . . . . . . . . . . 9 Purpose of Thesis . . . . . . . . . . . . . . . . . . . . . . 12 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . 15 2. COMPUTATIONAL METHOD . . . . . . . . . . . . . . . . . 16 3D HartreeFock Method . . . . . . . . . . . . . . . . . . 17 CLDM . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 Instrumentation and Collection of Data . . . . . . . . . . 20 Treatment of Data . . . . . . . . . . . . . . . . . . . . . 21 3. RESULTS AND DISCUSSION . . . . . . . . . . . . . . . . . 22 Data Analysis . . . . . . . . . . . . . . . . . . . . . . . . 22 viii 4. CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . . . 45 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . 45 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . 47 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . 47 REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 VITA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 ix LIST OF TABLES TABLE 1. Skyrme interactions used in this work . . . . . . . . . . . . . . . 22 x LIST OF FIGURES FIGURE 1. [SLy4, nb = 0:08, yp = 0:036] etot vs A. The dashed line is the result assuming the matter is at uniform density; the fact that is is slightly higher is due to the fact that matter isn’t quite uniform at this density. . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2. [SLy4, nb = 0:08, A=1000] Integrated number density over Zaxis. 25 3. [SLy4, nb = 0:04, yp = 0:03] etot vs A . . . . . . . . . . . . . . . 26 4. [SLy4, nb = 0:04]Integrated number density over Zaxis . . . . 27 5. [SLy4, nb = 0:04, yp = 0:03] average number density of unbound neutrons in the unit cell n vs A . . . . . . . . . . . . . . . . . . 29 6. [NRAPR] Fitted espurious vs nb with A=200 and 400 . . . . . . . 30 7. [NRAPR] Fitted espurious vs nb with A=600 and 800 . . . . . . . 31 8. [NRAPR] Fitted espurious vs nb with A=1000 and 1200. . . . . . 32 9. [NRAPR] Fitted espurious vs nb with A=1400 and 1600 . . . . . . 33 10. [NRAPR] Fitted espurious vs nb with A=1800 and 2000 . . . . . . 34 11. [NRAPR, nb = 0:03] etot and e0 vs A . . . . . . . . . . . . . . . 36 12. [NRAPR, nb = 0:04] etot and e0 vs A . . . . . . . . . . . . . . . 37 13. [SkIUFSUL30] Intergrated number density over Zaxis . . . . 38 14. [SkIUFSUL30] Intergrated number density over Zaxis . . . . 39 15. [SkIUFSUL90] Intergrated number density over Zaxis . . . . 40 16. [SkIUFSUL90] Intergrated number density over Zaxis . . . . 41 xi 17. [SkIUFSUL30] Amin vs nb from CLDM and 3DHF . . . . . . . . 43 18. [SkIUFSUL90] Amin vs nb from CLDM and 3DHF . . . . . . . . 44 1 Chapter 1 INTRODUCTION As one of the remnants of the collapse after supernovae, a neutron star is made up of superdense nuclear matter. The mass of a neutron star is between 1.4 to 2 solar masses, and the radius of it is about 12 to 13 kilometers (Kiziltan, 2011). Based on the current understanding of nuclear physics, the structure of a neutron star has been hypothesized as consisting of different layers according to the mass density of matter. Considering a neutron star with zero temperature, the surface of the neutron star at zero pressure will consist of an iron lattice. The layer of the neutron star from the surface down to a mass density of about 1011g=cm3 is referred to as the outer crust. In the outer crust, inverse beta decay (the capture of electrons by protons resulting in a neutron and neutrino) is allowed to occur, which results in the creation of more neutronrich nuclei. As the density increases in the inner crust, there will be increasingly more neutrons than protons in the nuclei, and the nuclei will become larger with weaker bound neutrons. When the ratio of neutrons to protons in the nuclei reaches a critical level, some neutrons become unbound, and neutrons “drip” out from the nuclei. This occurs at a density of about 1011g=cm3 and marks the start of the inner crust, which extends down to a mass density of around 1014g=cm3. The bottom part of this layer will be our focus. In the inner crust, there are two phases of nuclear matter coexisting: free neutrons and nuclei. Between 1 2 1014g=cm3 up to 1015g=cm3 is the core of the neutron star 2 (Newton, 2007). In the core region, the nuclei merge into a fluid of neutrons and protons. In addition, there are muons, and possibly pions and kaons, existing in the core region. Furthermore, at the highest densities, the matter in the core region might contain quarkgluon matter. Our knowledge of the core region is limited, observational data is scant, and we currently do not have a good model to describe the matter at these high densities. It should be noted that we used the number density to refer to the density at any point in the neutron star. Number density is simply the division of the total number of particles  in our case the neutrons and protons  by volume. A massdensity of 2:5 1014g=cm3 is equivalent to a number density of 0.16 fm3. This is the density inside terrestrial nuclei. The neutron star was first observed by Bell, Hewish et al. in 1967 as a “pulsar,” which was detected as a periodic source of radio waves with period of 1.337 s (Hewish et al., 1968). In the following year, the Crab pulsar was discovered with a shorter period 33 ms (Staelin & Reifenstein, 1968). The CrustCore Transition and Nuclear Pasta At the bottom of the inner crust region are the “pasta phases.” These are a variety of complex structures of nuclear matter, with the geometrical structures of nuclear matter resembling different types of pasta (Pons et al., 2013). In the inner crust, with increasing density, nucleons are expected to 3 arrange themselves into the following five shapes: spheres (ordinary nuclei), cylinders (spaghetti), slabs (lasagna), cylindrical holes (tubes), and spherical holes. These basic shapes of subnuclear matter were first established by Ravenhall et al. (1983), Hashimoto et al. (1984), and Oyamatsu et al. (1984), using a Liquid Drop Model (LDM) framework. These configurations develop independently of the nuclear force used, which means these shapes are from purely geometrical considerations. Astronomers collect observational data on several different neutron star phenomena which have the potential to probe the properties of the pasta phases. During the process of neutron star cooling, there is a cooling wave that moves towards the surface of the neutron star from the core. During this time, the effective surface temperature keeps constant. As the cooling wave arrives at the surface, the temperature drops sharply, from 250 eV to 30100 eV. The magnitude and duration of the drop are sensitive to the microscopic properties of crust matter, such as its heat capacity, thermal conductivity, and the superfluidity of the free neutrons (Gnedin et al., 2001), all of which could be different in the pasta phases. Neutron stars enter a spindown phase for the million years after their birth and are observed to exhibit glitches in the measurements of the spin down rate. In order to explain the glitches, researchers need to study the properties of nuclei and pasta in the inner crust and their interaction with superfluid 4 neutrons and vortices (Crawford et al., 2003; Horvath, 2004; Larson & Link, 2002; Newton, 2007). Recently, a nuclear physics group at Indiana University, Bloomington with their collaborator in Canada, presented their research outcomes on disordered nuclear pasta (Horowitz et al., 2015). They used the large scale molecular (MD) simulations and showed the existence of longliving topological defects in pasta regions. These defects could increase electron scattering and lead to a decrease in both thermal and electrical conductivities. In their MD simulations, topological defects are formed in different shapes, which always quickly assemble and stay for a long time. Then the researchers determined the effect of these defects on thermal conductivity by defining an effective impurity parameter Qimp, which is estimated to be around 40 in their paper (a number bigger than 1 indicates that matter is quite disordered). After inserting the resulting low conductivity pasta layer in their model with a bigger Qimp, the results were able to better satisfy the late time Chandra observation for the temperature of MXB 165929, which is a neutron star in a lowmass Xray binary. Mature neutron stars in lowmass binaries can easily accrete enough matter to replace its entire crust (Brown, 2000). The new crust will be melted by heat diffusion at the surface. Also, the Xradiation from the crust could help us constrain the thermal conductivity in this region. Furthermore, the accreted 5 material, which is distributed inhomogeneously on the surface of the neutron star, becomes a gravitational wave source (Haskell et al. (2006); Haensel & Zdunik (2003)), the strength of which depends on the size of the “mountain” which builds up on the surface of the star from the accreted material. The gravitational waves should then be sensitive to the mechanical properties of the subnuclear matter in the inner crust, and the examination of these should help determining the properties of the crust and, in particular, the pasta phases (Andersson & Kokkotas, 2005). By studying these observational results, we can learn more about the properties of crustal matter, but to do so we need accurate models of crustal matter, and the pasta phases in particular. Models of The Inner Crust and Nuclear Pasta In this thesis, we focused on the bottom layer of the inner crust, where the pasta phases are expected. The nuclear matter in the inner crust is assumed to have periodic structure, which allowed us to identify a repeating unit cell. The study of one unit cell could then establish the physical properties of the layer in the inner crust composed of that unit cell. At a given density, the composition of the crustal lattice is characterized by variables such as the number of total nucleons in one unit cell, proton fraction, and the size of the nucleus or pasta structure within the unit cell. Identifying these quantities allow determination of the total energy of the unit cell and structure of the geometrical shape of the nuclear matter in the unit cell. 6 In order to compute these properties, we used two different models of the unit cell of matter. Both models use the same fundamental physical principle: the actual structure of matter at a given density is the one that minimizes the total energy density of matter. The two models are: The Compressible Liquid Drop Model (CLDM) and ThreeDimensional HartreeFock (3DHF) method. Model Inputs Both models require a type of the interaction between neutrons and protons as input. In this thesis, we use the Skyrme interaction. The Skyrme interaction is a type of nonrelativistic effective potential which was firstly proposed by Skyrme and then applied on the calculation of finite nuclei by Vautherin and Brink (1972). Currently, there are over 200 parameterizations in Skyrme interaction that are fitted to results of different experiments. The main uncertainty in the Skyrme model is from the symmetry energy, which is the change in nuclear energy associated with changing neutronproton asymmetry: it means as the ratio of neutrons to protons goes away from the symmetry point (equal neutrons and protons), the binding energy of nuclear matter raises as a result of the increase of symmetry energy. The basic input for our models is the energy function of number density and proton fraction  E(n; yp). Generally, it can be expanded: 7 E(n; yp) = E0(n) + S(n) 2 + (1.1) S(n) = J + L + 1 2 Ksym 2 + (1.2) The symmetry energy is calculated from the second order differential of total energy and is 1 2yp: S(n) = 1 2 @2E @ 2 (1.3) The second term in equation 1.2 gives the most uncertain part of symmetry energy, which is denoted by L. L = nn0 3n0 @S(n) @n and is named the slope of symmetry energy; n0 is nuclear saturation density and equal 0.16 fm3. Saturation density is the baryon number density where the infinite symmetric nuclear matter has the minimal energy. Also, at the saturation density, the nuclear force that binds nucleons together is saturated. L can only be constrained by experiments to between 30 and 100 MeV. Recently, researchers found an anticorrelation between L and crustcore transition density (Oyamatsu & Iida, 2007). For larger L, they found that no “pasta” appears; for smaller L, the various “pasta” phases emerge with increasing number density. The Skyrme interaction gives us an expression for the energy density above as discussed in Chapter 2. 8 Model 1: The Compressible Liquid Drop Model (CLDM) The CLDM treats nuclei as a droplet filled with homogenous nuclear matter. The droplet has surface with an associated surface energy and is surrounded by free neutrons in the unit cell. The CLDM was first proposed by Baym et al. (Baym et al., 1971) as a model to describe nuclear matter in the inner crust of the neutron star. In the CLDM, nuclear matter in this region is described as the sum of the nuclear component, the free neutron component (both treated as fluids), and a surface separating them. As density increases in the crust, the inhomogeneous nuclear matter becomes homogeneous; the presence of the free neutron gas reduces the surface energy of nuclei, and the Coulomb repulsion between nuclei becomes significant when the distance between nuclei is in the same order with nuclei radius. Then CLDM is improved to take spherical, rodlike, slablike nuclei, as well as cylindrical and spherical nuclear bubbles into account (Watanabe et al., 2000). Nuclear physics focuses on the properties of the neutron star inner crust, which are provided by the computation of the Compressible Liquid Drop Model (CLDM) (Newton et al., 2013). A comprehensive survey of the inner crust using the CLDM was undertaken by researchers at TAMUCommerce in 2012. They explored the impact of experimental and theoretical uncertainties of symmetric 9 nuclear matter EoS and the symmetry energy and strength of surface energy, t on the crustal composition. In this thesis, we will use the CLDM developed in the latter work. A calculation minimizes total energy density, written as energy of nuclear matter in droplet, surface energy, and energy of free neutrons. At each density, the calculation can produce the radius of the nucleus and proton fraction, as well as the total nucleons, of the unit cell. The CLDM requires parameters specifying the strength of the surface and curvature tensions in addition to the nuclear interaction, in contrast to the 3DSHF method discussed next, which calculates the surface energies from the nuclear interaction input. The CLDM is a fast method for computing the structure of the inner crust, but its semiclassical nature means that it misses a description of the essential physical effects necessary for calculating the properties of crust matter, including shell effects. Model 2: ThreeDimensional HartreeFock (3DHF) Method The 3DHF model describes the nuclear matter in the unit cell through solving the Schrodinger equation for each nucleon in the cell. The solution is simplified by assuming that the total manybody wavefunction in the unit cell can be written as a Slater determinant. This is the HartreeFock approximation. A Slater determinant is an expression that describes the wavefunction of a multifermionic system that satisfies antisymmetry requirements and the Pauli 10 principle by changing sign upon exchange of two electrons (Atkins & Friedman, 2011). In the calculation, at a given density, we obtain the energy for one unit cell. We have to run the calculation over a range of nucleon number A (corresponding to cell size) and yp and to find the one that gives minimum energy. In this model, our unit cell is a cubic WignerSeitz (WS) cell. The WignerSeitz cell is a type of unit cell used in the study of crystalline structure in solid state physics. In general, the unit cell will have a complicated shape. The WS cell is an approximation to that using a simpler shape. The work published by P. Magierski and P. H. Heenen (2002) was the first to use the 3DSHF method. They demonstrated the shell effects associated with unbound neutrons in the inner crust of a neutron star. As a result, their study suggests that the inner crust has a quite complicated structure led by the shell effects, and the number of phase transitions may increase since the same phases may appear in different density ranges. Also, they claim that several phases from different lattice types could coexist in the crust, which is much different from liquiddropbased approach. Finally, they conclude the inner crust may be close to a disordered phase(Magierski & Heenen, 2002). A similar method was used to investigate the existence of the “pasta phases” in 2007 (Gögelein & Müther, 2007). They also used the relativistic meanfield method with ThomasFermi approximation as comparison in their work. Their calculation includes a pairingcorrection, which is evaluated by the 11 standard BCS theory expressing the tendency of neutrons and protons to pair up. W. Newton and J. Stone presented their work in the paper published in 2009 (Newton & Stone, 2009). They used the 3DHF model, which takes Skyrme interaction and BCS theory into account. Comparing former researches, this work extended the simulation for inhomogeneous nuclear matter to a lager range of temperature, baryon number density and proton fraction. Their work focused on the how inhomogeneous nuclear matter transitioned to uniform nuclear matter with increasing temperature and density. They also discussed the geometry and thermodynamical properties of nuclear matter. Furthermore, the researchers explored spurious shell effects and paid attention to the local minima in the energydeformation surface. Currently, the 3DHF method, which takes a microscopic description of each nucleon into account, is the most accurate approach. However, it is timeconsuming (a typical run lasts up to 24 hours)  it needs much more time and computational resources than the calculations of CLDM, which limits its application. In the calculation, we have to run lots of times at different nucleon numbers and protons fractions at each density to find the minimum energy configuration. It also has spurious numerical effects, the biggest problem being the spurious shell effect. Shell effects arise when the Schrodinger equation is solved for a finite system  e.g. if the Schrodinger equation is solved for a 12 Hydrogen atom, the electron energies will be arranged in shells, which we know from chemistry. These are real shell effects. In our case, the free neutrons in the unit cell are not finite in real life  they‘re free to move throughout the crystal lattice. But we solve the Schrodinger equation for the free neutrons in the finite unit cell in the 3DHF model. This gives rise to spurious shell effects  the free neutron energies are arranged in shells, when they shouldn‘t be. The spurious shell effects show up as fluctuations when the total energy is plotted versus the number of nucleons per unit cell. Since we are using the fundamental principle that everything is organized to minimize the total energy, the physical minima of matter at a given density is obscured by the spurious shell effects. Therefore, they must be removed. Purpose of Thesis The aim of this thesis was to attempt to combine the advantages of both CLDM and 3DHF models (the speed of the former and the more physical description of the latter). In order to do this, we tried two things. The first was to test one method of eliminating spurious shell effects from the results of 3DHF calculations. This allowed us to predict the energetically preferred nucleon numbers and proton fractions at a particular number density in the inner crust of the neutron star. Based on the results from the former problem (the getting rid of spurious shell effects from 3DHF), we adjusted the parameters in CLDM which determine surface energy, to produce the same variables, like the 13 number density of neutrons, total number density, and proton fraction as 3DHF calculations. We were then able to use the CLDM method to quickly figure out the nucleon number and proton fraction corresponding to the minimum energy at a given density, utilizing the timesaving benefits which that method offers, and then using the 3DHF model at that nucleon number density and proton fraction to calculate accurately the properties of matter in that configuration. This cond The method we used to eliminate the spurious shell effect was as follows. We aimed to find out how the spurious effects have an influence on uniform matter (which exists at higher densities than the crustcore transition density), where we could calculate the exact energy semianalytically in a way that was free of spurious shell effects. The difference between 3DHF calculations of uniform matter and the exact result gave us the size of the spurious shell effect. This was extrapolated down in density to inhomogeneous matter and subtracted from the results in the region of densities where the pasta phases existed. The two different models of the inner crust, CLDM and 3DHF, were implemented in two different codes. In addition, a third code was used to calculate the energy density of uniform nuclear matter. The data were generated by the 3DHF and the CLDM computational models, respectively, implemented as computer codes. Analysis of the data collected from these numerical simulations was analyzed and some basic linear 14 fitting to data was carried out. In order to remove spurious shell effect, we needed to find some functions or quantities to define the relation between that effect with density and proton fraction. In order to improve the CLDM, we needed to adjust factors in the computational model of the CLDM to produce the results which could satisfy the data generated by the 3DHF model. The following procedure was done: 1. We collected the data generated by the 3DHF model. This included the spurious shell effect which is shown as a fluctuation when the energy densities with varying nucleon numbers are plotted. As the spurious shell effect impedes the definition of the minimal energy, which is the way we find how matter to be arranged in nature, we sought to remove its effect from the data. 2. To remove the spurious shell effect, we calculated the properties of uniform nuclear matter. We used an exact calculation of the energy density of uniform matter and subtracted these from the 3DHF model‘s results. The differences of these subtractions defined the spurious shell energy. The spurious energies have roughly linear relations with number densities  using this relation we were able to predict the size of the spurious energies in the pasta phases of the inner crust and subtract them away, leaving behind only the physical energies, and an easier determination of the minimum energy configuration. 15 3. After getting rid of the spurious shell effect, it was possible for us to find out the actual quantities, such as nucleon number and proton fraction at a given density in the inner crust. Adjusting the CLDM parameters to fit the outcomes of the 3DHF increased the accuracy of the CLDM simulation and allowed a quicker determination of the composition of inner crust matter from the 3DHF method. Assumptions 1. The basic assumption in this thesis work was that 3D Hartree Fock (3DHF) method is the most accurate way to simulate the properties for nuclear matter. 2. We assumed the nuclear matter was arranged in a periodic structure, so the whole range of properties could be given on the studies for one unit cell. This also was the reason why we employed WignerSeitz (WS) approximation in our computational models. 16 Chapter 2 COMPUTATIONAL METHOD In this work, we applied both 3DHF model and CLDM method in the calculation of nuclear matter located in the inner crust of a neutron star. Both models used the Skyrme forces to represent the interaction between neutrons and protons. The HartreeFock procedure was used to take the twobody Skyrme interaction and averages over interactions between all pairs of nucleons to produce a mean field potential. The total energy density derived this way from Skyrme forces is written as: Skyrme = h2 2m + 1 4 t0[(2 + x0) 2 (2x0 + 1)( 2 p 2 n)] + 1 32 [3t1(2 + x1) t2(2 + x2)](r )2 1 32 [3t1(2x1 + 1) t2(2x2 + 1)][(r p)2 + (r n)2] + 1 24 t3 [(2 + x3) 2 (2x3 + 1)( 2 p 2 n)] + 1 8 [3t1(2 + x1) t2(2 + x2)] 1 8 [3t1(2x1 + 1) t2(2x2 + 1)]( p p + n n) + 1 2 t4[J r + Jp r p + Jn r n] 1 16 (t1x1 + t2x2)J2 + 1 16 (t1 t2)[J2 p + J2 n] (2.1) 17 In equation 2.1, n is the number density of neutrons, p is the number density of protons, and is the number density of baryosn, which eaquals to n + p. is the total kinetic energy; p and n are kinetic energy densities for protons and neutrons, respectively. In our simulation, we didn’t take the last two terms  so and sg, which are related to spin, into account. The parameters t0; t1; t2; t3; x0; x1; x2; x3 and are fitted by different experiments. For uniform matter, is constant, so all the terms that included r became zero. Also, all the terms included J disappeared. 3D HartreeFock Method The HartreeFock theory produces a mean field by approximating the ground state wave function of particles as a single Slater determinant. The mean field (one body) Skyrme potential, is: uq = t0[1 + 1 2 x0] t0[ 1 2 + x0] q + 1 12 t3 3[(2 + )(1 + 1 2 x3) 2( 1 2 + x3) q ( 1 2 + x3) ( p)2 + ( n)2 ] + 1 4 [t1(1 + 1 2 x1) + t2(1 + 1 2 x2] 1 4 [t1( 1 2 + x1) t2( 1 2 + x2)] q 1 8 [3t1(1 + 1 2 x1) t2(1 + 1 2 x2]r2 + 1 8 [3t1( 1 2 + x1) + t2( 1 2 + x2)]r2 q 1 2 t4(r J + r Jq) (2.2) 18 In the 3DHF model, we assumed that the structure of nuclear matter in the inner crust was periodic and we just needed one unit cell identified. In this way, we began our study with the unit cell. In the model, at a given number density nb, there are A nucleons including Z protons in the unit cell. These three variables are the parameters that were given to the computational model as inputs. The model calculates the wavefunctions for each of the nucleons through the Schrodinger equation and produces the energy densities as a function of number density (nb), proton fraction (yp = Z A) and total nucleons number (A). In other words, for the specific nb and A, as well as yp, the 3DHF model gave the total energy of the unit cell. We ran the code for a range of different nb, A and yp and stored the various energy densities. CLDM 3DHF is a microscopic model  matter is treated as made up of neutrons and protons, and their wave functions calculated. In the Compressible Liquid Drop Model (CLDM), by contrast, treats matter as made up of two liquids: a nuclear matter liquid that makes up nuclei and pasta, and the free neutron liquid surrounding the nuclei, or pasta. These are separated by the nuclear surface. The energy density is made up by three parts: the nuclei, the surface of nuclei and free neutrons. So the energy for one unit cell includes the energies from each part and is dependent on four variables, which are proton fraction, number density inside nuclei, number density of free neutrons, as well as the size of the unit cell. The CLDM will find the minima of energy through solving 19 differential equations, the solution of which gives us the four parameters. The four equations come from the minimization of the total energy with respect to the four parameters. We formulated the energy for the unit cell as (Newton et al., 2013): "cell(rc; x; n; nn) = v[nE(n; x) + "exch + "thick] + u("surf + "curv) (2.3) + u"Coul + (1 v)nnE(nn; 0) + "e(ne); (2.4) The first term v[nE(n; x)+"exch+"thick] gives the energy of bulk matter in the nucleus, which is assumed to be at uniform density. The term u("surf + "curv) is the energy corresponding to the surface of nucleus, which is also the part we focused on in this thesis work. "surf represents the strength of surface energy and "curv provides the strength of curvature energy. The definition of these two are given as: "surf = d s rN ; "curv = d(d 1) c r2N (2.5) The term u"Coul gives the electrostatic energy. The term (1 v)nnE(nn; 0) is the energy of neutron gas outside of nucleus, which are also considered to be at uniform density. The last term, "e(ne), is the energy of electrons. 20 For a given number density (nb), the number of nucleons in the unit cell (A) and proton fraction can be found from the CLDM simulation, which can be compared with 3DHF method’s outcomes. In summary, both the two models are built on the WS unit cell and treat it as the basis to study the entire region, the inner crust. The 3DHF model is solving A Schrodinger equations for A nucleons to find the wavefunctions. Through these wavefunctions, we can determine the total energy in one unit cell. The CLDM treats the unit cell in the inner crust as a spherical WS cell, which contains nuclei and free neutrons. Then the model solves the semiclassical equations to get the total energy in terms of the bulk nuclei, surface, and electrons. After we found the energy for different baryon densities nb, we were able to find the derivative of the energy with respect to number density to get the pressure. The pressure as a function of nb is Equation of State (EoS), which is a necessary input into the equation of hydrostatic equilibrium which is solved to find physical information, such as mass, radius and density, of neutron star. Instrumentation and Collection of Data This work was purely theoretical and based on simulation, which required a large amount of computational resources. For the 3DHF model, we used Titan, which is the second most powerful supercomputer in the world with 560,640 cores and 18,000 Tera Flops per second. Except Titan, which is located 21 in Oak Ridge National Laboratory (ORNL), we applied computational codes of CLDM on a 16core cluster at TAMUC. Both the computational models are written in FORTRAN codes. We collected data and stored them on local servers for further analysis. For 3DHF, we collected results varying the parameters A, which is the nucleon number in one unit cell, between100 to 2500, yp; which is proton fraction, between 0.005 and 0.04 and nb, which is the baryon density and runs from 0.01 to 0.1 fm3. We used two temperatures: T=0 and 1 MeV. Treatment of Data The data generated by computational models were analyzed using PYTHON codes. We wrote programs to plot the energy as a function of A for the results of the 3DHF model, and after basic plotting, we fit these functions (which at higher densities contain the spurious shell effects) and extrapolated to lower densities, where we used the fit functions to subtract off the spurious shell effects and then found the minimum energy configuration. In addition, we compared the results, specifically nucleon numbers per unit cell and proton fraction, given by the two models and used the 3DHF model to improve the CLDM method. Using PYTHON codes, the parameters of the CLDM that gave the best fit to the results of the 3DHF models were found. Combining the advantages of the two computational methods was the central aim of this thesis. 22 Chapter 3 RESULTS AND DISCUSSION Data Analysis We applied four Skyrme forces: SLy4, NRAPR, SkIUFSUL30 and L90 in this work. Both the SLy4 and NRAPR have the slope of symmetry energy (L) around 60 MeV; SkIUFSUL30 has L = 30 MeV; and SkIUFSUL90 has L = 90 MeV (see Table 1). Table 1 Skyrme interactions used in this work Parameter SLy4 NRAPR SkIUFSUL30 SkIUFSUL90 t0(MeV fm3) 2488.9 2719.7 2244.684 2244.684 t1(MeV fm5) 486.8 417.64 431.976 431.976 t2(MeV fm5) 546.4 66.687 138.738 138.738 t3(MeV fm3(1+ )) 13777.0 15042.0 12323.91 12323.91 x0 0.83 0.16154 0.676554 0.202581 x1 0.34 0.047986 0.047986 0.4537011 x2 1.0 0.027170 0.644156 0.644156 x3 1.35 0.13611 1.147068 0.231331 0.16667 0.14416 0.1922489 0.1922489 L (MeV) 60 60 30 90 Even though 3DHF is the most accurate method to describe nuclear matter now, it is limited by the spurious shell effect. To illustrate this, see Figure 1, a plot we generated for number density equal to 0.08 fm3 of en 23 ergy density versus the number of nucleons in our computational cell. For the uniform matter, the total energy density should not depend on the number of nucleons in our unit cell  it is a constant (see Figure 1). However, we found the data varies upanddown  these variations contributed to the energy of the spurious shell effects, and are what we aimed to remove. To show that matter at this density is uniform, we show in Figure 2 the density integrated over the Z axis of the cell, for A=1000: z(x; y) = Z Lz Lz (x; y; z) dz (3.1) where 2Lz is the length on one side of the unit cell. In this plot, the density is integrated over Z direction and XY presents the geometric location in the unit cell. The next plots show how the spurious shell energy espurious influences the 3DHF simulations at a lower number density (nb). For nb = 0.04, we see that the energy versus A is still dominated by the spurious shell effect as shown in Figure 3. In Figure 4 we show the Zintegrated density in the unit cell, which clearly reveals the presence of a nucleus at the center of the cell, and a close to uniform neutron gas surrounding it, out to the edges of the cell. As a result of the distortion of the energy by spurious shell effects, which appears as characteristic fluctuations when energy is plotted versus A, we couldn’t determine the baryon number of each unit cell (A) that leads to 24 0 500 1000 1500 2000 2500 3000 3500 4000 A 0.70 0.72 0.74 0.76 0.78 etot (MeV/fm3 ) etot vs A (for nb=0.08 fm−3 ) etot euniform Figure 1. [SLy4, nb = 0:08, yp = 0:036] etot vs A. The dashed line is the result assuming the matter is at uniform density; the fact that is is slightly higher is due to the fact that matter isn’t quite uniform at this density. 25 x (fm) 15 10 5 0 5 10 15 y (fm) 15 10 5 0 5 10 15 zintegrated (fm−2 ) 0.0 0.5 1.0 1.5 2.0 [ SLy4 ] Integrated Neutron Density over Z (nb=0.080 fm−3 ) 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 0.0035 0.0040 0.0045 +1.787 Figure 2. [SLy4, nb = 0:08, A=1000] Integrated number density over Zaxis. 26 0 500 1000 1500 2000 2500 3000 3500 A 0.262 0.264 0.266 0.268 0.270 0.272 0.274 etot (MeV/fm3 ) etot vs A (for nb=0.04 fm−3 ) etot Figure 3. [SLy4, nb = 0:04, yp = 0:03] etot vs A 27 x (fm) 15 10 5 0 5 10 15 y (fm) 15 10 5 0 5 10 15 zintegrated (fm−2 ) 0.0 0.5 1.0 1.5 2.0 [ SLy4 ] Integrated Neutron Density over Z (nb=0.040 fm−3 ) 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 Figure 4. [SLy4, nb = 0:04]Integrated number density over Zaxis 28 minimal energy and has physical meaning. We can only do that when we remove the spurious shell effect. The spurious shell effects arise from the energy spectrum of the neutron gas outside the nucleus; in order to subtract off the shell effects, we needed to calculate the average densities of the free neutrons over three dimensions for each A, which is the estimate of the unbound neutron densities in the unit cell. We illustrate the results of that in Figure 5. The approach we used to try to remove the spurious shell effects is as follows. We began with the energy densities that are above the crustcore transition density, where the nuclear matter is uniform. We performed calculations over a wide range of densities above the crustcore transition and a wide range of proton fractions. Then we calculated espurious = etot  euniform and plotted esuprious vs nb and yp for each given A. We then fit the variation of esuprious vs nb and yp with linear functions. Based on the best fit lines, we extrapolated the results from higher number densities (nb) to low number densities in the pasta region (down to 0.02fm3, and from higher yp to yp = 0 as well. The reason why we needed to use the data of yp = 0 to predict the spurious shell effect in nuclear matter in pasta region is that the spurious shell effect is caused by the free neutrons surrounded around nuclear cluster. A neutron gas means yp=0, so espurious at those points are what we needed to remove from the total energy given by 3DHF calculations. We illustrate this procedure using the NRAPR Skyrme interaction. 29 0 500 1000 1500 2000 2500 3000 3500 A 0.024 0.026 0.028 0.030 0.032 0.034 0.036 ½n (MeV/fm3 ) ½n vs A (for nb=0.04 fm−3 ) ½n Figure 5. [SLy4, nb = 0:04, yp = 0:03] average number density of unbound neutrons in the unit cell n vs A 30 0.00 0.02 0.04 0.06 0.08 0.10 nb 0.015 0.010 0.005 0.000 0.005 0.010 0.015 espurious (Mev) (0.0300,0.0057) (0.0400,0.0033) espurious and fits vs nb for fixed yp (for A=0200) at T=1 MeV yp=0.020 yp=0.030 esp (A=200; yp=0.0) = 0.0131 + ( 0.2457 ) * nb 0.00 0.02 0.04 0.06 0.08 0.10 nb 0.004 0.003 0.002 0.001 0.000 0.001 0.002 0.003 0.004 0.005 espurious (Mev) (0.0300,0.0015) (0.0400,0.0007) espurious and fits vs nb for fixed yp (for A=0400) at T=1 MeV yp=0.020 yp=0.025 yp=0.030 esp (A=400; yp=0.0) = 0.0040 + ( 0.0816 ) * nb Figure 6. [NRAPR] Fitted espurious vs nb with A=200 and 400 31 0.00 0.02 0.04 0.06 0.08 0.10 nb 0.015 0.010 0.005 0.000 0.005 0.010 0.015 espurious (Mev) (0.0300,0.0003) (0.0400,0.0017) espurious and fits vs nb for fixed yp (for A=0600) at T=1 MeV yp=0.020 yp=0.030 esp (A=600; yp=0.0) = 0.0065 + ( 0.2043 ) * nb 0.00 0.02 0.04 0.06 0.08 0.10 nb 0.002 0.001 0.000 0.001 0.002 0.003 espurious (Mev) (0.0300,0.0007) (0.0400,0.0003) espurious and fits vs nb for fixed yp (for A=0800) at T=1 MeV yp=0.020 yp=0.025 yp=0.030 esp (A=800; yp=0.0) = 0.0018 + ( 0.0363 ) * nb Figure 7. [NRAPR] Fitted espurious vs nb with A=600 and 800 32 0.00 0.02 0.04 0.06 0.08 0.10 nb 0.005 0.004 0.003 0.002 0.001 0.000 0.001 0.002 0.003 espurious (Mev) (0.0300,0.0012) (0.0400,0.0009) espurious and fits vs nb for fixed yp (for A=1000) at T=1 MeV yp=0.020 yp=0.022 yp=0.024 yp=0.026 yp=0.028 yp=0.030 esp (A=1000; yp=0.0) = 0.0021 + ( 0.0306 ) * nb 0.00 0.02 0.04 0.06 0.08 0.10 nb 0.006 0.004 0.002 0.000 0.002 0.004 0.006 0.008 espurious (Mev) (0.0300,0.0006) (0.0400,0.0003) espurious and fits vs nb for fixed yp (for A=1200) at T=1 MeV yp=0.020 yp=0.025 yp=0.030 esp (A=1200; yp=0.0) = 0.0032 + ( 0.0877 ) * nb Figure 8. [NRAPR] Fitted espurious vs nb with A=1000 and 1200. 33 0.00 0.02 0.04 0.06 0.08 0.10 nb 0.005 0.004 0.003 0.002 0.001 0.000 0.001 0.002 0.003 espurious (Mev) (0.0300,0.0029) (0.0400,0.0024) espurious and fits vs nb for fixed yp (for A=1400) at T=1 MeV yp=0.020 yp=0.030 esp (A=1400; yp=0.0) = 0.0042 + ( 0.0436 ) * nb 0.00 0.02 0.04 0.06 0.08 0.10 nb 0.003 0.002 0.001 0.000 0.001 0.002 espurious (Mev) (0.0300,0.0018) (0.0400,0.0015) espurious and fits vs nb for fixed yp (for A=1600) at T=1 MeV yp=0.020 yp=0.025 yp=0.030 esp (A=1600; yp=0.0) = 0.0025 + ( 0.0254 ) * nb Figure 9. [NRAPR] Fitted espurious vs nb with A=1400 and 1600 34 0.00 0.02 0.04 0.06 0.08 0.10 nb 0.0010 0.0005 0.0000 0.0005 0.0010 0.0015 0.0020 espurious (Mev) (0.0300,0.0002) (0.0400,0.0001) espurious and fits vs nb for fixed yp (for A=1800) at T=1 MeV yp=0.020 yp=0.030 esp (A=1800; yp=0.0) = 0.0005 + ( 0.0098 ) * nb 0.00 0.02 0.04 0.06 0.08 0.10 nb 0.0006 0.0004 0.0002 0.0000 0.0002 0.0004 0.0006 0.0008 0.0010 0.0012 espurious (Mev) (0.0300,0.0005) (0.0400,0.0003) espurious and fits vs nb for fixed yp (for A=2000) at T=1 MeV yp=0.020 yp=0.022 yp=0.023 yp=0.024 yp=0.025 yp=0.026 yp=0.027 yp=0.028 yp=0.029 yp=0.030 esp (A=2000; yp=0.0) = 0.0009 + ( 0.0138 ) * nb Figure 10. [NRAPR] Fitted espurious vs nb with A=1800 and 2000 35 Given the fitting functions, we calculated espurious for the pasta region, and subtracted it from the calculated energies, giving e0 tot = etot espurious where e0 tot are the corrected total energy densities, which is supposed to get rid of spurious shell effects. We plotted the e0 totvsA and etotvsA for nb = 0:03 and 0:04 in the same graphs as comparisons. After that, we extended our work to two new Skyrme forces: SkIUFSUL30 and L90. We show the Zintegrated densities to illustrate their respective predictions for the structure of nuclear matter in the neutron star as shown in Figures 13, 14, 15 and 16. From these graphs, we can determine the crust core transition density to be around 0.085 fm3 for L=30MeV and 0.06 fm3 for L=90MeV. Notably, L=90 MeV predicts no pasta at all  the crustcore transition happens straight from spherical nuclei to uniform nuclear matter. We calculated espurious in the same way for these two Skyrme forces and then subtracted them from the total energy. After we got the e0 tot, then we traced all the e0 tot for a fixed nb and found the values of A and yp, which correspond to the minimal e0 tot, noted as Amin and ypmin. As we already knew, the values corresponding to the minimal energy should be the physical ones, which are assumed to occur in nature. So the Amin and ypmin we found in the graphs are the predictions for the actual neutron star. These values are shown as the blue points in Figures 17 and 18. 36 0 200 400 600 800 1000 1200 1400 1600 1800 A 0.130 0.135 0.140 0.145 0.150 0.155 0.160 0.165 0.170 Energy (Mev) etot and e0 vs A (for nb=0.03) etot e0=etot−esp Figure 11. [NRAPR, nb = 0:03] etot and e0 vs A 37 0 500 1000 1500 2000 A 0.210 0.215 0.220 0.225 0.230 0.235 Energy (Mev) etot and e0 vs A (for nb=0.04) etot e0=etot−esp Figure 12. [NRAPR, nb = 0:04] etot and e0 vs A 38 x (fm) 20 15 10 5 0 5 10 15 20 y (fm) 20 15 10 5 0 5 10 15 20 zintegrated (fm−2 ) 0.0 0.5 1.0 1.5 [ L30 ] Integrated Neutron Density over Z (nb=0.025 fm−3 ) 0.75 0.90 1.05 1.20 1.35 1.50 1.65 x (fm) 15 10 5 0 5 10 15 y (fm) 15 10 5 0 5 10 15 zintegrated (fm−2 ) 0.0 0.5 1.0 1.5 2.0 [ L30 ] Integrated Neutron Density over Z (nb=0.035 fm−3 ) 0.90 1.05 1.20 1.35 1.50 1.65 1.80 x (fm) 15 10 5 0 5 10 15 y (fm) 15 10 5 0 5 10 15 zintegrated (fm−2 ) 0.0 0.5 1.0 1.5 2.0 [ L30 ] Integrated Neutron Density over Z (nb=0.045 fm−3 ) 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 x (fm) 15 10 5 0 5 10 15 y (fm) 15 10 5 0 5 10 15 zintegrated (fm−2 ) 0.0 0.5 1.0 1.5 2.0 [ L30 ] Integrated Neutron Density over Z (nb=0.055 fm−3 ) 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 Figure 13. [SkIUFSUL30] Intergrated number density over Zaxis 39 x (fm) 15 10 5 0 5 10 15 y (fm) 15 10 5 0 5 10 15 zintegrated (fm−2 ) 0.0 0.5 1.0 1.5 2.0 2.5 [ L30 ] Integrated Neutron Density over Z (nb=0.065 fm−3 ) 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 x (fm) 15 10 5 0 5 10 15 y (fm) 15 10 5 0 5 10 15 zintegrated (fm−2 ) 0.0 0.5 1.0 1.5 2.0 [ L30 ] Integrated Neutron Density over Z (nb=0.075 fm−3 ) 1.52 1.60 1.68 1.76 1.84 1.92 2.00 2.08 2.16 x (fm) 15 10 5 0 5 10 15 y (fm) 15 10 5 0 5 10 15 zintegrated (fm−2 ) 0.0 0.5 1.0 1.5 2.0 [ L30 ] Integrated Neutron Density over Z (nb=0.085 fm−3 ) 0.00004 0.00012 0.00020 0.00028 0.00036 0.00044 0.00052 0.00060 0.00068 +1.875 x (fm) 15 10 5 0 5 10 15 y (fm) 15 10 5 0 5 10 15 zintegrated (fm−2 ) 0.0 0.5 1.0 1.5 2.0 [ L30 ] Integrated Neutron Density over Z (nb=0.095 fm−3 ) 0.00050 0.00075 0.00100 0.00125 0.00150 0.00175 0.00200 0.00225 0.00250 +2.018 Figure 14. [SkIUFSUL30] Intergrated number density over Zaxis 40 x (fm) 20 15 10 5 0 5 10 15 20 y (fm) 20 15 10 5 0 5 10 15 20 zintegrated (fm−2 ) 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 [ L90 ] Integrated Neutron Density over Z (nb=0.025 fm−3 ) 0.80 0.88 0.96 1.04 1.12 1.20 1.28 1.36 1.44 x (fm) 15 10 5 0 5 10 15 y (fm) 15 10 5 0 5 10 15 zintegrated (fm−2 ) 0.0 0.5 1.0 1.5 [ L90 ] Integrated Neutron Density over Z (nb=0.035 fm−3 ) 1.04 1.12 1.20 1.28 1.36 1.44 1.52 1.60 x (fm) 15 10 5 0 5 10 15 y (fm) 15 10 5 0 5 10 15 zintegrated (fm−2 ) 0.0 0.5 1.0 1.5 [ L90 ] Integrated Neutron Density over Z (nb=0.045 fm−3 ) 1.20 1.26 1.32 1.38 1.44 1.50 1.56 1.62 1.68 x (fm) 15 10 5 0 5 10 15 y (fm) 15 10 5 0 5 10 15 zintegrated (fm−2 ) 0.0 0.5 1.0 1.5 [ L90 ] Integrated Neutron Density over Z (nb=0.055 fm−3 ) 1.40 1.44 1.48 1.52 1.56 1.60 1.64 1.68 Figure 15. [SkIUFSUL90] Intergrated number density over Zaxis 41 x (fm) 15 10 5 0 5 10 15 y (fm) 15 10 5 0 5 10 15 zintegrated (fm−2 ) 0.0 0.5 1.0 1.5 [ L90 ] Integrated Neutron Density over Z (nb=0.065 fm−3 ) 0.00080 0.00095 0.00110 0.00125 0.00140 0.00155 0.00170 0.00185 +1.583 x (fm) 15 10 5 0 5 10 15 y (fm) 15 10 5 0 5 10 15 zintegrated (fm−2 ) 0.0 0.5 1.0 1.5 [ L90 ] Integrated Neutron Density over Z (nb=0.075 fm−3 ) 0.00075 0.00100 0.00125 0.00150 0.00175 0.00200 0.00225 0.00250 0.00275 +1.741 Figure 16. [SkIUFSUL90] Intergrated number density over Zaxis 42 Given these results from the 3DHF model, we compared them with the CLDM. In addition to the Skyrme parameters, there are two parameters we were able to adjust  the correlation between the surface tension ( surf) and the surface symmetry energy, given by c; and the curvature tension ( curv)  in CLDM to generate results. After several trials, we used curv = 0.6 MeV fm3 and c = 2.0, 4.8 and 7.0 to present our results. In order to have a clear comparison with CLDM, we combined the results of Amin vs nb from two methods in one graph, with respective L. For the slope of symmetry energy L = 30 MeV, the results are shown in Figure 17. For L = 90 MeV, the results are shown in Figure 18. It can be seen that no one value of the surface parameter c fits all the data from the 3DHF model. 43 0.00 0.02 0.04 0.06 0.08 0.10 nb (fm−3 ) 0 500 1000 1500 2000 A [ L30 ] Amin and clustered A (curvature=0.6) vs nb (for all yp for e0) Amin(SkIUFSU−L303DHF) A (surface symmetry=2.0) A (surface symmetry=4.8) A (surface symmetry=7.0) Figure 17. [SkIUFSUL30] Amin vs nb from CLDM and 3DHF 44 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 nb (fm−3 ) 0 500 1000 1500 2000 2500 A [ L90 ] Amin and clustered A (curvature=0.6) vs nb (for all yp for e0) Amin(SkIUFSU−L903DHF) A (surface symmetry=2.0) A (surface symmetry=4.8) A (surface symmetry=7.0) Figure 18. [SkIUFSUL90] Amin vs nb from CLDM and 3DHF 45 Chapter 4 CONCLUSIONS Summary In this work, we tried to find several physical values  proton fraction (yp) and the number of baryons per unit cell (A)  that correspond to minimal energies in the pasta phases through the 3DHF method. In order to achieve this goal, we needed to calculate the spurious shell effect in the data generated from the 3DHF method and subtract it. In addition, we wanted to compare the results from CLDM and 3DHF by adjusting parameters in CLDM. It should be noted that while the 3DHF method is accurate, it needs a large amount of computational resources. On the other hand, the CLDM method runs quickly, but is oversimplified. The other goal of our work was to combine the advantages of these two methods and improve our simulation models to help future research into the inner crust properties. There were two main limitations in this work. One limitation came from the limited computation time. The 3DHF model is the best method to solve the problems in nuclear matter, however, it also takes the longest time to compute, which limits the application of 3DHF in different computational conditions. There was another bigger limitation existing in our work, which resulted from the lack of knowledge of the form of nuclear energy density functions which 46 describes the interactions between neutrons and protons. In our simulations we used the Skyrme interaction, which is parameterized by over 200 sets of parameters. Since we did not know which Skyrme interaction was the best to use  as they all give good descriptions of the existing experimental data  we picked one Skyrme interaction for this study. However, this choice cannot be guaranteed to give more realistic results compared with other Skyrme interactions. The reason for this dilemma is our scant knowledge of the symmetry energy and its slope with respect to the number density. The symmetry energy is the binding in nuclei to be symmetric, which will increase when the nuclear matter changes towards asymmetry. In the neutron star, there are more neutrons compared with protons, which means the symmetry energy would be much larger and influence the properties of the neutron star, like the radius of neutron star  the slope of symmetry energy would determine the pressure within the neutron star and the pressure should be balanced by the gravity which leads to the radius of neutron star directly. Based on experiments, the slope of the symmetry energy is constrained from 30 to 90 MeV. In our work we used Skyrme models that predict a slope of the symmetry energy to be 30, 60 and 90MeV, spanning the range of uncertainty. We found that they predict crustcore transition densities around 0.085, 0.08 and 0.06 fm3 respectively. 47 Conclusions The subtraction of spurious shell effects was only partially successful; we did not remove the spurious shell effect from the results completely, but we still had some meaningful trials. We calculated the structure of crustcore transition densities for the slope of symmetry energy L=30 and 90, then confirmed that there was no pasta phase for L=90, while pasta emerged in L=30’s simulation. Furthermore, we compared the particle numbers in each unit cell(A) from two different models to determine what surface parameters of the CLDM model could best reproduce the surface which could give the best results to match the 3DHF model. The results indicated that no one set of surface parameters fits the data at all densities, but the predicted values of A and yp from the 3DHF model may not be the true predicted physical values, as we were unable to subtract off the spurious shell effects completely. Future Work We will attempt more ways to remove the spurious shell effect. One such way will be to perform 3DHF calculations at larger proton fractions. At larger proton fractions, the neutron gas is lower, and the spurious shell effects are much less prominent. We might be able to locate the true energy minimum at large proton fractions, and extrapolate to lower proton fractions. We could also compare the predicted energy minima of these highproton fraction results with those predicted by the CLDM model, tuning the surface parameters to obtain 48 agreement, and use those values to predict the minimum energy at lower proton fractions. 49 REFERENCES Andersson, N., & Kokkotas, K. D. 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Physical Review C, 79 (5), 055801. Oyamatsu, K., Hashimoto, M.a., & Yamada, M. (1984). Further study of the nuclear shape in highdensity matter. Progress of Theoretical Physics, 72 (2), 373–375. Oyamatsu, K., & Iida, K. (2007). Symmetry energy at subnuclear densities and nuclei in neutron star crusts. Physical Review C, 75 (1), 015801. Pons, J. A., Viganò, D., & Rea, N. (2013). A highly resistive layer within the crust of xray pulsars limits their spin periods. Nature Physics, 9 (7), 431–434. Ravenhall, D., Pethick, C., & Wilson, J. (1983). Structure of matter below nuclear saturation density. Physical Review Letters, 50 (26), 2066. Staelin, D. H., & Reifenstein, E. C. (1968). Pulsating radio sources near the crab nebula. Science, 162 (3861), 1481–1483. Vautherin, D., & Brink, D. M. (1972). Hartreefock calculations with skyrme’s interaction. i. spherical nuclei. Physical Review C, 5 (3), 626. 52 Watanabe, G., Iida, K., & Sato, K. (2000). Thermodynamic properties of nuclear ”pasta” in neutron star crusts. Nuclear Physics A, 676 (1), 455–473. 53 VITA Shuxi Wang received her Bachelor of Science in Physics from Southeast University of Nanjing, China in 2013. She next pursued her Master of Science in Physics at Texas A&M UniversityCommerce, where she graduated in 2015. Permanent address: Department of Physics and Astronomy Texas A&M UniversityCommerce P.O. Box 3011 Commerce, Texas 75429 Email: swang8@leomail.tamuc.edu
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Title  Improving Microscopic Models of The Neutron Star Crust 
Author  Wang, Shuxi 
Subject  Physics 
Abstract  IMPROVING MICROSCOPIC MODELS OF THE NEUTRON STAR CRUST A Thesis by SHUXI WANG Submitted to the Office of Graduate Studies of Texas A&M UniversityCommerce in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE August 2015 IMPROVING MICROSCOPIC MODELS OF THE NEUTRON STAR CRUST A Thesis by SHUXI WANG Approved by: Advisor: William Newton Committee: Kent Montgomery Kurtis Williams Head of Department: Matt A. Wood Dean of the College: Brent Donham Dean of Graduate Studies: Arlene Horne Copyright © 2015 Shuxi Wang iv ABSTRACT IMPROVING MICROSCOPIC MODELS OF THE NEUTRON STAR CRUST Shuxi Wang, MS Texas A&M UniversityCommerce, 2015 Advisor: William Newton, PhD The task of understanding the properties of neutron stars has attracted the attention of researchers in different fields, such as astronomy and nuclear and condensed matter physics. In this thesis, we focus on a region of matter called the nuclear “pasta phases,” exotic nuclear geometries which occur at the transition from the bottom of the inner crust to the superdense core of the neutron star. There are two simulation models applied in this work; one is the Three Dimensional Hartree Fock (3DHF) model, and the other is the Compressible Liquid Drop Model (CLDM). The Hartree Fock method is the most accurate way to simulate nuclear matter in recent times but is timeconsuming and contains spurious numerical effects. On the other hand, the Liquid Drop Model is efficient but oversimplifies the physics. The CLDM method treats nuclear matter as a “droplet” of nuclear matter, which loses sight of the behavior of each nucleon and the interaction between v them. The 3DHF model provides a more microscopic description of crustal matter in the neutron star. In this thesis we aimed to combine the advantages of both methods. We aimed to (i) devise a way of subtracting the spurious numerical effects from the results of 3DHF calculations, and then (ii) fit the results from the CLDM model to those of the 3DHF model by varying the parameters that determine the surface tension of nuclei in the model. This allowed the CLDM to accurately predict the composition of the inner crust and the 3DHF model to do precise calculations of crustal properties given that composition. vi ACKNOWLEDGEMENTS For the thesis to be completed, I gained the continued support from my advisor: Dr. William Newton. His guidance helped me in all the time of research and writing thesis. I would like to express my sincere thanks to him. Besides this, my sincere appreciation goes to the rest of my committee: Dr. Kent Montgomery and Dr. Kurtis Williams, for their patience and suggestions they offered me. I also take this opportunity to express gratitude to all of the department faculty and staff for their help and support. In addition, I would like to thank my parents and my friends; their support has kept me on the way to pursue the goals in my life. vii TABLE OF CONTENTS LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . x CHAPTER 1. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . 1 The CrustCore Transition and Nuclear Pasta . . . . . . . 2 Models of The Inner Crust and Nuclear Pasta . . . . . . . 5 Model Inputs . . . . . . . . . . . . . . . . . . . . 6 Model 1: The Compressible Liquid Drop Model (CLDM) . . . . . . . . . . . . . . . . . . . . . . . 8 Model 2: ThreeDimensional HartreeFock (3DHF) Method . . . . . . . . . . . . . . . . . . . . . . . 9 Purpose of Thesis . . . . . . . . . . . . . . . . . . . . . . 12 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . 15 2. COMPUTATIONAL METHOD . . . . . . . . . . . . . . . . . 16 3D HartreeFock Method . . . . . . . . . . . . . . . . . . 17 CLDM . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 Instrumentation and Collection of Data . . . . . . . . . . 20 Treatment of Data . . . . . . . . . . . . . . . . . . . . . 21 3. RESULTS AND DISCUSSION . . . . . . . . . . . . . . . . . 22 Data Analysis . . . . . . . . . . . . . . . . . . . . . . . . 22 viii 4. CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . . . 45 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . 45 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . 47 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . 47 REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 VITA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 ix LIST OF TABLES TABLE 1. Skyrme interactions used in this work . . . . . . . . . . . . . . . 22 x LIST OF FIGURES FIGURE 1. [SLy4, nb = 0:08, yp = 0:036] etot vs A. The dashed line is the result assuming the matter is at uniform density; the fact that is is slightly higher is due to the fact that matter isn’t quite uniform at this density. . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2. [SLy4, nb = 0:08, A=1000] Integrated number density over Zaxis. 25 3. [SLy4, nb = 0:04, yp = 0:03] etot vs A . . . . . . . . . . . . . . . 26 4. [SLy4, nb = 0:04]Integrated number density over Zaxis . . . . 27 5. [SLy4, nb = 0:04, yp = 0:03] average number density of unbound neutrons in the unit cell n vs A . . . . . . . . . . . . . . . . . . 29 6. [NRAPR] Fitted espurious vs nb with A=200 and 400 . . . . . . . 30 7. [NRAPR] Fitted espurious vs nb with A=600 and 800 . . . . . . . 31 8. [NRAPR] Fitted espurious vs nb with A=1000 and 1200. . . . . . 32 9. [NRAPR] Fitted espurious vs nb with A=1400 and 1600 . . . . . . 33 10. [NRAPR] Fitted espurious vs nb with A=1800 and 2000 . . . . . . 34 11. [NRAPR, nb = 0:03] etot and e0 vs A . . . . . . . . . . . . . . . 36 12. [NRAPR, nb = 0:04] etot and e0 vs A . . . . . . . . . . . . . . . 37 13. [SkIUFSUL30] Intergrated number density over Zaxis . . . . 38 14. [SkIUFSUL30] Intergrated number density over Zaxis . . . . 39 15. [SkIUFSUL90] Intergrated number density over Zaxis . . . . 40 16. [SkIUFSUL90] Intergrated number density over Zaxis . . . . 41 xi 17. [SkIUFSUL30] Amin vs nb from CLDM and 3DHF . . . . . . . . 43 18. [SkIUFSUL90] Amin vs nb from CLDM and 3DHF . . . . . . . . 44 1 Chapter 1 INTRODUCTION As one of the remnants of the collapse after supernovae, a neutron star is made up of superdense nuclear matter. The mass of a neutron star is between 1.4 to 2 solar masses, and the radius of it is about 12 to 13 kilometers (Kiziltan, 2011). Based on the current understanding of nuclear physics, the structure of a neutron star has been hypothesized as consisting of different layers according to the mass density of matter. Considering a neutron star with zero temperature, the surface of the neutron star at zero pressure will consist of an iron lattice. The layer of the neutron star from the surface down to a mass density of about 1011g=cm3 is referred to as the outer crust. In the outer crust, inverse beta decay (the capture of electrons by protons resulting in a neutron and neutrino) is allowed to occur, which results in the creation of more neutronrich nuclei. As the density increases in the inner crust, there will be increasingly more neutrons than protons in the nuclei, and the nuclei will become larger with weaker bound neutrons. When the ratio of neutrons to protons in the nuclei reaches a critical level, some neutrons become unbound, and neutrons “drip” out from the nuclei. This occurs at a density of about 1011g=cm3 and marks the start of the inner crust, which extends down to a mass density of around 1014g=cm3. The bottom part of this layer will be our focus. In the inner crust, there are two phases of nuclear matter coexisting: free neutrons and nuclei. Between 1 2 1014g=cm3 up to 1015g=cm3 is the core of the neutron star 2 (Newton, 2007). In the core region, the nuclei merge into a fluid of neutrons and protons. In addition, there are muons, and possibly pions and kaons, existing in the core region. Furthermore, at the highest densities, the matter in the core region might contain quarkgluon matter. Our knowledge of the core region is limited, observational data is scant, and we currently do not have a good model to describe the matter at these high densities. It should be noted that we used the number density to refer to the density at any point in the neutron star. Number density is simply the division of the total number of particles  in our case the neutrons and protons  by volume. A massdensity of 2:5 1014g=cm3 is equivalent to a number density of 0.16 fm3. This is the density inside terrestrial nuclei. The neutron star was first observed by Bell, Hewish et al. in 1967 as a “pulsar,” which was detected as a periodic source of radio waves with period of 1.337 s (Hewish et al., 1968). In the following year, the Crab pulsar was discovered with a shorter period 33 ms (Staelin & Reifenstein, 1968). The CrustCore Transition and Nuclear Pasta At the bottom of the inner crust region are the “pasta phases.” These are a variety of complex structures of nuclear matter, with the geometrical structures of nuclear matter resembling different types of pasta (Pons et al., 2013). In the inner crust, with increasing density, nucleons are expected to 3 arrange themselves into the following five shapes: spheres (ordinary nuclei), cylinders (spaghetti), slabs (lasagna), cylindrical holes (tubes), and spherical holes. These basic shapes of subnuclear matter were first established by Ravenhall et al. (1983), Hashimoto et al. (1984), and Oyamatsu et al. (1984), using a Liquid Drop Model (LDM) framework. These configurations develop independently of the nuclear force used, which means these shapes are from purely geometrical considerations. Astronomers collect observational data on several different neutron star phenomena which have the potential to probe the properties of the pasta phases. During the process of neutron star cooling, there is a cooling wave that moves towards the surface of the neutron star from the core. During this time, the effective surface temperature keeps constant. As the cooling wave arrives at the surface, the temperature drops sharply, from 250 eV to 30100 eV. The magnitude and duration of the drop are sensitive to the microscopic properties of crust matter, such as its heat capacity, thermal conductivity, and the superfluidity of the free neutrons (Gnedin et al., 2001), all of which could be different in the pasta phases. Neutron stars enter a spindown phase for the million years after their birth and are observed to exhibit glitches in the measurements of the spin down rate. In order to explain the glitches, researchers need to study the properties of nuclei and pasta in the inner crust and their interaction with superfluid 4 neutrons and vortices (Crawford et al., 2003; Horvath, 2004; Larson & Link, 2002; Newton, 2007). Recently, a nuclear physics group at Indiana University, Bloomington with their collaborator in Canada, presented their research outcomes on disordered nuclear pasta (Horowitz et al., 2015). They used the large scale molecular (MD) simulations and showed the existence of longliving topological defects in pasta regions. These defects could increase electron scattering and lead to a decrease in both thermal and electrical conductivities. In their MD simulations, topological defects are formed in different shapes, which always quickly assemble and stay for a long time. Then the researchers determined the effect of these defects on thermal conductivity by defining an effective impurity parameter Qimp, which is estimated to be around 40 in their paper (a number bigger than 1 indicates that matter is quite disordered). After inserting the resulting low conductivity pasta layer in their model with a bigger Qimp, the results were able to better satisfy the late time Chandra observation for the temperature of MXB 165929, which is a neutron star in a lowmass Xray binary. Mature neutron stars in lowmass binaries can easily accrete enough matter to replace its entire crust (Brown, 2000). The new crust will be melted by heat diffusion at the surface. Also, the Xradiation from the crust could help us constrain the thermal conductivity in this region. Furthermore, the accreted 5 material, which is distributed inhomogeneously on the surface of the neutron star, becomes a gravitational wave source (Haskell et al. (2006); Haensel & Zdunik (2003)), the strength of which depends on the size of the “mountain” which builds up on the surface of the star from the accreted material. The gravitational waves should then be sensitive to the mechanical properties of the subnuclear matter in the inner crust, and the examination of these should help determining the properties of the crust and, in particular, the pasta phases (Andersson & Kokkotas, 2005). By studying these observational results, we can learn more about the properties of crustal matter, but to do so we need accurate models of crustal matter, and the pasta phases in particular. Models of The Inner Crust and Nuclear Pasta In this thesis, we focused on the bottom layer of the inner crust, where the pasta phases are expected. The nuclear matter in the inner crust is assumed to have periodic structure, which allowed us to identify a repeating unit cell. The study of one unit cell could then establish the physical properties of the layer in the inner crust composed of that unit cell. At a given density, the composition of the crustal lattice is characterized by variables such as the number of total nucleons in one unit cell, proton fraction, and the size of the nucleus or pasta structure within the unit cell. Identifying these quantities allow determination of the total energy of the unit cell and structure of the geometrical shape of the nuclear matter in the unit cell. 6 In order to compute these properties, we used two different models of the unit cell of matter. Both models use the same fundamental physical principle: the actual structure of matter at a given density is the one that minimizes the total energy density of matter. The two models are: The Compressible Liquid Drop Model (CLDM) and ThreeDimensional HartreeFock (3DHF) method. Model Inputs Both models require a type of the interaction between neutrons and protons as input. In this thesis, we use the Skyrme interaction. The Skyrme interaction is a type of nonrelativistic effective potential which was firstly proposed by Skyrme and then applied on the calculation of finite nuclei by Vautherin and Brink (1972). Currently, there are over 200 parameterizations in Skyrme interaction that are fitted to results of different experiments. The main uncertainty in the Skyrme model is from the symmetry energy, which is the change in nuclear energy associated with changing neutronproton asymmetry: it means as the ratio of neutrons to protons goes away from the symmetry point (equal neutrons and protons), the binding energy of nuclear matter raises as a result of the increase of symmetry energy. The basic input for our models is the energy function of number density and proton fraction  E(n; yp). Generally, it can be expanded: 7 E(n; yp) = E0(n) + S(n) 2 + (1.1) S(n) = J + L + 1 2 Ksym 2 + (1.2) The symmetry energy is calculated from the second order differential of total energy and is 1 2yp: S(n) = 1 2 @2E @ 2 (1.3) The second term in equation 1.2 gives the most uncertain part of symmetry energy, which is denoted by L. L = nn0 3n0 @S(n) @n and is named the slope of symmetry energy; n0 is nuclear saturation density and equal 0.16 fm3. Saturation density is the baryon number density where the infinite symmetric nuclear matter has the minimal energy. Also, at the saturation density, the nuclear force that binds nucleons together is saturated. L can only be constrained by experiments to between 30 and 100 MeV. Recently, researchers found an anticorrelation between L and crustcore transition density (Oyamatsu & Iida, 2007). For larger L, they found that no “pasta” appears; for smaller L, the various “pasta” phases emerge with increasing number density. The Skyrme interaction gives us an expression for the energy density above as discussed in Chapter 2. 8 Model 1: The Compressible Liquid Drop Model (CLDM) The CLDM treats nuclei as a droplet filled with homogenous nuclear matter. The droplet has surface with an associated surface energy and is surrounded by free neutrons in the unit cell. The CLDM was first proposed by Baym et al. (Baym et al., 1971) as a model to describe nuclear matter in the inner crust of the neutron star. In the CLDM, nuclear matter in this region is described as the sum of the nuclear component, the free neutron component (both treated as fluids), and a surface separating them. As density increases in the crust, the inhomogeneous nuclear matter becomes homogeneous; the presence of the free neutron gas reduces the surface energy of nuclei, and the Coulomb repulsion between nuclei becomes significant when the distance between nuclei is in the same order with nuclei radius. Then CLDM is improved to take spherical, rodlike, slablike nuclei, as well as cylindrical and spherical nuclear bubbles into account (Watanabe et al., 2000). Nuclear physics focuses on the properties of the neutron star inner crust, which are provided by the computation of the Compressible Liquid Drop Model (CLDM) (Newton et al., 2013). A comprehensive survey of the inner crust using the CLDM was undertaken by researchers at TAMUCommerce in 2012. They explored the impact of experimental and theoretical uncertainties of symmetric 9 nuclear matter EoS and the symmetry energy and strength of surface energy, t on the crustal composition. In this thesis, we will use the CLDM developed in the latter work. A calculation minimizes total energy density, written as energy of nuclear matter in droplet, surface energy, and energy of free neutrons. At each density, the calculation can produce the radius of the nucleus and proton fraction, as well as the total nucleons, of the unit cell. The CLDM requires parameters specifying the strength of the surface and curvature tensions in addition to the nuclear interaction, in contrast to the 3DSHF method discussed next, which calculates the surface energies from the nuclear interaction input. The CLDM is a fast method for computing the structure of the inner crust, but its semiclassical nature means that it misses a description of the essential physical effects necessary for calculating the properties of crust matter, including shell effects. Model 2: ThreeDimensional HartreeFock (3DHF) Method The 3DHF model describes the nuclear matter in the unit cell through solving the Schrodinger equation for each nucleon in the cell. The solution is simplified by assuming that the total manybody wavefunction in the unit cell can be written as a Slater determinant. This is the HartreeFock approximation. A Slater determinant is an expression that describes the wavefunction of a multifermionic system that satisfies antisymmetry requirements and the Pauli 10 principle by changing sign upon exchange of two electrons (Atkins & Friedman, 2011). In the calculation, at a given density, we obtain the energy for one unit cell. We have to run the calculation over a range of nucleon number A (corresponding to cell size) and yp and to find the one that gives minimum energy. In this model, our unit cell is a cubic WignerSeitz (WS) cell. The WignerSeitz cell is a type of unit cell used in the study of crystalline structure in solid state physics. In general, the unit cell will have a complicated shape. The WS cell is an approximation to that using a simpler shape. The work published by P. Magierski and P. H. Heenen (2002) was the first to use the 3DSHF method. They demonstrated the shell effects associated with unbound neutrons in the inner crust of a neutron star. As a result, their study suggests that the inner crust has a quite complicated structure led by the shell effects, and the number of phase transitions may increase since the same phases may appear in different density ranges. Also, they claim that several phases from different lattice types could coexist in the crust, which is much different from liquiddropbased approach. Finally, they conclude the inner crust may be close to a disordered phase(Magierski & Heenen, 2002). A similar method was used to investigate the existence of the “pasta phases” in 2007 (Gögelein & Müther, 2007). They also used the relativistic meanfield method with ThomasFermi approximation as comparison in their work. Their calculation includes a pairingcorrection, which is evaluated by the 11 standard BCS theory expressing the tendency of neutrons and protons to pair up. W. Newton and J. Stone presented their work in the paper published in 2009 (Newton & Stone, 2009). They used the 3DHF model, which takes Skyrme interaction and BCS theory into account. Comparing former researches, this work extended the simulation for inhomogeneous nuclear matter to a lager range of temperature, baryon number density and proton fraction. Their work focused on the how inhomogeneous nuclear matter transitioned to uniform nuclear matter with increasing temperature and density. They also discussed the geometry and thermodynamical properties of nuclear matter. Furthermore, the researchers explored spurious shell effects and paid attention to the local minima in the energydeformation surface. Currently, the 3DHF method, which takes a microscopic description of each nucleon into account, is the most accurate approach. However, it is timeconsuming (a typical run lasts up to 24 hours)  it needs much more time and computational resources than the calculations of CLDM, which limits its application. In the calculation, we have to run lots of times at different nucleon numbers and protons fractions at each density to find the minimum energy configuration. It also has spurious numerical effects, the biggest problem being the spurious shell effect. Shell effects arise when the Schrodinger equation is solved for a finite system  e.g. if the Schrodinger equation is solved for a 12 Hydrogen atom, the electron energies will be arranged in shells, which we know from chemistry. These are real shell effects. In our case, the free neutrons in the unit cell are not finite in real life  they‘re free to move throughout the crystal lattice. But we solve the Schrodinger equation for the free neutrons in the finite unit cell in the 3DHF model. This gives rise to spurious shell effects  the free neutron energies are arranged in shells, when they shouldn‘t be. The spurious shell effects show up as fluctuations when the total energy is plotted versus the number of nucleons per unit cell. Since we are using the fundamental principle that everything is organized to minimize the total energy, the physical minima of matter at a given density is obscured by the spurious shell effects. Therefore, they must be removed. Purpose of Thesis The aim of this thesis was to attempt to combine the advantages of both CLDM and 3DHF models (the speed of the former and the more physical description of the latter). In order to do this, we tried two things. The first was to test one method of eliminating spurious shell effects from the results of 3DHF calculations. This allowed us to predict the energetically preferred nucleon numbers and proton fractions at a particular number density in the inner crust of the neutron star. Based on the results from the former problem (the getting rid of spurious shell effects from 3DHF), we adjusted the parameters in CLDM which determine surface energy, to produce the same variables, like the 13 number density of neutrons, total number density, and proton fraction as 3DHF calculations. We were then able to use the CLDM method to quickly figure out the nucleon number and proton fraction corresponding to the minimum energy at a given density, utilizing the timesaving benefits which that method offers, and then using the 3DHF model at that nucleon number density and proton fraction to calculate accurately the properties of matter in that configuration. This cond The method we used to eliminate the spurious shell effect was as follows. We aimed to find out how the spurious effects have an influence on uniform matter (which exists at higher densities than the crustcore transition density), where we could calculate the exact energy semianalytically in a way that was free of spurious shell effects. The difference between 3DHF calculations of uniform matter and the exact result gave us the size of the spurious shell effect. This was extrapolated down in density to inhomogeneous matter and subtracted from the results in the region of densities where the pasta phases existed. The two different models of the inner crust, CLDM and 3DHF, were implemented in two different codes. In addition, a third code was used to calculate the energy density of uniform nuclear matter. The data were generated by the 3DHF and the CLDM computational models, respectively, implemented as computer codes. Analysis of the data collected from these numerical simulations was analyzed and some basic linear 14 fitting to data was carried out. In order to remove spurious shell effect, we needed to find some functions or quantities to define the relation between that effect with density and proton fraction. In order to improve the CLDM, we needed to adjust factors in the computational model of the CLDM to produce the results which could satisfy the data generated by the 3DHF model. The following procedure was done: 1. We collected the data generated by the 3DHF model. This included the spurious shell effect which is shown as a fluctuation when the energy densities with varying nucleon numbers are plotted. As the spurious shell effect impedes the definition of the minimal energy, which is the way we find how matter to be arranged in nature, we sought to remove its effect from the data. 2. To remove the spurious shell effect, we calculated the properties of uniform nuclear matter. We used an exact calculation of the energy density of uniform matter and subtracted these from the 3DHF model‘s results. The differences of these subtractions defined the spurious shell energy. The spurious energies have roughly linear relations with number densities  using this relation we were able to predict the size of the spurious energies in the pasta phases of the inner crust and subtract them away, leaving behind only the physical energies, and an easier determination of the minimum energy configuration. 15 3. After getting rid of the spurious shell effect, it was possible for us to find out the actual quantities, such as nucleon number and proton fraction at a given density in the inner crust. Adjusting the CLDM parameters to fit the outcomes of the 3DHF increased the accuracy of the CLDM simulation and allowed a quicker determination of the composition of inner crust matter from the 3DHF method. Assumptions 1. The basic assumption in this thesis work was that 3D Hartree Fock (3DHF) method is the most accurate way to simulate the properties for nuclear matter. 2. We assumed the nuclear matter was arranged in a periodic structure, so the whole range of properties could be given on the studies for one unit cell. This also was the reason why we employed WignerSeitz (WS) approximation in our computational models. 16 Chapter 2 COMPUTATIONAL METHOD In this work, we applied both 3DHF model and CLDM method in the calculation of nuclear matter located in the inner crust of a neutron star. Both models used the Skyrme forces to represent the interaction between neutrons and protons. The HartreeFock procedure was used to take the twobody Skyrme interaction and averages over interactions between all pairs of nucleons to produce a mean field potential. The total energy density derived this way from Skyrme forces is written as: Skyrme = h2 2m + 1 4 t0[(2 + x0) 2 (2x0 + 1)( 2 p 2 n)] + 1 32 [3t1(2 + x1) t2(2 + x2)](r )2 1 32 [3t1(2x1 + 1) t2(2x2 + 1)][(r p)2 + (r n)2] + 1 24 t3 [(2 + x3) 2 (2x3 + 1)( 2 p 2 n)] + 1 8 [3t1(2 + x1) t2(2 + x2)] 1 8 [3t1(2x1 + 1) t2(2x2 + 1)]( p p + n n) + 1 2 t4[J r + Jp r p + Jn r n] 1 16 (t1x1 + t2x2)J2 + 1 16 (t1 t2)[J2 p + J2 n] (2.1) 17 In equation 2.1, n is the number density of neutrons, p is the number density of protons, and is the number density of baryosn, which eaquals to n + p. is the total kinetic energy; p and n are kinetic energy densities for protons and neutrons, respectively. In our simulation, we didn’t take the last two terms  so and sg, which are related to spin, into account. The parameters t0; t1; t2; t3; x0; x1; x2; x3 and are fitted by different experiments. For uniform matter, is constant, so all the terms that included r became zero. Also, all the terms included J disappeared. 3D HartreeFock Method The HartreeFock theory produces a mean field by approximating the ground state wave function of particles as a single Slater determinant. The mean field (one body) Skyrme potential, is: uq = t0[1 + 1 2 x0] t0[ 1 2 + x0] q + 1 12 t3 3[(2 + )(1 + 1 2 x3) 2( 1 2 + x3) q ( 1 2 + x3) ( p)2 + ( n)2 ] + 1 4 [t1(1 + 1 2 x1) + t2(1 + 1 2 x2] 1 4 [t1( 1 2 + x1) t2( 1 2 + x2)] q 1 8 [3t1(1 + 1 2 x1) t2(1 + 1 2 x2]r2 + 1 8 [3t1( 1 2 + x1) + t2( 1 2 + x2)]r2 q 1 2 t4(r J + r Jq) (2.2) 18 In the 3DHF model, we assumed that the structure of nuclear matter in the inner crust was periodic and we just needed one unit cell identified. In this way, we began our study with the unit cell. In the model, at a given number density nb, there are A nucleons including Z protons in the unit cell. These three variables are the parameters that were given to the computational model as inputs. The model calculates the wavefunctions for each of the nucleons through the Schrodinger equation and produces the energy densities as a function of number density (nb), proton fraction (yp = Z A) and total nucleons number (A). In other words, for the specific nb and A, as well as yp, the 3DHF model gave the total energy of the unit cell. We ran the code for a range of different nb, A and yp and stored the various energy densities. CLDM 3DHF is a microscopic model  matter is treated as made up of neutrons and protons, and their wave functions calculated. In the Compressible Liquid Drop Model (CLDM), by contrast, treats matter as made up of two liquids: a nuclear matter liquid that makes up nuclei and pasta, and the free neutron liquid surrounding the nuclei, or pasta. These are separated by the nuclear surface. The energy density is made up by three parts: the nuclei, the surface of nuclei and free neutrons. So the energy for one unit cell includes the energies from each part and is dependent on four variables, which are proton fraction, number density inside nuclei, number density of free neutrons, as well as the size of the unit cell. The CLDM will find the minima of energy through solving 19 differential equations, the solution of which gives us the four parameters. The four equations come from the minimization of the total energy with respect to the four parameters. We formulated the energy for the unit cell as (Newton et al., 2013): "cell(rc; x; n; nn) = v[nE(n; x) + "exch + "thick] + u("surf + "curv) (2.3) + u"Coul + (1 v)nnE(nn; 0) + "e(ne); (2.4) The first term v[nE(n; x)+"exch+"thick] gives the energy of bulk matter in the nucleus, which is assumed to be at uniform density. The term u("surf + "curv) is the energy corresponding to the surface of nucleus, which is also the part we focused on in this thesis work. "surf represents the strength of surface energy and "curv provides the strength of curvature energy. The definition of these two are given as: "surf = d s rN ; "curv = d(d 1) c r2N (2.5) The term u"Coul gives the electrostatic energy. The term (1 v)nnE(nn; 0) is the energy of neutron gas outside of nucleus, which are also considered to be at uniform density. The last term, "e(ne), is the energy of electrons. 20 For a given number density (nb), the number of nucleons in the unit cell (A) and proton fraction can be found from the CLDM simulation, which can be compared with 3DHF method’s outcomes. In summary, both the two models are built on the WS unit cell and treat it as the basis to study the entire region, the inner crust. The 3DHF model is solving A Schrodinger equations for A nucleons to find the wavefunctions. Through these wavefunctions, we can determine the total energy in one unit cell. The CLDM treats the unit cell in the inner crust as a spherical WS cell, which contains nuclei and free neutrons. Then the model solves the semiclassical equations to get the total energy in terms of the bulk nuclei, surface, and electrons. After we found the energy for different baryon densities nb, we were able to find the derivative of the energy with respect to number density to get the pressure. The pressure as a function of nb is Equation of State (EoS), which is a necessary input into the equation of hydrostatic equilibrium which is solved to find physical information, such as mass, radius and density, of neutron star. Instrumentation and Collection of Data This work was purely theoretical and based on simulation, which required a large amount of computational resources. For the 3DHF model, we used Titan, which is the second most powerful supercomputer in the world with 560,640 cores and 18,000 Tera Flops per second. Except Titan, which is located 21 in Oak Ridge National Laboratory (ORNL), we applied computational codes of CLDM on a 16core cluster at TAMUC. Both the computational models are written in FORTRAN codes. We collected data and stored them on local servers for further analysis. For 3DHF, we collected results varying the parameters A, which is the nucleon number in one unit cell, between100 to 2500, yp; which is proton fraction, between 0.005 and 0.04 and nb, which is the baryon density and runs from 0.01 to 0.1 fm3. We used two temperatures: T=0 and 1 MeV. Treatment of Data The data generated by computational models were analyzed using PYTHON codes. We wrote programs to plot the energy as a function of A for the results of the 3DHF model, and after basic plotting, we fit these functions (which at higher densities contain the spurious shell effects) and extrapolated to lower densities, where we used the fit functions to subtract off the spurious shell effects and then found the minimum energy configuration. In addition, we compared the results, specifically nucleon numbers per unit cell and proton fraction, given by the two models and used the 3DHF model to improve the CLDM method. Using PYTHON codes, the parameters of the CLDM that gave the best fit to the results of the 3DHF models were found. Combining the advantages of the two computational methods was the central aim of this thesis. 22 Chapter 3 RESULTS AND DISCUSSION Data Analysis We applied four Skyrme forces: SLy4, NRAPR, SkIUFSUL30 and L90 in this work. Both the SLy4 and NRAPR have the slope of symmetry energy (L) around 60 MeV; SkIUFSUL30 has L = 30 MeV; and SkIUFSUL90 has L = 90 MeV (see Table 1). Table 1 Skyrme interactions used in this work Parameter SLy4 NRAPR SkIUFSUL30 SkIUFSUL90 t0(MeV fm3) 2488.9 2719.7 2244.684 2244.684 t1(MeV fm5) 486.8 417.64 431.976 431.976 t2(MeV fm5) 546.4 66.687 138.738 138.738 t3(MeV fm3(1+ )) 13777.0 15042.0 12323.91 12323.91 x0 0.83 0.16154 0.676554 0.202581 x1 0.34 0.047986 0.047986 0.4537011 x2 1.0 0.027170 0.644156 0.644156 x3 1.35 0.13611 1.147068 0.231331 0.16667 0.14416 0.1922489 0.1922489 L (MeV) 60 60 30 90 Even though 3DHF is the most accurate method to describe nuclear matter now, it is limited by the spurious shell effect. To illustrate this, see Figure 1, a plot we generated for number density equal to 0.08 fm3 of en 23 ergy density versus the number of nucleons in our computational cell. For the uniform matter, the total energy density should not depend on the number of nucleons in our unit cell  it is a constant (see Figure 1). However, we found the data varies upanddown  these variations contributed to the energy of the spurious shell effects, and are what we aimed to remove. To show that matter at this density is uniform, we show in Figure 2 the density integrated over the Z axis of the cell, for A=1000: z(x; y) = Z Lz Lz (x; y; z) dz (3.1) where 2Lz is the length on one side of the unit cell. In this plot, the density is integrated over Z direction and XY presents the geometric location in the unit cell. The next plots show how the spurious shell energy espurious influences the 3DHF simulations at a lower number density (nb). For nb = 0.04, we see that the energy versus A is still dominated by the spurious shell effect as shown in Figure 3. In Figure 4 we show the Zintegrated density in the unit cell, which clearly reveals the presence of a nucleus at the center of the cell, and a close to uniform neutron gas surrounding it, out to the edges of the cell. As a result of the distortion of the energy by spurious shell effects, which appears as characteristic fluctuations when energy is plotted versus A, we couldn’t determine the baryon number of each unit cell (A) that leads to 24 0 500 1000 1500 2000 2500 3000 3500 4000 A 0.70 0.72 0.74 0.76 0.78 etot (MeV/fm3 ) etot vs A (for nb=0.08 fm−3 ) etot euniform Figure 1. [SLy4, nb = 0:08, yp = 0:036] etot vs A. The dashed line is the result assuming the matter is at uniform density; the fact that is is slightly higher is due to the fact that matter isn’t quite uniform at this density. 25 x (fm) 15 10 5 0 5 10 15 y (fm) 15 10 5 0 5 10 15 zintegrated (fm−2 ) 0.0 0.5 1.0 1.5 2.0 [ SLy4 ] Integrated Neutron Density over Z (nb=0.080 fm−3 ) 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 0.0035 0.0040 0.0045 +1.787 Figure 2. [SLy4, nb = 0:08, A=1000] Integrated number density over Zaxis. 26 0 500 1000 1500 2000 2500 3000 3500 A 0.262 0.264 0.266 0.268 0.270 0.272 0.274 etot (MeV/fm3 ) etot vs A (for nb=0.04 fm−3 ) etot Figure 3. [SLy4, nb = 0:04, yp = 0:03] etot vs A 27 x (fm) 15 10 5 0 5 10 15 y (fm) 15 10 5 0 5 10 15 zintegrated (fm−2 ) 0.0 0.5 1.0 1.5 2.0 [ SLy4 ] Integrated Neutron Density over Z (nb=0.040 fm−3 ) 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 Figure 4. [SLy4, nb = 0:04]Integrated number density over Zaxis 28 minimal energy and has physical meaning. We can only do that when we remove the spurious shell effect. The spurious shell effects arise from the energy spectrum of the neutron gas outside the nucleus; in order to subtract off the shell effects, we needed to calculate the average densities of the free neutrons over three dimensions for each A, which is the estimate of the unbound neutron densities in the unit cell. We illustrate the results of that in Figure 5. The approach we used to try to remove the spurious shell effects is as follows. We began with the energy densities that are above the crustcore transition density, where the nuclear matter is uniform. We performed calculations over a wide range of densities above the crustcore transition and a wide range of proton fractions. Then we calculated espurious = etot  euniform and plotted esuprious vs nb and yp for each given A. We then fit the variation of esuprious vs nb and yp with linear functions. Based on the best fit lines, we extrapolated the results from higher number densities (nb) to low number densities in the pasta region (down to 0.02fm3, and from higher yp to yp = 0 as well. The reason why we needed to use the data of yp = 0 to predict the spurious shell effect in nuclear matter in pasta region is that the spurious shell effect is caused by the free neutrons surrounded around nuclear cluster. A neutron gas means yp=0, so espurious at those points are what we needed to remove from the total energy given by 3DHF calculations. We illustrate this procedure using the NRAPR Skyrme interaction. 29 0 500 1000 1500 2000 2500 3000 3500 A 0.024 0.026 0.028 0.030 0.032 0.034 0.036 ½n (MeV/fm3 ) ½n vs A (for nb=0.04 fm−3 ) ½n Figure 5. [SLy4, nb = 0:04, yp = 0:03] average number density of unbound neutrons in the unit cell n vs A 30 0.00 0.02 0.04 0.06 0.08 0.10 nb 0.015 0.010 0.005 0.000 0.005 0.010 0.015 espurious (Mev) (0.0300,0.0057) (0.0400,0.0033) espurious and fits vs nb for fixed yp (for A=0200) at T=1 MeV yp=0.020 yp=0.030 esp (A=200; yp=0.0) = 0.0131 + ( 0.2457 ) * nb 0.00 0.02 0.04 0.06 0.08 0.10 nb 0.004 0.003 0.002 0.001 0.000 0.001 0.002 0.003 0.004 0.005 espurious (Mev) (0.0300,0.0015) (0.0400,0.0007) espurious and fits vs nb for fixed yp (for A=0400) at T=1 MeV yp=0.020 yp=0.025 yp=0.030 esp (A=400; yp=0.0) = 0.0040 + ( 0.0816 ) * nb Figure 6. [NRAPR] Fitted espurious vs nb with A=200 and 400 31 0.00 0.02 0.04 0.06 0.08 0.10 nb 0.015 0.010 0.005 0.000 0.005 0.010 0.015 espurious (Mev) (0.0300,0.0003) (0.0400,0.0017) espurious and fits vs nb for fixed yp (for A=0600) at T=1 MeV yp=0.020 yp=0.030 esp (A=600; yp=0.0) = 0.0065 + ( 0.2043 ) * nb 0.00 0.02 0.04 0.06 0.08 0.10 nb 0.002 0.001 0.000 0.001 0.002 0.003 espurious (Mev) (0.0300,0.0007) (0.0400,0.0003) espurious and fits vs nb for fixed yp (for A=0800) at T=1 MeV yp=0.020 yp=0.025 yp=0.030 esp (A=800; yp=0.0) = 0.0018 + ( 0.0363 ) * nb Figure 7. [NRAPR] Fitted espurious vs nb with A=600 and 800 32 0.00 0.02 0.04 0.06 0.08 0.10 nb 0.005 0.004 0.003 0.002 0.001 0.000 0.001 0.002 0.003 espurious (Mev) (0.0300,0.0012) (0.0400,0.0009) espurious and fits vs nb for fixed yp (for A=1000) at T=1 MeV yp=0.020 yp=0.022 yp=0.024 yp=0.026 yp=0.028 yp=0.030 esp (A=1000; yp=0.0) = 0.0021 + ( 0.0306 ) * nb 0.00 0.02 0.04 0.06 0.08 0.10 nb 0.006 0.004 0.002 0.000 0.002 0.004 0.006 0.008 espurious (Mev) (0.0300,0.0006) (0.0400,0.0003) espurious and fits vs nb for fixed yp (for A=1200) at T=1 MeV yp=0.020 yp=0.025 yp=0.030 esp (A=1200; yp=0.0) = 0.0032 + ( 0.0877 ) * nb Figure 8. [NRAPR] Fitted espurious vs nb with A=1000 and 1200. 33 0.00 0.02 0.04 0.06 0.08 0.10 nb 0.005 0.004 0.003 0.002 0.001 0.000 0.001 0.002 0.003 espurious (Mev) (0.0300,0.0029) (0.0400,0.0024) espurious and fits vs nb for fixed yp (for A=1400) at T=1 MeV yp=0.020 yp=0.030 esp (A=1400; yp=0.0) = 0.0042 + ( 0.0436 ) * nb 0.00 0.02 0.04 0.06 0.08 0.10 nb 0.003 0.002 0.001 0.000 0.001 0.002 espurious (Mev) (0.0300,0.0018) (0.0400,0.0015) espurious and fits vs nb for fixed yp (for A=1600) at T=1 MeV yp=0.020 yp=0.025 yp=0.030 esp (A=1600; yp=0.0) = 0.0025 + ( 0.0254 ) * nb Figure 9. [NRAPR] Fitted espurious vs nb with A=1400 and 1600 34 0.00 0.02 0.04 0.06 0.08 0.10 nb 0.0010 0.0005 0.0000 0.0005 0.0010 0.0015 0.0020 espurious (Mev) (0.0300,0.0002) (0.0400,0.0001) espurious and fits vs nb for fixed yp (for A=1800) at T=1 MeV yp=0.020 yp=0.030 esp (A=1800; yp=0.0) = 0.0005 + ( 0.0098 ) * nb 0.00 0.02 0.04 0.06 0.08 0.10 nb 0.0006 0.0004 0.0002 0.0000 0.0002 0.0004 0.0006 0.0008 0.0010 0.0012 espurious (Mev) (0.0300,0.0005) (0.0400,0.0003) espurious and fits vs nb for fixed yp (for A=2000) at T=1 MeV yp=0.020 yp=0.022 yp=0.023 yp=0.024 yp=0.025 yp=0.026 yp=0.027 yp=0.028 yp=0.029 yp=0.030 esp (A=2000; yp=0.0) = 0.0009 + ( 0.0138 ) * nb Figure 10. [NRAPR] Fitted espurious vs nb with A=1800 and 2000 35 Given the fitting functions, we calculated espurious for the pasta region, and subtracted it from the calculated energies, giving e0 tot = etot espurious where e0 tot are the corrected total energy densities, which is supposed to get rid of spurious shell effects. We plotted the e0 totvsA and etotvsA for nb = 0:03 and 0:04 in the same graphs as comparisons. After that, we extended our work to two new Skyrme forces: SkIUFSUL30 and L90. We show the Zintegrated densities to illustrate their respective predictions for the structure of nuclear matter in the neutron star as shown in Figures 13, 14, 15 and 16. From these graphs, we can determine the crust core transition density to be around 0.085 fm3 for L=30MeV and 0.06 fm3 for L=90MeV. Notably, L=90 MeV predicts no pasta at all  the crustcore transition happens straight from spherical nuclei to uniform nuclear matter. We calculated espurious in the same way for these two Skyrme forces and then subtracted them from the total energy. After we got the e0 tot, then we traced all the e0 tot for a fixed nb and found the values of A and yp, which correspond to the minimal e0 tot, noted as Amin and ypmin. As we already knew, the values corresponding to the minimal energy should be the physical ones, which are assumed to occur in nature. So the Amin and ypmin we found in the graphs are the predictions for the actual neutron star. These values are shown as the blue points in Figures 17 and 18. 36 0 200 400 600 800 1000 1200 1400 1600 1800 A 0.130 0.135 0.140 0.145 0.150 0.155 0.160 0.165 0.170 Energy (Mev) etot and e0 vs A (for nb=0.03) etot e0=etot−esp Figure 11. [NRAPR, nb = 0:03] etot and e0 vs A 37 0 500 1000 1500 2000 A 0.210 0.215 0.220 0.225 0.230 0.235 Energy (Mev) etot and e0 vs A (for nb=0.04) etot e0=etot−esp Figure 12. [NRAPR, nb = 0:04] etot and e0 vs A 38 x (fm) 20 15 10 5 0 5 10 15 20 y (fm) 20 15 10 5 0 5 10 15 20 zintegrated (fm−2 ) 0.0 0.5 1.0 1.5 [ L30 ] Integrated Neutron Density over Z (nb=0.025 fm−3 ) 0.75 0.90 1.05 1.20 1.35 1.50 1.65 x (fm) 15 10 5 0 5 10 15 y (fm) 15 10 5 0 5 10 15 zintegrated (fm−2 ) 0.0 0.5 1.0 1.5 2.0 [ L30 ] Integrated Neutron Density over Z (nb=0.035 fm−3 ) 0.90 1.05 1.20 1.35 1.50 1.65 1.80 x (fm) 15 10 5 0 5 10 15 y (fm) 15 10 5 0 5 10 15 zintegrated (fm−2 ) 0.0 0.5 1.0 1.5 2.0 [ L30 ] Integrated Neutron Density over Z (nb=0.045 fm−3 ) 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 x (fm) 15 10 5 0 5 10 15 y (fm) 15 10 5 0 5 10 15 zintegrated (fm−2 ) 0.0 0.5 1.0 1.5 2.0 [ L30 ] Integrated Neutron Density over Z (nb=0.055 fm−3 ) 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 Figure 13. [SkIUFSUL30] Intergrated number density over Zaxis 39 x (fm) 15 10 5 0 5 10 15 y (fm) 15 10 5 0 5 10 15 zintegrated (fm−2 ) 0.0 0.5 1.0 1.5 2.0 2.5 [ L30 ] Integrated Neutron Density over Z (nb=0.065 fm−3 ) 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 x (fm) 15 10 5 0 5 10 15 y (fm) 15 10 5 0 5 10 15 zintegrated (fm−2 ) 0.0 0.5 1.0 1.5 2.0 [ L30 ] Integrated Neutron Density over Z (nb=0.075 fm−3 ) 1.52 1.60 1.68 1.76 1.84 1.92 2.00 2.08 2.16 x (fm) 15 10 5 0 5 10 15 y (fm) 15 10 5 0 5 10 15 zintegrated (fm−2 ) 0.0 0.5 1.0 1.5 2.0 [ L30 ] Integrated Neutron Density over Z (nb=0.085 fm−3 ) 0.00004 0.00012 0.00020 0.00028 0.00036 0.00044 0.00052 0.00060 0.00068 +1.875 x (fm) 15 10 5 0 5 10 15 y (fm) 15 10 5 0 5 10 15 zintegrated (fm−2 ) 0.0 0.5 1.0 1.5 2.0 [ L30 ] Integrated Neutron Density over Z (nb=0.095 fm−3 ) 0.00050 0.00075 0.00100 0.00125 0.00150 0.00175 0.00200 0.00225 0.00250 +2.018 Figure 14. [SkIUFSUL30] Intergrated number density over Zaxis 40 x (fm) 20 15 10 5 0 5 10 15 20 y (fm) 20 15 10 5 0 5 10 15 20 zintegrated (fm−2 ) 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 [ L90 ] Integrated Neutron Density over Z (nb=0.025 fm−3 ) 0.80 0.88 0.96 1.04 1.12 1.20 1.28 1.36 1.44 x (fm) 15 10 5 0 5 10 15 y (fm) 15 10 5 0 5 10 15 zintegrated (fm−2 ) 0.0 0.5 1.0 1.5 [ L90 ] Integrated Neutron Density over Z (nb=0.035 fm−3 ) 1.04 1.12 1.20 1.28 1.36 1.44 1.52 1.60 x (fm) 15 10 5 0 5 10 15 y (fm) 15 10 5 0 5 10 15 zintegrated (fm−2 ) 0.0 0.5 1.0 1.5 [ L90 ] Integrated Neutron Density over Z (nb=0.045 fm−3 ) 1.20 1.26 1.32 1.38 1.44 1.50 1.56 1.62 1.68 x (fm) 15 10 5 0 5 10 15 y (fm) 15 10 5 0 5 10 15 zintegrated (fm−2 ) 0.0 0.5 1.0 1.5 [ L90 ] Integrated Neutron Density over Z (nb=0.055 fm−3 ) 1.40 1.44 1.48 1.52 1.56 1.60 1.64 1.68 Figure 15. [SkIUFSUL90] Intergrated number density over Zaxis 41 x (fm) 15 10 5 0 5 10 15 y (fm) 15 10 5 0 5 10 15 zintegrated (fm−2 ) 0.0 0.5 1.0 1.5 [ L90 ] Integrated Neutron Density over Z (nb=0.065 fm−3 ) 0.00080 0.00095 0.00110 0.00125 0.00140 0.00155 0.00170 0.00185 +1.583 x (fm) 15 10 5 0 5 10 15 y (fm) 15 10 5 0 5 10 15 zintegrated (fm−2 ) 0.0 0.5 1.0 1.5 [ L90 ] Integrated Neutron Density over Z (nb=0.075 fm−3 ) 0.00075 0.00100 0.00125 0.00150 0.00175 0.00200 0.00225 0.00250 0.00275 +1.741 Figure 16. [SkIUFSUL90] Intergrated number density over Zaxis 42 Given these results from the 3DHF model, we compared them with the CLDM. In addition to the Skyrme parameters, there are two parameters we were able to adjust  the correlation between the surface tension ( surf) and the surface symmetry energy, given by c; and the curvature tension ( curv)  in CLDM to generate results. After several trials, we used curv = 0.6 MeV fm3 and c = 2.0, 4.8 and 7.0 to present our results. In order to have a clear comparison with CLDM, we combined the results of Amin vs nb from two methods in one graph, with respective L. For the slope of symmetry energy L = 30 MeV, the results are shown in Figure 17. For L = 90 MeV, the results are shown in Figure 18. It can be seen that no one value of the surface parameter c fits all the data from the 3DHF model. 43 0.00 0.02 0.04 0.06 0.08 0.10 nb (fm−3 ) 0 500 1000 1500 2000 A [ L30 ] Amin and clustered A (curvature=0.6) vs nb (for all yp for e0) Amin(SkIUFSU−L303DHF) A (surface symmetry=2.0) A (surface symmetry=4.8) A (surface symmetry=7.0) Figure 17. [SkIUFSUL30] Amin vs nb from CLDM and 3DHF 44 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 nb (fm−3 ) 0 500 1000 1500 2000 2500 A [ L90 ] Amin and clustered A (curvature=0.6) vs nb (for all yp for e0) Amin(SkIUFSU−L903DHF) A (surface symmetry=2.0) A (surface symmetry=4.8) A (surface symmetry=7.0) Figure 18. [SkIUFSUL90] Amin vs nb from CLDM and 3DHF 45 Chapter 4 CONCLUSIONS Summary In this work, we tried to find several physical values  proton fraction (yp) and the number of baryons per unit cell (A)  that correspond to minimal energies in the pasta phases through the 3DHF method. In order to achieve this goal, we needed to calculate the spurious shell effect in the data generated from the 3DHF method and subtract it. In addition, we wanted to compare the results from CLDM and 3DHF by adjusting parameters in CLDM. It should be noted that while the 3DHF method is accurate, it needs a large amount of computational resources. On the other hand, the CLDM method runs quickly, but is oversimplified. The other goal of our work was to combine the advantages of these two methods and improve our simulation models to help future research into the inner crust properties. There were two main limitations in this work. One limitation came from the limited computation time. The 3DHF model is the best method to solve the problems in nuclear matter, however, it also takes the longest time to compute, which limits the application of 3DHF in different computational conditions. There was another bigger limitation existing in our work, which resulted from the lack of knowledge of the form of nuclear energy density functions which 46 describes the interactions between neutrons and protons. In our simulations we used the Skyrme interaction, which is parameterized by over 200 sets of parameters. Since we did not know which Skyrme interaction was the best to use  as they all give good descriptions of the existing experimental data  we picked one Skyrme interaction for this study. However, this choice cannot be guaranteed to give more realistic results compared with other Skyrme interactions. The reason for this dilemma is our scant knowledge of the symmetry energy and its slope with respect to the number density. The symmetry energy is the binding in nuclei to be symmetric, which will increase when the nuclear matter changes towards asymmetry. In the neutron star, there are more neutrons compared with protons, which means the symmetry energy would be much larger and influence the properties of the neutron star, like the radius of neutron star  the slope of symmetry energy would determine the pressure within the neutron star and the pressure should be balanced by the gravity which leads to the radius of neutron star directly. Based on experiments, the slope of the symmetry energy is constrained from 30 to 90 MeV. In our work we used Skyrme models that predict a slope of the symmetry energy to be 30, 60 and 90MeV, spanning the range of uncertainty. We found that they predict crustcore transition densities around 0.085, 0.08 and 0.06 fm3 respectively. 47 Conclusions The subtraction of spurious shell effects was only partially successful; we did not remove the spurious shell effect from the results completely, but we still had some meaningful trials. We calculated the structure of crustcore transition densities for the slope of symmetry energy L=30 and 90, then confirmed that there was no pasta phase for L=90, while pasta emerged in L=30’s simulation. Furthermore, we compared the particle numbers in each unit cell(A) from two different models to determine what surface parameters of the CLDM model could best reproduce the surface which could give the best results to match the 3DHF model. The results indicated that no one set of surface parameters fits the data at all densities, but the predicted values of A and yp from the 3DHF model may not be the true predicted physical values, as we were unable to subtract off the spurious shell effects completely. Future Work We will attempt more ways to remove the spurious shell effect. One such way will be to perform 3DHF calculations at larger proton fractions. At larger proton fractions, the neutron gas is lower, and the spurious shell effects are much less prominent. We might be able to locate the true energy minimum at large proton fractions, and extrapolate to lower proton fractions. We could also compare the predicted energy minima of these highproton fraction results with those predicted by the CLDM model, tuning the surface parameters to obtain 48 agreement, and use those values to predict the minimum energy at lower proton fractions. 49 REFERENCES Andersson, N., & Kokkotas, K. D. 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Thermodynamic properties of nuclear ”pasta” in neutron star crusts. Nuclear Physics A, 676 (1), 455–473. 53 VITA Shuxi Wang received her Bachelor of Science in Physics from Southeast University of Nanjing, China in 2013. She next pursued her Master of Science in Physics at Texas A&M UniversityCommerce, where she graduated in 2015. Permanent address: Department of Physics and Astronomy Texas A&M UniversityCommerce P.O. Box 3011 Commerce, Texas 75429 Email: swang8@leomail.tamuc.edu 
Date  2015 
Faculty Advisor  Newton, William 
Committee Members 
Newton, William Montgomery, Kent Williams, Kurtis 
University Affiliation  Texas A&M UniversityCommerce 
Department  MSPhysics 
Degree Awarded  M.S. 
Pages  64 
Type  Text 
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Language  eng 
Rights  All rights reserved. 



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