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+{"text":"\\section{Introduction}\n\nUpon its discovery on 2004 June at Kitt Peak by R.A. Tucker, D.J. Tholen, and F. Bernardi \\citep{gar2004, tucker2004, Smalley2004}, 99942 Apophis (originally designated as 2004 MN4) has its orbit constantly monitored since it was reported a high probability of collision with Earth of more than 2\\% \\citep{chesley_2005}. Although this possibility was later discharged \\citep{Larsen2004}, other potential impact or approaches with Earth were reported in the following years, and Apophis has been considered as a potentially hazardous asteroid (PHA). The trajectory and the future close encounters parameters were improved with ephemeris derived from radar observations in 2005-2006 from Arecibo radiotelescope \\citep{giorgini2008predicting}. With this data, they were able to reduce orbital uncertainties and infer a nominal distance for the 2029 approach of about 6 Earth radii, predict a distance of 0.34 au for the 2036 encounter, and find that an impact probability still remains \\citep{giorgini2008predicting}. The next approach will be the closest encounter with the Earth and will occur on 13 April 2029. Apophis will pass at a distance near to the geostationary orbit and a distance about one-tenth the distance between the Earth and Moon.\n\nThe predictions for the pre-2029 orbit turn to be well-defined, however, for the post-2029 orbit, the predictions are not well determined due to the uncertainties caused by perturbations. The Yarkovsky effect is a considerable source for the orbital uncertainties of Apophis' orbit \\citep{giorgini2008predicting, farnocchia2013yarkovsky, vokrouhlicky2015yarkovsky}. Thus considering the Yarkovsky effect and using astrometric data from 2004-2008, \\citet{farnocchia2013yarkovsky} presented a new impact risk for Apophis. They infer an impact probability of about $\\sim$10$^{-9}$ for the 2036 encounter and $\\sim$10$^{-6}$ for 2068. \n\nObservations from the 2021 encounter provided measurements that improved the fit for the orbit of Apophis and eliminate the possibility of impact for the next 100 years. It led Apophis to be removed from the ESA's asteroid Risk List after remaining on this list for almost 17 years \\footnote{\\href{https:\/\/cneos.jpl.nasa.gov\/sentry\/removed.html}{https:\/\/cneos.jpl.nasa.gov\/sentry\/removed.html}}.\n\nThe 2029 flyby may provide an opportunity to improve the 3D shape model, investigate possible changes on the spin state, reshaping and effects of the encounter on the surface. \\citet{scheeres2005abrupt} predicted that the terrestrial torques caused by the encounter will change Apophis' spin state drastically, and consequently the Yarkovsky accelerations. Conversely, \\citet{souchay2018changes} showed that the changes in the spin rate may be small, and the larger effects may occur in the obliquity and precession in longitude. The effects of a close approach with the Earth also could cause material landslides and migration as is showed in \\citet{binzel2010earth}. However, the numerical simulations made for \\citet{yu2014numerical} using soft-sphere code implementation predicted that the effects of the tidal pertubations on the Apophis' surface may be small, but could produce small landslides.\n\nTherefore, the closest approach of 2029 with the Earth may provide measures through observations before, during, and after the encounter. The observational data from these observations may improve some physical characteristics, the understanding of the effects of the closest encounters with the Earth, validate models about the material, and other possibilities. In this work, we used the convex shape model from \\citet{Pravec2014} to analyse the surface and the nearby dynamics of the asteroid Apophis considering the effects caused by the 2029 closest approach with the Earth. The paper is composed of the following sections. In the next section, we present the asteroid model used in this work, discussing its polyhedral shape model, general characteristics and the 2029 encounter configuration. Section \\ref{surface} introduces the gravitational potential and geopotential considering the polyhedra method. We discuss physical features using the slope angle and its variation due to the Earth perturbation. In Section \\ref{stability}, we explore the nearby environment of Apophis by calculating the zero-velocity curves and equilibrium points. We also present a set of numerical simulations of a disc of particles around Apophis considering the gravitational field and two additional scenarios of perturbations: the solar radiation pressure and, in Section \\ref{instability}, the Earth perturbation on the 2029 trajectory. Finally, in the last section we provide our final comments.\n\n\\section{The asteroid model}\n\\label{asteroid_model}\n\nApophis was observed by the VLT (Very Large Telescope) and the data obtained from a polarimetric observing campaign in 2006 resulted in an approximate diameter of 270 metres for the asteroid \\citep{cellino2006albedo}. In 2012 and 2013, \\citet{Pravec2014} made a photometric observational campaign and discovered that Apophis is a tumbling asteroid. They computed the Apophis' tumbling spin state as a retrograde rotational period of 27.38 $\\pm$ 0.07 hours and a precession period of 263 $\\pm$ 6 hours. \n\nThe 2012-2013 Apophis' apparition also allowed the construction of a convex shape model. The Apophis' shape model is represented as an unscaled polyhedron with 1014 vertices and 2024 triangular faces \\citep{Pravec2014}. \\citet{BROZOVIC2018115} also provided a shape model for the asteroid Apophis, but they used radar observations from Goldstone and Arecibo to improve the previous model. This improved shape model is a 340 metres polyhedron with 2000 vertices and 3996 faces. \n\nFrom thermal infrared observations, \\citet{muller2014thermal} reported an approximated size of 375 metres for (99942) Apophis. \\citet{licandro2015canaricam} also provided an estimated size for Apophis using this technique, however, they combined their data with previous thermal observations. The resulting size for Apophis ranges from 380 to 393 metres.\n\nWe adopted the Apophis' diameter presented by \\citet{BROZOVIC2018115}, 340 metres, and the shape model provided by \\citet{Pravec2014}, hence the model by \\citet{BROZOVIC2018115} is not publicly available. We used the rotational period of 27.38 hours with an obliquity of 180$^{\\circ}$ to reproduce the Apophis' retrograde rotation \\citep{Pravec2014} and do not consider the precession period as it is 10 times larger than the rotational period. \n\n\\begin{figure}\n\\begin{center}\n\\subfloat{\\includegraphics[trim = 0mm 5.5cm 0mm 0mm,width=1\\columnwidth]{images\/apop_geom.png}}\n\\end{center}\n\\caption{\\label{fig:geom} Geometric altitude map computed across the\nsurface of Apophis under different views. The sea-level (the smaller distance between the geometric centre of the body and the barycentre among all triangular faces) is 130.54 metres.}\n\\end{figure}\n\n\nThe model for Apophis has an ellipsoid elongated format and tapered ends in the equatorial regions, as shown in its geometric map (Fig. \\ref{fig:geom}). To measure the geometric altitude, we determine the distance between the geometric centre of the body and the barycentre of each triangular face of the polyhedron. Then, we identify the smaller distance between these points and set it as ``sea-level\" \\citep{Scheeres2016}. The sea-level is 130.54 metres and it is almost half the value of the maximum distance calculated between the geometric centre and the barycentre among all the faces.\n\nAnalysing the geometric map, we notice that the maximum values of the geometric altitude are on the equatorial regions, while the minimum values are on the poles, which means that these regions are at the sea-level. The Top and Bottom views in Fig. \\ref{fig:geom} show the poles of the asteroid, and they shows that the north pole (Top view) has a concentrated region of the smaller altitudes on its middle. However, the south pole (Bottom view) presents a band of a larger distribution of small altitudes.\n\nApophis has a grain density in the range 3.4-3.6 g$\\cdot$cm$^{-3}$ and a total porosity with range of 4-62\\% because of its similar spectral characteristics with LL ordinary chondrite \\citep{BINZEL2009480}. This results in an approximate bulk density of 1.29-3.46 g$\\cdot$cm$^{-3}$. As the composition and some similarities suggest, Apophis could have a total porosity similar to the asteroid (25143) Itokawa. So \\citet{BINZEL2009480} adopted Apophis' radius as 135 metres, density of 3.2 g$\\cdot$cm$^{-3}$ and Itokawa's total porosity of 40\\% to estimate Apophis' mass as 2.0$\\times$10$^{10}$ kg. \\citet{dachwald2007head} assumed a spherical radius of 160 metres and a bulk density of 2.72 g$\\cdot$cm$^{-3}$ to determine the Apophis' mass as 4.67$\\times$10$^{10}$ kg. \\citet{muller2014thermal} also estimated a mass for Apophis, however they used a radius of 187.5 metres, density of 3.2 g$\\cdot$cm$^{-3}$ and total porosity of 30-50\\%, resulting in a total mass of 4.4-6.2$\\times$10$^{10}$ kg.  Therefore, for this work, we assumed three bulk densities of 1.29 g$\\cdot$cm$^{-3}$, 2.2  g$\\cdot$cm$^{-3}$ and 3.5 g$\\cdot$cm$^{-3}$.\n\nConsidering the density values adopted and the Apophis' size provided from \\citet{BROZOVIC2018115}, we constructed three models for the asteroid, one for each density. As the mass is not a known parameter, we set the volume of the asteroid as the same of a sphere with an equivalent radius of 170 metres. So, we preserved the volume and the size and estimate the mass for each model according to the density. The volume adopted was 0.0205 km$^3$ and the three calculated masses are 2.64$\\times$10$^{10}$ kg, 4.50$\\times$10$^{10}$ kg and 7.16$\\times$10$^{10}$ kg for the densities of 1.29 g$\\cdot$cm$^{-3}$, 2.2 g$\\cdot$cm$^{-3}$ and 3.5 g$\\cdot$cm$^{-3}$, respectively.\n\nWith the three models, we defined a coordinate system with the origin at the object centre of mass and align the system axes to the axes of the principal moment of inertia. The $x$, $y$, and $z$ axes correspond, respectively, to the smallest, intermediate, and largest moments of inertia. This process was made using the algorithm presented by \\citet{mir1996} and it was assumed that Apophis has a constant density and a uniform rotation about the largest moment of inertia. \n\nThe values of the principal moment of inertia normalized by the mass are: \n\n\\begin{equation}\nI_{xx}\/M = 559.23 \\ \\ensuremath{\\,\\textrm{m}}^{2},\n\\end{equation}\n\\begin{equation}\nI_{yy}\/M = 897.72 \\ \\ensuremath{\\,\\textrm{m}}^{2},\n\\end{equation}\n\\begin{equation}\nI_{zz}\/M = 969.39 \\ \\ensuremath{\\,\\textrm{m}}^{2}.\n\\end{equation}\n\nFrom the principal moment of inertia, we calculated the second-order degree terms $C_{20}$ and $C_{22}$ of the gravity expansion. The coefficient $J_2$ ($-C_{20}$) represents how oblate Apophis is and $C_{22}$ how elongated. The values are \\citep{macmillan1958theory, huscheeres2004}:\n\n\\begin{equation}\n C_{20}=-\\frac{1}{2R_{n}^2}(2I_{zz}-I_{xx}-I_{yy})= -0.00837 \\hspace{-0.4mm},\n\\label{eq:C20}  \n\\end{equation}\n\\begin{equation}\nC_{22}=\\frac{1}{4R_{n}^2}(I_{yy}-I_{xx})= 0.00294.\n\\label{eq: C22}  \n\\end{equation}\nwhere the equivalent radius, $R_{equivalent} = $ 170 metres, was used for normalization.\n\nNote that the coefficient $C_{20}$ is of the same order of magnitude as the coefficient $C_{22}$, but the absolute value of $C_{22}$ is smaller than $J_{2}$. We also modeled Apophis' shape as an ellipsoid with the parameters $a$, $b$ and $c$ representing the ellipsoid semi axes of the asteroid. The equivalent ellipsoid parameters for Apophis are:\n\n\\begin{equation}\na = 228.966631 \\ \\ensuremath{\\,\\textrm{m}},\n\\end{equation}\n\\begin{equation}\nb = 159.026432 \\ \\ensuremath{\\,\\textrm{m}},\n\\end{equation}\n\\begin{equation}\nc = 139.797960 \\ \\ensuremath{\\,\\textrm{m}}.\n\\end{equation}\n\nObserve that $a$ is about 64\\% larger than the parameter $c$, evidencing the elongated shape at the equator and flattened at the poles. So, considering the harmonic coefficients and the equivalent ellipsoid parameters, the shape of Apophis has a proportion of $5:3.5:3$ between the semi axes $a:b:c$, respectively.\n\n\\subsection{2029 Earth's Encounter}\n\nOn April 13 2029, Apophis will have the closest approach with the Earth at an approximate distance of $\\sim$38,000 km according to the JPL's HORIZONS ephemerides\\footnote{\\href{https:\/\/ssd.jpl.nasa.gov\/?horizons}{https:\/\/ssd.jpl.nasa.gov\/?horizons}}. The trajectory is shown in Fig. \\ref{fig:orbit_encounter}, where the black dot represents the Earth, the green circle and line illustrate the equatorial plane and Moon's orbit, respectively. The filled and the dashed blue lines represent the trajectory of the object above and below the equatorial plane, respectively, while the arrow illustrates the direction of the movement of Apophis. Note that Apophis approaches the Earth from left to right and it takes 34.30 hours to enter and leave the Moon's orbit. Apophis traverses the equatorial plane from the bottom-up and just crosses the equatorial plane near to the closest approach.\n\nSince we do not know the exact orientation at the encounter, we investigate several hypothetical orientations for the asteroid Apophis at the moment of the encounter with the Earth in 2029 and see if this approach could change the surface characteristics of the body (section \\ref{surface}). We also used the 2029 approach trajectory (Fig. \\ref{fig:orbit_encounter}) to compute the effects on the dynamical nearby environment (section \\ref{stability}). This analysis may provide insights to the 2029 observational campaign, and eventual space missions designed to study the asteroid.\n\n\\begin{figure}\n\\begin{center}\n\\subfloat{\\includegraphics[trim = 0mm 1.5cm 0mm 0mm,width=1\\columnwidth]{images\/ApophisTrajectory_xy.png}}\n\\end{center}\n\\caption{\\label{fig:orbit_encounter} Apophis' trajectory of the encounter with Earth in 2029 in the $xoy$ plane. The black dot represents the Earth and, the green line and circle represent the Moon's orbit and the equatorial plane, respectively. The blue line represents the trajectory above the equatorial plane and the blue dashed line represents the trajectory below the equatorial plane. The arrow represents the movement direction of Apophis.}\n\\end{figure}\n\n\\section{Effects on the Surface}\n\\label{surface}\nThe geopotential is an effective way to measure the relative energy on the body surface since it considers the gravitational and rotational potential \\citep{Scheeres2012, Scheeres2016}. It is possible to relate the geopotential energy with the motion of a cohesionless particle by associating the quantity of energy necessary to move this particle across the body surface. \n\nAssuming a constant angular velocity vector $\\pmb \\omega$, the reference frame centred at the centre of mass and the axes aligned to the principal inertia axes for the asteroid Apophis, the expression for the geopotential is \\citep{Scheeres2016}:\n\\begin{equation}\nV(\\pmb r) = -\\frac{1}{2} \\omega ^{2}(x^2+y^2) - U(\\pmb r),\n\\label{eq:geopotencial}\n\\end{equation}\nwhere $\\pmb r$ is the position vector of a massless particle in the body-fixed frame and $U(\\pmb r)$ the gravitation potential that is given by the method of polyhedra.\n\nTo model the object we used the polyhedra method, that computes the gravitational potential energy of an irregular body modeled by a uniform density polyhedron with a given number of faces and vertices. Then, the expression for the gravitational potential is \\citep{wernerscheeres1996}:\n\\begin{equation}\nU =\\frac{G\\rho}{2} \\sum_{e\\in edges}{\\pmb r_e} \\cdotp {\\pmb E_e} \\cdotp {\\pmb r_e} \\cdotp L_e -  \\frac{G\\rho}{2} \\sum_{f\\in faces}{\\pmb r_f} \\cdotp {\\pmb F_f} \\cdotp {\\pmb r_f} \\cdotp \\omega_f,\n \\label{eq: potencial}  \n\\end{equation}\nwhere $\\rho$ is the density of Apophis, $G$ is the gravitational constant; $\\pmb r_f$ and $\\pmb r_e$ represent, respectively, the position vectors from a point in the gravitational field to any point in the face $f$  and edge $e$ planes; $\\pmb F_f$ and $\\pmb E_e$ are the faces and edges tensors; $\\omega_f$ and $L_e$ are the signed angle viewed from the field point and the integration factor, respectively.\n\nFrom the geopotential, \\citet{Scheeres2016} derive the slope angle using its gradient (see Section \\ref{slope}), this measure considers the effects on the surface of the body, but we also can derive an approach for a particle in the nearby environment of the asteroid.\n\nSince it is considered that Apophis has a constant angular velocity ($\\pmb \\omega$) about its axis of maximum moment of inertia, the movement of a particle orbiting the nearby environment is described by \\citep{Scheeres2016}:\n\\begin{equation}\n\\ddot{\\pmb r} +2{\\pmb\\omega} \\times \\dot{\\pmb r} =-\\frac{\\partial V}{\\partial \\pmb {r}},\n\\label{eq: eqs do movimento}\n\\end{equation}\nwhere $\\pmb r$ is the position vector of the particle, $\\dot{\\pmb r}$ and $\\ddot{\\pmb r}$ are its velocity and acceleration vectors, respectively.\n\nEquation \\ref{eq: eqs do movimento} is time-invariant since the Apophis' spin-rate is assumed to be a constant value. Therefore this equation can be associated with a conserved quantity, $C_j$,  called ``Jacobi constant\". This constant is defined by \\citep {Scheeres2016}:\n\\begin{equation}\nC_j = \\frac{1}{2}v^2+V(\\pmb r),\n\\label{eq: jacobi}\n\\end{equation}\nwhere the magnitude of the velocity vector relative to the rotating frame of the Apophis is denoted by $v$.\n\n\n\\subsection{Slope}\n\\label{slope}\n\nThe slope angle is the supplement between the normal and total acceleration vectors \\citep{Scheeres2012, Scheeres2016}. The slope is an angle that quantifies how inclined is a region on the Apophis' surface with respect to its local acceleration vector. Physically, slope assists in understanding the movement of free particles on the Apophis surface.\n\nThe slope angle distribution across the Apophis' surface for the larger and smaller densities is shown in Fig. \\ref{fig:slope}. The variation between the models is small and the slope distribution is almost identical. The slope variation amplitude goes from 35.81$^{\\circ}$ for the smaller density, up to 36.16$^{\\circ}$ for the larger density, a difference of only 0.35$^{\\circ}$. \n\n\\begin{figure}\n\\begin{center}\n\\subfloat[$\\rho$ = 1.29 g$\\cdot$cm$^{-3}$]{\\includegraphics*[trim = 0mm 4.5cm 0mm 0mm,\nwidth=1\\columnwidth]{images\/apop_slope_1.29.png}\\label{fig:slope_129}}\\\\\n\\subfloat[$\\rho$ = 3.5 g$\\cdot$cm$^{-3}$]{\\includegraphics*[trim = 0mm 4.5cm 0mm 0mm, width=1\\columnwidth]{images\/apop_slope_3.5.png}\\label{fig:slope_35}}\\\\\n\\end{center}\n\\caption{\\label{fig:slope} Slope angle maps across the surface of Apophis under different views. The letters (a) and (b) represent, respectively, the slope distribution considering the densities of 1.29 g$\\cdot$cm$^{-3}$ and 3.5 g$\\cdot$cm$^{-3}$.}\n\\end{figure}\n\nFor all three cases, the maximum slope angles are smaller than 37$^{\\circ}$, and the minimum angles are larger than 0.5$^{\\circ}$. The maximum values of slope occur just in the small regions on the equatorial extremities of Apophis (see front, left, and bottom views in Fig. \\ref{fig:slope}). The minimum values of the slope are distributed preferentially at the central region of the body as we can see in the bottom view in Fig. \\ref{fig:slope}, the south pole has a strip of minimum values is passing through the middle of the surface.\n\nBy definition, slope angles varies between 0$^{\\circ}$ and 180$^{\\circ}$. When it is larger than 90$^{\\circ}$, the centrifugal force is larger than the gravitational force, causing cohesionless particles to escape from the body's surface. If the slope angle is smaller than 90$^{\\circ}$, the movement of the cohesionless particles can be associated with the repose angle of a given material. The angle of repose for geological material is about 35-40 degrees \\citep{lambe1969, apollo1974, al2018review}. \n\nAs already referred, the maximum value of the slope considering the three values of density is about 37$^{\\circ}$. So we can see two regimes, one for the lower limit value of the angle of repose, 35$^{\\circ}$, and the other for the upper limit value, 40$^{\\circ}$. For the lower limit, the flow of the cohesionless particles may occur, since we did not find a slope value higher than this limit.\n\n\\begin{figure}\n\\begin{center}\n\\subfloat{\\includegraphics[trim = 0mm 4.5cm 0mm 0mm,width=1\\columnwidth]{images\/apop_slope_seta_6.png}}\n\\end{center}\n\\caption{\\label{fig:slope_setas} Slope angle map computed across the surface of Apophis with the directions of the local acceleration tangent vectors and under different views for the density of 2.2 g$\\cdot$cm$^{-3}$.}\n\\end{figure}\n\nTo understand the possible flow of free particles on Apophis' surface, we compute the directions of the tangential component of the local acceleration vector \\citep{Scheeres2012, Scheeres2016}. Figure \\ref{fig:slope_setas} shows the tangential acceleration vectors considering a density of 2.2 g$\\cdot$cm$^{-3}$. \n\nAs expected, the vectors are pointing to the regions where the slope is smaller. The flow of loose material might occur from the small regions on the equatorial extremities to the middle of the Apophis' surface. The dark regions in Fig. \\ref{fig:slope_setas} represent the near-zero slopes, thus are stable resting areas. Those regions are propitious regions to accumulate cohesionless particles.\n\n\\begin{figure}\n\\begin{center}\n\\subfloat[$\\rho$ = 1.29 g$\\cdot$cm$^{-3}$]{\\includegraphics*[trim = 0mm 3cm 11cm 0mm,\nwidth=0.9\\columnwidth]{images\/delta_slope_129.png}\\label{fig:slope_129_terra}}\\\\\n\\subfloat[$\\rho$ = 3.5 g$\\cdot$cm$^{-3}$]{\\includegraphics*[trim = 0mm 3cm 11cm 0mm, width=0.9\\columnwidth]{images\/delta_slope_35.png}\\label{fig:slope_35_terra}}\\\\\n\\end{center}\n\\caption{\\label{fig:slope_terra} The $\\Delta$slope angle maps across the surface of Apophis under different views and considering the Earth perturbation in 2029 encounter. The letters (a) and (b) represent, respectively, the slope distribution considering the densities of 1.29 g$\\cdot$cm$^{-3}$ and 3.5 g$\\cdot$cm$^{-3}$.}\n\\end{figure}\n\nApophis will have a close approach with the Earth in 2029 and will be affected by time-varying forces. Thus, to evaluate the strongest possible effect on its surface due to the Earth`s gravity, we calculated a $\\Delta$slope that computes the slope variation caused due to the encounter at the closest distance of the whole approach.\n\nIn order to identify the most extreme condition that may occur in 2029, we assume a hypothetical configuration for the Apophis-Earth system. We set the Earth on the equatorial plane of Apophis and the distance between the centre of mass of Apophis and the Earth as the closest distance during the whole encounter, and  we calculate the slope on the Apophis' surface considering its geopotential including the Earth's gravitational force on the closest approach. Then, we rotate the shape model of Apophis all along the 360$^{\\circ}$, so that each Apophis' orientation experiences the extreme approach condition.\n\nThe difference between the total acceleration vectors, with and without the Earth's gravity, leads to a slope that we call a delta slope. The delta slope is the variation of the slope angle produced by the Earth's gravitational perturbation on the surface of Apophis at the extreme approach condition. The $\\Delta$slope map was also made over the rotation of 360$^{\\circ}$ of the Apophis' shape model, in order to compute the $\\Delta$slope on each Apophis' orientation.\n\nFigure \\ref{fig:slope_terra} shows the $\\Delta$slope for the smaller and larger density of Apophis (see the complete animation of the $\\Delta$slope angle maps in the videos of the complementary material). The top and bottom views are not shown since they are not directly pointed to the Earth at the closest approach, as we defined before. For the 1.29 g$\\cdot$cm$^{-3}$ density model, the slope variation was smaller than 4$^{\\circ}$, and for 3.5 g$\\cdot$cm$^{-3}$ density model was about 2$^{\\circ}$. This variation represents about 11\\% of the regular slope angle for the smaller density model and 5.5\\% for the larger density model.  \n\nWith the perturbation caused by the Earth, some slope angle values may exceed the angle of repose. \\citet{ballouz2019surface} showed that a variation around 2$^{\\circ}$ may sufficient to start a slow erosion process in regions with high-slope that experiment larger perturbations. However, the perturbations due to the 2029 encounter will not trigger drastic reshaping on Apophis' surface and shape. The numerical simulations provided by \\citet{yu2014numerical} revealed that the effects caused by the Earth are small and just can cause local landslides.\n\n\\section{Effects on the Nearby Stability}\n\\label{stability}\n\nIn this section, we present an investigation of the environment around Apophis. The analysis of the variation of the Jacobi constant, through zero-velocity curves, allows us to identify where the movement is permitted or not. These curves delimit the external equilibrium points of the body, the location and the topological classification of these points are presented. A set of numerical simulations with a disc of particles is also presented in order to analyse the stability around the asteroid considering the perturbation of its own gravitational field and two additional perturbations: the solar radiation pressure and the 2029 flyby scenario.\n\n\\subsection{Equilibrium Points and Zero-Velocity Curves}\n\\label{points}\n\nTo understand the stability in the vicinity of Apophis, we calculated the zero-velocity curves and equilibrium points. The zero-velocity curves limit the movement of a particle, delimiting where its movement is allowed or not. This delimited movement depends on the Jacobi constant, C$_j$, and,  it arises from the inequality\n\\begin{equation}\nC_j - V(\\pmb r) \\ge 0,\n\\label{eq: inequality}\n\\end{equation}\nsince the velocity term, $\\frac{1}{2}v^2$, from equation \\ref{eq: jacobi} shall be always positive. So, when $ C_j < V(\\pmb r)$ there will be forbidden regions to the movement of the particle, considering that the inequality is infringed. In the case of the regions where $ C_j > V(\\pmb r)$, there will be no a priori limitation to the movement of the particle, since the inequality is satisfied. When $ C_j = V(\\pmb r)$ we have the zero-velocity curves that separate the regions between the permitted and prohibited movement.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics*[trim = 0mm 0cm 0mm 0cm,\nwidth=1\\columnwidth]{images\/apop_cvz_2.2.png}\n\\end{center}\n\\caption{\\label{fig:cvz}  Zero-velocity curves contourplots in the $xoy$ plane considering the mean density of 2.2 g$\\cdot$cm$^{-3}$. The colorbox represents the value of the Jacobi constant.}\n\\end{figure}\n\nFig. \\ref{fig:cvz} illustrates the zero-velocity curves countourplots in the $xoy$ plane for the model with the mean density. The colour of the zero-velocity curves changes as the Jacobi constant value changes. At a certain value of C$_j$, the curves could form confined regions that will encompass an equilibrium point, similar to a ``banana\" shape in Fig. \\ref{fig:cvz}. The region between these confined curves also will include an equilibrium point.\n\nThe equilibrium points are critical points of geopotential where there is a balance between the gravitational and centrifugal forces. Since the resulting force is null at this points, the equilibrium points of the asteroid Apophis are calculated by the solution of:\n\\begin{equation}\n\\nabla V = 0,\n\\label{eq: eq_point}\n\\end{equation}\nwhere the $\\nabla V$ means the gradient of the geopotential.\n\n\nIn general, there is no fixed number of solutions for equation \\ref{eq: eq_point}, since the solution depends on the body shape model, its density, and rotational period. However, we could estimate the number of equilibrium points just by looking at the zero-velocity curves of the body. The zero-velocity curves of Apophis show it has at least four equilibrium points, two of them inside the curves with a ``banana\" shape  and two between the larger ``banana\" curves near to the $y=0$ line (Fig. \\ref{fig:cvz}).\n\nThe location of the equilibrium points also depends on the rotational period, density, and shape of the asteroid. Once defined the shape model and the spin, if we change its density, the location of the equilibrium points will change as well. If the density decreases, the location of the equilibrium points will be closer to the body due to the reduction of the gravitational force. Conversely, if the density is increased, the equilibrium points will be more distant from the body. \n\n\\begin{figure}\n\\begin{center}\n\\includegraphics*[trim = 0mm 2.5cm 0mm 6cm,\nwidth=1\\columnwidth]{images\/apop_pontos.png}\n\\end{center}\n\\caption{\\label{fig:pontos_eq}  Location of the equilibrium points in the $xoy$ plane for each density. The point's color represents the density of Apophis.}\n\\end{figure}\n\nConsidering the polyhedral model derived from \\citet{Pravec2014} and the densities of 1.29 g$\\cdot$cm$^{-3}$,  2.2 g$\\cdot$cm$^{-3}$, and 3.5 g$\\cdot$cm$^{-3}$, we found five equilibrium points for each density, being one in the centre of the body. The position of the equilibrium points are shown in table \\ref{table:autovalores}. We note they are close to the equatorial plane, so Fig. \\ref{fig:pontos_eq} shows the projection of the asteroid Apophis and the equilibrium points in the $xoy$ plane. \n\nNote that the number of equilibrium points does not change, differently from other systems \\citet{jiang2018annihilation}. Observe that although the model with a larger density is almost three times larger than the model with a smaller density, the difference in the radial distance between the equilibrium points of the model with larger and smaller densities is about 300 metres.\n\n\\begin{table*}\n\\centering\n\\caption{Coordinates and characteristic time of the equilibrium points of 99942 Apophis considering the three densities (1.29 g$\\cdot$cm$^{-3}$,  2.2 g$\\cdot$cm$^{-3}$, and 3.5 g$\\cdot$cm$^{-3}$) and their topological structures.}\n\\label{table:autovalores} \n\\begin{tabular}{c|r|r|r|r|c}\n\\hline\\hline\n\\multicolumn{1}{c}{{Point}} & \\multicolumn{1}{c}{{$x$ (km)}} & \\multicolumn{1}{c}{{$y$ (km)}} & \\multicolumn{1}{c}{{$z$ (km)}} & \\multicolumn{1}{c}{{Topological classification}} & \\multicolumn{1}{c}{{ \\begin{tabular}{@{}c@{}}Characteristic Time \\\\ (hours)\\end{tabular}}}\\\\\n\\hline\\hline\n\\multicolumn{6}{|c|}{$\\rho$ = 1.29 g$\\cdot$cm$^{-3}$}\\\\\n\\hline\n    $E_1$ & 0.764033 & -0.014211 & -0.001470 & Saddle-Centre-Centre & \\begin{tabular}{@{}c@{}}26.83192 \\\\ 26.97251\\end{tabular}\\\\\n     \\hline\n    $E_2$ & -0.038260& -0.752882 &  0.000598 & Centre-Centre-Centre & \\begin{tabular}{@{}c@{}}27.28828 \\\\ 28.90402 \\\\ 88.39612\\end{tabular}\\\\\n     \\hline\n    $E_3$ & -0.765131& -0.007504 & -0.001544 & Saddle-Centre-Centre & \\begin{tabular}{@{}c@{}}26.57328 \\\\ 26.90200\\end{tabular}\\\\  \n     \\hline\n    $E_4$ & -0.041584& 0.752713  & 0.000675 & Centre-Centre-Centre & \\begin{tabular}{@{}c@{}}27.30720 \\\\ 28.79664 \\\\ 90.94898\\end{tabular}\\\\ \n\\hline \\hline\n\\multicolumn{6}{c}{$\\rho$ = 2.2 g$\\cdot$cm$^{-3}$}\\\\\n\\hline\n    $E_1$ & 0.910268  & -0.013482 & -0.001033 & Saddle-Centre-Centre & \\begin{tabular}{@{}c@{}}26.98514 \\\\ 27.08051\\end{tabular}\\\\\n     \\hline\n    $E_2$ & -0.038718 & -0.900849 &  0.000424 & Centre-Centre-Centre & \\begin{tabular}{@{}c@{}}27.31676 \\\\ 28.38482 \\\\ 107.46649 \\end{tabular}\\\\\n     \\hline\n    $E_3$ & -0.911036 & -0.007877 & -0.001078 & Saddle-Centre-Centre & \\begin{tabular}{@{}c@{}}26.82109 \\\\ 27.04486\\end{tabular}\\\\  \n     \\hline\n    $E_4$ & -0.041482 & 0.900733  & 0.000470 & Centre-Centre-Centre & \\begin{tabular}{@{}c@{}}27.32790 \\\\ 28.32686 \\\\ 110.03305\\end{tabular}\\\\ \n\\hline \\hline\n\\multicolumn{6}{c}{$\\rho$ = 3.5 g$\\cdot$cm$^{-3}$}\\\\\n\\hline\n    $E_1$ & 1.060747  & -0.012966 & -0.000760 & Saddle-Centre-Centre & \\begin{tabular}{@{}c@{}}27.08273 \\\\ 27.15419\\end{tabular}\\\\\n     \\hline\n    $E_2$ & -0.039031 & -1.052625 & 0.000314 & Centre-Centre-Centre & \\begin{tabular}{@{}c@{}}27.33415 \\\\ 28.09159 \\\\ 126.74930\\end{tabular}\\\\\n     \\hline\n    $E_3$ & -1.061310 & -0.008169 & -0.000789 & Saddle-Centre-Centre & \\begin{tabular}{@{}c@{}}26.97445 \\\\ 27.13409\\end{tabular}\\\\  \n     \\hline\n    $E_4$ & -0.041388 & 1.052541  & 0.000343 & Centre-Centre-Centre & \\begin{tabular}{@{}c@{}}27.34118 \\\\ 28.05680 \\\\ 129.32771\\end{tabular}\\\\ \n\\hline \\hline\n\\end{tabular}\n\\end{table*}\n\nApplying the linearization method to the equations of motion, we analysed the six eigenvalues of the characteristic equation for each point in order to identify their topological classification \\citep{Jiang2014}. Table \\ref{table:autovalores} shows the coordinates and the topological classification of each equilibrium point for each density of Apophis. The odd indexed points, $E_1$ and $E_3$, are classified as a Saddle-Center-Center, while the even points ($E_2$ and $E_4$) are Center-Center-Center, implying that they are linearly stable points. \n\nAccording to \\citet{Jiang2014} and \\citet{yu2012orbital}, when analysing the eigenvalues of the characteristic equation, we identified the existence of three families of periodic orbits in the tangent space of the equilibrium points $E_2$ and $E_4$, which have characteristic times or oscillation periods of approximately 27.3 hr, 28.9 hr and 88.4 hr for $E_2$, and 27.3 hr, 28.8 hr and 90.9 hr for $E_4$, when we consider the density of 1.29 g$\\cdot$cm$^{-3}$. Only the period value of the third periodic orbit family underwent a significant change, with an increase of about 21\\% and 43\\%, respectively, for the Apophis intermediate and upper densities, when we analysed the $E_2$ equilibrium point eigenvalues. The same conclusion can be applied to point $E_4$. While for points $E_1$ and $E_3$, there are two families of periodic orbits, which have periods of oscillation around 26.8 hr and 26.9 for $E_1$, and 26.6 hr and 26.9 hr for $E_3$, and these values undergo small changes, less than 30 minutes, for other Apophis densities (Table \\ref{table:autovalores}).\n\nNote that despite of changing the density of Apophis, the topological classification of the equilibrium points $E_2$ and $E_4$ remains as linearly stable. Thus, we performed numerical simulations of a disc of particles encompassing the equilibrium points regions in order to identify possible stable zones around them.\n\n\n\\subsection{Stability Regions}\n\\label{regions}\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics*[trim = 0cm 0cm 0cm 0mm,\nwidth=1\\columnwidth]{images\/Parameters-Apophis_large-V2.png}\n\\end{center}\n\\caption{\\label{fig:param} Variation of the parameters oblateness (blue), Earth and Sun tide (black and green, respectively), the solar radiation pressure for a particle with a radius of 1 cm (pink) and 15 cm (red) and the Apophis gravity (yellow) as a function of the distance from Apophis. The shaded region indicates the variation of these parameters between the pericentre and apocentre. The vertical lines represent the limits of the radial location of the the equilibrium points for the density model of 1.29 g$\\cdot$cm$^{-3}$ and 3.5 g$\\cdot$cm$^{-3}$. The equivalent radius, $R_{equivalent}$, of Apophis is 170 metres.}\n\\end{figure}\n\nThere have been some studies \\citep{aljbaae2020close, aljbaae2021influence, lang2021spacecraft} investigating suitable orbits around Apophis for a spacecraft taking into account the irregular gravitational field of the asteroid and the solar radiation pressure for an area to mass ratio similar to the OSIRES-REX spacecraft. Some stable orbits were found, but during the 2029 close encounter, the majority of the orbits for a spacecraft suffer a collision or ejection.\n\nThe small particles around Apophis may be subject to a plethora of forces besides the gravitational potencial of the main body: the oblateness coefficient add an extra gravitational pull and there are the tides by the Sun and the Earth at the close encounter. If the grains are small they can also experience the disturbance due to the solar radiation force.\n\nTo estimate which perturbation may be relevant, we computed adimensional parameters that allow us to analyze the relative strength of each force (for a detailed definition of the parameters, see \\citet{hamilton1996circumplanetary, moura2020dynamical}). Figure \\ref{fig:param} shows the parameter strengths according to the distance from Apophis: the solar tide (green), Earth's gravity at the closest approach distance (black), the solar radiation force for grains with radius of 1 cm (pink) and 15 cm (red); it is also shown the oblateness effect (blue) and the Apophis gravity (yellow).  For each parameter the lines are calculated for the nominal density (2.2 g$\\cdot$cm$^{-3}$) and distance equal to the semimajor axis, while  the shaded region corresponds to the variation of these parameters between the pericentre and apocentre.\n\nIn Fig. \\ref{fig:param} the vertical lines indicate the limits of the radial location of the equilibrium points for the density model of 1.29 g$\\cdot$cm$^{-3}$ and 3.5 g$\\cdot$cm$^{-3}$. The Apophis gravity has the major magnitude among the other perturbations, about seven orders of magnitude larger than the Sun tide and five orders larger than the Earth tide at the closest approach distance. The oblateness of Apophis as the Sun tide reaches the same magnitude at a distance of about 9 radii of Apophis. The Earth tide is also lower than the oblateness near the asteroid and they both are comparable in the region of the equilibrium points. Beyond this distance the Earth tide dominates the dynamic over the oblateness. So, the Sun tide will not contribute with major perturbations to the system.\n\nThe solar radiation pressure for a particle with a 1 cm of radius gets a larger order of magnitude than the oblateness at about 3 radii of Apophis, what means that this particle will suffer a major influence of the solar radiation pressure. However, for a particle with a 15 cm of radius, the solar radiation pressure is about one order of magnitude smaller, and it is equivalent to the oblateness near the region of the equilibrium points. Thus, a particle of 15 cm is more likely to survive in the nearby environment.\n\nIn the current work we are concerned with natural objects, such as dust, boulders, fragments or even larger bodies that might be orbiting around Apophis. Therefore, aiming to identify the size and location of possible stable regions around Apophis and\/or around the equilibrium points, we performed sets of numerical simulations of particles around these regions. We numerically integrated a disc with 15,000 massless particles initially for 24 hours (in order to compare to the close encounter with the Earth in 2029 showed in Fig. \\ref{fig:orbit_encounter}) and subsequently for 30 years ($\\sim$10,000 times the rotation period of the asteroid). \n\nThe particles were distributed at an initial distance of 300 metres from the asteroid centre of mass and different widths for each density model, since the position of the equilibrium points changes as the density changes. For the 1.29 g$\\cdot$cm$^{-3}$, 2.2 g$\\cdot$cm$^{-3}$ and 3.5 g$\\cdot$cm$^{-3}$ densities, the final distances were 1.0 km, 1.15 km, and 1.3 km, respectively. All the particles were assumed to be initially with Keplerian angular velocity in the equatorial plane and circular orbits.\n\nTo perform these simulations, we used an N-body integrator package called $N$-$BoM$ \\citep{moura2020dynamical, Winter2020}. It considers the gravitational potential of an irregular body as a mass concentration model, MASCONS \\citep{Geissler1996}. \nTo reproduce the Apophis' gravitational potential field, we calculate the sum of the gravitational potential of all the masses points as \\citep{borderes2018}\n\\begin{equation}\n U (x, y, z)= \\sum_{i=1}^{N} \\frac{Gm}{r_i},\n \\label{eq: mascon}  \n\\end{equation}\nwhere $N$ is the number of mass points (which is 20,457 in our model), $r_i$ is the distance mascon-particle, and $m$ is the mass of each mascon. As the sum of all mascons is equal to the Apophis' total mass, $M_{Apophis}=Nm$, the mass of each mascon is different according to the density model. \n\nWe considered a particle as ejected if it reaches a distance larger than ten times the distance between the Apophis' centre of mass and the equilibrium point, this is almost equivalent to half of the Hill radius \\citep{hamilton1992orbital} of Apophis with respect to the Sun. For the 24 hours simulations, an additional criterion of ejection was considered since some particles may not have sufficient time to exceed the ejection distance due to the short time of the simulation. So we also considered as ejected those particles with positive energy. In this way, a region is called stable where the particles remain at the final integration time, obviously, without collision with the body or be ejected.\n\n\\begin{figure*}\n\\begin{center}\n\\subfloat[$\\rho$ = 1.29 g$\\cdot$cm$^{-3}$]{\\includegraphics*[trim = 0cm 7cm 7cm 0mm,\nwidth=0.66\\columnwidth]{images\/apop_24h_sem_Terra_1.29.png}\\label{afig:24h_129}}\n\\subfloat[$\\rho$ = 2.2 g$\\cdot$cm$^{-3}$]{\\includegraphics*[trim = 0mm 7cm 7cm 0mm,\nwidth=0.66\\columnwidth]{images\/apop_24h_sem_Terra_2.2.png}\\label{bfig:24h_22}}\n\\subfloat[$\\rho$ = 3.5 g$\\cdot$cm$^{-3}$]{\\includegraphics*[trim = 0mm 7cm 7cm 0mm, width=0.66\\columnwidth]{images\/apop_24h_sem_Terra_3.5.png}\\label{cfig:24h_35}}\n\\end{center}\n\\caption{\\label{fig:24h} Initial conditions of the 15 thousand particles around the asteroid Apophis in the $xoy$ plane. The green dots represent the equilibrium points. The black dots represent the particles that survived after the 24 hours of integration and the blue dots the particles that collide with the asteroid. The letters (a)-(c) represent, respectively, the densities of 1.29 g$\\cdot$cm$^{-3}$, 2.2 g$\\cdot$cm$^{-3}$ and 3.5 g$\\cdot$cm$^{-3}$.}\n\\end{figure*}\n\n\\begin{figure}\n\\begin{center}\n\\subfloat[$\\rho$ = 1.29 g$\\cdot$cm$^{-3}$]{\\includegraphics*[trim = 0mm 6cm 0cm 1cm,\nwidth=1\\columnwidth]{images\/129_30.png}\\label{afig:30y_129}}\\\\\n\\subfloat[$\\rho$ = 3.5 g$\\cdot$cm$^{-3}$]{\\includegraphics*[trim = 0mm 6.8cm 0cm 1.5cm, width=1\\columnwidth]{images\/35_30.png}\\label{cfig:30y_35}}\n\\end{center}\n\\caption{\\label{fig:30y} Initial conditions of the 15 thousand particles around the asteroid Apophis in the $xoy$ plane. The green dots represent the equilibrium points. The black dots represent the particles that survived after 30 years, the blue and pink dots are the particles that collide with the asteroid and are ejected from the system, respectively. The letters (a) and (b) represent, respectively, the densities of 1.29 g$\\cdot$cm$^{-3}$ and 3.5 g$\\cdot$cm$^{-3}$.}\n\\end{figure}\n\nFigure \\ref{fig:24h} shows the initial conditions of the 15,000 particles, colored according to their final outcome at the end of the 24 hours of integration. The blue dots represent the particles that collided with Apophis, while the black dots indicate the particles that have survived the entire simulation. We note that the distribution of the particles that collided with the surface is similar among the three density models, and they are located in two preferential regions closer to the surface of Apophis and near to the equatorial extremities of the body, so the gravitational field of the equatorial extremities is related to the cause of these collisions. \n\nMost of the particles survived, but we can note a slight difference of the collision percentage among the different density models. For the smaller density, the percentage of particles that collided was about 1.4\\%, while for the larger model was only 0.78\\%. Although these percentage of collisions, the majority of the disc is stable for the whole time of integration. In section \\ref{instability}, we will simulate these same conditions adding the perturbation of the Earth during the trajectory of the 2029 encounter (Fig. \\ref{fig:orbit_encounter}) to see its effects in the nearby environment of Apophis.\n\nExtending the simulation for 30 years, we obtain the results shown in Fig. \\ref{fig:30y}. The region of collided particles near the asteroid still remains the same for the large density, but an increase of collided particles is noticed for the smaller density. There is a random distribution of collided particles at the proximities of the equilibrium points and a thin ring of collided particles at about 500-600 metres from Apophis. Some of these particles remain in the system for about 10 years and are chaotic, presenting a large orbital radius variation and inclination. \n\nThe ejected particles of the larger density model are distributed in regions involving the equilibrium points, similar to a zero-velocity curve. Note that these particles are located in a limit region where the gravitational and centrifugal forces of the central body exerts influence on them. This means that outside these regions the Apophis influence is sufficiently weak to cause perturbations on the particles, so they survived as we can note in Fig. \\ref{cfig:30y_35}. The same behavior is presented in the smaller density model but, since the mass is smaller so the gravitational perturbation exerted on the particles, the ejected regions are larger.\n\nThe survived particles are the majority for the larger density model, about 96\\% of particles survived for the 30 years simulation, while 3.1\\% were ejected and 0.9\\% collided with Apophis. Meanwhile, for the smaller density model, 48\\% were ejected or collided with Apophis and 52\\% survived for 30 years. For both densities, the survived particles are located all over the disc, but there are three preferentially concentrated regions of them (Fig. \\ref{fig:30y}). The first one is the region inner the equilibrium points due to the gravitational influence of Apophis. The second one is around both linearly stable equilibrium points $E_2$ and $E_4$. The third region is the region outside the influence of the perturbation of Apophis near the ejected particles, as discussed before. The smaller density model presents some agglomerate of survived particles inside the region of ejected particles, these regions will be studied in detail in future works.\n\n\nSince the stability region of prograde orbits is about half the Hill radius \\citep{domingos2006stable, hunter1967motions}, we expand the radial distribution of the particles around Apophis for the 24 hours simulations. For the smaller, mean, and larger densities, the new radial distances, $r$, were 1.0 $< r <$ 7.5 km, 1.15 $< r <$ 9.0 km, and 1.3 $< r <$ 10.5 km, respectively.\n\nAll the 15,000 particles simulated for this new radial distance have survived for the entire 24 hours simulations. The particles are distant from the Apophis, so its irregular gravitational field is low and the system can be interpreted as a two-body problem, thus the region is stable as expected. Later we compare these simulations when the perturbation of the Earth during the 2029 encounter is taken into account.\n\n\\subsubsection{Solar Radiation Pressure}\n\\label{solar}\n\nThe previous simulations were performed considering only the gravitational potential of Apophis. However, the solar radiation pressure may cause a significant perturbation on the particles' evolution. To identify the size of the particles that may survive the effects of the solar radiation pressure, we assume particles with different sizes and simulate the same conditions described in section \\ref{regions} adding the solar radiation pressure perturbation in the system.\n\nWe have carried out numerical simulations with micrometric-sized particles, and the entire ensamble did not survive for the 30 years for particles smaller than 100 $\\mu$m. This allowed us to set a lower limit for the particle radius that could survive despite of the solar radiation disturbance. So, we proceed the simulations with the $N$-$BoM$ package for particles with radius larger than 100 $\\mu$m. Considering the area to mass ratio ($A$) of the particle and a solar constant in function of the solar luminosity and speed of light ($G^*=1\\times10^{17}$ kg$\\cdot$m$\\cdot$s$^{-2}$), the radiation pressure acceleration is given by \\citep{scheeres2002spacecraft}\n\n\\begin{equation}\n\\vec{a}_{srp} = \\frac{AG^{*}(1+\\eta)}{R^3}\\vec{R},\n \\label{eq: acel_medium}  \n\\end{equation}\nwhere $R$ is the module of the position vector Sun-particle ($\\vec{R}$) and $\\eta$ is the reflectance of the particle, which is assumed to be a unit for a totally reflective material. We do not consider the effects caused by shadowing since they are not significant to produce a considerable change in the surviving scenario, i.e particles that survive the entire simulation without ejection or collision with Apophis.\n\nSince our goal was to determine the order of magnitude of particle's size that could survive in the system despite the perturbation due to the solar radiation force, we simulate a set of particles with radius ($r_p$) in a discrete distribution of 1 cm, 5 cm, 10 cm, and 15 cm. The particles have the same density of Apophis considering each density model (1.29 g$\\cdot$cm$^{-3}$, 2.2 g$\\cdot$cm$^{-3}$ or 3.5 g$\\cdot$cm$^{-3}$).\n\n\\begin{figure*}\n\\begin{center}\n\\subfloat[$r_p$ = 15 cm and $\\rho$ = 1.29 g$\\cdot$cm$^{-3}$]{\\includegraphics*[trim = 0mm 7cm 7cm 0mm,\nwidth=0.66\\columnwidth]{images\/apop129_anel_15cm.png}\\label{afig:30y_129_srp}}\n\\subfloat[$r_p$ = 5 cm and $\\rho$ = 2.2 g$\\cdot$cm$^{-3}$]{\\includegraphics*[trim = 0mm 7cm 7cm 0mm,\nwidth=0.66\\columnwidth]{images\/apop22_anel_5cm.png}\\label{bfig:30y_22_srp}}\n\\subfloat[$r_p$ = 5 cm and $\\rho$ = 3.5 g$\\cdot$cm$^{-3}$]{\\includegraphics*[trim = 0mm 7cm 7cm 0mm, width=0.66\\columnwidth]{images\/apop35_anel_5cm.png}\\label{cfig:30y_35_srp}}\n\\end{center}\n\\caption{\\label{fig:30y_srp} Initial conditions of the 15 thousand particles around the equilibrium points of the asteroid Apophis in the $xoy$ plane considering the perturbation of the solar radiation pressure. The green dots represent the equilibrium points. The black dots represent the particles that survived after the 24 hours of integration and the blue dots the particles that collide with the asteroid. The letters (a)-(c) represent, respectively, the densities of 1.29 g$\\cdot$cm$^{-3}$, 2.2 g$\\cdot$cm$^{-3}$ and 3.5 g$\\cdot$cm$^{-3}$. For the densities of 2.2 g$\\cdot$cm$^{-3}$ and 3.5 g$\\cdot$cm$^{-3}$ the radius of the particles is 5 cm and for the density of 1.29 g$\\cdot$cm$^{-3}$, 15 cm.}\n\\end{figure*}\n\n\nFor the lower density, Figure \\ref{afig:30y_129_srp} shows the initial conditions of the particles where the blue dots represent the collided particles and the black dots the survivors. For the sizes of 1 cm and 5 cm, no particles survived, and for the radius of 10 cm, just one particle survived. Then, we simulated particles with radius of 15 cm, and for this, about 3\\% of the 15,000 particles survived 30 years. We notice that the survived particles are concentrated in three small regions near the inner edge of the disc and, consequently, close to Apophis.\n\nFor the density models of 2.2 g$\\cdot$cm$^{-3}$ and 3.5 g$\\cdot$cm$^{-3}$, there are no survivors for 1 cm particles, but for particles with radius of 5 cm it has about 3.5\\% of survivors for the mean density model and almost 18\\% for the larger one. The small regions with surviving particles for the mean density model are similar to the regions of the smaller model, also presenting three defined regions (Fig. \\ref{bfig:30y_22_srp}). However, the larger density model has a single ring of survived particles (Fig. \\ref{cfig:30y_35_srp}). Note that we can clearly see the same two preferential regions of collisions close to the  Apophis' surface existent in the 24 hours simulations without the Earth's perturbation (see Fig. \\ref{fig:24h}). Thus, just particles with cm-sized survive for 30 years of simulations considering the solar radiation pressure.\n\n\n\\begin{figure*}\n\\begin{center}\n\\subfloat[]{\\includegraphics*[trim = 0mm 0cm 0cm 0mm,\nwidth=1\\columnwidth]{images\/35comportada_3_14rxt.png}\\label{efig:rxt_zxt_sr}}\n\\subfloat[]{\\includegraphics*[trim = 0mm 0cm 0cm 0mm, width=1\\columnwidth]{images\/35comportada_3_14zxt.png}\\label{ffig:rxt_zxt_sr}}\\\\\n\\subfloat[]{\\includegraphics*[trim = 0mm 0cm 0cm 0mm,\nwidth=1\\columnwidth]{images\/22caotica_6_9_rxt.png}\\label{afig:rxt_zxt_sr}}\n\\subfloat[]{\\includegraphics*[trim = 0mm 0cm 0cm 0mm,\nwidth=1\\columnwidth]{images\/22caotica_6_9_zxt.png}\\label{bfig:rxt_zxt_sr}}\\\\\n\\subfloat[]{\\includegraphics*[trim = 0mm 0cm 0cm 0mm, width=1\\columnwidth]{images\/35_maismenos_comportada_54_24_rxt.png}\\label{cfig:rxt_zxt_sr}}\n\\subfloat[]{\\includegraphics*[trim = 0mm 0cm 0cm 0mm,\nwidth=1\\columnwidth]{images\/35_maismenos_comportada_54_24_zxt.png}\\label{dfig:rxt_zxt_sr}}\\\\\n\\end{center}\n\\caption{\\label{fig:rxt_zxt_srp} Three different selected particles among the simulations with solar radiation pressure representing the different orbital radius behavior. The letters (a)-(f) represent the orbital radius and $z$ coordinate for three survived selected particles. The letters (a), (b), (e) and (f) represent the plot for two selected particles with a density of 3.5 g$\\cdot$cm$^{-3}$. The letters (c) and (d) represent the plot for a selected particle with a density of 2.2 g$\\cdot$cm$^{-3}$.}\n\\end{figure*}\n\n\nAnalysing the particles that survived for the entire simulation, we note two different predominant behaviours of particles' orbital radius. The first and most common behaviour is a radial oscillation, that could happen either in a large or small scale as exemplified in Figures \\ref{cfig:rxt_zxt_sr} and \\ref{efig:rxt_zxt_sr}. The small radial oscillation occurs preferentially for the densities models of 1.29 g$\\cdot$cm$^{-3}$ and 2.2 g$\\cdot$cm$^{-3}$ since the initial conditions of the survived particles are near the Apophis' surface (Figs. \\ref{afig:30y_129_srp} and  \\ref{bfig:30y_22_srp}). Thus the orbit of these particles could not have a large amplitude, since it would implicate in large eccentricity, a pericentre closer to the surface, and consequently in a collision with Apophis. The small radial oscillation also occurs for the density of 3.5 g$\\cdot$cm$^{-3}$, but the frequency of large radial oscillation is larger (Fig. \\ref{cfig:30y_35_srp}). Since there is a large region of survived particles and most of them have initial conditions more distant from the body, the orbit of these particles can obtain larger amplitudes and eccentricities without collision with Apophis.\n\nThe second behavior is an abrupt radial variation and occurs for the three densities models. The irregular gravitational potential of the body could produce a variation of eccentricity. When the eccentricity increases the particle collides with Apophis and when it decreases, it produces the decays shown in Figure \\ref{afig:rxt_zxt_sr}. Despite different behavior, the orbital radius of all the particles that survived are restricted to the minimum value of 300 metres, approximately, which is expected since an inferior value could result in a collision with the body's surface. The $z$ coordinate also has a variation (Figs. \\ref{efig:rxt_zxt_sr},  \\ref{ffig:rxt_zxt_sr} and \\ref{dfig:rxt_zxt_sr}), implying the change in the orbits' inclination. The amplitude of the orbital inclination is related to the amplitude of the orbital variation, thus orbits with larger eccentricity allow larger inclinations.\n\n\n\n\\section{Instability Due to the Earth's Encounter}\n\\label{instability}\n\nTo understand how the Earth will affect the environment around Apophis in the 2029 encounter, we simulate the initial condition described in section \\ref{regions} for the disc encompassing the equilibrium points and the expanded disc for 24 hours adding the perturbation of the Earth and using the trajectory shown in Fig. \\ref{fig:orbit_encounter}. Those particles we simulate are at least cm-sized, otherwise they are removed by the solar radiation pressure. \n\nTo evaluate the largest possible perturbation of the Earth, we positioned Apophis in such a way that its orbital plane coincides with its equatorial plane of the Earth. Then, we interpolate this trajectory in the $N$-$BoM$ integrator as an additional external force and simulated the 30 thousand particles around Apophis for 24 hours.\n\n\\begin{figure*}\n\\begin{center}\n\\subfloat[]{\\includegraphics*[trim = 0mm 7cm 6.5cm 0mm,\nwidth=0.66\\columnwidth]{images\/surv_129.png}\\label{afig:24hT_129}}\n\\subfloat[]{\\includegraphics*[trim = 0mm 7cm 6.5cm 0mm,\nwidth=0.66\\columnwidth]{images\/coll_129.png}\\label{bfig:24hT_22}}\n\\subfloat[]{\\includegraphics*[trim = 0mm 7cm 6.5cm 0mm, width=0.66\\columnwidth]{images\/ejec_129.png}\\label{cfig:24hT_35}}\\\\\n\\subfloat[]{\\includegraphics*[trim = 0mm 7cm 6.5cm 0mm,\nwidth=0.66\\columnwidth]{images\/surv_22.png}\\label{dfig:24hT_129}}\n\\subfloat[]{\\includegraphics*[trim = 0mm 7cm 6.5cm 0mm,\nwidth=0.66\\columnwidth]{images\/coll_22.png}\\label{efig:24hT_22}}\n\\subfloat[]{\\includegraphics*[trim = 0mm 7cm 6.5cm 0mm, width=0.66\\columnwidth]{images\/ejec_22.png}\\label{ffig:24hT_35}}\\\\\n\\subfloat[]{\\includegraphics*[trim = 0mm 7cm 6.5cm 0mm,\nwidth=0.66\\columnwidth]{images\/surv_35.png}\\label{gfig:24hT_129}}\n\\subfloat[]{\\includegraphics*[trim = 0mm 7cm 6.5cm 0mm,\nwidth=0.66\\columnwidth]{images\/coll_35.png}\\label{ffig:24hT_22}}\n\\subfloat[]{\\includegraphics*[trim = 0mm 7cm 6.5cm 0mm, width=0.66\\columnwidth]{images\/ejec_35.png}\\label{hfig:24hT_35}}\n\\end{center}\n\\caption{\\label{fig:24hT} Initial conditions of the 15 thousand particles around the equilibrium points of the asteroid Apophis in the 2029 approach in the $xoy$ plane. The green dots represent the equilibrium points. The black dots represent the particles that survived after the 24 hours of integration and the blue and pink dots the particles that collide and ejected, respectively, with the asteroid. The letters (a)-(c), (d)-(f) and (g)-(i) represent, respectively, the densities of 1.29 g$\\cdot$cm$^{-3}$, 2.2 g$\\cdot$cm$^{-3}$ and 3.5 g$\\cdot$cm$^{-3}$.}\n\\end{figure*}\n\nThe initial positions of the 15,000 particles around the equilibrium points in the encounter are shown in Fig. \\ref{fig:24hT} for the three densities models. The larger density model has about 56\\% of surviving particles, while 16\\% and 28\\% collided and ejected, respectively. On the other hand, the smaller density model has about 59\\% of surviving particles, while 15\\% collided and 26\\% were ejected by the Earth perturbation.\n\nWithout the Earth perturbation, almost 99\\% of the particles survive for 24 hours (section \\ref{regions}). So, the encounter causes a large change in the Apophis nearby environment as the number of surviving particles decreases approximately 40\\% and 44\\% for the smaller and larger model of density, respectively. The number of collisions due to the approach increase about 19 times for the smaller model of density and 11 times for the larger model. The ejection of the particles did not occur for the previous simulation, however, considering the 2029 approach 26-28\\% of the particles were ejected.\n\n\\begin{figure}\n\\begin{center}\n\\subfloat[]{\\includegraphics*[trim = 0mm 0cm 0cm 0mm,\nwidth=1\\columnwidth]{images\/4_energy.png}\\label{afig:24h_energy4}}\\\\\n\\subfloat[]{\\includegraphics*[trim = 0mm 0cm 0cm 0mm,\nwidth=1\\columnwidth]{images\/20_energy.png}\\label{bfig:24h_energy20}}\n\\end{center}\n\\caption{\\label{fig:24h_energy} Two-body orbital energy of two different particles in the 24 hours simulation considering the 2029 encounter with the Earth. The orange line denotes the time that the closest approach occurs and the black line represents the line of zero energy. The letters (a) and (b) represent the particle that was ejected due to the Earth perturbation and the surviving one, respectively.}\n\\end{figure}\n\nSome of the ejections occur by the energy criterion, as the Apophis approaches the Earth the orbital energy of the particles increase, and some of them remain positive for the rest of the simulation. Figure \\ref{fig:24h_energy} shows the energy of two different particles, a particle that survived and an ejected one. The orange line represents the moment that occurs the closest distance between Apophis and the Earth and the black line represents the line of zero energy. The energy of the particle that was ejected is shown in Fig. \\ref{afig:24h_energy4}, observe that near the closest encounter with the Earth the orbital energy increases to approximately 0.0025 m$^2\\cdot$s$^{-2}$ and after the encounter, it decreases to 0.0005 m$^2\\cdot$s$^{-2}$ until the end of the simulation. Similar behaviour occurs for the particle that survived the 2029 approach (Fig. \\ref{bfig:24h_energy20}), but after the encounter, the particle energy decreases to $-$0.0005 m$^2\\cdot$s$^{-2}$.\n\n\\begin{figure*}\n\\begin{center}\n\\subfloat[]{\\includegraphics*[trim = 0mm 7cm 6.5cm 0mm,\nwidth=0.66\\columnwidth]{images\/surv_129_l.png}\\label{afig:24hT_129L}}\n\\subfloat[]{\\includegraphics*[trim = 0mm 7cm 6.5cm 0mm,\nwidth=0.66\\columnwidth]{images\/coll_129_l.png}\\label{bfig:24hT_22L}}\n\\subfloat[]{\\includegraphics*[trim = 0mm 7cm 6.5cm 0mm, width=0.66\\columnwidth]{images\/ejec_129_l.png}\\label{cfig:24hT_35L}}\\\\\n\\subfloat[]{\\includegraphics*[trim = 0mm 7cm 6.5cm 0mm,\nwidth=0.66\\columnwidth]{images\/surv_22_l.png}\\label{dfig:24hT_129L}}\n\\subfloat[]{\\includegraphics*[trim = 0mm 7cm 6.5cm 0mm,\nwidth=0.66\\columnwidth]{images\/coll_22_l.png}\\label{efig:24hT_22L}}\n\\subfloat[]{\\includegraphics*[trim = 0mm 7cm 6.5cm 0mm, width=0.66\\columnwidth]{images\/ejec_22_l.png}\\label{ffig:24hT_35L}}\\\\\n\\subfloat[]{\\includegraphics*[trim = 0mm 7cm 6.5cm 0mm,\nwidth=0.66\\columnwidth]{images\/surv_35_l.png}\\label{gfig:24hT_129L}}\n\\subfloat[]{\\includegraphics*[trim = 0mm 7cm 6.5cm 0mm,\nwidth=0.66\\columnwidth]{images\/coll_35_l.png}\\label{hfig:24hT_22L}}\n\\subfloat[]{\\includegraphics*[trim = 0mm 7cm 6.5cm 0mm, width=0.66\\columnwidth]{images\/ejec_35_l.png}\\label{ifig:24hT_35L}}\n\\end{center}\n\\caption{\\label{fig:24hTL} Initial conditions of the 15 thousand particles of the expanded ring in the 2029 approach in the $xoy$ plane. The green dots represent the equilibrium points. The black dots represent the particles that survived after the 24 hours of integration and the blue and pink dots the particles that collided with the asteroid and were ejected, respectively. The letters (a)-(c) represent, respectively, the densities of 1.29 g$\\cdot$cm$^{-3}$, 2.2 g$\\cdot$cm$^{-3}$ and 3.5 g$\\cdot$cm$^{-3}$.}\n\\end{figure*}\n\nFor the expanded radial distribution the number of ejections for the regular case (section \\ref{regions}) was null, but for the encounter simulation was high (Fig. \\ref{fig:24hTL}). About 75\\% and 77\\% of the particles were ejected for the smaller and larger model of density, respectively, being 1.3\\% of this ejection by the energy criterion for the smaller and larger models. The percentage of survived particles was approximately 24.2\\% for the model with the smaller density, while for the larger one was about 22.5\\%. The number of collisions was small, about 0.8\\% and 0.5\\% for the smaller and larger model of density. \n\nThe 2029 approach with the Earth shown to be significant in the nearby environment of Apophis for both disc cases. For the disc around the equilibrium points, the Earth's perturbation causes a high number of ejections, but also almost twice the value of collisions due to the gravitational field of this region being high. On the other hand for the expanded disc, the encounter causes a massive number of ejections and a smaller number of collisions and surviving particles.\n\n\\section{Final Comments}\n\\label{final}\n\nIn this study, we provided the exploration of the effects of the 2029 Earth's encounter on the surface and nearby dynamics of the asteroid Apophis. Firstly, we briefly discuss the Apophis' shape model \\citep{Pravec2014}, its physical properties as the density and size discordance \\citep{BINZEL2009480, muller2014thermal, licandro2015canaricam, BROZOVIC2018115}, and the 2029 Earth's encounter and its trajectory. Then, we define gravitational potential by the polyhedra method \\citep{wernerscheeres1996} and the geopotential on the Apophis' surface \\citep{Scheeres2012, Scheeres2016}.\n\nWe presented the slope angle and its implications on the body's surface. The slope computed on Apophis' surface was about 36$^{\\circ}$ for the three density models with small variations among the models, a difference of only 0.35$^{\\circ}$ between the smaller and larger density model. To comprehend the possible effect caused by the 2029 encounter, we calculated the variation of the slope angle produced by the Earth's gravitational perturbation called $\\Delta$slope. The $\\Delta$slope was smaller than 4$^{\\circ}$ and 2$^{\\circ}$ for the density of 1.29 g$\\cdot$cm$^{-3}$ and 3.5 g$\\cdot$cm$^{-3}$, respectively. Those variations may cause migration of cohesionless particles since they can reach the value of the repose angle of geological materials (35$^{\\circ}$-40$^{\\circ}$) in some regions \\citep{lambe1969, apollo1974, al2018review}. A variation of about 2$^{\\circ}$ may generate a slow erosion process in those regions with high-slope \\citep{ballouz2019surface}, but numerical simulations have shown that the perturbation of the Earth in the 2029 flyby may create just local landslides \\citep{yu2014numerical}.\n\n \nNext, we analysed the zero-velocity curves and computed the equilibrium points of the body. Apophis has four external equilibrium points and two of them are topologically classified as Centre-Centre-Centre for the three density models, implying that they are linearly stable points \\citep{Jiang2014}. Thus, we performed a set of numerical simulations of a disc of 15,000 particles in the nearby environment of Apophis. The first set of simulations was performed considering just the gravitational perturbation of Apophis for a period of 24 hours and 30 years. For a period of 24 hours, the majority of the ring encompassing the equilibrium points is stable with just 0.78-1.4\\% of collision and no ejection, while for the expanded disc all particles survived. For the period of 30 years, 96\\% of the particles survived for the larger density model, 0.9\\% collided and 3.1\\% ejected in a region that has limited the influence of Apophis, creating an external disc of survived particles (Fig. \\ref{cfig:30y_35}). The same phenomenon happens for the smaller density, however, 52\\% of the particles survived, 4\\% collided, and 44\\% ejected. For the density of 1.29 g$\\cdot$cm$^{-3}$, we found interesting regions of survived particles that are related to resonance regions and will be studied in future works.\n\nWe added the perturbation of the solar radiation pressure at the system and simulate the same initial conditions of the previous set of simulations, but now just for the period of 30 years. With those simulations, we identify that just centimetres particles survive for the whole period. For the density model of 2.2 g$\\cdot$cm$^{-3}$ and 3.5 g$\\cdot$cm$^{-3}$, we compute a significant amount of surviving particles with a radius of 5 cm and for the density model of 1.29 g$\\cdot$cm$^{-3}$, just particles with radius of 15 cm survived, considering our set of discrete sizes of particles.\n\nIn addition, we performed simulations considering the 2029 encounter and the Earth's perturbation but removing the solar radiation pressure. Since the solar radiation pressure removes small particles, for this set of simulations we implicitly assume that all of the particles must have a centimetre size of about 5-15 cm. At the end of the 24 hours simulation for the disc encompassing the equilibrium points, we compute a survival percentage of 56\\% for the larger density model and 59\\% for the smaller one. However, for the expanded disc, the majority of the particles were ejected and just 22.5-24.2\\% of particles survived. In general, the 2029 approach shows that the Earth perturbation is significant in the nearby environment of Apophis, but a considerable number of particles still remain at the end of the encounter.\n\nAll the analyses presented in this study, whose purpose was to identify the possible effects on Apophis' surface and nearby environment due to the 2029 flyby, may contribute to the 2029 observational campaign and also to validate our results.\n\n\n\n\\section*{Acknowledgements}\n\nThis study was financed in part by the Coordena\u00e7\u00e3o de Aperfei\u00e7oamento de Pessoal de N\u00edvel Superior - Brasil (CAPES) - Finance Code 001, Funda\u00e7\u00e3o de Amparo \u00e0 Pesquisa do Estado de S\u00e3o Paulo (FAPESP) - Proc. 2016\/24561-0 and Proc. 2019\/23963-5, Conselho Nacional de Desenvolvimento Cient\u00edfico e Tecnol\u00f3gico (CNPq) - Proc. 305210\/2018-1. \n\n\n\\section*{ORCID iDs}\nG. Valvano \\orcidicon{0000-0002-7905-1788} \\href{https:\/\/orcid.org\/0000-0002-7905-1788}{https:\/\/orcid.org\/0000-0002-7905-1788}\\\\\nO. C. Winter \\orcidicon{0000-0002-4901-3289} \\href{https:\/\/orcid.org\/0000-0002-4901-3289}{https:\/\/orcid.org\/0000-0002-4901-3289}\\\\\nR. Sfair \\orcidicon{0000-0002-4939-013X} \\href{https:\/\/orcid.org\/0000-0002-4939-013X}{https:\/\/orcid.org\/0000-0002-4939-013X}\\\\\nR. Machado Oliveira \\orcidicon{0000-0002-6875-0508} \\href{https:\/\/orcid.org\/0000-0002-6875-0508}{https:\/\/orcid.org\/0000-0002-6875-0508}\\\\\nG. Borderes-Motta \\orcidicon{0000-0002-4680-8414} \\href{https:\/\/orcid.org\/0000-0002-4680-8414}{https:\/\/orcid.org\/0000-0002-4680-8414}\\\\\nT. S. Moura \\orcidicon{0000-0002-3991-8738} \\href{https:\/\/orcid.org\/0000-0002-3991-8738}{https:\/\/orcid.org\/0000-0002-3991-8738}\\\\\n\n\\section*{Data availability}\nThe data underlying this article will be shared on reasonable request to the corresponding authors.\n\n\n\n\\bibliographystyle{mnras}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe 4{\\it f\\\/} electron in Cerium is energetically weakly bound\ndespite the fact that it resides deep within the core of the atom.\nThis is due to the relatively extended nature of the 4{\\it f\\\/}\nwave function. Strong correlations between the Ce 4{\\it f\\\/}\nelectron and hybridization between the 4{\\it f\\\/} state with\nthose of the ligand states arise in Ce compounds and hence the\nlocal environment of the Ce-atom dictates whether this electron\nwill remain bound within the core (Ce$^{3+}$ state), join the\nspatially extended valence electrons (Ce$^{4+}$ state), or reside\nwith certain probability in each. In the last case, the atom is\nsaid to be in an ``intermediate valent\" state with fluctuating\ncharge occupancy of the 4{\\it f\\\/} shell\nstate.\\cite{Sereni1982,Johansson1987,Malterre1989} The fragile\ncharacter of the 4{\\it f\\\/} electron due to its sensitivity on\ninteratomic distances, as this determines the hybridization\nstrength, is capitalized on for example investigating quantum\ncritical phenomena, where by either applying hydrostatic pressure\nor chemical substitution the unit-cell volume shrinks or expands\nwith the result that ground state of the material under\ninvestigation changes.\\cite{Loehn2007}\\\\\nKnowledge about the electronic structure and understanding its\nrelation to the physical properties observed in intermetallics, and\nin particular rare earth based compounds is an ambitious\nundertaking in condensed matter research. Investigation by varying\nthe properties utilizing pressure or chemical substitution is one\nway. In addition, some compounds exist in more than one\ncrystallographic structure.\\cite{Ghad1988} By means of pressure\nand\/or temperature it is possible to convey one into the other\nreversibly. Such polymorphic transitions allow for comparative\nstudy.\\cite{Mihalik2010} The chemical composition is retained but\nbonds of the individual atoms and therefore, the overall\nelectronic structure and related physical properties can differ\nsignificantly. An example is \\LIS . Polymorphism of \\LIS\\ between\na high-temperature phase of the primitive tetragonal\nCaBe$_2$Ge$_2$-type structure and a low-temperature phase of the\nbody-centered tetragonal ThCr$_2$Si$_2$-type structure has been\ndemonstrated by Braun {\\it et al.}.\\cite{Braun1983} Notably, the\nhigh-temperature modification displays superconductivity below\n1.6~K, while the low-temperature phase is\nnormal down to 1~K.\\\\\n\n\nThe polymorphic isostructural $\\gamma$ (fcc) $\\rightarrow \\alpha$\n(fcc) transition in elementary Cerium~\\cite{Bridgman1948} is\nillustrative for the difficulty in understanding the complexity\nbetween ``chemical bonds\", ``physical properties\" and ``structural\ntransition\" especially in materials with electrons near the\nboundary between itinerant and localized\nbehavior.\\cite{Koskenmake1978} The transition involves a large\nvolume collapse of $\\sim 17$~\\% at room temperature and pressure\n$\\sim 0.8$~GPa. A general consensus exists to attribute the\ntransition to an instability of the Ce 4{\\it f\\\/} electron.\nHowever, Johansson~\\cite{Johansson1974} explained the $\\gamma\n\\rightarrow \\alpha$ transition as sort of Mott transition in which\nthe localized 4{\\it f\\\/} electrons in the $\\gamma$-phase become\nitinerant and participate in bonding in the lower volume $\\alpha$-phase.\nThis model continues to compete with the\nKondo-volume-collapse scenario,\\cite{Allen1982,Allen1992} which\nassumes that the 4{\\it f\\\/} electron is localized in both the\n$\\gamma$- and $\\alpha$-phases. The loss of magnetic moment in the\n$\\alpha$-phase results from screening of the moments by the\nsurrounding conduction electrons. To complicate, latest neutron\nand X--ray diffraction studies acknowledge the importance\nof lattice vibrations as well.\\cite{Jeong2004,Lipp2008}\\\\\n\nIn many aspects the recently observed polymorphic transition in\nthe equiatomic stannide \\CRS\\ seems to have much in common with\nthe $\\gamma \\rightarrow \\alpha$\ntransition in Cerium. \\\\\n\\CRS\\ at room temperature crystallizes in a superstructure\nmodification of the monoclinic CeCoAl-type crystal structure (new\nmonoclinic type, space group C2\/$m$) with lattice parameters $a=\n11.561(4)$~\\AA\\,, $b= 4.759(2)$~\\AA\\,, $c= 10.233(4)$~\\AA\\,, and\n$\\beta = 102.89(3)^\\circ$.\\cite{Riecken2007} As a consequence of\nthis doubling of the original CeCoAl unit cell along the $c$ axis,\nthe compound possesses two crystallographic independent cerium\nsites labelled Ce1 and Ce2. Although topology of both Cerium sites\nis identical, five rhodium, six tin, and six cerium atoms in the\ncoordination shell, the tiny changes in interatomic distances,\nmost notably the Ce--Ru bonds (Ce1--Ru: ranging from 2.33 to\n2.46\\AA\\,; Ce2--Ru: ranging from 2.88 to 2.91~\\AA\\,) result in Ce1\nbeing in intermediate valent state while Ce2 shows strong\nlocalization of the {\\it f\\\/} electron as suggested by magnetic\nsusceptibility experiments.\\cite{Riecken2007,Mydosh2011} This\npresumption is borne out by electronic structure\ncalculations~\\cite{Matar2007} and proven by X--ray absorption\nnear-edge structures (XANES) data.\\cite{Feyerherm2012}\nLatest yields average valencies of 3.18 for Ce1 and Ce2. \\\\\nThe polymorphic transition in \\CRS\\ sets in just below room\ntemperature at $\\sim 290$~K and is completed at around $\\sim\n160$~K upon cooling. The reverse transformation occurs on heating\nwith $\\sim 170$~K and $\\sim 320$~K as the onset and end\ntemperatures, respectively. Initial measurements of the magnetic\nsusceptibility, specific heat, thermopower, and resistivity were\nperformed on polycrystalline samples.\\cite{Mydosh2011} The\ntransformation was smeared out and manifested as broad hysteresis\nwith a cusp-like structure in resistivity, a step-like decrease of\nthe susceptibility, a broad hump in the specific heat and a strong\nincrease in thermopower. A detailed analysis of the transition by\nmeans of synchotron X--ray diffraction experiments on a single\ncrystal revealed that the room temperature phase is replaced by a\nset of close to commensurate modulations along the $c$ axis,\nnamely quintupling ($\\sim 290$~K) and (dominant) quadrupling\n(below 210~K) before finalizing ($\\sim 180$~K) in an ill-defined\nmodulated ground state, which is close to a tripling of the basic\nmonoclinic CeCoAl-type structure.\\cite{Feyerherm2012} \\\\\n\nThe present work gives a detailed examination of the physical\nproperties of the polymorphic transition of \\CRS\\. For this\npurpose, measurements were performed on high quality single\ncrystals. The lower amount of crystal lattice defects, absence of\ngrain boundaries and the ability to perform experiments along\nspecific crystallographic orientations allows us to resolve\ndetails related to the transition and to attribute those\nsignatures in the experiments to the respective modulation in the\nstructure.\n\n\\section{Experimental Details}\n\\label{Sec1}\n\\subsubsection*{Sample preparation}\nSingle crystals of \\CRS\\ were prepared in two stages. First a\npolycrystalline button of the nominal 1:1:1 stoichiometry was\nsynthesized using elements of purity 3N Ce (Ce from Alpha Aesar\nwhich was additionally purified by solid state electrotransport\ntechnique\\cite{Carlson1977}), 4N Ru and 5N Sn as starting\nmaterials. The reaction of the stoichiometric mixture of the\nelements was performed on a water-cooled copper crucible in a\nmono-arc furnace under 6N Argon atmosphere. The mass difference\nbefore and after the reactions was negligible ($< 0.1$~\\%). The\ncrystal was than grown utilizing a modified Czochralski technique;\nthe button was remelted in a tri--arc furnace under 6N Argon\nprotection atmosphere and a tungsten rod was used as a seed.\\\\\nThe quality of the single crystal was checked by X--ray Laue\nback--scattering, which was also used for orienting the crystals\nlater on. The chemical composition was verified employing a Tescan\nMira I LMH scanning electron microscope (SEM). The instrument is\nequipped with a Bruker AXS energy dispersive X--ray detector\n(EDX). Within the accuracy of the device, no impurity phases were\nresolved and the measurement confirmed the correct\n1:1:1 stoichiometry. \\\\\nAfterwards, the crystal was cut for further analysis. One piece\nwas pulverized and examined at room temperature by means of powder\nX--ray diffraction (Bruker D8 Advance diffractometer with\nCu-K$_\\alpha$ radiation with $\\lambda = 1.5405$~\\AA\\,). The\nobtained diffraction patterns were refined by Rietveld analysis\nusing FULLPROF.\\cite{Fullprof} The analysis confirmed the CeCoAl\nsuperstructure and the corresponding lattice parameters agreed\nwell with those values reported in\nliterature.\\cite{Riecken2007}\\\\\nThe other piece of the crystal was annealed at 700~$^\\circ$C for\none week in vacuum ($p= 1 \\times 10^{-6}$~mbar) in order to\nimprove homogeneity. In the following, the whole characterization\nprocedure was repeated unveiling no significant differences.\\\\\n\n\\subsubsection*{Experimental setup}\nFrom the annealed single crystal a small piece was cut for\ninvestigating the crystal structure by X--rays at defined\ntemperatures. Therefore, the approximately $0.1 \\times 0.1 \\times\n0.1$~mm$^3$ piece was placed inside a Lindemann capillary. The\ncapillary itself was mounted into a Bruker Apex II diffractometer\nwith Mo--K$_\\alpha$ radiation ($\\lambda = 0.71073$~\\AA\\,). In\norder to reach lower temperatures, the capillary was inserted into\na flow of cold nitrogen gas. The crystal structure was resolved by\ndirect methods~\\cite{Sheldrick2007} and adjacent refinement was\ndone by\nfull--matrix least--squares based on $F^2$. \\\\\n\nBulk properties were retrieved employing standard equipment. The\nmagnetization was measured in a MPMS7 (Quantum Design). Data were\ncollected in the temperature range from 1.8 to 350~K and in fields\nup to 7~T. Resistivity, Hall resistivity, thermopower, thermal\nconductivity were measured in a PPMS14 (Quantum Design) using the\nrespective optional accessories of the device. The temperature was\nvaried between 1.8 and 350~K and magnetic fields up to 14~T were\napplied. The resistivity was measured using standard 4--point\ntechnique. In order to reduce contact resistance, the 25~$\\mu$m\ndiameter Au--wires were spot welded onto the sample. Measurements\nof the resistivity were performed at ambient and hydrostatic\npressure. For the later, the PPMS device was used only to control\ntemperature. The sample was loaded into a double cylinder\nCuBe\/NiCrAl pressure cell. Daphne 7373 oil was used as pressure\nmedium and the applied pressure was determined at\nroom temperature utilizing a manganin manometer. \\\\\nThe thermal expansion was measured in a temperature interval of\n180--340~K. The sample was built into a miniature capacitance\ncell.\\cite{Rotter1998} The capacity was read out by an Andeen\nHagerling 2500A capacitance bridge. The cell was inserted into the\nPPMS whose controlling was used to set temperature. \\\\\n\nMost of the experiments were conducted on both, the as cast and\nthe annealed single crystals. The quality of the crystals improved\nconsiderably by annealing. The resistivity\nbehavior of the as cast crystals to some extent resembled the\nresults of the polycrystalline sample presented in earlier\nwork\\cite{Mydosh2011} that is a single broad hysteresis, which\ndiffers in detail depending on the current\ndirection with respect to the crystallographic axis. On the\ncontrary the annealed crystals exhibit two sharp transitions\nevident for two distinct transitions as will be discussed below.\nIn addition, differences in the hysteresis of each of the\ntransitions could be resolved. Results presented in this report\nwere obtained on the annealed crystals. The bulk properties were\nmeasured with respect to the three principle crystallographic\naxes. Data shown have been collected on two batches. The\nresistivity and susceptibility experiments were performed on\nbatch I, while for thermal expansion a piece from batch II was used.\nHence, slight differences in the respective transition\ntemperatures are observed, which we attribute to sample\ndependencies.\n\n\\section{Results}\n\\subsection{Measurement of bulk properties}\n\\begin{figure}[t]\n\\centerline{\\includegraphics[width=0.8\\textwidth]{FikacekCeRuSnFigure1.eps}}\n\\caption{\\label{Fig1}Temperature dependences of the relative\nresistivity $\\rho(T)\/\\rho_{1.8\\mathrm{K}}$ of CeRuSn measured with\ncurrent applied along the 3 principle crystallographic directions\nfor cooling (solid lines) and warming (short--dashed lines). The\nmain panel focuses on the temperature range the discussed\ntransitions take place. The vertical dashed lines mark the\nestimated transition temperatures \\Tcu\\,, \\Tcl\\,, \\Twl\\, and\n\\Twu\\,, respectively (see text). The inset shows the whole\ntemperature range of the experiment for cooling regime.}\n\\end{figure}\n\nFigure~\\ref{Fig1} depicts the temperature dependence of the scaled\nelectrical resistivity $\\rho\/\\rho_{1.8\\mathrm{K}}$ for current\napplied along the $a$, $b$ and $c$ axis, respectively. The inset\nshows the full temperature range. The different corresponding\nvalues for $\\rho(T)$ document anisotropy of the electronic\ntransport in \\CRS\\,. Yielding\n$\\rho_{300\\mathrm{K}}\/\\rho_{1.8\\mathrm{K}} \\sim 18$ (for $I\n\\parallel a$) comparing to only $\\sim 1.75$ for a polycrystalline sample proofs the high quality of our single\ncrystals. More remarkably, the observed behavior in resistivity\ndiffers significantly from previously published work on a\npolycrystalline sample.\\cite{Mydosh2011} Cooling down the sample\n(solid lines in Fig.~\\ref{Fig1}) from above room temperature, the\nresistivity undergoes a sharp step-like increase by about 7~\\%\njust below 290~K. The anomaly, indicated by \\Tcu\\ in the main\npanel, is clearly seen in all applied current directions while\nabsent in the polycrystalline sample. Below 225~K a second\ntransition emerges marked by \\Tcl\\,, and the resistivity seems to\nfall back onto the original curve from before the first transition\n($I \\parallel c$ and $a$). The transition is weakly pronounced for\n$I \\parallel b$ while decrease of the resistivity is only a\nfraction of the increase at \\Tcu\\,. Interestingly, this \\Tcl\\\ntransition, although slightly shifted towards lower temperatures,\nis observed in the polycrystal as well. However, contrary to our\ndata, resistivity increases. Below 3~K, resistivity reveals a\nthird anomaly, which can be attributed to the onset of\nantiferromagnetic ordering reported earlier.\\cite{Mydosh2011} In\nthe following, discussion on the antiferromagntic order is omitted\nand focus is entirely on the\nrelevant temperature range of \\Tcu\\ and \\Tcl\\,. \\\\\nUpon warming up (dashed lines in Fig.~\\ref{Fig1}), both the lower,\n\\Twl\\ $\\sim 256$~K, and upper, \\Twu\\ $\\sim 307$~K, transitions of\n$\\rho(T)$ preserve shape and size of the step. However, they are\nobserved at much higher temperatures than their corresponding\nanomalies when cooling down, i.\\,e.\\,, exhibiting large\ntemperature hysteresis. In comparison, hysteresis of the\nlower-temperature transition yields \\Tcl\\ $-$ \\Twl\\ $\\sim 40$~K\nalmost double the hysteresis of the upper-temperature transition\n\\Tcu\\ $-$ \\Twu\\ $\\sim 20$~K. These remarkable features in \\CRS\\ remain intact even in magnetic fields up to 14~T. \\\\\n\n\n\n\\begin{figure}[t]\n\\centerline{\\includegraphics[width=0.8\\textwidth]{FikacekCeRuSnFigure2.eps}}\n\\caption{\\label{Fig2} The inverse magnetic susceptibility\n$\\chi^{-1}(T)= H\/M(T)$ of CeRuSn in magnetic field applied along\nthe $a$, $b$ and $c$ axis in the temperature interval of interest.\nSolid (short--dashed) lines denote measurements in cooling down\n(warming up). The applied magnetic field was 1~T for $H\\parallel\na$ and $H \\parallel c$ and 7~T for $H \\parallel b$, respectively.\nThe vertical lines are guides to the eye illustrating the\nestimated transition temperatures \\Tcu\\,, \\Tcl\\,, \\Twl\\, and \\Twu\\\nfrom temperatures of the resistivity anomalies. Mind the breaks in\nthe scale on the vertical axis.}\n\\end{figure}\n\nFigure~\\ref{Fig2} plots the temperature dependence (solid lines\nrefer to cooling down, dashed lines for warming up sequence,\nrespectively) of the inverse dc magnetic susceptibility,\n$\\chi^{-1}$, in fields applied along the principle crystallographic\naxes. The magnetic susceptibility is strongly anisotropic\napparently due to the influence of the very low--symmetry crystal\nelectric field (CEF) on the orbitals of the Ce--ion. The\ntransitions at \\Tcu\\ and \\Tcl\\ when cooling down, and \\Twl\\ and\n\\Twu\\ when warming up \\CRS\\ are clearly witnessed by a small\nnegative step in the magnetization, i.\\,e., a positive jump in\n$\\chi^{-1}$. Note that the polycrystalline sample shows only a\nsingle step upon cooling at $T \\sim\n180$~K.\\cite{Riecken2007,Mydosh2011} The lower transitions, \\Tcl\\\nand \\Twl\\ are much weaker than the upper ones.\\\\\nShort temperature intervals that are above \\Tcu\\ and between \\Tcu\\\nand \\Tcl\\, prevent a meaningful qualitative analysis of the\ntemperature dependence of each of the separate paramagnetic\nphases. Quantitatively, assuming the effective moment remains\nconserved across the transitions, a reasonable assumption\nrecalling that XANES unveil no chance in Ce\nvalency,\\cite{Riecken2007} the change in $\\chi(T)$ implies a\nshift of the paramagnetic Curie temperatures towards larger\nnegative values for each field direction objecting statements on\nthe polycrystalline sample.\\cite{Riecken2007}\\\\\n\n\n\\begin{figure}[t]\n\\centerline{\\includegraphics[width=\\textwidth]{FikacekCeRuSnFigure3.eps}}\n\\caption{\\label{Fig3} The temperature evolution of the relative\nchange of length $\\Delta L\/L_{340 \\mathrm{K}}$ (left axis)\nresolved for each of the principle crystallographic directions\nwhen the sample was in cool down (solid line) and warm up run\n(short-dashed line) through the transitions. On the right axis,\nthe calculated relative volume change  $V\/V_{340 \\mathrm{K}}$ for\ncooling (solid line) and warming (short--dashed line) is depicted.\n}\n\\end{figure}\n\nDetailed dilatometric measurements performed on a well-defined\nsingle crystal provide important information on the evolution of\nthe lattice parameters. The thermal expansion was measured along\neach of the three principle crystallographic directions. As\npresented in Fig.~\\ref{Fig3}, two steps are observed along each of\nthe axes at similar temperatures to the anomalies recorded in\n$\\rho(T)$ and $\\chi(T)$, respectively. When cooling (solid lines\nin Fig.~\\ref{Fig3}) from room temperature, the crystal contracts\nconsiderably along the $c$ axis by almost 0.8~\\% between 340 and\n180~K. Contrary, the tiny positive jumps disclosed for $a$ and $b$\ndirections represent very small expansion of $a$ and nearly\nnegligible increase of the lattice parameter $b$, respectively.\nConsequently, the volume changes at \\Tcu\\ and \\Tcl\\ express\nmainly the $c$ axis behavior, i.\\,e.\\,, the crystal shrinks in two\nsteps with decreasing temperature. The corresponding reverse\ntransitions appear at \\Twl\\ and \\Twu\\ corroborating the hysteretic\nbehavior of the phases as inferred by resistivity and\nmagnetization experiments already.\n\n\\begin{figure}[t]\n\\centerline{\\includegraphics[width=0.8\\textwidth]{FikacekCeRuSnFigure4.eps}}\n\\caption{\\label{Fig4} Comparative plot of the temperature\ndependences of (a) the temperature derivative of the electrical\nresistivity $\\partial\\rho\/\\partial T$ for $I \\parallel c$, (b) the\ntemperature derivative of the magnetic susceptibility $\\partial\n\\chi\/\\partial T$ with $H \\parallel c$, and (c) the linear thermal\nexpansion coefficient $\\alpha(T)$ along the $c$ axis. Solid\n(short--dashed) lines show data taken when cooling down (warming\nup) the sample. The vertical lines are guides to the eye\nillustrating the estimated transition temperatures \\Tcu\\,, \\Tcl\\,,\n\\Twl\\, and \\Twu\\,, respectively, from temperatures of the\nresistivity anomalies.}\n\\end{figure}\n\nIn figure~\\ref{Fig4}, a comparison of the temperatures of the\nanomalies disclosed in aforementioned bulk experiments is made. To\nvisualize the location of the transition more clearly, the\ntemperature derivative of the resistivity ($\\partial \\rho\/\\partial\nT$), dc magnetic susceptibility ($\\partial \\chi\/\\partial T$) and\nthermal expansion coefficient ($\\alpha$) are displayed. The maxima\nin ($\\partial \\rho\/\\partial T$) and ($\\alpha$) well coincide\nexcept for the \\Twu\\ transition. This difference likely arose because a sample from batch II had been used,\nas mentioned in\nsection~\\ref{Sec1}.\n\n\n\\subsection{X--ray single crystal study of polymorphs}\n\\begin{figure*}[t]\n\\centerline{\\includegraphics[width=0.8\\textwidth,angle=-90]{FikacekCeRuSnFigure5.eps}}\n\\caption{\\label{Fig5} Illustration of CeRuSn crystal structures\nabove, between and below the transitions at 300, 275 and 120~K,\nrespectively. For all types of atoms, numbers label two\nnon-equivalent crystallographic sites. Ce sites are additionally\nmarked on the right-hand site by $S$, $L$ and $I$ letters\naccording to the similarity of their environment shown in table\nII. Black and white lines emphasize the typical Ce--Ru shortest\ninteratomic distances and tin network, respectively. The dashed\nline illustrates Bravais unit cell.}\n\\end{figure*}\n\n\\begin{table*}\n    \\caption{{Lattice parameters determined from SC X--ray diffraction. The lattice\n    parameters are sorted according to the temperature evolution during our experiment.\n    The last column shows the ratio between $\\Delta c_{\\mathrm{red}}$ standing for the $c$ axis modified to room\n    temperature size and the room temperature $c$ axis value.\n    It estimates the real change of crystal dimension along the $c$\n    direction.}}\n     \\vspace*{2mm}\n    \\label{Table1}\n    \\renewcommand{\\arraystretch}{1.2}\n  \\begin{tabular}{ccccccccccccc}\n  \\hline \\hline\n    $T$~(K)&$\\quad$ &$a$~(\\AA) & $\\quad$& $b$~(\\AA)&$\\quad$ &$c$~(\\AA) & $\\quad$&$\\beta$~($\\deg$) & $\\quad$&$V$~(\\AA$^3$) &$\\quad$ & $\\Delta\n    c_{\\mathrm{red}}\/c_{300\\mathrm{K}}$ (\\%)\\\\ [0.5ex]\n   \\hline\n   297 & & 11.565(1) & & 4.7529(5) & & 10.2299(9) & & 103.028(2) & & 547.85(9) & & -- \\\\\n   290 & & 11.560(3) & & 4.751(1) & & 10.227(3) & & 103.081(7) & & 547.1(2) & & 0.028(1) \\\\\n   275 & & 11.576(2) & & 4.7556(7) & & 25.454(3) & & 102.959(4) & & 1365.6(3) & & 0.47(1) \\\\\n   120 & & 11.566(2) & & 4.7477(6) & & 15.229(2) & & 103.554(4) & & 813.0(2) & & 0.70(2)\n   \\\\ [1ex]\n   200 & & 11.569(2) & & 4.7505(6) & & 15.237(2) & & 103.496(4) & & 814.3(2) & & 0.76(2) \\\\\n   \\hline \\hline\n  \\end{tabular}\n\\end{table*}\n\n\n\n\\begin{table*}\n    \\caption{{Closest ruthenium neighbors of cerium atoms at different temperatures. ($S$), ($L$) and ($I$)\n    denote bond type {\\it short}, {\\it long} and {\\it intermediate} respectively (see also text).}}\n     \\vspace*{2mm}\n    \\label{Table2}\n    \\renewcommand{\\arraystretch}{1.2}\n  \\begin{tabular}{llllllllllllllllllll}\n  \\hline \\hline\n  \\multicolumn{6}{c}{$T=300$~K} & $\\qquad$ & \\multicolumn{6}{c}{$T=275$~K} & $\\qquad$ & \\multicolumn{6}{c}{$T=120$~K} \\\\ [0.5ex]\n  \\hline\n  Ce1 & $\\quad$ & 2.3277(9) & $\\quad$ & Ru1 &                          & & Ce1 & $\\quad$ & 2.8882(19) & $\\quad$ & Ru5 &                         & & Ce1 & $\\quad$ & 2.4325(18) & $\\quad$ & Ru2 &  \\\\ [-1ex]\n      & $\\quad$ & 2.4661(11) & $\\quad$ & Ru2 & \\raisebox{1.5ex}{($S$)} & &     & $\\quad$ & 2.9247(19) & $\\quad$ & Ru2 & \\raisebox{1.5ex}{($L$)} & &     & $\\quad$ & 2.4345(18) & $\\quad$ & Ru3 & \\raisebox{1.5ex}{($S$)}\n\\\\ [0.5ex]\n  Ce2 & $\\quad$ & 2.8783(10) & $\\quad$ & Ru1 &  & &  Ce2  & $\\quad$ & 2.4331(18) & $\\quad$ & Ru1 &  & & Ce2\n& $\\quad$ & 2.9153(17) & $\\quad$ & Ru1 &  \\\\ [-1ex]\n      & $\\quad$ & 2.9081(10) & $\\quad$ & Ru2 & \\raisebox{1.5ex}{($L$)} & &    & $\\quad$ & 2.4380(19) & $\\quad$ & Ru3 & \\raisebox{1.5ex}{($S$)} & &\n& $\\quad$ & 2.9257(18) & $\\quad$ & Ru2 & \\raisebox{1.5ex}{($L$)}\n\\\\ [0.5ex]\n      & $\\quad$ &  & $\\quad$ &  &  & &  Ce3  & $\\quad$ & 2.271(2) & $\\quad$ & Ru2 & & &  Ce3\n& $\\quad$ & 2.2676(17) & $\\quad$ & Ru1 &  \\\\ [-1ex]\n      & $\\quad$ &  & $\\quad$ &  &  & &    & $\\quad$ & 2.7575(19) & $\\quad$ & Ru3 & \\raisebox{1.5ex}{($I$)} & &\n& $\\quad$ & 2.7482(17) & $\\quad$ & Ru3 & \\raisebox{1.5ex}{($I$)}\n\\\\ [0.5ex]\n      & $\\quad$ &  & $\\quad$ &  &  & &  Ce4  & $\\quad$ & 2.3219(19) & $\\quad$ & Ru4 &  & &\n& $\\quad$ &  & $\\quad$ &  &  \\\\ [-1ex]\n      & $\\quad$ &  & $\\quad$ &  &  & &    & $\\quad$ & 2.4779(18) & $\\quad$ & Ru5 & \\raisebox{1.5ex}{($S$)} & &\n& $\\quad$ &  & $\\quad$ &  & \\\\ [0.5ex]\n      & $\\quad$ &  & $\\quad$ &  &  & &  Ce5  & $\\quad$ & 2.8800(18) & $\\quad$ & Ru4 & & &\n& $\\quad$ &  & $\\quad$ &  &  \\\\ [-1ex]\n      & $\\quad$ &  & $\\quad$ &  &  & &    & $\\quad$ & 2.9453(19) & $\\quad$ & Ru1 & \\raisebox{1.5ex}{($L$)} & &\n& $\\quad$ &  & $\\quad$ &  & \\\\ [0.5ex]\n  \\hline \\hline\n  \\end{tabular}\n\\end{table*}\n\n\n\n\\begin{table*}\n    \\caption{{Fraction atomic coordinates \\mbox{$^x$\/$_a$}, \\mbox{$^y$\/$_b$} and \\mbox{$^z$\/$_c$} at 300~K and 120~K.}}\n     \\vspace*{2mm}\n    \\label{Table3}\n    \\renewcommand{\\arraystretch}{1.2}\n  \\begin{tabular}{llllllllllllllll}\n  \\hline \\hline\n   \\multicolumn{7}{c}{ $T=300$~K} & $\\qquad$ & $\\qquad$ & \\multicolumn{7}{c}{ $T=120$~K}  \\\\ [0.5ex]\n  \\hline\n  Ce1 & $\\quad$ & 0.1396 & $\\quad$ & 0 & $\\quad$ & 0.4146 & $\\qquad$& $\\qquad$ & Ce1 & $\\quad$ & 0.6306 & $\\quad$ & 0 & $\\quad$ & 0.1075 \\\\\n  Ce2 & $\\quad$ & 0.1225 & $\\quad$ & 0 & $\\quad$ & 0.9063 & $\\qquad$& $\\qquad$ & Ce2 & $\\quad$ & 0.3567 & $\\quad$ & 0 & $\\quad$ & 0.2216 \\\\\n      & $\\quad$ &        & $\\quad$ &   & $\\quad$ &        & $\\qquad$& $\\qquad$ & Ce3 & $\\quad$ & 0.6226 & $\\quad$ & 0 & $\\quad$ & 0.4375\n      \\\\[0.5ex]\n  Sn1 & $\\quad$ & 0.4265 & $\\quad$ & 0 & $\\quad$ & 0.3469 & $\\qquad$& $\\qquad$ & Sn1 & $\\quad$ & 0.0858 & $\\quad$ & 0 & $\\quad$ & 0.9376 \\\\\n  Sn2 & $\\quad$ & 0.4043 & $\\quad$ & 0 & $\\quad$ & 0.8486 & $\\qquad$& $\\qquad$ & Sn2 & $\\quad$ & 0.0906 & $\\quad$ & 0 & $\\quad$ & 0.6008 \\\\\n      & $\\quad$ &        & $\\quad$ &   & $\\quad$ &        & $\\qquad$& $\\qquad$ & Sn3 & $\\quad$ & 0.0659 & $\\quad$ & 0 & $\\quad$ & 0.2688\n      \\\\ [0.5ex]\n  Ru1 & $\\quad$ & 0.1827 & $\\quad$ & 0 & $\\quad$ & 0.6481 & $\\qquad$& $\\qquad$ & Ru1 & $\\quad$ & 0.6784 & $\\quad$ & 0 & $\\quad$ & 0.2605 \\\\\n  Ru2 & $\\quad$ & 0.1983 & $\\quad$ & 0 & $\\quad$ & 0.1973 & $\\qquad$& $\\qquad$ & Ru2 & $\\quad$ & 0.2992 & $\\quad$ & 0 & $\\quad$ & 0.3654 \\\\\n      & $\\quad$ &        & $\\quad$ &   & $\\quad$ &        & $\\qquad$& $\\qquad$ & Ru3 & $\\quad$ & 0.3102 & $\\quad$ & 0 & $\\quad$ & 0.0572 \\\\\n  \\hline \\hline\n  \\end{tabular}\n\\end{table*}\n\n\nThe presented results on resistivity, magnetization and thermal\nexpansion conflict in many ways with earlier\nstudies.\\cite{Riecken2007,Mydosh2011} Moreover, it was mentioned\nin the introduction that in \\CRS\\ several polymorphic transitions\ngradually emerged on cooling, which gave rise to additional\nreflections in the diffraction patterns obtained by synchotron\nexperiment.\\cite{Feyerherm2012} Those superstructure reflections\ncan be described by nearly inverse--integer folded propagation\nvectors having non-zero $c$ components only. The first transition\ntakes place just below room temperature, changing from a\n\\mbox{$^1$\/$_2$}-- to a \\mbox{$^1$\/$_5$}--like modulation. Upon\ncooling, these become partially suppressed and replaced by\n\\mbox{$^1$\/$_4$}--like ones, which are dominant at 210~K. Finally,\na \\mbox{$^1$\/$_3$}--kind modulation develops having the most\nintensive reflections below 180~K. This one coexists with the\naforementioned modulations down to at least\n100~K.\\cite{Feyerherm2012} While \\Tcu\\ can be attributed to the\nfirst polymorphic transition (\\mbox{$^1$\/$_2$} $\\rightarrow$\n\\mbox{$^1$\/$_5$}) no evidence is found in the data for the\nstructural change \\mbox{$^1$\/$_5$} $\\rightarrow$ \\mbox{$^1$\/$_4$},\nwhich at least is accepted to manifest in thermal expansion being\nan extremely sensitive experiment on lattice changes. To\nanticipate speculations about the structure, a detailed X--ray\nsingle crystal diffraction study over the majority of the\nreciprocal lattice and at defined temperatures was conducted. The\nfollowing temperature sequence was applied: 300~K, 290~K, 275~K,\n120~K and 200~K matching the regions for determining the structure\nabove \\Tcu\\, between \\Tcu\\ and \\Tcl\\, below \\Tcl\\ and below\n\\Twl\\, respectively.\\\\\nIn all cases, the space group $C2\/m$ for the unit cell has been\nobserved. However, the size of the unit cell varies significantly\nbecause of formation of superstructures as displayed in\ntable~\\ref{Table1}. In comparison, the simultaneous changes of the\n$a$ and $b$ cell parameters are rather negligible and the crystal\nunit cell size change is related to integer multiplications of the\noriginal CeCoAl-type unit along the $c$ axis in agreement with\nRef.~\\onlinecite{Feyerherm2012}. Within this process, the number of\ninequivalent Ce lattice sites in the intermediate (275~K) and low\ntemperature (120, 200~K) phase, respectively, is larger than in\nthe room temperature polymorph (300, 290~K). The structure at\n300~K (Fig.~\\ref{Fig5}) is practically identical to that one at\n290~K and in accordance to previous\nreports.\\cite{Riecken2007,Feyerherm2012} Upon cooling below the\nfirst transition to $T=275$~K, a tremendous prolongation of the\n$c$ axis is observed. The resulting superstructure can be\ndescribed as a quintuple of the CeCoAl subcell with five\ncrystallographically independent cerium sites (see\nFig.~\\ref{Fig5}). With further cooling down to 120~K, \\CRS\\ passes\nthrough the second polymorphic transition at \\Tcl\\,. The final,\nand only existing structure can be viewed as a tripling of the\nCeCoAl unit--cell exhibiting three different cerium sites\n(Fig.~\\ref{Fig5}). The nearest Ce--Ru distances\nfor all phases are summarized in table~\\ref{Table2}. With closer\ninspection of the crystallographic parameters, one can see that\nall the inter--atomic distances were more or less modified. But\nthe crucial aspect seems to be the increased number of Ce\npositions with short Ce--Ru pairs (cf. table~\\ref{Table2}). This\nexplains the $c$ axis contraction, since Ce--Ru bonds are oriented\nalmost entirely along this direction and scales the distances\nalong the $c$ axis. Calculating the relative change of the average\nCeCoAl subcell from the obtained lattice parameters yields a\nshrinking of about 0.47~\\% (between room temperature and 275~K)\nand 0.70~\\% (between room temperature\nand 120~K) along the $c$ axis, which is in line with the thermal expansion results.\\\\\n\n\n\n\n\\section{Discussion}\n\n\\begin{figure}[ht]\n\\centerline{\\includegraphics[width=0.8\\textwidth]{FikacekCeRuSnFigure6.eps}}\n\\caption{\\label{Fig6} Temperature dependency of the resistivity of\nCeRuSn at ambient and under hydrostatic pressure $p = 0.4$~GPa and\n0.8~GPa. The current is applied along the $b$ axis. Solid\n(short--dashed) lines show measurements in cooling down (warming\nup). For $p = 0.8$~GPa, the upper transition is shifted beyond the\nmaximum temperature of our experiment already.}\n\\end{figure}\n\n\n\\begin{figure}[ht]\n\\centerline{\\includegraphics[width=0.8\\textwidth]{FikacekCeRuSnFigure7.eps}}\n\\caption{\\label{rho} $p$--$T$ phase diagram of CeRuSn determined\nby resistivity experiments under hydrostatic pressure. The full\nlines are the guides for the eye separating the high-temperature,\nintermediate and low-temperature polymorphic phase, respectively,\nfor the warming regime. The dashed lines represent the alternative\nfor the cooling regime.}\n\\end{figure}\n\nSignatures of the polymorphic transitions were presented for\nresistivity, magnetization and thermal expansion. We did not show the\ndata on the thermoelectric power, temperature evolution of the\nHall effect and thermal conductivity in which also a step-like\nstructure was observed at the at \\Tcu\\,, \\Tcl\\,, \\Twl\\, and \\Twu\\,.\nAll these findings strongly indicate that the polymorphic\ntransition involves a Fermi surface reconstruction accompanied by\na change of the electronic structure. A strong electron--lattice\ncoupling is to be expected. Moreover, as pointed out in the\nintroduction, because of mutual interplay between electronic\nstructure and interatomic forces, various lattice vibration\nproperties can be expected according to whether the 4{\\it f\\\/}\nmoment is localized in an intermediate state. It was discussed by\nMydosh {\\it et. al.}~\\cite{Mydosh2011} that the step--like\nreduction of the magnetization signals a decrease of the density\nof states, i.\\,e.\\,, change of the electronic structure. However,\none might interpret the observed reduction in $\\chi(T)$ (see\nFig.~\\ref{Fig2}) as manifestation of a sudden (partial) Kondo\nscreening of the localized Ce2 moments in \\CRS\\. The slightly\nhigher than expected resistivity values for $I \\parallel b$ below\n\\Tcl\\, therefore would be a result of a reduction of conduction\nelectrons involved in screening. Such scenario applies to Cerium\nwhere lattice vibrations are suggested to play an important role\nin the Ce $\\gamma \\rightarrow \\alpha$ transition together with\nspin and charge degrees of freedom.\\cite{Jeong2004} \\\\\nSpeculation about an analogy with elementary Cerium are further\ninspired by hydrostatic pressure experiments on \\CRS\\. The upper\naccessible temperature (380~K) is limited by the properties of the\nStyCast epoxy used for sealing of the wires in the plug of the\npressure cell. In Fig.~\\ref{Fig6}, the resistivity data in\narbitrary units are depicted against temperature at ambient and at\npressures of $p = 0.4$ and $0.8$~GPa. The character of the\ntransitions remains qualitatively the same even at highest applied\npressure when they are still observable within the temperature\nrange of the experiment. Interestingly, within the applied\npressure range and the resolution of the experiment \\Tcu\\,,\n\\Tcl\\,, \\Twl\\, and \\Twu\\ increase roughly linear with an identical\nrate. The slopes of the respective polymorphic transition shift\namounts to approximately 125~K\/GPa. This linear increase of the\ntransition temperature is reminiscent to the linear shifting of\nthe $\\gamma \\rightarrow \\alpha$ phase line of\nCerium.\\cite{Gschneider1962} Here the slope is roughly\n250~K\/GPa~\\cite{Koskenmake1978} twice the ones in \\CRS\\,.\n\nThe results of the structural investigation can be understood as a\nsubsequent annealing and consecutive evolution of polymorph\nphases. In the determined phases, the cerium position (see\ntable~\\ref{Table2}) can be divided into three groups -- those\nexhibiting two short $(S)$ Ce--Ru distances (close Ru nearest\nneighbors), those with two long $(L)$ Ce--Ru distances (far\nnearest neighbors) and those $(I)$ with one close and one far Ru\nnearest neighbor (within each group, there is a variation of the\nCe--Ru distance across the polymorphs but in comparison to the\nshort-long distance the change is minimal). With this in mind (see\nFig.~\\ref{Fig5}), the low temperature structure can be described\nas $SLLS$ base building block (two CeCoAl subcells, same sequence\nas at room temperature) alternating with the $II$ cerium sequence\n(leading to the tripling of the CeCoAl subcell observed at low\ntemperatures) along the $c$ axis. With increasing temperature,\na rearrangement by displacive transformation leads to the extinction\nof half of the $II$ spacers leaving the structure with quintuple\nCeCoAl subcell (two base blocks, one spacer). Further heating up\nremoves rest of the $II$ spacers and resulting in the appearance\nof the base building block at high temperatures. Within this\ncontext, the existence of the $II$ spacer can be understood as a\ndeformation of the CeCoAl--sized cell driven by cohesion forces in\norder to stabilize the whole structure. The results of the XANES\nexperiment~\\cite{Feyerherm2012} seem to be in contradiction with\nthe above presented structural data (with decreasing temperature\nthe number of sites with short Ce--Ru distance is increased).\nHowever, it is necessary to keep in mind, that the dependence of\nthe valence on the Ce--Ru distance is not simple and that there\nare several different short Ce--Ru distances at lower\ntemperatures. This nonlinearity together with an increased number\nof crystallographically inequivalent of Ce sites and presumable Kondo\nscreening leads to shrinking of the $c$ axis concurrently and to\nan unchanged overall Ce valence, which is different from the\nstatement~\\cite{Feyerherm2012} that the valence of the Ce ions\nthrough the transition remains conserved. Further experiments\nresolving this issue are desired.\\\\\n\nTo find the true nature, i.\\,e.\\,, the driving mechanism behind\nthe polymorphic transitions is a challenging task for future work.\nThe transitions in \\CRS\\ show to some extend similarities to the\ncerium case, which might serve as reference point. In order to\nenlighten the role of lattice vibrations, inelastic neutron\nscattering experiments are envisaged.\n\n\\section{Summary}\nInvestigation of the polymorphic transitions by means of\nresistivity, magnetization, thermal expansion and X--ray\ndiffraction on single crystals of \\CRS\\ was carried out.\nMeasurements were conducted along all principle axes. In all\nphysical properties, upon cooling, two subsequent anomalies at\n\\Tcu\\ $\\sim 285$~K and \\Tcl\\ $\\sim 185$~K were detected. These\nsignatures can be attributed to polymorphic transitions, i.\\,e.\\,,\nfrom the room temperature double CeCoAl-type superstructure to a\nquintuple at \\Tcu\\ and from the quintuple to a triple CeCoAl\nunit-cell superstructure at the lower transition temperature. The\nrefined superstructures are characterized by an increased number\nof crystallographically inequivalent Ce sites. Simultaneously, the\nratio between the number of short and long Ce--Ru bonds, which are\nessentially aligned along $c$ direction, is increased. As\nconsequence, the lattice gradually contracts mainly along the $c$\naxis as observed in thermal expansion eliciting an overall\nshrinking of the sample volume. The transitions exhibit\nlarge hysteric behavior. \\\\\nThe strong response of the polymorphic transitions in transport\nand magnetic properties infers a close connection to variations in\nthe electronic structure of \\CRS\\. Slight jumps in the\nmagnetization as well as unexpected behavior in resistivity\nsuggest influence of Kondo interaction to play a role in the\nstructural change. Together with lattice vibrations, it might be\nthe driving mechanism behind the polymorphic transitions similar to the one in\nelementary Cerium. This scenario is partially rooted in\nresistivity data on \\CRS\\ under hydrostatic pressure revealing an\nalmost linear increase of the transition temperatures upon\npressure as had been observed for the $\\gamma \\rightarrow \\alpha$\ntransition in Ce as well.\n\n\n\\section{Acknowledgments}\nThis work was supported by the Czech Science Foundation (Project\n\\# 202\/09\/1027) and Charles University grants GAUK440811 and UNCE\n11.\n\n\n\n\\vspace{0.5cm}\n\n\\noindent${\\ast}$ corresponding author: fikacekjan@seznam.cz\n\n\\bibliographystyle{prsty}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\chapter*{\\contentsname}%\n    \\@mkboth{\\MakeUppercase\\contentsname}{\\MakeUppercase\\contentsname}%\n    \\@starttoc{toc}%\n    \\if@restonecol\\twocolumn\\fi\n    }\n\\makeatother\n\n\\begin{document}\n\n\\AddToShipoutPicture*{\\BackgroundPic}\n\\maketitle\n\n\n\\fancyhead[RE, LO]{  }\n\\pagenumbering{roman}\n\\thispagestyle{plain}\n\\begin{center}\n\\bf{ {\\myfont Neutron-rich matter in atomic nuclei and neutron stars}}\\\\\n\\vspace{2cm}\n{\\Large Mem\\`oria presentada per optar al grau de doctor per la Universitat de Barcelona}\\\\\n\\vspace{2cm}\n{\\Large Programa de Doctorat en F\\'isica}\\\\\n\\vspace{2cm}\n{ {\\Large Autora} \\\\ { \\large Claudia Gonz\\'alez Boquera}}\\\\\n\\vspace{2cm}\n{ {\\Large Directors} \\\\ { \\large Mario Centelles Aixal\\`a i Xavier Vi{\\~n}as Gaus\\'i}}\\\\\n\\vspace{2cm}\n{ {\\Large Tutor} \\\\ { \\large Dom\\`enec Espriu Climent}}\\\\\n\\vspace{1.5cm} \n{\\Large Departament de F\\'isica Qu\\`antica i Astrof\\'isica}\\\\\n\\vspace{1cm}\n{\\Large Setembre 2019}\\\\\n\\vspace{0.5cm}\n{\\includegraphics[clip=true,width=0.4\\paperwidth]{.\/grafics\/marca_pos_cmyk}\n\\hspace{1cm}\n{\\includegraphics[clip=true,width=0.25\\paperwidth]{.\/grafics\/LogoICCUB_MdM_prefered_negative_EN_cmyk}}}\n\\vspace{10cm}\n\\end{center}\n\n\\thispagestyle{empty}\n\\begin{center}\n\\newpage\n\\end{center}\n\n\\thispagestyle{plain}\n\\begin{center}\n\\vspace*{5cm}\n\n\\hspace{8cm}{\\it Dedicat a vosaltres, els que sempre}\\\\\n\\hspace{8cm}{\\it heu estat al meu costat}\\\\\n\\end{center}\n\n\\blankpage\n\\thispagestyle{plain}\n\\begin{center}\n\\small{\n\\hspace{7cm}You gave me the best of me\\\\\n\\hspace{7cm}So I'll give you the best of you\\\\\n\\hspace{7cm}You found me. You knew me\\\\\n\\hspace{7cm}You gave me the best of me\\\\\n\\hspace{7cm}So you'll give you the best of you\\\\\n\\hspace{7cm}You'll find it, the galaxy inside you\\\\\n\\hspace{9cm} {\\it Magic Shop, BTS}}\n\n\\small{\n\\hspace{-7cm}Starlight that shines brighter in the darkest night\\\\\n\\hspace{-7cm}Starlight that shines brighter in the darkest night\\\\\n\\hspace{-7cm}The deeper the night, the brighter the starlight\\\\\n\\hspace{-7cm}One history in one person\\\\\n\\hspace{-7cm}One star in one person\\\\\n\\hspace{-7cm}7 billion different worlds\\\\\n\\hspace{-7cm}Shining with 7 billion lights\\\\\n\\hspace{-7cm}7 billion lives, the city's night view\\\\\n\\hspace{-7cm}Is possibly another city's night\\\\\n\\hspace{-7cm}Our own dreams, let us shine\\\\\n\\hspace{-7cm}You shine brighter than anyone else\\\\\n\\hspace{-7cm}One\\\\\n\\hspace{-5cm} {\\it Mikrokosmos, BTS}}\n\n\\small{\n\\hspace{7cm}When you're standing on the edge\\\\\n\\hspace{7cm}So young and hopeless\\\\\n\\hspace{7cm}Got demons in your head\\\\\n\\hspace{7cm}We are, we are\\\\\n\\hspace{7cm}No ground beneath your feet\\\\\n\\hspace{7cm}Now here to hold you\\\\\n\\hspace{7cm}'Cause we are, we are\\\\\n\\hspace{7cm}The colors in the dark\\\\\n\\hspace{9cm} {\\it We are, One OK Rock}}\n\n\n\\small{\n\\hspace{-7cm}We'll fight fight till there's nothing left to say\\\\\n\\hspace{-7cm}(Whatever it takes)\\\\\n\\hspace{-7cm}We'll fight fight till your fears, they go away\\\\\n\\hspace{-7cm}The light is gone and we know once more\\\\\n\\hspace{-7cm}We'll fight fight till we see another day\\\\\n\\hspace{-5cm} {\\it Fight the night, One OK Rock}}\n\n\n\\small{\n\\hspace{7cm}So you can throw me to the wolves\\\\\n\\hspace{7cm}Tomorrow I will come back\\\\\n\\hspace{7cm}Leader of the whole pack\\\\\n\\hspace{7cm}Beat me black and blue\\\\\n\\hspace{7cm}Every wound will shape me\\\\\n\\hspace{7cm}Every scar will build my throne\\\\\n\\hspace{9cm} {\\it Throne, Bring me the horizon}}\n\n\n\\small{\n\\hspace{-7cm}Please just once\\\\\n\\hspace{-7cm}If I can just see you\\\\\n\\hspace{-7cm}I'm OK with losing everything I have\\\\\n\\hspace{-7cm}I'll meet you, even if it's in a dream\\\\\n\\hspace{-7cm}And we can love again\\\\\n\\hspace{-7cm}Just as we are\\\\\n\\hspace{-5cm} {\\it UNTITLED, 2014, G-Dragon}}\n\\end{center}\n\n\n\\blankpage\n\n\\chapter*{Abstract}\nThe proper understanding of the equation of state (EoS) of highly asymmetric nuclear matter is essential \nwhen studying systems such as neutron stars (NSs). Using zero-range Skyrme interactions and finite-range \ninteractions such as Gogny forces, momentum-dependent \ninteractions (MDI) and simple effective interactions (SEI), we analyze the properties of the EoS and the\ninfluence they may have on the calculations for NSs. \n\nWe start by studying the convergence properties of the Taylor series expansion of the EoS in powers of the isospin asymmetry.\nNext, we analyze the accuracy of the results for $\\beta$-stable nuclear matter, which is found in the interior of NSs, \nwhen it is computed using the Taylor expansion of the EoS. The agreement with the results obtained using the full expression of the EoS \nis better for interactions with small-to-moderate values of the slope of the symmetry energy, $L$.\nWe also obtain the results for the $\\beta$-equilibrated matter when the Taylor expansion of the EoS is performed up to \nsecond order in the potential part of the interaction, while the kinetic part is used in its full form. In this case, \none almost recovers the exact results. \n\nThe mass and radius relation for an NS is obtained by integrating the so-called Tolman-Oppenheimer-Volkoff (TOV) equations, where the\ninput is the EoS of the system. We have studied the mass-radius relation for Skyrme and Gogny interactions, and we see  \nthat very soft forces are not able to give stable solutions of the TOV equations and only the stiff enough\nparametrizations can provide $2$ solar mass ($M_\\odot$) NSs.\nWe also notice that none of the existing parametrizations of the  \nstandard Gogny D1 interaction are able to provide an NS inside the observational constraints. \nBecause of that, we propose a new parametrization, which we name D1M$^*$, that is able to \nprovide NSs of $2 M_\\odot$ while still providing the same good description of finite nuclei as D1M. A parametrization\nD1M$^{**}$ is also presented, which is fitted in the same way as D1M$^*$ and provides NSs up to $1.91 M_\\odot$.\n\nAn accurate determination of the core-crust transition point, which is intimately related to the isospin dependence \nof the nuclear force at low baryon densities, \nis necessary for the modeling of NSs for astrophysical \npurposes. We estimate the core-crust transition in NSs by finding \nwhere the nuclear matter in the core is unstable against fluctuations of the density. \nTo do that, we employ two methods, the thermodynamical method and the dynamical method. \nThe first one considers the mechanical and chemical stability conditions for the core, and neglects \nthe surface and Coulomb effects in the stability conditions.\nMoreover, we obtain the core-crust transition using the dynamical method, \nwhere one considers bulk, surface and Coulomb effects when studying \nthe stability of the uniform matter.\nIn the case of finite-range interactions, such as the Gogny forces, we have had to derive the explicit expression of the energy curvature matrix in \nmomentum space for this type of interactions. \nWe observe a decreasing trend of the transition density with the slope $L$ of the symmetry energy, while \nthe correlation between the transition pressure and $L$ is much lower.\nThe results of the core-crust transition properties\nobtained with the Taylor expansion of the EoS are close to the exact results only in the case of soft EoSs. \nFor interactions with large values of $L$ and stiff EoSs, the results \ncomputed using the Taylor expansion, even after adding terms beyond the second-order in the expansion, are far from the exact values. \n\nFinally, different NS properties are studied. The crustal properties, such as the crustal mass, crustal thickness and crustal \nfraction of the moment of inertia, have lower values if one computes them using the core-crust transition density \nobtained with the dynamical method instead of the one obtained with the thermodynamical method,\npointing out the importance of the accurate evaluation of the transition density when studying observational phenomena. \nWe have also studied  \nthe moment of inertia of NSs, which is compared to some constraints proposed in the literature. \nFinally, the tidal deformability for NSs is also calculated and compared with the constraints coming from the \nGW170817 event detected by the LIGO and Virgo observatories and which accounts for the merger of two NSs in a binary system. \n\n\n\\chapter*{Resum}\\label{resum}\nAquesta tesi doctoral pret\\'en estendre els estudis de l'equaci\\'o d'estat (EoS)\nde mat\\`eria nuclear altament assim\\`etria, utilitzant models de camp mig no relativistes, com per \nexemple les interaccions de contacte tipus Skyrme~\\cite{skyrme56, vautherin72,sly41}, o models d'abast finit com les \ninteraccions de Gogny~\\cite{decharge80, berger91}, \nles anomenades {\\it\nmomentum dependent interactions}, (MDI)~\\cite{das03,li08} i les {\\it simple effective\ninteractions} (SEI)~\\cite{behera98, Behera05}. \n\nEl Cap\\'itol~\\ref{chapter1} recull un breu resum de l'aproximaci\\'o\nde camp mig, a on un assumeix el sistema nuclear com a un conjunt\nde quasi-part\\'icules no interactuants que es mouen independentment\ndins d'un potencial de camp mig efectiu. \nEl mateix cap\\'itol recull els conceptes b\\`asics del m\\`etode de \nHartree-Fock utilitzat per tal de trobar l'energia del sistema.\nA m\\'es a m\\'es, s'introdueixen els diferents potencials \nfenomenol\\`ogics que s'utilitzaran al llarg d'aquesta tesi, tals com\nles interaccions de Skyrme, Gogny, MDI i SEI.\nTots aquests funcionals, especialment els de Skyrme i els de Gogny,\nreprodueixen amb bona qualitat les propietats dels nuclis finits. \nEn aquest treball tamb\\'e estudiarem \nmat\\`eria nuclear a elevades densitats i a elevades \nassimetries d'isosp\\'i.\nLes definicions de diverses propietats de \nl'EoS de mat\\`eria nuclear sim\\`etrica\ni de mat\\`eria nuclear assim\\`etrica tamb\\'e estan incloses en \naquest cap\\'itol. \n\n\nEs dedica el Cap\\'itol~\\ref{chapter2} a l'estudi de les\npropietats de la mat\\`eria nuclear assim\\`etrica utilitzant un conjunt d'~interaccions de Skyrme i de Gogny.\nPrimerament s'analitza el comportament dels coeficients de \nl'energia de simetria que apareixen en l'expansi\\'o de Taylor\nde l'energia per part\\'icula en termes de l'assimetria \nd'isosp\\'i. L'EoS s'expandeix fins al des\\`e ordre en el cas de les \ninteraccions de Skyrme i fins al sis\\`e ordre en el cas de les\nforces de Gogny~\\cite{gonzalez17}. \nEl comportament del coeficient de segon ordre de l'energia de simetria, \nel qual se'l coneix com a energia de simetria, divideix les interaccions\nde Skyrme (i tamb\u00e9 les de Gogny) en dos grups. \nEl primer grup recull les parametritzacions que tenen una energia\nde simetria que desapareix a una certa densitat sobre la saturaci\\'o,\nimplicant una inestabilitat d'isosp\\'i. \nEl segon grup est\\`a format per aquelles interaccions, normalment\namb un pendent $L$ a la densitat de saturaci\\'o major, les\nenergies de simetria de les quals tenen sempre un pendent creixent. \nTamb\\'e s'estudia l'energia de simetria si s'ent\\'en com la \ndifer\\`encia entre l'energia per part\\'icula en mat\\`eria\nneutr\\`onica i en mat\\`eria sim\\`etrica, la qual s'anomenar\\`a\nenergia de simetria parab\\`olica. \nAquesta definici\\'o tamb\\'e coincideix amb la suma infinita \nde tots els coeficients de l'expansi\\'o de Taylor de \nl'energia per part\\'icula en termes de l'assimetria si aquesta\npren valors de la unitat. Al voltant del punt de saturaci\\'o, \nles difer\\`encies entre el coeficient de segon ordre i l'energia\nde simetria parab\\`olica es redueixen si es\nconsideren m\\'es termes de l'expansi\\'o de l'EoS~\\cite{gonzalez17}.\nA m\\'es a m\\'es, tamb\\'e  s'evaluen \nels seus respectius pendents de l'energia de simetria, \ni veiem que poden sorgir algunes discrep\\`ancies entre ells. \n\nL'interior de les estrelles de neutrons (NSs) est\\`a format\nper mat\\`eria en $\\beta$-equilibri. \nEn el Cap\\'itol~\\ref{chapter2} d'aquesta tesi s'analitza la converg\\`encia de l'expansi\\'o\nen s\\`erie de Taylor de l'EoS en pot\\`encies de l'assimetria\nd'isosp\\'i quan estudiem mat\\`eria nuclear en $\\beta$-equilibri.\nL'acord de l'assimetria d'isosp\\'i i de la pressi\\'o al llarg de totes les \ndensitats calculades amb l'expansi\\'o de l'EoS millora si es consideren m\\'es ordres, \nsent la millora m\\'es lenta per interaccions amb par\\`ametre de pendent $L$ m\\'es gran. \nAquestes difer\\`encies s\\'on rellevants en l'estudi de NSs, en el qual s'utilitza \nl'EoS de mat\\`eria nuclear infinita per descriure el nucli d'una NS.\nSi es duu a terme el desenvolupament en s\\`erie de Taylor nom\\'es en la part \npotencial de la for\\c{c}a i s'utilitza l'expressi\\'o completa per a la part cin\\`etica, \npr\\`acticament es recobren els mateixos valors per l'assimetria i per la pressi\\'o \nque en el cas que s\\'on calculades amb l'expressi\\'o completa de l'EoS~\\cite{gonzalez17}.\n\nLa relaci\\'o entre la massa i el radi de les NSs tamb\\'e ha estat estudiada \nal Cap\\'itol~\\ref{chapter2}\nconsiderant models de Skyrme i de Gogny. Es troba que forces que s\\'on \nmolt {\\it soft}, i.e., \namb un baix valor del pendent de l'EoS, no s\\'on capaces de donar \nsolucions estables de les equacions \nTOV, i nom\\'es les interaccions suficientment {\\it stiff}, i.e., amb \nun valor alt del pendent\nde l'EoS, poden proveir NSs de $2$ masses solars ($M_\\odot$).\nEn particular, notem que cap de les interaccions de Gogny que s'engloben dins de \nla familia D1 d\\'ona NSs dins dels l\\'imits observacionals~\\cite{Sellahewa14, gonzalez17}. \nLa converg\\`encia de l'EoS tamb\\'e \\'es testejada quan s'estudien propietats de les NSs. \nUn troba que, si l'expansi\\'o de Taylor es talla al segon ordre, els resultats \npoden quedar lluny dels obtinguts utilitzant l'EoS completa. \nAquesta converg\\`encia \\'es m\\'es lenta contra m\\'es elevat \\'es\nel pendent de l'energia de simetria de la \ninteracci\\'o .\nAquest comportament senyala la necessitat d'utilitzar l'expressi\\'o completa de l'EoS sempre\nque es pugui. \n\nTal i com s'ha esmentat, la fam\\'ilia D1 d'interaccions Gogny no inclou cap for\\c{c}a que\nsigui capa\\c{c} de donar una NS que arribi a $2 M_\\odot$, ja que totes les parametritzacions\ntenen energies de simetria {\\it soft}~\\cite{Sellahewa14, gonzalez17}. \nDins del Cap\\'itol~\\ref{chapter3} proposem dues interaccions de Gogny noves, \nles quals anomenem D1M$^*$ i D1M$^{**}$, que s\\'on capaces de donar una NS dins dels\nlligams observacionals a la vegada que proveeixen una bona descripci\\'o dels nuclis \nfinits semblant a la de la interacci\\'o D1M~\\cite{gonzalez18, gonzalez18a,Vinas19}. \nLa interacci\\'o D1M$^{*}$ \\'es capa\\c{c} de donar una NS de $2M_\\odot$, \nmentre que la interacci\\'o D1M$^{**}$ \\'es capa\\c{c} de descriure NSs de fins a $1.91 M_\\odot$.\nAltres propietats estudiades amb les interaccions D1M$^*$ i D1M$^{**}$ estan en acord amb els resultats obtinguts\nutilitzant l'EoS de SLy4~\\cite{douchin01}.\nEn aquest cap\\'itol s'analitzen algunes propietats de l'estat fonamental de nuclis finits, com per \nexemple energies de lligam, els radis de neutrons i protons, la resposta al moment quadrupolar i barreres de fissi\\'o.\nAquestes dues noves parametritzacions D1M$^{*}$ i D1M$^{**}$ duen a terme igual de b\\'e que D1M aquests estudis relacionats \namb nuclis finits~\\cite{gonzalez18, gonzalez18a}. Es pot dir que les interaccions D1M$^{*}$ i D1M$^{**}$ s\\'on bones alternatives per descriure \nsimult\\`aniament els nuclis finits i les NSs, donant resultats molt bons en harmonia amb dades experimentals i observacionals. \n\nLa determinaci\\'o correcte de la transici\\'o entre el nucli i l'escor\\c{c}a en les NSs \\'es clau en la comprensi\\'o\nde fen\\`omens en les NSs, com per exemple {\\it glitches} en els p\\'ulsars, els quals depenen de la mida de l'escor\\c{c}a~\\cite{Link1999,Fattoyev:2010tb,Chamel2013,PRC90Piekarewicz2014,Newton2015}. \nEn el Cap\\'itol~\\ref{chapter4} s'estima sistem\\`aticament el punt de transici\\'o entre el nucli i l'escor\\c{c}a\nbuscant la densitat en que la mat\\`eria nuclear del nucli estel\u00b7lar \\'es inestable contra fluctuacions de densitat. \nLes inestabilitats s\\'on determinades utilitzant dos m\\`etodes. Primer, utilitzem l'anomenat m\\`etode termodin\\`amic, \na on s'estudien les estabilitats mec\\`anica i qu\\'imica del nucli, i ho fem per interaccions de Skyrme i de Gogny. \nTal i com s'ha esmentat en literatura anterior, es troba una tend\\`encia a disminuir la densitat de transici\\'o \nquan el pendent $L$ augmenta. \nPer altra banda, no es troben correlacions fortes entre la pressi\\'o de transici\\'o i $L$~\\cite{gonzalez17}.\nTamb\\'e s'ha estudiat la converg\\`encia de les propietats de transici\\'o quan s'utilitza el desenvolupament de Taylor de l'EoS. \nEn general, quan s'afegeixen m\\'es termes al desenvolupament, la densitat de transici\\'o s'aproxima als resultats trobats amb l'EoS exacte. \nNo obstant, es continuen trobant difer\\`encies significatives quan s'utilitzen fins i tot termes d'ordre majors que dos, \nespecialment en casos a on el pendent de l'energia de simetria \\'es elevat. \nLa densitat de transici\\'o tamb\\'e s'ha obtingut amb el m\\`etode din\\`amic, a on un considera,\na l'hora d'estudiar l'estabilitat del sistema, efectes de volum, superf\\'icie i de Coulomb.\nAl Cap\\'itol~\\ref{chapter4} es duen a terme els c\\`alculs per a interaccions de Skyrme, i per diferents \nforces d'abast finit que s\\'on, en el nostre cas, les interaccions de Gogny, MDI i SEI. \nEn general els resultats per la densitat de transici\\'o utilitzant el m\\`etode din\\`amic s\\'on inferiors als que es troben quan \ns'utilitza el m\\`etode termodin\\`amic. \nLa converg\\`encia \\'es millor per a EoSs {\\it soft}.\nPrimer obtenim els resultats per a interaccions de Skyrme, i analitzem la converg\\`encia de les propietats de transici\\'o \nsi s'utilitza el desenvolupament de Taylor quan es calculen. \nLa converg\\`encia de les propietats de transici\\'o entre el nucli i l'escor\\c{c}a \\'es la mateixa que es troba quan s'utilitza \nel m\\`etode termodin\\`amic, \\'es a dir, els resultats s\\'on m\\'es propers als exactes si s'utilitzen m\\'es termes a l'expansi\\'o de Taylor de l'EoS.\nSi la transici\\'o \\'es obtinguda aplicant el desenvolupament de Taylor de l'EoS nom\\'es en la part potencial i utilitzant \nl'energia cin\\`etica exacte, els resultats tornen a ser pr\\`acticament els mateixos que els exactes. \n\nFinalment, al Cap\\'itol~\\ref{chapter4} s'obtenen els valors de les propietats de transici\\'o utilitzant el m\\`etode din\\`amic\namb interaccions d'abast finit. \nContr\\`ariament al cas de les interaccions de Skyrme, s'ha de derivar expl\\'icitament l'expressi\\'o de la matriu de curvatura de l'energia\nen espai de moments per aquest tipus de forces~\\cite{gonzalez19}. \nLes contribucions al terme de superf\\'icie s'han pres tant de la part d'interacci\\'o com de la part cin\\`etica, fent aquesta derivaci\\'o\nm\\'es autoconsistent comparada a la d'estudis previs. Les contribucions provinents de la part directa s\\'on obtingudes a partir de \nl'expansi\\'o en termes de distribucions dels seus factors de forma, i les contribucions provinents dels termes d'intercanvi i cin\\`etics \nes troben expressant les seves energies com una suma del terme de volum m\\'es una correcci\\'o $\\hbar^2$ en el marc de l'aproximaci\\'o\n{\\it Extended Thomas Fermi}. Es troba que els efectes de la part d'abast finit de la interacci\\'o sobre la matriu de curvatura venen\nmajorit\\`ariament del terme directe de l'energia. \nPer tant, en l'aplicaci\\'o del m\\`etode din\\`amic amb interaccions d'abast finit, utilitzar nom\\'es la contribuci\\'o del terme directe \\'es una\naproximaci\\'o acurada, al menys per les interaccions utilitzades en aquesta tesi. \nTamb\\'e s'ha analitzat el comportament global de la densitat de transici\\'o i de la pressi\\'o de transici\\'o en funci\\'o del\npendent de l'energia de simetria a la saturaci\\'o. Els resultats per les interaccions MDI estan en acord amb resultats previs~\\cite{xu10b} i tamb\\'e, per MDI i SEI, \nla densitat de transici\\'o i la pressi\\'o de transici\\'o estan altament correlacionades amb $L$. Tot i aix\\'i, si els models tenen \ndiferents propietats de saturaci\\'o, com per exemple en el cas del grup de forces de Gogny que hem utilitzat en aquest treball, les correlacions es deterioren. \n\nEl Cap\\'itol~\\ref{chapter5} de la tesi inclou l'an\\`alisi de diferents propietats de les NSs.\nPrimer s'estudia la influ\\`encia de l'EoS a l'escor\\c{c}a interna quan s'estudien propietats globals, com per exemple masses i radis~\\cite{gonzalez17, gonzalez19}. \nS'analitzen algunes propietats de l'escor\\c{c}a, com poden ser la massa de  l'escor\\c{c}a o el seu gruix. \nAqu\\'i es veu una altra vegada la import\\`ancia de la bona determinaci\\'o de la localitzaci\\'o de la transici\\'o entre el nucli i \nl'escor\\c{c}a, ja que els resultats de les propietats de l'escor\\c{c}a s\\'on menors si la transici\\'o s'ha estimat a dins de l'aproximaci\\'o\ndin\\`amica en comptes de la termodin\\`amica. \nAquestes propietats de l'escor\\c{c}a juguen un paper crucial a l'hora de predir molts fen\\`omens observacionals, com per exemple {\\it glitches}, \noscil{\u00b7}lacions {\\it r-mode}, etc. \nPer tant, una bona estimaci\\'o de les propietats de l'escor\\c{c}a \\'es clau en la comprensi\\'o de les NSs. \n\nLa detecci\\'o d'ones gravitacionals ha obert una nova finestra a l'Univers. La senyal GW170817 detectada per la \ncol{\u00b7}laboraci\\'o LIGO i Virgo provinent d'un {\\it merger} (fusi\\'o) de dues NSs ha donat peu a un seguit de nous lligams tant en \nastrof\\'isica com en f\\'isica nuclear~\\cite{Abbott2017, Abbott2018, Abbott2019}.\nUn lligam directament observat de la senyal \\'es el relacionat amb el que s'anomena la {\\it tidal deformability}, o la \ndeformaci\\'o deguda a les forces de marea, representada per $\\tilde{\\Lambda}$, a una certa {\\it chirp mass} del sistema binari. \nDespr\\'es de l'an\\`alisi de les dades, es van donar lligams a altres propietats, com per exemple a la {\\it tidal deformability} d'una \nNS can\\`onica de $1.4 M_\\odot$ ($\\Lambda_{1.4}$), a les masses, als radis, etc.~\\cite{Abbott2017, Abbott2018, Abbott2019}. \nDins del Cap\\'itol~\\ref{chapter5} s'han analitzat els valors de $\\tilde{\\Lambda}$ i $\\Lambda_{1.4}$, i es veu que EoSs molt {\\it stiff}\nno s\\'on capaces de predir valors dins de les restriccions observacionals. Utilitzant un grup de diverses interaccions de \ncamp mig, s'estima el radi d' una NS de $1.4M_\\odot$, els quals estan en conson\\`ancia amb els valors donats per la col{\u00b7}laboraci\\'o LIGO i Virgo.\nFinalment, el moment d'in\\`ercia tamb\\'e \\'es analitzat, trobant, una altra vegada, que EoSs molt {\\it stiff} no proveeixen moments d'in\\`ercia\ndins dels lligams predits per Landry i Kumar per el sistema doble p\\'ulsar PSR J0737-3039~\\cite{Landry18}.\nLes noves interaccions D1M$^{*}$ i D1M$^{**}$ donen molt bons resultats tant per les estimacions de la {\\it tidal deformability} com \nper les estimacions del moment d'in\\`ercia, confirmant el seu bon rendiment en el domini astrof\\'isic. \nHem analitzat la fracci\\'o del moment d'in\\`eria encl\\`os en l'escor\\c{c}a, utilitzant les densitats de transici\\'o obtingudes \namb els m\\`etodes termodin\\`amic i din\\`amic. \nTal i com passa amb la massa i el gruix de l'escor\\c{c}a, la fracci\\'o del moment d'in\\`ercia encl\\`os en l'escor\\c{c}a \n\\'es menor si la transici\\'o s'ha obtingut amb el m\\`etode din\\`amic~\\cite{gonzalez19}. \n\nLes conclusions de la tesi estan incloses al Cap\\'itol~\\ref{conclusions}.\nS'afegeixen tres Ap\\`endixs al final de la tesi. L'Ap\\`endix~\\ref{appendix_thermal} inclou les expressions expl\\'icites \nde les derivades que es necessiten per tal d'obtenir la transici\\'o entre el nucli i l'escor\\c{c}a. L'Ap\\`endix~\\ref{app_taules}\n cont\\'e els resultats de les propietats de transici\\'o obtinguts utilitzant tant el m\\`etode termodin\\`amic com el din\\`amic.\nFacilitem a l'Ap\\`endix~\\ref{app_vdyn} detalls t\\`ecnics sobre l'aproximaci\\'o {\\it Extended Thomas Fermi}, la qual \ns'utilitza per derivar la teoria del m\\`etode din\\`amic per a interaccions d'abast finit. \n\n\\chapter*{Acknowledgements}\nM'agradaria comen\\c{c}ar donant les gr\\`acies als meus dos directors de tesi, en Mario Centelles i en Xavier Vi{\\~n}as, els quals des d'un primer moment \nem van instar i animar a comen\\c{c}ar aquesta aventura. Els vull donar les gr\\`acies per tota la paci\\`encia que \nhan tingut amb mi, i per tot el coneixement que m'han transm\\`es al llarg d'aquests cinc anys (quatre de doctorat i \nun de m\\`aster). Vaig fer b\\'e d'escollir-los a ells com a mentors, gr\\`acies a ells he apr\\`es, no nom\\'es de f\\'isica, \nsin\\'o tamb\\'e com a persona. \nTamb\\'e m'agradaria fer esment aqu\\'i a l'Elo\\\"isa, la qual sempre m'ha obert les portes de casa seva. Moltes gr\\`acies per les teves \ns\\`avies paraules. \n\n A tota la gent del Departament, en especial al grup de F\\'isica Nuclear i At\\`omica, moltes gr\\`acies per brindar-me \nla m\\`a cada vegada que ho he necessitat. M'agradaria fer esment als Professors  Artur Polls, \\`Angels Ramos, Assumpta Parre{\\~n}o, \nVolodymir Magas, Bruno Juli\\`a, Manuel Barranco, Jos\\'e Maria Fern\\'andez i Francesc Salvat per aportar amenes converses i ajut en tot el possible. \nNo puc deixar de fer esment a totes les persones que he anat coneixent durant aquests anys, cada una aportant el seu gra de sorra en el meu creixement. \nEn especial, m'agradaria remarcar l'ajut que sempre m'ha brindat el Professor Isaac Vida{\\~n}a.\n\nI would also like to thank my external collaborators with whom I have worked. Thank you Professors Arnau Rios, Luis Robledo, T.R. Tusar, L.Tolos,\nO. Louren\\c{c}\u0327o, M. Bhuyan, C. H. Lenzi, M. Dutra, for your very valuable inputs in the expansion of my knowledge. \n\nAls meus companys de despatx, els que ja han partit i els que es quedaran quan jo marxi: els Antonios, els Alberts, l'Ivan, la Clara, en Rahul i la Maria. \nGr\\`acies per crear un bon ambient de treball en el que un pot estar com a casa dins del seu despatx. \n\nAl grup del {\\~n}am {\\~n}am: Albert, Adri\\`a, Alejandro, Andreu, Chiranjib, Gl\\`oria, Ivan, Javi, Marc Illa i Marc Oncins. Gr\\`acies per fer les hores de \ndinar tant divertides i que passin com si fossin cinc minuts. Que l' amistat que ha creat compartir quatre metres quadrats entre vint persones no la \ntrenqui la dist\\`ancia. Per moltes excursions m\\'es a Montserrat i cal\\c{c}otades al febrer. Recordeu que tenim pendent escapades a Osca, Finl\\`andia, \nCant\\`abria, Nova Zelanda, India i a l'Ant\\`artida. \n\nMoltes gr\\`acies al Departament de F\\'isica Qu\\`antica i Astrof\\'isica per brindar-me un lloc en un acollidor despatx i tot el material i eines \nnecess\\`aries per dur a terme aquesta tesi. \nTamb\\'e agraeixo el suport de l'ajut \nFIS2017-87534-P provinent del MINECO i de FEDER,\nel projecte MDM-2014-0369 de l'ICCUB\n(Unidad de Excelencia Mar\u0131\u0301a de Maeztu) del MINECO i la beca FPI BES-2015-074210 provinent del Ministerio de Ciencia, Innovaci\\'on y Universidades.\n\nJa cap al final, per\\`o no menys important, m'agradaria donar les gr\\`acies al meu pilar fonamental durant aquest anys: la meva fam\\'ilia. \nMoltes gr\\`acies per estar sempre amb mi, tant en els bons moments com en els no tant bons. Moltes gr\\`acies per evitar que m'enfons\\'es en els \nmoments que pensava que ja no podia m\\'es. Gr\\`acies per aconsellar-me, per posar-me tot el m\\'es f\\`acil possible, encara que us perjudiqu\\'es a vosaltres. \nPer riure amb mi, per\\`o tamb\\'e per haver plorat amb mi al llarg d'aquests anys.\nUn remarc especial ha d'anar cap a la meva mare, la qual sempre ha estat al meu costat, tant quan s'ha tingut el vent a favor, com quan anava en contra.\n Espero que hagi valgut la pena. Gr\\`acies per tot. \n\n\nFinally, I would like to thank you, the reader, to take the time to read the work of four years of my life.\n\nA tots els que heu passat en algun moment per la meva vida durant aquests anys, per tot el que hi ha hagut i pel que vindr\\`a, gr\\`acies. \n\n\\tableofcontents\n\n\n\\chapter{Introduction}\\label{intro}\n\\fancyhead[RE, LO]{Chapter \\thechapter}\n\\pagenumbering{arabic}\nThe presence of neutrons, neutral-charged particles, inside atomic nuclei \nwas proposed by Ernest Rutherford in 1920 and experimentally proved in 1932 by James Chadwick, who \nreceived the Nobel prize for it in 1935. \nDuring that time, the existence of compact stars with a density comparable to the one of an atomic nucleus \nwas first discussed by Landau, Bohr, and Rosenfeld. \nIn 1934, Walter Baade and Fritz Zwicky wrote~\\cite{Baade34}:\n{\\it ``With all \nreserve we advance the view that supernovae represent the transition from ordinary stars into \nneutron stars, which in their final stages consist of extremely closely packed neutrons.''}\nWith this sentence, they pointed out the origin of neutron stars to be supernova explosions. \nThis led Richard Tolman~\\cite{Tolman39} and, independently, Robert Oppenheimer and his student George Volkoff~\\cite{Oppenheimer39} to perform \nthe first neutron star calculations by proposing a set of equations describing static spherical stars in General Relativity. \nIn order to find the relation between the pressure and the energy density, i.e., the equation of state (EoS) of the system,\nthey considered neutron stars as spheres of a degenerate gas of free neutrons. This led Oppenheimer and Volkoff to \nfind that static neutron stars could not have masses larger than $\\sim 0.7$ solar masses ($M_\\odot$), a value that is\nmuch lower than the Chandrasekhar\nmass limit of white dwarfs $\\sim 1.44 M_\\odot$. This result pointed out the \nimportance of considering nuclear forces in the description of the neutron star interior. \nAround 1960, John Weeler and collaborators presented~\\cite{Wheeler58} the first results for neutron stars considering their interiors composed of \nneutron, proton, and electron Fermi gases. \nIn 1959, Cameron used Skyrme interactions to study the effect of nuclear interactions on the structure of neutron stars, finding \nsolutions of maximum masses around $2 M_\\odot$~\\cite{Cameron59}. More works related to the possible new ingredients to the neutron star EoS followed, where \nother particles like muons, mesons, hyperons, or even deconfined quark matter were considered~\\cite{Vidana18}. \n\nNeutron stars were expected to be seen in X-rays, but the observations were inconclusive until the detection of pulsars. \nIn 1967, Jocelyn Bell, a Ph.D. student under the supervision of Anthony Hewish, was observing quasars with a radio telescope at Cambridge University\nwhen she detected an extremely regular pulsating signal of $81.5$ MHz and a period of $1.377$s~\\cite{Hewish68}.\nAfter eliminating possible man-made sources for those regularly-spaced bursts of radio source, she realized that this emission \nhad come from outer space. \nOne possible explanation she and their collaborators jokingly gave for the signal was that they perhaps had observed extraterrestrial life, \nand named the signal as LGM: Little Green Men. Later on, they realized that the source of the signal had to come from a rapidly spinning neutron star. \nFor that, Anthony Hewish was awarded the Nobel Prize in 1974. \n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[clip=true, width=0.7\\linewidth]{.\/grafics\/intro\/Star_laters.jpg}\n\\caption{Scheme of the onion-like structure of a burning star.}\\label{fig:Slayers}\n\\end{figure}\n\nDuring the majority of their lives, stars are in thermal and gravitational equilibrium, \nfusing hydrogen~\\cite{shapiro83,Glendenning2000, haensel07}. When the hydrogen is all burned, the core of the star, which is now mostly composed of helium, contracts until it reaches such \ntemperatures that the helium can be burned, leading the star to a new gravitational and thermal equilibrium. \nSurrounding the core, the star has a hydrogen shell, which is also burning, and the envelope is expanded to such dimensions that the star becomes a red giant. \nWhen the helium in the core is exhausted, the star has a carbon core, and the same process of finding a new equilibrium where this element can be used as fuel is started. \nFor stars of mass greater than about $10 M_\\odot$, this process is repeated several times, obtaining each time a core composed of a heavier element, \nobtaining an onion-like structure inside the star (see Fig.~\\ref{fig:Slayers}). \nThis process is stopped when the core is formed of iron, which is the most tightly bound element in the universe. \nBecause of that, the star cannot produce energy through iron fusion. As the other shells are still burning lighter elements, more matter \nwill be falling to the core, \nuntil there is a point that the electrons become ultrarelativistic. The mass of the core will continue growing until reaching \nthe Chandrasekhar mass when the electrons cannot avoid the gravitational collapse. \nInside the core, which has reached temperatures of $ \\sim 10^{10}$ K, highly energetic photons are able to photodissociate \nthe iron nuclei and the core starts to cool and further contract, increasing its density. \nMoreover, there are inverse $\\beta$-decay processes, in which the electrons are captured by protons, releasing neutrinos and forming neutrons, which at such densities \nbecome degenerate. \nContrary to the electrons, which cannot leave the core, the neutrinos can escape, giving an additional energy loss to the system and speeding the \ncollapse. When reaching densities around \\mbox{$\\sim 4 \\times 10^{11}-10^{12}$ g cm$^{-3}$}, the core becomes opaque to the neutrinos. The energy \ncannot be freed, reheating nuclei that start to burn again. The collapse continues until the core reaches densities of \n$\\sim 2-3$ times the saturation density, which is of the order of $ \\sim 10^{14}$ g cm$^{-3}$. At this stage, the radius of the core\nis around $10$ km and the core consists of $A \\sim 56$ nuclei, neutrons, protons, and electrons. The material falling in the core bounces \nreleasing a shock-wave outwards from the interior of the proto-neutron star and the material produced in the previous stages is expelled at very high energies\nin the form of a supernova explosion. \nFor stars of a mass of about $10 M_\\odot \\lesssim M \\lesssim 40 M_\\odot$, the result of the supernova explosion will be a neutron star. These stellar objects will have \nradii of $\\sim 10-16$ km, masses of the order of $\\sim 1-2 M_\\odot$ and average densities around $\\sim 10^{14}-10^{15}$ g cm$^{-3}$. Some of them can present strong \nmagnetic fields and highly precise rotational periods. \nPulsars are magnetized neutron stars that emit focused beams of electromagnetic radiation through their magnetic axis. \nIf the rotational axis is not aligned with its magnetic axis, a ``lighthouse effect'' will arise which from the Earth will be seen \nas radio pulses.\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[clip=true, width=0.7\\linewidth]{.\/grafics\/intro\/NS_layers_difernt_fons_ampliat.jpg}\n\\caption{Scheme of the structure of a neutron star.}\\label{fig:NSlayers}\n\\end{figure}\n\nA neutron star can be divided into different regions, namely the atmosphere, the crust, and the core. \nThe atmosphere contains a negligible amount of mass compared to the crust and the core. \nIt influences the photon spectrum and the thermal energy released from the surface of the star~\\cite{Lattimer2004}. \nAfter the atmosphere, we find the crust, which can be separated into two different regions, the outer crust and the inner crust. \nThe outer crust consists of nuclei distributed in a solid body-centered cubic (bcc) lattice of \npositive-charged clusters permeated by a \nfree electron gas. It goes from a density around the one of the terrestrial iron $\\sim 7.5$ g cm$^{-3}$, to densities \naround $\\sim 4.3 \\times 10^{11}$ g cm$^{-3}$, where the density and the pressure are so high that the nuclear force repels the neutrons inside the nuclei\n and they start to drip, i.e., the system has reached the neutron drip line. \n The transition between the outer and inner crust parts is essentially determined by nuclear masses~\\cite{baym71}, \n which are experimentally known up to average densities of $\\sim 4 \\times 10^{11}$ g cm$^{-3}$ of the outer crust. \n From this density on, masses are predicted theoretically by using finite-tuned mass formulas~\\cite{Duflo95, Moller95} or successful mean-field models \n of nuclear masses~\\cite{ruster06, XaviRoca08, Pearson11, sharma15}. \n After these densities, one enters the region of the inner crust, where the lattice structure of nuclear clusters remains, \n but now is embedded in free neutron and electron gases. The fraction of free neutrons grows when the density increases up to \n about one half of the nuclear matter saturation density. At this density, the transition to the core\n occurs because it is energetically favorable for the system to change from a solid \n to a liquid phase. In the deepest layers of the inner crust, the nuclear clusters may adopt shapes different from the spherical one, i.e., \n the so-called ``pasta phases'', in order to minimize the Coulomb energy. \nSince the inner crust is largely dominated by the neutron gas and shell effects are to a certain extent marginal, semiclassical approaches \n are very useful to describe the inner crust of neutron stars including non-spherical shapes. \n Finally, we have the core, which can be separated into the outer core and the inner core.\n It constitutes around the $99\\%$ of the neutron star mass. The outer core is formed of uniform matter composed of neutrons, protons, \n electrons and eventually muons in $\\beta$-equilibrium. The composition of the core is yet to be fully determined. Due to energetic \n reasons, more exotic particles, such as hyperons, which contain strange quarks, may appear. Also, at those densities and pressures, the transition\n to a phase of hadronic and deconfined quark matter could be feasible~\\cite{Lattimer2004}. \n  \nThe study of the EoS is one of the central issues in nuclear physics as well as in astrophysics. The EoS of symmetric nuclear matter has been studied, \nthrough experiments based on nuclear masses and sizes, giant resonances of finite nuclei, heavy-ion collisions, etc., \nfor more than half a century, becoming relatively well-determined. \nOn the other hand, the EoS of asymmetric nuclear matter, which characterizes the \nisospin-dependent part of the EoS, is less known. \nMany facilities have been constructed, or are under construction, around the world with the purpose of constraining the asymmetric nuclear matter properties. \nSome of them are the Radioactive Ion Beam (RIB) Factory at RIKEN in Japan, the FAIR\/GSI in Germany, the SPIRAL2\/GANIL in France, the Facility \nfor Rare Isotope Beams (FRIB), the FRIB\/NSCL, the T-REX\/TAMU and the Jefferson Lab in the USA, the CSR\/Lanzou and BRIF-II\/Beijing in China, the SPES\/LNL in Italy, the RAON in Korea, etc.\nThese facilities aim to extract information of the isovector part of effective interactions, as well as of the EoS of \nasymmetric nuclear matter, studying nuclear matter at high densities through radioactive beam physics, heavy-ion collisions, giant  \nresonances, isobar analog states, parity-violating phenomena, etc. \n\\begin{figure}[!t]\n\\centering\n\\includegraphics[clip=true, width=0.5\\linewidth]{.\/grafics\/intro\/Pulsars}\n\\caption{Masses measured from pulsar timing. Vertical dashed (dotted) lines indicate category error-weighted\n(unweighted) averages. Figure extracted from Ref.~\\cite{Lattimer19}. }\\label{fig:Latimerpulsars}\n\\end{figure}\n\n\nIn the astrophysical domain, it is known that various properties of neutron stars, such as the mass-radius relation, the moment of inertia or the tidal deformability, \nare very sensitive to the properties of nuclear matter at saturation and at supra-nuclear densities. \nAs the present facilities available for laboratory experiments still cannot reach high densities such as the ones found in the interior of\nneutron stars, it is important to study theoretically the EoS inside them, especially in their core, which in the center can attain densities several \ntimes the saturation density. \nA variety of different functionals (and many-body theories) have been used to \ndetermine the properties of neutron stars, including Brueckner--Hartree--Fock interactions~\\cite{Wiringa95,Vidana2009,Ducoin11,Li2016},\nSkyrme forces \\cite{xu09a,ducoin07,Pearson12,Newton2014}, \nfinite-range functionals \\cite{routray16}, relativistic mean-field (RMF) models \n\\cite{horowitz01a,carriere03,Klahn06,Moustakidis10,Fattoyev:2010tb,Cai2012,Newton2014} and\nmomentum-dependent interactions \\cite{xu09a,Moustakidis12, routray16}.  \n \nThe total mass is one of the best well-established observables of neutron stars from \nmany observational studies. Among them, there are the recent accurate observations of highly massive \nneutron stars, corresponding to $(1.97 \\pm 0.04)M_{\\odot}$ and $(2.01 \\pm 0.04)M_{\\odot}$ for the \n\\mbox{PSR J1614-2230} and \\mbox{PSR J0348+0432} pulsars, respectively~\\cite{Demorest10,Antoniadis13}.\nThe mass of \\mbox{PSR J1614-2230} has been revised in Ref.~\\cite{Fonseca_2016} constraining it to $(1.928 \\pm 0.017)M_{\\odot}$.\nA very recent observation~\\cite{Cromartie19} of $(2.14^{+0.10}_{-0.09})M_{\\odot}$ for the pulsar PSR J0740+6620 \n would correspond to the heaviest neutron star detected up to date. \nStill, these are preliminary results with high error bars, and the earlier Shapiro delay mass measurements \nare being revised~\\cite{Zhang_2019}. Hence, in our work, we will restrict ourselves to the observational mass constraints of Refs.~\\cite{Demorest10,Antoniadis13}.\nThese masses revoke many of the proposed theoretical EoSs for neutron stars if the calculated maximum neutron star mass does\nnot reach the observed values.\nAs a result, a great effort has been addressed to \nderive nuclear models able to generate EOSs that predict such massive objects (see~\\cite{Oertel:2016bki,Li2014} \nand references therein). However, a precise mass measurement is not enough to completely constrain \nthe underlying EoS. One would also need a precise measurement of the radius of the neutron star whose mass has \nbeen obtained. The uncertainties in the determination of the neutron star radius are still an open question \nfor observational studies~\\cite{Stein_2014,Nattila16,De2018}. The Neutron Star Interior Composition Explorer \n(NICER) mission is already set up with the aim to provide a measurement of the radius with an accuracy \nof order 5\\%. \n\nThe detection of gravitational waves (GWs) has opened a new window to explore the Universe and, specifically, neutron stars, with the \nhelp of the new generation of gravitational observatories like the Laser Interferometer Gravitational-Wave Observatory (LIGO), \nthe Virgo laboratory from the European Gravitational Observatory or the future European Space Agency mission LISA (Laser Interferometer \nSpace Antenna), planned to be launched around 2034. \nThe LIGO and Virgo collaboration detected GWs from the GW170817 event~\\cite{Abbott2017}, which accounted for the \nfirst time or a merger of two neutron stars. This detection led to a whole new set of constraints in both astrophysics and nuclear physics, as it has \nenhanced the present interest to examine the sensitivity of the EoS at large values of the density and of the isospin asymmetry.\n\nIn this thesis we further extend the studies of the EoS of highly asymmetric nuclear matter, using non-relativistic\nmean-field models, such as zero-range density-dependent Skyrme interactions~\\cite{skyrme56, vautherin72,sly41}, or \nfinite-range forces like Gogny interactions~\\cite{decharge80, berger91}, momentum-dependent interactions (MDI)~\\cite{das03,li08} and simple effective interactions (SEI)~\\cite{behera98, Behera05}.\nThe theoretical calculations obtained using these models are compared to experimental data from finite nuclei and from\nastrophysical observations. \n\nChapter~\\ref{chapter1} collects a brief summary of the mean-field approximation, where one assumes the nuclear system as a set \nof non-interacting quasi-particles that move independently inside an effective mean-field potential. \nWe also collect the basic concepts of the Hartree-Fock method used to find the energy of the system. \nMoreover, we introduce the different phenomenological potentials we are going to use through this thesis, namely the Skyrme, \nGogny, MDI and SEI interactions. All these functionals, especially Skyrme and Gogny forces, reproduce with \ngood quality the properties of finite nuclei. In this work, we will also study nuclear matter at large densities and isospin asymmetries\nwith them.\nThe definition of some properties of the EoS of symmetric nuclear matter and of asymmetric nuclear matter that we will use in the following\nchapters are also given for the different models.\n\n\nWe devote Chapter~\\ref{chapter2} to study the properties of asymmetric nuclear matter\nusing a set of Skyrme and Gogny interactions~\\cite{gonzalez17}. Firstly, we analyze the behaviour of the different symmetry energy \ncoefficients appearing in the Taylor expansion of the energy per particle in terms of even powers of the isospin asymmetry.\nWe expand the EoS up to tenth-order for Skyrme interactions and up to sixth-order for Gogny forces. \nThe behaviour of the second-order coefficient, commonly known as the symmetry energy, divides the Skyrme (and also the Gogny)\ninteractions in two groups. \nThe first group contains the parametrizations that have a symmetry energy that vanishes at some density above saturation, implying \nan isospin instability. The second group is formed by those interactions, usually with a larger slope parameter $L$, that have an increasing \ntrend for the symmetry energy.\nWe also study the symmetry energy understood as the difference between the \nenergy per particle in pure neutron matter and in symmetric nuclear matter, which we call parabolic symmetry energy. \nThis definition also coincides with the infinite sum of all the coefficients of the Taylor expansion of the energy per particle in terms of the asymmetry\nif the isospin asymmetry is equal to one.\nAround saturation, the differences between the parabolic and the second-order symmetry energy  coefficients are reduced when more terms of the expansion \nare considered~\\cite{gonzalez17}. \nMoreover, the corresponding slopes of the symmetry energy are also evaluated, and we see that some discrepancies can arise between them. \nThe interior of neutron stars is composed of matter that is in $\\beta$-equilibrium. We test the convergence of the Taylor expansion of the EoS in \npowers of the isospin asymmetry when studying $\\beta$-stable nuclear matter~\\cite{gonzalez17}. \nThe agreement of the isospin asymmetry and pressure along all densities calculated with the EoS expansion improves if more orders are considered, \nbeing the improvement slower for interactions with larger slope parameter $L$.\nThese differences are relevant when studying neutron star properties, where one uses the EoS of infinite nuclear matter to describe the neutron star core. \nIf the Taylor expansion is performed only in the potential part of the force and using the full expression for the kinetic part, \none almost recovers the same values for the \nisospin asymmetry and for the pressure as if they are calculated using the full expression of the EoS. \n\nThe mass and radius relation of neutron stars has been studied also in Chapter~\\ref{chapter2} considering the EoSs of Skyrme and Gogny models. \nWe find that very soft forces are not able to give stable solutions of the TOV equations and only the stiff enough\nparametrizations can provide $2 M_\\odot$ neutron stars.\nIn particular, remark that none of the Gogny interactions of the D1 family can provide a neutron star inside the observational bounds~\\cite{Sellahewa14, gonzalez17}. \nThe convergence of the EoS is also tested when studying neutron star properties. One finds that, if the Taylor expansion is cut at second order, the \nresults may lay quite far from the ones obtained using the full EoS. This convergence is slower as larger is the slope of the symmetry \nenergy of the interaction.\nThis behaviour points out the necessity of using the full expression of the EoS whenever possible. \n\nAs said previously, the Gogny D1 family does not include any parametrization able to provide a neutron star that reaches $2 M_\\odot$, as they \nhave very soft symmetry energies~\\cite{Sellahewa14, gonzalez17}. \nIn Chapter~\\ref{chapter3} we propose two new Gogny forces, which we name D1M$^{*}$ and D1M$^{**}$, that are able to \nprovide neutron stars inside the observational constraints while still providing the same good description of finite nuclei as D1M~\\cite{gonzalez18, gonzalez18a,Vinas19}.\nThe D1M$^{*}$ interaction is able to provide a neutron star of $2M_\\odot$, while the D1M$^{**}$ is able to describe up to $1.91 M_\\odot$ neutron stars.\nOther stellar properties studied with the D1M$^{*}$ and D1M$^{**}$ are in agreement with the Douchin-Haensel SLy4 EoS~\\cite{douchin01}. \nWe analyze some ground state properties of finite nuclei, such as binding energies, neutron and proton radii, response to \nquadrupole deformation and fission barriers. \nThe two new parametrizations D1M$^{*}$ and D1M$^{**}$ perform as well as D1M in all these studies of finite nuclei~\\cite{gonzalez18, gonzalez18a}. \nWe can say that the D1M$^{*}$ and D1M$^{**}$ are good alternatives to describe simultaneously finite nuclei and neutron stars\nproviding excellent results in harmony with the experimental and observational data. \n\n\nThe correct determination of the transition between the core and the crust in neutron stars is key in the understanding of \nneutron star phenomena, such as pulsar glitches, which depend on the size of the crust~\\cite{Link1999,Fattoyev:2010tb,Chamel2013,PRC90Piekarewicz2014,Newton2015}.\nIn Chapter~\\ref{chapter4} we estimate systematically the core-crust transition searching \nfor the density where the nuclear matter in the core is unstable against fluctuations of the density. \nThe instabilities are determined using two methods. First, we use the so-called thermodynamical method, \nwhere one studies the mechanical and chemical stabilities of the core. We find the corresponding results for Skyrme and Gogny\ninteractions. As stated in previous literature, we find a downward trend when the transition density is plotted against the slope $L$.\nOn the other hand, we do not find strong correlations between the transition pressure and $L$~\\cite{gonzalez17}.\nWe have also studied the convergence of the transition properties when the Taylor expansion of the EoS is used. In general, adding \nmore terms to the expansion brings the transition density closer to the values found with the exact EoS. \nHowever, we still find significant differences even using terms of order higher than two, especially in cases where the \nslope of the symmetry energy is large. \nThe transition density is also obtained by the dynamical method, where one considers bulk, surface, and Coulomb effects when studying \nthe stability of the uniform matter. \nWe perform the calculations for Skyrme interactions, and for different finite-range forces, which, in our case, are the Gogny, MDI and SEI models. \nIn general, the results for the transition density using the dynamical method are lower than the ones obtained with the thermodynamical method.\nThe convergence is better for softer EoSs. \nWe first obtain the results for Skyrme interactions, and we analyze the convergence of the transition properties \nif the Taylor expansion of the EoS is used to calculate them. \nThe convergence of the core-crust transition properties is the same as the one obtained using the thermodynamical method, that is, \nthe results are closer to the exact ones as higher-order terms in the expansion are considered.\nIf the transition density is obtained using the Taylor expansion of the EoS only in the potential part of the interaction\nand the exact kinetic energy, the results obtained \nare almost the exact ones. \n\nFinally, in Chapter~\\ref{chapter4} we obtain the values of the transition properties using the dynamical method with finite-range interactions. \nContrary to the case of Skyrme interactions, we have had to derive the explicit expression of the energy curvature matrix in \nmomentum space for these types of forces~\\cite{gonzalez19}. \nThe contributions to the surface term have been taken from both the interaction part and the kinetic part, making this \nderivation more self-consistent compared to earlier studies. The contributions coming from the direct part are obtained through the \nexpansion of their finite-range form factors in terms of distributions, and the contributions coming from the exchange and \nkinetic parts are found expressing their energies as a sum of a bulk term plus a $\\hbar^2$ correction within the Extended Thomas Fermi approximation. \nWe find that the effects of the finite-range part of the interaction on the curvature matrix arise mostly from the direct part of the energy. \nTherefore, in the application of the dynamical method with finite-range forces, it is an accurate approximation to use only the direct contribution to the energy, \nat least for the forces used in this thesis.\nWe have also analyzed the global behaviour of the core-crust transition density and pressure as a function of the slope of the symmetry energy at saturation\nfor these finite-range interactions. The results for MDI are in agreement with previous literature~\\cite{xu10b}, and for MDI and SEI, the transition density and pressure\nare highly correlated with $L$. However, if the models have different saturation properties, as the set of Gogny interactions we have used in this work, \nthe correlations are deteriorated. \n\n\n\nChapter~\\ref{chapter5} encloses the analysis of different neutron star properties. \nWe first study the influence of the inner crust part of the EoS when analyzing global properties such as the mass and radius~\\cite{gonzalez17, gonzalez19}. \nWe analyze some crustal properties, such as the crustal mass and the crustal radius. We see again the importance of the \ngood determination of the location of the core-crust transition, as the results for the crust are way lower \nif the core-crust transition is estimated within the dynamical approach instead of within the thermodynamical approach. \nThese crustal properties play a crucial role when predicting several observed phenomena, like glitches, r-mode oscillations, etc. \nHence, a good estimation of these properties is key in the understanding of neutron stars. \n\nAs said previously in this Introduction, the detection of GWs opened a new window to look at the Universe. \nThe GW170817 signal detected by the LIGO and Virgo collaboration coming from a merger of two neutron stars has established a new set of constraints in \nastrophysics and in nuclear physics~\\cite{Abbott2017, Abbott2018, Abbott2019}.\nOne constraint directly measured form the signal is on the dimensionless mass-weighted tidal deformability, $\\tilde{\\Lambda}$, at a certain chirp mass of the binary system. \nAfter a data analysis, constraints on other properties, such as on the dimensionless tidal deformability of a canonical neutron star ($\\Lambda_{1.4}$), on masses, on radii, etc. \nwere provided~\\cite{Abbott2017, Abbott2018, Abbott2019}. \nWe have analyzed the values of $\\tilde{\\Lambda}$ and $\\Lambda_{1.4}$, and we see that very stiff EoSs are not able to predict values \ninside the observational bounds. With a set of mean-field interactions, we are also able to roughly estimate a radius of a neutron star of $1.4M_\\odot$\nin consonance with the values given by the LIGO and Virgo collaboration. \nFinally, the moment of inertia is also analyzed, finding that, again, very stiff EoSs are not able to provide moments of inertia\ninside the constraints predicted by Landry and Kumar for the binary double pulsar PSR J0737-3039~\\cite{Landry18}. \nThe newly D1M$^{*}$ and D1M$^{**}$ interactions provide very good results for both the tidal deformability and the moment of inertia, confirming their\ngood performance in the astrophysical domain. \nWe have analyzed the fraction of the moment of inertia enclosed in the crust, using the core-crust transition density obtained either \nwith the thermodynamical and the dynamical methods. As happened with the crustal thickness and crustal mass, the crustal fraction of the \nmoment of inertia is lower if the transition is obtained with the dynamical approach~\\cite{gonzalez19}. \n\nFinally, the conclusions of this work are given in Chapter~\\ref{conclusions}. \nThree Appendices are added at the end of the thesis. Appendix~\\ref{appendix_thermal} collects the explicit \nexpressions of the derivatives needed to obtain the core-crust transition using either the thermodynamical or the \ndynamical methods. Appendix~\\ref{app_taules} contains the numerical results of the core-crust transition properties \nfor Skyrme and finite-range interactions computed using both the thermodynamical and the dynamical methods. \nWe provide in Appendix~\\ref{app_vdyn} technical details about the Extended Thomas Fermi \napproximation, which is used to derive the theory of the dynamical method for finite-range interactions. \n\n\\chapter{Non-Relativistic Mean Field Models}\\label{chapter1}\n\\section{Mean field approximation within the Hartree-Fock framework}\nThe system of a nucleus composed of $A$ nucleons (neutrons and protons) can be described by a many-body Hamiltonian $H$, which\nconsists of a kinetic part plus a potential part. \nThere are different approaches to compute the nuclear structure~\\cite{Bender03}. \nThe ab-initio calculation of the properties of a nuclear system starts with a nucleon-nucleon potential, \nwhich describes the nucleon-nucleon scattering data~\\cite{Machleidt_2001}. It is characterized for having a highly\nrepulsive core, and for reproducing the basic features of nuclear saturation. However, if only the nucleon-nucleon\npotential is employed to reproduce the properties of the system, the ab-initio approach fails to determine \nquantitatively the saturation point, and additional three-body forces have to be considered. \nThe study of a nuclear many-body system with ab-initio methods requires highly-developed many-body theories\nlike, e.g., the Brueckner-Hartree-Fock~\\cite{Serot:1984ey, Brockmann90, Dickhoff_1992}, \ncorrelated basis functions~\\cite{pandh81, Heiselberg00} or self-consistent Green's functions~\\cite{MarioArtur2, Artur2,Artur1, \nMarioArtur4, MarioArtur3, MarioArtur1}. Thus, it is a highly complicated and challenging\nendeavour. \n\nIn the other extreme of the existing approaches for describing nuclear systems, one finds \nthe macroscopic nuclear liquid-drop model~\\cite{myers82}. \nIn this case, the energy of the system is phenomenologically parametrized in terms of global properties of nuclei, \nsuch as volume energy, asymmetry energy, surface energy, etc. \nUsually, shell correction energies that approximate quantal effects are added to these macroscopic models, \ngiving rise to the so-called microscopic-macroscopic (mic-mac) models. \n\nIn between these two approaches one finds, on the one hand, the shell model, in which one considers a phenomenological\nsingle-particle potential and performs a configuration mixing calculation involving all many-body\nstates that can be constructed using a band of the possible single-nucleon states around the Fermi energy~\\cite{brown88}.\nOn the other hand, between the ab-initio and macroscopic models, there is the mean-field approximation,\nwhich we will use in this thesis.\nA way to circumvent the determination of the full potential describing the whole system is to assume the nucleus as a set of \nquasiparticles that do no interact between them and where each nucleon moves independently within an \neffective mean-field (MF) created by the nucleons themselves. \nWe will restrict ourselves to an effective two-body Hamiltonian, whose potential part $V$ can be decomposed \nas the sum of a single-particle potential for each nucleon $i$ plus a \nresidual potential,\n\\begin{equation}\\label{eq:pot}\n V = \\sum_i v ({\\bf r}_i) + V^\\mathrm{res}=V^\\mathrm{MF} + V^\\mathrm{res}.\n\\end{equation}\nThe sum of all $ v ({\\bf r}_i)$ will be denoted as the mean-field potential $V^\\mathrm{MF}$ of the whole system.\nThe contribution of the residual potential $V^\\mathrm{res}$ is supposed to be much weaker than the  \ncontribution of the original potential $V$, \nand usually is treated in perturbation theory.\nTaking Eq.~(\\ref{eq:pot}) into account, we can rewrite the Hamiltonian as\n\\begin{equation}\n H= T+V^\\mathrm{MF} + V^\\mathrm{res}= H^\\mathrm{MF}+ V^\\mathrm{res},\n\\end{equation}\nwhere the sum of the kinetic term $T$ and the mean-field potential $V^\\mathrm{MF}$ is denoted as the mean-field Hamiltonian $H^\\mathrm{MF}$:\n\\begin{equation}\\label{HH}\n H^\\mathrm{MF} = T+ V^\\mathrm{MF} = \\sum_{i} t({\\bf r}_i) + \\sum_i  v({\\bf r}_i) = \\sum_i h({\\bf r}_i),\n\\end{equation}\nwhere ${\\bf r}_i$ is the coordinate of the $i$-th nucleon, with $i,j=1\\cdots A$ and $h({\\bf r}_i)$ are the single-particle Hamiltonians. \n\nIf we write the Hamiltonian $H$ in second quantization, the Hamiltonian operator becomes \\cite{rin80}\n\\begin{equation}\\label{eq:Ham}\nH = \\sum_{ij} t_{ij} \\hat{c}^\\dagger_i \\hat{c}_j + \\frac{1}{4} \\sum_{ijkl} \\bar{v}_{ijkl} \\hat{c}^\\dagger_i \\hat{c}^\\dagger_j \\hat{c}_l \\hat{c}_k,\n\\end{equation}\nwhere $\\hat{c}^\\dagger_i$ and $\\hat{c}_i$ are the single-particle creation and annihilation operators in a single-particle state $i$, \nand \n\\begin{equation}\n \\bar{v}_{ijkl} = v_{ijkl}- v_{ijlk}\n\\end{equation}\nis the antisymmetrized two-body interaction matrix elements.\nThe indexes ${i,j,k,l}$ run over a complete set of states.\n\nThe Schr\\\"{o}dinger equation associated to the Hamiltonian can be written as\n\\begin{equation}\\label{eq:sch}\n H |\\phi \\rangle = E  |\\phi \\rangle,\n\\end{equation}\nwhere $ |\\phi \\rangle$ is the total wave-function of the system and $E$ the corresponding energy. \nThe solution of Eq.~(\\ref{eq:sch}) will be found using the variational principle, which states that the \nexact  Schr\\\"{o}dinger equation is equivalent to the variational equation~\\cite{rin80}\n\\begin{equation}\\label{eq:var1}\n \\delta E[\\phi] = 0,\n\\end{equation}\nwhere \n\\begin{equation}\\label{eq:var2}\n E[\\phi] = \\frac{ \\langle \\phi | H | \\phi \\rangle}{ \\langle \\phi| \\phi \\rangle}\n\\end{equation}\nis the expectation value of the energy.\nThe variation equation~(\\ref{eq:var1}) can be expanded as\n\\begin{equation}\\label{eq:var3}\n \\delta E[\\phi] = \\langle \\delta \\phi | H - E |\\phi \\rangle + \\langle \\phi | H - E | \\delta \\phi \\rangle = 0,\n\\end{equation}\nwhere $E$ is a Lagrange multiplier which can be understood as the energy corresponding to $|\\phi \\rangle$.\nThe wave-function $|\\phi \\rangle$ can be a complex function, and therefore the variation has to be performed in the \nreal and imaginary parts independently, which is equivalent to carry out the variation over $|\\delta \\phi \\rangle$\nand $ \\langle \\delta \\phi|$ independently. \nThis yields Eq.~(\\ref{eq:var3}) to be reduced to\n\\begin{equation}\n \\langle \\delta \\phi | H- E |\\phi \\rangle =0\n\\end{equation}\nand its respective complex conjugate equation. \nThe determination of the ground state, with corresponding energy $E_0$, and for a trial $|\\phi \\rangle$ satisfies\n\\begin{equation}\n E [\\phi] \\geq E_0.\n\\end{equation}\nTherefore, the goal of the variational principle will be to find a wave-function $|\\phi \\rangle$ that minimizes the value of $E[\\phi]$.\n\nUp to now, it has been assumed that the Hamiltonian $H$ does not depend on the wave function $|\\phi \\rangle$.\nHowever, many effective interactions do depend on the density and therefore on $|\\phi \\rangle$. Hence, the problem has to be \nsolved in a self-consistent way until the solution converges. \n\nIn the Hartree-Fock (HF) approach, one solves self-consistently the variational problem considering the many-body wave-functions \nof the type of Slater determinants\n\\begin{equation}\\label{Slater}\n |\\phi^\\mathrm{HF} \\rangle = | \\phi (1 \\cdots A) \\rangle= \\prod_{i=1}^A \\hat{c}_i^\\dagger |0 \\rangle,\n\\end{equation}\nwhere $c_k^\\dagger$ and $c_k$ correspond to the single-particle wave-functions $\\phi_k$ which, at the\nsame time are eigenfunctions of the single-particle Hamiltonian $h$, i.e., \n\\begin{equation}\\label{hMF}\n h(\\mathbf{r}, \\sigma, \\tau) \\phi_k (\\mathbf{r}, \\sigma, \\tau) = \\varepsilon_k \\phi_k(\\mathbf{r}, \\sigma, \\tau).\n\\end{equation}\nEq.~(\\ref{hMF}) is determined with the variational condition (\\ref{eq:var1}), where $\\phi_k(\\mathbf{r}, \\sigma, \\tau)$ are the\nrespective eigenstates.\nThe $|\\phi^\\mathrm{HF} \\rangle$ wave-function will describe the fermions in the nuclear system and therefore it has to be antisymmetrized.\n\nThe Hartree-Fock energy will be given by \\cite{rin80}\n\\begin{equation}\n E^\\mathrm{HF} =  \\langle \\phi^\\mathrm{HF} | H | \\phi^\\mathrm{HF} \\rangle , \n\\end{equation}\nwhere the Hamiltonian $H$ is given in Eq.~(\\ref{eq:Ham}). Using Wick's theorem, \nthe energy can be calculated as a functional of the single-particle density \n$\\rho_{ij} =  \\langle \\phi^\\mathrm{HF} |  c_j^\\dagger c_i | \\phi^\\mathrm{HF} \\rangle$: \n\\begin{eqnarray}\\label{EHF}\n E^\\mathrm{HF} &=& \\sum_{ij} t_{ij}  \\langle \\phi^\\mathrm{HF} | c_i^\\dagger c_j | \\phi^\\mathrm{HF} \\rangle + \n \\frac{1}{4} \\sum_{ijkl} \\bar{v}_{ijkl} \\langle \\phi^\\mathrm{HF} | c_i^\\dagger c_j^\\dagger c_l c_k |\\phi^\\mathrm{HF} \\rangle \\nonumber\\\\\n &=& \\sum_{ij} t_{ij} \\rho_{ji} + \\frac{1}{2} \\sum_{ijkl} \\rho_{ki} \\bar{v}_{ijkl} \\rho_{lj}.\n \\end{eqnarray}\nIf the HF energy in Eq.~(\\ref{EHF}) is solved using the Slater wave-functions (\\ref{Slater}), it can be written \nin coordinate space as \n\\begin{eqnarray}\n E^\\mathrm{HF} &=& - \\sum_{i} \\frac{\\hbar^2}{2m} \\int \\phi^*_ i (\\mathbf{r}) \\nabla^2 \\phi_i (\\mathbf{r}) d \\mathbf{r}\\nonumber\\\\\n &+& \\frac{1}{2} \\sum_{ij} \\int \\phi_i^* (\\mathbf{r}) \\phi_j^* (\\mathbf{r}') V(\\mathbf{r}, \\mathbf{r}') \\phi_i (\\mathbf{r}) \\phi_j (\\mathbf{r}')\n d \\mathbf{r} d \\mathbf{r}'\\nonumber\\\\\n &-& \\frac{1}{2} \\sum_{ij} \\int \\phi_i^* (\\mathbf{r}) \\phi_j^* (\\mathbf{r}') V(\\mathbf{r}, \\mathbf{r}') \\phi_j (\\mathbf{r}) \\phi_i (\\mathbf{r}')\n d \\mathbf{r} d \\mathbf{r}',\n\\end{eqnarray}\nwhere $m$ is the nucleon mass and $V(\\mathbf{r}, \\mathbf{r}') $ is the two-body interaction between two nucleons at $\\mathbf{r}$ and $\\mathbf{r}'$.\n\nFrom Eqs.~(\\ref{HH}) and (\\ref{hMF}), the Hartree-Fock equations in coordinate space for the single-particle wave functions\nwill be given by\n\\begin{equation}\n \\frac{-\\hbar^2}{2m} \\nabla^2 \\phi_k({\\bf r}) + V_H ({\\bf r}) \\phi_k ({\\bf r})+\n \\int d {\\bf r}' V_{F} ({\\bf r}, {\\bf r}')\\phi_k ({\\bf r}') = \\varepsilon_k \\phi_{k}({\\bf r}).\n\\end{equation}\nThe Hartree term is the local part of the potential, depends on the one-body density\n\\begin{equation}\n \\rho ({\\bf r}) = \\sum_k \\phi_k^* ({\\bf r}) \\phi_k ({\\bf r}) \n\\end{equation}\nand is defined as\n\\begin{equation}\n V_H({\\bf r})=\\int d {\\bf r}' v({\\bf r},{\\bf r}') \\rho({\\bf r}').\n\\end{equation}\nOn the other hand, the Fock potential gives the non-locality of this type of systems. It depends \non the non-local one-body density matrix\n\\begin{equation}\n \\rho ({\\bf r},{\\bf r}') = \\sum_k \\phi_k^* ({\\bf r}') \\phi_k ({\\bf r}) \n\\end{equation} \nand is given by\n \\begin{equation}\n  V_{F}  ({\\bf r},{\\bf r}')= -v({\\bf r},{\\bf r}') \\rho({\\bf r},{\\bf r}') .\n \\end{equation}\n\n The potential term $ V({\\bf r}, {\\bf r}')$ includes all possible nucleon-nucleon forces, as well Coulomb interactions.\n In our case, the Coulomb interaction in the system will be represented by\n\\begin{equation}\n V_\\mathrm{Coul} ({\\bf r})= \\frac{e^2}{2} \\int \\frac{\\rho_p ({\\bf r}') d^3 r'}{|{\\bf r}-{\\bf r}'| } - \\frac{e^2}{2}\n \\left( \\frac{3}{\\pi}\\right)^{1\/3} \\rho_p^{1\/3} ({\\bf r}'),\n\\end{equation}\nwhere the Slater approximation has been used in the exchange part \\cite{sly42}.\n\n\\section{Infinite matter properties with phenomenological potentials}\nIn the MF approach, the interaction between nucleons is characterized by a phenomenological \npotential $V(\\mathbf{r}, \\mathbf{r}') $ which depends on several free parameters which will be fitted\nto reproduce the experimental data of some nuclear properties. \nTypically, these properties are the observables related to nuclear masses, radii, binding energies, \nshell structure properties, etc., or to infinite nuclear matter properties, such as the saturation energy, the \nnuclear matter incompressibility, etc.\nThese phenomenological potentials can be of zero-range type, such as \nthe Skyrme interactions \\cite{skyrme56, vautherin72,sly41}, or may include finite-range terms, such as the \nGogny interactions~\\cite{decharge80, berger91}, the MDI forces~\\cite{das03,li08} and the SEI functionals~\\cite{behera98, Behera05},\nwhich we will use in the present thesis.\n\n In the fitting procedure of phenomenological potentials, it is very usual to consider the saturation density $\\rho_0$ in symmetric nuclear matter (SNM)\n and the energy per particle $E_b$ in SNM at $\\rho_0$,  which have values of $\\rho_0 \\simeq 0.16$ fm$^{-3}$ and $E_b (\\rho_0)\\simeq-16$ MeV, respectively.\nThe saturation density $\\rho_0$ is given by the minimum of the energy per particle in SNM, \n\\begin{equation}\n \\left.\\frac{\\partial E_b (\\rho)}{\\partial \\rho}\\right|_{\\rho_0}=0 .\n\\end{equation}\nMoreover, the pressure and the incompressibility are given, respectively, by \n\\begin{equation}\\label{eq:press0}\n  P (\\rho)= \n\\rho^2 \\frac{\\partial E_b (\\rho)}{\\partial \\rho}\n\\end{equation}\nand \n\\begin{equation}\\label{eq:K00}\n K (\\rho)= 9\\rho^2 \\frac{\\partial^2 E_b (\\rho)}{\\partial \\rho^2}.\n\\end{equation}\nAt the saturation density $\\rho_0$ there is a cancellation of the pressure, i.e., \n\\begin{equation}\\label{eq:press}\n  \\hspace{0.5cm}P_0 (\\rho_0)= \n \\left.\\rho_0^2 \\frac{\\partial E_b (\\rho)}{\\partial \\rho}\\right|_{\\rho_0}=0,\n\\end{equation}\nand the nuclear matter incompressibility at the saturation point, \n\\begin{equation}\\label{eq:K0}\n K_0 (\\rho_0)= \\left.9\\rho_0^2 \\frac{\\partial^2 E_b (\\rho)}{\\partial \\rho^2}\\right|_{\\rho_0},\n\\end{equation}\nis usually considered as a constraint over the equation of state, as well as the effective mass of the system,\n\\begin{equation}\\label{eq:effmass}\n \\frac{m^*}{m}= \\left[1+ \\frac{m}{\\hbar k} \\frac{\\partial V_{F} (k)}{\\partial k} \\right]^{-1}_{k_F},\n\\end{equation}\nwhere $V_F (k)$ is the Fock potential in momentum space and $k_F = (3 \\pi^2 \\rho\/2)^{1\/3}$ is the Fermi momentum of the system.\n\nOn the other hand, in asymmetric nuclear matter, where the neutron and proton densities take different values\n$\\rho_n \\neq \\rho_p$, the energy per particle will be a function of them, $E_b (\\rho_n, \\rho_p)$.\nAlso, it can be rewritten as a function of the total density \n\\begin{equation}\n \\rho= \\rho_n+\\rho_p\n\\end{equation}\nand of the isospin asymmetry\n\\begin{equation}\n \\delta= (\\rho_n-\\rho_p)\/\\rho,\n\\end{equation}\n i.e., $E_b (\\rho, \\delta)$.\nThis way, the energy density is given by \n\\begin{equation}\n \\mathcal{H}_b ( \\rho, \\delta) = \\rho E_b( \\rho, \\delta), \n\\end{equation}\nand the neutron and proton chemical potentials are defined \nas the derivative of the baryon\nenergy density \n with respect to the neutron and proton densities:\n \\begin{equation}\\label{chempot}\n  \\mu_n = \\frac{\\partial \\mathcal{H}_b}{\\partial \\rho_n} \\hspace{2cm} \n  \\mu_p = \\frac{\\partial \\mathcal{H}_b}{\\partial \\rho_p}.\n \\end{equation}\n Finally, the pressure of the system can be defined either as a function of the derivative of the energy per particle or as a function of the chemical potentials, i.e.,  \n\\begin{equation}\\label{eq:pre}\n P (\\rho, \\delta) = \\rho^2 \\frac{\\partial E_b (\\rho, \\delta)}{\\partial \\rho} = \\mu_n \\rho_n + \\mu_p \\rho_p - \\mathcal{H}_b (\\rho, \\delta).\n\\end{equation} \n\nIf one expands the EoS around isospin asymmetry $\\delta=0$, the energy per particle of a nuclear system can be rewritten as \n\\begin{equation}\\label{eq:eosexp}\n E_b(\\rho, \\delta) = E_b(\\rho, \\delta=0) + E_{\\mathrm{sym}}(\\rho) \\delta^2 +\\mathcal{O}(\\delta^{4}),\n\\end{equation}\nwhere the lowest term $E_b(\\rho, \\delta=0)$ is the energy of the system in SNM and \n$E_{\\mathrm{sym}}(\\rho)$ is the symmetry energy of the system, which reads\n\n\\begin{equation}\\label{eq:esym}\n E_\\mathrm{sym} (\\rho) = \\left.\\frac{1}{2} \\frac{\\partial^2 E_b (\\rho, \\delta)}{\\partial \\delta^2} \\right|_{\\delta=0}.\n\\end{equation}\nNotice that, due to the charge symmetry, assumed in the nuclear interactions, \nonly even powers of $\\delta$ can appear in the expansion of the symmetry energy~(\\ref{eq:eosexp}).\nIf we expand the symmetry energy around the \nsaturation density $\\rho_0$ one obtains the expression\n\\begin{equation}\\label{esymexp}\nE_{\\mathrm{sym}} (\\rho)= E_{\\mathrm{sym}} (\\rho_0) + L \\epsilon + K_\\mathrm{sym} \\epsilon^2+\\mathcal{O}(\\epsilon^3),\n\\end{equation}\nwhere the density displacement from the saturation density $\\rho_0$ is given by\n\\begin{equation}\n \\epsilon = (\\rho - \\rho_ 0)\/3\\rho_0\n\\end{equation}\nand $L$ is the slope of the symmetry energy at saturation, defined as \n\\begin{eqnarray}\\label{eq:L}\nL\\equiv  L (\\rho_0)&=& 3\\rho_0 \\left.\\frac{\\partial E_{\\mathrm{sym}} (\\rho)}{\\partial \\rho} \\right|_{\\rho_0} \n\\end{eqnarray}\nand which gives information about the stiffness of the equation of state.\nIn Eq.~(\\ref{esymexp}), the coefficient $K_\\mathrm{sym}$ is the symmetry energy curvature, defined as\n\\begin{eqnarray}\nK_\\mathrm{sym}\\equiv K_\\mathrm{sym} (\\rho_0) &=& 9 \\rho_0^2 \\left.\\frac{\\partial^2 E_{\\mathrm{sym}} (\\rho) }{\\partial \\rho^2} \\right|_{\\rho_0}.\n\\end{eqnarray}\n\n\nIn the following sections, we will introduce the phenomenological interactions we have used through all this work.\nFirst, in Section \\ref{Skyrme}, we will introduce the Skyrme zero-range forces, in Section \\ref{Gogny} we introduce the Gogny finite-range \ninteractions and finally, in Section \\ref{MDISEI} we will introduce the finite-range momentum-dependent interactions\n(MDI) and simple effective interactions (SEI).\n\n\n \\section{Skyrme interactions}\\label{Skyrme}\nSkyrme interactions were first proposed considering that the functional of the energy could be expressed \nas a minimal expansion in momentum space compatible with the underlying symmetries\nin terms of a zero-range expansion~\\cite{skyrme56, Skyrme58}.\nThe standard Skyrme two-body effective nuclear interaction in coordinate space reads as \\cite{Skyrme58, skyrme56, vautherin72,sly41,sly42}\n \\begin{eqnarray}\\label{VSkyrme}\n  V (\\mathbf{r}_1 , \\mathbf{r}_2) &=& t_0 (1+x_0 P_\\sigma) \\delta (\\mathbf{r})\\nonumber\n   \\\\\n &+& \\frac{1}{2} t_1 (1 + x_1 P_\\sigma) \\left[ \\mathbf{k'}^2 \\delta (\\mathbf{r}) + \n\\delta (\\mathbf{r}) \\mathbf{k}^2 \\right]\\nonumber\n \\\\ \n  &+& t_2 (1+ x_2 P_\\sigma) \\mathbf{k'} \\cdot \\delta (\\mathbf{r}) \\mathbf{k} \\nonumber\n \\\\\n  &+&\\frac{1}{6} t_3 (1+ x_3 P_\\sigma) \\rho^\\alpha (\\mathbf{R}) \\delta (\\mathbf{r}) \\nonumber\n \\\\\n  &+& i W_0 (\\bm{\\sigma}_1 + \\bm{\\sigma}_2) [\\mathbf{k'} \\times \\delta (\\mathbf{r}) \\mathbf{k}], \n\\end{eqnarray}\nwhere $\\mathbf{r} =\\mathbf{r}_1 - \\mathbf{r}_2 $ is the relative distance between two nucleons and\n$\\mathbf{R}= (\\mathbf{r}_1 + \\mathbf{r}_2)\/2$ is their  center of mass coordinate. \nThe two-body spin-exchange operator is defined as \n$P_\\sigma = (1+ \\bm{\\sigma}_1 \\cdot\\bm{\\sigma}_2)\/2$, \n$\\mathbf{k}= (\\overrightarrow{\\nabla}_1-\\overrightarrow{\\nabla}_2)\/2i$ is the relative momentum between \ntwo nucleons, and $\\mathbf{k}'$ is its complex conjugate,\n$\\mathbf{k}'= -(\\overleftarrow{\\nabla}_1-\\overleftarrow{\\nabla}_2)\/2i$. \n The first term in Eq.~(\\ref{VSkyrme}) is the central term and the second and third ones are \nthe non-local contributions, which simulate the finite range. \nThe three-body force is also assumed as a zero-range force, which provides a simple \nphenomenological representation of many-body effects, and describes how\nthe interaction between two nucleons is influenced by the presence of others~\\cite{vautherin72}.\nFinally, the last term in Eq.~(\\ref{VSkyrme}) is the spin-orbit \ncontribution, which depends on the gradients of the density, and does not contribute in the case of homogeneous systems. \n\nIn an infinite symmetric nuclear system the energy per baryon is given by~\\cite{sly41}\n\\begin{equation}\\label{eq:ebsnm}\n E_b (\\rho) = \\frac{3 \\hbar^2}{10m} \\left( \\frac{3 \\pi^2}{2}\\right)^{2\/3} \\rho^{2\/3} + \n \\frac{3}{8} t_0 \\rho + \\frac{3}{80} \\left[3 t_1 + (5+4 x_2)t_2 \\right] \\left( \\frac{3 \\pi^2}{2}\\right)^{2\/3}\n \\rho^{5\/3} + \\frac{1}{16} t_3 \\rho^{\\alpha+1}.\n\\end{equation}\n Using the definition in Eq.~(\\ref{eq:pre}) the pressure in SNM reads\n \\begin{eqnarray}\n  P(\\rho)&=& \n  \\frac{\\hbar^2}{5m} \\left(\\frac{3 \\pi^2}{2} \\right)^{2\/3} \\rho^{5\/3} +\\frac{3}{8} t_0 \\rho^2 +  \\frac{1}{16} t_3 (\\alpha + 1) \\rho^{\\alpha+2}\\nonumber\\\\\n  &+& \\frac{1}{16}  \n  \\left[3 t_1 + (5+4 x_2)t_2 \\right] \\left(\\frac{3 \\pi^2}{2} \\right)^{2\/3} \\rho^{8\/3} ,\n \\end{eqnarray}\nand the incompressibility for a Skyrme interaction is given by \n\\begin{eqnarray}\n K(\\rho)&=& \n -\\frac{3\\hbar^2}{5m} \\left(\\frac{3 \\pi^2}{2} \\right)^{2\/3} \\rho^{2\/3}+ \\frac{9}{16}\\alpha (\\alpha+1) t_3 \\rho^{\\alpha+1} \n \\nonumber\\\\\n &+& \\frac{3}{8} \\left[3 t_1 + (5+4 x_2)t_2 \\right]\n \\left( \\frac{3 \\pi^2}{2}\\right)^{2\/3} \\rho^{5\/3} .\n\\end{eqnarray}\n\nIn asymmetric nuclear matter of density $\\rho$ and isospin asymmetry $\\delta$, the energy per particle for\na Skyrme interaction becomes\n\\begin{eqnarray}\\label{eq:Ebanm}\nE_b( \\rho, \\delta) &=& \\frac{ 3 \\hbar^2}{10m} \\left(\\frac{3 \\pi^2}{2}\\right)^{2\/3} \\rho^{2\/3} \nF_{5\/3} + \\frac{1}{8} t_0 \\rho \\left[ 2 (x_0 +2) - (2 x_0 +1) F_2 \\right] \\nonumber\\\\\n&+& \\frac{1}{48} t_3 \\rho^{\\alpha + 1} \\left[ 2 (x_3 +2) \n- (2 x_3 +1) F_2  \\right] + \\frac{3}{40} \\left(\\frac{3 \\pi^2}{2} \\right)^{2\/3} \n\\rho^{5\/3} \\\\\n&\\times& \\left[ \\vphantom{\\frac{1}{2}} \\left[ t_1 (x_1 + 2) + t_2 (x_2 + 2) \\right] F_{5\/3} \n+ \\frac{1}{2} \\left[ t_2 (2 x_2 + 1 ) - t_1 (2 x_1 + 1) \\right] F_{8\/3}  \\right], \\nonumber\n\\end{eqnarray}\nwhere the function $F_m$ is defined as\n\\begin{equation}\nF_m = \\frac{1}{2} \\left[ (1 + \\delta)^m + (1- \\delta)^m \\right].\n\\end{equation}\nFor Skyrme interactions of the type (\\ref{VSkyrme}), the neutron ($n$) and proton ($p$) chemical\npotentials, defined as the derivative of the energy density $\\mathcal{H}_b$ with respect to \nthe density of each kind of nucleons take form of\n\\begin{eqnarray}\\label{eq:chempotskyrme}\n \\mu_\\tau (\\rho_\\tau, \\rho_{\\tau'})&=& \\frac{\\hbar^2}{2m}  \n (3 \\pi^2)^{2\/3} \\rho_\\tau^{2\/3} + \\frac{1}{2} t_0 \\rho_\\tau \\left[ 2(x_0+2) - (2x_0+1) \\right]\n \\nonumber\\\\\n &+& \\frac{1}{24} t_3 \\left[ 2 (x_3+2) - (2 x_3+1) \\right] \\left[ \\alpha \\left( \\rho_\\tau^2+ \\rho_{\\tau'}^2 \\right)\n \\rho^{\\alpha-1} + 2 \\rho^\\alpha \\rho_\\tau\\right]\\nonumber\\\\\n &+& \\frac{3}{40} (3 \\pi^2)^{2\/3} \\left\\{  \\left[t_1 (x_1+2)+t_2 (x_2+1) \\right] \\left[\n  \\rho_\\tau^{5\/3}+ \\rho_{\\tau'}^{5\/3} + \\frac{5}{3} \\rho \\rho_\\tau^{2\/3}\\right] \\right. \\nonumber\\\\\n &+& \\left. \\frac{8}{3} \\left[  t_2 (2 x_2 + 1) - t_1 (2 x_1+1)\\right] \\rho_\\tau^{5\/3}  \\right\\},\n\\end{eqnarray}\nbeing $\\tau, \\tau' = n,p$, $\\tau \\neq \\tau'$ the isospin indexes.\nMoreover, the pressure in asymmetric nuclear matter, obtained through the derivative of $E_b (\\rho, \\delta)$ in \nEq.~(\\ref{eq:Ebanm}) with respect to the density is defined as \n\\begin{eqnarray}\\label{eq:press_skyrme}\n P(\\rho, \\delta)&=& \\frac{\\hbar^2}{5m} \n \\left(\\frac{3 \\pi^2}{2}\\right)^{2\/3} \\rho^{5\/3} \nF_{5\/3} +  \\frac{1}{8} t_0 \\rho^2 \\left[ 2 (x_0 +2) - (2 x_0 +1) F_2 \\right]\\nonumber\\\\\n&+&\\frac{1}{48} (\\alpha+1) t_3 \\rho^{\\alpha + 2} \\left[ 2 (x_3 +2) \n- (2 x_3 +1) F_2  \\right] + \\frac{1}{8} \\left(\\frac{3 \\pi^2}{2} \\right)^{2\/3} \n\\rho^{8\/3} \\\\\n&\\times& \\left[ \\vphantom{\\frac{1}{2}} \\left[ t_1 (x_1 + 2) + t_2 (x_2 + 2) \\right] F_{5\/3} \n+ \\frac{1}{2} \\left[ t_2 (2 x_2 + 1 ) - t_1 (2 x_1 + 1) \\right] F_{8\/3}  \\right]. \\nonumber\n\\end{eqnarray}\n\nIf one expands the EoS in Eq.~(\\ref{eq:Ebanm}) around isospin asymmetry $\\delta=0$ [see Eq.~(\\ref{eq:eosexp})],  the \nsymmetry energy for the case of Skyrme forces is found to be of the form~\\cite{sly41}\n\\begin{eqnarray}\\label{eq:esym2skyrme}\n E_{\\mathrm{sym}} (\\rho)&=&\n  \\frac{\\hbar^2}{6m} \\left(  \\frac{3 \\pi^2}{2}\\right)^{2\/3} \\rho^{2\/3} - \\frac{1}{8} t_0 \\rho (2x_0 +1) \n  \\\\\n&-& \\frac{1}{48} t_3 \\rho^{\\alpha+1} (2x_3 +1) \n+ \\frac{1}{24} \\left(  \\frac{3 \\pi^2}{2}\\right)^{2\/3}   \n \\times \\rho^{5\/3} \\left[ -3 x_1 t_1 + t_2 (5 x_2 + 4) \\right],  \\nonumber\n\\end{eqnarray}\nwith a slope parameter $L$\n\\begin{eqnarray}\n L &=& \\frac{\\hbar^2}{3m}\n \\left(  \\frac{3 \\pi^2}{2}\\right)^{2\/3} \\rho_0^{2\/3} -\\frac{3}{8} t_0 \\rho_0 (2 x_0+1) \n-\\frac{1}{16}(\\alpha+1) t_3 \\rho_0^{\\alpha+1} (2 x_3+1) \n\\\\\n&+& \\frac{5}{24} \n\\left(  \\frac{3 \\pi^2}{2}\\right)^{2\/3} \\rho_0^{5\/3} \\left[ -3 x_1 t_1 + t_2 (5 x_2+4)\\right]\\nonumber\n\\end{eqnarray}\nand symmetry energy curvature \n\\begin{eqnarray}\n K_\\mathrm{sym} &=& \n -\\frac{\\hbar^2}{3m}\n \\left(  \\frac{3 \\pi^2}{2}\\right)^{2\/3} \\rho_0^{2\/3} -\\frac{3}{16}\\alpha(\\alpha+1) t_3 \\rho_0^{\\alpha+1} (2 x_3+1)\n\\\\\n &+& \\frac{5}{12} \n\\left(  \\frac{3 \\pi^2}{2}\\right)^{2\/3} \\rho_0^{5\/3} \\left[ -3 x_1 t_1 + t_2 (5 x_2+4)\\right].\\nonumber\n\\end{eqnarray}\n\nFinally, if we consider pure neutron matter with isospin asymmetry $\\delta=1$, the energy per particle for \nSkyrme interactions becomes\n\\begin{eqnarray}\n  E_b(\\rho, \\delta=1)&=&  \\frac{3\\hbar^2}{10m} \\left( 3 \\pi^2\\right)^{2\/3} \\rho^{2\/3} + \\frac{1}{4}\\rho t_0 (1-x_0)\n  +\\frac{1}{24} \\rho^{\\alpha+1} t_3 (1-x_3) \\nonumber\\\\\n  &+& \\frac{3}{40} \\left(3 \\pi^2 \\right)^{2\/3} \\rho^{5\/3} \\left[ t_1 (1-x_1) + 3 t_2 (1+x_2) \\right].\n\\end{eqnarray}\n\nIn the following chapters, we will use several Skyrme parametrizations, that have been fitted to different\ninfinite nuclear matter properties and to properties of finite nuclei.\nThey are collected in Table~\\ref{table:Skyrmeprops}, along with some of their properties of symmetric and asymmetric matter, \nand the references where \ntheir fittings are explained. We can see that all of the considered parametrizations have saturation densities around $\\rho_0\\simeq 0.16$ fm$^{-3}$, \nwith energies of SNM at saturation around $E_b (\\rho_0) \\simeq -16$ MeV. Their incompressibilities are mostly\nbetween $220 \\lesssim K (\\rho_0) \\lesssim 270$ MeV.\nThe symmetry energy of these forces are within $ 26 \\lesssim E_\\mathrm{sym}  (\\rho_0)\\lesssim 37$ MeV, \nand their slopes within $9 \\lesssim L \\lesssim 130$ MeV,\ncovering the range given by some estimates coming from experimental data~\\cite{BaoAnLi13,Vinas14}.\nFinally, we see that the symmetry energy incompressibility ($K_\\mathrm{sym}$) is the less constrained parameter, as also\nthere are not many constraints on it coming from experiments. The range of $K_\\mathrm{sym} $\nwhen using the interactions in Table \\ref{table:Skyrmeprops} is $ -275 \\lesssim K_\\mathrm{sym}  \\lesssim 71$ MeV.\nAll interactions in this Table~\\ref{table:Skyrmeprops} are of the type described in Eq.~(\\ref{VSkyrme}), \nexcepting the Skyrme interactions Sk$\\chi$414, Sk$\\chi$450 and Sk$\\chi$500, which\nhave an additional density-dependent term\n\\begin{equation}\nV (\\mathbf{r}_1 , \\mathbf{r}_2) \\rightarrow V (\\mathbf{r}_1 , \\mathbf{r}_2)\n+\\frac{1}{6} t_4 (1+ x_4 P_\\sigma) \\rho^{\\alpha'} (\\mathbf{R}) \\delta (\\mathbf{r})\n\\end{equation}\nthat takes into account the chiral N3LO asymmetric matter equation of state~\\cite{skyrmechiral}. In these cases, the \nequations of the energy per particle, pressure and chemical potentials are changed accordingly, \nadding the new zero-range density-dependent term.\n\\begin{table}[t!]\n\\centering\n\\begin{tabular}{c|cdddddd}\n\\hline\n\\multirow{2}{*}{Skyrme force} &\\multirow{2}{*}{Ref.}& \\multicolumn{1}{c}{$\\rho_0$}    & \n\\multicolumn{1}{c}{$E_b (\\rho_0)$} & \\multicolumn{1}{c}{$K (\\rho_0)$}& \n\\multicolumn{1}{c}{$E_\\mathrm{sym} (\\rho_0)$}& \\multicolumn{1}{c}{$L$ }   &  \n\\multicolumn{1}{c}{$K_\\mathrm{sym}$} \\\\\n             & & \\multicolumn{1}{c}{(fm$^{-3}$)} & \\multicolumn{1}{c}{(MeV)}  \n             & \\multicolumn{1}{c}{(MeV)}                     &\\multicolumn{1}{c}{(MeV)} \n             & \\multicolumn{1}{c}{(MeV)}  & \\multicolumn{1}{c}{(MeV)}              \\\\\\hline\\hline\nMSk7                      &\\cite{Msk7}    & 0.158       & -15.80         & 231.21         & 27.95              & 9.41   & -274.62                   \\\\\nSIII                      &\\cite{SIII}     & 0.145       & -15.85         & 355.35         & 28.16              & 9.91   & -393.72                   \\\\\nSkP                      &\\cite{SkP}      & 0.163       & -15.95         & 200.96         & 30.00              & 19.68  & -266.59                   \\\\\nHFB-27                  &\\cite{HFB27}      & 0.159       & -16.05         & 241.63         & 30.00              & 28.50  & -221.41                   \\\\\nSKX                      &\\cite{SkX}     & 0.156       & -16.05         & 271.05         & 31.10              & 33.19  & -252.11                   \\\\\nHFB-17                   &\\cite{HFB17}     & 0.159       & -16.06         & 241.68         & 30.00              & 36.29  & -181.83                   \\\\\nSGII                     & \\cite{sgii}     & 0.158       & -15.59         & 214.64         & 26.83              & 37.63  & -145.90                   \\\\\nUNEDF1                    &\\cite{unedf1}    & 0.159       & -15.80         & 220.00         & 28.99              & 40.01  & -179.46                   \\\\\nSk$\\chi$500                    &\\cite{skyrmechiral}    & 0.168       & -15.99         & 238.14         & 29.12              & 40.74  & -77.40                    \\\\\nSk$\\chi$450                    &\\cite{skyrmechiral}    & 0.156       & -15.93         & 239.51         & 30.64              & 42.06  & -142.68                   \\\\\nUNEDF0                    &\\cite{unedf0}     & 0.161       & -16.06         & 230.00         & 30.54              & 45.08  & -189.67                   \\\\\nSkM*                     &\\cite{skms}     & 0.161       & -15.77         & 216.60         & 30.03              & 45.78  & -155.93                   \\\\\nSLy4                     &\\cite{sly41, sly42}     & 0.160       & -15.97         & 229.90         & 32.00              & 45.96  & -119.70                   \\\\\nSLy7                     &\\cite{sly41, sly42}     & 0.158       & -15.90         & 229.69         & 31.99              & 47.22  & -113.32                   \\\\\nSLy5                     &\\cite{sly41, sly42}     & 0.160       & -15.98         & 229.92         & 32.03              & 48.27  & -112.34                   \\\\\nSk$\\chi$414                   &\\cite{skyrmechiral}     & 0.170       & -16.20         & 243.17         & 32.34              & 51.92  & -95.71                    \\\\\nMSka                     &\\cite{mska}     & 0.154       & -15.99         & 313.32         & 30.35              & 57.17  & -135.34                   \\\\\nMSL0                     &\\cite{msl0}     & 0.160       & -16.00         & 229.99         & 30.00              & 60.00  & -99.33                    \\\\\nSIV                      &\\cite{SIII}     & 0.151       & -15.96         & 324.54         & 31.22              & 63.50  & -136.71                   \\\\\nSkMP                      &\\cite{skmp}     & 0.157       & -15.56         & 230.86         & 29.89              & 70.31  & -49.82                    \\\\\nSKa                      &\\cite{ska}     & 0.155       & -15.99         & 263.14         & 32.91              & 74.62  & -78.45                    \\\\\nR$_\\sigma$               &\\cite{rsgs}     & 0.158       & -15.59         & 237.35         & 30.58              & 85.69  & -9.14                     \\\\\nG$_\\sigma$               &\\cite{rsgs}     & 0.158       & -15.59         & 237.22         & 31.37              & 94.01  & 13.98                     \\\\\nSV                       &\\cite{SIII}      & 0.155       & -16.08         & 305.68         & 32.83              & 96.09  & 24.18                     \\\\\nSkI2                    &\\cite{ski2}      & 0.158       & -15.78         & 240.92         & 33.38              & 104.33 & 70.68                     \\\\\nSkI5                    &\\cite{ski2}       & 0.156       & -15.78         & 255.78         & 36.64              & 129.33 & 70.68         \\\\\\hline            \n\\end{tabular}\n\\caption{Compilation of the Skyrme interactions used through this work, some of their SNM properties, such as the \nsaturation density $\\rho_0$, energy per particle $E_b (\\rho_0)$ and incompressibility $K (\\rho_0)$ at saturation density, and some ANM properties, \nsuch as the symmetry energy at the saturation point, $E_\\mathrm{sym} (\\rho_0)$, and its slope $L$ and curvature $K_\\mathrm{sym}$ at saturation.\n\\label{table:Skyrmeprops}}\n\\end{table}\n\\newpage\n\n\n\\section{Gogny interactions}\\label{Gogny}\nGogny interactions were proposed by D. Gogny with the aim of describing the mean-field and the pairing field in the same interaction.\nThe standard Gogny two-body effective nuclear interaction reads as\\cite{decharge80, berger91, chappert08, goriely09,Sellahewa14}\n\\begin{eqnarray}\\label{VGogny}\n  V (\\mathbf{r}_1 , \\mathbf{r}_2) &=&  \\sum_{i=1,2} \\left( W_i + B_i P_\\sigma - H_i P_\\tau - M_i P_\\sigma P_\\tau \\right)e^{-r^2 \/\\mu_i^2} \\nonumber\n   \\\\\n &+& t_3 \\left( 1+x_3 P_\\sigma \\right) \\rho^\\alpha(\\mathbf{R})\\delta(\\mathbf{r}) \\nonumber\n \\\\\n  &+ & i W_0 \\left( \\bm{\\sigma}_1 + \\bm{\\sigma}_2 \\right) \\left[ \\mathbf{k}' \\times \\delta(\\mathbf{r}) \\mathbf{k} \\right].\n\\end{eqnarray}\nThe first term in Eq.~(\\ref{VGogny}) is the finite-range part of the interaction, and it is modulated by two\nGaussian form-factors of long- and short-ranges. \nThe following term is the zero-range density-dependent contribution to the interaction. The last term \nof the interaction corresponds to the spin-orbit force, which is also zero-range as in the \ncase of Skyrme interactions and does not contribute in infinite nuclear matter. \nGogny forces describe nicely ground-state systematics of finite nuclei, nuclear excitation properties, and fission phenomena.\n\nThe energy per baryon $E_b(\\rho, \\delta)$ in the Hartree--Fock approximation \nin asymmetric infinite nuclear matter for Gogny forces\nas a function of the total baryon number density $\\rho $ and \nof the isospin asymmetry $\\delta$ can be decomposed as a \nsum of four different contributions, namely, a kinetic and a zero-range contributions, \nand the direct and exchange finite-range terms:\n\\begin{eqnarray}\n E_b (\\rho, \\delta)=  E_b^{\\mathrm{kin}} (\\rho, \\delta)+ E_b^{\\mathrm{zr}} (\\rho, \\delta)  \n \\mbox{} + E_b^{\\mathrm{dir}} (\\rho, \\delta) + E_b^{\\mathrm{exch}} (\\rho, \\delta) \\label{eq:eb.terms} \\, ,\n\\end{eqnarray}\nwhich read as\n\\begin{eqnarray}\n E_b^{\\mathrm{kin}} (\\rho, \\delta)&=& \\frac{ 3 \\hbar^2}{20m} \\left(\\frac{3 \\pi^2}{2}\\right)^{2\/3} \\rho^{2\/3}\n \\left[ (1+\\delta)^{5\/3} + (1-\\delta)^{5\/3} \\right] \\label{eq:eb.kin}\n\\\\\nE_b^{\\mathrm{zr}} (\\rho, \\delta)&=&  \\frac{1}{8} t_3 \\rho^{\\alpha+1} \\left[ 3-(2x_3+1)\\delta^2 \\right] \\label{eq:eb.zr}\n\\\\\nE_b^{\\mathrm{dir}} (\\rho, \\delta)&=&  \\frac{1}{2} \\sum_{i=1,2} \\mu_i^3 \\pi^{3\/2} \\rho  \\left[ {\\cal A}_i \n+{\\cal B}_i \\delta^2 \\right] \\label{eq:eb.dir}\n\\\\\nE_b^{\\mathrm{exch}} (\\rho, \\delta)&= & -\\sum_{\\mathrm{i}=1,2}\\frac{1}{2  k_F^3 \\mu_i^3} \n\\Big\\{ {\\cal C}_i \\left[ {\\mathsf  e} (k_{Fn} \\mu_i ) + {\\mathsf  e} (k_{Fp} \\mu_i ) \\right]\n-  {\\cal D}_i  \\bar {\\mathsf  e}( k_{Fn} \\mu_i,k_{Fp} \\mu_i )  \\Big\\}, \\label{eq:eb.exch}\n\\end{eqnarray}\n with \n \\begin{equation}\n{\\mathsf e}(\\eta) = \\frac{\\sqrt{\\pi}}{2} \\eta^3 \\mathrm{erf}(\\eta) \n+ \\left(\\frac{\\eta^2}{2} - 1 \\right) e^{-\\eta^2} - \\frac{3 \\eta^2}{2} +  1 \\, ,\n\\end{equation}\nand \n \\begin{eqnarray}\n\\bar {\\mathsf  e}(\\eta_1,\\eta_2)&=& \\sum_{s=\\pm 1} s \n\\left[ \n\\frac{ \\sqrt{\\pi}}{2} (\\eta_1 + s \\eta_2 ) \\left( \\eta_1^2 + \\eta_2^2 - s \\eta_1 \\eta_2 \\right) \n\\mathrm{erf} \\left( \\frac{\\eta_1 + s \\eta_2 }{2}  \\right) \\right. \\nonumber \\\\\n&+& \\left. \\left( \\eta_1^2 + \\eta_2^2  - s \\eta_1 \\eta_2  -2 \\right) e^{ - \\frac{1}{4} (\\eta_1 + s \\eta_2)^2 } \n\\right] ,\n\\end{eqnarray}\nwhere \n\\begin{equation}\n \\displaystyle \\mathrm{erf}(x) =  \\frac{2}{\\sqrt{\\pi}} \\int_0^x  e^{-t^2} dt\n\\end{equation}\n is the error function.\nThe function $\\bar {\\mathsf  e}(\\eta_1,\\eta_2)$ is a symmetric function of its arguments, satisfying \n$\\mathsf{\\bar e}(\\eta,\\eta)= 2 \\mathsf{e}(\\eta)$ and $\\mathsf{\\bar e}(\\eta,0) = 0$.\nThe value of the parameter $x_3$ is considered equal to one, $x_3=1$, for all Gogny parametrizations of the D1 family, in order to \navoid the zero-range contributions to the pairing field~\\cite{decharge80}.\n\nThe term $E_b^{\\mathrm{kin}} (\\rho, \\delta)$ is the sum of the contributions of the neutron and proton kinetic energies, and \n$E_b^{\\mathrm{zr}} (\\rho, \\delta)$ comes from the zero-range interaction. The term in Eq.~(\\ref{eq:eb.dir}) defines\nthe direct contribution of the finite-range part of the force, whereas the term in Eq.~(\\ref{eq:eb.exch}) defines its exchange contribution. \nThe kinetic, zero-range and finite-range direct terms can be expressed as functions of the density $\\rho$ and the asymmetry $\\delta$, \nwhereas the finite-term exchange contribution is defined as a function of the neutron, $k_{Fn} = k_F (1+\\delta)^{1\/3}$ ,\nand proton, $k_{Fp} = k_F (1-\\delta)^{1\/3}$, Fermi momenta. The Fermi momentum \nof symmetric nuclear matter is given by $k_F = (3 \\pi^2 \\rho\/2 )^{1\/3}$.\nMoreover, the combinations of parameters appearing in the different terms of Eq.~(\\ref{eq:eb.terms}) are the following:\n\\begin{eqnarray}\n{\\cal A}_i &=& \\frac{1}{4} \\left( 4 W_i + 2 B_i - 2H_i -M_i \\right) \\label{Ai}\n\\\\\n{\\cal B}_i&=&  -\\frac{1}{4}\\left( 2 H_i + M_i \\right)\\label{Bi}\n\\\\\n{\\cal C}_i&=& \\frac{1}{\\sqrt{\\pi}} \\left( W_i + 2 B_i - H_i -2 M_i\\right)\\label{Ci}\n\\\\\n{\\cal D}_i&=& \\frac{1}{\\sqrt{\\pi}} \\left( H_i + 2 M_i \\right).\\label{Di}\n\\end{eqnarray}\nThe constants $ {\\cal A}_i $ and ${\\cal B}_i$ define, respectively, the isoscalar and isovector \npart of the direct term. For the exchange terms, the matrix elements ${\\cal C}_i $ relate to \nneutron-neutron and proton-proton interactions, whereas \nthe matrix elements ${\\cal D}_i$ take care of neutron-proton interactions. \n\nIf the energy per particle in asymmetric nuclear matter is expanded \nin terms of the isospin asymmetry $\\delta$ [see Eq.~(\\ref{eq:eosexp})] the symmetry energy coefficient \nfor the Gogny parametrization is defined as~\\cite{Sellahewa14}\n\\begin{eqnarray}\nE_{\\mathrm{sym}} (\\rho) &=& \n\\frac{\\hbar^2}{6m} \\left(  \\frac{3 \\pi^2}{2}\\right)^{2\/3} \\rho^{2\/3}   -\n\\frac{1}{8} t_3 \\rho^{\\alpha+1} (2x_3 +1) \\nonumber\n\\\\\n&+& \\frac{1}{2} \\sum_{i=1,2} \\mu_i^3 \\pi^{3\/2}  {\\cal B}_i  \\rho  \n+ \\mbox{} \\frac{1}{6}\\sum_{i=1,2}  \\left[-{\\cal C}_i  G_1 ( k_F \\mu_i)+ {\\cal D}_i G_2 ( k_F \\mu_i)  \\right] \\label{eq:esym2gogny}\n\\end{eqnarray}\nwith\n\\begin{eqnarray}\n G_1 (\\eta)&=& \\frac{1}{\\eta} -\\left( \\eta + \\frac{1}{\\eta} \\right) e^{-\\eta^2} \\label{G1}\n\\\\\n G_2 (\\eta)&=& \\frac{1}{\\eta} -\\bigg( \\eta + \\frac{e^{-\\eta^2}}{\\eta} \\bigg), \\label{G2}\n\\end{eqnarray}\nand its symmetry energy slope $L$ at saturation density $\\rho_0$ is given by \n \\begin{eqnarray}\n L &=& 3 \\frac{\\hbar^2}{3m} \\left(\\frac{3 \\pi^2}{2}\\right)^{2\/3} \\rho_0^{2\/3} -\n  \\frac{3(\\alpha + 1)}{8} t_3 \\rho_0^{\\alpha+1} (2 x_3 +1)\\nonumber \\\\\n  &+& \\frac{3}{2} \\sum_{i=1,2} \\mu_i^3 \\pi^{3\/2} \\mathcal{B}_i \\rho_0 \n  + \\mbox{}  \\frac{1}{6} \\sum_{i=1,2} \\left[ - \\mathcal{C}_i L_1(\\mu_i k_{F0}) + \\mathcal{D}_i L_2(\\mu_i k_{F0}) \\right] ,\n \\end{eqnarray}\nwhere $k_{F0} = (3 \\pi^2 \\rho_0\/2)^{1\/3}$ is the Fermi momentum at saturation and the $L_n (\\eta)$ functions are\n\\begin{eqnarray}\n L_1 (\\eta) &=& -\\frac{1}{\\eta} +e^{-\\eta^2} \\left( \\frac{1}{\\eta} + \\eta + 2 \\eta^3\\right) \\label{G1'}\n \\\\\n L_2 (\\eta) &=&  -\\frac{1}{\\eta} + e^{-\\eta^2} \\left( \\frac{1}{\\eta} +2\\eta \\right) - \\eta.\n\\end{eqnarray}\nAlso, for Gogny interactions, the symmetry energy curvature is defined as\n\\begin{eqnarray}\n K_\\mathrm{sym}  &=&  \n  -\\frac{\\hbar^2}{3m} \\left(\\frac{3 \\pi^2}{2}\\right)^{2\/3} \\rho_0^{2\/3} -\n  \\frac{9(\\alpha + 1)}{8} t_3 \\rho_0^{\\alpha+1} (2 x_3 +1)\\nonumber \\\\\n  &-& \\mbox{}  \\frac{2}{3} \\sum_{i=1,2} \\left[ - \\mathcal{C}_i K_1(\\mu_i k_{F0}) + \\mathcal{D}_i K_2(\\mu_i k_{F0}) \\right] ,\n\\end{eqnarray}\nwith the functions $K_n (\\eta)$ reading as\n\\begin{eqnarray}\n K_1 (\\eta) &=& -\\frac{1}{\\eta} + \\left( \\frac{1}{\\eta} + \\eta + \\frac{\\eta^3}{2} + \\eta^5\\right)e^{-\\eta^2}\n \\\\\n K_2 (\\eta) &=&  -\\frac{1}{\\eta} - \\frac{\\eta}{2} + \\left( \\frac{1}{\\eta} + \\frac{3 \\eta}{2} + \\eta^3 \\right)e^{-\\eta^2}.\n\\end{eqnarray}\n\nAs stated previously, the proton and chemical potentials in asymmetric matter will be given by the \nderivatives of the nuclear energy density $\\mathcal{H}_b = \\rho E_b$ with respect to the neutron ($\\rho_n$) and\nproton ($\\rho_p$) densities. For Gogny interactions, the neutron ($\\tau = +1$) and proton ($\\tau = -1$) chemical potentials are \n\\begin{eqnarray}\n \\mu_\\tau (\\rho_\\tau, \\rho_{\\tau'})&=&  \\frac{\\hbar^2}{2m} \\left( 3 \\pi^2 \\right)^{2\/3} \\rho_\\tau^{2\/3} + \\frac{t_3}{8} \\rho^{\\alpha + 1} \n \\left[ 3 \\left( \\alpha + 2\\right) - 2 \\tau \\left( 2 x_3 +1\\right) \\delta - \\alpha \\left( 2 x_3 + 1\\right) \n \\delta^2\\right] \\nonumber\n \\\\\n &+&  \\sum_{i=1,2} \\mu_i^3 \\pi^{3\/2} \\rho \\left( {\\cal A}_i + \\tau {\\cal B}_i \\delta \\right) \n - \\sum_{i=1,2} \\left[ {\\cal C}_i\\,  \\bar {\\mathsf w}( k_F^\\tau \\mu_i , k_F^\\tau \\mu_i  ) - \n {\\cal D}_i\\, \\bar {\\mathsf  w} (k_F^\\tau \\mu_i , k_F^{-\\tau} \\mu_i )\\right],\\nonumber\\\\ \n\\label{mu-tau-gog}\n\\end{eqnarray}\nwhere $\\bar {\\mathsf w} \\left( \\eta_1 , \\eta_2\\right)$ is the dimensionless function \n\\begin{equation}\n\\bar {\\mathsf w} \\left( \\eta_1 , \\eta_2\\right) =\n\\sum_{s=\\pm 1} s \\left[ \\frac{\\sqrt{\\pi}}{2} \\mathrm{erf}\\left( \\frac{\\eta_1 + s \\eta_2}{2}\\right) \n+ \\frac{1}{\\eta_1} e^{- \\frac{1}{4} ( \\eta_1 + s \\eta_2 )^2} \\right] \\, .\n\\end{equation}\n\nFrom the derivative of the energy per particle with respect to the density, one can obtain the pressure of the baryons \nin the system, which for Gogny forces reads\n\\begin{eqnarray}\\label{eq:pressure_bars}\nP_b(\\rho,\\delta) &=& \n\\frac{\\hbar^2}{10m} \\left(\\frac{3 \\pi^2}{2}\\right)^{2\/3} \\rho^{5\/3} \\left[ (1+\\delta)^{5\/3} + (1-\\delta)^{5\/3} \n\\right] \\nonumber \\\\\n&+& \\frac{(\\alpha+1)}{8} t_3 \\rho^{\\alpha+2} \\left[ 3 - (2x_3+1) \\delta^2 \\right] \n + \\frac{\\rho^2}{2} \\sum_{i=1,2} \\pi^{3\/2} \\mu_i^3 \\left( {\\cal A}_i + {\\cal B}_i \\delta^2 \\right)  \\\\\n& -&  \\frac{\\rho}{2}  \\sum_{i=1,2} \\left\\{ \n{\\cal C}_i\n\\left[ (1+\\delta) \\mathsf{p} ( k_{Fn} \\mu_i ) \n     + (1-\\delta) \\mathsf{p} ( k_{Fp} \\mu_i ) \\right]   \n- {\\cal D}_i \\mathsf{ \\bar p} ( k_{Fn} \\mu_i,k_{Fp}\\mu_i )  \n     \\right\\} \\, .\\nonumber\n\\end{eqnarray}\nThe function $\\mathsf{p}(\\eta)$ contains the density dependence of the pressure in both symmetric and neutron \nmatter \\cite{Sellahewa14}:\n\\begin{align}\n\\mathsf{p}\\left(\\eta \\right) &= -\\frac{1}{\\eta^3} + \\frac{1}{2 \\eta} + \\left( \\frac{1}{\\eta^3} + \n\\frac{1}{2 \\eta} \\right) e^{- \\eta^2} \\, .\n\\end{align}\nIn asymmetric matter, the double integral on the exchange terms leads to the appearance of a term that \ndepends on the two Fermi momenta: \n\\begin{align}\n \\mathsf{\\bar p} \\left( \\eta_1, \\eta_2 \\right)  = &\n\\frac{2}{\\eta_1^3 + \\eta_2^3}\n\\sum_{s=\\pm 1}  (\\eta_1 \\eta_2 +2s ) e^{- \\frac{1}{4} \\left(\\eta_1 + s\\eta_2\\right)^2 } .\n \\label{eq:p_function}\n \\end{align}\n\nIn SNM, where the neutron and proton densities are equal ($\\delta=0$),\nthe energy per baryon (\\ref{eq:eb.terms}) can be \nrewritten as~\\cite{Sellahewa14}\n\\begin{equation}\\label{Ebgognysnm}\n E_{b} (\\rho) = \\frac{3}{5} \\frac{\\hbar^2}{2 m} k_F^2 + \\frac{3}{8} t_3 \\rho^{\\alpha+1} + \\frac{1}{2} \\sum_{i=1,2}\\left[ \\mu_i^3\n \\pi^{3\/2} \\rho \\mathcal{A}_i + \\mathcal{B}_{0i}  g (\\mu_i k_F)\\right], \n\\end{equation}\nwhere the coefficient $\\mathcal{A}_i$ is defined in Eq.~(\\ref{Ai}) and $\\mathcal{B}_{0i}$ is\n\\begin{equation}\\label{Boigogny}\n \\mathcal{B}_{0i} = -\\frac{1}{\\sqrt{\\pi}} \\left( W_i + 2 B_i - 2 H_i -4 M_i \\right).\n\\end{equation}\nSimilarly to asymmetric nuclear matter, one can find the pressure in SNM as the derivative of the energy per particle\nwith respect to the density, obtaining\n\\begin{equation}\\label{PbsnmGogny}\n P (\\rho) = \n \\frac{2}{5} \\frac{\\hbar^2}{2 m} k_F^2 \\rho + \\frac{3}{8} t_3 (\\alpha + 1)\\rho^{\\alpha+2} + \\frac{1}{2} \\sum_{i=1,2}\\left[ \\mu_i^3\n \\pi^{3\/2} \\rho^2 \\mathcal{A}_i + \\rho \\mathcal{B}_{0i}   p (\\mu_i k_F)\\right].\n\\end{equation}\nThe nuclear matter incompressibility is given by the curvature of the energy per particle in SNM. \nIn the case of Gogny interactions, the expression for the nuclear matter incompressibility is\n\\begin{equation}\\label{K0Gogny}\n K (\\rho) = \n -\\frac{6}{5} \\frac{\\hbar^2}{2 m} k_F^2  + \\frac{9}{8} t_3 (\\alpha + 1)\\alpha\\rho^{\\alpha+1} - 3 \\sum_{i=1,2}\n  \\mathcal{B}_{0i}   k (\\mu_i k_F).\n\\end{equation}\nThe functions $g(\\eta)$, $p(\\eta)$ and $k(\\eta)$ appearing in the expressions of the energy per particle, pressure and \nincompressibility in SNM are given, respectively, by \n\\begin{eqnarray}\n g(\\eta) &=& \\frac{2}{\\eta^3} - \\frac{3}{\\eta} - \\left( \\frac{2}{\\eta^3}-\\frac{1}{\\eta}\\right)e^{-\\eta^2} +\n \\sqrt{\\pi} \\mathrm{erf} (\\eta)\n\\\\\n p(\\eta) &=& - \\frac{1}{\\eta^3} + \\frac{1}{2\\eta} + \\left( \\frac{1}{\\eta^3} + \\frac{1}{2\\eta} \\right) e^{-\\eta^2}\n\\\\\n k(\\eta) &=& - \\frac{6}{\\eta^3} + \\frac{2}{\\eta} + \\left( \\frac{6}{\\eta^3} + \\frac{4}{\\eta}+\\eta \\right) e^{-\\eta^2}.\n\\end{eqnarray}\nIf one considers pure neutron matter, with $\\delta=1$, the expression of the energy per particle can be\nwritten as~\\cite{Sellahewa14}\n\\begin{equation}\n E_{b} (\\rho, \\delta=1) = \\frac{3}{5} \\frac{\\hbar^2}{2 m} k_{Fn}^2 + \\frac{1}{4} t_3 \\rho^{\\alpha+1} \n (1-x_3)+ \\frac{1}{2} \\sum_{i=1,2}\\left[ \\mu_i^3\n \\pi^{3\/2} \\rho \\mathcal{A}_i - \\mathcal{C}_{i}  g (\\mu_i k_{Fn})\\right].\n\\end{equation}\n\\begin{table}[!t]\n\\centering\n\\begin{tabular}{c|ccccccc}\n\\hline\n\\multirow{2}{*}{Gogny Force} & \\multirow{2}{*}{Ref.} & $\\rho_0$    & $E_b$ ($\\rho_0$) & $K(\\rho_0)$  & $E_\\mathrm{sym} (\\rho_0)$ & $L$ & $K_\\mathrm{sym} $ \\\\\n                             &                       & (fm$^{-3}$) & (MeV)                            & (MeV)  & (MeV)                     & (MeV)        & (MeV)                     \\\\\\hline\\hline\nD260                         &   \\cite{NPA591Blaizot1995}                    & 0.1601      & -16.26                           & 259.49 & 30.11                     & 17.57        & 259.49                    \\\\\nD1                           &    \\cite{decharge80}                   & 0.1665      & -16.31                           & 229.37 & 30.70                     & 18.36        & 229.37                    \\\\\nD1S                          &     \\cite{berger91}                  & 0.1633      & -16.01                           & 202.88 & 31.13                     & 22.43        & 202.88                    \\\\\nD1M                          &    \\cite{goriely09}                  & 0.1647      & -16.03                           & 224.98 & 28.55                     & 24.83        & 224.98                    \\\\\nD250                         &    \\cite{NPA591Blaizot1995}                   & 0.1577      & -15.80                           & 249.41 & 31.54                     & 24.90        & 249.41                    \\\\\nD300                         &    \\cite{NPA591Blaizot1995}                   & 0.1562      & -16.22                           & 299.14 & 31.22                     & 25.84        & 299.14                    \\\\\nD1N                          &     \\cite{chappert08}                  & 0.1612      & -15.96                           & 225.65 & 29.60                     & 33.58        & 225.65                    \\\\\nD1M$^{**}$                        &     \\cite{gonzalez18a}                  & 0.1647      & -16.02                           & 225.38 & 29.37                     & 33.91        & 224.98                    \\\\\nD1M$^*$                         &     \\cite{gonzalez18}                  & 0.1650      & -16.06                           & 224.98 & 30.25                     & 43.18        & 225.38                    \\\\\nD2                           &   \\cite{chappert15}                    & 0.1628      & -16.00                           & 209.26 & 31.11                     & 44.83        & 209.26                    \\\\\nD280                         &    \\cite{NPA591Blaizot1995}                   & 0.1525      & -16.33                           & 285.20 & 33.14                     & 46.53        & 285.20                   \\\\\\hline\n\\end{tabular}\n\\caption{Compilation of the Gogny interactions used through this work, some of their SNM properties, such as the \nsaturation density $\\rho_0$, energy per particle $E_b (\\rho_0)$ and incompressibility $ K  (\\rho_0)$ at saturation density, and some ANM properties, \nsuch as the symmetry energy at the saturation point $E_\\mathrm{sym} (\\rho_0)$, and its slope $L$ and curvature $K_\\mathrm{sym}$ at saturation.\\label{table:Gognyprops}}\n\\end{table}\nWe collect in Table~\\ref{table:Gognyprops} some isoscalar and isovector properties of the different Gogny interactions we have used in this work.\nThe first proposed Gogny interaction was D1 \\cite{decharge80}, fitted to some properties of nuclear matter properties and of few closed-shell nuclei.\nAfter D1, D1S \\cite{chappert08} was proposed with the aim of getting a better description of nuclear fission \\cite{berger91}. \nThe interactions  D250, D260, D280, and D300 \\cite{NPA591Blaizot1995} were devised to have different \nnuclear matter incompressibility for calculations of the breathing mode in nuclei. \nIn order to improve the isovector part of the Gogny interactions, the D1N \\cite{chappert08} and D1M \n\\cite{goriely09} interactions were fitted to reproduce the microscopic neutron matter EoS of  \nFriedman and Pandharipande~\\cite{Friedman81}. \nThe interactions D1M$^*$~\\cite{gonzalez18} and D1M$^{**}$~\\cite{gonzalez18a} will be introduced later in \nChapter~\\ref{chapter3}. They are fitted in such a way that, \nwhile preserving the description of finite nuclei similar to the one obtained with D1M, they are able to provide NSs inside \nthe observational constraints for the neutron star mass~\\cite{Demorest10, Antoniadis13}. \nFinally, the D2 \\cite{chappert15} interaction is a recent Gogny interaction where the zero-range density-dependent  \nterm in Eq.~(\\ref{VGogny}) has \nbeen replaced by a finite-range density-dependent term of Gaussian type, i.e., \n\\begin{eqnarray}  \n V_\\mathrm{dens}^{D1} &=& t_3 \\left( 1+x_3 P_\\sigma \\right) \\rho^\\alpha(\\mathbf{R})\\delta(\\mathbf{r}) \\nonumber\\\\\n &&\\downarrow\\nonumber\\\\\n V_\\mathrm{dens}^{D2} &=& \\left( W_3 + B_3 P_\\sigma - H_3 P_\\tau - M_3 P_\\sigma P_\\tau \\right)\\times \\frac{e^{-r^2 \/\\mu_3^2}}{(\\mu_3 \\sqrt{\\pi})^3}\n \\frac{\\rho^\\alpha ({\\bf r}_1) + \\rho^\\alpha ({\\bf r}_2)}{2},\n \\end{eqnarray}\nand therefore the equations listed above are switched accordingly. \nFrom Table~\\ref{table:Gognyprops} we see that Gogny interactions have similar saturation densities around $\\rho_0 \\simeq 0.16$ fm$^{-3}$, \nand SNM energy per particle  at saturation around \\mbox{$E_b (\\rho_0) \\simeq-16$ MeV}. Their incompressibilities vary in the range \n$202$ MeV $ \\lesssim K_0  (\\rho_0)\\lesssim 286$ MeV and their symmetry energies lay within \n$ 28$ MeV $   \\lesssim E_\\mathrm{sym}(\\rho_0) \\lesssim 34$ MeV. Finally, Gogny interactions have \nslope parameters $L$ in the low-moderate regime between $17$ and $47$ MeV, and asymmetric incompressibilities \n$K_\\mathrm{sym}$ that go from $202$ to $300$ MeV. \n\n\n\n\\section{Momentum-dependent interactions and Simple effective interactions}\\label{MDISEI}\n\nThe momentum-dependent interactions (MDI)~\\cite{das03} have been extensively used to study  transport calculations in heavy-ion \ncollisions~\\cite{das03,li08}, and have also been applied to other different scenarios,\nin particular to neutron stars \\cite{xu09a,xu09b,xu10a,xu10b,li08,Krastev19}.\nThe  potential energy density that one uses for MDI interactions in an asymmetric nuclear matter system \nis \\cite{das03}\n\\begin{eqnarray}\n V (\\rho, \\delta) &=& \\frac{A_1}{2 \\rho_0} \\rho^2 + \\frac{A_2}{2 \\rho_0} \\rho^2 \\delta^2 + \n \\frac{B}{\\sigma+1} \\frac{\\rho^{\\sigma+1}}{\\rho_0^\\sigma} (1-x \\delta^2) \\nonumber\\\\\n &+& \\frac{1}{\\rho_0} \\sum_{\\tau, \\tau'} C_{\\tau, \\tau'} \\int \\int d^3p d^3p' \\frac{f_\\tau ({\\bf r}, {\\bf p})\n f_{\\tau'} ({\\bf r}, {\\bf p}')}{1+({\\bf p} - {\\bf p}')^2\/\\Lambda^2},\n\\end{eqnarray}\nwhere $\\tau$ and $\\tau'$ refer to the nucleon isospin and $f_\\tau ({\\bf r}, {\\bf p}) $ is the nucleon phase-space\ndistribution function at the position ${\\bf r}$. The parameters $A_1= (A_l+A_u)\/2$, $A_2= (A_l-A_u)\/2$, $B$, \n$\\sigma$, $\\Lambda$, $C_l=C_{\\tau, \\tau}$ and $C_u=C_{\\tau, \\tau'}$ are fitted as explained in Refs. \\cite{das03, chen14},\nsubject to the constraint that the \nmomentum-dependence of the single-particle potential reproduces the behaviour predicted by the Gogny\ninteraction, which gives a reasonable parametrization of the real part of the optical potential in nuclear\nmatter as a function of the incident energy. \nIn the fitting protocol of the MDI parameters, it is  \nrequired that in SNM the saturation density has values of $\\rho_0=0.16$ fm$^{-3}$, the binding energy of $E_b(\\rho_0)= 16$ MeV \nand the incompressibility\nof $K (\\rho_0)=211$ MeV. Moreover, it is imposed that the symmetry energy at saturation takes a value of \n$E_\\mathrm{sym}(\\rho_0)=30.5$ MeV \\cite{das03, chen14}. \nIn the original MDI interaction~\\cite{das03}, the parameter $x$ could only take values of $x=0$ and $x=1$.\nIn order to be able to have a larger range of the dependence of the symmetry energy with the density\nwhile keeping its fitted value $E_\\mathrm{sym} (\\rho_0)=30.5$ MeV at saturation, the parameters $A_l$ and $A_u$ \\cite{Chen05} are\nredefined as\n\\begin{equation}\n A_l=-120.57 + x \\frac{2B}{\\sigma+1} \\hspace{2cm} A_u=-95.98-x \\frac{2B}{\\sigma+1}.\n\\end{equation}\nHence, a change in the parameter $x$ of the MDI interaction will modify the density \ndependence of the symmetry energy and of the neutron matter EoS, without changing the EoS of \nSNM and the symmetry energy at the saturation density.\n\nThe MDI interaction can be rewritten as a zero-range contribution plus a finite range term\nwith a single Yukawa form factor \\cite{das03}, \n\\begin{eqnarray}\\label{VMDI}\nV (\\mathbf{r}_1 , \\mathbf{r}_2) &=& \\frac{1}{6} t_3 \\left( 1+x_3 P_\\sigma \\right) \\rho^\\sigma \\left({\\bf R} \\right) \\delta({\\bf r})\n+ \\left( W+B P_\\sigma -H P_\\tau- M P_\\sigma P_\\tau \\right)\\frac{e^{-\\mu r}}{ r}.\n\\end{eqnarray}\n\nThe simple effective interaction (SEI) is an effective nuclear force constructed with a minimum number of parameters to \nstudy the momentum and density dependence of the nuclear mean-field,\ndefined as \\cite{behera98, Behera05}\n\\begin{eqnarray}\\label{VSEI}\nV (\\mathbf{r}_1 , \\mathbf{r}_2) &=& t_0(1+x_0P_\\sigma) \\delta({\\bf r})\n+\\frac{1}{6} t_3 \\left( 1+x_3 P_\\sigma \\right) \\left(\n\\frac{\\rho({\\bf R})}{1+b \\rho ({\\bf R}) }\\right)^\\gamma  \\delta({\\bf r})\\nonumber\\\\\n&+& \\left( W+B P_\\sigma -H P_\\tau- M P_\\sigma P_\\tau \\right)v (r) .\n\\end{eqnarray}\nThis interaction has a finite-range part, which can be of any conventional form factor $v(r)$ of Gaussian, Yukawa or exponential types, and \ntwo zero-range terms, one of them density-dependent, containing altogether eleven parameters plus the range $\\mu$ of the form factor\nand the spin-orbit strength parameter $W_0$ which enters in the description of finite nuclei. \nWe see that the density-dependent\nterm of SEI contains the factor (1+b$\\rho$)$^\\gamma$ in the denominator, \nwhere the parameter $b$ is fixed to prevent the supra-luminous behavior in nuclear matter\nat high densities~\\cite{Behera_1997}.\nThe study of asymmetric \nnuclear matter involves altogether nine parameters, \nnamely, $\\gamma$, $b$, $\\varepsilon_{0}^{l}$, $\\varepsilon_{0}^{ul}$,\n$\\varepsilon_{\\gamma}^{l}$,$\\varepsilon_{\\gamma}^{ul}$, $\\varepsilon_{ex}^{l}$,\n$\\varepsilon_{ex}^{ul}$ and $\\alpha$, whose connection to the interaction \nparameters is given in the following pages.\nHowever, the description of SNM requires only the following combinations\nof the strength parameters in the like and unlike channels\n\\begin{eqnarray}\n\\left(\\frac{\\varepsilon_{0}^{l}+\\varepsilon_{0}^{ul}}{2}\\right)=\\varepsilon_0,\n\\left(\\frac{\\varepsilon_{\\gamma}^{l}+\\varepsilon_{\\gamma}^{ul}}{2}\\right)=\\varepsilon_{\\gamma},\n\\left(\\frac{\\varepsilon_{ex}^{l}+\\varepsilon_{ex}^{ul}}{2}\\right)=\\varepsilon_{ex},\n\\label{eq.18}\n\\end{eqnarray}\ntogether with $\\gamma$, $b$ and $\\alpha$, \ni.e., altogether six parameters.\nThe coefficients are fitted, considering $\\gamma$ as a free parameter, to properties of finite nuclei, such as \nthe nucleon mass, saturation density or \nbinding energy per particle at saturation.\nMoreover, it is required that \n the nuclear mean-field in SNM at saturation density \nvanishes for a kinetic energy of the nucleon of $300$ MeV, a result extracted from optical \nmodel analysis\nof nucleon-nucleus data~\\cite{behera98}. \nThe splitting of $\\varepsilon_{ex}$ into\n$\\varepsilon_{ex}^{l}$ and $\\varepsilon_{ex}^{ul}$ is decided from the \ncondition that the entropy density in pure neutron matter should not exceed \nthat of the symmetric nuclear matter, prescribing a critical value of\n$\\varepsilon_{ex}^{l}=2\\varepsilon_{ex}\/3$~\\cite{Behera_2011}. \nThe splitting of the remaining two strength parameters \n$\\varepsilon_{\\gamma}$ and $\\varepsilon_{0}$,\nis obtained from the values of the symmetry energy and \nits derivative with respect to the density\nat saturation density $\\rho_0$.\nIn our study, we will also consider the slope parameter $L$ as a free parameter. \n\n\nIn this work we will mostly use the Yukawian form factor $v(r)=\\frac{e^{-\\mu r}}{ \\mu r}$ for the SEI force.\nIn this case, if we compare the MDI interaction defined in Eq.~(\\ref{VMDI}) and the SEI interaction given in Eq.~(\\ref{VSEI}),\nwe can write the expressions of the \nenergy in a general way to useful for both MDI and SEI model with the following caveats in mind:\n\\begin{itemize}\n \\item Firstly, MDI interactions only have a zero-range density-dependent term, contrary to SEI interactions that have two \nzero-range terms. Therefore, in the following expressions, we will consider, for MDI the parameters $t_0=0$ and $x_0=0$.\n\\item The expression for the SEI the zero-range density-dependent term is defined differently with \nrespect to the MDI one:\n\\begin{equation}\n V_\\mathrm{SEI}^{zr} (\\mathbf{r}_1 , \\mathbf{r}_2) = \\frac{V_\\mathrm{MDI}^{zr} (\\mathbf{r}_1 , \\mathbf{r}_2)}{(1+b\\rho({\\bf R}))^\\gamma}.\n\\end{equation}\nFor this reason, from here on, we will consider the value of $b=0$ when using the MDI force.\n\\item The Yukawian form factor for SEI interaction is defined in dimensionless form, \n\\begin{equation}\n v(r) = \\frac{e^{-\\mu r}}{ \\mu r}.\n\\end{equation}\nOn the other hand, the MDI form factor has units of $\\mathrm{fm}^{-1}$, i.e., \n\\begin{equation}\n v(r) = \\frac{e^{-\\mu r}}{r}.\n\\end{equation}\nThus, the contribution of the potential part of the SEI interactions will be divided with an additional parameter $\\mu$ \nwith respect to the contribution given by the MDI interaction. We will further emphasize this point \nwhen we define the energy, pressure and chemical potentials in the following lines.  \n\\end{itemize} \n\nHence, from here on, we present the expressions of different properties of both symmetric and asymmetric nuclear matter for the case of SEI interactions. \nTaking into account the stated differences between the SEI and MDI functionals, one can use the same \nequations to describe the corresponding properties for the case of MDI interactions\n\\footnote{Notice that a similar procedure could be applied for Gogny interactions and for \nSEI interactions with a Gaussian form factor instead of the Yukawa form factor considered here.}.\n\nIn the limit of an infinite symmetric nuclear system $\\rho_n=\\rho_p$, the energy per particle of SEI (MDI) interactions can be written as~\\cite{routray16}\n\\begin{eqnarray}\n E_b (\\rho) &=& \\frac{3 \\hbar^2 k_F^2}{10 m } + \\frac{\\left(\\varepsilon_0^l + \\varepsilon_0^{ul}\\right)}{4 \\rho_0} \\rho +\n \\frac{\\left(\\varepsilon_\\gamma^l + \\varepsilon_\\gamma^{ul} \\right)}{4 \\rho^{\\gamma+1}} \\rho \\left(\\frac{\\rho}{1+ b \\rho} \\right)^\\gamma\n + \\frac{(\\varepsilon_{ex}^l+\\varepsilon_{ex}^{ul})}{4 \\rho_0} \\rho\\mathcal{A} (x_F), \\nonumber\n\\end{eqnarray}\nwhere the exchange term $\\mathcal{A}(x_F)$ is\n\\begin{equation}\n \\mathcal{A} (x_F)=  \\left(\\frac{3}{32 x_F^6} + \\frac{9}{8 x_F^4} \\right)\\ln (1+4 x_F^2)\n -\\frac{3}{8x_F^4} + \\frac{9}{4 x_F^2} - \\frac{3}{x_F ^3} \\tan^{-1} (2 x_F).\n\\end{equation}\n The dimensionless quantity $x_F$ is defined as $x_F = k_F\/\\mu$, being $k_F=(3 \\pi^2\\rho\/2 )^{1\/3}$ the Fermi momentum of the system.\n \nOn the other hand, in an infinite pure neutron matter system, that is, with $\\rho= \\rho_n$ and $\\rho_p=0$,\nthe energy per particle will be defined as\n\\begin{eqnarray}\\label{SEIe0}\n E_b (\\rho, \\delta=1) &=& \\frac{3 \\hbar^2 k_{Fn}^2}{10 m} + \\frac{\\varepsilon_0^l}{2 \\rho_0} \\rho + \\frac{\\varepsilon_\\gamma^l}{2 \\rho_0^{\\gamma+1}}\n \\rho \\left( \\frac{\\rho}{1+ b \\rho} \\right)^\\gamma\n + \\frac{\\varepsilon_{ex}^l}{2 \\rho_0} \\rho  \\mathcal{B} (x_F),\n\\end{eqnarray}\nwith\n\\begin{equation}\\label{SEIen}\n \\mathcal{B} (x_n) =  \\left( \\frac{3}{32 x_n^6}  + \\frac{9}{8 x_n^4}\n \\right) \\ln (1+4 x_n^2) - \\frac{3}{8 x_n^4} + \\frac{9}{4 x_n^2 } - \\frac{3}{x_n}^3 \\mathrm{tan}^{-1} (2x_n).\n\\end{equation}\nIn this case, $x_n$ is defined as $x_n= k_{Fn}\/\\mu$, with $k_{Fi}= (3 \\pi^2 \\rho_i)^{1\/3}$ ($i=n,p$) the Fermi momentum of each type of nucleon.\n\nThe coefficients $\\varepsilon_i^{l (ul)}$ include the parameters of the interaction in the following way\n\\begin{eqnarray}\n \\varepsilon_{0}^l &=& \\frac{t_0}{2} \\rho_0 (1-x_0) +\\left( W+ \\frac{B}{2} -H-\\frac{M}{2} \\right)\\rho_0 \\int v(r) d^3 r\n \\\\\n \\varepsilon_{0}^{ul} &=& \\frac{t_0}{2} \\rho_0 (2+x_0)+ \\left( W+ \\frac{B}{2} \\right)\\rho_0 \\int v(r) d^3 r\n \\\\\n \\varepsilon_{\\gamma}^l &=&  \\frac{t_3}{12} \\rho_0^{\\gamma+1} (1-x_3)\n \\\\\n \\varepsilon_{\\gamma}^{ul} &=& \\frac{t_3}{12} \\rho_0^{\\gamma+1} (2+x_3)\n \\\\\n \\varepsilon_{ex}^l &=&   \\left( M + \\frac{H}{2} -B - \\frac{W}{2} \\right)\\rho_0 \\int v(r) d^3 r\n \\\\\n \\varepsilon_{ex}^{ul} &=&  \\left(M + \\frac{H}{2} \\right) \\rho_0 \\int v(r) d^3 r.\n\\end{eqnarray}\nNotice that the integration of the form factor of MDI interactions is\n\\begin{equation}\\label{ffMDI}\n \\int v(r) d^3 r = \\int \\frac{e^{-\\mu r}}{r} d^3r= \\frac{4\\pi^2}{\\mu^2} ,\n\\end{equation}\nwhereas for SEI interactions is\n\\begin{equation}\\label{ffS}\n \\int v(r) d^3 r =\\int \\frac{e^{-\\mu r}}{\\mu r} d^3r= \\frac{4\\pi^2}{\\mu^3}.\n\\end{equation}\nHence, defining the expressions of the energy using the  parameters $\\varepsilon_i^{l (ul)}$\nallows one to write the properties of nuclear matter in a general form for both MDI and SEI  without the worry\nof the definition of the form factor, which is \ntaken into account only when obtaining the $\\varepsilon_i^{l (ul)}$ parameters.\n\nThe pressure in SNM will be given by \n\\begin{eqnarray}\n P (\\rho) &=& \\frac{\\hbar^2 k_F^2\\rho}{5m } + \\rho^2\\frac{\\varepsilon_0^l+\\varepsilon_0^{ul}}{4 \\rho_0} + \n \\frac{\\varepsilon_\\gamma^l+\\varepsilon_\\gamma^{ul}}{4 \\rho_0^{\\gamma+1}}\\rho^2 \\left[  \\frac{(\\gamma+1) \\rho^{\\gamma}}{(1+b\\rho)^{\\gamma}} \n-\\frac{\\gamma b \\rho^{\\gamma+1}}{(1+b\\rho)^{\\gamma+1}}\\right] \n \\nonumber\\\\\n &+&\\frac{\\varepsilon_{ex}^l+\\varepsilon_{ex}^{ul}}{4 \\rho_0} \\rho^2 \\left[ \\mathcal{A}(x_F) + \\frac{k_F}{3 \\mu} \\frac{\\partial A(x_F)}{\\partial x_F}  \\right]\n\\end{eqnarray}\nand its nuclear matter incompressibility is defined as\n\\begin{eqnarray}\n K (\\rho) &=&  -\\frac{3\\hbar^2 k_F^2}{5 m} +  \\frac{\\varepsilon_\\gamma^l +\\varepsilon_\\gamma^{ul}}{4 \\rho_0^{\\gamma+1}}  \\rho^2\n \\left[ \\frac{\\gamma(\\gamma+1) \\rho^{\\gamma-1}}{(1+b \\rho)^{\\gamma}} - \\frac{2 b \\gamma (\\gamma+1) \\rho^\\gamma}{(1+b \\rho)^{\\gamma+1}} + \n \\frac{\\gamma b^2 (\\gamma+1) \\rho^{\\gamma+1}}{(1+b \\rho)^{\\gamma+2}}\\right] \\nonumber\\\\\n &+& \n \\frac{\\varepsilon_{ex}^l + \\varepsilon_{ex}^{ul}}{4 \\rho_0} \\left[ \\frac{4 k_F \\rho}{\\mu}\\frac{\\partial \\mathcal{A}(x_F)}{\\partial x_F}\n + \\frac{k_F^2 \\rho}{\\mu^2}  \\frac{\\partial^2 \\mathcal{A}(x_F)}{\\partial x_F^2}\\right].\n\n\n\n\\end{eqnarray}\nThe derivatives of $\\mathcal{A} (x_F)$  appearing in the expressions for the pressure $ P (\\rho) $ and incompressibility \n$ K (\\rho)$ are given by\n\\begin{eqnarray}\n\\frac{\\partial \\mathcal{A}}{\\partial x_F} (x_F)&=& -9 \\left[ \\left( \\frac{1}{16 x^7} + \\frac{1}{2 x_F^5} \\right) \n\\ln (1+4x_F^2) - \\frac{1}{4 x_F^5} + \\frac{1}{2 x_F^3} - \\frac{1}{x_F^4} \\mathrm{tan}^{-1} (2 x_F)\n\\right]\\nonumber\n\\\\\n\\\\\n \\frac{\\partial^2 \\mathcal{A}}{\\partial x_F^2} (x_F) &=& 9 \\left[ \n \\left( \\frac{7}{16x_F^8} + \\frac{5}{2 x_F^6}\\right) \\ln (1+4 x_F^2) - \\frac{7}{16 x_F^6} + \\frac{3}{2 x_F^4} - \n \\frac{4}{x_F^5} \\mathrm{tan}^{-1} (2 x_F)\n \\right].\\nonumber\\\\\n\\end{eqnarray}\n\nOn the other hand, in an asymmetric nuclear system with $\\rho_n \\neq \\rho_p$, \nthe energy per particle is given by~\\cite{routray16}\n\\begin{eqnarray}\n E_b (\\rho_n, \\rho_p) &=& \\frac{3 \\hbar^2}{10 m } \\left( k_n^2 \\frac{\\rho_n}{\\rho} + k_p^2 \\frac{\\rho_p}{\\rho} \\right) + \n \\frac{\\varepsilon_0^l}{2\\rho_0 } \\left( \\frac{\\rho_n^2}{\\rho} + \\frac{\\rho_p^2}{\\rho} \\right) + \\frac{\\varepsilon_0^{ul}}{\\rho_0} \\frac{\\rho_n \\rho_p}{\\rho}\n \\nonumber \\\\\n &+& \\left[ \\frac{\\varepsilon_\\gamma^l}{2\\rho_0^{\\gamma +1} } \\left( \\frac{\\rho_n^2}{\\rho} + \\frac{\\rho_p^2}{\\rho} \\right) + \n \\frac{\\varepsilon_\\gamma^{ul}}{\\rho_0} \\frac{\\rho_n \\rho_p}{\\rho} \\right] \\left( \\frac{\\rho}{1+ b \\rho} \\right)^\\gamma\n + \\frac{\\varepsilon_{ex}^l}{2 \\rho_0} \\left( \\frac{\\rho_n^2}{\\rho} J(k_{Fn}) +  \\frac{\\rho_p^2}{\\rho} J(k_{Fp}) \\right)\n \\nonumber\\\\\n &+& \\frac{\\varepsilon_{ex}^{ul}}{2 \\rho_0} \\frac{1}{\\pi^2} \\left[\\frac{\\rho_n}{\\rho} \\int_0^{k_{Fp}} I (k, k_{Fn}) k^2 dk +\n \\frac{\\rho_p}{\\rho} \\int_0^{k_{Fn}} I (k, k_{Fp}) k^2 dk\\right] ,\n\\end{eqnarray}\nwith\n\\begin{equation}\n J(k_{Fi}) = \\left( \\frac{3}{32 x_i^6} + \\frac{9}{8 x_i^4}\\right)\\ln (1+4 x_i^2) - \\frac{3}{8 x_i^4} + \\frac{9}{4 x_i^2}\n - \\frac{3}{x_i^3} \\mathrm{tan}^{-1} (2 x_i)\n\\end{equation}\nand\n\\begin{equation}\n I (k, k_{Fi}) = \\frac{3 (1+ x_i^2 - x^2)}{8 x_i^3 x} \\ln \\left[ \\frac{1 + (x+x_i)^2}{1+ (x-x_i)^2} \\right]\n + \\frac{3}{2 x_i^2} - \\frac{3}{2 x_i^3} \\left[ \\mathrm{tan}^{-1} (x+x_i) - \\mathrm{tan}^{-1} (x-x_i) \\right].\n\\end{equation}\n\n\nThe neutron and proton chemical potentials are given by~\\cite{routray16}\n\\begin{eqnarray}\n \\mu_\\tau (\\rho_\\tau, \\rho_{\\tau'})&=& \\frac{\\hbar^2 k_{F \\tau}}{2m} \\left[  \\frac{\\varepsilon_0^l}{\\rho_0} + \\frac{\\varepsilon_\\gamma^l}{\\rho_0^{\\gamma+1}} \\left( \n \\frac{\\rho}{1+b \\rho}\\right)^\\gamma\\right] \\rho_\\tau + \\left[  \\frac{\\varepsilon_0^{ul}}{\\rho_0} + \\frac{\\varepsilon_\\gamma^{ul}}{\\rho_0^{\\gamma+1}} \\left( \n \\frac{\\rho}{1+b \\rho}\\right)^\\gamma\\right] \\rho_{\\tau'} \\nonumber \\\\\n &+& \\varepsilon_{ex}^l \\frac{\\rho_\\tau}{\\rho_0} \\left[ \\frac{3 \\ln (1+4 x_{\\tau}^2)}{8 x_\\tau^4} + \\frac{3}{2 x_\\tau^2} -\n \\frac{3 \\mathrm{tan}^{-1} (2 x_\\tau)}{2 x_\\tau^3}\\right] \\nonumber\\\\\n &+& \\varepsilon_{ex}^{ul} \\frac{\\rho_{\\tau'}}{\\rho_0} \\left[ \\frac{3 (1+x_{\\tau'}^2-x_\\tau^2)}{8 x_\\tau x_{\\tau'}^3}\n \\ln \\left[ \\frac{1+ (x_{\\tau} + x_{\\tau'})^2}{1+ (x_{\\tau} - x_{\\tau'})^2}\\right] + \\frac{3}{2 x_{\\tau'}^2} \\right.\n \\nonumber\\\\\n &-& \\left.\\frac{3}{2 x_{\\tau'}^3} \\left( \\mathrm{tan}^{-1} (x_\\tau+x_{\\tau'}) -\\mathrm{tan}^{-1} (x_\\tau-x_{\\tau'}) \\right)\\right] \\nonumber\\\\\n &+& \\left[ \\frac{\\varepsilon_\\gamma^l (\\rho_n^2+\\rho_p^2)}{2 \\rho_0^{\\gamma+1}} + \\frac{\\varepsilon_\\gamma^{ul}(\\rho_n \\rho_p)}{\\rho_0^{\\gamma+1}} \\right]\n \\frac{\\gamma \\rho^{\\gamma-1}}{(1+b \\rho)^(\\gamma+1)},\n\\end{eqnarray}\nwhere $\\tau$ and $\\tau'$ refer to either protons or neutrons.\n\nIf the energy per particle of asymmetric matter, rewritten in terms of the total density $\\rho$ and the isospin \nasymmetry $\\delta$, is expanded around $\\delta=0$ [see Eq.~(\\ref{eq:eosexp})], the symmetry energy coefficient of the system\nfor SEI (MDI) interactions is defined as \n\\begin{eqnarray}\n E_\\mathrm{sym} (\\rho) &=& \n \\frac{\\hbar^2 k_F^2}{6 m} +\n  \\frac{\\rho}{4} \\left( \\frac{\\varepsilon_0^l - \\varepsilon_0^{ul}}{\\rho_0} \\right)\n  \\nonumber\\\\\n&+& \\frac{\\rho}{4} \\left( \\frac{\\varepsilon_\\gamma^l - \\varepsilon_\\gamma^{ul}}{\\rho_0^{\\gamma+1}} \\right)\n \\left( \\frac{\\rho}{1+b\\rho} \\right)^\\gamma \n +\\frac{\\rho}{4} \\left(\\frac{\\varepsilon_{ex}^l - \\varepsilon_{ex}^{ul}}{\\rho_0} \\right) \\mathcal{C} (x_F)\n \\nonumber\\\\\n &-& \\frac{\\rho}{4} \\left(  \\frac{\\varepsilon_{ex}^l + \\varepsilon_{ex}^{ul}}{\\rho_0}\\right) \\mathcal{D} (x_F),\n\\end{eqnarray}\nwhere \n\\begin{equation}\n \\mathcal{C} (x_F) =   \\frac{\\ln(1+4x_F^2)}{4 x_F^2}\n\\end{equation}\nand\n\\begin{equation}\n \\mathcal{D} (x_F)=\n  \\left( \n \\frac{1}{4 x_F^2} + \\frac{1}{8 x_F^4}\\right) \\ln(1+4 x_F^2) - \\frac{1}{2x_F^2} .\n\\end{equation}\nThe slope of the symmetry energy $L$ for SEI (MDI) interactions reads\n\\begin{eqnarray}\n L  &=&  \n \\frac{\\hbar^2 k_F^2}{3 m} + \\frac{3}{4} \\rho_0 \\left( \\frac{\\varepsilon_0^l- \\varepsilon_0^{ul}}{\\rho_0} \\right) \\nonumber\\\\\n  &+& \\frac{3}{4} \\rho_0 \\left( \\frac{\\varepsilon_\\gamma^l- \\varepsilon_\\gamma^{ul}}{\\rho_0^{\\gamma+1}} \\right)  \n  \\left[ \\frac{(\\gamma+1) \\rho_0^\\gamma}{(1+b\\rho_0)^\\gamma}-\n  \\frac{b \\gamma \\rho_0^{\\gamma+1}}{(1+b\\rho_0)^{\\gamma+1}}\\right] \\nonumber\\\\\n  &+& \\frac{3}{4} \\rho_0\\left( \\frac{\\varepsilon_{ex}^l- \\varepsilon_{ex}^{ul}}{\\rho_0} \\right) \\left[ \\mathcal{C}(x_F) \n  + \\left.\\frac{k_F}{3 \\mu}\\frac{\\partial \\mathcal{C}(x_F)}{\\partial x_F}\\right|_{\\rho_0}\n \\right] \\nonumber\\\\\n &-&  \n  \\frac{3}{4} \\rho_0\\left( \\frac{\\varepsilon_{ex}^l+ \\varepsilon_{ex}^{ul}}{\\rho_0} \\right) \\left[ \\mathcal{D}(x_F) +  \n  \\left.\\frac{k_F}{3 \\mu} \\frac{\\partial \\mathcal{D}(x_F)}{\\partial x_F}\\right|_{\\rho_0}\n  \\right]\\nonumber\\\\\n\\end{eqnarray}\nand the isospin curvature of the system is \n\\begin{eqnarray}\n K_\\mathrm{sym}  &=& -\\frac{\\hbar^2 k_F^2}{3m } + \\frac{9}{4} \\rho_0^2 \n \\left( \\frac{\\varepsilon_\\gamma^l- \\varepsilon_\\gamma^{ul}}{\\rho_0} \\right)\\nonumber\\\\\n &\\times&\\left[  \\frac{\\gamma (\\gamma+1) \\rho_0^{\\gamma-1}}{(1+b \\rho_0)^{\\gamma}} - \n \\frac{2 \\gamma b (\\gamma+1) \\rho_0^\\gamma}{(1+b \\rho_0)^{\\gamma+1}} + \\frac{b^2 \\gamma (\\gamma+1) \n \\rho_0^{\\gamma+1}}{(1+b\\rho_0)^{\\gamma+2}}\\right]\\nonumber\\\\\n &+& \\frac{1}{4} \\left( \\frac{\\varepsilon_{ex}^l- \\varepsilon_{ex}^{ul}}{\\rho_0} \\right) \\left[ \\frac{4 k_{F0} \\rho_0 }{\\mu}\n \\left.\\frac{\\partial \\mathcal{C}(x_F)}{\\partial x_F}\\right|_{\\rho_0}\n + \\frac{k_{F0}^2 \\rho_0}{\\mu^2} \\left.\\frac{\\partial^2 \\mathcal{C}(x_F)}{\\partial x_F^2}\\right|_{\\rho_0} \\right]\\nonumber\\\\\n  &+& \\frac{1}{4} \\left( \\frac{\\varepsilon_{ex}^l- \\varepsilon_{ex}^{ul}}{\\rho_0} \\right)\n  \\left[ \\frac{4 k_{F0} \\rho_0 }{\\mu} \\left.\\frac{\\partial \\mathcal{D}(x_F)}{\\partial x_F}\\right|_{\\rho_0}\n + \\frac{k_{F0}^2 \\rho_0}{\\mu^2} \\left.\\frac{\\partial^2 \\mathcal{D}(x_F)}{\\partial x_F^2}\\right|_{\\rho_0} \\right],\n\n\n\n\n\n\n\\end{eqnarray}\nwhere $k_{F0}$ is the Fermi momentum of the system calculated at the saturation density $\\rho_0$.\nThe derivatives of the expressions $\\mathcal{C} (x_F)$ and $\\mathcal{D} (x_F)$ needed to implement $L$ and $K_\\mathrm{sym}$\nare, respectively, \n\\begin{eqnarray}\n \\frac{\\partial \\mathcal{C}(x_F)}{\\partial x_F}  &=& \\frac{2}{4 x_F^3 + x_F} - \\frac{1}{2 x_F^3} \\ln (1+4 x_F^2)\n\\\\\n \\frac{\\partial^2 \\mathcal{C}(x_F)}{\\partial x_F^2}  &=& \\frac{3}{2 x_F^4} \\ln (1+4x_F^2) - \\frac{40 x_F^2 +6}{(4 x_F^3+x_F)^2}\n\\\\\n \\frac{\\partial \\mathcal{D}(x_F)}{\\partial x_F}  &=& \\frac{6 x_F^2+2}{4 x_F^5+x_F^3} - \\frac{x_F^2+1}{2x_F^5} \\ln (1+4x_F^2)\n\\\\\n \\frac{\\partial^2\\mathcal{D}(x_F)}{\\partial x_F^2}  &=& -\\frac{2 (5+33x_F^2+44 x_F^4)}{x_F^4 (1+4x_F^2)^2} + \\frac{5+3x_F^2}{2 x_F^6} \\ln (1+4x_F^2).\n\\end{eqnarray}\n\n\\chapter{Asymmetric nuclear matter studied with  Skyrme and Gogny interactions}\\label{chapter2}\nThe EoS of asymmetric nuclear matter (ANM) is not completely established\nand, sometimes, it is not trivial to compute as, for example, in the case\nof finite-range interactions, such as Gogny, MDI or SEI forces.\nIn order to prove the main features of the isospin dependence of the EoS, it can useful to expand the energy per particle\ngiven by such interactions, which may be written in terms of \nthe total density $\\rho$ and the isospin asymmetry $\\delta$\nof the system,\naround asymmetry $\\delta=0$ \\cite{gonzalez17}:\n\\begin{eqnarray}\\label{eq:EOSexpgeneral}\n E_b(\\rho, \\delta) &=& E_b(\\rho, \\delta=0) + E_{\\mathrm{sym}, 2}(\\rho)\\delta^{2} +... \n + E_{\\mathrm{sym}, 2k}(\\rho)\\delta^{2k} + \\mathcal{O}(\\delta^{2k+2}), \\nonumber\\\\\n\\end{eqnarray}\nwith $k \\geq  1 $, and where each symmetry energy coefficient is defined as  \n\\begin{equation}\\label{eq:esymgen}\n \\left. E_{\\mathrm{sym}, 2k} (\\rho) = \\frac{1}{(2k)!} \\frac{\\partial^{2k} E_b(\\rho, \\delta)}\n{\\partial \\delta^{2k}}\\right|_{\\delta=0}.\n\\end{equation} \nThe coefficients in Eq.~(\\ref{eq:EOSexpgeneral}) are directly connected to the \nproperties of the single-nucleon potential \nin ANM~\\cite{Xu11,Chen12}. At low densities, such as the ones found in \nterrestrial nuclei, the system has   \nsmall isospin asymmetry values, \nand the expansion~(\\ref{eq:EOSexpgeneral}) can be cut at second order,  \nneglecting higher-order terms [see Eq.~(\\ref{eq:eosexp}) of Chapter~\\ref{chapter1}].\nThe coefficient $E_\\mathrm{sym,2} (\\rho)$ is the quantity we have previously \ndefined in Eq.~(\\ref{eq:esym}) as\nthe symmetry energy of the system. \nSometimes in the literature, the second-order\nsymmetry energy is also denoted as $S(\\rho)$~\\cite{Sellahewa14,Piekarewicz08}.\nThe second-order symmetry energy coefficient, which we will also refer to as symmetry energy, has been relatively well constrained both \nexperimentally and theoretically at low values of the density below and around the saturation density~\\cite{Danielewicz13,Tsang08,Zhang15, Chen_15}. \n\nOn the other hand, when studying systems with high isospin asymmetry, such as the \ninterior of neutron stars (NSs), where $\\beta$-stable \nnuclear matter shows large differences in its neutron and proton contributions,  \nhigher-order terms in the expansion asymmetry $\\delta$ may play a significant role when obtaining the equation of state \nby adding corrections to the parabolic law \\cite{xu09a,Xu11, Cai2012, Moustakidis12,Seif14, gonzalez17, Liu18}. \n\nTo analyze the density dependence of the symmetry energy, \none can expand the coefficients $E_{\\mathrm{sym},2k} (\\rho)$\naround the saturation density $\\rho_0$ as follows:\n\\begin{equation}\\label{esymexpgen}\nE_{\\mathrm{sym},2k} (\\rho)= E_{\\mathrm{sym}, 2k} (\\rho_0) + L_{2k} \\epsilon + \\mathcal{O}(\\epsilon^2),\n\\end{equation}\nwhere $ \\epsilon = (\\rho - \\rho_ 0)\/3\\rho_0$ is the density displacement from the saturation density $\\rho_0$ and \nthe coefficients $L_{2k}$ are the slope parameters of the symmetry energy coefficients, which are computed at the\nsaturation density as\n\\begin{eqnarray}\\label{eq:L2k} \nL_{2k}\\equiv  L_{2k} (\\rho_0)&=& 3\\rho_0 \\left.\\frac{\\partial E_{\\mathrm{sym}, 2k} (\\rho)}{\\partial \\rho} \\right|_{\\rho_0}.\n\\end{eqnarray}\nRecalling Eq.~(\\ref{eq:EOSexpgeneral}) and the saturation condition of nuclear forces, we see that the density slope at \nsaturation of the energy per particle $E_b (\\rho, \\delta)$ of asymmetric matter can be parametrized as~\\cite{gonzalez17}\n\\begin{equation}\n\\left. \\frac{\\partial E_b (\\rho, \\delta)}{\\partial \\rho} \\right|_{\\rho_0} =\n\\frac{1}{3\\rho_0} \\left( L_2 \\delta^2 + L_4 \\delta^4 + L_6 \\delta^6 + \\cdots \\right) .\n \\label{eq:slope2k}\n\\end{equation}\nIn this notation, $L_2$ is the commonly known slope of the symmetry energy coefficient and we will refer to it as $L$ [cf. Eq.~(\\ref{eq:L})].\n\nOn the other hand, if the $\\delta$-expansion is truncated at second order and one considers an asymmetry $\\delta=1$,  we can \ndefine the symmetry energy as the difference between the energy per particle in neutron matter and in symmetric matter:\n\\begin{equation}\\label{eq:esympa}\n E_{\\mathrm{sym}}^{PA}(\\rho) = E_b(\\rho, \\delta=1)- E_b(\\rho, \\delta=0).\n\\end{equation}\nThis definition of the parabolic symmetry energy can be understood as the energy \ncost for converting all protons into neutrons in SNM, which can be \nuseful in some cases as for instance in microscopic calculations of \nBrueckner-Hartree-Fock type.\nIf one considers the Taylor expansion of the EoS~(\\ref{eq:EOSexpgeneral}), one gets that the parabolic symmetry energy \ncorresponds to the sum of the whole series of the symmetry energy coefficients when the isospin asymmetry is $\\delta=1$, \nassuming that this series is convergent:\n\\begin{equation}  \nE_{\\mathrm{sym}}^{PA}(\\rho)= \\sum_{k=1}^{\\infty} E_{\\mathrm{sym}, 2k} (\\rho).\n\\label{eq:PAgen}\n\\end{equation}\nAnalogously to the definition in Eq.~(\\ref{eq:L2k}), the slope parameter to the symmetry energy using the PA can be computed as \n\\begin{eqnarray}\n   L_{PA} \\equiv L_{PA}(\\rho_0)&=& \\left.3 \\rho_0 \\frac{\\partial E_{\\mathrm{sym}}^{PA} (\\rho)}{\\partial \\rho} \\right|_{\\rho_0} .\n\\end{eqnarray}\n\nRecent calculations in many-body perturbation theory have shown  \nthat the isospin asymmetry expansion~(\\ref{eq:EOSexpgeneral}) may not be convergent at zero temperature \nwhen the many-body corrections beyond the Hartree-Fock mean-field level are incorporated~\\cite{Wellenhofer2016}. \nWe do not deal with this complication here since we will be working at the Hartree-Fock level, \nwhere no non-analyticities are found in the equation of state.\n\nIn this Chapter, we will analyze the behaviour of higher-order terms in the Taylor expansion  \nof different Skyrme and Gogny EoSs, and their influence when studying $\\beta$-equilibrated matter. \nAlso, we will check how the isovector characteristics of the EoS affect the \nrelation between the mass and the radius of NSs. \n\\section{Convergence of the isospin Taylor expansion of the EoS for Skyrme interactions}\nIn this Section, we compute the contributions to the symmetry energy up to \n$10^\\mathrm{th}$ order in the expansion of the energy per particle~(\\ref{eq:EOSexpgeneral}) for Skyrme interactions, and we \nstudy their influence on the ANM EoS. \nApplying Eq.~(\\ref{eq:esymgen}) to Skyrme forces, the coefficients up to second-, fourth-, sixth-, eighth- and tenth- order in the expansion of the EoS\nfor Skyrme interactions are obtained as\n\\begin{eqnarray}\nE_{\\mathrm{sym}, 2} (\\rho)&=&\\frac{\\hbar^2}{6m} \\left(  \\frac{3 \\pi^2}{2}\\right)^{2\/3} \\rho^{2\/3} - \\frac{1}{8}\nt_0 \\rho (2x_0 +1) \n- \\frac{1}{48} t_3 \\rho^{\\sigma+1} (2x_3 +1) \\nonumber \n\\\\\n&+& \\frac{1}{24} \\left(  \\frac{3 \\pi^2}{2}\\right)^{2\/3}   \n\\times \\rho^{5\/3} \\left[ -3 x_1 t_1 + t_2 (5 x_2 + 4) \\right], \\label{eq:esym2skyrme2}\n\\\\\nE_{\\mathrm{sym}, 4}(\\rho) &=& \\frac{\\hbar^2}{162m} \\left(  \\frac{3 \\pi^2}{2}\\right)^{2\/3} \\rho^{2\/3} +\n\\frac{1}{648} \\left(  \\frac{3 \\pi^2}{2}\\right)^{2\/3}  \n \\nonumber\\\\\n &\\times& \\rho^{5\/3} \\left[ 3 t_1 (x_1 +1) - t_2 ( x_2 -1) \\right], \n\\\\\nE_{\\mathrm{sym}, 6} (\\rho)&=&  \\frac{7\\hbar^2}{4374m} \\left(  \\frac{3 \\pi^2}{2}\\right)^{2\/3} \\rho^{2\/3} +\n\\frac{7}{87840} \\left(  \\frac{3 \\pi^2}{2}\\right)^{2\/3}  \n\\nonumber\\\\\n&\\times& \\rho^{5\/3} \\left[  t_1 (9x_1 +12) + t_2 ( x_2 +8) \\right], \n\\\\\n E_{\\mathrm{sym}, 8}(\\rho) &=&  \\frac{13\\hbar^2}{19683m} \\left(  \\frac{3 \\pi^2}{2}\\right)^{2\/3} \\rho^{2\/3}+ \n \\frac{13}{314928} \\left(  \\frac{3 \\pi^2}{2}\\right)^{2\/3} \n\\nonumber \\\\\n &\\times&  \\rho^{5\/3} \\left[ 3 t_1 (2x_1 +3)  + t_2 (2  x_2 +7) \\right],\n\\\\\nE_{\\mathrm{sym},10} (\\rho)&=&  \\frac{2717\\hbar^2}{7971615m} \\left(  \\frac{3 \\pi^2}{2}\\right)^{2\/3} \\rho^{2\/3}+\n\\frac{247}{31886460} \\left(  \\frac{3 \\pi^2}{2}\\right)^{2\/3} \\nonumber\n\\\\\n& \\times& \\rho^{5\/3} \\left[t_1 (15x_1 +24) + t_2 (7  x_2 +20) \\right]. \\label{eq:esym10skyrme}\n\\end{eqnarray}\nNote that, in contrast to $E_{\\mathrm{sym},2} (\\rho)$, the higher-order symmetry energy coefficients, i.e, \n$E_{\\mathrm{sym}, 4,6,8,10} (\\rho)$, arise exclusively from the kinetic term and from the momentum-dependent term \nof the interaction, which in Skyrme forces is the term with the usual $t_1$ and $t_2$ parameters \\cite{sly41,sly42}.\n\nWe plot in Fig.~\\ref{fig:esym2} the second-order symmetry energy coefficient against the density of the system \nfor the MSk7, UNEDF0, SkM$^*$, SLy4 and SkI5 Skyrme interactions. To carry out our study, we have \nchosen these parametrizations as representative ones\nbecause they cover a wide range of values of the slope of the symmetry energies $L$, from $L=10$ MeV to $L=130$ MeV. \nIn the same figure, we plot various existing empirical constraints for the symmetry energy at\nsubsaturation density \\cite{Danielewicz13,Zhang15,Tsang08, Chen_15}.\nAt subsaturation densities, the considered Skyrme forces fit inside the majority of the \nconstraints. On the other hand, we see that at higher densities, neither the MSk7 interaction, with \na very small value of $L$ and with a soft EoS, nor the SkI5 interaction, with a very large $L$ value and a stiff EoS,\nfit inside the bands coming from systematics at high density~\\cite{Chen_15}.  \n\nAt low densities below $\\rho \\sim0.1$ fm$^{-3}$ the coefficients $E_{\\mathrm{sym},2} (\\rho)$ of all interactions behave in a\nsimilar way. The reason behind it is the fact that this is the density regime where experimental \ndata of finite nuclei exist, and which has been used to fit the majority of the interactions. \nThe second-order symmetry energy of all interactions intersect at a density around $\\rho \\sim0.1$ fm$^{-3}$\nand from this density onwards they show a more model-dependent behaviour. \nSome of them show an increasing trend as a function of the density, such as the ones \ncalculated with the SkI5 and SLy4 interactions. However, \nthere are other parametrizations, such as the MSk7, UNEDF0 and SkM$^*$ forces,\nwhose symmetry energy coefficients $E_{\\mathrm{sym},2} (\\rho)$ \nreach a maximum and then decrease until vanishing. This implies an isospin instability, as \nthe energy per particle in neutron matter becomes more bound than in SNM. \n\\begin{figure}[t]\n \\centering\n \\includegraphics[width=0.8\\linewidth, clip=true]{.\/grafics\/chapter2\/skyrme\/Esym2_vs_rho}\n  \\caption{Density dependence of the second-order symmetry energy coefficient \n  $E_{\\mathrm{sym}, 2}(\\rho)$ for the Skyrme \nforces MSk7 ($L=9.41$ MeV), UNEDF0 ($L=45.08$ MeV), SkM$^*$ ($L=45.78$ MeV), SLy4 ($L=45.96$ MeV) and SkI5 ($L=129.33$ MeV).\nAlso represented are the symmetry energy constraints extracted from the analysis of data on isobaric analog states (IAS)~\\cite{Danielewicz13}\nand of IAS data combined with neutron skins (IAS+n.skin)~\\cite{Danielewicz13}, \nthe constraints from the electric dipole polarizability in lead ($\\alpha_D$ in $^{208}$Pb)~\\cite{Zhang15},\n from transport simulations of heavy-ion collisions of tin isotopes (HIC)~\\cite{Tsang08} and from systematics of the symmetry energy at high densities~\\cite{Chen_15}.}\n \\label{fig:esym2}\n\\end{figure}\n\nIn Fig.~\\ref{fig:esym2k} we plot the fourth-, sixth-, eighth- and tenth-order symmetry energy coefficients \nfor the same interactions as in Fig.~\\ref{fig:esym2}. As happens with the second-order\nsymmetry energy coefficient, the behaviour of the higher-order coefficients is model-dependent.\nThe interaction SkI5 is, of the ones plotted, the one that has the stiffest second-order symmetry energy at saturation, with \na slope parameter of $L=129.33$ MeV. \nHowever, its fourth-order symmetry energy coefficient bends and vanishes at a density\n$\\rho \\sim0.2$ fm$^{-3}$, giving a small or even negative contribution to the expansion of the symmetry energy.\nThe $E_{\\mathrm{sym},6} (\\rho)$ coefficient for SkI5 has a similar behaviour as its $E_{\\mathrm{sym},4} (\\rho)$.\nIt bends at $\\rho \\sim0.4$ fm$^{-3}$\nand becomes negative at densities larger than the ones considered in the plot. \nIn this case, the sixth-order contribution may have a larger impact on the expansion of the EoS, at least for \nvalues of isospin asymmetry $\\delta$ close to the unity. \nThe interactions SLy4, SkM$^*$ and UNEDF0 have similar values of the slope of the symmetry energy $L$, which are,\nrespectively, $L=45.96$ MeV, $L=45.78$ MeV and $L= 45.08$ MeV. However, \nthe $E_{\\mathrm{sym},4} (\\rho)$ of the UNEDF0 and SkM$^*$ interactions are very stiff having similar trends,\nwhile the $E_{\\mathrm{sym},4} (\\rho)$ coefficient of the \nSLy4 interaction bends at a certain density $\\rho \\sim 0.45$ fm$^{-3}$ and then \ndecreases, becoming negative at a density larger than the ones shown in the figure. This scenario is \ndifferent from the one found in Fig.~\\ref{fig:esym2}, \nwhere the $E_{\\mathrm{sym},2} (\\rho)$ of the SLy4 interaction does not bend, while the ones of the UNEDF0 and SkM$^*$\nmodels bend at relatively small densities, presenting isospin instabilities. \nThe sixth-order symmetry energy coefficients of these three interactions increase at all values of the density \nthat are considered in the plot. \nFor the MSk7 interaction, which has a small value of the slope parameter of $L=9.41$ MeV, we find a fourth-order \nsymmetry energy coefficient that is rather stiff inside the range \nof densities considered, and a sixth-order coefficient that is also positive in this same density regime. \nThe $E_{\\mathrm{sym},8} (\\rho)$ and $E_{\\mathrm{sym},10} (\\rho)$ coefficients of all the above interactions are \npositive and do not bend inside the range of densities up to $ \\rho =1$ fm$^{-3}$.\nAt subsaturation densities the $E_{\\mathrm{sym},8} (\\rho)$ coefficient has values below $\\sim 0.08$ MeV and the $E_{\\mathrm{sym},10} (\\rho)$ \ndo not exceed values of $\\sim 0.05$ MeV.\n\\begin{figure}[t]\n \\centering\n \\includegraphics[width=1\\linewidth, clip=true]{.\/grafics\/chapter2\/skyrme\/esym_orders_2x2}\n  \\caption{Density dependence of the fourth-, sixth-, eight- and tenth-order symmetry energy coefficients, \n  $E_{\\mathrm{sym}, 4}(\\rho)$, $E_{\\mathrm{sym}, 6}(\\rho)$, $E_{\\mathrm{sym}, 8}(\\rho)$ and $E_{\\mathrm{sym}, 10}(\\rho)$ respectively, for the Skyrme \nforces MSk7, UNEDF0, SkM$^*$, SLy4 and SkI5.}\n \\label{fig:esym2k}\n\\end{figure}\n\n\\begin{figure}[t]\n \\centering\n \\includegraphics[width=1\\linewidth, clip=true]{.\/grafics\/chapter2\/skyrme\/Esym4Esym6Esym8Esym10Esym2_Skyrme}\n  \\caption{Ratios $E_{\\mathrm{sym},2k}(\\rho)\/E_{\\mathrm{sym},2}(\\rho)$ ($k=2,3,4,5$) as a function of the \n  density for the Skyrme forces MSk7, UNEDF0, SkM$^*$, SLy4 and SkI5.}\n \\label{fig:esym2ratio}\n\\end{figure}\nThe density dependence of the \nratios of E$_{\\mathrm{sym}, 4} (\\rho)$,  E$_{\\mathrm{sym}, 6}(\\rho)$, E$_{\\mathrm{sym},8}(\\rho)$ and \nE$_{\\mathrm{sym},10}(\\rho)$ with respect to E$_{\\mathrm{sym}, 2}(\\rho)$ is plotted in Fig.~\\ref{fig:esym2ratio} \nfor the Skyrme forces MSk7, UNEDF0,\nSkM$^*$, SLy4 and SkI5\nup to a density $\\rho=0.4$ fm$^{-3}$.\nAt low densities $\\rho \\sim 0.1$ fm$^{-3}$, the fourth-order symmetry energy \nis not bigger than 3$\\%$ of the symmetry energy at second order, \nand the sixth-, eighth- and tenth-order terms are, respectively, less than $0.8\\%$, $0.3\\%$ and $0.15\\%$.\nHowever, as we go to higher densities, the contributions of these coefficients increase, \nup to the point that for some interactions they may not be negligible. If they are not considered, \nthe calculations of the equation of state may lead to non-realistic results far from the ones \nobtained if one uses the exact expression of the EoS. \nNotice that the sudden increase of the ratios $E_{\\mathrm{sym},2k}(\\rho)\/E_{\\mathrm{sym},2}(\\rho)$ of the MSk7 interaction \nat densities close to $\\rho=0.4$ fm$^{-3}$ is caused by the low values of the second-order symmetry energy coefficient \nat that density regime.\n\n\n\\subsection{Comparison between the parabolic approximation $E_{\\mathrm{sym}}^{PA}(\\rho)$ and the $E_{\\mathrm{sym},2k}(\\rho)$\ncoefficients}\nIn order to analyze up to which extent the parabolic approximation compares to the EoS expansion in asymmetry,\nwe plot in Fig.~\\ref{fig:esympaskyrme} the \nsymmetry energy coefficient $E_{\\mathrm{sym}}^{PA}(\\rho)$ \ncalculated within the parabolic approximation \n with the same Skyrme interactions MSk7, UNEDF0, SkM$^*$, SLy4 and SkI5 against the baryon density.\nWe observe that the behaviour of $E_{\\mathrm{sym}}^{PA}(\\rho)$ is considerably similar in general trends \nto the one we find in Fig.~\\ref{fig:esym2} for $E_{\\mathrm{sym},2}(\\rho)$. \n\\begin{figure}[!t]\n \\centering\n \\includegraphics[width=0.8\\linewidth, clip=true]{.\/grafics\/chapter2\/skyrme\/EsymPA_skyrme}\n  \\caption{Density dependence of the symmetry energy coefficient calculated within the parabolic approximation\n  $E_{\\mathrm{sym}}^{PA}(\\rho)$ for the Skyrme \nforces MSk7, UNEDF0, SkM$^*$, SLy4 and SkI5.}\n \\label{fig:esympaskyrme}\n\\end{figure}\nIn the case of $E_{\\mathrm{sym}}^{PA}(\\rho)$ we see again that the SkI5 and SLy4 interactions show stiff trends, while the \nsofter SkM$^*$, UNEDF0 and MSk7 interactions bend at a certain point, presenting isospin instabilities.  \nTo study the convergence of the series in Eq.~(\\ref{eq:PAgen}), \nwe plot in Fig.~\\ref{fig:esymPAesym2skyrme}\nthe differences for the SLy4 and SkI5 interactions between the symmetry energy calculated with the parabolic approximation, \nand the sum of the symmetry energy coefficients up to a given order, i.e., \n\\begin{equation}\\label{eq:ccoeff}\n C \\left(  \\rho \\right) =  E_\\mathrm{sym}^{PA} (\\rho) - \\sum_{k} E_{\\mathrm{sym}, 2k} (\\rho).\n\\end{equation}\nThe different symmetry energy coefficients entering in the sum of the \nright hand side of Eq.~(\\ref{eq:PAgen}) are calculated using the definition~(\\ref{eq:esymgen}), and the difference between the\nenergies of neutron and symmetric nuclear matter is obtained using the exact energy per particle, which in the case of \nSkyrme interactions is defined in Eq.~(\\ref{eq:Ebanm}) of Chapter~\\ref{chapter1}.\nThe differences $C(\\rho)$ should go to zero when the number of symmetry energy coefficients considered in the sum increases. \n\\begin{figure}[!b]\n\\centering     \n\\subfigure{\\label{fig:esymPAesym2skyrme}\\includegraphics[width=0.4\\linewidth]{.\/grafics\/chapter2\/skyrme\/Ed1-Ed0-Esym-SLy4-SkI5}}\n\\subfigure{\\label{fig:esymPaesym205skyrme}\\includegraphics[width=0.422\\linewidth]{.\/grafics\/chapter2\/skyrme\/Ed05-Ed0-Esym-SLy4-SkI5}}\n\\caption{Panel a: Density dependence of the parabolic approximation $E_{\\mathrm{sym}}^{PA}=E(\\delta=1)-E(\\delta=0)$ \nminus the sum of the symmetry energy contributions at \n  second, fourth and tenth-order in neutron matter ($\\delta=1$).\n  Panel b: Same as in Panel (a) but for $\\rho_n=3\\rho_p$ matter ($\\delta=0.5$).}\n\\end{figure}\n\nIn the case where one only considers the second-order symmetry energy coefficient, the differences between $E_\\mathrm{sym}^{PA}(\\rho)$\nand $E_{\\mathrm{sym}, 2}(\\rho)$ can reach up to $2-4$ MeV in the range of the considered densities. \nThese differences are reduced to values of $0.5$ MeV when higher-order contributions are added to the sum of the symmetry energy coefficients.\nFrom Fig.~\\ref{fig:esymPAesym2skyrme} we can extract some conclusions. Firstly, we see that the \nconvergence of the series as a function of the density shows a strong model dependence. Moreover, we observe that \nthe full convergence of the series is not completely achieved, even taking up to tenth-order \ncoefficients in the expansion~(\\ref{eq:EOSexpgeneral}), at least using the SLy4 and SkI5 Skyrme interactions. This result points out that \nthe expansion~(\\ref{eq:EOSexpgeneral}) is slowly convergent in the case of neutron matter computed with these two forces.   \nTo further explore the convergence of the asymmetry expansion of the EoS, we now \nconsider the energy per particle of a system with an asymmetry $\\delta$ intermediate between symmetric matter and neutron matter. From Eq.~(\\ref{eq:EOSexpgeneral}) we \nobtain \n   \\begin{equation} \\label{Ed0g2}\n E_\\mathrm{\\delta - \\delta=0} (\\rho,\\delta) \\simeq \\sum_{k} E_{\\mathrm{sym}, 2k}  (\\rho) \\delta^{2k},\n \\end{equation}\nwhere $E_{\\delta - \\delta=0} (\\rho,\\delta) \\equiv E_b(\\rho, \\delta) - E_b(\\rho, \\delta=0)$ is the difference between the exact energy per particle calculated \nin asymmetric nuclear matter and in symmetric matter. \nWe plot in Fig.~\\ref{fig:esymPaesym205skyrme} the differences \n \\begin{equation}\\label{eq:ccoeff05}\n C \\left(  \\rho \\right) =  E_\\mathrm{\\delta - \\delta=0} (\\rho) - \\sum_{k} E_{\\mathrm{sym}, 2k} (\\rho) \\delta^{2k}\n\\end{equation}\nfor the same forces as in Fig.~\\ref{fig:esymPAesym2skyrme} considering a system where $\\delta=0.5$, which corresponds to $\\rho_n=3 \\rho_p$.\nIn this case, the differences between the two sides of \nEq.~(\\ref{Ed0g2}) are much smaller than the ones obtained in pure neutron matter, and \nbecome almost zero using the expansion of the energy per particle up to tenth order. This points out that the convergence of the \nexpansion (\\ref{eq:EOSexpgeneral}) becomes slower as the isospin asymmetry $\\delta$ of the system increases.\n\n\\subsection{Convergence of the expansion of the slope of the symmetry energy for Skyrme interactions}\nIn previous Sections, we have discussed two possible definitions for the slope of the symmetry energy. \nThe first one, which we call $L$, results from considering the symmetry energy as the second-order coefficient of the Taylor expansion~(\\ref{eq:EOSexpgeneral})\nand defined in Eq.~(\\ref{eq:esymgen}).\nThe second definition (see Eq.~(\\ref{eq:esympa})) comes from assuming a parabolic expansion of the EoS and defining the symmetry energy as the \ndifference between the energy per particle in pure neutron matter and in symmetric nuclear matter. \nThis last definition can be also understood as the infinite sum of the $L_{2k}$ (\\ref{eq:L2k}).\nThe slopes of the fourth-, sixth-, eighth- and tenth- order symmetry energy coefficients \nare reported in Table~\\ref{table_L}, together with the slope $L_{\\mathrm{PA}}$ of the parabolic symmetry energy. \nTo test the convergence of this sum, we also include in the same table the results of the sum of the $L_{2k}$ series up to the\ntenth order. We observe that, indeed, there is a good convergence of the $L_{2k}$ sum to $L_\\mathrm{PA}$ if the slopes \nof the coefficients with higher order than two are considered.\n\n\n\\begin{table}[t]\n\\centering\n\\begin{tabular}{c|ppppppp}\n\\hline\n\\multirow{2}{*}{Force}  & \\multicolumn{1}{c}{$L$}      & \\multicolumn{1}{c}{$L_4$}   & \\multicolumn{1}{c}{$L_6$}   & \\multicolumn{1}{c}{$L_8$}  & \\multicolumn{1}{c}{$L_{10}$} &\n\\multicolumn{1}{c}{$\\sum_{k=1}^5 L_{2k}$} & \\multicolumn{1}{c}{$L_\\mathrm{PA}$} \\\\ \n  & \\multicolumn{1}{c}{(MeV)}      & \\multicolumn{1}{c}{(MeV)}   & \\multicolumn{1}{c}{(MeV)}   & \\multicolumn{1}{c}{(MeV)}  & \\multicolumn{1}{c}{(MeV)} &\n\\multicolumn{1}{c}{(MeV)} & \\multicolumn{1}{c}{(MeV)} \\\\ \\hline\\hline\nMSk7   & 9.41  & 0.79  & 0.21 & 0.08 & 0.04  & 10.53  & 10.63\\\\\nSIII   & 9.91   & 2.89  & 0.60 & 0.24 & 0.12  & 13.76 & 14.02\\\\\nSkP    & 19.68  & 3.33  & 0.61 & 0.23 & 0.11  & 23.96 & 24.20\\\\\nHFB-27 & 28.50 & 2.44  & 0.53 & 0.21 & 0.11  & 31.78 & 32.02\\\\\nSkX    & 33.19 & 3.10  & 0.57 & 0.21 & 0.11  & 37.18  & 37.40\\\\\nHFB-17 & 36.29 & 1.66  & 0.41 & 0.17 & 0.09  & 38.61 & 38.81\\\\\nSGII   & 37.63   & 3.01  & 0.62 & 0.24 & 0.12 & 41.63 & 41.90\\\\\nUNEDF1 & 40.00  & 2.63  & 0.50 & 0.19 & 0.09  & 43.42 & 43.62\\\\\nSk$\\chi$500 & 40.74   & -0.58 & -0.01 & -0.01& 0.01 &40.17  &40.20 \\\\\nSk$\\chi$450 & 42.06   & 1.30 & 0.29 &0.12 & 0.06 &43.83  &43.96 \\\\\nUNEDF0 & 45.08  & 3.08  & 0.55 & 0.20 & 0.10  & 49.00 & 49.21\\\\\nSkM*   & 45.78  & 3.32  & 0.67 & 0.26 & 0.13  & 50.16 & 50.44\\\\\nSLy4   & 45.96  & 0.61  & 0.29 & 0.13 & 0.07  & 47.08 & 47.25\\\\\nSLy7   & 47.22  & 0.54  & 0.28 & 0.13 & 0.07  & 48.25 & 48.42\\\\\nSLy5   & 48.27  & 0.64  & 0.30 & 0.14 & 0.07  & 49.41 & 49.59\\\\\nSk$\\chi$414 &  51.92 & 0.84 & 0.21 & 0.09& 0.04 &53.11  &53.21 \\\\\nMSka   & 57.17  & 2.98  & 0.61 & 0.24 & 0.12  & 61.12 & 61.38\\\\\nMSL0   & 60.00  & 2.70  & 0.57 & 0.22 & 0.11  & 63.60 & 63.85\\\\\nSIV    & 63.50   & 5.51  & 1.20 & 0.47 & 0.24 & 70.92 & 71.45\\\\\nSkMP   & 70.31   & 3.30  & 0.73 & 0.29 & 0.15 & 74.77 & 75.10\\\\\nSKa    & 74.62  & 4.33  & 0.91 & 0.36 & 0.18  & 80.40 & 80.79\\\\\nR$_\\sigma$     & 85.69 & 2.88  & 0.60 & 0.24 & 0.12 & 89.53  & 89.79 \\\\\nG$_\\sigma$     & 94.01  & 2.87  & 0.60 & 0.24 & 0.12 & 97.84 & 98.10 \\\\\nSV     & 96.09  & 7.18  & 1.58 & 0.62 & 0.32 & 105.78 & 106.49\\\\\nSkI2   & 104.33 & 0.48  & 0.28 & 0.13 & 0.07 & 105.29 & 105.46\\\\\nSkI5   & 129.33 & -0.72 & 0.15 & 0.10 & 0.06 & 128.91 & 129.06\\\\\\hline\n\\end{tabular}\n\\caption{Values of the slope of the symmetry energy coefficients appearing in the Taylor expansion of the \nenergy per particle up to tenth order in the isospin asymmetry $\\delta$, and the parabolic approximation.}\n\\label{table_L}\n\\end{table}\n\n\n\n\\newpage\n\\section{Convergence of the isospin Taylor expansion of the EoS for Gogny interactions}\nWe proceed to study the contributions to the symmetry energy up to 6$^{\\mathrm{th}}$ order for Gogny interactions\nin the energy per particle Taylor expansion~(\\ref{eq:EOSexpgeneral})~\\cite{gonzalez17}. \nThe calculation of the second-, fourth- and sixth-order symmetry energy coefficients for Gogny interactions \nis much more involved than in the case of the zero-range Skyrme interactions. We have obtained the \nfollowing expressions for $ E_{\\mathrm{sym}, 2} (\\rho)$, $E_{\\mathrm{sym}, 4} (\\rho)$ and $E_{\\mathrm{sym}, 6} (\\rho)$\ncoefficients for Gogny forces~\\cite{gonzalez17}:\n\\begin{eqnarray}\nE_{\\mathrm{sym}, 2} (\\rho) &=& \n\\left. \\frac{1}{2!} \\frac{\\partial^{2} E_b(\\rho, \\delta)}{\\partial \\delta^{2}}\\right|_{\\delta=0} =\n\\frac{\\hbar^2}{6m} \\left(  \\frac{3 \\pi^2}{2}\\right)^{2\/3} \\rho^{2\/3}  - \n\\frac{1}{8} t_3 \\rho^{\\alpha+1} (2x_3 +1)\n\\nonumber\n\\\\\n&+& \\frac{1}{2} \\sum_{i=1,2} \\mu_i^3 \\pi^{3\/2}  {\\cal B}_i  \\rho  \n \\mbox{} +\\frac{1}{6}\\sum_{i=1,2}  \\left[-{\\cal C}_i  G_1 ( k_F \\mu_i)+ {\\cal D}_i G_2 ( k_F \\mu_i)  \\right] , \\label{eq:esym2gog}\n\\\\\nE_{\\mathrm{sym}, 4} (\\rho) &=&\n\\left. \\frac{1}{4!} \\frac{\\partial^{4} E_b(\\rho, \\delta)}{\\partial \\delta^{4}}\\right|_{\\delta=0} =\n\\frac{\\hbar^2}{162m} \\left(  \\frac{3 \\pi^2}{2}\\right)^{2\/3} \\rho^{2\/3} \n\\nonumber\n\\\\\n&+&\n\\frac{1}{324} \\sum_{i=1,2}\n\\left[ {\\cal C}_i  G_3 ( k_F \\mu_i)  + {\\cal D}_i G_4 ( k_F \\mu_i) \\right] \\, ,\\label{eq:esym4gog}\n\\\\\nE_{\\mathrm{sym}, 6} (\\rho) &=&\n\\left. \\frac{1}{6!} \\frac{\\partial^{6} E_b(\\rho, \\delta)}{\\partial \\delta^{6}}\\right|_{\\delta=0} =\n\\frac{7\\hbar^2}{4374m} \\left(  \\frac{3 \\pi^2}{2}\\right)^{2\/3} \\rho^{2\/3} \n\\nonumber\n\\\\\n&+& \n\\frac{1}{43740} \\sum_{i=1,2}\n \\left[ {\\cal C}_i  G_5(k_F \\mu_i )  - {\\cal D}_i  G_6( k_F \\mu_i) \\right] , \\label{eq:esym6gog}\n\\end{eqnarray}\nwith $G_1 (\\eta)$ and $G_2(\\eta)$ already given, respectively, in Eqs.~(\\ref{G1}) and (\\ref{G2}) and with \n\\begin{eqnarray}\n G_3 (\\eta)&=&  -\\frac{14}{\\eta}  + e^{-\\eta^2} \\left( \\frac{14}{\\eta} + 14 \\eta + 7 \\eta^3 + 2\\eta^5 \\right) \\label{G3}\n\\\\\n G_4 (\\eta)&=&   \\frac{14}{\\eta} - 8 \\eta + \\eta^3 - 2 e^{-\\eta^2} \\left( \\frac{7}{\\eta}+3\\eta\\right)\\label{G4}\n\\\\\n G_5 (\\eta)&=&  -\\frac{910}{\\eta} + e^{-\\eta^2}\\left( \\frac{910}{\\eta} +   910\\eta  + 455 \\eta^3 + 147 \\eta^5 \n + 32\\eta^7 + 4 \\eta^9 \\right) \\label{G5}\n\\\\\n G_6 (\\eta)&=&  -\\frac{910}{\\eta} + 460 \\eta - 65 \\eta^3+ 3 \\eta^5 +e^{-\\eta^2} \\left( \\frac{910}{\\eta} \n + 450 \\eta + 60\\eta^3    \\right). \\label{G6}\n\\end{eqnarray}\nThe expressions for the constants ${\\cal B}_i$, ${\\cal C}_i$, and ${\\cal D}_i$ have been given in Section~\\ref{Gogny} of Chapter~\\ref{chapter1}.\n\\begin{figure}[t!]\n \\centering\n \\includegraphics[width=0.8\\linewidth, clip=true]{.\/grafics\/chapter2\/Gogny\/Esym2-Gogny}\n  \\caption{Density dependence of the second-order symmetry energy coefficient $E_{\\mathrm{sym}, 2}(\\rho)$ for different Gogny interactions. \nAlso represented are the constraints on the symmetry energy extracted from the analysis of data on isobaric analog states (IAS)~\\cite{Danielewicz13}\nand of IAS data combined with neutron skins (IAS+n.skin)~\\cite{Danielewicz13}, \nthe constraints from the electric dipole polarizability in lead ($\\alpha_D$ in $^{208}$Pb) \\cite{Zhang15},\nand from transport simulations of heavy-ion collisions of tin isotopes (HIC) \\cite{Tsang08}.}\n \\label{fig:esym2gog}\n\\end{figure}\n\nIn Fig.~\\ref{fig:esym2gog} we show the density dependence of the second-order symmetry energy coefficient $E_{\\mathrm{sym}, 2} (\\rho)$ for a set of\ndifferent Gogny interactions. As happened with Skyrme forces, at low densities $\\rho \\lesssim 0.1$ fm$^{-3}$, the\n$E_{\\mathrm{sym}, 2} (\\rho)$ coefficient has similar trends for all interactions and \nincreases with density. \nThis is because most of the Gogny interactions are fitted to properties of finite nuclei, which are found in this subsaturation density regime.\nOn the other hand, from $\\rho \\gtrsim 0.1$ fm$^{-3}$ onwards, there are substantial differences between the trends of the different \nparametrizations. In comparison with existing empirical constraints for the symmetry energy \nat subsaturation density \\cite{Danielewicz13, Zhang15, Tsang08}, \none finds that the Gogny functionals in general respect them.\nThe symmetry energy coefficients of all considered interactions bend at values $E_{\\mathrm{sym}, 2} (\\rho) \\sim 30-40$ MeV right above saturation density, \nand from this maximum value all parametrizations decrease with density.\nIn all interactions, their $E_{\\mathrm{sym}, 2} (\\rho)$ eventually becomes \nnegative beyond $0.4$ fm$^{-3}$ (in D1M this happens only at a very large density of $1.9$ fm$^{-3}$), signaling the onset of an isospin instability. \nWe show the symmetry energy coefficients of fourth-order, $E_{\\mathrm{sym}, 4}(\\rho)$, and sixth-order, \n$E_{\\mathrm{sym}, 6}(\\rho)$, in Figs.~\\ref{fig:esym4gog} and \\ref{fig:esym6gog}, respectively.\nOn the one hand, at subsaturation densities both terms are relatively small: below saturation density, $E_{\\mathrm{sym}, 4}(\\rho)$ is below $\\approx$\\,1 MeV and \n$E_{\\mathrm{sym}, 6}(\\rho)$ does not go above $\\approx$\\,0.3 MeV. These values can be compared with the larger values of \n$E_{\\mathrm{sym}, 2}(\\rho) > 10$ MeV in the same density regime. \n\\begin{figure}[t]\n\\centering     \n\\subfigure{\\label{fig:esym4gog}\\includegraphics[width=0.6\\linewidth]{.\/grafics\/chapter2\/Gogny\/Esym4-Gogny}}\n\\subfigure{\\label{fig:esym6gog}\\includegraphics[width=0.6 \\linewidth]{.\/grafics\/chapter2\/Gogny\/Esym6-Gogny}}\n\\caption{Density dependence of the fourth-order symmetry energy coefficient $E_{\\mathrm{sym}, 4}(\\rho)$ \n(panel (a)) and of the sixth-order symmetry energy coefficient $E_{\\mathrm{sym}, 6}(\\rho)$ (panel (b))\n for different Gogny interactions.}\n\\end{figure}\nOne should also consider that in the expansion of Eq.~(\\ref{eq:EOSexpgeneral}) \nthe terms $E_{\\mathrm{sym}, 4}(\\rho)$ and $E_{\\mathrm{sym}, 6}(\\rho)$ carry additional factors $\\delta^2$ and $\\delta^4$ with respect to \n$E_{\\mathrm{sym}, 2}(\\rho)$, and their overall magnitude will therefore be smaller.\nOn the other hand, above saturation density, we observe two markedly different behaviours for the density \ndependence of $E_{\\mathrm{sym}, 4}(\\rho)$ and $E_{\\mathrm{sym}, 6} (\\rho)$. \nFor both $E_{\\mathrm{sym}, 4}(\\rho)$ and $E_{\\mathrm{sym}, 6}(\\rho)$, we find a group of parametrizations \n(D1S, D1M, D1N, and D250) that reach a maximum and then decrease with density. \nWe call this set of forces as  ``group~1\" from now on. A second set of forces, ``group~2'', \nis formed of D1, D260, D280, and D300,\nwhich yield $E_{\\mathrm{sym}, 4}(\\rho)$ and $E_{\\mathrm{sym}, 6}(\\rho)$ terms\nthat do not reach a maximum and increase steeply in the range of the studied densities.\n\\begin{figure}[t!]\n \\centering\n \\includegraphics[width=0.7\\linewidth, clip=true]{.\/grafics\/chapter2\/Gogny\/Esym4Esym6Esym2_Gogny}\n  \\caption{Density dependence of the ratios $E_{\\mathrm{sym}, 4} (\\rho)\/E_{\\mathrm{sym}, 2} (\\rho)$ (panel (a)) \n  and $E_{\\mathrm{sym}, 6} (\\rho)\/E_{\\mathrm{sym}, 2} (\\rho)$ (panel (b)) for different Gogny interactions.}\n\\label{fig:E46E2_Gogny}\n\\end{figure}\n\nThe difference in density dependence between the second-order symmetry energy and its higher-order corrections can be understood by \ndecomposing them into terms associated to the different contributions from the nuclear Hamiltonian.\nAll three coefficients $E_{\\mathrm{sym}, 2}(\\rho)$, $E_{\\mathrm{sym}, 4}(\\rho)$ and $E_{\\mathrm{sym}, 6}(\\rho)$ include a \nkinetic component, which decreases substantially as the order increases. The\n$E_{\\mathrm{sym}, 2}(\\rho)$ coefficient also receives contributions \nfrom the zero-range term of the force [Eq.~(\\ref{eq:eb.zr})] as well as from the finite-range direct and exchange terms\n[Eqs.~(\\ref{eq:eb.dir})--(\\ref{eq:eb.exch})].\nWe note that the direct terms of the finite-range contribution to $E_{\\mathrm{sym}, 2}(\\rho)$ are directly proportional to the constants ${\\cal B}_i$ and to the density $\\rho$. \nThe functions $G_n (\\mu_i k_F)$ are due solely to the exchange contribution in the matrix elements of the Gogny force. \nOne can equally say that they reflect the contribution of the momentum dependence of the interaction to the symmetry energy. \nAs discussed in Ref.~\\cite{Sellahewa14}, the zero-range term, the direct term, and the exchange (momentum-dependent) term contribute \nwith similar magnitudes to the determination of $E_{\\mathrm{sym}, 2}(\\rho)$ with Gogny forces. However, they contribute with different signs, which leads to \ncancellations in $E_{\\mathrm{sym}, 2}(\\rho)$ between the power-law zero-range term, the linear density-dependent direct term, and the exchange term. \nDepending on the parametrization, the sum of the zero-range and direct terms is positive and the exchange term is negative, \nor the other way around. In any case, there is a balance between terms, \nwhich gives rise to a somewhat similar density dependence of the symmetry energy coefficient $E_{\\mathrm{sym}, 2}(\\rho)$ for all parameter sets. \n\nIn contrast to the case of the $E_{\\mathrm{sym}, 2}(\\rho)$ coefficient, neither the zero-range nor the direct term contribute to \nthe $E_{\\mathrm{sym}, 4}(\\rho)$ and $E_{\\mathrm{sym}, 6}(\\rho)$ coefficients, see Eqs.~(\\ref{eq:esym4gog}) and (\\ref{eq:esym6gog}).\nThe cause is that both the zero-range and the direct components of the energy per particle [cf.\\ Eqs.~(\\ref{eq:eb.zr})--(\\ref{eq:eb.dir})] \ndepend on the square of the isospin asymmetry, $\\delta^2$. \nIn other words, the higher-order corrections to the symmetry energy are only sensitive to the kinetic term and to \nthe momentum-dependent term, i.e., the exchange term of the Gogny force. \nWe note that the same pattern is found in zero-range Skyrme forces, but in that case, \nthe functional dependence of the momentum-dependent contribution to the symmetry energy coefficients is   \nproportional to $\\rho^{5\/3}$, whereas in the Gogny case it has a more intricate density dependence due to the finite range of the interaction,\nwhich is reflected in the $G_n (\\mu_i k_F)$ functions~\\cite{gonzalez17}.\n\nIn both $E_{\\mathrm{sym}, 4}(\\rho)$ and $E_{\\mathrm{sym}, 6}(\\rho)$ of Gogny forces, cf.\\ Eqs.~(\\ref{eq:esym4gog}) and (\\ref{eq:esym6gog}), \nthe exchange term is given by the product of two parametrization-dependent constants, ${\\cal C}_i$ and ${\\cal D}_i$, \nand two density-dependent functions, \n$G_3$ and $G_4$, or $G_5$ and $G_6$. Because the density dependence of these functions is similar, one does expect \nthat comparable density dependences arise for the fourth- and the sixth-order symmetry energy coefficients, as observed in Figs.~\\ref{fig:esym4gog} and \\ref{fig:esym6gog}.\nThis simple structure also provides an explanation for the appearance of two distinct groups of forces in terms of the density dependence \n of $E_{\\mathrm{sym}, 4}(\\rho)$ and $E_{\\mathrm{sym}, 6}(\\rho)$. In group~1 forces, the fourth- and sixth-order contributions to the symmetry \nenergy change signs as a function of density, whereas group~2 forces produce monotonically increasing functions of density. The change of sign\n is necessarily due to the exchange contribution, which in the case of group~1 forces must also be attractive \n enough to overcome the kinetic term. \n\nFor further insight into the relevance of $E_{\\mathrm{sym}, 4} (\\rho)$ and $E_{\\mathrm{sym}, 6} (\\rho)$ for \nthe Taylor expansion of the EoS at each density, we plot in Fig.~\\ref{fig:E46E2_Gogny} their ratios with respect \nto $E_{\\mathrm{sym}, 2} (\\rho)$. In the zero density limit, we see that both ratios tend to a constant value. \nThis is expected in the non-interacting case, although the actual values of these ratios are modified by the \nexchange contributions. In this limit, we find $E_{\\mathrm{sym}, 4}(\\rho)\/E_{\\mathrm{sym}, 2}(\\rho) \\approx 1.5 \\%$ and \n$E_{\\mathrm{sym}, 6}(\\rho)\/E_{\\mathrm{sym}, 2}(\\rho) \\approx 0.4 \\%$. \nAt low, but finite densities, $\\rho \\lesssim 0.1$ fm$^{-3}$, the ratio $E_{\\mathrm{sym}, 4}(\\rho)\/E_{\\mathrm{sym}, 2}(\\rho)$ \nis relatively flat and not larger than $3\\%$. The ratio for the sixth-order term is also mildly density-dependent \nand less than $0.6 \\%$. Beyond saturation, both ratios increase in absolute value, to the point that for some \nparametrizations the ratio of the fourth- (sixth-) order term to the second-order term is not negligible and \nof about $10-30 \\%$ ($2-8 \\%$) or even more. In particular, this is due to the decreasing trend of \n$E_{\\mathrm{sym}, 2} (\\rho)$ with increasing density for several interactions when $\\rho$ is above saturation.\nWe may compare these results for the ratios with previous literature. For example, the calculations of \nRef.~\\cite{Moustakidis12} with the momentum-dependent interaction (MDI) and with the Skyrme forces \nSLy4, SkI5 and Ska find values of $ \\left| E_{\\mathrm{sym}, 4} (\\rho)\/E_{\\mathrm{sym}, 2} (\\rho) \\right| < 8 \\%$ \nat $\\rho \\sim 0.4 $ fm$^{-3}$. \nOur results for Skyrme interactions give similar ratios as the ones presented in Ref.~\\cite{Moustakidis12}.\nIn the same reference~\\cite{Moustakidis12}, the Thomas-Fermi model of Myers and Swiatecki \nyields a ratio $ \\left| E_{\\mathrm{sym}, 4} (\\rho)\/E_{\\mathrm{sym}, 2} (\\rho) \\right|$ reaching 60\\% already at a\nmoderate density $\\rho= 1.6\\rho_0$.\nWith RMF models such as FSUGold or IU-FSU, at densities $\\rho \\sim 0.4 $ fm$^{-3}$ one has ratios of $ \\left| \nE_{\\mathrm{sym}, 4} (\\rho)\/E_{\\mathrm{sym}, 2} (\\rho) \\right| < 4 \\%$ \\cite{Cai2012}. All in all, \nit appears that Gogny parametrizations provide ratios that are commensurate with previous literature. \n\n\n\n\\subsection{Parabolic approximation for Gogny interactions}\nIn Fig.~\\ref{fig:esymPA} we show the results for $E_\\mathrm{sym}^{PA}(\\rho)$ from the different Gogny\nfunctionals. In general we find a similar picture to that of Fig.~\\ref{fig:esym2gog} for the second-order symmetry energy $E_{\\mathrm{sym},2} (\\rho)$.  \nAt subsaturation densities, the symmetry energies $E_{\\mathrm{sym}}^{PA}(\\rho)$ of all the forces are \nquite close to each other presenting similar trends. On the other hand, at densities \nabove $\\rho \\gtrsim 0.1$ fm$^{-3}$, there are markedly different behaviours between the interactions. \nUsually, $E_{\\mathrm{sym}}^{PA}(\\rho)$ reaches a maximum and then starts to decrease up to a given density \nwhere it becomes negative, presenting an isospin instability.\n\\begin{figure}[!b]\n \\centering\n \\includegraphics[width=0.8\\linewidth, clip=true]{.\/grafics\/chapter2\/Gogny\/ESYM_PA_GOGNY}\n  \\caption{Density dependence of the symmetry energy coefficient in the parabolic approximation [Eq.~(\\ref{eq:PAgen})] for different Gogny interactions.}\n\\label{fig:esymPA}\n\\end{figure}\n\nThe correspondence between the second-order symmetry energy coefficient and the symmetry \nenergy calculated within the parabolic approach may be spoiled by the influence of the\nhigher-order terms in the expansion of the energy per particle~(\\ref{eq:EOSexpgeneral}).\nIn order to analyze better the differences between $E_\\mathrm{sym}^{PA}(\\rho)$ and $E_{\\mathrm{sym}, 2}(\\rho)$,\nwe plot in Fig.~\\ref{fig:esymPAesym2} the ratio $E_{\\mathrm{sym}}^{PA}(\\rho)\/E_{\\mathrm{sym}, 2}(\\rho)$.  \nAt low densities $\\rho \\lesssim 0.1$ fm$^{-3}$, the symmetry energy calculated with the parabolic law is always a little\nlarger than calculated with Eq.~(\\ref{eq:esym4gog}) and (\\ref{eq:esym6gog}) for $k=1$. The ratio is approximately 1.025 irrespective of the functional. \nThis is relatively consistent with the zero-density limit of a free Fermi gas,  which has a ratio \n$E_{\\mathrm{sym}}^{PA}(\\rho)\/E_{\\mathrm{sym}, 2}(\\rho) = \\frac{9}{5}(2^{2\/3}-1) \\approx 1.06$. \nAt densities $\\rho \\gtrsim 0.1$ fm$^{-3}$, the ratios change depending on the Gogny force. \nHere, group~1 and group~2 parametrizations again show two distinct behaviours. In group~1 (D1S, D1M, D1N, D250), \nthe ratio becomes smaller than $1$ at large densities, whereas in group~2 (D1, D260, D280, D300), it increases with density.\nThere is a clear resemblance between Fig.~\\ref{fig:esymPAesym2} and Fig.~\\ref{fig:E46E2_Gogny}(a). \nIndeed, Eq.~(\\ref{eq:PAgen}) suggests that the two ratios are connected~\\cite{gonzalez17},\n\\begin{equation}\n\\frac{ E_{\\mathrm{sym}}^{PA} (\\rho) }{ E_{\\mathrm{sym},2} (\\rho) } = 1 + \n\\frac{E_{\\mathrm{sym}, 4} (\\rho) }{E_{\\mathrm{sym}, 2} (\\rho) } + \\cdots \\, ,\n\\end{equation} \nas long as the next-order term $\\frac{E_{\\mathrm{sym}, 6} (\\rho) }{E_{\\mathrm{sym}, 2} (\\rho) }$ \nis small. The behaviour of the ratio $\\frac{ E_{\\mathrm{sym}}^{PA} (\\rho) }{ E_{\\mathrm{sym},2} (\\rho) }$ \ncan therefore be discussed in similar terms as the ratios shown in Fig.~\\ref{fig:E46E2_Gogny}. \nAs discussed earlier in the context of Eqs.~Eq.~(\\ref{eq:esym4gog}) and (\\ref{eq:esym6gog}), $E_{\\mathrm{sym}, 4}(\\rho)$ and $E_{\\mathrm{sym}, 6}(\\rho)$\nare entirely determined by the exchange contributions that are proportional to the \nconstants ${\\cal C}_i$ and ${\\cal D}_i$ and the functions $G_n (\\mu_i k_F)$.\n\\begin{figure}[t!]\n \\centering\n \\includegraphics[width=0.8\\linewidth, clip=true]{.\/grafics\/chapter2\/Gogny\/EsymPAoverEsym2_Gogny}\n  \\caption{Density dependence of the ratio $E_{\\mathrm{sym}}^{PA} (\\rho)\/E_{\\mathrm{sym}, 2} (\\rho)$ for different Gogny interactions.}\n\\label{fig:esymPAesym2}\n\\end{figure}\n\n\n\n\\subsection{Convergence of the expansion of the slope of the symmetry energy for Gogny interactions}\n\\begin{table}[!b]\n\\centering\n\\resizebox{\\columnwidth}{!}{%\n\\begin{tabular}{ddddddddd}\n\\hline\n \\multicolumn{1}{c}{Force}  &  \\multicolumn{1}{c}{D1}  &  \\multicolumn{1}{c}{D1S}  &  \\multicolumn{1}{c}{D1M}  &  \\multicolumn{1}{c}{D1N}  &  \\multicolumn{1}{c}{D250}  &  \\multicolumn{1}{c}{D260}  &  \\multicolumn{1}{c}{D280}  &  \\multicolumn{1}{c}{D300}  \\\\\n \\hline\\hline\n \\multicolumn{1}{c}{$E_{\\mathrm{sym}, 2}$($\\rho_0$)}  &   30.70    &   31.13    &   28.55    &   29.60    &   31.54    &   30.11    &   33.14    &   31.23    \\\\\n \\multicolumn{1}{c}{$E_{\\mathrm{sym}, 4}$($\\rho_0$)}  &   0.76     &   0.45     &   0.69     &   0.21     &   0.43     &   1.20     &   1.18     &   0.80     \\\\\n \\multicolumn{1}{c}{$E_{\\mathrm{sym}, 6}$($\\rho_0$)}  &   0.20     &   0.16     &   0.24     &   0.15     &   0.16     &   0.27     &   0.29     &   0.20     \\\\\n \\multicolumn{1}{c}{$L$}  &   18.36    &   22.43    &   24.83    &   33.58    &   24.90    &   17.57    &   46.53    &   25.84    \\\\\n \\multicolumn{1}{c}{$L_4$}  &   1.75     &   -0.52    &   -1.04    &   -1.96    &   -0.33    &   4.73     &   4.36     &   2.62     \\\\\n \\multicolumn{1}{c}{$L_6$}  &   0.46     &   0.08     &   0.42     &   0.08     &   0.09    &   0.99     &   1.19     &   0.63     \\\\\n \\multicolumn{1}{c}{$\\sum_{k=1}^3 L_{2k}$} &   20.57    &   21.99     &   24.21    &   31.7     &   24.66   &   23.26     &   52.08     &   29.09    \\\\\n \\hline\n \\multicolumn{1}{c}{$E_\\mathrm{sym}^{PA}$($\\rho_0$)}  &   31.91    &   31.95    &   29.73    &   30.14    &   32.34    &   31.85    &   35.89    &   32.44    \\\\\n \\multicolumn{1}{c}{$L_{PA}$}  &   21.16    &   22.28    &   24.67    &   31.95    &   24.94    &   24.33    &   53.25    &   29.80    \\\\\\hline\n\\end{tabular}}\n\\caption{Values of the $E_{\\mathrm{sym}, 2k}$($\\rho_0$) symmetry energy coefficients and their corresponding \nslopes $L_{2k}$ parameters at the saturation density $\\rho_0$ for Gogny interactions. The values for the parabolic approximation, $E_\\mathrm{sym}^{PA}$($\\rho_0$) and $L_{PA}$,\nare also included.}\n\\label{Table-saturation}\n\\end{table}\nThe values of $L_{2k}$ \nprovide a good handle on the density dependence of the corresponding $E_{\\mathrm{sym}, 2k} (\\rho)$ contributions~\\cite{gonzalez17}.\nAt second order, the slope \nparameter $L$ is positive in all the considered Gogny interactions. It goes from $L=17.57$ MeV in D260 to $46.53$ MeV in D280.\nThis large variation of the $L$ value indicates that the density dependence of the symmetry energy is poorly \nconstrained with these forces \\cite{Sellahewa14}. \nWe also emphasize that all forces in Table \\ref{Table-saturation} have a low slope parameter, under $50$ MeV, \nand thus correspond to soft symmetry energies \\cite{Tsang2012,Lattimer2013,BaoAnLi13,Vinas14,Roca-Maza15,Lattimer2016}.\nIndeed, we see that the $L$ values in Table \\ref{Table-saturation}  are below or on the \nlow side of recent results proposed from microscopic calculations, such as the ones coming from the study of the\nelectric dipole polarizability of $^{48}$Ca,\n$L=43.8$--48.6 MeV \\cite{Birkhan16}, or the ones coming from chiral effective field theory, $L=20$--65 MeV \\cite{Holt2017} and \n$L=45$--70 MeV \\cite{Drischler1710.08220}.\nThe higher-order slope parameters $L_4$ and $L_6$ are in keeping with the density dependence of \n$E_{\\mathrm{sym}, 4}(\\rho)$ and $E_{\\mathrm{sym}, 6}(\\rho)$, respectively.\n$L_4$ goes from about $-2$ MeV (D1N) to \n$4.7$ MeV (D260) and $L_{6}$ is in the range of $0.1-1.2$ MeV for the different forces. \nInterestingly, we find a one-to-one correspondence between group 1 and group 2 forces and the sign of $L_4$. \nFor group 1 forces, $E_{\\mathrm{sym}, 4} (\\rho)$ has already reached a maximum at saturation density and tends \nto decrease with density (cf.~Fig.~\\ref{fig:esym4gog});\nconsequently, $L_4$ is negative. On the contrary, group 2 forces have positive $L_4$, reflecting the increasing nature \nof $E_{\\mathrm{sym}, 4} (\\rho)$ with density. In contrast to $L_4$, \nthe values of $L_{6}$ are always positive. This is a reflection \nof the fact that the maximum of $E_{\\mathrm{sym}, 6} (\\rho)$ occurs somewhat above saturation density, as shown \nin Fig.~\\ref{fig:esym6gog}. \nIt is worth noting that in absolute terms the value of the $L_{2k}$ parameters decreases with increasing order \nof the expansion, i.e., we have $|L_6| < |L_4| < |L|$. This indicates that the dominant density dependence of the isovector part of the \nfunctional is accounted for by the second-order parameter $L$.\nThe $L_{PA}$ values are displayed in the last row of Table~\\ref{Table-saturation}.\nThere are again differences between the two groups of functionals. \nIn group 1 forces, such as D1S, D1M, D1N, or D250, the $L_{PA}$ values are fairly close to the slope parameter $L$. \nIn contrast, group 2 forces have $L_{PA}$ values that are substantially larger than $L$. For example, the relative \ndifferences between $L_{PA}$ and $L$ are of the order of $40 \\%$ for D260 and $15 \\%$ for D280. This again may be \nexplained in terms of the higher-order $L_{2k}$ \ncontributions, which add up to give $L_{PA}$ analogously to Eq.~(\\ref{eq:PAgen}).\nFor group 1 interactions, the addition of the higher-order terms to the $L$ parameter tends to disrupt a little the similarities\nwith $L_{PA}$. Nevertheless, the relative differences between $\\sum_{k=1}^3 L_{2k}$ and $L_{PA}$ do not exceed values of $2\\%$.\nFor group 2 interactions, on the other hand, the addition of higher-order terms to $L$ tend to reduce the differences \nbetween $\\sum_{k=1}^3 L_{2k}$ and $L_{PA}$, achieving relative differences up to a maximum of $5\\%$.\n\nOn the whole, for Gogny interactions, the parabolic approximation seems \nto work relatively well at the level of the symmetry energy. For the slope parameter, however, the contribution of \n$L_4$ can be large and spoil the agreement between the approximated $L_{PA}$ and $L$. \n$L_4$ is a density derivative of $E_{\\mathrm{sym},4}(\\rho)$, which, as shown in Eq.~(\\ref{eq:esym4gog}), is entirely determined \nby the exchange finite-range terms in the Gogny force.\nThe large values of \n$L_4$ are given by the isovector finite-range exchange contributions. We therefore conclude that exchange contributions play \na very important role in the slope parameter. These terms can provide substantial (in some cases of order $30 \\%$) \ncorrections and should be explicitly considered when it is possible to do so \\cite{Vidana2009}. \n\n\n\\section{Beta-stable nuclear matter}\nWe now proceed to study the impact of the higher-order symmetry energy terms on the equation of state\nof $\\beta$-equilibrated matter. This condition is found in the interior \nof NSs, where the URCA reactions\n\\begin{align}\n n \\rightarrow p + e^- + \\bar{\\nu}_e \\qquad\n p+ e^- \\rightarrow n + \\nu_e \n\\end{align}\ntake place simultaneously.\nIf one assumes that the neutrinos leave the system, the $\\beta$-equilibrium leads to the condition \n\\begin{equation}\\label{betaeq}\n \\mu_n - \\mu_p = \\mu_e ,\n\\end{equation}\nwhere $\\mu_n$, $\\mu_p$, and $\\mu_e$ are the chemical potentials of neutrons, protons, and electrons, respectively.\nThe expressions for the neutron, proton chemical potentials are given in Chapter~\\ref{chapter1}.\nThe contribution given by the leptons, i.e., the electrons ($e$) and muons ($\\mu$) in the system, to the \nenergy density is\n\\begin{eqnarray} \\label{He}\n \\mathcal{H}_l (\\rho, \\delta) &=& \\frac{m_l^4}{8\\pi^2 \\rho} \\left[ x_F \\sqrt{1+x_F^2} \\left(1+ 2 x_F^2 \n\\right) - \\mathrm{ln}\\left(x_F + \\sqrt{1+x_F^2}\\right) \\right],\n\\end{eqnarray}\nwhere $m_l$ is the mass of each type of leptons $l=e, \\mu$ and the dimensionless Fermi momentum is \n$x_F \\equiv k_{Fl}\/m_l = (3 \\pi^2 \\rho_l)^{1\/3}\/m_l$. \nThe density $\\rho_l$ defines the density\n of electrons ($\\rho_e$) or muons ($\\rho_m$).\n Due to charge neutrality, the density of the leptons is the same as the density of \nprotons, i.e., $\\rho_l=\\rho_p$.\nIf the density regime allows the electron chemical potential to be larger than the \nmass of muons, $\\mu_e \\geq m_\\mu$, the  appearance of muons in the system is energetically favorable.\nIn this case, the $\\beta$-stability condition is given by \n\\begin{equation}\n \\mu_n - \\mu_p = \\mu_e =\\mu_\\mu\n\\end{equation}\nand the charge neutrality establishes that $\\rho_p=\\rho_e+\\rho_\\mu$, \nwhere $\\mu_\\mu$ and $\\rho_\\mu$ are the muon chemical potential and muon density, respectively. \n \nThe chemical potential of each type of leptons is defined as \n\\begin{equation}\\label{mu_lepton}\n \\mu_l = \\sqrt{k_{Fl}^2 + m_l^2}\n\\end{equation}\nand their the pressure is given either by \n\\begin{equation}\\label{eq:P_lepton}\n P_l (\\rho, \\delta) = \\rho_l^2 \\frac{\\partial E_{l} (\\rho, \\delta)}{\\partial \\rho_l} \\hspace{1cm}\n\\mathrm{or~ by} \\hspace{1cm}\n P_l (\\rho, \\delta)= \\mu_l \\rho_l-\\mathcal{H}_l(\\rho, \\delta),\n\\end{equation}\nwhere $E_{l} (\\rho, \\delta)$ is the energy per particle of each type of leptons $E_{l} (\\rho, \\delta)= \\mathcal{H}_l (\\rho, \\delta)\/\\rho_l$.\n\n\n\n\nUsing the definitions of the baryon chemical potentials placed in Eqs.~(\\ref{chempot}) and expressed in \nterms of the density and asymmetry of the system, the \n$\\beta$-equilibrium condition can be rewritten as\n\\begin{eqnarray}\\label{betamatter-full}\n\\mu_n - \\mu_p=2 \\frac{\\partial E_b(\\rho,\\delta)}{\\partial \\delta} = \\mu_e=\\mu_\\mu,\n\\end{eqnarray}\nwhere $E_b(\\rho,\\delta)$ is the baryon energy per particle.\nIf one uses the Taylor expansion of the EoS [cf. Eq.~(\\ref{eq:EOSexpgeneral})] instead of its exact expression, \nthe $\\beta$-equilibrium condition is also expressed in terms of the symmetry energy coefficients, and reads \n \\begin{eqnarray}\\label{betamatter}\n \\mu_n - \\mu_p &=& 2 \\frac{\\partial E_b (\\rho, \\delta)}{\\partial \\delta} = 4 \\delta E_{\\mathrm{sym}, 2}(\\rho) + 8 \\delta^3 E_{\\mathrm{sym}, 4} (\\rho)\n + 12 \\delta^5 E_{\\mathrm{sym}, 6}(\\rho) + 16 \\delta^7 E_{\\mathrm{sym}, 8} (\\rho)\\nonumber\\\\\n &+& 20\\delta^9 E_{\\mathrm{sym},10}(\\rho) +\\mathcal{O}(\\delta^{11}) =\\mu_e=\\mu_\\mu.  \n \\end{eqnarray}\nMoreover, if one uses the parabolic expression to calculate the energy per particle, the \n$\\beta$-equilibrium condition takes the form\n\\begin{eqnarray}\\label{betamatter-PA}\n  \\mu_{n} -\\mu_p &=& 4 \\delta E_{\\mathrm{sym}}^{PA}(\\rho) = \\mu_e = \\mu_\\mu  .\n\\end{eqnarray}\n\n\\begin{figure}[t]\n\\centering     \n\\subfigure{\\label{fig:delta_skyrme}\\includegraphics[width=0.49\\linewidth]{.\/grafics\/chapter2\/skyrme\/d_vs_r_Skyrme}}\n\\subfigure{\\label{fig:delta_gogny}\\includegraphics[width=0.49 \\linewidth]{.\/grafics\/chapter2\/Gogny\/d_vs_r_Gogny}}\n\\caption{Panel a: Density dependence of the asymmetry in $\\beta$-stable matter calculated using the\n exact expression of the energy per particle or the expression in Eq.~(\\ref{eq:EOSexpgeneral}) up to second, fourth\n  and tenth order for the Skyrme forces MSk7, SLy4 and SkI5.\n  Panel b: Density dependence of the isospin asymmetry in $\\beta$-stable matter calculated using the exact \n  expression of the EoS or the expression in Eq.~(\\ref{eq:EOSexpgeneral})\n  up to second, fourth, and sixth order for the D1S and D280 interactions. The results of the parabolic \n  approximation are also included in both panels, as well as the results using the second-order expansion of \n  for the potential part of the EoS up to second-order and \n  the full expression for the kinetic energy (label ``Exact E$_\\mathrm{kin}$'').\\label{fig:delta}}\n\\end{figure}\n\nWe display in Fig.~\\ref{fig:delta_skyrme} the isospin asymmetry $\\delta$ corresponding to the \n$\\beta$-equilibrium of {\\it npe} matter as a function of the density calculated\nwith the expansion of the EoS up to second- and tenth-order\nfor three representative Skyrme interactions, MSk7, with a very soft symmetry energy \nof slope $L=9.41$ MeV, SLy4, with with an intermediate slope of $L=45.96$ MeV and SkI5, with a stiff EoS of $L=129.33$ MeV.\nMoreover, we plot in Fig.~\\ref{fig:delta_gogny}~\\cite{gonzalez17} the density dependence of the isospin asymmetry $\\delta$\nfor two illustrative Gogny interactions, namely D1S ($L=18.36$ MeV) and D280 ($L=46.53$ MeV), if the density per particle is expanded up \nto second-, fourth- and sixth-order. \nA general trend for both types of functionals is that if one takes into account more terms in the expansion of the energy per particle,\nthe results are closer to the ones obtained using the exact \nEoS.\nWhen one calculates the $\\beta$-stability condition at second order in the case of Skyrme interactions,\none finds that the results for the isospin asymmetry $\\delta$ calculated with the MSk7, SLy4, and SkI5 interactions do not exceed a\n$5\\%$ of relative differences with respect to the exact results. On the other hand, for interactions that find larger\nrelative differences, the second-order Taylor expansion is not enough to reproduce the exact values, and terms of \na higher order than two are needed. \nIn the case of Gogny interactions, we present the results for the D1S force\nthat has a low slope parameter $L=22.4$~MeV and the results for D280\nthat has $L=46.5$~MeV, the largest $L$ value of the analyzed Gogny forces.\nFor Gogny interactions, there is a trend of having an overall larger isospin \nasymmetry  at densities above $\\sim 0.1$ fm$^{-3}$ for models with softer symmetry energies, \nthat is, the system is more neutron-rich for these interactions. We can see this in Fig.~\\ref{fig:delta_gogny},\nwhere the isospin asymmetry of D1S is larger than the one of D280. This is \nin \nconsonance with the fact that for the same density range the symmetry energy of D1S is smaller than in D280, \nas can be seen in Fig.~\\ref{fig:esym2gog}.\nFor both D1S and D280 interactions, the results obtained when using the EoS up to second order are quite far \nfrom the exact results, hence, higher-order coefficients are needed to lessen these differences. Still, \neven using (\\ref{eq:EOSexpgeneral}) up to sixth-order, the isospin asymmetries found with these interactions are not in line with \nthe values obtained with the exact EoS.\nThe convergence of the symmetry parameter corresponding to $\\beta$-equilibrium obtained starting from the \nexpansion~(\\ref{eq:EOSexpgeneral}) is also model dependent. \n \nIn both panels of Fig.~\\ref{fig:delta} we have also added the results obtained using the parabolic \napproximation~(\\ref{eq:PAgen}) for the EoS.\nIt is interesting to note that they are significantly different from those obtained in the second-order \napproximation. \nIn fact, for the functionals under consideration, the PA asymmetries are overall closer to the exact asymmetries \nthan the second-order values. \n\n\\begin{figure}[t]\n\\centering     \n\\subfigure{\\label{fig:press_skyrme}\\includegraphics[width=0.49\\linewidth]{.\/grafics\/chapter2\/skyrme\/Skyrme_Pressure}}\n\\subfigure{\\label{fig:press_gogny}\\includegraphics[width=0.49 \\linewidth]{.\/grafics\/chapter2\/Gogny\/P_vs_rho_Gogny}}\n\\caption{Panel a: Density dependence of the pressure in $\\beta$-stable matter calculated using the exact expression \n  of the energy per particle \nor the expression in Eq.~(\\ref{eq:EOSexpgeneral}) up to second, fourth, \n and tenth order for three Skyrme forces, MSk7, SLy4, and SkI5.\n  Panel b: Density dependence of the pressure in $\\beta$-stable matter calculated using the \n  exact expression of the EoS or the expression in Eq.~(\\ref{eq:EOSexpgeneral})\n  up to second, fourth, and sixth order for the D1S and D280 interactions. The results of the parabolic \n  approximation are also included in both panels, as well \n  as the results obtained if applying the Taylor expansion only in the potential part of the EoS and using the full expression for its\n  kinetic part (label ``Exact E$_\\mathrm{kin}$''). In both panels the vertical axis is in logarithmic scale.\\label{fig:press}}\n\\end{figure}\n\nWe display in Fig.~\\ref{fig:press_skyrme} the total pressure of the system [cf. Eq.~(\\ref{eq:press_skyrme})] in logarithmic scale  \nat $\\beta$-equilibrium as a function of the density calculated for the same \nthree Skyrme forces for which we have analyzed the density dependence of the isospin asymmetry, namely the MSk7, SLy4, and SkI5 forces. \nMoreover, the same results for the D1S and D280 Gogny interactions are shown in Fig.~\\ref{fig:press_gogny}~\\cite{gonzalez17}, \nwhose pressure is given in Eq.~(\\ref{eq:pressure_bars}).\nWe plot the results calculated using the exact EoS and\nthe pressure obtained starting from  \nthe expansion~(\\ref{eq:EOSexpgeneral}) up to tenth order for Skyrme models and up to sixth order for the Gogny ones. We observe that,\nas happens for the isospin asymmetry $\\delta$, all \nresults obtained with the expanded EoS are closer to the exact values when more terms are included in\nthe calculation.\nAt a density $\\rho \\sim 0.1$ fm$^{-3}$, the relative difference between the pressure obtained\nwith the second-order approximation and the one calculated with the full EoS using \nthe SkI5 interaction is of $1\\%$, while if the expansion is pushed up to the tenth-order in \n$\\delta$, the relative difference is only of $0.05\\%$. The other forces displayed in\nFig.~\\ref{fig:press_skyrme} show a similar behavior. The values of the relative differences \nbetween the exact and approximated pressures computed with the SLy4 interaction are, \n of $4\\%$ and $0.07\\%$ when the expansions are pushed up to second- and tenth-order, respectively,\nand the same differences are $6\\%$ and $0.6\\%$ when the pressure is computed using the MSk7 force.\nAs for the Gogny interactions, the relative differences for the D1S force between the pressure calculated at second \norder and the pressure of the exact EoS are of $30 \\%$ at the largest density ($0.4$ fm$^{-3}$) of the figure.\nWith the corrections up to sixth order included, the differences reduce to $1 \\%$. For D280, these \ndifferences are of $10 \\%$ and $1.5 \\%$, respectively.\nIn all cases, adding more terms to the expansion brings the results closer to the pressure of the exact EoS.\nThe results for the pressure are in keeping with the pure neutron matter predictions of \nRef.~\\cite{Sellahewa14} \nand the $\\beta$-stable calculations of Ref.~\\cite{Loan2011}.\nMoreover, the $\\beta$-stable nuclear matter has also been studied using the PA, and the results are\nplotted in the same Fig.~\\ref{fig:press_skyrme} and Fig.~\\ref{fig:press_gogny} for Skyrme and Gogny forces, respectively. \nIn all cases, the PA clearly improves the results calculated up to quadratic \nterms in the energy per baryon expansion, \nproviding values close to the ones estimated when adding up to the tenth- order in $\\delta$ in the \nTaylor series~(\\ref{eq:EOSexpgeneral}). \n\nSome time ago, it was proposed in Ref.~\\cite{Ducoin11} an improvement of \nthe $E_b(\\rho, \\delta)$ expansion, \nconsisting of using the expansion in powers of the asymmetry $\\delta$ only in the potential energy \npart up to $\\delta^2$, \nwhile using the exact kinetic energy. \nThe underlying reason for this approach is the following. Although the quadratic expansion of the\nenergy per particle in asymmetric nuclear matter is quite accurate to describe the EoS even at high \nisospin asymmetry, this \nexpansion fails in reproducing the spinodal contour in neutron-rich matter~\\cite{Ducoin11} \nbecause the energy \ndensity curvature in the proton density direction diverges at small values of the proton density owing \nthe kinetic energy term \n\\cite{Baldo09}. In addition, we have also seen that a simple quadratic expansion of the \nenergy per particle is not enough to predict accurately the exact isospin asymmetry corresponding to \n$\\beta$-stable NS \nmatter in all the range of considered densities, at least for some of the considered interactions. \nWe plot in Fig.~\\ref{fig:delta} and in Fig.~\\ref{fig:press}, respectively, the isospin asymmetry $\\delta$ and the \ntotal pressure for Skyrme and Gogny interactions if we only expand up to second-order the potential part of the \ninteraction and we keep the full expression for the kinetic part. We have labeled these results as ``Exact E$_\\mathrm{kin}$''.\nThis approximation works very well for most of the interactions, pointing out that the majority of the differences \nbetween the results obtained with the Taylor expansion~(\\ref{eq:EOSexpgeneral}) and the full expression of the EoS come \nfrom the kinetic contribution to the energy.\n\n\\section{Influence of the symmetry energy on neutron star bulk properties}\\label{MRbeta}\n\nWith access to the analytical expressions for the pressure and the energy density \nin asymmetric matter, one can compute the mass-radius relation \nof NSs by integrating the Tolman-Oppenheimer-Volkoff~(TOV) equations~\\cite{shapiro83,Glendenning2000,haensel07}, given by\n\\begin{eqnarray}\n \\frac{dP (r)}{dr} &=& \\frac{G}{r^2 c^2} \\left[\\epsilon (r) + P(r) \\right] \\left[ m(r) + 4 \\pi r^3 P (r)\\right] \n \\left[ 1-\\frac{2Gm(r)}{rc^2}\\right]^{-1}\\label{eq:TOV}\\\\\n \\frac{dm(r)}{dr} &=& 4 \\pi r^2 \\epsilon(r),\\label{eq:TOV2}\n\\end{eqnarray}\nwhere $\\epsilon(r)$, $P(r)$ and $m(r)$ are, respectively, the energy density, pressure and mass at each radius $r$ inside the NS. \nOne starts from the center of the star at a given central energy density $\\epsilon (0)$, central pressure $P(0)$ and mass $m(0)=0$\nand integrates outwards until reaching the surface of the star, where the pressure is $P(R)=0$. The location of the \nsurface of the NS will determine the total radius of the star $R$ and its total mass $M=m(R)$.\n\nWe have solved the TOV equations for a set of Skyrme and Gogny~\\cite{SellahewaPhD, gonzalez17} forces using the $\\beta$-equilibrium \nEoS with the exact isospin asymmetry dependence in the NS core.\nNote that for interactions with very soft symmetry energies, at high densities these conditions may yield a pure NS with $\\delta=1$, and we ignore the effects of an \nisospin instability at and beyond that point. At very low densities, we use the Haensel-Pichon EoS \nfor the outer crust \\cite{douchin01}. \n\nIn the absence of microscopic calculations of the EoS of the inner crust for many Skyrme interactions\nand for Gogny forces, \nwe adopt the prescription of previous works \\cite{Link1999, carriere03,xu09a,Zhang15, gonzalez17} by taking \nthe EoS of the inner crust to be of the polytropic form \n\\begin{equation}\n P=a+b\\epsilon^{4\/3},\n\\end{equation}\nwhere $\\epsilon$ \ndenotes the mass-energy density.\nThe constants $a$ and $b$ are adjusted by demanding continuity at the inner-outer crust interface \nand at the core-crust transition point~\\cite{carriere03,xu09a,Zhang15, gonzalez17}:\n\\begin{eqnarray}\n a&=& \\frac{P_\\mathrm{out} \\epsilon_t^{4\/3}-P_t \\epsilon_\\mathrm{out}^{4\/3}}{\\epsilon_t^{4\/3}-\\epsilon_\\mathrm{out}^{4\/3}}\n\\label{eq:coefa}\\\\\nb&=& \\frac{P_t -P_\\mathrm{out}}{\\epsilon_t^{4\/3}-\\epsilon_\\mathrm{out}^{4\/3}},\\label{eq:coefb}\n \\end{eqnarray}\n where $P_\\mathrm{out}$ ($P_t$) and $\\epsilon_\\mathrm{out}$ ($\\epsilon_t$) are the corresponding\n pressure and energy density at the outer crust-inner crust (core-crust) transition.\n We have calculated the transition density using the thermodynamical method,\n which will be explained later on in Chapter~\\ref{chapter4}~\\cite{gonzalez17}. \nAt the subsaturation densities of the inner crust, the pressure of matter is dominated by the \nrelativistic degenerate electrons. Hence, a polytropic form with an index of average value of about $4\/3$ is found\nto be a good approximation to the EoS in this region \\cite{Link1999,Lattimer01,Lattimer2016}.\nFor more accurate predictions of the crustal properties, it would be of great interest to determine the \nmicroscopic EoS of the crust with the same interaction used for describing the core \\cite{Than2011}.\n\n\\begin{figure}[!b]\n\\centering     \n\\subfigure{\\label{fig:MR_skyrme}\\includegraphics[width=0.49\\linewidth]{.\/grafics\/chapter2\/MR\/MR_Skyrme}}\n\\subfigure{\\label{fig:MR_gogny}\\includegraphics[width=0.49 \\linewidth]{.\/grafics\/chapter2\/MR\/MR_Gogny}}\n\\caption{Mass-radius relation for the NSs for 4 stable Skyrme (panel a) and 4 Gogny (panel b) functionals. \nWe show, with horizontal bands, the accurate \n  $M \\approx 2 M_\\odot$ mass measurements of highly massive NS~\\cite{Demorest10,Antoniadis13}.\nThe vertical green band shows the \\mbox{M-R} region deduced from chiral nuclear interactions up to normal density \nplus astrophysically constrained high-density EoS extrapolations \\cite{Hebeler13}. \nThe brown dotted band is the zone constrained by the cooling tails of type-I X-ray bursts \nin three low-mass X-ray binaries and a Bayesian analysis \\cite{Nattila16}, and the beige \nconstraint at the front is from five quiescent low-mass X-ray binaries and five photospheric\nradius expansion X-ray bursters after a Bayesian analysis~\\cite{Lattimer14}.\nFinally, the squared blue band accounts for \na Bayesian analysis of the data coming from the GW170817 detection of gravitational waves from a binary NS merger~\\cite{Abbott2018}.\n  \\label{fig:MR}\n  }\n\\end{figure}\n\nThe results for the mass-radius relation computed with \nSkyrme and Gogny interactions are presented in Fig.~\\ref{fig:MR}. \nFirst, we compute the mass-radius relation for the same representative Skyrme models, and \nthe results obtained are plotted in Fig.~\\ref{fig:MR_skyrme}. \nWe observe that only four of the five selected Skyrme interactions, namely UNEDF0, SkM$^*$,\nSLy4 and SkI5 provide a stable solution \nof the TOV equations. Moreover, of these ones, only SLy4 and SkI5 can produce an NS\nwith a mass higher than the lower bound $M = 2 M_\\odot$, given by observations of \nhighly massive NSs~\\cite{Demorest10,Antoniadis13}, and the UNEDF0 interaction cannot\nconverge to an NS of a canonical mass of $M = 1.4 M_\\odot$. In order to study the radii \nobtained with Skyrme interactions, we add in Fig.~\\ref{fig:MR_skyrme} boundary conditions for the NS radius coming from different analyses\n\\cite{Hebeler13, Nattila16, Lattimer14, Abbott2018}. \nFrom the plot, we see that the SLy4 Skyrme interaction, \nwhich gives  a radius of $\\sim 10$ km for an NS with maximum mass and $\\sim 11.8$ km for a \ncanonical NS of mass $1.4 M_\\odot$, is the only force that \nfits inside all constraints for the radii coming from low-mass X-ray binaries, X-ray burst sources, and gravitational waves,\nwhich provide radii below 13 km for canonical \nmass stars~\\cite{Nattila16, Lattimer14}. \nThe good behaviour of SLy4 can be understood knowing that it was fitted to both properties of \nfinite nuclei and neutron matter~\\cite{douchin01}.\nOther recent extractions of stellar \nradii from quiescent low-mass x-ray binaries and x-ray burst sources have suggested values in \nthe range of $9-13$ km \\cite{Guillot14,Heinke14,Ozel15,Ozel16}, which would include also the \nresults for SkM$^*$.\n\n\\begin{table}[!t]\n\\resizebox{\\columnwidth}{!}{%\n\\begin{tabular}{ccccccccc}\n\\hline \n\\multirow{2}{*}{Force} & $L$    & $M_\\mathrm{max}$ & $R(M_\\mathrm{max})$ & $\\rho_c(M_\\mathrm{max})$ & $\\epsilon_c(M_\\mathrm{max})$ & $R(1.4M_\\odot)$ & $\\rho_c(1.4M_\\odot)$ & $\\epsilon_c(1.4M_\\odot)$ \\\\\n                       & (MeV)  & ($M_\\odot$)      & (km)                & (fm$^{-3}$)              & ($10^{15}$ g cm$^{-3}$)         & (km)            & (fm$^{-3}$)          & ($10^{15}$ g cm$^{-3}$)                                                                       \\\\\\hline \\hline\nMSk7                   & 9.41   & ---              & ---                 & ---                      & ---                             & ---             & ---                  & ---                                                                                           \\\\\nSIII                   & 9.91   & 1.185            & 5.68                & 3.68                     & 8.81                            & ---             & ---                  & ---                                                                                           \\\\\nSkP                    & 19.68  & ---              & ---                 & ---                      & ---                             & ---             & ----                 & ---                                                                                           \\\\\nHFB-27                 & 28.50  & 1.530            & 7.91                & 2.04                     & 4.81                            & 9.00            & 1.29                 & 2.57                                                                                          \\\\\nSKX                    & 33.19  & 1.396            & 7.99                & 2.16                     & 4.97                            & ---             & ----                 & ---                                                                                           \\\\\nHFB-17                 & 36.29  & 1.767            & 8.96               & 1.56                     & 3.68                            & 10.65           & 0.77                 & 1.46                                                                                          \\\\\nSGII                   & 37.63  & 1.663            & 8.88                & 1.65                     & 3.82                            & 10.49           & 0.84                 & 1.61                                                                                          \\\\\nUNEDF1                 & 40.01  & 1.157            & 8.90                & 1.89                     & 3.94                            & ---             & ---                  & ---                                                                                           \\\\\nSk$\\chi$500             & 40.74  & 2.142            & 10.47               & 1.09                     & 2.52                            & 11.73           & 0.49                 & 0.88                                                                                          \\\\\nSk$\\chi$450             & 42.01  & 2.098            & 10.05               & 1.18                     & 2.80                            & 11.82           & 0.53                 & 0.97                                                                                          \\\\\nUNEDF0                 & 45.08  & 1.029            & 9.97                & 1.47                     & 2.89                            & ---             & ---                  & ---                                                                                           \\\\\nSkM*                   & 45.78  & 1.618            & 9.01                & 1.66                     & 3.81                            & 10.69           & 0.86                 & 1.65                                                                                          \\\\\nSLy4                   & 45.96  & 2.056             & 10.03               & 1.20                     & 2.84                            & 11.82           & 0.53                 & 0.98                                                                                          \\\\\nSLy7                   & 47.22  & 2.080            & 10.16               & 1.17                     & 2.77                            & 11.97           & 0.52                 & 0.94                                                                                          \\\\\nSLy5                   & 48.27  & 2.060            & 10.09               & 1.19                     & 2.81                            & 11.91           & 0.53                 & 0.96                                                                                          \\\\\nSk$\\chi$414             & 51.92  & 2.110            & 10.37               & 1.13                    & 2.64                            & 12.01           & 0.50                 & 0.91                                                                                          \\\\\nMSka                   & 57.17  & 2.320            & 11.23               & 0.95                     & 2.24                            & 13.14           & 0.39                 & 0.70                                                                                          \\\\\nMSL0                   & 60.00  & 1.955            & 10.15               & 1.24                     & 2.87                            & 12.21           & 0.53                 & 0.97                                                                                          \\\\\nSIV                    & 63.50  & 2.380            & 11.60               & 0.90                     & 2.11                            & 13.61           & 0.36                 & 0.65                                                                                          \\\\\nSkMP                   & 70.31  & 2.120            & 10.72               & 1.09                     & 2.54                            & 12.94           & 0.44                 & 0.79                                                                                          \\\\\nSKa                    & 74.62  & 2.217            & 11.07               & 1.01                     & 2.38                            & 13.44           & 0.40                 & 0.73                                                                                          \\\\\nR$_\\sigma$             & 85.69  & 2.140            & 11.01               & 1.04                     & 2.42                            & 13.40           & 0.40                 & 0.72                                                                                          \\\\\nG$\\sigma$              & 94.01  & 2.151            & 11.14               & 1.02                     & 2.38                            & 13.62           & 0.38                 & 0.69                                                                                          \\\\\nSV                     & 96.09  & 2.452            & 11.97               & 0.85                     & 1.99                            & 14.24           & 0.32                 & 0.57                                                                                          \\\\\nSkI2                   & 104.33 & 2.201            & 11.46               & 0.97                     & 2.24                            & 13.96           & 0.35                 & 0.63                                                                                          \\\\\nSkI5                   & 129.33 & 2.283            & 11.89               & 0.90                     & 2.09                            & 14.63           & 0.31                 & 0.55                                                                                          \\\\\n\\hline \nD260                   & 17.57  & ---              & ---                 & ---                      & ---                            & ---             & ---                  & ---                                                                                           \\\\\nD1                     & 18.36  & ---              & ---                 & ---                      & ---                            & ---             & ---                  & ---                                                                                           \\\\\nD1S                    & 22.43  & ---              & ---                 & ---                      & ---                           & ---             & ---                  & ---                                                                                           \\\\\nD1M                    & 24.83  & 1.745            & 8.84                & 1.58                     & 3.65                            & 10.14           & 0.80                 & 1.51                                                                                          \\\\\nD250                   & 24.90  & ---              & ---                 & ---                      & ---                            & ---             & ---                  & ---                                                                                           \\\\\nD300                   & 25.84  & 0.724            & 6.04                & 4.71                     & 10.07                           & ---             & ---                  & ---                                                                                           \\\\\nD1N                    & 33.58  & 1.230            & 7.77                & 2.38                     & 5.27                            & ---             & ---                  & ---                                                                                           \\\\\nD280                   & 46.53  & 1.662            & 9.78                & 1.45                     & 3.27                            & 11.71           & 0.69                 & 1.30\t\t\t\t\t\t\t\t\t\t\t\\\\ \\hline                                                                                         \n\\end{tabular}\n}\n\\caption{Properties (mass $M$, radius $R$, central density $\\rho_c$ and central mass-energy density $\\epsilon_c$) \nfor the maximum mass and the $1.4 M_\\odot$ configurations of NSs predicted by different Skyrme and Gogny functionals.\n\\label{Table-NSs}}\n\\end{table}\n\nFig.~\\ref{fig:MR_gogny} contains the results of the mass versus radius only for \nthe four Gogny functionals (of the ones we have studied) that provide \nsolutions for NSs. \nAlso, we see that all Gogny EoSs predict maximum NS masses that are well below \nthe observational limit of $M \\approx 2 M_\\odot$ from Refs.~\\cite{Demorest10,Antoniadis13}. \nAs a matter of fact, only D1M and D280 are able to generate masses above the canonical $1.4 M_\\odot$ value. \nThe NS radii from these two EoSs are considerably different, however, with D1M producing \nstars with radii $R\\approx 9-10.5$ km, and D280 stars with radii $R \\approx 10-12$ km. \nThese small radii for a canonical NS would be in line with recent extractions of stellar \nradii from quiescent low-mass X-ray binaries, X-ray burst sources and gravitational waves, that have suggested values in \nthe range of $9-13$ km \\cite{Hebeler13,Lattimer14, Guillot14,Heinke14,Ozel15,Lattimer2016,Ozel16,Nattila16}. \nIt appears that a certain degree of softness of the nuclear symmetry energy is necessary in order \nto reproduce small radii for a canonical mass NS~\\cite{Chen15,Jiang15,Tolos16}. \nThe parameterizations D1N and D300, in contrast to D1M and D280, generate NSs which are\nunrealistically small in terms of both mass and radius. \nOne should of course be cautious in \ninterpreting these results. Most Skyrme and Gogny forces have not been fitted to reproduce high-density, neutron-rich \nsystems and it is not surprising that some parametrizations do not yield realistic NSs. \nOne could presumably improve these results by guaranteeing that, \nat least around the saturation region, \nthe pressure of neutron-rich matter is compatible with NS observations \\cite{Lattimer2013}. \n\nWe provide in Table~\\ref{Table-NSs} data on the \nmaximum mass and $1.4 M_\\odot$ configurations \nof NSs produced by the different sets of Skyrme and Gogny models. The majority of\nmaximum mass \nconfigurations are reached at central baryon number densities close to $\\sim 7-8 \\rho_0$ for \nSkyrme interactions and $\\sim 10 \\rho_0$ for Gogny forces, \nwhereas $1.4 M_\\odot$ NSs have central baryon densities close to around $2-3 \\rho_0$\nand $4-5 \\rho_0$ for Skyrme and Gogny forces, respectively. \nThese large central density values  for Gogny are in keeping with the fact that the neutron matter Gogny \nEoSs are relatively soft, which requires larger central densities to produce realistic NSs.\n\n\\begin{figure}[!t]\n\\centering     \n\\subfigure{\\label{fig:MR_skyrme_orders}\\includegraphics[width=0.49\\linewidth]{.\/grafics\/chapter2\/MR\/MR_Skyrme_orders}}\n\\subfigure{\\label{fig:MR_gogny_orders}\\includegraphics[width=0.49 \\linewidth]{.\/grafics\/chapter2\/MR\/MR_Gogny_orders}}\n\\caption{Mass-radius relation computed with two Skyrme (panel a) and two Gogny (panel b) forces. The results have been \nobtained using the EoS computed through its Taylor expansion up to second- and up to tenth- (Skyrme) or \nup to sixth- (Gogny) order. Moreover, the results calculated with the full EoS are also included. \nWe show the constraints for the maximum mass of $2 M_\\odot$ from the mass measurements of Refs.~\\cite{Demorest10, Antoniadis13}.\n  \\label{fig:MRorders}}\n\\end{figure}\n\nTo analyze the impact on the expansion of the EoS of higher-order terms in its Taylor expansion, \nwe plot in Fig.~\\ref{fig:MRorders} the mass-radius relation for the SLy4 and SkI5 Skyrme interactions [see Fig.~\\ref{fig:MR_skyrme_orders}]\nand for the D1M and D280 Gogny forces [see Fig.~\\ref{fig:MR_gogny_orders}].\nIn black solid lines, we have plotted the results when the full EoS of each type of interaction is used. These values\nare the same as the ones in Fig.~\\ref{fig:MR}. To test the accuracy of the Taylor expansion of the EoS, \nwe first compute the TOV equations when the EoSs are expanded up to second order, and the mass-radius \nrelation is plotted in the same Fig.~\\ref{fig:MRorders} with red dashed-dotted lines. We also include \nthe results if they are obtained using the expansion up to the highest order we have considered \nfor each type of model. For Skyrme interactions, this is up to the tenth order and, for Gogny interactions,\nit is up to the sixth order, and it is represented with blue dashed-double dotted lines.\nIn general, if we use the second-order EoS expansion, we get results that are relatively far from the ones\nobtained using the full EoS, which has been labeled them as ``Exact'' in Fig.~\\ref{fig:MRorders}.\nThe results using the approximated SLy4 EoS up to second-order are close to their corresponding exact mass-radius relation, \nonly differing in the low-mass regime, giving higher values of the radii, and if the EoS up to tenth-order\nis used, the results  are even closer. For the SLy4 interaction, \nwe obtain relative differences in the radius of a canonical NS mass of $1.4 M_\\odot$ of \n$0.6\\%$ if we compare the second-order approach and the exact results and of $0.4\\%$ if we compare the values\nobtained with a tenth-order expansion of the EoS and the exact ones.\nOn the other hand, for the SkI5 interaction, the scenario is very different. \nIf we plot the results using the EoS expanded even up to the tenth order, the results of the mass-radius relation\nare quite different from the ones obtained using the full EoS, especially at the low-mass regime. In this case, the relative differences \nobtained between the approximated results for the mass and the exact ones are of $11\\%$ if we consider a second-order\nexpansion and of $6\\%$ if we consider a tenth-order approximation.\nSuch differences for low-mass NSs could be explained given the fact that the transition density \ngiven by the different EoS expansions may be quite different from the one obtained with the full EoS. \nThis is more prominent for interactions with large values of the slope of the symmetry energy (see Chapter~\\ref{chapter4}). \nTherefore, the polytropic equation of state used to reproduce the inner crust may bring some uncertainties given these\ndifferences in the transition point. \nIn the case of Gogny interactions, we see that the second-order EoS expansion gives a \nhigher maximum mass for NSs at a higher radius for the case of the D1M interaction. Then, if instead of the second-order we consider\nthe sixth-order EoS expansion, the results are almost the same as the exact ones. For D1M we get \nrelative differences for a canonical NS of $4\\%$ if we use the EoS cut at second-order in asymmetry\nand of $0.2\\%$ if we cut the expansion at sixth-order. \nIn the case of the D280 Gogny interaction, if we use the second-order \napproach, we get results for the mass-radius relation that are below the exact ones. On the other hand, \nusing the sixth-order expansion, the mass-radius relation becomes very close to the exact one at high densities, \nwhile at low densities it gives higher values for the radii. \nThe relative differences we get for the radius of a $1.4 M_\\odot$ NS are of $4\\%$ while \ncomparing the EoS of second-order in $\\delta$ and the exact results and of $0.8\\%$ if comparing \nthe values computed with the full EoS and with its sixth-order expansion. \n\n\\chapter{D1M$^*$ and D1M$^{**}$ Gogny parametrizations}\\label{chapter3}\nIn spite of their accurate description of ground-state properties of finite nuclei,\nthe extrapolation of Gogny interactions to the neutron star (NS) domain, as we have seen in Chapter~\\ref{chapter2}, is not completely satisfactory. \nThe successful Gogny D1S, D1N and D1M forces of the D1 family, which nicely reproduce the ground-state\nproperties of finite nuclei are unable to reach a maximal \nNS mass of 2$M_{\\odot}$ as required by recent astronomical observations \n\\cite{Demorest10,Antoniadis13}. Actually, only the D1M interaction predicts an NS mass \nabove the canonical value of 1.4$M_{\\odot}$ \\cite{Sellahewa14, gonzalez17}.\nAs we have pointed out in the previous chapter, the basic underlying\nproblem lies in the fact that the symmetry energy, \nwhich determines the NS EoS and hence the maximal mass predicted by the model, \nis too soft in the high-density regime.\n\nWe have tried to reconcile this problem introducing two new Gogny \ninteractions, which we call D1M$^*$\\cite{gonzalez18} and D1M$^{**}$ \\cite{gonzalez18a}, \nby reparametrizing one of \nthe most recognized Gogny interactions, the  D1M force, to be able to explain the properties of \nNSs and, at the same time, preserving its \nsuccessful predictions in the domain of finite nuclei.\n\n\n\\section{Fitting procedure of the new D1M$^*$ and D1M$^{**}$ Gogny interactions and properties of their EoSs}\n\nTo determine the new Gogny interactions D1M$^*$ and D1M$^{**}$, we have modified the values of the parameters that control  \nthe stiffness of the symmetry energy while retaining as much as possible the quality of D1M \nfor the binding energies and charge radii of nuclei~\\cite{gonzalez18, gonzalez18a}. \nThe fitting procedure is similar to \nprevious literature where families of Skyrme and RMF parametrizations were generated starting from accurate models, \nas for example the \\mbox{SAMi-J}~\\cite{Roca13}, KDE0-J~\\cite{Agrawal05} or FSU-TAMU~\\cite{Piekarewicz11,Fattoyev13} families.\nThe basic idea to obtain these  families is the following. Starting from a well calibrated and successful mean-field\nmodel, one modifies the values of some parameters, which determine the symmetry energy, around their optimal values\nretaining as much as possible values of the binding energies and radii of finite nuclei of the original model.\n\nIn our case, we readjust the eight parameters $W_i$, $B_i$, $H_i$, $M_i$ ($i=1,2$) of the \nfinite-range part of the Gogny interaction (\\ref{VGogny}),\nwhile the other parameters, namely the ranges of the Gaussians, the zero-range part of the\ninteraction and the spin-orbit force, are kept fixed to the values of D1M.\nThe open parameters are constrained by requiring the same saturation density, energy per particle, incompressibility and effective mass \nin symmetric nuclear matter as in the original D1M force, and, in order to have a correct description of \nasymmetric nuclei, the same value of $E_{\\rm sym}(0.1)$, i.e., the symmetry energy at density 0.1 fm$^{-3}$.\nThe last condition is based on the fact that the binding energies of finite nuclei constrain \nthe symmetry energy at an average density of nuclei of about 0.1 fm$^{-3}$ more tightly than \nat the saturation density $\\rho_0$ \\cite{horowitz01a,Centelles09}.\nTo preserve the pairing properties in the $S=0$, $T=1$ channel,\nwe demand in the new forces the same value of D1M for the two combinations of parameters $W_i-B_i-H_i+M_i$ ($i=1,2$). \nThus, we are able to obtain seven of the eight free parameters of D1M$^*$ and of D1M$^{**}$ as a function of a \nsingle parameter, which we choose to be $B_1$. This parameter is used to modify the slope $L$ of the symmetry energy at \nsaturation and, therefore, the behavior of the neutron matter EoS above saturation, which in turn determines the \nmaximum mass of NSs by solving the TOV equations [see Eqs.(\\ref{eq:TOV}) and (\\ref{eq:TOV2}) in Chapter~\\ref{chapter2}]. \nWe adjust $B_1$ so that the maximum NS mass predicted by\nthe D1M$^*$ force is 2$M_{\\odot}$. With the same strategy, we have also fitted the \nD1M$^{**}$ force, but imposing a constraint on the maximum\nNS mass of  1.91$M_{\\odot}$, which is close to the lower limit, within the error \nbars, of the heaviest observed masses of NS at the date of the fitting\n\\cite{Demorest10,Antoniadis13}. \nFinally, in the case of D1M$^*$, we perform a small readjustment of the zero-range strength $t_3$ of about 1 MeV fm$^4$ to optimize \nthe results for nuclear masses, which\ninduces a slight change in the values of the saturation properties of uniform matter~\\cite{gonzalez18,gonzalez18a}.\n\n\\begin{table}[t!]\n\\centering\n\\begin{tabular}{c|rrrrr}\n\\hline\nD1M      & \\multicolumn{1}{c}{$W_i$} & \\multicolumn{1}{c}{$B_i$} & \\multicolumn{1}{c}{$H_i$} & \\multicolumn{1}{c}{$M_i$} & \\multicolumn{1}{c}{$\\mu_i$}  \\\\ \\hline\\hline\n$i$=1    & -12797.57  & 14048.85   & -15144.43  & 11963.81   & 0.50 \\\\\n$i$=2    &    490.95  &  -752.27   &    675.12  &  -693.57   & 1.00  \\\\\\hline\nD1M$^*$     & \\multicolumn{1}{c}{$W_i$} & \\multicolumn{1}{c}{$B_i$} & \\multicolumn{1}{c}{$H_i$} & \\multicolumn{1}{c}{$M_i$} & \\multicolumn{1}{c}{$\\mu_i$}  \\\\ \\hline\\hline\n$i$=1    & -17242.0144 & 19604.4056  & -20699.9856 & 16408.3344 & 0.50 \\\\\n$i$=2    &    712.2732 &  -982.8150  &    905.6650 &  -878.0060 & 1.00 \\\\\\hline\nD1M$^{**}$     & \\multicolumn{1}{c}{$W_i$} & \\multicolumn{1}{c}{$B_i$} & \\multicolumn{1}{c}{$H_i$} & \\multicolumn{1}{c}{$M_i$} & \\multicolumn{1}{c}{$\\mu_i$}  \\\\ \\hline\\hline\n$i$=1    & -15019.7922 & 16826.6278 & -17922.2078 & 14186.1122 & 0.50 \\\\\n$i$=2    &    583.1680 &  -867.5425  &   790.3925 &  -785.7880 & 1.00 \\\\\\hline\n\\end{tabular}\n\\caption{Parameters of the D1M, D1M$^*$ and D1M$^{**}$ Gogny interactions, where $W_i$, $B_i$, $H_i$ and\n$M_i$ are in MeV and $\\mu_i$ in fm. The coefficients $x_3=1$, $\\alpha=1\/3$ and $W_{LS}=115.36$ MeV fm$^5$ \nare the same in the three interactions, and $t_3$ has values of $t_3=1562.22$ MeV fm$^4$ for the Gogny D1M and D1M$^{**}$\n forces and $t_3=1561.22$ MeV fm$^4$ for the D1M$^*$ interaction.}\n\\label{param}\n\\end{table}\n \n\\begin{table}[t!]\n\\centering\n\\begin{tabular}{lccccccc}\n\\hline\n         &$\\rho_0$ & $E_0$ & $K_0$ & $m^*\/m$   & $E_{\\rm sym}(\\rho_0)$ & $E_{\\rm sym}(0.1)$ & $L$  \\\\\n         & (fm$^{-3}$) &  (MeV)  &  (MeV) &        & (MeV) &(MeV) & (MeV)\\\\  \\hline \\hline\nD1M$^*$  &       0.1650          & $-$16.06  & 225.4   &  0.746  & 30.25   & 23.82 & 43.18  \\\\\nD1M$^{**}$  &    0.1647          & $-$16.02  & 225.0   &  0.746  &29.37   & 23.80 & 33.91  \\\\\nD1M      &       0.1647          & $-$16.02  & 225.0   &  0.746  & 28.55   & 23.80 & 24.83  \\\\\nD1N      &       0.1612          & $-$15.96  & 225.7   &  0.697  & 29.60   & 23.80 & 33.58 \\\\\nD1S      &       0.1633          & $-$16.01  & 202.9   &  0.747  & 31.13   & 25.93 & 22.43 \\\\\nD2       &       0.1628          & $-$16.00  & 209.3   &  0.738  & 31.13   & 24.32 & 44.85  \\\\\nSLy4     &       0.1596          & $-$15.98  & 229.9   &  0.695  & 32.00   & 25.15 & 45.96 \\\\\\hline\n\\end{tabular}\n\\caption{Nuclear matter properties predicted by the D1M$^{*}$, D1M$^{**}$, D1M, D1N, D1S and D2 Gogny\ninteractions and the SLy4 Skyrme force.}\n\\label{inm}\n\\end{table}\n \nThe parameters of the new forces D1M$^*$ and D1M$^{**}$ are collected in Table \\ref{param} and several nuclear\nmatter properties predicted by these forces are collected in Table \\ref{inm} \\cite{gonzalez18, gonzalez18a}. \nTable~\\ref{inm} also collects the properties predicted by other Gogny forces, namely the D1M, D1N, D1S, and D2 parametrizations,\nand by the Skyrme SLy4 interaction.\nAs stated previously, the SLy4 force \\cite{sly42} is a Skyrme force specially designed to predict results in agreement with experimental\ndata of finite nuclei as well as with astronomical observations. We have used this force as a benchmark for comparison with the results \nobtained with the D1M$^*$ and D1M$^{**}$ models. \n\nWe observe that the change of the finite-range parameters, as compared with the original\nones of the D1M force, is larger for the D1M$^*$ force than for the D1M$^{**}$ interaction because the variation in the\nisovector sector is more important in the former than in the latter force. \nThough the change in the $W_i$, $B_i$, $H_i$, $M_i$ values is relatively large with respect to the D1M values \\cite{goriely09}, \nthe saturation properties of symmetric nuclear matter obtained with D1M$^*$ and D1M$^{**}$, {\\it e.g.}\nthe saturation density $\\rho_0$, the binding energy per nucleon at\nsaturation $E_0$, the incompressibility $K_0$ and the effective mass $m^*\/m$, as well as the symmetry energy at 0.1~fm$^{-3}$  \nare basically the same as in D1M (see Table~\\ref{inm}). \nThe mainly modified property is the density dependence of the symmetry energy, with a change in the slope from $L=24.83$ MeV \nto $L=43.18$ MeV for D1M$^*$ and to $L=33.91$ MeV for D1M$^{**}$, in order to provide a stiffer neutron matter \nEoS and limiting NS masses of $2 M_\\odot$ and $1.91 M_\\odot$, \nrespectively.\nThe different $L$ value, as we fixed $E_{\\rm sym}(0.1)$, implies that \nthe symmetry energy $E_{\\rm sym}(\\rho_0)$ at saturation differs in D1M$^*$ and D1M$^{**}$ from D1M, but in a much less extent.\nThe D2 interaction, which differs from the other Gogny interactions used in this thesis for its finite-range density \ndependent term (see Section~\\ref{Gogny} of Chapter~\\ref{chapter1}), has a slope parameter of $L=44.85$ MeV. This value is fairly larger than \nthe values predicted by the D1 family and close to $L$ obtained for D1M$^*$. \n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.75\\columnwidth,clip=true]{.\/grafics\/chapter3\/Esym_Pressure.pdf}\n\\caption{a) Symmetry energy versus density from the D1S, D1N, D1M, D1M$^*$, D1M$^{**}$ and D2 Gogny forces \nand from the SLy4 Skyrme force. The inset is a magnified view of the low-density region.\nAlso plotted are the constraints from isobaric analog states (IAS) and from IAS and neutron skins (IAS+n.skin) \\cite{Danielewicz13}, \nfrom the electric dipole polarizability in lead ($\\alpha_D$ in $^{208}$Pb) \\cite{Zhang15}\nand from transport in heavy-ion collisions (HIC) \\cite{Tsang08}.\nb) Pressure in $\\beta$-stable nuclear matter in logarithmic scale\nas a function of density for the same interactions of panel~a). The shaded area depicts the\nregion compatible with collective flow in HICs~\\cite{Danielewicz:2002pu}.\\label{fig:essympres}}\n\\end{figure}\n\n\nThe symmetry energy as a function of density is displayed in Fig.~\\ref{fig:essympres}(a) for several\nGogny forces and for the SLy4 Skyrme interaction. \nAt subsaturation densities, the symmetry energy of the considered forces displays a similar behavior  \nand takes a value of about 30 MeV at saturation (see Table~\\ref{inm}).\nThe subsaturation regime is also the finite nuclei regime, where the parameters of the nuclear forces are fitted to.\nIndeed, we observe in Fig.~\\ref{fig:essympres}(a) that at subsaturation the present forces fall within, or are very close, to the region \ncompatible with recent constraints on $E_{\\rm sym} (\\rho)$ deduced from several nuclear observables \\cite{Danielewicz13,Zhang15,Tsang08}.\nIn contrast, above saturation density, the behavior of the calculated symmetry energy shows a strong model dependence. \nFrom this figure, two different patterns can be observed. On the one hand,\nas seen in Chapter~\\ref{chapter2},\nthe symmetry energy computed with the  D1S, D1N, and D1M interactions increases till reaching a maximum\nvalue around $30$--$40$ MeV and then bends and decreases with increasing density until vanishing at some\ndensity where the isospin instability starts. \nAlthough this happens at large densities for terrestrial phenomena, it is critical for NSs, \nwhere larger densities occur in the star's interior. \nOn the other hand, the other forces, namely D1M$^*$, D1M$^{**}$\nand D2, predict a symmetry energy with a well defined increasing trend with growing density. This different\nbehaviour of the symmetry energy strongly influences the EoS of the NS matter as it can be seen in Fig~\\ref{fig:essympres}(b). \nIn this figure, \nthe EoS (total pressure against density) of $\\beta$-stable, globally charge-neutral NS matter \\cite{Sellahewa14,gonzalez17, gonzalez18} \ncalculated with the given functionals is displayed.\nThe new Gogny forces D1M$^*$ and D1M$^{**}$, and the D2 force predict a high-density EoS with a similar stiffness to \nthe SLy4 EoS and they agree well with the region constrained by collective flow in energetic \nheavy-ion collisions (HIC) \\cite{Danielewicz:2002pu}, shown as the shaded area in \nFig.~\\ref{fig:essympres}(b).\\footnote{Though the constraint of \\cite{Danielewicz:2002pu} was proposed for neutron matter,\nat these densities the pressures of $\\beta$-stable matter and neutron matter are very similar.}\nThe EoSs from the original D1M parametrization and from D1N are significantly softer, stating that even though\n interactions may have increasing EoSs with the density, they may not be able to provide NSs of $2 M_\\odot$.\nThe D1S force yields a too soft EoS soon after saturation density, which implies it is not \nsuitable for describing NSs. \n\n\nA few recent bounds on $E_{\\rm sym}(\\rho_0)$ and $L$ proposed  \nfrom analyzing different laboratory data and astrophysical observations \n\\cite{BaoAnLi13,Lattimer2013,Roca-Maza15} and from ab initio nuclear calculations using\nchiral interactions \\cite{Hagen:2015yea,Birkhan16}\nare represented in Fig.~\\ref{esym_l}. The prediction of D1M$^*$ is seen to overlap with the various constraints.\nWe note this was not incorporated in the fit of D1M$^*$ (nor of D1M$^{**}$).\nIt follows as a consequence of having tuned the density dependence of the symmetry energy of the interaction  \nto be able to reproduce heavy NS masses simultaneously with the properties of nuclear matter and nuclei.\nOn the other hand, the predictions for D1M$^{**}$ fall in between the ones of D1M and D1M$^*$, \nvery near the values for D1N, outside the constraint bands. \nD2 and SLy4 also show good agreement with the constraints of Fig.~\\ref{esym_l}. \nWe observe that the three interactions D1M$^*$, D2 and SLy4 have $E_{\\rm sym}(\\rho_0)$ values of 30--32 MeV\nand $L$ values of about 45 MeV. A similar feature was recently found in the frame of RMF \nmodels if the radii of canonical NSs are to be no larger than $\\sim$13~km \n\\cite{Chen:2014mza,Tolos16}. It seems remarkable that \nmean-field models of different nature (Gogny, Skyrme, and RMF) converge\nto specific values $E_{\\rm sym}(\\rho_0)\\sim$30--32 MeV and $L\\sim$\\,45 MeV for the nuclear symmetry energy\nwhen the models successfully predict the properties of nuclear matter and finite nuclei \nand heavy NSs with small stellar radii. \n\n\\begin{figure}[t!]\n\\centering\n\\includegraphics[width=0.65\\columnwidth,clip=true]{.\/grafics\/chapter3\/Esym_L.pdf}\n\\caption{Slope $L$ and value $E_{\\rm sym}(\\rho_0)$ of the symmetry energy at saturation density\nfor the discussed interactions. The hatched regions are the experimental and theoretical constraints \nderived in \\cite{BaoAnLi13,Lattimer2013,Roca-Maza15,Hagen:2015yea,Birkhan16}.}\n\\label{esym_l}\n\\end{figure}\n\\begin{figure}[!b]\n \\centering\n \\includegraphics[width=0.65\\linewidth, clip=true]{.\/grafics\/chapter3\/Landau_parameters}\n  \\caption{Density dependence of the Landau parameters $F_0^\\mathrm{ST}$ for the D1S, D1M, D1N, \nD1M$^*$ and D1M$^{**}$ Gogny interactions.\\label{landau0}}\n\\end{figure}\nIn order to further study the behavior of the new interactions D1M$^*$ and D1M$^{**}$ we \nanalyze their corresponding Landau parameters $F_l^\\mathrm{ST}$.\nThe Landau parameters for Gogny interactions corresponding to the spin-isospin channels ST, \n$f_l^\\mathrm{ST}$, of $l=0$ and $l=1$ are the following~\\cite{Ventura94}:\n\\begin{eqnarray}\n f_0^{00} (k_F, k_F, 0) &=& \\frac{3}{8} t_3 (\\alpha + 1) (\\alpha +2) \\rho^\\alpha + \\frac{\\pi^{3\/2}}{4} \\sum_i\n \\mu_i^3 \\left\\{ \\left[ 4 W_i + 2 \\left( B_i - H_i\\right) - M_i\\right] \\vphantom{\\frac{1}{1}}\\right. \\nonumber\\\\\n &&\\left.- \\left[ W_i - 2 \\left( B_i - H_i \\right) - 4 M_i \\right] e^{-\\gamma_i} \\frac{\\sinh (\\gamma_i)}{\\gamma_i} \\right\\}\n\\\\\n f_0^{01} (k_F, k_F, 0) &=& -\\frac{1}{4} t_3 \\rho^\\alpha (1+ 2 x_3)- \\frac{\\pi^{3\/2}}{4} \\sum_i\n \\mu_i^3 \\left\\{ \\left( 2 H_i + M_i \\right) \\vphantom{\\frac{1}{1}}\\right. \\nonumber\\\\\n &&\\left. +  \\left( W_i + 2 B_i \\right) e^{-\\gamma_i} \\frac{\\sinh (\\gamma_i)}{\\gamma_i} \\right\\}\n\\\\\n f_0^{10} (k_F, k_F, 0) &=& -\\frac{1}{4} t_3 \\rho^\\alpha (1- 2 x_3)+ \\frac{\\pi^{3\/2}}{4} \\sum_i\n \\mu_i^3 \\left\\{ \\left( 2 B_i - M_i \\right) \\vphantom{\\frac{1}{1}}\\right. \\nonumber\\\\\n &&\\left. -  \\left( W_i - 2 H_i \\right) e^{-\\gamma_i} \\frac{\\sinh (\\gamma_i)}{\\gamma_i} \\right\\}\n\\\\\n f_0^{11} (k_F, k_F, 0) &=& -\\frac{1}{4} t_3 \\rho^\\alpha - \\frac{\\pi^{3\/2}}{4} \\sum_i\n \\mu_i^3 \\left\\{M_i \n +  W_i e^{-\\gamma_i} \\frac{\\sinh (\\gamma_i)}{\\gamma_i} \\right\\}\n\\\\\nf_1^{00} (k_F, k_F, 0) &=& - \\frac{3\\pi^{3\/2}}{4} \\sum_i\n \\mu_i^3 \\left[ W_i + 2 \\left(  B_i - H_i \\right) - 4 M_i \\right] e^{-\\gamma_i} \\nonumber\\\\\n && \\times \\left( \\frac{ \\cosh (\\gamma_i)}{\\gamma_i} - \\frac{\\sinh (\\gamma_i)}{\\gamma_i^2} \\right) \n \\\\\n f_1^{01} (k_F, k_F, 0) &=& - \\frac{3\\pi^{3\/2}}{4} \\sum_i\n \\mu_i^3 \\left( 2 B_i + W_i\\right) e^{-\\gamma_i} \\left( \\frac{ \\cosh (\\gamma_i)}{\\gamma_i} - \\frac{\\sinh (\\gamma_i)}{\\gamma_i^2} \\right) \n \\\\\n  f_1^{10} (k_F, k_F, 0) &=& - \\frac{3\\pi^{3\/2}}{4} \\sum_i\n \\mu_i^3 \\left(  W_i- 2 H_i\\right) e^{-\\gamma_i} \\left( \\frac{ \\cosh (\\gamma_i)}{\\gamma_i} - \\frac{\\sinh (\\gamma_i)}{\\gamma_i^2} \\right) \n \\\\\n f_1^{11} (k_F, k_F, 0) &=& - \\frac{3\\pi^{3\/2}}{4} \\sum_i\n \\mu_i^3W_i e^{-\\gamma_i} \\left( \\frac{ \\cosh (\\gamma_i)}{\\gamma_i} - \\frac{\\sinh (\\gamma_i)}{\\gamma_i^2} \\right) \n\\end{eqnarray}\nwhere \n\n\\begin{equation}\n \\gamma_i = \\frac{\\mu_i^2 k_F^2}{2}.\n\\end{equation}\nThe dimensionless Landau parameters are given by\n\\begin{equation}\n F_l^{\\mathrm{ST}} = \\frac{2 m^* (k_F) k_F}{\\pi^2 \\hbar^2} f_l^{\\mathrm{ST}},\n\\end{equation}\nwhere $m^*$ is the effective mass which is given by\n\\begin{eqnarray}\n \\frac{m}{m^*} &=& 1- \\sum_i \\left\\{ \\vphantom{\\frac{1}{1}} \\frac{C_{1i}}{\\mu_i^3 k_F^3 } \\left[ 2 - \n \\mu_i^2 k_F^2 - \\left( 2 + \\mu_i^2 k_F^2 \\right) e^{-\\mu_i^2 k_F^2 }\\right] \\right.\\nonumber\\\\\n && \\left.- \\frac{C_{2i}}{\\mu_i^3 k_F^3 } \\left[ \\left( 2 + \\mu_i^2 k_F^2  \\right)  e^{-\\mu_i^2 k_F^2 } - \n \\left( 2 - \\mu_i^2 k_F^2 \\right) \\right]\\right\\}.\n\\end{eqnarray}\n Moreover, the coefficients $C_{1i}$ and $C_{2i}$ are, respectively,\n\\begin{equation}\n C_{1i} = \\frac{m \\mu_i^2}{\\hbar^2 \\sqrt{\\pi}} \\left( B_i + \\frac{W_i}{2} - M_i -  \\frac{H_i}{2} \\right)\n\\end{equation}\nand\n\\begin{equation}\n C_{2i} = \\frac{m \\mu_i^2}{\\hbar^2 \\sqrt{\\pi}} \\left( M_i + \\frac{H_i}{2}\\right).\n\\end{equation}\n\n\nWe plot in Fig.~\\ref{landau0} the density dependence of the Landau parameters $F_0^\\mathrm{ST}$ and  in Fig.~\\ref{landau1} the density dependence of the \nLandau parameters $F_1^\\mathrm{ST}$ \nfor the Gogny D1S, D1M, D1N, D1M$^*$ and D1M$^{**}$ interactions. \nAs expected, the behaviours of $F_0^\\mathrm{00}$ and $F_1^\\mathrm{00}$ calculated with D1M$^*$ and D1M$^{**}$ are \nthe same of D1M, because in the \nfitting of these new interactions we have not changed the \nsymmetric nuclear matter properties. \nIn the other spin-isospin (ST) channels we have small differences if we compare D1M$^*$ and D1M$^{**}$ with D1M, \nmore prominent beyond $\\rho=0.2$ fm$^{-3}$ for $F_0^\\mathrm{S1}$ and beyond $\\rho=0.1$ fm$^{-3}$ for $F_1^\\mathrm{S1}$.\nThese differences take into account the changes we have applied in the isospin sector when fitting D1M$^*$ and D1M$^{**}$.\nNotice that, as we have tried to preserve the properties of the $S=0$ $T=1$ channel, the $F_0^\\mathrm{01}$ and $F_1^\\mathrm{01}$\nparameters obtained with D1M$^*$ and D1M$^{**}$ have similar behaviors as the ones given by D1M.\n\n\n\n\\begin{figure}[t!]\n \\centering\n \\includegraphics[width=0.65\\linewidth, clip=true]{.\/grafics\/chapter3\/Landau_parameters_l1}\n  \\caption{Same as Fig.~\\ref{landau0} but for $F_1^\\mathrm{ST}$. \\label{landau1}}\n\n\\end{figure}\n\n\n\n\\section{Neutron star mass-radius relation computed with the D1M$^*$ and D1M$^{**}$ interactions}\n\\begin{figure}[!b]\n\\centering\n\\includegraphics[width=0.8\\columnwidth,clip=true]{.\/grafics\/chapter3\/MR.pdf}\n\\caption{Mass-radius relation in NSs from the D1N, D1M, D1M$^*$, D1M$^{**}$, D2 Gogny forces and the SLy4 Skyrme \nforce.~The horizontal bands depict the heaviest observed NS masses \\cite{Demorest10, Antoniadis13}.~The \nvertical green band shows the \\mbox{M-R} region deduced from chiral nuclear interactions up to normal density \nplus astrophysically constrained high-density EoS extrapolations \\cite{Hebeler13}. \nThe brown dotted band is the zone constrained by the cooling tails of type-I X-ray bursts \nin three low-mass X-ray binaries and a Bayesian analysis \\cite{Nattila16}, and the beige \nconstraint at the front is from five quiescent low-mass X-ray binaries and five photospheric\nradius expansion X-ray bursters after a Bayesian analysis~\\cite{Lattimer14}. Finally, the squared blue band accounts for \na Bayesian analysis of the data coming from the GW170817 detection of gravitational waves from a binary NS merger~\\cite{Abbott2018}.}\n \\label{fig:MR2}\n\\end{figure}\n\nThe mass-radius (M-R) relation in NSs is dictated by the corresponding EoS, which is the essential\ningredient to solve the TOV equations \\cite{shapiro83}. \nIn Fig.~\\ref{fig:MR2} we display the mass of an NS as a \nfunction of its size for the D1M$^*$, D1M$^{**}$, D1M, D1N, and D2 Gogny interactions.\nAs explained in Section~\\ref{MRbeta} of Chapter~\\ref{chapter2}, to solve the TOV equations for an NS, knowledge of the EoS from the center to the surface \nof the star is needed. At present we do not have microscopic calculations of the EoS of the inner crust with \nGogny forces.\nFollowing previous literature \\cite{Link1999,carriere03,xu09a,Zhang15, gonzalez17}, we interpolate \nthe inner-crust EoS by a polytropic form  $P = a + b \\epsilon^{4\/3}$ ($\\epsilon$ is the \nmass-energy density), where the index $4\/3$ assumes that the pressure at these densities \nis dominated by the relativistic degenerate electrons [see Section~\\ref{MRbeta} of Chapter~\\ref{chapter2}]. \nWe match this formula continuously to our Gogny EoSs of the homogeneous core\nand to the Haensel-Pichon EoS of the outer crust \\cite{douchin01}. \nThe core-crust transition density is selfconsistently computed for each Gogny force by the thermodynamical method \\cite{gonzalez17} [see the following Chapter~\\ref{chapter4}]. \nWe also plot as a benchmark the M-R curve calculated with the unified NS EoS proposed by Douchin and Haensel \\cite{douchin01},\nwhich uses the Skyrme SLy4 force.\nIt can be seen that standard Gogny forces, such as D1M and D1N, predict too low maximum stellar masses, with D1N being unable to  \ngenerate masses above $1.4M_\\odot$. We note that this common failure of conventional\nGogny parametrizations \\cite{Loan2011,Sellahewa14,gonzalez17} has been cured in the new D1M$^*$ and D1M$^{**}$ forces, which, as well as D2 and SLy4, \nare successful in reaching the masses around $2M_\\odot$ observed in NSs \\cite{Demorest10,Antoniadis13}.\nThis fact is directly related to the behavior of the EoS in $\\beta$-stable matter, which we have plotted in Fig.~\\ref{fig:essympres}. As can be seen by\nlooking at Figs.~\\ref{fig:essympres}(b) and \\ref{fig:MR2}, the stiffer the EoS at high density, the larger the maximum NS mass.\nWe notice, however, that the maximum mass predicted by a \ngiven model does not only depend on the value of the parameter $L$ but also on the behaviour of \nthe EoS of NS matter at high density. For example, the D1N and D1M$^{**}$ forces have almost \nthe same value of $L$ (see Table \\ref{inm}), but the maximum mass predicted by D1N is much\nsmaller than the one predicted by D1M$^{**}$. The maximum mass predicted\nby the D1N interaction is around $1.23 M_{\\odot}$ and $1.91M_{\\odot}$ for the D1M$^{**}$ force.\nThis fact points out that, in spite of the same\nslope of the symmetry energy at saturation, the behaviour of the symmetry energy above saturation (see Fig.~\\ref{fig:essympres}(a))\nstrongly determines the EoS at high density and therefore the maximum NS mass predicted by each force.\n\nAs mentioned at the beginning of the chapter, we have included in the fitting procedure of D1M$^*$ \n(D1M$^{**}$) the constraint of a maximum mass of $2 M_{\\odot}$ ($1.91 M_{\\odot}$). The solution of the TOV equations \nalso provides the radius of the NS. As seen in Table~\\ref{Table-NSs2}, \nthe radii corresponding to the NS of maximum mass are $10.2$ km for D1M$^*$ and $9.6$ km for D1M$^{**}$. Moreover, \nfor an NS of a canonical mass of 1.4$M_{\\odot}$ the radii obtained are of 11.6 km  for D1M$^*$ and of 11.1 km for D1M$^{**}$.\nThese values are similar to \nthe predictions of the SLy4 model and are in harmony with recent extractions of the NS radius from low-mass X-ray binaries, \nX-ray bursters, and gravitational waves \\cite{Nattila16,Lattimer14, Abbott2018}.\n\n\\begin{table}[t]\n\\resizebox{\\columnwidth}{!}{%\n\\begin{tabular}{lcccccccc}\n\\hline \n\\multirow{2}{*}{Force} & $L$    & $M_\\mathrm{max}$ & $R(M_\\mathrm{max})$ & $\\rho_c(M_\\mathrm{max})$ & $\\epsilon_c(M_\\mathrm{max})$ & $R(1.4M_\\odot)$ & $\\rho_c(1.4M_\\odot)$ & $\\epsilon_c(1.4M_\\odot)$ \\\\\n                       & (MeV)  & ($M_\\odot$)      & (km)                & (fm$^{-3}$)              & ($10^{15}$ g cm$^{-3}$)         & (km)            & (fm$^{-3}$)          & ($10^{15}$ g cm$^{-3}$)                                                                       \\\\\\hline \\hline\nD1M                    & 24.83  & 1.745            & 8.84                & 1.58                     & 3.65                            & 10.14           & 0.80                 & 1.51                                                                                          \\\\\nD1M$^*$                & 43.18  &  1.997           & 10.18                & 1.19                     & 2.73                           & 11.65            & 0.53                 & 0.96                                                                                           \\\\\nD1M$^{**}$             & 33.91  &  1.912           & 9.58               &  1.33                   &  3.09                          &    11.04        &    0.62             &\t1.14\t\t\\\\ \\hline                                                                                         \n\\end{tabular}\n}\n                                                                              \n\\caption{Properties (mass $M$, radius $R$, central density $\\rho_c$ and central mass-energy density $\\epsilon_c$) \nof NS maximum mass and $1.4 M_\\odot$ configurations for the D1M Gogny interaction and the new D1M$^*$ and D1M$^{**}$ parametrizations.\n\\label{Table-NSs2}}\n\\end{table}\n\n\\begin{table}[!b]\n\\centering\n\\resizebox{0.5\\columnwidth}{!}{%\n\\begin{tabular}{lccc}\n\\hline\n           & $A \\leq 80$    &  $80 < A \\leq 160$    & $A > 160$ \\\\ \\hline\\hline\nD1M$^{*}$  & 1.55        &  1.31                  &   1.26    \\\\\nD1M        & 1.82        &  1.12                &   1.29    \\\\ \\hline\n\\end{tabular}\n}\n\\caption{Partial rms deviation (in MeV) \nfrom the experimental binding energies \\cite{Audi12} in even-even nuclei, computed in the given mass-number intervals.}\n\\label{prms}\n\\end{table}\n\\section{Properties of nuclei}\nOne of the goals of the D1M$^*$ and D1M$^{**}$ parametrizations is to reproduce nuclear structure \nproperties of finite nuclei with the same global quality as the original D1M force \\cite{gonzalez18, gonzalez18a, Vinas19}. \n We have checked that the basic bulk properties of \nD1M$^*$, such as binding energies or charge radii of even-even nuclei, remain \nglobally unaltered as compared to D1M. The finite nuclei \ncalculations have been carried out with the \\mbox{HFBaxial} code\n\\cite{robledo02} using an approximate second-order gradient method to \nsolve the HFB equations \\cite{rob11} in a harmonic oscillator (HO) \nbasis. The code preserves axial symmetry but is allowed to break \nreflection symmetry. It has already been used in large-scale \ncalculations of nuclear properties with the D1M force, as e.g.\\ in Ref.\\ \\cite{Rob11b}.\nAlthough the calculations of finite nuclei properties with the D1M$^{**}$ \nforce have not been performed as extensively as with D1M$^{*}$, looking at \ntheir parameters reported in Table \\ref{param}, it is expected that \nthe predictions of D1M$^{**}$ will lie between the ones of D1M and D1M$^{*}$. \nOur preliminary investigations confirm this expectation.\n\n\\subsection{Binding energies and neutron and proton radii}\n\\begin{figure}[b!]\n\\centering\n\\includegraphics[width=0.7\\columnwidth]{.\/grafics\/chapter3\/DeltaE_D1M_S.pdf}\n\\caption{Binding energy differences in 620 even-even nuclei between computed and \nexperimental values \\cite{Audi12}\nfor the new D1M$^*$ force, as a function of neutron number $N$.\\label{Binding}}\n\\end{figure}\n\\begin{figure}[t!]\n\\centering\n\\includegraphics[width=0.7\\columnwidth]{.\/grafics\/chapter3\/DeltaEDiff.pdf}\n\\caption{Binding energy differences between the theoretical predictions of D1M$^{*}$ and D1M\nfor 818 even-even nuclei.\\label{Bfn}}\n\\end{figure}\n\nThe binding energy is obtained by subtracting to the HFB energy the rotational energy correction,\nas given in Ref.\\ \\cite{RRG00}. The ground-state calculation is repeated with an\nenlarged basis containing two more HO major shells and an extrapolation\nscheme to an infinite HO basis is used to obtain the final binding energy \\cite{Hilaire.07,Baldo13a}.\nIn this framework, the zero-point energy (ZPE)\nof quadrupole motion used in the original fitting of D1M \\cite{goriely09} is not taken into account\nbecause it requires considering $\\beta$-$\\gamma$ potential energy\nsurfaces (PES) and solving\nthe five-dimensional collective Hamiltonian for all the nuclei. This is still an enormous task\nand we follow a different strategy where the quadrupole ZPE is replaced by a constant binding energy shift. This is \nsomehow justified as in general the ZPE shows a weak mass dependence\n(see \\cite{Rob15} for an example with the octupole degree of freedom). The energy shift is fixed\nby minimizing the global rms deviation, $\\sigma_{E}$, for the \nknown binding energies of 620 even-even nuclei \\cite{Audi12}.\n\nWith a shift of 2.7 MeV we obtain for D1M a $\\sigma_{E}$ of 1.36 MeV,\nwhich is larger than the 798 keV reported for D1M in \\cite{goriely09} including also odd-even and odd-odd nuclei. \nThe result is still satisfying and gives us confidence in the procedure followed.\nUsing the same approach for D1M$^{*}$ we obtain a $\\sigma_{E}$ of\n1.34 MeV (with a shift of 1.1 MeV), which compares favorably with our $\\sigma_{E}$ of 1.36 MeV for D1M.\nThis indicates a similar performance of both parametrizations in the average \ndescription of binding energies along the periodic table.\n\nThe differences between the binding energies of\nD1M$^*$ and the experimental values, $\\Delta B=B_\\textrm{th}-B_\\textrm{exp}$, for 620 even-even nuclei belonging to different isotopic \nchains are displayed in Fig.~\\ref{Binding} against the neutron number $N$. \nThe $\\Delta B$ values are scattered around zero and show no drift with increasing $N$.\nThe agreement between theory and experiment is especially good for medium-mass and heavy\nnuclei away from magic numbers and deteriorates for light nuclei, \nas may be seen from the partial $\\sigma_{E}$ deviations given in Table~\\ref{prms}.\nFrom the partial $\\sigma_{E}$ values of Table~\\ref{prms}, we also conclude that the\ncloseness in the total $\\sigma_{E}$ of D1M and D1M$^{*}$ involves subtle\ncancellations that take place all over the nuclear chart.\nWe plot the differences in binding energy predictions between D1M$^*$ and D1M in Fig.~\\ref{Bfn} \nagainst $N$ for 818 even-even nuclei.\nThe differences between the binding\nenergies computed with the D1M$^*$ and D1M are never larger than $\\pm 2.5$ MeV and show a clear shift along\nisotopic chains because of the different density dependence of the symmetry energy in both forces. A similar \nbehavior can be observed in a recent comparison \\cite{Pillet17} between D2 and D1S. It is also interesting to note that\nthe results for neutron radii show a similar isotopic drift as the\n binding energies. Namely, the difference $r_\\mathrm{D1M^{*}}-r_\\mathrm{D1M}$\n(where $r$ is the rms radius computed from the HFB\nwave function) increases linearly with $N$ for the neutron radii, \nwhereas it remains essentially constant with $N$ for the proton radii. This is again\na consequence of the larger slope $L$ of the symmetry energy in D1M$^{*}$ \\cite{Brown00,Centelles09}. \n\n\\begin{figure}[t!]\n\\centering\n\\includegraphics[clip=true,width=0.9\\columnwidth]{.\/grafics\/chapter3\/MFPES_Er2_v2}\n\\caption[]{Potential energy surfaces of the Er isotopic chain computed with the D1M (red) and \nD1M$^*$ (black) interactions as a function of the quadrupole deformation parameter $\\beta_2$.\\label{Er2}}\n\\end{figure}\n\n\\begin{figure}[b!]\n\\centering\n\\includegraphics[clip=true,width=0.65\\columnwidth]{.\/grafics\/chapter3\/240PuFISS_v2}\n\\caption[]{Fission barrier of the nucleus $^{240}$Pu \nas a function of the quadrupole moment $Q_2$ calculated with the same Gogny forces. The evolution of the mass \nparameter, octupole and hexadecapole moments and neutron and proton pairing energies along the fission path \nare also displayed in the same figure.\\label{Fis}}\n\\end{figure}\n\\subsection{Potential Energy Surfaces}\nAn important aspect of any nuclear interaction is the way it determines the response of the\nnucleus to shape deformation, in particular to the quadrupole deformation. To know if a nucleus is quadrupole \ndeformed or not plays a crucial role in the determination of the low energy-spectrum. To study the \nresponse of the D1M$^*$ force to the quadrupole deformation, we have performed constrained HFB calculations \nin finite nuclei fixing the quadrupole moment $Q_{20}$ to given values, which allows one to obtain the PES. \nAs an example, in Fig.~\\ref{Er2} the PES along the Er (Z=68) isotopic chain is displayed as a function of the deformation parameter \n$\\beta_2$ for the original D1M interaction and for the modified D1M$^*$ force.\nIt can be observed that the curves corresponding to the calculations performed with the D1M \nand D1M$^*$ forces follow basically the same trends  with a small displacement of one curve with respect to the \nother~\\cite{gonzalez18, gonzalez18a}.\n\n\n\\newpage\n\n\\subsection{Fission Barriers}\nFinally, we plot in Fig.~\\ref{Fis} the fission barrier of the paradigmatic nucleus $^{240}$Pu~\\cite{gonzalez18a}.\nWe see that the inner fission barrier predicted by D1M and D1M$^*$ is the same \nin both models with a value $B_I$=9.5 MeV. This value is a little bit large compared with the experimental \nvalue of 6.05 MeV. However, it should be pointed out that triaxiality effects, not accounted for in the present \ncalculation, might lower the inner barrier by 2-3 MeV. The excitation energy of the fission isomer $E_{II}$ is\n3.36 MeV computed with D1M and 2.80 MeV with D1M$^*$. The outer fission barrier $B_{II}$ height are \n8.58 and 8.00 MeV calculated with the D1M and D1M$^*$ forces, respectively. These values clearly \noverestimate the empirical value, which is 5.15 MeV. In the other panels of Fig.~\\ref{Fis} we have  \ndisplayed as a function of the quadrupole deformation the neutron and proton pairing energies, the octupole \nmoment (responsible for asymmetric fission) and the hexadecapole moment of the mass distributions.\nAll these quantities take very similar values computed with both interactions.\n \n\\chapter{Core-crust transition in neutron stars}\\label{chapter4}\nOur aim in this chapter is to study the properties of the transition between the core and the crust inside neutron stars (NSs). \nThe structure of an NS consists of a\nsolid crust at low densities encompassing a homogeneous core in a liquid phase. The density is maximum at the center, several times\nthe nuclear matter density, which has a value of $\\sim$ 2.3 $\\times$ 10$^{14}$ g cm$^{-3}$, and decreases with the distance from the center\nreaching a value of the terrestrial\niron around $\\sim$ 7.5 g cm$^{-3}$ at the surface of the star.\nThe external part of the star, i.e., its outer crust, consists of nuclei \ndistributed in a solid body-centered-cubic (bcc) lattice permeated by a free electron gas. \nWhen the density increases, the nuclei in the crust become so neutron-rich that neutrons start to drip from \nthem. \nIn this scenario, the lattice structure of nuclear clusters is embedded in free \nelectron and neutron gases. When the average density reaches a value of about half of the nuclear  \nmatter saturation density, the lattice structure disappears due to energetic reasons and the system \nchanges to a liquid phase. The boundary between the outer and inner crust is determined by the nuclear masses, \nand corresponds to the neutron drip out density around $\\sim 4\\times 10^{11}$ g cm$^{-3}$ \\cite{ruster06,hampel08}.\nHowever, the transition density from the inner crust to\nthe core is much more uncertain and is strongly model dependent \n\\cite{baym70,kubis06,ducoin07,xu09a,xu09b,xu10a,xu10b,Moustakidis10,Moustakidis12,Seif14,routray16,gonzalez17}. \nTo determine the core-crust transition density from the crust side it is required to have a precise \nknowledge of the EoS in this region of the star, as the boundary between the liquid core and the \ninhomogeneous solid crust is connected to the isospin dependence of nuclear models below saturation density.\nHowever, this is a very challenging \ntask owing to the presence of the neutron gas and the possible existence of complex structures in the\ndeep layers of the inner crust, where the nuclear clusters may adopt shapes differently\nfrom the spherical one (i.e., the so-called ``pasta phases\") in order to minimize the energy\n\\cite{baym71, lattimer95, shen98a, shen98b, douchin01, sharma15, Carreau19}.\nTherefore, it is easier to investigate the core-crust transition from the core side. To this\nend, one searches for the violation of the stability conditions of the homogeneous core under\nsmall-amplitude oscillations, which indicates the appearance of nuclear clusters and,  \nconsequently, the transition to the inner crust. There are different ways to determine the transition \ndensity from the core side, namely, the thermodynamical method \\cite{kubis04,kubis06,xu09a,Moustakidis10,Moustakidis12,Cai2012,Seif14,routray16,gonzalez17}, \nthe dynamical method\n\\cite{baym71,xu09a,xu09b,xu10a,xu10b,pethick95,ducoin07,Tsaloukidis19, gonzalez19}, the random phase approximation \n\\cite{horowitz01a, horowitz01b,carriere03} or the Vlasov equation method \\cite{chomaz04,providencia06,\nducoin08a,ducoin08b,pais10}. \n Also, a variety of different functionals (and many-body theories) have been used to \ndetermine the properties of the core-crust transition, including Skyrme forces \n\\cite{xu09a,ducoin07,Pearson12,Newton2014}, \nfinite-range functionals \\cite{routray16, gonzalez17}, relativistic mean-field (RMF) models \n\\cite{horowitz01a,carriere03,Klahn06,Moustakidis10,Fattoyev:2010tb,Cai2012,Newton2014}, \nmomentum-dependent interactions \\cite{xu09a,Moustakidis12, gonzalez19} and \nBrueckner--Hartree--Fock theory~\\cite{Vidana2009,Ducoin11,Li2016}.  \n\nIn the low-density regime of the core near the crust, the NS matter is composed of\nneutrons, protons, and electrons. \nTo find the instabilities in the core, in this thesis we restrict ourselves to two approaches:\nthe thermodynamical method and the dynamical method.\nIn the thermodynamical approach, the stability is discussed in terms of the bulk properties of \nthe EoS by imposing mechanical and chemical stability conditions, which set the boundaries of\n the core in the homogeneous  case.  \nOn the other hand, in the dynamical method, one introduces density fluctuations\nfor neutrons, protons, and electrons, which\ncan be expanded in plane waves of amplitude $A_i$ ($i=n, p, e$) and of wave-vector ${\\bf k}$.\nThese fluctuations may be induced by collisions, which transfer some momentum to the system.\nFollowing Refs.~\\cite{baym71,pethick95,ducoin08a,ducoin08b,xu09a} one writes the variation of \nthe energy density generated by these fluctuations in terms of the energy curvature matrix \n$C^f$. \nThe system is stable when the curvature matrix is convex, that is, the transition density is \nobtained from the condition $\\vert C^f \\vert$=0. \nThis condition determines the dispersion relation $\\omega(k)$ of the collective excitations \nof the system due to the perturbation and the \ninstability will appear when this frequency $\\omega(k)$ becomes imaginary. \nThe dynamical method is more realistic than the thermodynamical approach, as it incorporates \nsurface and Coulomb effects in the stability condition that are not taken into account in the \nthermodynamical method.\n\nIn Section~\\ref{Theory_thermo} we introduce the thermodynamical method to find the core-crust transition and\nin Section~\\ref{Results_thermo} we obtain the corresponding core-crust transition properties for Skyrme and Gogny interactions. \nIn Section~\\ref{Theory_dyn} we give the theory for the more sophisticated dynamical method, \nand in Section~\\ref{Results_dyn} we present the results of the transition properties obtained with it. \n\n\\section{The thermodynamical method}\\label{Theory_thermo}\nWe will first focus on the search of the transition between the core and the crust of NSs \nusing the so-called thermodynamical method, \nwhich has been widely used in the literature \\cite{kubis04,kubis06,xu09a,Moustakidis10,Cai2012,\nMoustakidis12,Seif14,routray16}.\nWithin this approach, the stability of the NS core is discussed in terms of its bulk properties. \nThe following mechanical and chemical stability conditions set the boundaries of the homogeneous core:\n\\begin{eqnarray}\\label{cond1}\n -\\left( \\frac{\\partial P}{\\partial v} \\right)_{\\mu_{np}} & > & 0 ,\n\\\\[2mm]\n\\label{cond2}\n -\\left( \\frac{\\partial \\mu_{np}}{\\partial q} \\right)_v & > & 0 .\n\\end{eqnarray}\nHere, $P$ is the total pressure of $\\beta$-stable matter, \n$\\mu_{np}= \\mu_n-\\mu_p$ is the difference between the neutron and proton chemical potentials [related to the $\\beta$-equilibrium \ncondition in~Eq.~(\\ref{betaeq})],\n$v=1\/\\rho$ is the volume per baryon and $q$ is the charge per baryon. \n\nFirst, we consider the mechanical stability condition in Eq.~(\\ref{cond1}). \nThe electron pressure does not contribute to this term, due to the fact that\nthe derivative is performed at a constant $\\mu_{np}$. \nIn $\\beta$-equilibrium, this involves a constant electron chemical potential $\\mu_e$ and, \nbecause the electron pressure \nin Eq.~(\\ref{eq:P_lepton}) is a function only of $\\mu_e$, the derivative of $P_e$ with respect to $v$ vanishes.\nEquation~(\\ref{cond1}) can therefore be rewritten as\n\\begin{equation}\\label{cond1Pb}\n -\\left( \\frac{\\partial P_b}{\\partial v} \\right)_{\\mu_{np}} >0,\n\\end{equation}\nwhere $P_b$ is the baryon pressure.\nMoreover, the isospin asymmetry of the $\\beta$-stable system is a function of density, $\\delta(\\rho)$. \nWith $\\mu_{np} = 2 \\partial E_b(\\rho, \\delta)\/\\partial \\delta$, and using Eq.~(\\ref{eq:pre}) for baryons,\nwe can express the mechanical stability condition as \\cite{xu09a, Cai2012,  Moustakidis10, Moustakidis12}\n\\begin{eqnarray}\\label{cond11}\n  -\\left( \\frac{\\partial P_b}{\\partial v} \\right)_{\\mu_{np}} =\\rho^2 \\left[ 2 \\rho \n  \\frac{\\partial E_b (\\rho, \\delta)}{\\partial \\rho} + \\rho^2 \\frac{\\partial^2 E_b (\\rho , \\delta)}{\\partial \\rho^2}  \\right. \n\\left. -\\frac{\\left( \\rho \\frac{\\partial^2 E_b (\\rho , \\delta)}{\\partial \\rho \\partial \\delta} \\right)^2}{\\frac{\\partial^2 \n E_b (\\rho , \\delta)}{\\partial \\delta^2}}\\right] >0 .\n\\end{eqnarray}\nIn the chemical stability condition of Eq.~(\\ref{cond2}), the charge $q$ can be written as $q=x_p - \\rho_e\/\\rho$, \nwhere \\mbox{$x_p = (1- \\delta)\/2$} is the proton fraction. \nIn the ultrarelativistic limit, the electron number density is related to the chemical potential by $\\rho_e= \\mu_e^3\/(3\\pi^2)$. \nWe can thus recast (\\ref{cond2}) as\n\\begin{equation}\\label{cond22}\n -\\left( \\frac{\\partial q}{\\partial \\mu_{np}} \\right)_v= \\frac{1}{4} \\left[ \\frac{\\partial^2 E_b (\\rho, \\delta)}{\\partial \\delta^2}\n \\right]^{-1} + \\frac{\\mu_e^2}{\\pi^2 \\rho}>0. \n\\end{equation}\nIn the low-density regime of interest for the core-crust transition, the first term on the right-hand side is positive \nfor the Skyrme and Gogny parameterizations studied here. With a second term that is also positive, we conclude that the \ninequality of Eq.~(\\ref{cond22}) is fulfilled. \nHence, the stability condition for $\\beta$-stable matter can be expressed in terms of Eq.~(\\ref{cond11}) alone, \nwith the result \n\\cite{kubis04, kubis06,xu09a,Moustakidis10}\n\\begin{equation}\\label{Vthermal}\n  V_{\\mathrm{ther}} (\\rho) = 2 \\rho \\frac{\\partial E_b (\\rho, \\delta)}{\\partial \\rho} + \\rho^2 \n  \\frac{\\partial^2 E_b (\\rho , \\delta)}{\\partial \\rho^2}\n -\n \\left( \\rho \\frac{\\partial^2 E_b (\\rho , \\delta)}{\\partial \\rho \\partial \\delta} \\right)^2 \\left(\\frac{\\partial^2 \n E_b (\\rho , \\delta)}{\\partial \\delta^2} \\right)^{-1}>0,\n\\end{equation}\nwhere we have introduced the so-called thermodynamical potential, $V_\\mathrm{ther} (\\rho)$. \n\nIf the condition for $V_\\mathrm{ther} (\\rho)$ is rewritten using the Taylor expansion of \n$E_b (\\rho, \\delta)$ given in Eq.~(\\ref{eq:EOSexpgeneral}), one finds\n\\begin{eqnarray}\\label{eq:Vtherapprox}\n V_{\\mathrm{ther}}  (\\rho) &=& \\rho^2 \\frac{\\partial^2 E_b (\\rho, \\delta=0)}{\\partial \\rho^2} + 2 \\rho \n \\frac{\\partial E_b (\\rho, \\delta=0)}{\\partial \\rho} \\nonumber\n \\\\\n&&+ \\sum_{k} \\delta^{2k} \\left( \\rho^2 \\frac{\\partial^2 E_{\\mathrm{sym}, 2k}(\\rho)}{\\partial \\rho^2} + 2 \\rho \n\\frac{\\partial E_{\\mathrm{sym}, 2k}(\\rho)}{\\partial \\rho}\\right) \\nonumber\n\\\\\n&&-2\\rho^2 \\delta^2 \\left( \\sum_{k} k \\delta^{2k-2}  \\frac{\\partial E_{\\mathrm{sym}, 2k}(\\rho)}{\\partial \\rho} \\right)^2 \\nonumber\n\\\\\n&&\\times \\left[ \\sum_{k} (2k-1)k\\delta^{2k-2} E_{\\mathrm{sym}, 2k}(\\rho)\\right]^{-1} >0 . \\nonumber \n\\\\\n\\end{eqnarray}\nThis equation can be solved order by order, together with the $\\beta$-equilibrium condition, Eq.~(\\ref{betamatter})\n(or Eq.~(\\ref{betamatter-PA}) in the PA case), to evaluate the influence on\nthe predictions for the core-crust transition of truncating the Taylor expansion of the EoS of asymmetric nuclear matter.\nWe collect in Appendix~\\ref{appendix_thermal} the expressions for the derivatives of $E_b(\\rho, \\delta)$ for Skyrme and Gogny forces \nthat are needed to \ncalculate $V_\\mathrm{ther} (\\rho)$ in both Eqs.~(\\ref{Vthermal}) and (\\ref{eq:Vtherapprox}). \n\n\n\\section{Core-crust transition studied within the thermodynamical method}\\label{Results_thermo}\n\\begin{figure}[t!]\n \\centering\n \\subfigure{\\label{fig:vtherskyrme}\\includegraphics[width=0.49\\linewidth]{.\/grafics\/chapter4\/vther\/SkI5-MSk7-Vthermal-vs-rho}}\n\\subfigure{\\label{fig:vthergogny}\\includegraphics[width=0.49 \\linewidth]{.\/grafics\/chapter4\/vther\/Vtherm-D1S-D280}}\n  \\caption{Density dependence of the thermodynamical potential in $\\beta$-stable nuclear matter calculated using the \n  exact expression of the EoS or the expression in Eq. (\\ref{eq:EOSexpgeneral}) up to second and \n  tenth order for two Skyrme forces (panel (a)) and up to second, fourth and sixth order for two Gogny forces (panel (b)).\n  The results for the parabolic approximation are also included in both panels.\\label{fig:vther}}\n\\end{figure}\n\n\nWe show in Fig.~\\ref{fig:vther} the density dependence of the thermodynamical potential $V_\\mathrm{ther} (\\rho)$ \nin $\\beta$-stable matter for Skyrme (Fig.~\\ref{fig:vtherskyrme}) and Gogny (Fig.~\\ref{fig:vthergogny}) interactions, \ncalculated with the exact expression of the EoS (solid \nlines), with its Taylor expansion up to second and tenth order for Skyrme forces and up to \nsecond, fourth and sixth order for Gogny models, and with the \nPA \\cite{gonzalez17}. An instability region characterized by negative $V_\\mathrm{ther} (\\rho)$ is found \nbelow $\\rho \\approx 0.13$ fm$^{-3}$ for both types of interactions. The condition $V_\\mathrm{ther}(\\rho_t) = 0$ \ndefines the density $\\rho_t$ of the transition from the homogeneous core to the crust.\nWe see in Fig.~\\ref{fig:vther} that adding more terms to Eq.~(\\ref{eq:EOSexpgeneral}) brings the \nresults for $V_\\mathrm{ther} (\\rho)$ closer to the exact values. \nAt densities near the core-crust transition, for the Skyrme MSk7 and the Gogny D1S and D280 interactions,\nthe higher-order results are rather \nsimilar, but differ significantly from the exact ones. The differences are larger for the SkI5 interaction, which \nis the interaction of the ones considered here that has the \nlargest slope of the symmetry energy.\nWe note that, all in all, the order-by-order convergence of the $\\delta^2$ expansion in \n$V_\\mathrm{ther} (\\rho)$ is slow. This indicates that the non-trivial isospin and \ndensity dependence arising from exchange terms needs to be considered in a complete \nmanner for realistic core-crust transition physics \n\\cite{Chen09,Vidana2009,Seif14, gonzalez17}.\nIf we look at the unstable low-density zone, both the exact and the approximated \nresults for $V_\\mathrm{ther} (\\rho)$ go to zero for vanishing density, but they keep a \ndifferent slope. In this case, we have found that the discrepancies are largely \nexplained by the differences in the low-density behaviour of the approximated \nkinetic energy terms, in consonance with the findings of Ref.~\\cite{routray16}.\n\nWe next analyze more closely the properties of the core-crust transition, using both exact and order-by-order \npredictions. The complete results for the sets of Skyrme and Gogny functionals are provided, respectively, in numerical \nform in Appendix~\\ref{app_taules}.\n\nFor a better understanding, we discuss each one of the key physical properties of the transition (asymmetry, density, and pressure) \nin separate figures. We plot our predictions as a function of the slope parameter $L$ of each functional, which does not necessarily \nprovide a stringent correlation with core-crust properties \\cite{Ducoin11}. \nThe slope parameter, however, can be constrained in terrestrial experiments and astrophysical observations \n\\cite{Tsang2012,Lattimer2013,Lattimer2016,BaoAnLi13,Vinas14,Roca-Maza15}\nand is, therefore, an informative parameter in terms of the isovector properties of the functional.\n\n\\begin{figure}[t!]\n \\centering\n \\subfigure{\\label{fig:deltatskyrme}\\includegraphics[width=0.49\\linewidth]{.\/grafics\/chapter4\/vther\/Skyrme_delta_t_vs_L_vther}}\n\\subfigure{\\label{fig:deltatgogny}\\includegraphics[width=0.49 \\linewidth]{.\/grafics\/chapter4\/vther\/delta_t_vs_rho_Gogny_vther}}\n   \\caption{Panel a: Core-crust transition asymmetry, $\\delta_t$, as a function of the slope parameter $L$ for a set of Skyrme forces\n   calculated with the thermodynamical approach\n   using the exact expression of the EoS (solid circles), \n   and the approximations up to second (solid squares) and tenth order (solid triangles). \n   The parabolic approximation is also included (empty squares).\n   Panel b: Same as panel (a) for a set of Gogny forces \n   calculated \n   using the exact expression of the EoS (solid circles), \n   and the approximations up to second (solid squares), fourth (solid diamonds) and sixth order (solid triangles). \n   The parabolic approximation is also included (empty squares).}\n \\label{fig:deltat}\n\\end{figure}\n\nIn Fig.~\\ref{fig:deltat}, we display the results for the transition asymmetry, $\\delta_t$ \\cite{gonzalez17}. Black dots \ncorrespond to the calculations with the exact EoS. We find that the set of Skyrme forces collected in Table~\\ref{table:Skyrmeprops} predict a range\nof $0.924 \\lesssim \\delta_t \\lesssim 0.978$ for the asymmetry at the transition point (see Fig.~\\ref{fig:deltatskyrme}), \nwhile the set of Gogny forces found in Table~\\ref{table:Gognyprops} plus the new D1M$^*$ and D1M$^{**}$ parametrizations predict a range of\n$0.909 \\lesssim \\delta_t \\lesssim 0.931$ (see Fig.~\\ref{fig:deltatgogny}).\nFor Skyrme interactions, we see that $\\delta_t$ presents an increasing tendency as the slope of the \nsymmetry energy of the interaction is larger. Of the interactions we have considered, G$_\\sigma$, SkI5, SkI2, and R$_\\sigma$\nare the ones with higher values, over $\\delta_t \\gtrsim 0.97$.\nIn the case of Gogny interactions, the D1N, D1M, D1M$^*$ and D1M$^{**}$ forces are the ones providing\ndistinctively large transition asymmetries,\nwhereas the other interactions predict very similar values $\\delta_t \\approx 0.91$ in spite of having different \nslope parameters.\nWhen using the Taylor expansion of the EoS up to second order (shown by red squares in both panels), \nthe predictions \nfor $\\delta_t$ are generally far from the exact result. For interactions (in both cases of Skyrme and Gogny functionals) \nwith the slope of the symmetry energy below $L\\lesssim 70$ MeV the results are well above the exact result,\nwhereas for interactions with higher slope $L$ the results obtained with the Taylor expansion up to second\norder remain below the exact results. \nFor Skyrme interactions, the results obtained with the tenth-order expansion (blue triangles) are close to the \nones obtained with the exact EoS for interactions that have a relatively low slope of the symmetry energy, \nwhile they remain quite far from them when $L$ is larger.\nThe results obtained with the PA (empty orange squares) provide a very good approximation if the slope $L$ is small, but works \nless well as $L$ becomes larger. Also, if $L\\gtrsim 60$ MeV, the values obtained with the PA are farther from the \nexact results than the results obtained with the second-order approximation.\nFor Gogny interactions, the fourth-order values (green diamonds) are still above the exact \nones but closer, and the sixth-order \ncalculations (blue triangles) produce results that are very close to the exact $\\delta_t$. \nThe $\\delta_t$ values obtained with the PA (empty orange squares) differ from the second-order approximation \nand turn out to be closer to the exact results.\n\n\n\\begin{figure}[t!]\n \\centering\n \\subfigure{\\label{fig:rhotskyrme}\\includegraphics[width=0.49\\linewidth]{.\/grafics\/chapter4\/vther\/Skyrme_rho_t_vs_L_vther}}\n\\subfigure{\\label{fig:rhotgogny}\\includegraphics[width=0.49 \\linewidth]{.\/grafics\/chapter4\/vther\/rho_t_gogny_vther}}\n  \\caption{Core-crust transition density, $\\rho_t$, in the thermodynamical approach as a function of the slope parameter $L$\n  for a set of Skyrme interactions (panel (a)) and for a set of Gogny forces (panel (b)). Symbols are the same as in Fig.~\\ref{fig:deltat}.}\n  \\label{fig:rhot}\n\\end{figure}\n\n\\begin{figure}[b!]\n \\centering\n \\subfigure{\\label{fig:ptskyrme}\\includegraphics[width=0.49\\linewidth]{.\/grafics\/chapter4\/vther\/P_t_vs_L_vther_skyrme}}\n\\subfigure{\\label{fig:ptgogny}\\includegraphics[width=0.49 \\linewidth]{.\/grafics\/chapter4\/vther\/P_t_vs_L_GOGNY_vther}}\n   \\caption{Core-crust transition pressure, $P_t$, in the thermodynamical approach as a function of the slope parameter $L$\n  for a set of Skyrme interactions (panel (a)) and for a set of Gogny forces (panel (b)). Symbols are the same as in Fig.~\\ref{fig:deltat}.}\n \\label{fig:Pt}\n\\end{figure}\n\nWe show in Fig.~\\ref{fig:rhot} the predictions for the density of the core-crust transition, $\\rho_t$, for the same of Skyrme \n(Fig.\\ref{fig:rhotskyrme}) and Gogny (Fig.~\\ref{fig:rhotgogny}) forces~\\cite{gonzalez17}. The calculations \nwith the exact EoS of the models give a window $0.060 \\text{ fm}^{-3} \\lesssim \\rho_t \\lesssim 0.125\\text{ fm}^{-3}$  for the Skyrme \nparametrizations and $0.094 \\text{ fm}^{-3} \\lesssim \\rho_t \\lesssim 0.118\\text{ fm}^{-3}$ for the set of Gogny interactions. \nWe find that the approximations of the EoS only provide upper bounds to the exact values. \nFor Skyrme interactions, the relative differences between the transition densities predicted using the $\\delta^2$ approximation of the EoS\nand the exact densities are about $3\\%-55 \\%$,\nand when one uses the EoS expanded up to $\\delta^{10}$, the differences are reduced \nto $1\\%-30 \\%$. This rather large window for the value for the relative \ndifferences comes from the fact that the approximated values are closer to the ones calculated \nwith the full EoS if the slope of the symmetry energy of the interactions is smaller. \nOn the other hand, for the Gogny functionals, \nthe relative differences between the transition densities predicted using the $\\delta^2$ approximation of the EoS\nand the exact densities are about $4\\%-7 \\%$.\nWhen the EoS up to $\\delta^4$ is used, the differences are slightly reduced \nto $3\\%-6 \\%$, and the sixth-order results remain at a similar level of accuracy, within $3\\%-5 \\%$. \nIn other words, the order-by-order convergence for the transition density is very slow.\nAs mentioned earlier in the discussion of Fig.~\\ref{fig:vther}, the non-trivial density and isospin asymmetry \ndependence of the thermodynamical potential arising from the exchange contributions is likely to be the underlying \ncause of this slow convergence pattern. The results for $\\rho_t$ of the PA do not exhibit a regular trend with respect \nto the other approximations. In some cases, the PA is the closest approximation to the results obtained with \nthe full EoS, like in the cases of the SIII Skyrme interaction or the D260 Gogny parametrization.\n\nWe find that there is a decreasing quasi-linear correlation between the transition \ndensity $\\rho_t$ and the slope parameter $L$. In fact, it is known from previous literature that the transition \ndensities calculated with Skyrme interactions and RMF models have an anticorrelation with $L$ \n\\cite{horowitz01a,xu09a, Moustakidis12,Ducoin11,Providencia14,Fattoyev:2010tb,Pais2016}. We confirm \nthis tendency and find that the transition densities calculated with Skyrme and Gogny functionals are in consonance with other \nmean-field models. Moreover, if we take into account the slope parameter of these interactions, the Gogny results are \nwithin the expected window of values provided by the Skyrme and RMF models \\cite{xu09a,Vidana2009}. \n\nIn Fig.~\\ref{fig:Pt}, we present the pressure at the transition point, $P_t$, for the same interactions of Figs.~\\ref{fig:deltat} and \\ref{fig:rhot}. \nThe results of the exact Skyrme EoSs (Fig.~\\ref{fig:ptskyrme}) lie in the range\n$0.269 \\text{ MeV fm}^{-3} \\lesssim P_t \\lesssim 0.668 \\text{ MeV fm}^{-3}$ and for the exact \nGogny EoSs (Fig.~\\ref{fig:ptgogny}) they lie in the range $0.339 \\text{ MeV fm}^{-3} \\lesssim P_t \\lesssim 0.665 \\text{ MeV fm}^{-3}$ \\cite{gonzalez17}.\nAccording to Ref.~\\cite{Lattimer01}, in general, the transition pressure for realistic EoSs \nvaries over a window $0.25 \\text{ MeV fm}^{-3} \\lesssim P_t \\lesssim 0.65 \\text{ MeV fm}^{-3}$. Skyrme and \nGogny forces,\ntherefore, seem to deliver reasonable predictions.\nIf we look at the accuracy of the isospin Taylor expansion of the EoS for predicting $P_t$, we find that the \nsecond-order approximation gives transition pressures  above the values of the exact EoS in almost \nall of the forces. \nFor Skyrme interactions, the differences are of $3\\%-54\\%$ if the interaction is in the range of $L \\lesssim 60$ MeV. \nHowever, the relative differences can reach up to $\\sim 300 \\%$ if the slope of the symmetry energy is large. \nThese differences reduce to $0.6\\%-25\\%$ and $\\sim 124\\%$, respectively, if the EoS expansion up to tenth order is\nused.\nFor Gogny functionals, the differences are of about $2\\%-17 \\%$ and become $3\\%-12 \\%$ at fourth order of the expansion, and $4\\%-10 \\%$ at sixth order. \nOn the whole, Fig.~\\ref{fig:Pt} shows that the order-by-order convergence for the transition pressure is \nnot only slow but actually erratic at times. \nFor some parametrizations, like the Gogny D1 or D300 forces, the fourth- and sixth-order predictions for $P_t$ differ more \nfrom the exact value than if we stop at second order. \nWe also see that the PA overestimates the transition pressure for all parametrizations---in fact, the PA \nprovides worse predictions for the transition pressure than any of the finite-order approximations.\n\nWe note that we do not find a general trend with the slope parameter $L$ in our results for the pressure of \nthe transition, i.e., forces with similar $L$ may have quite different pressure values at the border between \nthe core and the crust. As in the case of the transition density, the transition pressure has been studied in \nprevious literature. However, the predictions on the correlation between the transition pressure and $L$ \ndiverge \\cite{xu09a,Moustakidis12,Fattoyev:2010tb,PRC90Piekarewicz2014,Moustakidis10}.\nIn our case, we obtain that the transition pressure is uncorrelated with the slope parameter~$L$. \nThe same was concluded in Ref.~\\cite{Fattoyev:2010tb} in an analysis with RMF models.\n\nWith these values we can conclude that for interactions with soft symmetry energy, if we want to use the\napproximations of the EoS, we do not have to go to larger orders than the second, whereas if we are using \ninteractions with a stiff symmetry  energy, especially with $L$ larger than $L \\gtrsim 60$ MeV, if possible, one should \nperform the calculations using the exact expression of the EoS. \n\n\\section{The dynamical method}\\label{Theory_dyn}\nWe proceed to study the stability of NS matter against the formation of nuclear clusters using the dynamical method. \nAs mentioned before, at least at low densities near the transition to the crust, the matter of the NS core is \ncomposed of neutrons, protons, and electrons. It is globally charge neutral and satisfies the $\\beta$-equilibrium \ncondition \\cite{shapiro83, haensel07}.\nFollowing~\\cite{baym71}, we write the particle density as an unperturbed constant part, $\\rho_U$, plus a \nposition-dependent fluctuating contribution:\n\\begin{equation} \n\\rho({\\bf R})= \\rho_U + \\delta \\rho ({\\bf R}),\n\\label{eq6}\n\\end{equation}\nwhere  ${\\bf R} = ({\\bf r}+{\\bf r'})\/2$ is the center of mass coordinate and the small variations are of sinusoidal type, i.e.,\n\\begin{equation}\n\\delta \\rho_i ({\\bf r}) = \\int \\frac{d{\\bf k}}{(2\\pi)^3} \\delta n_i({\\bf k})e^{i {\\bf k} \\cdot {\\bf r}},\n\\label{density}\n\\end{equation} \nbeing $i = {n,p,e}$ and $n({\\bf k})$ the density in momentum space.\nThese fluctuations of the densities fulfill $n_i({\\bf k}) = n_i^*(-{\\bf k})$ in order to ensure that the variations\n of the particle densities are real. Next, the total energy of the system is expanded up to second order in the variations of \nthe densities, which implies\n\\begin{equation}\nE = E_0 + \\frac{1}{2} \\sum_{i,j} \\int \\frac{d{\\bf k}}{(2\\pi)^3}\n\\frac{\\delta^2 E}{\\delta n_{i}({\\bf k}) \\delta n_{j}^*({\\bf k})} \\delta n_{i}({\\bf k}) \\delta n_{j}^*({\\bf k}),\n\\label{eq10}\n\\end{equation}\nwhere $E_0$ is the energy of the uniform phase and the subscripts $i$ and $j$ concern to the different types of particle.\nThe first-order variation of the energy vanishes due to the particle number conservation for each type of particle.\nThe second-order variation of the energy, that is, its curvature matrix: \n\\begin{equation}\nC^f =\\frac{\\delta^2 E}{\\delta n_{i}({\\bf k}) \\delta n_{j}^*({\\bf k})},\n\\label{eq11}\n\\end{equation}\ncan be written as a sum of three matrices as\n\\begin{eqnarray}\nC^f &=& \\left( \\begin{array}{ccc}\n\\partial \\mu_n \/ \\partial \\rho_n & \\partial \\mu_n \/ \\partial \\rho_p & 0 \\\\\n \\partial \\mu_p \/ \\partial \\rho_n& \\partial \\mu_p \/ \\partial \\rho_p & 0 \\\\\n0 & 0 & \\partial \\mu_e \/ \\partial \\rho_e \\end{array} \\right)\n\\nonumber \\\\ &&\n +  \\left( \\begin{array}{ccc}\nD_{nn}(\\rho, k) & D_{np}(\\rho, k) & 0 \\\\\n D_{pn}(\\rho,k) & D_{pp}(\\rho, k) & 0 \\\\\n0 &  0& 0 \\end{array} \\right) + \\frac{4 \\pi e^2}{k^2} \\left( \\begin{array}{ccc}\n0 & 0 &  0 \\\\\n0 & 1 & -1 \\\\\n0 &-1 &  1 \\end{array} \\right), \\nonumber\n\\\\\n\\label{eq12}\n\\end{eqnarray}\nwhere it is understood that all quantities are evaluated at the unperturbed density $\\rho_U$.\nThe curvature matrix $C^f$ is composed of three different pieces. The first one, which is the dominant term, corresponds\nto the bulk contribution. It defines the stability of uniform NS matter and corresponds to\nthe equilibrium condition of the thermodynamical method for locating the core-crust transition point. The second piece describes the \ncontributions due to the gradient expansion of the energy density functional. Finally, the last piece\n is due to the direct Coulomb interactions of protons and electrons. These last two terms\n tend to stabilize the system reducing the instability region predicted by the bulk contribution alone.\n \n \nThe zero-range Skyrme forces directly provide an energy density functional which \nis expressed as a sum of a homogeneous bulk part, an inhomogeneous term\ndepending on the gradients of the neutron and proton densities and the direct Coulomb energy.\nTherefore, the coefficients $D_{qq'}$ ($q,q'=n,p$) in the surface term, which correspond to the terms of the nuclear energy \ndensity coming from the momentum-dependent\npart of the interaction, which in the case of Skyrme forces reads as \\cite{baym71, pethick95,ducoin07}\n\\begin{equation}\n \\mathcal{H}^\\nabla= C_{nn} \\left( \\nabla \\rho_n\\right)^2 + C_{pp} \\left( \\nabla \\rho_p\\right)^2 + 2  C_{np} \\nabla \\rho_n \\cdot  \\nabla \\rho_p, \n\\end{equation}\ncan be found explicitly from the energy density functional.\nFor Skyrme interactions, the $D_{qq'}$ coefficients are expressed in terms of the interaction parameters $x_1$, $x_2$, $t_1$ and $t_2$ \nas \\cite{sly41}:\n \\begin{eqnarray}\n     D_{qq'} = k^2 C_{qq'},\n \\end{eqnarray}\n where\n\\begin{eqnarray}\n C_{nn} &=&C_{pp} = \\frac{3}{16} \\left[ t_1 (1-x_1) - t_2 (1+x_2)\\right]\\\\\n C_{np} &=&C_{pn} = \\frac{1}{16} \\left[ 3t_1 (2+x_1) - t_2 (2+x_2)\\right].\n\\end{eqnarray} \nNotice that the coefficients $D_{qq'}$ for Skyrme interactions are quadratic functions\nof the momentum $k$ with constant coefficients.\n\nWith finite-range forces, to study the core-crust transition density using the dynamical method, \n and obtaining the corresponding $D_{qq'} (\\rho, k)$ coefficients, \n is much more involved than in Skyrme forces. This approach has been discussed, to our \nknowledge, only in the particular case of the MDI interaction in Refs.~\\cite{xu09a,xu10b}.\nIn the dynamical method using finite-range interactions, one needs to extract the gradient \ncorrections, which are encoded in the force but do not appear explicitly in their energy \ndensity functional. \nTo solve this problem, the authors in Ref.~\\cite{xu09a} adopted the phenomenological approach of approximating the gradient contributions\nwith constant coefficients whose values are taken as the respective average values of the contributions\nprovided by 51 Skyrme interactions. \nA step further in the application of the dynamical method to estimate the core-crust \ntransition density is discussed in Ref.~\\cite{xu10b}, also for the case of MDI\ninteractions. In this work, the authors use the density matrix (DM) expansion proposed by \nNegele and Vautherin \\cite{negele72a, negele72b} to derive a Skyrme-type energy density functional \nincluding gradient contributions, but with\ndensity-dependent coefficients. Using this functional, the authors study the core-crust transition\nthrough the stability conditions \nprovided by the linearized Vlasov equations in NS matter.\n In our case, to generalize the dynamical method to finite-range forces, we perform an expansion of the direct energy in terms of the \ngradients of the nuclear densities which, expressed in momentum space, can be summed up at all orders. \nThis exact calculation goes beyond the phenomenological approach of Ref.~\\cite{xu09a} to deal with these\ncorrections. It also goes beyond the calculation of Ref.~\\cite{xu10b}, where the gradient expansion expressed\nin momentum space was truncated at second order in momentum, therefore remaining in the long-wavelength limit. \nAs in Ref.~\\cite{xu10b}, we include the effects coming from the exchange energy, not considered explicitly in Ref.~\\cite{xu09a},\nwith the help of a DM expansion. This allows us to obtain a quadratic combination of the gradients of the nuclear\ndensities, which in momentum space becomes a quadratic function of the momentum. In this thesis we use a DM \nexpansion based on the Extended Thomas-Fermi (ETF) approximation introduced in Ref.~\\cite{soubbotin00} and applied\nto finite nuclei in Refs.~\\cite{soubbotin03,krewald06,behera16}. \nThis DM expansion allows one to estimate the contribution \nof the kinetic energy due to inhomogeneities of the nuclear densities not taken into consideration in the earlier works.\nWe have formulated the method in a general form and it can be applied to different effective finite-range interactions, \nsuch as the Gogny, MDI and SEI interactions used in this work. \nIn Appendix~\\ref{app_vdyn} we collect the steps of the expansion of the direct part of the interaction in terms of \nthe gradients of the nuclear densities. Moreover, we show in the same Appendix~\\ref{app_vdyn} the use of the DM expansion \nin the ETF approach when looking for the contributions coming from the exchange and kinetic energies to the \nsurface term of the curvature matrix~(\\ref{eq12}).\n\nUsing Eq.~(\\ref{eq6}), the total energy of the system can be expanded as:\n\\begin{eqnarray}\nE &=& E_0 + \\int d{\\bf R} \\left[\\frac{\\partial {\\cal H}}{\\partial \\rho_{n}}\\delta \\rho_n +\n\\frac{\\partial {\\cal H}}{\\partial \\rho_{p}}\\delta \\rho_p + \\frac{\\partial {\\cal H}}{\\partial \\rho_{e}}\\delta \\rho_e \\right]_{\\rho_U}\\nonumber\\\\\n&+& \\frac{1}{2}\\int d{\\bf R} \\left[\\frac{\\partial^2 {\\cal H}}{\\partial \\rho_{n}^2}(\\delta \\rho_n)^2 \n+ \\frac{\\partial^2 {\\cal H}}{\\partial \\rho_{p}^2}(\\delta \\rho_p)^2 + \\frac{\\partial^2 {\\cal H}}{\\partial \\rho_{e}^2}\n(\\delta \\rho_e)^2 + 2\\frac{\\partial^2 {\\cal H}}{\\partial \\rho_{n} \\partial \\rho_{p}}\\delta \\rho_n \\delta \\rho_p \\right]_{\\rho_U}\n\\nonumber \\\\ \n&+& \\int d{\\bf R}\\left[B_{nn}(\\rho_{n},\\rho_{p})\\left({\\bf \\nabla}\\delta \\rho_n\\right)^2 \n+ B_{pp}(\\rho_{n},\\rho_{p})\\left({\\bf \\nabla}\\delta \\rho_p\\right)^2 \\right.\\nonumber\\\\\n&+& \\left.B_{np}(\\rho_{n},\\rho_{p}){\\bf \\nabla}\\delta \\rho_n\\cdot{\\bf \\nabla}\\delta \\rho_p \n+ B_{pn}(\\rho_{p},\\rho_{n}){\\bf \\nabla}\\delta \\rho_p\\cdot{\\bf \\nabla}\\delta \\rho_n \\right]_{\\rho_U}\n\\nonumber \\\\\n&+& \\int d{\\bf R}\\left[{\\cal H}_{dir}(\\delta \\rho_n,\\delta \\rho_p) + {\\cal H}_{Coul}(\\delta \\rho_p,\\delta \\rho_e)\\right],\n\\label{eq7}\n\\end{eqnarray}\nwhere $E_0$ contains the contribution to the energy from the unperturbed parts of the neutron, \nproton and electron densities, $\\rho_{Un}$, $\\rho_{Up}$ and $\\rho_{Ue}$, respectively. The subscript \n$\\rho_U$ labeling the square brackets in Eq.~(\\ref{eq7}) implies that the derivatives of the energy \ndensity ${\\cal H}$ as well as the coefficients $B_{qq}$ are \nevaluated at the unperturbed nucleon and electron densities. The last integral in Eq.~(\\ref{eq7}) is the \ncontribution from the nuclear direct and Coulomb parts arising out of the fluctuation in \nthe particle densities. The two terms of this last integral in Eq.~(\\ref{eq7}) are explicitly given by\n\\begin{equation}\\label{Hdir}\n {\\cal{H}}_{dir}  (\\delta \\rho_n, \\delta \\rho_p) = \\frac{1}{2} \\sum_q \\delta \\rho_q ({\\bf R}) \\int d {\\bf s} \\left[  \\sum_i D_{L,dir}^{i} v_i ({\\bf s}) \n \\delta \\rho_q ({\\bf R} - {\\bf s}) + \\sum_i D_{U,dir}^i v_i ({\\bf s}) \\delta \\rho_{q'} ({\\bf R} - {\\bf s}) \\right]\n\\end{equation}\nand \n \\begin{equation}\n  {\\cal{H}}_{Coul}  (\\delta \\rho_n, \\delta \\rho_p) = \\frac{e^2}{2}  (\\delta \\rho_p  ({\\bf R})- \\delta \\rho_e  ({\\bf R})) \\int d {\\bf s} \n  \\frac{\\delta \\rho_p ({\\bf R} - {\\bf s} ) -\\delta \\rho_e ({\\bf R} - {\\bf s} ) }{s}.\n \\end{equation}\nLinear terms in the $\\delta\\rho$ fluctuation vanish in Eq.~(\\ref{eq7}) by the following reason. We are assuming that neutrons, protons and\nelectrons are in $\\beta$-equilibrium. Therefore, the corresponding chemical potentials, defined as\n$\\mu_i={\\partial {\\cal H}}\/{\\partial \\rho_{i}}\\vert_{\\rho_U}$ for each kind of particle ($i=n,p,e$), \nfulfill $\\mu_n - \\mu_p = \\mu_e$. Using this fact, the linear terms in Eq.~(\\ref{eq7}) can be written as:\n\\begin{equation}\n\\mu_n \\delta \\rho_n + \\mu_p \\delta \\rho_p + \\mu_e \\delta \\rho_e =\n\\mu_n( \\delta \\rho_n + \\delta \\rho_p ) + \\mu_e( \\delta \\rho_e - \\delta \\rho_p ).\n\\label{eq7a}\n\\end{equation}\nThe integration of this expression over the space vanishes owing to the charge neutrality of the matter \n(i.e., $\\int d{\\bf R} (\\delta \\rho_e - \\delta \\rho_p)=0$) and to the conservation of the baryon number \n(i.e., $\\int d{\\bf R} (\\delta \\rho_n + \\delta \\rho_p)=0$).\n\nNext, we write the varying particle densities as the Fourier transform of \nthe corresponding momentum distributions $\\delta n_q({\\bf k})$ as \\cite{baym71}\n\\begin{equation}\n\\delta \\rho_q({\\bf R}) = \\int \\frac{d{\\bf k}}{(2\\pi)^3} \\delta n_q({\\bf k}) e^{i {\\bf k} \\cdot {\\bf R}}.\n\\label{eq8}\n\\end{equation}\nOne can transform this equation to momentum space due \nto the fact that the fluctuating densities are the only quantities in Eq.~(\\ref{eq7}) that depend on the position.\nConsider for example the crossed gradient term ${\\bf \\nabla}\\delta \\rho_n\\cdot{\\bf \\nabla}\\delta \\rho_p$\nin Eq.~(\\ref{eq7}).\nTaking into account Eq.~(\\ref{eq8}) we can write \n\\begin{eqnarray}\n\\int d{\\bf R}{\\bf \\nabla}\\delta \\rho_n\\cdot{\\bf \\nabla}\\delta \\rho_p &=&\n-\\int \\frac{d{\\bf k_1}}{(2 \\pi)^3}\\frac{d{\\bf k_2}}{(2 \\pi)^3}{\\bf k_1}\\cdot{\\bf k_2} \\delta n_n({\\bf k_1}) \\delta n_p({\\bf k_2})\n\\int d{\\bf R}e^{i({\\bf k_1}+{\\bf k_2})\\cdot{\\bf R}}\\nonumber\\\\\n&=&-\\int \\frac{d{\\bf K}d{\\bf k}}{(2 \\pi)^3} \\left(\\frac{{\\bf K}}{2}+{\\bf k}\\right) \\cdot \\left(\\frac{{\\bf K}}{2}-{\\bf k}\\right)\n\\delta n_n\\left(\\frac{{\\bf K}}{2}+{\\bf k}\\right) \\delta n_p\\left(\\frac{{\\bf K}}{2}-{\\bf k}\\right)\n\\delta({\\bf K})\\nonumber \\\\\n&=& \\int \\frac{d{\\bf k}}{(2 \\pi)^3}\\delta n_n({\\bf k}) \\delta n_p(-{\\bf k})k^2 =\n\\int \\frac{d{\\bf k}}{(2 \\pi)^3} \\delta n_n({\\bf k})\\delta n_p^*({\\bf k})k^2, \n\\label{eqB1}\n\\end{eqnarray}\nwhere we have used the fact that $\\delta \\rho_q = \\delta\\rho^* _q$ and, therefore, due to (\\ref{eq8}), \n$\\delta n_q(-{\\bf k}) = \\delta n^*_q({\\bf k})$. Similarly,\nthe other quadratic terms in the fluctuating density in Eq.~(\\ref{eq7}) can also be transformed \ninto integrals in momentum space of quadratic combinations of fluctuations of the momentum distributions (\\ref{eq10}).\nAfter some algebra, one obtains\n\\begin{eqnarray}\nE &=& E_0 +  \\frac{1}{2}\\int \\frac{d{\\bf k}}{(2\\pi)^3} \n\\left\\{\\left[\\frac{\\partial \\mu_n}{\\partial \\rho_n}\\delta n_n({\\bf k})\\delta n^*_n({\\bf k})\n+ \\frac{\\partial \\mu_p}{\\partial \\rho_p}\\delta n_p({\\bf k})\\delta n^*_p({\\bf k})\n+ \\frac{\\partial \\mu_n}{\\partial \\rho_p}\\delta n_n({\\bf k})\\delta n^*_p({\\bf k})\\right.\\right.\\nonumber\\\\\n&+&\\left. \\frac{\\partial \\mu_p}{\\partial \\rho_n}\\delta n_p({\\bf k})\\delta n^*_n({\\bf k})\n+ \n \\frac{\\partial \\mu_e}{\\partial \\rho_e}\\delta n_e({\\bf k})\\delta n^*_e({\\bf k})\\right]_{\\rho_0}\n\\nonumber \\\\\n&+& 2k^2\\left[B_{nn}(\\rho_{n},\\rho_{p})\\delta n_n({\\bf k})\\delta n^*_n({\\bf k})\n+ B_{pp}(\\rho_{n},\\rho_{p})\\delta n_p({\\bf k})\\delta n^*_p({\\bf k})\\right.\\nonumber\\\\\n&+&\\left. B_{np}(\\rho_{n},\\rho_{p})\\left(\\delta n_n({\\bf k})\\delta n^*_p({\\bf k}) \n+ \\delta n_p({\\bf k})\\delta n^*_n({\\bf k})\\right)\\right]_{\\rho_0}\n\\nonumber \\\\\n&+& \\sum_i\\left[D_{L,dir}^i\\left(\\delta n_n({\\bf k})\\delta n^*_n({\\bf k}) + \\delta n_p({\\bf k})\\delta n^*_p({\\bf k})\\right)\\right.\\nonumber \\\\\n&+&\\left. D_{U,dir}^i\\left(\\delta n_n({\\bf k})\\delta n^*_p({\\bf k})+\\delta n_p({\\bf k})\\delta n^*_n({\\bf k})\\right)\n\\right]({\\cal F}_i(k)- {\\cal F}_i(0)) \n\\nonumber \\\\\n&+&\\left. \\frac{4 \\pi e^2}{k^2}\\left(\\delta n_p({\\bf k})\\delta n^*_p({\\bf k}) + \\delta n_e({\\bf k})\\delta n^*_e({\\bf k})\n- \\delta n_p({\\bf k})\\delta n^*_e({\\bf k}) - \\delta n_e({\\bf k})\\delta n^*_p({\\bf k})\\right)\\right\\}.\n\\label{eq9}\n\\end{eqnarray}\nThe factors ${\\cal F}_i(k)$ which enter in the contributions of the direct potential \nin Eq.~(\\ref{eq9}) are the Fourier transform of the form factors\n $v_i(s)$.\n For Gaussian type interactions, like Gogny forces, the form factor is\\footnote{Notice that the notation \n of the range in the Gaussian form factor for Gogny interactions has changed from $\\mu$ in Chapter~\\ref{chapter1} to $\\alpha$ to not confuse it with the \n corresponding range parameter of the Yukawa form factors found in the SEI and MDI interactions.}\n \\begin{equation}\n  v_i(s)= e^{-s^2\/\\alpha_i^2}\n \\end{equation}\n and for Yukawa-type interactions we will have, for SEI forces\n \\begin{equation}\n  v_i(s) = \\frac{e^{-\\mu_i s}}{\\mu_i s}\n \\end{equation}\nand for MDI models\n \\begin{equation}\n  v_i(s) = \\frac{e^{-\\mu_i s}}{ s}.\n \\end{equation}\n The form factors $v_i(s)$ can be expanded in a sum of distributions, which in momentum \n space can be ressumated as ${\\cal F}_i(k)$. They are given by\n\\begin{equation}\n {\\cal F}_i(k)=\\pi^{3\/2} \\alpha_i^3 e^{-\\alpha_i^2 k^2\/4}\n\\end{equation}\nfor Gogny forces and\n\\begin{equation}\n {\\cal F}_i(k)=\\frac{4\\pi}{\\mu_i(\\mu_i^2 + k^2)} \\hspace{2mm}\\mathrm{(SEI)}\\hspace{1cm} \\mathrm{or} \\hspace{1cm} {\\cal F}_i(k)=\\frac{4\\pi}{\\mu_i^2 + k^2} \\hspace{2mm}\\mathrm{(MDI)}\n\\end{equation}\nfor Yukawa-type interactions. The full derivation can be found in Appendix~\\ref{app_vdyn}. \n\nThe functions $D_{qq'}(\\rho,k)$ ($qq'=n,p$) for finite range interactions \ncontain the terms of the nuclear energy density coming from the form factor\nof the nuclear interaction plus the $\\hbar^2$ contributions of the kinetic \nenergy and exchange energy densities, see Appendix~\\ref{app_vdyn}. They\ncan be written as:\n\\begin{small}\n\\begin{eqnarray}\n D_{nn}(\\rho, k)&=& \\sum_i D_{L,dir}^i\\big({\\cal F}_i(k)-{\\cal F}_i(0)\\big) + 2k^2 B_{nn}(\\rho_{n},\\rho_{p})\\nonumber \\\\\n D_{pp}(\\rho, k)&=& \\sum_i D_{L,dir}^i\\big({\\cal F}_i(k)-{\\cal F}_i(0)\\big) + 2k^2 B_{pp}(\\rho_{n},\\rho_{p})\\nonumber \\\\\n D_{np}(\\rho, k)&=&  D_{pn}(\\rho, k) \\nonumber \\\\\n&=&\\sum_m D_{U,dir}^m\\big({\\cal F}_m(k)-{\\cal F}_m(0)\\big) + 2k^2 B_{np}(\\rho_{n},\\rho_{p}),\\nonumber\\\\\n\\label{eq13}\n\\end{eqnarray}\n\\end{small}\nThe stability of the system against small density fluctuations requires that the curvature matrix $C^f$ has to be convex for all values\nof $k$, and if this condition is violated, the system becomes unstable. This is guaranteed if the 3$\\times$3 determinant of the matrix is positive, provided that \n$\\partial \\mu_n\/\\partial \\rho_n$ (or $\\partial \\mu_p\/\\partial \\rho_p$) and the 2$\\times$2 minor of the nuclear sector in (\\ref{eq12}) \nare also positive \\cite{xu09a}:\n\\begin{eqnarray}\n a_{11}> 0, \\hspace{0.3cm} a_{22}> 0,  \\hspace{0.3cm}  \\left| \\begin{array}{cc}\na_{11} & a_{12}  \\\\\n a_{21}& a_{22}   \\end{array} \\right| > 0 ,\\hspace{0.3cm}\\left| \\begin{array}{ccc}\na_{11} & a_{12} & a_{13} \\\\\n a_{21}& a_{22} &a_{23}  \\\\\na_{31} &a_{32} &a_{33} \\end{array} \\right| > 0. \\nonumber\n\\\\\n\\label{minors}\n\\end{eqnarray}\nThe two first conditions in~(\\ref{minors}) demand the system to be stable under fluctuations of proton and neutron densities separately, \nthe third minor implies \nthe stability against simultaneous modifications of protons and neutrons, and\nthe last one implies stability under simultaneous proton, neutron and electron density variations.\n\nTherefore, the stability condition against cluster formation, which indicates the transition from\nthe core to the crust, is given by the condition that the dynamical potential, defined as \n\\begin{equation}\n V_{\\mathrm{dyn}} (\\rho, k) = \\left(\\frac{\\partial \\mu_p}{\\partial \\rho_p} + D_{pp}(\\rho, k) + \\frac{4 \\pi e^2}{k^2}\\right) \n - \\frac{\\left( \\partial \\mu_n \/ \\partial \\rho_p + D_{np}(\\rho, k)\\right)^2}{\\partial \\mu_n \/ \\partial \\rho_n + D_{nn}(\\rho, k)} \n- \\frac{(4\\pi e^2 \/ k^2)^2}{\\partial \\mu_e \/ \\partial \\rho_e + 4 \\pi e^2 \/ k^2},\n\\label{eq14}\n\\end{equation}\nhas to be positive. For a given baryon density $\\rho$, the dynamical potential $V_\\mathrm{dyn}(\\rho, k)$ is calculated at the $k$ value that \nminimizes Eq.~(\\ref{eq14}), i.e., that fulfills\n$\\left.\\partial V_\\mathrm{dyn}(\\rho, k) \/ \\partial k \\right|_\\rho = 0$ \\cite{baym71,ducoin07,xu09a}.\nThe first term of (\\ref{eq14}) gives the stability of protons, and the second and third ones give the\nstability when the protons interact with neutrons and electrons, respectively.\nFrom the condition $\\left.\\partial V_\\mathrm{dyn}(\\rho, k) \/ \\partial k \\right|_\\rho =0$ \none gets the dependence of the momentum on the density, $k (\\rho)$,\nand therefore Eq.~(\\ref{eq14}) becomes a function only depending on the unperturbed density $\\rho=\\rho_U$. \nThe core-crust transition takes place at the density where the dynamical \npotential vanishes, i.e., $V_\\mathrm{dyn} (\\rho, k(\\rho))=0$.\n\nThe derivatives of the chemical potentials of neutrons and protons are related with the derivatives of the baryon \nenergy per particle by \\cite{xu09a}\n\\begin{eqnarray}\n \\frac{\\partial \\mu_n}{\\partial \\rho_n} &=& \\rho  \\frac{\\partial^2 E_b}{\\partial \\rho^2} \n + 2\\frac{\\partial E_b}{\\partial \\rho}\n  +  \\frac{\\partial^2 E_b}{\\partial x_p^2}\\frac{ x_p^2}{\\rho} - 2 x_p  \n  \\frac{\\partial^2 E_b}{\\partial \\rho \\partial x_p}\n  \\\\\n \\frac{\\partial \\mu_n}{\\partial \\rho_p} &=&  \\frac{\\partial \\mu_p}{\\partial \\rho_n}=  \n \\frac{\\partial^2 E_b}{\\partial \\rho \\partial x_p}  (1-2 x_p)\n  + \\rho \\frac{\\partial^2 E_b}{\\partial \\rho^2} + 2 \\frac{\\partial E_b}{\\partial \\rho} +\n  \\frac{\\partial^2 E_b}{\\partial x_p^2}  \\frac{ x_p}{\\rho} (x_p -1)\\nonumber\n  \\\\\n  \\frac{\\partial \\mu_p}{\\partial \\rho_p} &=& \\frac{1}{\\rho} \\frac{\\partial^2 E_b}{\\partial x_p^2} \n  (1-x_p)^2 + 2 \\frac{\\partial^2 E_b}{\\partial \\rho \\partial x_p} (1-x_p) \n   +  \\rho  \\frac{\\partial^2 E_b}{\\partial \\rho^2} + \\frac{\\partial E_b}{\\partial \\rho},\n\\label{dermu}\n \\end{eqnarray}\n where the Coulomb interaction is not considered in nuclear matter and, \ntherefore, the cross derivatives are equal $\\partial \\mu_n \/ \\partial \\rho_p =  \\partial \\mu_p \/ \\partial \\rho_n$. \nThe explicit expressions of the derivatives of the energy per particle are given in Appendix~\\ref{appendix_thermal}.\n\nWe conclude this section by mentioning that dropping the Coulomb contributions in Eq.~(\\ref{eq14})\nand taking the $k\\to0$ limit leads to the so-called thermodynamical potential used in the thermodynamical method\nfor looking for the core-crust transition (see Section~\\ref{Theory_thermo}):\n\\begin{equation}\nV_{\\mathrm{ther}}(\\rho) = \\frac{\\partial \\mu_p}{\\partial \\rho_p} \n - \\frac{\\left( \\partial \\mu_n \/ \\partial \\rho_p \\right)^2}{\\partial \\mu_n \/ \\partial \\rho_n } ,\n\\label{eqB7}\n\\end{equation}\nwhere the condition of stability of the uniform matter of the core corresponds to a value of the thermodynamical potential \n$V_{\\mathrm{ther}}(\\rho)~>~0$.\n\n\\section{Core-crust transition within the dynamical method}\\label{Results_dyn}\n\\subsection{Core-crust transition properties obtained with Skyrme interactions}\n\\begin{figure}[t!]\n \\centering\n \\includegraphics[width=0.8\\linewidth, clip=true]{.\/grafics\/chapter4\/vdyn\/MSk7-SkI5-V_dynamic}\\\\\n  \\caption{ Density dependence of the dynamical potential V$_\\mathrm{dyn}$ in $\\beta$-stable matter calculated using the exact expression \nof the energy per particle or the expression in Eq.~(\\ref{eq:EOSexpgeneral}) up to second and tenth order for two Skyrme forces, MSk7 and SkI5.\nThe results for the PA and for the case where the expansion up to second order is considered only in the potential part of the interaction \nare also included (label E$_\\mathrm{kin}$ exact).}\n \\label{fig:dynamic}\n\\end{figure}\nWe plot in Fig.~\\ref{fig:dynamic}  the density dependence of V$_\\mathrm{dyn} (\\rho)$ computed with \nthe MSk7 ($L=9.41$~MeV) and SkI5 ($L=129.33$ MeV) Skyrme interactions. These two models have the smallest and largest value of the slope parameter of the symmetry energy \nat saturation within the sets of \nSkyrme forces considered, implying that the isovector properties predicted by these two functionals are actually  very different, \nat least around saturation. From Fig.~\\ref{fig:dynamic} we see that, in general,  $V_\\mathrm{dyn} (\\rho)$ as a function of the density shows a \nnegative minimum value at $\\rho \\lesssim 0.04$ fm$^{-3}$, and then increases cutting the $V_\\mathrm{dyn} (\\rho) = 0$ line at \na density that corresponds to the transition density, which separates the unstable and stable $npe$ systems.\nTaking into account higher-order terms in the expansion~(\\ref{eq:EOSexpgeneral}) of the EoS, the corresponding\ntransition densities approach the value obtained with the full EoS in Eq.~(\\ref{eq:Ebanm}). In particular, taking the expansion up to \n$\\delta^{10}$, the exact transition density is almost reproduced using the MSk7 interaction. However, this is not the situation \nwhen the transition density is calculated with the SkI5 force. In this case, although the expansion~(\\ref{eq:EOSexpgeneral}) is pushed until \n$\\delta^{10}$, the approximate estimate of the transition density overcomes the exact density by more than 0.01 fm$^{-3}$. \nThe transition density computed with the PA is similar to the values obtained using the Taylor expansion~(\\ref{eq:EOSexpgeneral}) \nup to $\\delta^4$ and $\\delta^2$ and with the MSk7 and SkI5 forces, respectively.\nFig.~\\ref{fig:dynamic} also contains the values of the dynamical potential computed expanding up to \nsecond order only the potential part of the interaction and using the full expression for the \nkinetic energy.\nFrom Fig.~\\ref{fig:dynamic} we notice that $V_\\mathrm{dyn} (\\rho)$\ncomputed exactly with the  MSk7 force predicts a larger transition density than calculated with the SkI5 interaction. This result suggests\nthat Skyrme models with small slope parameters tend to cut the  V$_\\mathrm{dyn}(\\rho) = 0$ line at larger densities than \nthe interactions with a larger slope parameter, and, therefore, predict larger core-crust transition densities.  \n\nThe exact transition properties, obtained as a solution of V$_\\mathrm{dyn} = 0$ with the additional constraint \n$\\partial V_\\mathrm{dyn}(\\rho, k) \/ \\partial k |_\\rho = 0$ and calculated through Eqs.~(\\ref{eq14})-(\\ref{dermu}) using the exact energy \nper particle given by Eq.~(\\ref{eq:Ebanm}), are reported in Appendix~\\ref{app_taules} for a set of Skyrme interactions available \nin the literature and characterized by a different slope parameter $L$. In the same \ntable we also give the isospin asymmetry and the pressure corresponding to the  \ntransition density calculated assuming $\\beta$-stable nuclear matter.\n\n\\begin{figure}[t!]\n \\centering\n \\includegraphics[width=0.8\\linewidth, clip=true]{.\/grafics\/chapter4\/vdyn\/rho_t_vs_L_Skyrme_Vdyn}\\\\\n  \\caption{Transition density versus the slope parameter $L$ for a set of Skyrme interactions, calculated \n  using the exact expression of the energy per particle and \n  the approximations up to second and tenth order. \n  The results for the PA and for the case where the expansion up to second order is considered only in the potential part of the interaction \nare also included.}\n \\label{fig:rhotvdynskyrme}\n\\end{figure}\n\nNext, we analyze in detail any possible correlation of the transition density, isospin asymmetry and pressure with the slope \nparameter $L$. In Fig.~\\ref{fig:rhotvdynskyrme} we plot the transition density as a function of the slope of the symmetry energy for \nthe Skyrme interactions reported in Appendix~\\ref{app_taules}, calculated using the exact expression of the energy \nper particle as well as with the different approximations of this quantity considered through this work. \nThe transition densities follow a quasi-linear decreasing tendency as a function of the slope parameter $L$.\nThe values of the core-crust transition densities obtained with the full EoS \nlie in the range $0.050$ fm$^{-3}\\lesssim \\rho_t \\lesssim 0.115 $ fm$^{-3}$.\nThe gap between the asymmetry for the exact results and the approximated ones increases with growing values of the slope parameter $L$. \nWe observe that, in general, the exact \ntransition densities are better reproduced by the different approximations discussed here if the slope parameter $L$ of the \ninteraction is smaller than $L\\lesssim60$ MeV. For larger values of $L$ the disagreement between the exact and approximated transition \ndensities increases. \nAs expected, the agreement between the exact \nand the approximate results improves when more terms of the expansion are considered.   \nFor example, if we consider the MSk7 interaction ($L$=9.41 MeV), the relative difference between the result calculated with \nthe expansion of Eq.~(\\ref{eq:EOSexpgeneral}) up to tenth order in $\\delta$\nand the exact EoS is $\\sim 1\\%$, while if we consider the SkI5 interaction ($L$ = 129.33 MeV), the value of the relative \ndifference is $\\sim 30\\%$. If the PA is used to estimate the transition densities, the results are quite similar to those calculated \nstarting from the second order expansion of the energy per particle. As we have pointed out before, considering \nonly the expansion up to send-order only to the potential part of the energy per particle\nreproduces very well the exact values of the transition density. \n\\begin{figure}[t!]\n \\centering\n \\includegraphics[width=0.8\\linewidth, clip=true]{.\/grafics\/chapter4\/vdyn\/delta_t_vs_L_Skyrme_Vdyn}\\\\\n   \\caption{Transition asymmetry versus the slope parameter $L$ for a set of Skyrme interactions, calculated \n  using the dynamical method with the exact expression of the energy per particle and \n  the approximations up to second and tenth order. \n  The results for the PA and for the case where the expansion up to second order is considered only in the potential part of the interaction \nare also included.}\n \\label{fig:deltatvdynskyrme}\n\\end{figure}\n\n\nWe have also obtained the asymmetry corresponding to the transition density, i.e., the transition asymmetry, assuming $\\beta$-stable matter.\nThe values calculated from the exact energy per particle with the set of Skyrme forces considered\nlie in the range \\mbox{$0.916 \\lesssim \\delta_t \\lesssim 0.982$}.\nThe results of the transition density plotted against the slope of the symmetry energy show\nroughly an increasing trend.\nWe plot in the same figure the transition asymmetries computed with the \ndifferent approximations considered before. \nThe relative differences between the values up to second order and the\nexact ones are $\\lesssim 1\\%$ for all the considered Skyrme interactions. Notice that a small difference in the isospin asymmetry can \ndevelop to large differences in the estimation of other properties, such as the density and the pressure. \nMoreover, the transition asymmetries predicted by the PA are similar to those obtained from the expanded energy per particle\nup to the second order in $\\delta$. If one expands up to second-order only the potential part of the energy per particle, \nthe transition asymmetry obtained with the full EoS is almost reproduced for all the analyzed interactions.   \n \\begin{figure}[t!]\n \\centering\n \\includegraphics[width=0.8\\linewidth, clip=true]{.\/grafics\/chapter4\/vdyn\/P_t_vs_L_Skyrme_Vdyn}\\\\\n   \\caption{Transition pressure versus the slope parameter $L$ for a set of Skyrme interactions, calculated \n  using the dynamical method with the exact expression of the energy per particle and \n  the approximations up to second and tenth order. \n  The results for the PA and for the case where the expansion up to second order is considered only in the potential part of the interaction \nare also included.}\n \\label{fig:Ptvdynskyrme}\n\\end{figure}\nIn Fig.~\\ref{fig:Ptvdynskyrme} we display the core-crust transition pressure as a function of the slope parameter $L$ for the same Skyrme forces.\nThe transition pressure calculated with the full EoS has a somewhat decreasing trend with $L$.\nThis correlation is practically destroyed when the pressure is obtained in an approximated way, in particular, if the expansion of the\nenergy per particle is terminated at second order or the PA are used to this end. As a consequence of the weak correlation between transition pressure\nand $L$, forces with similar slope parameters may predict quite different pressures at the transition density. \nThe exact values of the transition pressure lie in the range $0.141 $~MeV~fm$^{-3}\\lesssim P_t \\lesssim 0.541$~MeV~fm$^{-3}$,\ndecreasing with increasing values of $L$.\nAs for the transition densities and asymmetries, the differences between the exact pressure and the ones obtained\nstarting from the different approximations to the energy per particle are nearly inexistent for models with small\nslope parameter $L$, but the gaps can become huge for models with $L$ larger than $\\sim 60$ MeV.\nFor example, the relative difference between the exact transition pressure and the estimate obtained using the expansion~(\\ref{eq:EOSexpgeneral})\nof the energy per particle up to tenth order is only $0.65\\%$ computed with the MSk7 interaction, but this relative difference \nbecomes $144\\%$ calculated with the SkI5 force.\nFrom the same Fig.~\\ref{fig:Ptvdynskyrme} we notice that the PA clearly overestimates the transition pressure for all the forces that we \nhave considered, giving the largest differences with respect to the exact pressure as compared with the other approximations\nto the energy per particle. Once again, the case where only the potential part is expanded \nreproduces very accurately the exact transition pressures\nfor all the interactions.\n\n\\subsection{Core-crust transition properties obtained with finite-range interactions}\nWe will discuss first the impact of the finite-range terms of the nuclear effective force\non the dynamical potential. Next, we will focus on the study of the core-crust transition density \nand pressure calculated from the dynamical and thermodynamical methods using three different finite-range nuclear \nmodels, namely Gogny, MDI and SEI interaction. Finally, we will examine the influence of the core-crust transition point on the crustal properties in NSs~\\cite{gonzalez19}.\n\nWe show in Fig.~\\ref{fig:Vdynfr}~\\cite{gonzalez19} the dynamical potential as a function of the unperturbed density computed using the Gogny\nforces D1S \\cite{berger91}, D1M \\cite{goriely09}, D1M$^*$ \\cite{gonzalez18} and D1N \\cite{chappert08}.\nThe density at which $V_\\mathrm{dyn}(\\rho, k(\\rho))$ vanishes depicts the transition density. \nBelow this point a negative value of $V_\\mathrm{dyn}$  implies instability. Above this density, \nthe dynamical potential is positive, meaning that the curvature matrix in Eq.~(\\ref{eq12}) is convex at all \nvalues of $k$ and therefore the system is stable against cluster formation.\nThe Gogny forces D1S and D1M predict larger values of the transition density compared to D1N and D1M$^*$. \nThis is due to the different density slope of the symmetry energy predicted \nby these forces.\nThe values of the slope parameter $L$ of the symmetry energy \nare $22.43$ MeV, $24.83$ MeV, $35.58$ MeV and $43.18$ MeV for the D1S, D1M, D1N and D1M* forces. \nA lower $L$ value implies a softer symmetry energy around the saturation density.\nThe prediction of higher core-crust\ntransition density for lower $L$ found in our study is in agreement with the previous \nresults reported in the literature (see \\cite{xu09a,gonzalez17} and references therein).\n\\begin{figure}[t]\n\\centering\n\\includegraphics[clip=true,width=0.8\\linewidth]{.\/grafics\/chapter4\/vdyn\/Vdynamic_vs_rho}\n\\caption{Dynamical potential as a function of the density for the D1S, D1M, D1M$^*$ and D1N Gogny interactions. }\\label{fig:Vdynfr}\n\\end{figure}\n\nAs has been done in previous works \\cite{baym70,baym71,xu09a}, the dynamical potential (\\ref{eq12}) can be\napproximated up to order $k^2$ by performing a Taylor expansion of the coefficients $D_{nn}$, $D_{pp}$ and $D_{np}=D_{pn}$ in Eqs.~(\\ref{eq13}) \\cite{gonzalez19}. \nIn this case the dynamical potential can be formally written as \\cite{pethick95,ducoin07,xu09a}\n\\begin{equation}\n{\\tilde V}_{\\mathrm{dyn}}(\\rho, k) = V_{\\mathrm{ther}}(\\rho) + \\beta(\\rho)k^2 + \n\\frac{4\\pi e^2}{k^2 + \\frac{4\\pi e^2}{\\partial \\mu_e\/ \\partial \\rho_e}},\n\\label{eq14a}\n\\end{equation}\nwhere $V_{\\mathrm{ther}}(\\rho)$ has been defined in Eq.~(\\ref{eqB7})\nand the expression of $\\beta (\\rho)$ is given in Eq.~(\\ref{eqB8}) of \nAppendix \\ref{app_vdyn}. \nThe practical advantage of Eq.~(\\ref{eq14a}) is that, for finite-range interactions, the $k$-dependence is separated from the $\\rho$-dependence,  \nwhereas in the full expression for the dynamical potential in Eq.~(\\ref{eq14}) they are not separated.\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[clip=true, width=0.6\\linewidth]{.\/grafics\/chapter4\/vdyn\/Com_Vdyn_Gogny}\n\\caption{Momentum dependence of the dynamical potential at two different densities, $\\rho=0.10$ fm$^{-3}$ (panel a)\nand $\\rho=0.08$ fm$^{-3}$ (panel b), for the D1S, D1M and D1M$^{*}$ Gogny interactions. The results plotted with solid \nlines are obtained using the full expression of the dynamical potential, given in Eq.~(\\ref{eq14}), while the dashed lines are \nthe results of its $k^2$-approximation, given in Eq.~(\\ref{eq14a}).}\\label{fig:Vdyncomp}\n\\end{figure}\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[clip=true, width=0.8\\linewidth]{.\/grafics\/chapter4\/vdyn\/beta}\n\\caption{Coefficient $\\beta(\\rho)$ of Eq.~(\\ref{eq14a}) as a function of the density for a set of Gogny interactions. The dashed line  \nincludes only contributions coming from the direct energy, the dash-dotted lines include contributions coming \nfrom the direct energy and from the from the gradient effects of the exchange energy. \nFinally, the solid lines include all contributions to $\\beta(\\rho)$.}\\label{fig:beta}\n\\end{figure}\nIn order to examine the validity of the $k^2$-approximation (long-wavelength limit) of the dynamical potential,\nwe plot in Fig.~\\ref{fig:Vdyncomp} the dynamical potential at a given density as a function of the momentum $k$ \nfor both Eq.~(\\ref{eq14}) (solid lines)\nand its $k^2$-approximation in Eq.~(\\ref{eq14a}) (dashed lines) for the Gogny forces D1S, \nD1M and D1M$^*$ \\cite{gonzalez19}. \nThe momentum dependence \nof the dynamical potential at density $\\rho=0.10$ fm$^{-3}$ is shown in the upper panel of Fig.~\\ref{fig:Vdyncomp}. \nOne sees that at this density the core\nhas not reached the transition point for any of the considered Gogny forces, as the minima of the \n$ V_{\\mathrm{dyn}} (\\rho, k)$ curves are positive. This implies that the system is stable against formation of clusters.\nIn the lower panel of Fig.~\\ref{fig:Vdyncomp} the same results but at a density $\\rho=0.08$ fm$^{-3}$ are shown.\nAs the minimum of the dynamical potential for all forces is negative, the matter is unstable against \ncluster formation. From the results in Fig.~\\ref{fig:Vdyncomp} it can be seen\nthat at low values of $k$ the agreement between the results of Eq.~(\\ref{eq14}) and its approximation Eq.~(\\ref{eq14a}) \nis very good. However, at large momenta beyond $k$ of the minimum of the dynamical potential, \nthere are increasingly larger differences between both calculations of the dynamical potential. \nThis is consistent with the fact that Eq.~(\\ref{eq14a}) is the $k^2$-approximation of Eq.~(\\ref{eq14}).\n\n\n \nTo investigate the impact of the direct and $\\hbar^2$ contributions of the finite-range part of the interaction\non the dynamical potential, in Fig.~\\ref{fig:beta} we plot for Gogny forces\nthe behaviour of the coefficient $\\beta(\\rho)$, which accounts for the finite-size effects\nin the stability condition of the core using the dynamical potential ${\\tilde V}_{\\mathrm{dyn}}(\\rho, k)$~\\cite{gonzalez19}.\nThe dashed lines are the result for $\\beta (\\rho)$ when only the direct contribution from the\nfinite range is taken into account.\nThe dash-dotted lines give the value of $\\beta(\\rho)$ where, along with the direct \ncontribution, the gradient effects from the \nexchange energy are included. Finally, solid lines correspond to \nthe total value of $\\beta(\\rho)$, which includes the direct contribution and\nthe complete $\\hbar^2$ corrections, coming from both the exchange and kinetic energies.\n We see that for Gogny forces the gradient correction due to the \n$\\hbar^2$ expansion of the exchange energy reduces the result of $\\beta (\\rho)$ from the direct contribution, at most, by 5\\%. When the  \ngradient corrections from both the exchange and kinetic energies are taken together, \nthe total result for $\\beta (\\rho)$ increases by 10\\% at most with respect to the direct contribution. The effects of the \nkinetic energy $\\hbar^2$ corrections are around three times larger than the effects from the exchange energy $\\hbar^2$ corrections \nand go in the opposite sense. \nTherefore, we conclude that the coefficient $\\beta(\\rho)$ is governed dominantly by the gradient expansion of the\ndirect energy (see Appendix \\ref{app_vdyn}), whereas the contributions from the $\\hbar^2$ corrections of the \nexchange and kinetic energies are small corrections. \n  \n\\begin{figure}[!b]\n\\centering\n\\includegraphics[clip=true,width=0.8\\linewidth]{.\/grafics\/chapter4\/vdyn\/rho_t_P_t_L_gogny_Vdyn}\n\\caption{Transition density (panel a) and transition pressure (panel b) as a function of the slope of the symmetry \nenergy calculated using the thermodynamical and the dynamical methods for \na set of different Gogny interactions.\\label{fig:rhotPtGogny}}\n\\end{figure}\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[clip=true, width=0.8\\linewidth]{.\/grafics\/chapter4\/vdyn\/rhot_P_t_L_MDI}\n\\caption{Same as Fig.~\\ref{fig:rhotPtGogny} but for a family of MDI interactions.\\label{fig:rhotPtMDI}}\n\\end{figure}\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[clip=true, width=0.8\\linewidth]{.\/grafics\/chapter4\/vdyn\/SEI_gamma12_rhot_Pt}\n\\caption{Same as Fig.~\\ref{fig:rhotPtGogny} but for a family of SEI interactions of $\\gamma=1\/2$ ($K_0=237.5$ MeV).\\label{fig:rhotPtSEI}}\n\\end{figure}\n\n\n\\begin{figure}[b!]\n\\centering\n\\includegraphics[clip=true, width=0.8\\linewidth]{.\/grafics\/chapter4\/vdyn\/rhot_L_all_opt_dyn_SEI}\n\\caption{Transition density as a function of the slope of the symmetry energy calculated using the dynamical method for \na set of Skyrme, Gogny, MDI and SEI interactions.\\label{fig:rhotall}}\n\\end{figure}\n\\begin{figure}[t!]\n\\centering\n\\includegraphics[clip=true,width=0.8\\linewidth]{.\/grafics\/chapter4\/vdyn\/Pt_L_all_opt_vdyn}\n\\caption{Transition density as a function of the slope of the symmetry energy calculated using the dynamical method for \na set of Skyrme, Gogny, MDI and SEI interactions.\\label{fig:Ptall}}\n\\end{figure}\n\n\nIn Figs.~\\ref{fig:rhotPtGogny}, \\ref{fig:rhotPtMDI} and \\ref{fig:rhotPtSEI} we display, as a \nfunction of the slope of the symmetry energy $L$,\nthe transition density (upper panels) and the transition pressure (lower panels) calculated with the \nthermodynamical ($V_\\mathrm{ther}$) and the dynamical ($V_\\mathrm{dyn}$) methods for three different types of finite-range interactions~\\cite{gonzalez19}. \nThe expression (\\ref{eq14}) with the complete momentum dependence of the dynamical potential has been used for the dynamical calculations.\nIn the study of the transition properties we have employed several Gogny interactions~\\cite{decharge80,Sellahewa14,gonzalez17,gonzalez18} \n(Fig.~\\ref{fig:rhotPtGogny}), a \nfamily of MDI forces \\cite{das03,xu10b}, with $L$ from $7.8$ to $124.0$ MeV (corresponding to\nvalues of the $x$ parameter ranging from\n$1.15$ to $-1.4$) (Fig.~\\ref{fig:rhotPtMDI}), and a family of SEI interactions \\cite{behera98} with $\\gamma=1\/2$ and different \nvalues of the slope parameter $L$ (Fig.~\\ref{fig:rhotPtSEI}). \nNotice that the Gogny interactions have different nuclear matter saturation properties, \nwhereas the parametrizations of the MDI or SEI families displayed in Figs.~\\ref{fig:rhotPtMDI} and \\ref{fig:rhotPtSEI} have the\nsame nuclear matter saturation properties within each family.\nIn particular, all parametrizations of the MDI family considered here have the same nuclear matter incompressibility  \n$K_0=212.6$ MeV, whereas the SEI parametrizations with $\\gamma=1\/2$ have $K_0= 237.5$ MeV.\nConsistently with previous investigations \n(see e.g. \\cite{xu09a, xu10b} and references therein) we find that for a given parametrization, the transition density \nobtained with the dynamical method is \nsmaller than the prediction of the thermodynamical approach, as the surface and Coulomb contributions tend to \nfurther stabilize the uniform matter in the core against the formation of clusters. The relative differences between the \ntransition densities obtained with the thermodynamical and dynamical approaches are found to vary within the ranges\n$8-14 \\%$ for Gogny interactions, \n$15-30 \\%$ for MDI, and $10-25 \\%$ for SEI $\\gamma=1\/2$. This clearly points out that important differences arise\nin the predicted core-crust transition point between the dynamical method and the simpler thermodynamical method.\nFor the MDI interactions, the trends of the transition density with $L$ are comparable to those found in Ref.~\\cite{xu09a} \nwhere the finite-size effects in the dynamical calculation\nwere taken into account phenomenologically through assumed constant values of 132 MeV fm$^{5}$\nfor the $D_{nn}$, $D_{pp}$ and $D_{np}=D_{pn}$ coefficients.\n The transition densities calculated with $D_{nn}=D_{pp}=D_{np}=132$ MeV fm$^{5}$ lie between the results\n obtained with the thermodynamical method and\nthose obtained using the dynamical approach in Fig.~\\ref{fig:rhotPtMDI}. As far as the results for the transition pressure \nare concerned, from Figs.~\\ref{fig:rhotPtGogny}--\\ref{fig:rhotPtSEI} it is evident again that the dynamical \ncalculation predicts smaller transition pressures than its thermodynamical counterpart, which is also consistent \nwith the findings in earlier works (see \\cite{xu09a, xu10b} and references therein).\n\nIt is to be observed that we have performed the dynamical calculations of the transition properties in two different \nways. On the one hand, we have considered the contributions to Eqs.~(\\ref{eq13}) from the direct energy only \n(the results are labeled as ``$V_\\mathrm{dyn}$ direct'' in Figs.~\\ref{fig:rhotPtGogny}--\\ref{fig:rhotPtSEI}).\nThis approximation corresponds to neglecting the terms $B_{nn}$, $B_{pp}$ and $B_{np}$ in Eqs.~(\\ref{eq13}) that \ndefine the $D_{qq'}$ coefficients.\nThen, we have used the complete expression of Eqs.~(\\ref{eq13}) (the results are labeled as ``$V_\\mathrm{dyn}$ \ndirect$+ \\hbar^2$'' in Figs.~\\ref{fig:rhotPtGogny}--\\ref{fig:rhotPtSEI}).\nAs with our findings for the dynamical potential in the previous subsection, we see that for the three types of \nforces the finite-range \neffects on $\\rho_t$ and $P_t$ coming from the $\\hbar^2$ corrections of the exchange and kinetic energies are \nalmost negligible\ncompared to the effects due to the direct energy.\n\nFigs.~\\ref{fig:rhotPtGogny}--\\ref{fig:rhotPtSEI} also provide information about the \ndependence of the transition properties with the slope of the symmetry energy. \nWe can see that within the MDI and SEI families the transition density and pressure\nshow a clear, nearly linear decreasing trend as a function of the slope parameter $L$. In contrast, the results\nof the several Gogny forces in Fig.~\\ref{fig:rhotPtGogny} show a weak trend with $L$ for $\\rho_t$ and almost \nno trend with $L$ for $P_t$ (as also happened with the different Skyrme forces in the previous subsection).\nA more global analysis of the eventual dependence of the transition properties with the slope of \nthe symmetry energy is displayed in Figs.~\\ref{fig:rhotall} and \\ref{fig:Ptall}. These figures include not only \nthe results for the core-crust transition \ncalculated using the considered sets of finite-range interactions (Gogny, MDI and SEI of $\\gamma=1\/2$ interactions) \nbut also the values for the Skyrme interactions found in the previous section.\nUsing this \nlarge set of nuclear models of different nature, it can be seen that the \ndecreasing trend of the transition density and pressure with a rising in the slope $L$ of the symmetry energy\nis a general feature.\nHowever, the  correlation of the transition properties with the slope parameter $L$ using all interactions is\nweaker than within a family of parametrizations where the saturation properties do not change. \nThe correlation in the case of the transition density is found to be slightly better than in the case of\nthe transition pressure. The reason for the weaker correlation obtained in the\ndifferent Gogny sets (Fig.~\\ref{fig:rhotPtGogny}) and Skyrme sets (Figs.~\\ref{fig:rhotall} and \\ref{fig:Ptall}) \nis attributed to the fact that these sets have different nuclear matter saturation properties apart from the $L$ values.\nFurther, we have also examined the correlation of the transition density and transition pressure with the curvature of the \nsymmetry energy ($K_\\mathrm{sym}$ parameter)\nfor the models used in this work. The transition density and pressure show a decreasing trend with increasing \n$K_\\mathrm{sym}$, in agreement with the findings in earlier works \\cite{xu09a, Carreau19}. \nThe correlations of the transition density and pressure with the value of $K_\\mathrm{sym}$ have a similar quality \nto the case with the $L$ parameter. \n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[clip=true, width=0.8\\linewidth]{.\/grafics\/chapter4\/vdyn\/SEI_Vdyn_diferents_gamma_rhot_Pt}\n\\caption{Transition density (panel a) and transition pressure (panel b) as a function of the slope of the symmetry energy,\ncalculated using the dynamical method for three SEI families with different nuclear matter incompressibilities.}\\label{fig:SEIK0}\n\\end{figure}\n\n\nIn order to test the impact of the nuclear matter incompressibility on the correlations\nbetween the core-crust transition properties and the slope of the symmetry energy \nfor a given type of finite-range interactions, we plot \nin Fig.~\\ref{fig:SEIK0} the transition density (panel a) and the transition pressure (panel b) against the $L$ parameter for \nSEI interactions of $\\gamma=1\/3$, $\\gamma=1\/2$ and $\\gamma=1$, which correspond to nuclear matter incompressibilities $K_0$ of $220.0$ MeV,\n$237.5$ MeV and $282.3$ MeV, respectively, covering an extended \nrange of $K_0$ values~\\cite{gonzalez19}.\nFrom this figure, we observe a high correlation with $L$ for the different sets of a given SEI force having the same incompressibility\nand differing only in the $L$ values. The comparison of the results\nobtained with the SEI forces of different incompressibility shows that higher transition density and pressure are predicted for the \nforce sets with higher incompressibility.\nThis also demonstrates, through the example of the nuclear matter incompressibility, a dependence of the features of \nthe core-crust transition with the nuclear matter saturation properties of the force,\non top of the dependence on the stiffness of the symmetry energy.\n\n\\chapter{Neutron star properties}\\label{chapter5}  \nThis chapter is devoted to the study of different properties of neutron stars (NSs). First of all, we analyze how the choice of the EoS in the inner \nlayer of the NS crust can affect the results for the NS mass and radius. \nThe EoS in the inner crust is not as widely studied as in the case of the core, due to the \ndifficulty of computing the neutron gas and the presence of nuclear clusters that may adopt non-spherical shapes in order to minimize the energy\nof the system. \nWe discuss possible alternatives one can use when the EoS of the inner crust is not known for the interaction used to describe the core. \n\nMoreover, we analyze the influence of the core-crust transition point when studying crustal properties, such as the crustal mass, \ncrustal thickness and crustal fraction of the moment of inertia. The proper determination of the different properties of the crust \nis important in the understanding of observed phenomena such as pulsar glitches, r-mode oscillations, cooling of isolated NSs, \netc.~\\cite{Link1999,Chamel2008,Fattoyev:2010tb,Chamel2013,PRC90Piekarewicz2014,Newton2015}.\n\nThe ratio between the crustal fraction of the moment of inertia and the total moment of inertia is essential when studying \nphenomena such as pulsar glitches. With the same interactions we have used up to now, we integrate the moment of inertia \nand study the performance of these interactions when compared to different theoretical and observational constraints~\\cite{Link1999, Andersson2012, Landry18}. \n\n\nThe recent GW170817 detection~\\cite{Abbott2017} of gravitational waves from the merger of two NSs by the LIGO and Virgo collaboration (LVC) has helped to \nfurther constrain the EoS of $\\beta$-stable nuclear matter. From the analysis of the data obtained from the detection, specific constraints \non the mass-weighted tidal deformability at a certain chirp mass of the binary system have been extracted by the LVC~\\cite{Abbott2017, Abbott2019}.\nMoreover, with further analyses, the LVC was able to provide\nconstraints on the masses, radii, tidal deformabilities of an NS of $1.4 M_\\odot$, etc.~\\cite{Abbott2017, Abbott2018, Abbott2019}.\nWe have applied the interactions we have used in this thesis, paying special attention to the stiffness of their symmetry energy, \nfor describing the tidal deformability of NSs and we compare them with the constraints coming from the \nobservations.\n\n\n\\section{Crustal properties}\nThe solution of the TOV equations, see Eqs.~(\\ref{eq:TOV}) and (\\ref{eq:TOV2}), \ncombined with the determination of the core-crust transition, allows one\nto separate the crust and the core regions inside an NS~\\cite{Chamel2008}. \nIn particular, one can compute the thickness ($R_\\mathrm{crust}$) and mass ($M_\\mathrm{crust}$) of the crust. The \ncrustal thickness is defined as the radial coordinate that goes from the core-crust transition to the \nsurface of the star, and the mass enclosed in this region is the crustal mass. \nThe study of the crustal properties depends on the EoS used to describe the crust, especially in its inner layers. \nIdeally, one should use a unified EoS, which is obtained with the same interaction in all regions of the NS\n(outer crust, inner crust, and core). However, the computation of the EoS in the inner crust is tougher \nthan the computation in the core due to the presence of the neutron gas and nuclear clusters with shapes different from the \nspherical one. \nHence, the EoS of the inner crust has been studied with fewer interactions than the core and, therefore, \nthe EoS in the crust region is still not computed for many forces. \n\nThere are different alternatives to characterize the inner crust when the used interaction to study the NS core\ndoes not have its equivalent in the crust. \nOne alternative is to use a polytropic EoS matching the core EoS with the outer crust EoS. \nThe outer crust, which goes from the surface of the star, which has densities of $\\sim 10^{-12}$ fm$^{-3}$, to densities around $\\sim 10^{-4}$ fm$^{-3}$, \ncontributes in a very small fraction to the total mass and thickness of the NS, and it is mostly determined by nuclear masses experimentally known. \nHence, the EoS that describes the outer crust of the star will have a relatively low influence on the determination of the properties of the NS.\nTherefore, to describe the outer crust, one can use an already existing EoS given in the literature. \nHaving the EoS for the outer crust and for the core, the inner crust can be obtained as a polytrope of the type $P=a+b \\epsilon^{4\/3}$, where $P$ is the pressure and $\\epsilon$ is the energy density. \nThis prescription has been widely used in previous works\\cite{Link1999, carriere03,xu09a,Zhang15, gonzalez17}, where one adjusts the coefficients $a$ and $b$\n by demanding continuity at the outer crust-inner crust and core-crust interfaces. \n The expressions to obtain $a$ and $b$ are given in Eqs.~(\\ref{eq:coefa}) and (\\ref{eq:coefb}) of Chapter~\\ref{chapter2}.\n A polytropic form with an index of average value of about $4\/3$ is found to be a good approximation to the EoS in this region~\\cite{Link1999,Lattimer01,Lattimer2016}, as \n the pressure of matter at these densities is largely influenced by the relativistic degenerate electrons. \n The assumption of a polytropic EoS for the inner crust is the prescription we have been using until now through all this thesis. \n\n For some interactions, the unified EoS for all the NS has been computed. For these forces, we can fortunately compare the \n behaviour of the polytropic EoS for the inner crust and their respective unified EoS computed taking into \n account the physics of the inner crust with the same interaction. \n We plot in Fig.~\\ref{fig:inner} in double logarithmic scale the EoS (total pressure against the total energy density) \n of two interactions for which their unified EoSs exist in the literature. These two models are the SLy4~\\cite{douchin01} Skyrme \n interaction and the BCPM~\\cite{sharma15} density functional. For each one of them, we plot with straight lines the unified EoS, which\n has all regions of the NS (outer crust, inner crust, core) computed with the same interaction. \n Moreover, in the same figure, we plot with dashed lines the EoS resulting from using a polytrope in the inner crust. \n We see that the unified EoSs of BCPM and the SLy4 forces are very similar, and that their respective polytropic approximations are quite accurate with the unified ones.\n On the other hand, it may happen that the agreement between the unified EoS and the polytropic approximation for the inner crust is not \n as good as in the cases of the BCPM and the SLy4 functionals. Thus, some differences between the microscopic calculations of the EoS and the \n polytropic prescription may lead to differences in the computation of NS properties, in particular in the case of the crustal properties, such\n as the crustal mass and crustal thickness. \n \n \nAnother way to circumvent the lack of the crust EoS for a specific nuclear force is to use the energy density and pressure \ngiven by a different interaction in this region of the star and, at the core-crust transition interface, match the \nEoS of the core with the one coming from the crust. \nThe choice of a polytropic EoS has the advantage of providing a continuous EoS along all the NS, while, if using an already computed EoS for the inner \ncrust layers which is different from the one used in the core region, the values of the energy density and pressure\nat the core-crust transition may not coincide. \nOn the other hand, already computed EoSs for the inner crust have the advantage of having considered the physics of the inner \ncrust and, in a sense, are more realistic. \n\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[clip=true, width=0.8\\linewidth]{.\/grafics\/chapter5\/EOS_inner_crust}\n\\caption{Total pressure against the total energy density computed with the SLy4 Skyrme interaction\nand the BCPM functional. The results plotted with straight lines account for the respective \nunified EoS, while the results plotted with dashed lines account for the use of a polytropic EoS in the \ninner crust region.}\\label{fig:inner}\n\\end{figure}\n\n\nThe choice of the inner crust EoS may not have large consequences when studying\nglobal properties of the star, such as the total mass and the total radius, but can influence the values obtained for the \ncrustal properties.\nA method recently proposed by Zdunik, Fortin, and Haensel~\\cite{Zdunik17} allows one to accurately\nestimate the total and crustal masses and radii of the NSs without the explicit knowledge of the EoS of the crust. \nThese authors have presented an approximate description of the NS crust structure in terms of the function relating the chemical potential and the pressure. \nIt only requires the knowledge of the chemical potential at the boundaries of a given layer in the crust. In particular, if one wants to \nstudy the thickness of the whole crust, one needs the chemical potential at the core-crust transition and at the surface of the star. \nOne integrates the TOV \nequations from the center of the NS up to the core-crust transition and, \nwith this approximation, using the fact that the crustal mass is small compared to the total mass of the NS, $M_\\mathrm{crust} << M$, and that \n$4 \\pi r^3 P \/mc^2 << 1$, the crustal thickness\n of an NS of mass $M$ is then given by~\\cite{Zdunik17}:\n\\begin{equation}\\label{rcrustz}\n R_\\mathrm{crust} = \\Phi R_\\mathrm{core} \\frac{1-2 G M \/ R_\\mathrm{core} c^2}{1- \\Phi \\left( 1- 2 G M \/R_\\mathrm{core} c^2\\right)},\n\\end{equation}\nwith   \n\\begin{equation}\n \\Phi \\equiv \\frac{(\\alpha-1) R_\\mathrm{core} c^2}{2 G M}\n\\end{equation}\nand \n\\begin{equation}\n \\alpha = \\left( \\frac{\\mu_t}{\\mu_0}\\right)^2,\n\\end{equation}\nwhere $\\mu_t = (P_t + {\\cal H}_t c^2)\/\\rho_t$ is the chemical potential at the core-crust transition, \n$\\mu_0$ is the chemical potential at the surface of the NS and $R_\\mathrm{core}$ is\nthe thickness of the core. The crustal mass is obtained as \n\\begin{equation}\\label{mcrustz}\n M_\\mathrm{crust} = \\frac{4 \\pi P_t R_\\mathrm{core}^4}{G M_\\mathrm{core}} \\left(1- \\frac{2 G M_\\mathrm{core}}{R_\\mathrm{core}c^2} \\right),\n\\end{equation}\nwhere $M_\\mathrm{core}$ is the mass of the NS core. The total mass of the NS is $M=M_\\mathrm{crust}+M_\\mathrm{core}$.\nThis approximate approach predicts the radius and mass of the crust with\naccuracy better than $\\sim 1\\%$ in $R_\\mathrm{crust}$ and $\\sim 5\\%$ in $M_\\mathrm{crust}$ for typical\nNS masses larger than $1 M_\\odot$~\\cite{Zdunik17}.\n\n\\begin{figure}[!b]\n\\centering\n\\includegraphics[clip=true, width=0.9\\linewidth]{.\/grafics\/chapter5\/SLy4_SkI5_ordres\/Sly4_ski5_crust}\n\\caption{Mass-radius relation calculated using the SLy4 and SkI5 Skyrme interactions for the NS core. \nWith black solid lines we plot the results if using the method proposed by Zdunik et al. to describe the crust, \nwith red dashed lines we plot the results if the SLy4 EoS~\\cite{douchin01} for the inner crust is used\nand with blue dashed-dotted lines we include the values for the M-R relation obtained \nusing a polytropic approximation for the inner crust. The constraints for the maximum value of the mass\ncoming from Refs.~\\cite{Demorest10, Antoniadis13}\nare also included.\\label{fig:MR_crust}} \n\\end{figure}\nWe plot in Fig.~\\ref{fig:MR_crust} the results for the mass-radius (M-R) relation computed with the SLy4 and SkI5 Skyrme interactions.\nWe use these two interactions as SLy4 has a moderate slope of the symmetry energy $L=46$ MeV, while the SkI5 force \nproduces a stiffer EoS with slope parameter $L=129$ MeV. We do not plot the results for the BCPM functional as they \nhave a very similar behaviour compared to the ones predicted by SLy4.\nWe represent with black solid lines the results obtained using the Zdunik et al. method~\\cite{Zdunik17} to compute the crustal thickness and crustal mass. \nMoreover, we include in the same Fig.~\\ref{fig:MR_crust} with red dashed lines the results obtained if using the crust of the SLy4 EoS. \nThis case corresponds to the use of the SLy4 unified EoS while for the SkI5 interaction, the crust EoS will be microscopically calculated with a different interaction.\nFinally, in the same figure, we plot with blue dashed-dotted lines the results for the M-R relation obtained using a polytropic EoS for the inner crust.\nIn the three approaches, the core-crust transition has been obtained using the dynamical method, and the outer crust is the one of Haensel-Pichon computed\nwith the SLy4 force~\\cite{HaenselPichon}.\nWe use as a benchmark the results obtained with the Zdunik et al. method, as in Ref.~\\cite{Zdunik17}.\nIn the case of the SLy4 interaction, the three prescriptions for the crust EoS provide almost the same results for the NS M-R relation. Let us remind that \nin this case, the line labeled as ``SLy4 crust'' in Fig.~\\ref{fig:MR_crust} is obtained with the SLy4 unified EoS. The results we find \nfor this interaction are in agreement with the ones found in Ref.~\\cite{Zdunik17}.\nOn the other hand, the SkI5 interaction, with a stiff EoS, presents more differences in the results computed with different\ndescriptions of the inner crust. We see that, in the case of the SkI5 force, the use of a crust computed with a different interaction, \nin this case the SLy4 force, predicts results that may be quite far from the ones obtained using the Zdunik et al. method for the crust. \nThis behavior may come from the fact that the symmetry energy of the SLy4 interaction is much softer than in the case of the SkI5 force. \nIn the case of the SkI5 parametrization, the M-R values given by the TOV equations if using the polytropic approximation for the inner crust EoS are closer \nto the ones of Zdunik et al. than if the SLy4 crust is used.  \nThe differences between the results using different definitions of the inner crust EoS are more prominent in the low-mass regime about $M\\lesssim 1 M_\\odot$, \nand for interactions with  stiff symmetry energy.\n\nLet us remark that, as we have seen, for interactions such as the SLy4 force, which has a moderate value of the slope parameter,\nthe differences between the M-R relation obtained using \neither of the three prescriptions of the crust are minimal.\nVery recently we have been able to calculate the EoS of the crust for Gogny forces. In particular, we have obtained firsts results for the EoS of the inner crust calculated with\nthe D1M$^*$ interaction, which, similarly to the SLy4 interaction, has a moderate value of the slope parameter of the symmetry energy ($L=43$ MeV). \nThe preliminary results for the inner crust EoS provide M-R relations very similar to the ones obtained if the \ninner crust is reproduced with the polytropic approximation, which we have used to fit the interaction.\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[clip=true, width=1\\linewidth]{.\/grafics\/chapter5\/MRcrust_com_Vther_Vdyn_Gogny_MDI_SEI_v2}\n\\caption{Neutron star crustal thickness (upper panels) and crustal mass (lower panels) \nagainst the total mass of the NS for Skyrme (left), Gogny (center-left), MDI (center-right) and SEI (right) interactions. The core-crust \ntransition has been determined using the thermodynamical potential (dashed lines) and the dynamical potential \n(solid lines).}\\label{fig:crust_MR}\n\\end{figure}\nOn the whole, we can conclude that the global properties of an NS depend, up to a certain extend, on the treatment of the crust EoS, and this is more relevant \nin the low-mass regime of the M-R relation plot, where the \ncrust of the NS is more prominent than its core.  \n\nWe proceed to study the crustal properties of NSs, such as the crustal thickness and crustal mass. \nTo obtain the results, we use the method proposed by Zdunik et al. as it does not require any prescription \nfor the crust EoS and it only uses information related to the core. \nWe compute the results by solving the TOV equations from the center of the star up to the core-crust transition, and \nwe use Eqs.~(\\ref{rcrustz}) and (\\ref{mcrustz}) to get the values of the crustal thickness and crustal mass. \nWe plot in Fig.~\\ref{fig:crust_MR} the crustal thickness and crustal mass against the total mass of \nthe NS. As representative examples, we show the results provided by the SLy4 and SkI5 Skyrme forces, the D1M and D1M$^{*}$ Gogny interactions, \nthree MDI and three SEI forces with different $L$ values. \nAll considered forces in this figure, except the Gogny D1M and the MDI force with $L=42$ MeV,\npredict NSs of mass about $2 M_\\odot$, in agreement with astronomical observations \\cite{Demorest10, Antoniadis13}.\nThe whole set of interactions is located in a wide range of the values of the slope of the symmetry energy.\nAs stated in Chapter~\\ref{chapter4}, one can obtain the core-crust transition interface using the thermodynamical method\nor the more sophisticated dynamical method.\nFor each interaction, we have obtained the crustal properties using the transition point given by the thermodynamical and the dynamical approaches.\nIn general, the global behaviour of these properties is similar for all four types of interactions. \nThe values of both the crustal thickness $R_\\mathrm{crust}$ and crustal mass $M_\\mathrm{crust}$\ndecrease as the NS is more massive, being the contribution of the crust smaller. This is true for either the case\nwhere one uses the core-crust thermodynamical transition density and the case\nwhere one uses the dynamical method. The first case, that is, when using the thermodynamical transition density, \npredicts higher values of the crustal properties than the case where the dynamical transition density is used, as the \nvalues predicted for the transition density and transition pressure are higher in the first case than in the second one (see Chapter~\\ref{chapter4}). \nThe influence of the transition point is smaller on the crustal radius, and has a larger impact on \nthe calculation of the crustal mass.\nMoreover, we see that the differences between the two calculations are larger for interactions with a large $L$ parameter and stiff EoS.\nThe crustal properties play a crucial role in the description of several phenomena such as \npulsar glitches, r-mode oscillations, cooling of isolated NSs, etc.~\\cite{Link1999,Chamel2008,Fattoyev:2010tb,Chamel2013,PRC90Piekarewicz2014,Newton2015}. \nHence, one should use the core-crust transition estimated using the dynamical method when looking for the results of NS\ncrustal properties because, as said in Chapter~\\ref{chapter4}, the dynamical method includes surface and Coulomb \ncontributions when one studies the stability of the NS core. \n\nFinally, we present in Table~\\ref{tablecrust} some quantitative values of the crustal properties \nfor the same interactions appearing in Fig.~\\ref{fig:crust_MR}. We provide the results computed \neither using the core-crust transition estimated with the thermodynamical approach ($V_\\mathrm{ther}$)\nand with the dynamical approach ($V_\\mathrm{dyn}$), for the NS configurations of maximum mass ($M_\\mathrm{max}$) and for a \ncanonical NS of $1.4 M_\\odot$.\nAs already seen in Fig.~\\ref{fig:crust_MR}, the values for the crustal properties are higher for the $V_\\mathrm{ther}$\ncase than for the $V_\\mathrm{dyn}$ case. \nMoreover, the values for a canonical NS are higher than in the cases of maximum mass configurations.\nThe differences between the results obtained using the thermodynamical core-crust transition\n and the ones obtained using the dynamical core-crust transition are larger as stiffer is the EoS describing the core. \nFor example, for the SLy4 interaction, one obtains differences of $0.0012 M_\\odot$ and $0.01$ km for the crustal \nmass and thickness in the maximum mass configuration. On the other hand, for the SEI $L=115$ MeV parametrization these differences are of \n$0.0067 M_\\odot$ and $0.03$ km, respectively. For a canonical NS, the differences for the crustal mass and \nthickness are of $0.005 M_\\odot$ and $0.04$ km in the case of SLy4 and \n$0.008 M_\\odot$ and $0.09$ km in the case of the  SEI $L=115$ MeV force.\n\\begin{table}[t!]\n\\centering\n\\resizebox{\\columnwidth}{!}{\n\\begin{tabular}{c|lcccccccc}\n\\hline\n\\multicolumn{2}{c|}{\\multirow{2}{*}{Force}} & \\multicolumn{2}{c|}{$M_\\mathrm{crust}(M_\\mathrm{max})$}                        & \\multicolumn{2}{c|}{$R_\\mathrm{crust}(M_\\mathrm{max})$}                         & \\multicolumn{2}{c|}{$M_\\mathrm{crust}(1.4M_\\odot)$}       & \\multicolumn{2}{c|}{$R_\\mathrm{crust}(1.4M_\\odot)$}       \\\\ \\cline{3-10} \n\\multicolumn{2}{c|}{}                       & \\multicolumn{1}{c|}{$V_\\mathrm{ther}$} & \\multicolumn{1}{c|}{$V_\\mathrm{dyn}$} & \\multicolumn{1}{c|}{$V_\\mathrm{ther}$} & \\multicolumn{1}{c|}{$V_\\mathrm{dyn}$} & \\multicolumn{1}{c|}{$V_\\mathrm{ther}$} & \\multicolumn{1}{c|}{$V_\\mathrm{dyn}$} & \\multicolumn{1}{c|}{$V_\\mathrm{ther}$} & \\multicolumn{1}{c|}{$V_\\mathrm{dyn}$} \\\\ \\hline\\hline\n\\multirow{2}{*}{Skyrme}    & SLy4 ($L=46$ MeV)         & 0.0058                                 & 0.0046                                & 0.31                                   & 0.30                                  & 0.022                                  & 0.017            & 0.96                                   & 0.92             \\\\\n                           & SkI5 ($L=129$ MeV)         & 0.0066                                 & 0.0035                                & 0.30                                   & 0.26                                  & 0.032                                  & 0.017            & 1.13                                   & 0.99             \\\\ \\hline\n\\multirow{2}{*}{Gogny}     & D1M  ($L=25$ MeV)         & 0.0033                                 & 0.0027                                & 0.28                                   & 0.27                                  & 0.009                                  & 0.008            & 0.63                                   & 0.62             \\\\\n                           & D1M$^*$ ($L=43$ MeV)       & 0.0049                                 & 0.0040                                & 0.32                                   & 0.31                                  & 0.013                                  & 0.015            & 0.87                                   & 0.85             \\\\ \\hline\n\\multirow{3}{*}{MDI}       & $L=42$ MeV     & 0.0045                                 & 0.0029                                & 0.32                                   & 0.29                                  & 0.011                                  & 0.008            & 0.62                                   & 0.57             \\\\\n                           & $L=60$ MeV     & 0.0050                                 & 0.0028                                & 0.29                                   & 0.26                                  & 0.018                                  & 0.010            & 0.81                                   & 0.73             \\\\\n                           & $L=88$ MeV     & 0.0031                                 & 0.0012                                & 0.24                                   & 0.21                                  & 0.012                                  & 0.005            & 0.72                                   & 0.63             \\\\ \\hline\n\\multirow{3}{*}{SEI}       & $L=86$ MeV     & 0.0078                                 & 0.0046                                & 0.35                                   & 0.31                                  & 0.028                                  & 0.017            & 0.98                                   & 0.86             \\\\\n                           & $L=100$ MeV    & 0.0059                                 & 0.0028                                & 0.31                                   & 0.26                                  & 0.022                                  & 0.011            & 0.89                                   & 0.76             \\\\\n                           & $L=115$ MeV    & 0.0096                                 & 0.0029                                & 0.24                                   & 0.21                                  & 0.012                                  & 0.004            & 0.72                                   & 0.63            \\\\ \\hline\n\\end{tabular}\n}\n\\caption{Crustal mass ($M_\\mathrm{crust}$) and crustal thickness ($R_\\mathrm{crust}$) for an\nNS of maximum mass ($M_\\mathrm{max}$) and for a canonical NS mass of $1.4 M_\\odot$ evaluated\ntaking into account the core-crust transition obtained using the thermodynamical method ($V_\\mathrm{ther}$)\nor the dynamical method ($V_\\mathrm{dyn}$) for a set of mean-field models.\nThe values of the mass are given in solar masses and the results for the radii in km. \\label{tablecrust}}\n\\end{table}\n\n\n\\section{Moment of inertia}\nOne property of interest, due to potential observational evidence in pulsar glitches as well as \nthe connection to the core-crust transition, is the ratio \nbetween the fraction of the moment of inertia enclosed in the NS crust, $\\Delta I_\\mathrm{crust}$, and the star's total moment of inertia, \n$I$~\\cite{Ravenhall1994,haensel07,Lattimer2005,Lattimer2016}.\nTo lowest order in angular velocity, the moment of inertia of the star can be computed \nfrom the static mass distribution and gravitational potentials encoded in the TOV equations \n\\cite{Hartle1967}. \nIn the slow-rotation limit, the moment of inertia of a spherically symmetric NS is given by~\\cite{Fattoyev:2010tb}\n\\begin{equation}\\label{inertia1}\n I\\equiv \\frac{J}{\\Omega} = \\frac{8 \\pi}{3} \\int_0^R r^4 e^{-\\nu(r)} \\frac{\\bar{\\omega} (r)}{\\Omega}\n \\frac{(\\varepsilon(r) + P(r))}{\\sqrt{1-2Gm(r)\/rc^2}}dr,\n\\end{equation}\nwhere $J$ is the angular momentum, $\\Omega$ is the stellar rotational frequency, $\\nu(r)$ and $ \\bar{\\omega}$ are \nradially dependent metric functions and $m(r)$, $\\varepsilon(r)$ and $P(r)$ are, respectively, the NS mass, energy density\nand total pressure enclosed in a radius $r$.\n\nThe metric function $\\nu(r)$ satisfies\n\\begin{equation}\n \\nu(r) = \\frac{1}{2} \\mathrm{ln} \\left(1-\\frac{2 G M}{R c^2} \\right) - \n \\frac{G}{c^2} \\int_r^R \\frac{(M(x) + 4 \\pi x^3 P(x)}{x^2 (1-2 G M(x)\/xc^2)}dx,\n\\end{equation}\nand the relative frequency $\\bar{\\omega}(r)$, which is defined as \n\\begin{equation}\n \\bar{\\omega}(r) \\equiv \\Omega - \\omega(r),\n\\end{equation}\nrepresents the angular velocity of the fluid as measured in a local reference frame~\\cite{Fattoyev:2010tb}.\nThe frequency $\\omega(r)$ is the frequency appearing due to the slow rotation. \nOne also can define the relative frequency $\\tilde{\\omega}(r) \\equiv \\bar{\\omega}(r)\/\\Omega$, \nwhich is obtained solving \n\\begin{equation}\\label{inertia2}\n \\frac{d}{dr} \\left( r^4 j(r) \\frac{d\\tilde{\\omega}(r)}{dr}\\right) + 4 r^3 \\frac{dj(r)}{dr} \\tilde{\\omega}(r)=0,\n\\end{equation}\nwith\n\\begin{equation}\n j(r) =\\left \\{ \\begin{matrix} e^{\\nu(r)} \\sqrt{1-2Gm(r)\/rc^2} & \\mbox{if } r \\leq R\n\\\\ 1 & \\mbox{if }r >R\\end{matrix}\\right.  .\n\\end{equation}\nThe boundary conditions defining the relative frequency $\\tilde{\\omega}(r)$ are:\n\\begin{equation}\\label{constraintI}\n \\tilde{\\omega}'(r) (0)=0 \\hspace{1cm} \\mathrm{and} \\hspace{1cm} \\tilde{\\omega}(r) + \\frac{R}{3} \\tilde{\\omega}'(r) =1.\n\\end{equation}\nOne integrates~Eq.~(\\ref{inertia2}) considering an arbitrary value of $\\tilde{\\omega}(0)$. The second \nboundary condition at the surface of the NS usually will not be satisfied for the given $\\tilde{\\omega}(0)$. \nHence, one must rescale the solution of (\\ref{inertia2}) with an appropriate constant in order to fulfill~(\\ref{constraintI}).\nNotice that in this slow-rotation regime the solution of the moment of inertia does not depend on the stellar \nfrequency $\\Omega$.\nA further check one can implement to ensure the accuracy of the full calculation is that the equation\n\\begin{equation}\n \\tilde{\\omega}'(R) =\\frac{6GI}{R^4c^2}\n\\end{equation}\nis fulfilled~\\cite{Fattoyev:2010tb}.\n\n\\begin{figure}[t!]\n\\centering\n\\includegraphics[clip=true, width=0.9\\linewidth]{.\/grafics\/chapter5\/I_vs_M}\n\\caption{Total NS moment of inertia against the total mass for a set of Skyrme (panel (a)) and Gogny (panel (b)) interactions. \nThe two constraints from Ref.~\\cite{Landry18} at $1.338 M_\\odot$ for the primary component of the double pulsar \nPSR J0737-303 are also included.}\\label{fig:momI}\n\\end{figure}\n\n\n\nWe show in Fig.~\\ref{fig:momI} \\cite{gonzalez17} the results for \nthe moment of inertia against the total mass of the NS for a set of Skyrme (panel (a)) and Gogny (panel (b)) parametrizations of interest.\nTo integrate the TOV equations and to find the total moment of inertia we have used the outer crust of Haensel-Pichon and a polytropic EoS \nfor the inner crust. One cannot use the method of Zdunik et al. to describe the crust, as one needs the EoS in all the NS to integrate the moment of inertia. \nIn agreement with the findings of the behaviour of the EoS, the moments of inertia are larger as stiffer are their respective EoSs.\nIn particular, we find that the moments of inertia of the Gogny parametrizations are below \nthe predictions of the SLy4 Skyrme interaction. \nAs expected, the maximum value of $I$ is reached slightly below the maximum \nmass configuration for all forces \\cite{haensel07}. \nFor Gogny interactions, we observe~\\cite{gonzalez17} that the D1M and D280 interactions give maximum values of \n$I_\\text{max} \\approx 1.3-1.4 \\times 10^{45}$ g cm$^2$, below the typical \nmaximum values of $\\approx 2 \\times 10^{45}$ g cm$^2$ obtained with stiffer EoSs~\\cite{haensel07}. \nOur results for D1N are commensurate with those of Ref.~\\cite{Loan2011}.\nThe new recently fitted D1M$^*$ and D1M$^{**}$ forces provide maximum values of \n$I_\\text{max} = 1.97 \\times 10^{45}$ g cm$^2$ and $I_\\text{max} = 1.70 \\times 10^{45}$ g cm$^2$ respectively.\n\nBinary pulsar observations may provide in the future information on the moment of inertia of NSs, \ngiving new constraints to the EoS of NS matter~\\cite{Lattimer2005}. \nThe binary PSR J0737-3039 is the only double-pulsar system known to date, and it is expected a \nprecise measurement in the near future of the moment of inertia of its primary component PSR J0737-3039A (or pulsar A) from\nradio observations~\\cite{burgay03, Lyne04}.\nIn Ref.~\\cite{Landry18}, Landry and Kumar estimate a range of $I=1.15^{+0.38}_{-0.24} \\times 10^{45}$ g cm$^{2}$\nfor pulsar A, which has a mass of $M=1.338 M_\\odot$. \nTo obtain these constraints, Landry and Kumar have combined the values for the tidal deformability of a $1.4 M_\\odot$\nNS reported by the LVC obtained from the GW170817 event~\\cite{Abbott2019} (see next Section~\\ref{seclambda}) with \napproximately universal relations\namong NS observables~\\cite{Yagi13a, Yagi13b}, known\nas the binary-Love and I-Love relations.\nIn the same Ref.~\\cite{Landry18}, a wider range of \\mbox{$I \\leq 1.67 \\times 10^{45}$ g cm$^2$} is given for the moment of inertia of pulsar A, \nobtained from the less restrictive limit on the tidal deformability, $\\tilde{\\Lambda} \\leq 800$,  obtained from the first analysis\nof the GW170817 event~\\cite{Abbott2017}.\nWe have plotted in Fig.~\\ref{fig:momI} the two constraints for pulsar A of the PSR J0737-3039 binary system.\nWe observe that not all Skyrme forces that provide NSs above the $2M_\\odot$ constraint limit may fit inside these boundaries.\nThis is the case of for example the SkI5 Skyrme force, where the EoS is so stiff that it gives too large values for the \nmoment of inertia. Interactions with smaller $L$ value, such as the new D1M$^*$\nand D1M$^{**}$ Gogny interactions that we have constructed, fit inside the constraints from Landry and Kumar\nfor the moment of inertia of pulsar PSR J0737-3039 A. \nWe also observe that the SLy4 Skyrme interaction fits well inside both boundary limits.\nA useful comparison with the systematics of other NS EoSs is provided by the \ndimensionless quantity $\\frac{I}{MR^2}$. This has been found to scale with the NS \ncompactness $\\chi=GM\/Rc^2$.\nIn fact, in a relatively wide region of $\\chi$ values, the dimensionless ratio $\\frac{I}{MR^2}$ \nfor the mass and radius combinations of several EoSs can be fitted by universal \nrelations~\\cite{Ravenhall1994,Lattimer2005,Breu2016}. \n\n\n\\begin{figure}[t!]\n\\centering\n\\includegraphics[clip=true, width=0.9\\linewidth]{.\/grafics\/chapter5\/IMR2_Xi}\n\\caption{Dimensionless quantity $\\frac{I}{MR^2}$ against the compactness $\\chi$ of an NS obtained\nfor a set of Skyrme (panel (a)) and Gogny (panel (b)) forces. The constraining bands from Refs.~\\cite{Lattimer2005, Breu2016} are also included.}\\label{fig:momI2}\n\\end{figure}\n\nWe show in Fig.~\\ref{fig:momI2}~\\cite{gonzalez17}\nthe dimensionless ratio $\\frac{I}{MR^2}$ as a function of the compactness $\\chi$ for the same set of Skyrme and\nGogny forces as in the previous Fig.~\\ref{fig:momI}. Our results are compared to the recent fits from Breu and \nRezzolla~\\cite{Breu2016} (red shaded area) and the older results from\nLattimer and Schutz~\\cite{Lattimer2005} (blue shaded region).\n These fits have been obtained from a very wide range of different theoretical EoS predictions.\nFor compactness $\\chi > 0.1$, in the case of Skyrme interactions, only the SLy4, Ska and SkI5 parametrizations \nfit inside both bands. On the other hand, the SkM$^*$ and UNEDF0, not being able to provide large enough moments \nof inertia, lie outside the constraint from Lattimer and Schutz in the case of SkM$^*$ and outside\nboth constraints in the case of UNEDF0. \nFor Gogny interactions, we see that the D1M, D1M$^*$ and D1M$^{**}$ forces give practically identical values of \nthe $I\/MR^2$ value if plotted against the NS compactness. In the three cases, the results fit inside both bands. \nOn the other hand, D280 is close to the lower limit of this \nfit, but well below the lower bounds of the fit in Lattimer and Schutz~\\cite{Lattimer2005}. We \nfind that, in spite of the significant differences in their absolute moments of inertia, \nD1M, D1M$^*$, D1M$^{**}$ and SLy4 produce dimensionless ratios that agree well with each other. In contrast, and as \nexpected, D1N produces too small moments of inertia for a given mass and radius, \nand systematically falls below the fits. \n\nWhen studying the ratio between the crustal fraction of the moment of inertia and the \ntotal moment of inertia $\\Delta I_\\mathrm{crust}\/I$, one way to circumvent the problem of \nnot having a good definition of the EoS in the inner crust is to use the \nthe approximation given by \\cite{Lattimer00, Lattimer01, Lattimer07},\nwhich allows one to express this quantity as\n\\begin{eqnarray}\\label{eq:Iaprox}\n \\frac{\\Delta I_\\mathrm{crust}}{I} = \\frac{28 \\pi P_t R^3}{3 M c^2} \\frac{\\left(1-1.67 \\chi-0.6 \\chi^2 \\right)}{\\chi} \n \\times \\left[ 1+ \\frac{2 P_t \\left( 1+ 5 \\chi -14 \\chi^2\\right)}{\\rho_t m c^2 \\chi^2}\\right]^{-1}, \n\\end{eqnarray}\nwhere $m$ is the baryon mass. \nThe reason for this choice is that, as stated previously, we do not have unified EoSs for all the interactions we are considering \nin these studies. The approximate expression for the crustal fraction of the moment of inertia given in Eq.~(\\ref{eq:Iaprox}),\ncombined with the Zdunik et al. method~\\cite{Zdunik17} to obtain the total mass and radius of an NS, requires only of the core EoS\nto find the crustal fraction of the moment of inertia. \n\n\\begin{figure}[b!]\n\\centering\n\\includegraphics[clip=true, width=0.9\\linewidth]{.\/grafics\/chapter5\/Icrust_vs_M}\n\\caption{Crustal fraction of the moment of inertia against the total NS mass for a set of Skyrme (panel (a)) and Gogny (panel (b))\ninteractions computed using a polytropic inner crust EoS (straight lines) or the approximation for \n$\\Delta I_\\mathrm{crust}\/I$ in Eq.~(\\ref{eq:Iaprox}) (dashed lines). The core-crust transition has been \nobtained using the dynamical approach.\nThe constraints from the Vela pulsar in Refs.~\\cite{Link1999, Andersson2012} are also included.}\\label{fig:Icrust2}\n\\end{figure}\nWe compare in Fig.~\\ref{fig:Icrust2}, for a set of Skyrme (panel (a)) and Gogny (panel (b)) interactions, the results \nof the crustal fraction moment of inertia obtained if using a polytropic \nEoS for the inner crust (straight lines) or obtained if using the combination of \nEq.~(\\ref{eq:Iaprox}) and the method proposed by Zdunik et al. to find the total thickness and mass of an NS (dashed lines).\nIn both cases, the core-crust transition has been obtained for each particular interaction using the dynamical method.\nWe find a very good agreement between the approximated formula (\\ref{eq:Iaprox})\nand the full results if using a polytropic EoS above $1-1.2 M_\\odot$, and the agreement improves for both types of interactions\nas the mass of the \npulsar increases. This is in keeping with the findings of Ref.~\\cite{xu09a, gonzalez17}. \n\\begin{figure}[!t]\n\\centering\n\\includegraphics[clip=true, width=1\\linewidth]{.\/grafics\/chapter5\/Crust_com_Vther_Vdyn_momI}\n\\caption{Neutron star crustal fraction of the moment of inertia\nagainst the total mass of the NS for Skyrme (left), Gogny (center-left), MDI (center-right) and SEI (right) interactions. The core-crust \ntransition has been determined using the thermodynamical potential (dashed lines) and the dynamical potential \n(solid lines).}\\label{fig:Icrust1}\n\\end{figure}\n\n\\begin{table}[!b]\n\\centering\n\\begin{tabular}{c|lcccc}\n\\hline\n\\multicolumn{2}{c|}{\\multirow{2}{*}{Force}} & \\multicolumn{2}{c|}{$\\Delta I_\\mathrm{crust}\/I (M_\\mathrm{max})$ ($\\%$)}                                        & \\multicolumn{2}{c}{$\\Delta I_\\mathrm{crust}\/I(1.4M_\\odot)$ $\\%$}            \\\\ \\cline{3-6} \n\\multicolumn{2}{c|}{}                       & \\multicolumn{1}{c|}{$V_\\mathrm{ther}$} & \\multicolumn{1}{c|}{$V_\\mathrm{dyn}$} & \\multicolumn{1}{c|}{$V_\\mathrm{ther}$} & \\multicolumn{1}{c}{$V_\\mathrm{dyn}$}  \\\\ \\hline\\hline\n\\multirow{2}{*}{Skyrme}    & SLy4  ($L=46$ MeV)          &                0.76   &             0.60       &                  3.61     &   2.96                                              \\\\\n                           & SkI5  ($L=129$ MeV)         &                    0.73      &        0.41       &                   4.60   &     2.85                        \\\\ \\hline\n\\multirow{2}{*}{Gogny}     & D1M     ($L=25$ MeV)       &                     0.52     &         0.44    &                        1.70  &     1.45                                   \\\\\n                           & D1M$^*$  ($L=43$ MeV)      &                   0.67        &      0.56       &                     2.81   &     2.37                                                  \\\\ \\hline\n\\multirow{3}{*}{MDI}       & $L=42$ MeV     &                    0.73        &    0.49       &                      1.95   &     1.34                                               \\\\\n                           & $L=60$ MeV     &                    0.67    &        0.39        &                         2.86   &     1.76                                                 \\\\\n                           & $L=88$ MeV     &                       0.41   &        0.16      &                      2.07      &       0.90                                               \\\\ \\hline\n\\multirow{3}{*}{SEI}       & $L=86$ MeV     &                     0.99      &        0.60       &                    4.08     &     2.68                                                       \\\\\n                           & $L=100$ MeV    &                      0.75      &           0.37   &                   3.36      &    1.84                                                 \\\\\n                           & $L=115$ MeV    &                       0.38      &    0.13       &                     1.96      &    0.78                                                    \\\\ \\hline\n\\end{tabular}\n\\caption{Crustal fraction of the moment of inertia (in \\%) for an\nNS of maximum mass ($M_\\mathrm{max}$) and for a canonical NS mass of ($1.4 M_\\odot$) evaluated\ntaking into account the core-crust transition obtained using the thermodynamical method ($V_\\mathrm{ther}$)\nor the dynamical method ($V_\\mathrm{dyn}$) for a set of mean-field models. \\label{tablecrustI}}\n\\end{table}\nTo account for the sizes of observed glitches, the widely used pinning model requires that a certain amount of angular momentum is \ncarried by the crust. This can be translated into constraints on the crustal fraction of the moment of inertia. \nInitial estimates suggested that $\\Delta I_\\text{crust}\/I>1.4 \\, \\%$ to explain  Vela and other glitching sources \n\\cite{Link1999}. We show this value as the bottom horizontal line in Fig.~\\ref{fig:Icrust2}.\nWe note that this does not pose \nmass limitations on the UNEDF0 and D280 interactions, which have minimum values of $\\Delta I_\\text{crust}\/I$ above that limit. \nFor other interactions that cross the limit, in contrast, glitching sources that satisfy this constraint should have below a certain mass. \nFor the SkM$^*$, SLy4, Ska and SkI5 interactions, these masses are, respectively, $M< 1.5$, $1.8$, $2.1$ and $1.8$ $M_\\odot$, \nand for Gogny interactions the limits are $M< 1.4$, $1.7$, $1.6$ and $1.2$ $M_\\odot$ for the D1M, D1M$^*$, D1M$^{**}$ and D1N interactions, respectively.\nMore recently, a more stringent constraint has \nbeen obtained by accounting for the entrainment of neutrons in the crust \\cite{Andersson2012}. With entrained neutrons, a \nlarger crustal fraction of moment of inertia, $\\Delta I_\\text{crust}\/I>7 \\, \\%$ (top horizontal line in the same figure), is needed \nto explain glitches.  Of course, a more realistic account \nof nuclear structure and superfluidity in the crust will modify the estimates. \nThen, all interactions would need \nsignificantly lower masses to account for glitching phenomena.\n\n\nWe present in Fig.~\\ref{fig:Icrust1} the results for the \ncrustal fraction of the moment of inertia against the total mass of the NS computed with the same interactions \nas in Fig.~\\ref{fig:crust_MR}, namely the SLy4 and SkI5 Skyrme forces, the D1M and D1M$^{*}$ Gogny interactions, \nthree MDI and three SEI forces with different $L$ values. \nThe results enclosed in these figures have been obtained using the Zdunik et al. method to find the core properties and \nemploying Eq.~(\\ref{eq:Iaprox}) to find the ratio $\\Delta I_\\mathrm{crust}\/I$.\nSimilarly to the analysis made for the crustal mass and thickness of an NS, \nwe have obtained for each interaction the crustal fraction of the moment of inertia\nusing the transition point given by the thermodynamical and the dynamical approaches.\nMoreover, we include in Table~\\ref{tablecrustI} the values of the crustal fraction\nof the moment of inertia $\\Delta I_\\mathrm{crust}\/I$  for an NS of maximum mass $M_\\mathrm{max}$ given by the interaction\nand for a canonical NS of $1.4 M_\\odot$, obtained either using the thermodynamical ($V_\\mathrm{ther}$) or the dynamical ($V_\\mathrm{dyn}$)\napproaches.\n\nThe global behaviour of the moment of inertia is, again, akin to all four types of interactions,\n$\\Delta I_\\text{crust}\/I$ decreasing with the NS mass.\nThe location of the transition point has a large impact on \nthe calculation of the crustal fraction of the moment of inertia, similar to the case when one studies the crustal mass.\nIn the case of the crustal fraction of the moment of inertia, the differences \nbetween the predictions using the core-crust transition found with the thermodynamical method \nor with the more realistic dynamical method are larger for interactions with a larger value of $L$. \nThese differences are very prominent in the typical NS mass region and could influence \nthe properties where the crust has an important role, such as pulsar glitches~\\cite{Link1999,Fattoyev:2010tb,Chamel2013,PRC90Piekarewicz2014,Newton2015, gonzalez17}. \n\n\n\n\n\n\\section{Tidal deformability}\\label{seclambda}\n\nIn a binary system composed of two NSs, each component star induces a\n perturbing gravitational tidal field on its companion, leading to a mass-quadrupole \ndeformation in each star. \nTo linear order, the tidal deformation of each component of the binary system is \ndescribed by the so-called tidal deformability $\\Lambda$, which is defined as \nthe ratio between the \ninduced quadrupole moment and the external tidal field~\\cite{Flanagan08, Hinderer08}.\nFor a single NS, the tidal deformability can be written in terms of the dimensionless tidal \nLove number $k_2$, and the mass and radius of the NS~\\cite{Flanagan08, Hinderer08, Hinderer2010}:\n\\begin{equation}\n \\Lambda= \\frac{2}{3}k_2 \\left(\\frac{R c^2}{G M} \\right)^5,\n\\end{equation}\nwhere $G$ is the gravitational constant and $c$ the speed of the light.\nThe mass and radius of an NS are determined by the solution of the TOV \nequations, and the Love number $k_2$ is given by\n\\begin{eqnarray}\n k_2&=& \\frac{8\\chi^5}{5} \\left(1-2\\chi \\right)^2 \\left[ 2+2\\chi (y-1) -y\\right]\n \\times \\left\\{ 2\\chi \\left[6-3y + 3\\chi(5y-8) \\right] \\right.\\nonumber \\\\\n &+&  4 \\chi^3 \\left[ 13-11y+\\chi (3 y -2) + 2 \\chi^2 (1+y)\\right]\\nonumber\\\\\n &+&3 \\left.(1-2\\chi)^2 \\left[2-y+2\\chi(y-1) \\right] \\mathrm{ln}(1-2\\chi)\\right\\}^{-1} ,\n\\end{eqnarray}\nwhere \n\\begin{equation}\\label{compactness}\n\\chi=\\frac{GM}{Rc^2},\n\\end{equation} \nis the compactness of the star and \n\\begin{equation}\ny=\\frac{R \\beta(R)}{H (R)}.\n\\end{equation}\nThe values of the functions $\\beta(R)$ and $H(R)$ can be obtained by solving \nthe following set of coupled differential equations~\\cite{Hinderer08, Hinderer2010}:\n\\begin{eqnarray}\n \\frac{dH(r)}{dr} &=& \\beta (r) \\\\\n \\frac{d\\beta (r)}{d r} &=& \\frac{2G}{c^2} \\left(1-\\frac{2Gm(r)}{r c^2} \\right)^{-1} H (r)\n \\left\\{ -2 \\pi \\left[ 5 \\epsilon + 9p + \\frac{d \\epsilon}{d p} \n(\\epsilon+p) \\right] + \\frac{3 c^2}{r^2 G} \\right.\\nonumber \\\\\n &+&\\left. \\frac{2G}{c^2} \\left(1-\\frac{2Gm(r)}{r c^2} \\right)^{-1} \\left( \\frac{m(r)}{r^2} \n+ 4 \\pi r p\\right)^2 \\right\\}\\nonumber \\\\\n &+& \\frac{2 \\beta (r)}{r} \\left(1-\\frac{2Gm(r)}{r c^2} \\right)^{-1} \\left\\{-1+\\frac{Gm(r)}{r c^2} \n+ \\frac{2 \\pi r^2 G}{c^2} \\left( \\epsilon -p\\right) \\right\\},\n\\label{eq.14a}\n\\end{eqnarray}\nwhere $m(r)$ is the mass enclosed inside a radius $r$, and $\\epsilon$ and $p$ are the corresponding \nenergy density and pressure. \nOne solves this system, together with the TOV \nequations given in Eqs.~(\\ref{eq:TOV}) and (\\ref{eq:TOV2}) integrating outwards and considering as boundary conditions\n$H(r)= a_0 r^2$ and $\\beta(r)= 2 a_0 r$\nas $r \\rightarrow 0$. The constant $a_0$ is arbitrary, as it cancels in the expression for \nthe Love number~\\cite{Hinderer2010}.\n\n\nFor a binary NS system, the mass weighted tidal deformability $\\tilde{\\Lambda}$, \ndefined as\n\\begin{equation}\\label{eq:wLambda}\n \\tilde{\\Lambda} = \\frac{16}{13} \\frac{(M_1 + 12M_2)M_1^4 \\Lambda_1 +(M_2 + 12M_1)M_2^4 \n\\Lambda_2 }{(M_1+M_2)^5},\n\\end{equation}\ntakes into account the contribution from the tidal effects to the phase evolution of the \ngravitational wave spectrum of the inspiraling NS binary. \nIn the definition of $\\tilde{\\Lambda}$ in Eq.~(\\ref{eq:wLambda}), $\\Lambda_1$ and $\\Lambda_2$\nrefer to the tidal deformabilities of each NS in the system and $M_1$ and $M_2$ are their corresponding masses. \nThe definition of $\\tilde{\\Lambda}$ fulfills $\\tilde{\\Lambda}=\\Lambda_1=\\Lambda_2$ \nwhen $M_1=M_2$.\n\nThe recent event GW170817 accounting for the detection of GWs coming \nfrom the merger of an NS \nbinary system has allowed the LVC to \nobtain constraints on the mass-weighted \ntidal deformability $\\tilde{\\Lambda}$ and on the chirp mass $\\mathcal{M}$, \nwhich for a binary NS system conformed of two stars of masses $M_1$ and $M_2$ is defined as \n\\begin{equation}\n \\mathcal{M}= \\frac{(M_1 M_2)^{3\/5}}{(M_1+M_2)^{1\/5}}.\n\\end{equation}\nIn the first data analysis of GW170817 by the LIGO and Virgo collaboration, values of \n$\\tilde{\\Lambda} \\leq 800$ and $\\mathcal{M} = 1.188^{+0.004}_{-0.005} M_\\odot$ were \nreported~\\cite{Abbott2017}. Moreover, they estimated the masses of the two NSs to be in the \nrange $M_1 \\in (1.36,1.60) M_\\odot$ and  $M_2 \\in (1.17,1.36) M_\\odot$. In a recent \n reanalysis of the data~\\cite{Abbott2019},\nthe values have been further constrained to $\\tilde{\\Lambda} =300^{+420}_{-230}$, \n$\\mathcal{M} = 1.186^{+0.001}_{-0.001} M_\\odot$,\n$M_1 \\in (1.36,1.60) M_\\odot$ and  $M_2 \\in (1.16,1.36) M_\\odot$.\n\n\\begin{figure}[t!]\n\\centering\n\\includegraphics[clip=true, width=0.9\\linewidth]{.\/grafics\/chapter5\/tidal_k2}\n\\caption{Tidal Love number $k_2$ and dimensionless tidal deformability $\\Lambda$\nagainst the NS compactness $\\chi=\\frac{GM}{Rc^2}$ and the total NS mass ($M$) for the SLy4 Skyrme force, \nthe D1M$^*$ interaction, the MDI with $L=88$ MeV model and the SEI with $L=100$ MeV parametrization. For these cases, the inner \ncrust has been computed using a polytropic EoS for the inner crust. Moreover, the \nresults using the SLy4 unified EoS are also included.\nThe constraint for the tidal deformability at $M=1.4 M_\\odot$ obtained from the GW170817 event detection~\\cite{Abbott2019} is also shown in panel (d).}\\label{fig:tidal1}\n\\end{figure}\n\nWe plot in Fig.~\\ref{fig:tidal1} the tidal Love number $k_2$ against the star compactness $\\chi=\\frac{GM}{Rc^2}$ (panel (a)) and against the \ntotal NS mass (panel (c)) for a set of representative mean-field interactions.\nMoreover, we plot in the same Fig.~\\ref{fig:tidal1} the values for the dimensionless \ntidal deformability $\\Lambda$ against the star compactness (panel (b)) and against the total NS mass (panel (d)).\nThe results have been obtained considering the outer crust EoS of Haensel-Pichon computed with the SLy4 interaction~\\cite{HaenselPichon}\nand a polytropic EoS for the \ninner region. In this case, one cannot use the method proposed by Zdunik et al., as the EoS along all the NS is needed\nwhen calculating the tidal deformability. Moreover, the core-crust transition has been obtained using the dynamical method. \nWe first center our discussion on the behaviour of the Love number $k_2$. We see an increase of its value up to\n$k_2 \\sim 0.10-0.15$ at around compactness $\\chi \\sim 0.05-0.10$ or mass $M \\sim 0.5-1 M_\\odot$. Afterwards, it decreases\nuntil vanishing at the black hole compactness $\\chi=0.5$ for all EoSs~\\cite{Krastev19}.\nThis can be understood  knowing that $k_2$ gives an estimation of how easy is for the bulk NS matter to deform. \nTherefore, more centrally condensed stellar models will have smaller $k_2$ and smaller $\\Lambda$.\nAlso, for low values of the compactness or the mass, the crust part of the EoS is more prominent than the core. Hence, \nthe NS becomes more centrally condensed and $k_2$ becomes smaller~\\cite{Krastev19}.\n\nWe proceed to study the dependence of the dimensionless tidal deformability $\\Lambda$ with the compactness and the NS mass.\nWe see in panel (b) of Fig.~\\ref{fig:tidal1} that the dependence of the tidal deformability with the compactness\nis almost the same for all EoSs, having a decreasing tendency as larger is $\\chi$. \nOn the other hand, if one plots $\\Lambda$ against the total mass $M$, the results separate between them. \nFor a given mass, the $\\Lambda$ results obtained with a stiffer EoS are larger than if obtained with softer equations of state.\nWe plot in panel (d) of Fig.~\\ref{fig:tidal1} the constraint for the tidal deformability of a canonical NS of $1.4 M_\\odot$, \n$\\Lambda_{1.4}= 190^{+390}_{-120}$ obtained from the analysis of the GW170817 data in Ref.~\\cite{Abbott2018}.\nWe see that the SLy4 and D1M$^*$ interactions fit very well inside the constraint, while models with stiffer EoSs, \nlike the MDI with $L=88$ MeV, reach the upper limit of the GW170817 data, and the SEI with $L=100$ MeV does not provide \nvalues of the tidal deformability inside the observed bands. \n\nWe have to mention that the results found for the Love number $k_2$ are rather sensitive to the EoS of the crust. \nWe see that the results of $k_2$ computed with the SLy4 unified EoS are a little bit different with respect to the ones\nobtained using the polytropic inner crust. The relative differences between them reach up to values of $5\\%$ in this case. \nOn the other hand, the results for $\\Lambda$ do not depend on this choice, as the changes in $k_2$ are compensated by the \npossible differences between the values of the NS radius~\\cite{Piekarewicz19}. Hence, \none can compare the results for the tidal deformability obtained with the polytropic inner crust, \nas it is our case, with the constraint for $\\Lambda_{1.4}$ coming from the data analysis of the \nGW170817 event.\n\n\\begin{figure}[t!]\n\\centering\n\\includegraphics[clip=true, width=1\\linewidth]{.\/grafics\/chapter5\/Crust_com_tidal}\n\\caption{Dimensionless mass-weighted tidal deformability plotted against the chirp mass $\\mathcal{M}$\nof a binary NS system\nfor a set of mean-field interactions. The constraint for $\\tilde{\\Lambda}$ at $\\mathcal{M} = 1.186 M_\\odot$~\\cite{Abbott2019}\nis also included.}\\label{fig:tidal2}\n\\end{figure}\n\nWe plot in Fig.~\\ref{fig:tidal2} the mass-weighted tidal deformability \nof a binary NS system against the chirp mass $\\mathcal{M}$ for a set of Skyrme (left),\nGogny (center-left), MDI (center-right) and SEI (right) interactions. \nIn all cases, we see that $\\tilde{\\Lambda}$ has a decreasing behaviour as the chirp mass becomes larger. \nWe have followed the same prescription for the crust as the results obtained in Fig.~\\ref{fig:tidal1}. In this case, the changes for \n$\\tilde{\\Lambda}$\nshould not differ taking into account other descriptions for the crust~\\cite{Piekarewicz19}.\nWe have also plotted in each panel the constraint for $\\tilde{\\Lambda}$ at $\\mathcal{M}=1.186 M_\\odot$ coming from the detection \nof the GW170817 event~\\cite{Abbott2019}. \nMost interactions that have been considered here lay \ninside the constraint. However, the SkI5 and the SEI with respective slopes $L=100$ MeV and $L=115$ MeV, and very stiff EoSs are not able to \nprovide results for $\\tilde{\\Lambda}$ inside the observational bounds. \nThe GW constraint gives some upper and lower bounds for the slope of the symmetry energy. \nAt the same time, one can use bounds to $L$ to constraint other NS properties, such as the \nradius of a $1.4 M_\\odot$ NS~\\cite{Lourenco19, Lourenco19a}. \nSome main properties of NSs such as the total mass and radius for the maximum mass configuration and for a \n$1.4 M_\\odot$ NS are contained in Table~\\ref{tabletidal}. \nNotice that minor changes with the values in Table~\\ref{Table-NSs} of Chapter~\\ref{chapter2} are due to the \nprescription used for the core-crust transition.\nWe include in the same table the \nvalues of the dimensionless tidal deformability for a canonical NS and the mass-weighted tidal deformability \nof a binary NS at a chirp mass $\\mathcal{M}= 1.186 M_\\odot$.\nWe find in the lower limit of the constraint the results obtained with the D1M and MDI with $L=42$ MeV interactions, which \nprovide radius $R_{1.4}$ for a $1.4 M_\\odot$ NS of $10.05$ km and $10.11$ km, respectively.\nOn the other hand, the MDI with $L=88$ MeV and the SEI with $L=100$ MeV lay close to the upper limit of the constraint, \nproviding, respectively, $R_{1.4}$ of $12.76$ km and $13.18$ km.\nTherefore, the predictions using these few interactions for the radius of a canonical star lay in the \nrange $10$ km $\\lesssim R_{1.4} \\lesssim 13$ km, which is in consonance with the values obtained by the LIGO and Virgo \ncollaboration for the value of the radius of a canonical NS of $1.4 M_\\odot$, i.e., $R_{1.4} = 11.9^{+1.4}_{-1.4}$ km~\\cite{Abbott2018}.\n\n\n\\begin{table}[t!]\n\\centering\n\\resizebox{\\columnwidth}{!}{\n\\begin{tabular}{c|lccccc}\n\\hline\n\\multicolumn{2}{c|}{Force}     & $M_\\mathrm{max}$  &$R (M_\\mathrm{max})$       & $R (1.4M_\\odot)$       & $\\Lambda (1.4M_\\odot)$    &  $\\tilde{\\Lambda} (\\mathcal{M}=1.186 M_\\odot)$  \\\\  \\hline\\hline\n\\multirow{2}{*}{Skyrme}    & SLy4 ($L=46$ MeV)      &     2.06   &     10.00    &   11.74     &          304       &        365                \\\\\n                           & SkI5 ($L=129$ MeV)       &     2.28   &     11.85    &   14.48     &          1185      &        1402      \\\\ \\hline\n\\multirow{2}{*}{Gogny}     & D1M   ($L=25$ MeV)     &    1.74   &     8.80     &   10.05      &         121       &         149      \\\\\n                           & D1M$^*$ ($L=43$ MeV)      &     2.00  &      10.13   &    11.52     &           310     &         370       \\\\ \\hline\n\\multirow{3}{*}{MDI}       & $L=42$ MeV    &     1.60   &     8.62    &    10.11     &           94      &          124    \\\\\n                           & $L=60$ MeV    &       1.91  &      9.91  &      11.85   &            312     &           380 \\\\\n                           & $L=88$ MeV    &       1.99  &      10.59 &      12.76   &            567      &          686  \\\\ \\hline\n\\multirow{3}{*}{SEI}       & $L=86$ MeV   &       1.95  &      10.67 &      12.90    &           511      &          618    \\\\\n                           & $L=100$ MeV  &        1.98 &       10.89 &      13.18   &            640     &             773      \\\\\n                           & $L=115$ MeV    &       1.99 &       11.05 &      13.38   &            789     &          954   \\\\ \\hline\n\\end{tabular}\n}\n\n\n\\caption{Neutron star maximum mass $M_\\mathrm{max}$, radius at the maximum mass $R (M_\\mathrm{max})$ and for a $1.4 M_\\odot$\nNS. The table also includes the values of the dimensionless tidal deformability $\\Lambda$ for a canonical $1.4M_\\odot$ NS\nand of the mass-weighted tidal deformability $\\tilde{\\Lambda}$ at a chirp mass of $\\mathcal{M}=1.186 M_\\odot$.\nThe mass results are given in units of solar masses and the radii in units of km. \\label{tabletidal}}\n\\end{table}\n\n\\chapter{Summary and Conclusions}\\label{conclusions}\n\\fancyhead[RE, LO]{Chapter 7}\nIn this thesis we have further extended the analysis of the properties of neutron stars (NSs) and of finite\nnuclei through relating them to microphysical predictions associated to the isospin dependence of the equation of state (EoS)\nused to characterize both the nuclei and the NS core. \n \nWe have recalled in Chapter~\\ref{chapter1} the basic idea of the mean-field approximation, \nwhere the system is described as a set of non-interacting quasiparticles moving independently inside an effective \nmean-field potential. \nWe have summarized the main features of the Hartree-Fock method, and we have reminded the concept\nof phenomenological potentials. The definition of some properties of the EoS of symmetric nuclear matter \nand of asymmetric nuclear matter that we have used in the following chapters are also collected there.\nIn our work, we have used the zero-range\ndensity-dependent Skyrme~\\cite{skyrme56, vautherin72,sly41} interactions, and the finite-range \nGogny~\\cite{decharge80, berger91}, MDI~\\cite{das03,li08} and SEI~\\cite{behera98, Behera05} models. We provide for each one of them \nthe explicit interaction, as well as the corresponding expressions for different isoscalar and isovector nuclear matter \nproperties. All these interactions perform fairly well in the finite-nuclei density regime, and we \nuse them to study nuclear matter at larger densities, like the ones found in systems such as NSs. \nSkyrme interactions have been already widely used to \nstudy stellar objects as, for example, the celebrated SLy4 force, as well as some MDI and SEI interactions. On the other hand, our study with \nGogny forces has been one of the few to date where these interactions have been applied to NSs. \n\nWe have studied some properties of asymmetric nuclear matter, and we have presented our results in \nChapter~\\ref{chapter2} for a large set of Skyrme models with different nuclear matter properties and a set of \ndifferent Gogny interactions. \nWe first analyze in detail the impact on different nuclear and NS properties of the Taylor expansion of the energy per particle in asymmetric nuclear matter in\neven powers of the isospin asymmetry $\\delta$. \nThe lowest order is the contribution in \nsymmetric nuclear matter and the next term, quadratic in $\\delta$, corresponds to the usual symmetry energy coefficient. \nThis truncation of the EoS at second order is widely used in the literature when performing microscopic calculations. Terms of a higher order than two\n in the Taylor expansion provide additional corrections that account for the departure of the energy from a quadratic law in $\\delta$. \nIn our work, we have expanded the energy per particle up to tenth order when working with Skyrme interactions and up to \nsixth order when working with Gogny forces.\n \nFrom our study of the symmetry energy, we notice that the considered Skyrme forces can be separated into two different groups. In one of them\nthe symmetry energy, defined as the second-order coefficient of the expansion of the energy per particle, vanishes at some suprasaturation \ndensity several times the saturation one, which implies that for larger densities the asymmetric nuclear matter obtained with these forces \nbecomes isospin unstable. This is a general trend exhibited by the Skyrme models with a slope parameter smaller than about $46$ MeV. The other group of \nSkyrme interactions have larger slope parameters and the corresponding symmetry energy has an increasing behaviour as a function of the density.\nTerms of order higher than two in the expansion will contain contributions coming only from the kinetic and from non-local terms.\nThe coefficients in the expansion from fourth to tenth order in $\\delta$ do not show any well defined common trend as a function of the density \nand are strongly model-dependent. \n \nOn the other hand, in the analyzed Gogny interactions, the second-order symmetry energy coefficient shows an\nisospin instability at large values of the density, above $0.4 - 0.5$~fm$^{-3}$~\\cite{gonzalez17}.\nThe fourth- and sixth-order symmetry energy coefficients contain contributions from the kinetic and\nexchange terms exclusively. The results indicate that Gogny parametrizations also fall into two different \ngroups according to the density behaviour of these coefficients above saturation.\nIn the first group (D1S, D1M, D1N, and D250), the fourth- and sixth-order coefficients reach a maximum \nand then decrease with growing density. In the second group (D1, D260, D280, and D300), these coefficients \nare always increasing functions of density in the range analyzed. The different behaviour\nof the two groups can be traced back to the density dependence of the exchange terms, which  \nadd to the kinetic part of the fourth- and sixth-order coefficients. \n\nAt saturation density, the higher-order symmetry energy coefficients\nare relatively small for both Skyrme and Gogny interactions. This supports the accuracy \nof the Taylor expansion at second order in calculations of the energy in asymmetric nuclear matter \naround the saturation density.  \n \nAn alternative definition of the symmetry energy is provided by the difference  between the energy per\nparticle in neutron matter and in symmetric matter, and we have called it parabolic symmetry energy.\nThis difference also coincides with the infinite sum of all the coefficients of the Taylor expansion\nof the energy per particle in powers of the isospin asymmetry if one considers isospin asymmetry equal to one~\\cite{gonzalez17}.\nWe find that around saturation, the difference between the PA estimate $E_{\\mathrm{sym}}^{PA} (\\rho)$ \nand the $E_{\\mathrm{sym},2} (\\rho)$ coefficient is largely accounted by the sum of higher-order contributions.\nAnother important quantity in studies of the symmetry energy is the slope parameter $L$, which is \ncommonly used to characterize the density dependence of the symmetry energy near saturation.\nWe find, either using Skyrme or Gogny models, that large discrepancies of several MeV can arise between the $L$ value calculated with \n$E_{\\mathrm{sym},2} (\\rho)$ or with $E_{\\mathrm{sym}}^{PA} (\\rho)$. \nAgain, adding higher-order contributions accounts for most of these differences.\n\nTo study several properties of NSs one needs to consider $\\beta$-stable stellar \nmatter first. To proceed with this study, we take into account neutrons, protons, and leptons in chemical equilibrium.  \nTo test the accuracy of this approach at high densities, we have performed a systematic study with Skyrme and Gogny functionals~\\cite{gonzalez17}. \nBy \nsolving the equations with the exact EoS and with the Taylor expansion of Eq.~(\\ref{eq:EOSexpgeneral}) at increasing \norders in $\\delta$, we are able to analyze the convergence of the solutions with the expansion. \n The agreement between the $\\beta$-equilibrium asymmetries obtained using the exact EoS \nand the truncated Taylor expansion improves order by order.\nHowever, the convergence of this expansion is rather slow, in particular for forces with large slope parameter~$L$~\\cite{gonzalez17}. \nIn this scenario, to reproduce the exact asymmetry of $\\beta$-stable matter with approximations based on the Taylor expansion of\nthe energy per particle requires to include contributions of higher-order than two.\nThe parabolic approximation case, in where the energy per particle for a given asymmetry and density comes \nfrom a $\\delta^2$ interpolation between the \ncorresponding values in neutron matter and in symmetric matter, is also unable to reproduce the exact asymmetry of \n$\\beta$-stable matter in the range of densities considered. \nThese differences will make an impact when one tries to obtain other NS properties, and remark on the importance of the knowledge of the exact\nEoS. \nAnother approximation, used sometimes in the past, \nfor example in cases when it is complicated to obtain the full expression of the EoS, \nconsists in considering the exact\nkinetic energy part and performing the Taylor expansion only in the potential contribution to the energy per particle. This approach\nworks very well, reproducing closely the exact asymmetry in $\\beta$-stable matter for all \nthe range of densities considered. \n\nWe have also studied in Chapter~\\ref{chapter2} the mass-radius relation of NSs by solving the TOV equations and\nusing Skyrme and Gogny interactions~\\cite{gonzalez17}. In this case, we have used a polytropic EoS for the inner crust\nand the transition to the core is obtained by the thermodynamical method~\\cite{xu09a, Moustakidis12,Ducoin11,Providencia14,Fattoyev:2010tb,Pais2016,routray16,gonzalez17}.\nWe find that interactions with soft symmetry energy are not able to \nprovide numerically stable solutions of the TOV equations, and that only stiff enough \nEoSs may give an NS of $2 M_\\odot$~\\cite{Demorest10, Antoniadis13}.\nWe have considered a large set of Skyrme interactions, and we have closely analyzed a sample of 5 of them, namely the \nMSk7, UNEDF0, SkM$^*$, SLy4 and SkI5 parameterizations. Of this subset,  \nonly the SLy4 and the SkI5 interactions fit inside the astrophysical constraints of $2 M_\\odot$\nMoreover, the SLy4 force is the only one that, giving a radius of $\\sim 10$ km at the maximum mass configuration \nand a radius of $11.8$ km for a canonical NS of $1.4M_\\odot$ fits inside the\nconstraints for the radii coming from low-mass X-ray binaries and\nX-ray bursters~\\cite{Nattila16, Lattimer14}.\nOf the considered Gogny interactions, i.e., D1, D1S, D1M, D1N, D250, D260, D280 and D300,\nonly the D1M and D280 parameterizations are able to generate  \nNSs above the $1.4 M_\\odot$ value, reaching maximum masses of $M=1.74 M_\\odot$ and $1.66 M_\\odot$, respectively. \nHowever, one has to take into account that these interactions have not been fitted to \nreproduce highly asymmetric nuclear matter like the one found in the interior of NSs,\neven though in the fitting of D1N and D1M it was imposed the reproduction of the equation of state of neutron matter of \nFriedman and Pandharipande~\\cite{Friedman81}.\nFinally, in Chapter~\\ref{chapter2}, we have analyzed the convergence of the mass-radius relation if the \nTaylor expansion of the EoS is used instead of its full form. The prescription for the inner crust and the transition is \nthe same as before. In general, if the second-order expansion of the EoS is used, one finds results quite far from the \nexact ones. If higher-orders are used, the results approach the ones obtained using the full expression of the \nEoS. However, for interactions with very stiff EoSs, the convergence of the solution is slower than for softer interactions. \nThis again points out the necessity of using the full expression of the EoS or, in its absence, its Taylor expansion cut at an order \nhigher than $\\delta^2$.\n\n\nThe non-existence of Gogny interactions of the D1 type that are able to provide \nNSs with large masses around $2M_\\odot$ because of their soft symmetry energies~\\cite{Sellahewa14, gonzalez17} leads us to\nintroduce in Chapter~\\ref{chapter3} a new Gogny parametrization, dubbed D1M$^*$ \nthat, while preserving \nthe description of nuclei similar to the one obtained with D1M, the slope of \nthe symmetry energy is\nmodified to make the EoS of $\\beta$-stable matter stiff enough to \nobtain NS masses of $2 M_\\odot$ inside the observational constraints~\\cite{gonzalez18}. \nWe have also introduced a second new parametrization, which we name D1M$^{**}$, following the \nsame fitting procedure as D1M$^{*}$ and which is able to provide a $1.91 M_\\odot$ NS in the lower \nregion of the observational bounds~\\cite{gonzalez18a, Vinas19}.\nThe D1M force \\cite{goriely09} is susceptible to being used in this procedure,\nbut not D1S and D1N, as they are too far from the $2M_\\odot$ target. \nWe find that the new sets of parameters also perform at the same level \nas D1M in all aspects of finite nuclei analyzed in this work. Stellar properties from \nD1M$^*$, such as the \\mbox{M-R} relation and the moment of inertia,\nare in good agreement with the results from the Douchin-Haensel SLy4 EoS~\\cite{douchin01}, which is \ndesigned especially for working in the astrophysical scenario.\n\nWith these modified interactions we also study \nsome ground-state properties of finite nuclei, such as binding energies, neutron and proton radii, response to quadrupole \ndeformation and fission barriers. We find that both D1M$^*$ and D1M$^{**}$ interactions perform as well as D1M in all the \nconcerned properties of finite nuclei~\\cite{gonzalez18, gonzalez18a}. On the whole, we can say\nthat the D1M$^*$ and D1M$^{**}$ forces presented in this work are a good alternative to describe simultaneously finite nuclei and NSs providing \nvery good results\nin harmony with the experimental and observational data. Moreover, \nthese two interactions are much less demanding than the D2 force in\nterms of computational resources \\cite{chappert15}.\n\nThe correct determination of the transition between the core and the crust in NSs is essential in the study \nof different NS properties, such as the properties of the crust, or pulsar glitches. \nIn Chapter~\\ref{chapter4} we have calculated the core-crust transition searching\nfor the density where the uniform $\\beta$-stable matter becomes unstable \nagainst small fluctuations of the neutron, proton and electron distributions. \nThe instabilities in the core have been determined following two methods, namely the thermodynamical method~\\cite{kubis04,kubis06,xu09a,Moustakidis10,Cai2012,\nMoustakidis12,Seif14,routray16, gonzalez17} and the dynamical method~\\cite{baym71, pethick95,ducoin07, gonzalez19}. \nFirst, we evaluate the core-crust transition using the thermodynamical method, \nwhere one requires the mechanical and chemical stability of the NS core.\nWe have obtained the results for Skyrme and Gogny interactions~\\cite{gonzalez17}. As \nnoted in earlier literature, \nthe core-crust transition density is anticorrelated with the slope parameter $L$ of the models for both types of interactions. \nOn the other hand, in contrast to the transition density, the transition pressure is not seen \nto have a high correlation with~$L$.\nMoreover, we have studied the convergence of the core-crust transition properties, i.e., density, pressure, and isospin \nasymmetry at the core-crust boundary, when the Taylor expansion of the EoS is used instead of the full EoS. \nIn general, adding more terms to the Taylor expansion of the EoS brings the \ntransition density closer to the exact values.\nHowever, there can be still significant differences even when the Taylor expansion is pushed \nto tenth (Skyrme) or sixth (Gogny) order. This points out that the convergence for the transition properties is slow.\nIn fact, the order-by-order convergence for the transition\npressure is  sometimes not only slow, but actually erratic~\\cite{gonzalez17}. \n\nWe also compute the transition density using the so-called dynamical \nmethod, where the stable densities correspond to the values for which the curvature matrix, i.e., the second variation of the total energy, \nincluding bulk, finite size and Coulomb effects, is convex with respect to the density.\nIn this case, we have performed the calculations using Skyrme interactions and three types of finite-range interactions, Gogny, MDI and SEI~\\cite{gonzalez19}. \nThe values of the transition properties show the same global trends as exhibited when they are obtained using the thermodynamical method, \nbut they are shifted to lower values, as the addition of the surface and Coulomb terms help to stabilize more the system. \n\nWe have first obtained the results for Skyrme interactions, where we have analyzed the convergence of the results when one \nuses the Taylor expansion of the EoS. \nAs seen before, the agreement of these core-crust transition\nproperties estimated using the energy per particle expanded as a Taylor series with the exact predictions improves if more terms are included\nin the truncated sum. However, the quality of this agreement is model dependent as far as it depends on the slope parameter $L$.\nRetaining up to $\\delta^{10}$-terms in the expansion, the gap between the exact and approximated transition densities, asymmetries, and pressures\nfor interactions with $L$ below $\\sim 60$ MeV is nearly inexistent. However, for forces with $L$ above $\\sim 60$ MeV, these gaps are not negligible and\nmay be quite large. Moreover, the parabolic approach, very useful for estimating the energy per particle in asymmetric nuclear matter, does not predict\nvery accurate results for the crust-core properties compared with the exact results, in particular for the transition pressures.   \nOn the other hand, we have tested the case where one considers the full expression for the kinetic part of the interaction and \nthe Taylor expansion up to second order is performed only in its potential part. \nIn this case, one almost recovers the results obtained with the exact EoS.\n\nA global conclusion of our analysis is that the use of \nthe Taylor expansion up to quadratic terms in isospin asymmetry of the energy per particle \nmay be relatively sufficient to study the crust-core transition for models with soft symmetry energy. However, models with slope\nparameter $L$ above $\\sim 60$ MeV require the use of the exact energy per particle or its improved delta expansion to describe properly the \nrelevant physical quantities in the crust-core transition.\n\nIn this Chapter~\\ref{chapter4} we have also analyzed the core-crust transition in NSs with the dynamical method\nusing several finite-range interactions of Gogny, MDI and SEI types. Contrary to the case of Skyrme interactions that have the contributions   \nto the dynamical potential explicitly differentiated, we have to derive the energy curvature matrix in momentum space.\nOur study extends previous works available\nin the literature for finite-range forces, which were basically performed using MDI interactions and where the \nsurface effect was considered by the gradient contributions coming from the interaction part only. \nWe have taken into account contributions to the surface from both the interaction part and the kinetic \nenergy part. In this context, it can be considered\nas more self-contained compared to the earlier studies on the subject. \nWe have obtained the contribution coming from the direct energy through the\nexpansion of its finite-range form factors in terms of distributions, which allows one to write the direct contribution beyond\nthe long-wavelength limit (expansion till $k^2$-order). We have used the ETF approximation \nto write the kinetic and exchange energies as the sum of a bulk term plus a $\\hbar^2$ correction. This $\\hbar^2$\nterm can be written as a linear combination of the square of the gradients of the neutron and proton distributions, with density-dependent coefficients,\nwhich in turn provide a $k^2$-dependence in the energy curvature matrix.\n\nWe have found that the effects of the finite-range part of the nuclear interaction on the energy curvature matrix mainly\narise from the direct part of the energy, where the $\\hbar^2$-contributions \nfrom the kinetic and exchange part of the finite-range interaction can be considered as a small correction.\nWe have concluded that in the application of the dynamical method with finite-range forces it is an\naccurate approximation, at least in the many cases we have studied here, to compute the $D_{qq'}$ coefficients \nof the dynamical potential by considering only\nthe contributions from the direct energy, namely~\\cite{gonzalez19}:\n\\begin{small}\n\\begin{eqnarray}\n D_{nn}(k)&=&  D_{pp}(k) \\nonumber \\\\\n&=&\\sum_m \\bigg[W_m + \\frac{B_m}{2} - H_m - \\frac{M_m}{2}\\bigg] \\big({\\cal F}_m(k)-{\\cal F}_m(0)\\big) , \\nonumber \\\\\n D_{np}(k)&=&  D_{pn}(k) \\nonumber \\\\\n&=&\\sum_m \\bigg[W_m + \\frac{B_m}{2}\\bigg] \\big({\\cal F}_m(k)-{\\cal F}_m(0)\\big) .\n\\end{eqnarray}\n\\end{small}\nTherefore, as the ${\\cal F}_m(k)$ factors are just the Fourier transforms of the form factors\n $v_m(s)$, i.e., ${\\cal F}_m(k)=\\pi^{3\/2} \\alpha_m^3 e^{-\\alpha_m^2 k^2\/4}$ for Gaussians and ${\\cal F}_m(k)= 4\\pi \\mu_m^{-1}(\\mu_m^2 + k^2)^{-1}$ for Yukawians,\nthe dynamical calculation of the core-crust transition with the dynamical method using finite-range\nforces becomes almost as simple as with Skyrme forces.\n\nWe have also analyzed the global behaviour of the core-crust transition density and pressure as a function of the slope of\nthe symmetry energy at saturation for a large set of nuclear models, which include finite-range\ninteractions as well as Skyrme forces. \nThe results for MDI sets found under the present formulation are in good agreement with the earlier ones \nreported by Xu and Ko~\\cite{xu10b}, where they use a different\ndensity matrix expansion to deal with the exchange energy and adopted the Vlasov equation method to obtain the curvature\nenergy matrix.\nWe also observe that within the MDI and SEI families of interactions, \nthe transition density and pressure are highly correlated with the slope of the symmetry\nenergy at saturation. However, when nuclear models with different saturation properties are considered, these correlations are deteriorated,\nin particular the one related to the transition pressure. \nFrom our analysis, we have also found noticeable differences in the transition density and pressure calculated \nwith the thermodynamical method and the dynamical method, and later we point out its consequences when studying properties of the \nNS crust.\n\n\nWe have devoted Chapter~\\ref{chapter5} to study global NS properties. First, we study stellar masses and radii \npaying special attention to the properties related to the crust, that is, the  crustal thickness and crustal mass, which can influence\nobservational properties. We calculate them using different sets of mean-field models.\nThese properties are directly related to the core-crust transition and to the choice of the inner crust. Many mean-field interactions used to \ncompute the core EoS have not been used to find the corresponding EoS in the inner crust\nas this region is particularly difficult to describe owing to the presence of the neutron gas and the possible existence of nuclear clusters \nwith non-spherical shapes.\nAs it is usual in the literature, we use either a polytropic EoS in the inner crust \nor an EoS computed with a different model in the crust region. \nThe use of different inner crust prescriptions may not have a large impact when studying global properties of the NS such as the total mass or radius, if the \nstar is massive enough. On the other hand, if the NS is light with masses $M \\lesssim 1 M_\\odot$, the \n crust region has a noticeable influence on the determination of these properties. If one studies\nthe crustal properties of the NS, the choice of the inner crust may lead to large uncertainties. \nBecause of that, we have studied the mass and thickness of the crust using the recent approximation derived by Zdunik et al.~\\cite{Zdunik17} who solved the \nTOV equations from the center of the star to the point corresponding to the transition density. \nThis points out again the importance of an accurate determination of the core-crust transition. Until now, in previous literature, the \nthermodynamical method has been widely used in all types of interactions. These calculations are relatively easy to perform because only the derivatives of the energy \nper particle are needed, and it takes into account neither surface nor Coulomb effects. On the other hand, to perform the calculation with the dynamical method is \nmore complicated, specifically when using finite-range forces, where the exchange contributions in the surface terms are also considered. \nWe analyze how the determination of the core-crust transition affects the results obtained for the crustal mass and crustal thickness, and it is found \nthat the differences in the transition point determined with the thermodynamical and dynamical methods have a relevant impact on the \nconsidered crustal properties, in particular for models with stiff symmetry energy~\\cite{gonzalez19}.\nThese crustal \nproperties play a crucial role in the description of several observed phenomena, like glitches, r-mode oscillation, etc.\nTherefore, the core-crust transition density needs to be ascertained as precisely as possible by \ntaking into account the associated physical conditions in order to have a realistic estimation of the observed phenomena.\n\nThe detection of gravitational waves has opened a new era in astrophysics, cosmology, and nuclear physics. The GW170817 detection \nby the LIGO and Virgo collaboration~\\cite{Abbott2017, Abbott2018, Abbott2019}, accounting for the merger of a binary NS system, led to \na new set of constraints in both the astrophysical and nuclear sector. \nOne of these constraints, directly measured from the GW signal of the merger, is the dimensionless mass-weighted tidal deformability $\\tilde{\\Lambda}$\nat a certain chirp mass $\\mathcal{M}$. Other constraints coming from the data analysis of the detection are on the dimensionless tidal \ndeformability of a single NS, $\\Lambda$, for a canonical NS of $1.4 M_\\odot$, on the masses of the two NSs, on the radii, etc.~\\cite{Abbott2017, Abbott2018, Abbott2019}.\nIn our work, we have calculated the tidal deformability $\\Lambda$ using different mean-field models. \nWe observe that very stiff EoSs will provide $\\Lambda$ at $1.4 M_\\odot$ that are not inside the boundary limits. \nIf one computes $\\tilde{\\Lambda}$ for an NS binary system one finds the same behaviour: very stiff EoSs with $L \\gtrsim 90-100$ MeV\nare not able to provide a system that gives small enough $\\tilde{\\Lambda}$.\nWith this limitation on $L$ coming from the constraints on the tidal deformability, \none can also restrict other NS properties, such as the radius of a canonical NS. In our case, with the interactions we have used, \nwe find that this radius should be in the range $10 \\lesssim R_{1.4} \\lesssim 13$ km, which is in consonance with the values \nobtained by the LIGO and Virgo collaboration of $R_{1.4} = 11.9 \\pm 1.4$ km.\n\nIn this work, the moment of inertia of NSs has also been studied in Chapter~\\ref{chapter5}. \nWe find large values of the moment of inertia when the considered EoS to describe the core is stiff~\\cite{gonzalez17}. \nUsing universal relations between $\\Lambda$ and the moment of inertia, and using the data extracted from the GW170817 event, \nLandry and Kumar extracted some constraints for the moment of inertia of the double binary pulsar PSR J0737-3039. \nWith these constraints, we find that very stiff interactions are not suitable to describe the moment of inertia of these observables. \nIn particular, the two new interactions D1M$^*$ and D1M$^{**}$ that we have formulated fit inside the constraint, as well as inside constraining bands from Refs.~\\cite{Lattimer2005, Breu2016}\nfor the plots of the dimensionless \nquantity $I\/MR^2$ against the compactness. \nThis points out again the good performance of these interactions when studying NS systems. \nFinally, the fraction of the moment of inertia enclosed in the crust is analyzed. Again, we see the importance of the determination \nof the core-crust transition, as the results calculated if considering the core-crust transition given by the dynamical method are much lower than the ones obtained if using the transition \ndensity given by the \ntransition density given by the\nthermodynamical approach. This may play a crucial role when predicting other phenomena like glitches or r-mode oscillations. \n \nTo conclude, we mention some future prospects:\n\n\\begin{itemize}\n \\item A more extensive and systematical application to the study of the properties of finite nuclei with the newly proposed D1M$^*$ and D1M$^{**}$ interactions is \n to be performed. \n This work is in progress and will be presented in the near future. \n \\item When studying NS properties, we have emphasized the necessity of having a unified EoS describing with the same interaction all regions of the NS. \n Because of that, there is work under progress in our group aimed to obtain the inner crust and outer crust EoSs using \n Gogny interactions and, in particular, with the new D1M$^*$ and D1M$^{**}$ interactions.\n \\item As stated previously, the detection of gravitational waves has opened a new window in physics. New observations on mergers of binary NS\n systems or binary NS+black hole systems will provide new constraints on NS observables, such as the tidal deformability, the stellar radii, the moment of inertia, etc. \n This will help to better determine the EoS of highly dense and asymmetric nuclear matter. Moreover, the NICER mission will aim to establish better boundaries on the \n radii of NSs. Universal relations, for example between $\\Lambda$ and the moment of inertia, may also be of great help to further constraint the nuclear \n matter EoS. \n\\end{itemize}\n\n\\fancyhead[RE, LO]{Chapter 7}\n\n\\begin{appendices}\n\\addtocontents{toc}{\\protect\\setcounter{tocdepth}{0}}\n\\fancyhead[RE, LO]{Chapter \\thechapter}\n\\chapter{Derivatives of the equation of state to study the stability of uniform nuclear matter}\\label{appendix_thermal}\nThe stability conditions for the thermodynamical potential $V_\\mathrm{ther} (\\rho)$ and the dynamical potential $V_\\mathrm{dyn} (\\rho)$ \ndiscussed in Chapter~\\ref{chapter4} require the calculation of\nthe first and second derivatives of the energy per baryon $E_b (\\rho, \\delta)$ with respect to the density $\\rho$ and the isospin asymmetry $\\delta$. \nIn this Appendix, we provide the corresponding expressions obtained with the exact EoS and with the Taylor expansion of the EoS up to order $\\delta^{10}$\nfor Skyrme forces and up to order $\\delta^6$ for Gogny interactions.\n\n\\section{Derivatives using the exact expression of the EoS} \nWe collect here the derivatives of $E_b (\\rho, \\delta)$ involved in the study of the stability of homogeneous matter, obtained using \nthe full expression of the EoS. \nThe derivative $\\partial E_b (\\rho, \\delta)\/\\partial \\rho$ is immediately obtained from the expression for the pressure $P_b(\\rho,\\delta)$ we \nhave given in Eq.~(\\ref{eq:press_skyrme}) for Skyrme and in Eq.~(\\ref{eq:pressure_bars}) for Gogny interactions, \ntaking into account that $\\partial E_b (\\rho, \\delta)\/\\partial \\rho = P_b(\\rho,\\delta)\/\\rho^2$. \nThe other derivatives that are needed to compute the thermodynamical and dynamical potentials for Skyrme interactions are the following:\n\\begin{eqnarray}\n\\frac{\\partial^2 E_b (\\rho, \\delta)}{\\partial \\rho^2}&=& -\\frac{1}{15} \\frac{\\hbar^2}{2m} \n\\left(\\frac{3 \\pi^2}{2} \\right)^{2\/3} \\rho^{-4\/3}\\left[ (1+\\delta)^{5\/3} + (1-\\delta)^{5\/3} \\right] \\nonumber\\\\\n&+& \\frac{(\\sigma + 1) \\sigma}{48} t_3 \\rho^{\\sigma-1} \\left[ 2(x_3+2) - \\frac{1}{2} (2x_3 + 1) \n\\left[ (1+\\delta)^2 + (1-\\delta)^2\\right]\\right]\\nonumber\\\\\n&+& \\frac{1}{24} \\left( \\frac{3 \\pi^2}{2}\\right)^{2\/3} \\rho^{-1\/3}\\left\\{  \\vphantom{\\frac{1}{2}}\n\\left[ t_1(x_1+2) + t_2(x_2+2)\\right] \\left[ (1+\\delta)^{5\/3} + (1-\\delta)^{5\/3}\\right] \\right. \\nonumber\\\\ \n&+&\\left.  \\frac{1}{2} \\left[ t_2(2x_2 + 1) - t_1 (2x_1 + 1)\\right] \\left[ (1+\\delta)^{8\/3} + (1-\\delta)^{8\/3}\\right] \\right\\}\n\\end{eqnarray}\n\n\\begin{eqnarray}\n\\frac{\\partial^2 E_b (\\rho, \\delta)}{\\partial \\rho \\partial \\delta}&=& \\frac{\\hbar^2}{6m} \n\\left(\\frac{3 \\pi^2}{2} \\right)^{2\/3} \\rho^{-1\/3}\\left[ (1+\\delta)^{2\/3} - (1-\\delta)^{2\/3} \\right]\\nonumber\\\\\n&-& \\frac{1}{4} t_0 (2x_0+1) \\delta - \\frac{(\\sigma +1)}{24} t_3 \\rho^{\\sigma} (2x_3+1) \\delta  \\nonumber\\\\\n&+& \\frac{1}{48} \\left( \\frac{3 \\pi^2}{2}\\right)^{2\/3} \\rho^{2\/3}\\left\\{5 \\left[ t_1(x_1+2) + t_2(x_2+2)\\right] \n\\left[ (1+\\delta)^{2\/3} - (1-\\delta)^{2\/3}\\right] \\right.\\nonumber \\\\ \n&+&\\left. 4 \\left[ t_2(2x_2 + 1) - t_1 (2x_1 + 1)\\right] \\left[ (1+\\delta)^{5\/3} - (1-\\delta)^{5\/3}\\right] \\right\\}\n\\end{eqnarray}\n\n\\begin{eqnarray}\n\\frac{\\partial^2 E_b (\\rho, \\delta)}{\\partial \\delta^2}&=& \\frac{\\hbar^2}{6m}\n\\left(\\frac{3 \\pi^2}{2} \\right)^{2\/3} \\rho^{2\/3}\\left[ (1+\\delta)^{-1\/3} + (1-\\delta)^{-1\/3} \\right]\\nonumber\\\\\n&-&  \\frac{t_0}{4} \\rho (2x_0+1) - \\frac{1}{24} t_3 \\rho^{\\sigma+1} (2x_3+1) \\nonumber \\\\\n&+& \\frac{1}{24} \\left( \\frac{3 \\pi^2}{2}\\right)^{2\/3} \\rho^{5\/3}\\left\\{ \\left[ t_1(x_1+2) + \nt_2(x_2+2)\\right] \\left[ (1+\\delta)^{-1\/3} + (1-\\delta)^{-1\/3}\\right] \\right.\\nonumber \\\\ \n&+&\\left.  2 \\left[ t_2(2x_2 + 1) - t_1 (2x_1 + 1)\\right] \\left[ (1+\\delta)^{2\/3} + (1-\\delta)^{2\/3}\\right] \\right\\}.\n\\end{eqnarray}\n\nOn the other hand, for Gogny interactions we have:\n\n\\begin{eqnarray}\n\\frac{\\partial^2 E_b (\\rho, \\delta)}{\\partial \\rho^2}&=& - \\frac{\\hbar^2}{30m} \\left(\\frac{3 \\pi^2}{2} \n\\right)^{2\/3} \\rho^{-4\/3}\\left[ (1+\\delta)^{5\/3} + (1-\\delta)^{5\/3} \\right]\\nonumber\n\\\\\n&+& \\frac{(\\alpha + 1) \\alpha}{8} t_3 \\rho^{\\alpha-1} \\left[ 3 - \\left( 2 x_3 + 1 \\right) \\delta^2 \\right] \n\\nonumber\n\\\\\n&+&\\sum_{i=1,2}\\frac{1}{6 \\rho^2 k_F^{3} \\mu_i^3}  \\Bigg\\{  {\\cal C}_i \\left[ \n\\vphantom{\\frac{1}{2}}2 \\left( -6 + \nk_{Fn}^2\\mu_i^2 + k_{Fp}^2\\mu_i^2\\right) +  e^{- k_{Fn}^2\\mu_i^2} \\left( 6  + 4  k_{Fn}^2\\mu_i^2 + \nk_{Fn}^4\\mu_i^4\\right)  \\right.  \\nonumber\n\\\\\n&+& \\left.\n  e^{- k_{Fp}^2\\mu_i^2} \\left( 6  + 4  k_{Fp}^2\\mu_i^2 + k_{Fp}^4\\mu_i^4\\right) \\vphantom{\\frac{1}{2}} \\right]\n + {D}_i e^{-\\frac{1}{4} \\left( k_{Fn}^2 + k_{Fp}^2\\right)\\mu_i^2} \\nonumber\n\\\\\n&\\times& \\left[  \\left( -12  k_{Fn} k_{Fp}\\mu_i^2- k_{Fn}^3 k_{Fp}\\mu_i^4 - k_{Fn} k_{Fp}^3\\mu_i^4 \\right) \n\\mathrm{cosh} \\left[ \\frac{k_{Fn} k_{Fp}\\mu_i^2}{2}  \\right] \\right.  \\nonumber\n\\\\\n&+&  2 \\left( 12 + k_{Fn}^2\\mu_i^2  +  k_{Fp}^2\\mu_i^2 + k_{Fn}^2 k_{Fp}^2\\mu_i^4\\right) \n\\left.  \\mathrm{sinh} \\left[ \\frac{k_{Fn} k_{Fp}\\mu_i^2}{2} \\right] \\right] \\Bigg\\} ,\n\\end{eqnarray}\n\\begin{eqnarray}\n\\frac{\\partial^2 E_b (\\rho, \n\\delta)}{\\partial \\rho \\partial \\delta} &=&  \\frac{\\hbar^2}{6m} \\left(\\frac{3 \\pi^2}{2} \\right)^{2\/3} \n\\rho^{-1\/3}\\left[ (1+\\delta)^{2\/3} - (1-\\delta)^{2\/3} \\right]\\nonumber\n\\\\\n&-& \\frac{(\\alpha +1)}{4} t_3 \\rho^{\\alpha} (2x_3+1) \\delta  +  \\sum_{i=1,2} \\mu_i^3 \\pi^{3\/2} {\\cal B}_i\\delta \\nonumber\n\\\\\n&-& \\sum_{i=1,2}\\frac{1}{6 \\rho} \\left\\{{\\cal C}_i \\left[ \\frac{-1 + e^{-k_{Fp}^2 \\mu_i^2} \n\\left( 1+ k_{Fp}^2 \\mu_i^2\\right)}{k_{Fp} \\mu_i} - \\frac{-1 + e^{-k_{Fn}^2 \\mu_i^2} \n\\left( 1+ k_{Fn}^2 \\mu_i^2\\right)}{k_{Fn} \\mu_i} \\right]\\nonumber \\right. \\nonumber\n\\\\\n&-& {\\cal D}_i e^{-\\frac{1}{4} \\left(k_{Fn}^2+ k_{Fp}^2\\right)\\mu_i^2} \n\\left[ \\left(k_{Fn} \\mu_i - k_{Fp} \\mu_i \\right)\n\\cosh \\left[ \\frac{k_{Fn} k_{Fp}\\mu_i^2}{2} \\right] \\right. \\nonumber\n\\\\\n&-& \\left.  \\frac{2}{k_{Fn} k_{Fp} \\mu_i^2}  \\left( k_{Fn} \\mu_i - k_{Fp} \\mu_i + \\delta k_F^3 \\mu_i^3\\right)\n\\sinh \\left[  \\frac{k_{Fn} k_{Fp}\\mu_i^2}{2} \\right] \\right] \\Bigg\\} ,\n\\end{eqnarray}\n\\begin{eqnarray}\n\\frac{\\partial^2 E_b (\\rho, \\delta)}{ \\partial \\delta^2} &=& \n\\frac{\\hbar^2}{6m} \\left(\\frac{3 \\pi^2}{2} \\right)^{2\/3} \\rho^{2\/3}\\left[ (1+\\delta)^{-1\/3} + (1-\\delta)^{-1\/3} \\right]\n\\nonumber\\\\\n&-&  \\frac{t_3}{4} \\rho^{\\alpha+1} (2x_3+1) + \\frac{1}{4}\\sum_{i=1,2} \\mu_i^3 \\pi^{3\/2}  {\\cal B}_i \\rho \\nonumber\n\\\\\n&-& \\frac{1}{6} \\sum_{i=1,2} \\left\\{ {\\cal C}_i \\left [  \\frac{1 - e^{- k_{Fp}^2 \\mu_i^2}\n\\left( 1 + k_{Fp}^2 \\mu_i^2 \\right)}{(1-\\delta) k_{Fp} \\mu_i} + \n\\frac{1 - e^{- k_{Fn}^2 \\mu_i^2} \n\\left( 1 + k_{Fn}^2 \\mu_i^2 \\right)}{(1+\\delta) k_{Fn} \\mu_i} \\right] \\right.\\nonumber\n\\\\\n&+&  {\\cal D}_i  e^{-\\frac{1}{4} \\left( k_{Fp}^2 +k_{Fn}^2 \\right) \\mu_i^2}\n \\left[ \\left(  k_{Fn} \\mu_i\\left(1-\\delta \\right)^{-1} +  k_{Fp} \\mu_i\\left(1+\\delta \\right)^{-1}\\right)  \n\\cosh \\left[\\frac{k_{Fn} k_{Fp}\\mu_i^2}{2}\\right] \\right. \\nonumber\n\\\\\n&-&   \\frac{2}{\\left( 1- \\delta^2 \\right) k_{Fn} k_{Fp} \\mu_i^2} \\left( k_{Fn} \\mu_i + k_{Fp} \\mu_i \n- k_F^3 \\mu_i^3  \\right. \\nonumber\n\\\\\n&+&\\left.   \\left. \\delta \\left(  k_{Fn} \\mu_i - k_{Fp} \\mu_i   \n+ \\delta k_F^3 \\mu_i^3 \\right) \\right)\\sinh \\left[\\frac{k_{Fn} k_{Fp}\\mu_i^2}{2}\\right] \\right] \\Bigg\\} .\n\\end{eqnarray}\n\n\\section{Derivatives using the Taylor expansion of the EoS}\nThe derivatives of the energy per particle taking into account the Taylor expansion of the EoS can be rewritten as \n\\begin{eqnarray}\n \\frac{\\partial E_b (\\rho, \\delta)}{\\partial \\rho}&=& \\frac{\\partial E_b(\\rho, \\delta=0)}{\\partial \\rho}+\n \\sum_k \\frac{\\partial E_{\\mathrm{sym}, 2k}}{\\partial \\rho} \\delta^{2k}\\\\\n \\frac{\\partial^2 E_b (\\rho, \\delta)}{\\partial \\rho \\partial \\delta}&=& \\frac{\\partial^2 E_b(\\rho, \\delta=0)}{\\partial \\rho^2}+\n \\sum_k \\frac{\\partial^2 E_{\\mathrm{sym}, 2k}}{\\partial \\rho^2} \\delta^{2k}\n \\\\\n \\frac{\\partial^2 E_b (\\rho, \\delta)}{\\partial \\rho^2}&=& 2 \\sum_k k \\delta^{2k-1} \\frac{\\partial E_{\\mathrm{sym}, 2k}}{\\partial \\rho} \n \\\\\n \\frac{\\partial^2 E_b (\\rho, \\delta)}{ \\partial \\delta^2} &=& 2 \\sum_k k (2k-1) \\delta^{2k-2} E_{\\mathrm{sym}, 2k}\n\\end{eqnarray}\n\n\nFor the Skyrme interaction, the density derivatives of the energy per baryon in symmetric \nnuclear matter $E_b(\\rho, \\delta=0)$ given by\n\n\\begin{eqnarray}\n\\frac{\\partial E_b(\\rho, \\delta=0)}{\\partial \\rho} &=& \\frac{\\hbar^2}{5m} \\left(  \\frac{3 \\pi^2}{2}\\right)^{2\/3} \\rho^{-1\/3} + \n\\frac{3}{8} t_0  + \\frac{(\\sigma +1)}{16} t_3 \\rho^{\\sigma}   \\nonumber\\\\\n &+& \\frac{1}{16} \\left(  \\frac{3 \\pi^2}{2}\\right)^{2\/3} \\rho^{2\/3} \\left[ 3 t_1 + t_2 (4 x_2 +5) \\right]\n\\end{eqnarray}\n\n\\begin{eqnarray}\n\\frac{\\partial^2 E_b (\\rho, \\delta=0)}{\\partial \\rho^2} &=& -\\frac{\\hbar^2}{15m} \\left(  \\frac{3 \\pi^2}{2}\\right)^{2\/3} \\rho^{-4\/3} \n+ \\frac{(\\sigma +1) \\sigma}{16} t_3 \\rho^{\\sigma-1}   \\nonumber\\\\\n &+& \\frac{1}{24} \\left(  \\frac{3 \\pi^2}{2}\\right)^{2\/3} \\rho^{-1\/3} \\left[ 3 t_1 + t_2 (4 x_2 +5) \\right]\n\\end{eqnarray}\n\nand for the Gogny interaction they read as\n\\begin{eqnarray}\n\\frac{\\partial E_b(\\rho, \\delta=0)}{\\partial \\rho} &=& \\frac{\\hbar^2}{5m} \\left(  \\frac{3 \\pi^2}{2}\\right)^{2\/3} \\rho^{-1\/3} \n+ \\frac{3(\\alpha +1)}{8} t_3 \\rho^{\\alpha}   +\n \\frac{1}{2} \\sum_{i=1,2} \\mu_i^3 \\pi^{3\/2}  {\\cal A}_i \\nonumber\n\\\\\n &-&\\sum_{i=1,2}\\frac{1}{2 \\rho k_F^3 \\mu_i^3}  \\left( {\\cal C}_i - {\\cal D}_i \\right)\n \\left[ -2 + k_F^2 \\mu_i^2 + e^{-k_F^2 \\mu_i^2} \\left(2 + k_F^2 \\mu_i^2 \\right) \\right] ,\n\\\\\n\\frac{\\partial^2 E_b(\\rho, \\delta=0)}{\\partial \\rho^2} &=& -\\frac{\\hbar^2}{15m} \\left(  \\frac{3 \\pi^2}{2}\\right)^{2\/3} \\rho^{-4\/3} \n+ \\frac{3(\\alpha +1) \\alpha}{8} t_3 \\rho^{\\alpha-1}\n\\\\  \n&-& \\sum_{i=1,2}\\frac{1}{3 \\rho^2 k_F^3 \\mu_i^3}  \\left( {\\cal C}_i - {\\cal D}_i \\right)\n\\left[ 6 -2 k_F^2 \\mu_i^2 - e^{-k_F^2 \\mu_i^2} \\left(6 +4 k_F^2 \\mu_i^2 +k_F^4 \\mu_i^4\\right) \\right] .\\nonumber\n\\end{eqnarray}\nThe first and second derivatives with respect to the density of the symmetry energy coefficients \ncan be readily computed from Eqs.~(\\ref{eq:esym2skyrme2})---(\\ref{eq:esym10skyrme}) \nfor Skyrme interactions and from Eqs.~(\\ref{eq:esym2gog})---(\\ref{eq:esym6gog}) for Gogny interactions,\nby \ntaking derivatives of the $G_n (\\eta)$ functions defined in Eqs.~(\\ref{G1}), (\\ref{G2}) and (\\ref{G3})---(\\ref{G6}), and using \n$\\displaystyle \\frac{\\partial G_n (\\eta)}{\\partial \\rho} = \\frac{\\partial G_n (\\eta)}{\\partial \\eta} \\, \\frac{\\partial \\eta}{\\partial \\rho}$,\nwhere $\\displaystyle \\frac{\\partial \\eta}{\\partial \\rho} = \\frac{\\pi^2 \\mu_i}{2 k_F^2}$ for $\\eta = \\mu_i k_F$.\nAs this is relatively straightforward, we omit here the explicit results for these derivatives.\n\n\\chapter{Core-crust transition properties}\\label{app_taules}\nIn this Appendix we collect the values of the transition density, pressure and asymmetry \nfound using the thermodynamical and the dynamical methods (Chapter~\\ref{chapter4}) for a set of different Skyrme \nparametrizations. The results have been obtained if using the full expression of the EoS,\nits Taylor expansion to different orders, and the parabolic approximation. \nThe results for Gogny interactions when using the thermodynamical method are also included, obtained \nusing both the exact and Taylor expanded EoSs. Finally, the values of the transition properties \nobtained using the exact expression of the EoS for the finite-range Gogny, MDI and SEI interactions \nare also collected. \n\n\\begin{table}[htb]\n\\begin{tabular}{cccccccccc}\n\\hline\n\\multicolumn{9}{c}{THERMODYNAMICAL METHOD} \\\\ \\hline\nForce & $L$ & \\begin{tabular}[c]{@{}c@{}}$\\delta_t$ \\\\ ($\\delta^2$)\\end{tabular} & \n\\begin{tabular}[c]{@{}c@{}}$\\delta_t$\\\\ ($\\delta^4$)\\end{tabular} & \\begin{tabular}[c]{@{}c@{}}$\\delta_t$\\\\ \n($\\delta^6$)\\end{tabular} & \\begin{tabular}[c]{@{}c@{}}$\\delta_t$\\\\ ($\\delta^8$)\\end{tabular} & \n\\begin{tabular}[c]{@{}c@{}}$\\delta_t$\\\\ ($\\delta^{10}$)\\end{tabular} & \\begin{tabular}[c]{@{}c@{}}$\\delta_t$\\\\ \n(Exact)\\end{tabular} & \\begin{tabular}[c]{@{}c@{}}$\\delta_t$\\\\ (PA)\\end{tabular} \\\\\\hline \\hline\nMSk7 & 9.41 & 0.9317 & 0.9275 & 0.9260 & 0.9254 & 0.9250 & 0.9243 & 0.9278 \\\\\nSIII & 9.91 & 0.9271 & 0.9195 & 0.9173 & 0.9164 & 0.9159 & 0.9151 & 0.9206 \\\\\nSkP & 19.68 & 0.9252 & 0.9182 & 0.9164 & 0.9158 & 0.9155 & 0.9152 & 0.9196 \\\\\nHFB-27 & 28.50 & 0.9299 & 0.9245 & 0.9230 & 0.9224 & 0.9222 & 0.9221 & 0.9252 \\\\\nSKX & 33.19 & 0.9248 & 0.9187 & 0.9172 & 0.9167 & 0.9166 & 0.9167 & 0.9195 \\\\\nHFB-17 & 36.29 & 0.9357 & 0.9315 & 0.9304 & 0.9300 & 0.9298 & 0.9301 & 0.9318 \\\\\nSGII & 37.63 & 0.9555 & 0.9510 & 0.9500 & 0.9497 & 0.9497 & 0.9509 & 0.9516 \\\\\nUNEDF1 & 40.01 & 0.9452 & 0.9408 & 0.9400 & 0.9397 & 0.9398 & 0.9411 & 0.9412 \\\\\nSk$\\chi$500 & 40.74 & 0.9452 & 0.9432 & 0.9424 & 0.9421 & 0.9420 & 0.9419 & 0.9429 \\\\\nSk$\\chi$450 & 42.06 & 0.9348 & 0.9311 & 0.9302 & 0.9299 & 0.9298 & 0.9301 & 0.9312 \\\\\nUNEDF0 & 45.08 & 0.9400 & 0.9353 & 0.9345 & 0.9344 & 0.9346 & 0.9361 & 0.9355 \\\\\nSkM* & 45.78 & 0.9440 & 0.9392 & 0.9383 & 0.9382 & 0.9383 & 0.9400 & 0.9395 \\\\\nSLy4 & 45.96 & 0.9305 & 0.9275 & 0.9266 & 0.9263 & 0.9262 & 0.9265 & 0.9272 \\\\\nSLy7 & 47.22 & 0.9311 & 0.9282 & 0.9273 & 0.9270 & 0.9269 & 0.9272 & 0.9278 \\\\\nSLy5& 48.27 & 0.9327 & 0.9297 & 0.9289 & 0.9286 & 0.9285 & 0.9289 & 0.9295 \\\\\nSk$\\chi$414 &51.92  & 0.9358 & 0.9328 & 0.9320 & 0.9318 & 0.9317 & 0.9322 & 0.9326 \\\\\nMSka & 57.17 & 0.9449 & 0.9403 & 0.9395 & 0.9393 & 0.9394 & 0.9410 & 0.9403 \\\\\nMSL0 & 60.00 & 0.9536 & 0.9503 & 0.9500 & 0.9503 & 0.9506 & 0.9538 & 0.9490 \\\\\nSIV & 63.50 & 0.9425 & 0.9358 & 0.9347 & 0.9347 & 0.9349 & 0.9377 & 0.9356 \\\\\nSkMP & 70.31 & 0.9596 & 0.9568 & 0.9568 & 0.9573 & 0.9578 & 0.9628 & 0.9556 \\\\\nSKa & 74.62 & 0.9440 & 0.9397 & 0.9395 & 0.9400 & 0.9405 & 0.9447 & 0.9386 \\\\\nR$_\\sigma$ & 85.69 & 0.9631 & 0.9622 & 0.9632 & 0.9643 & 0.9653 & 0.9736 & 0.9596 \\\\\nG$_\\sigma$ & 94.01 & 0.9626 & 0.9626 & 0.9643 & 0.9656 & 0.9669 & 0.9772 & 0.9590 \\\\\nSV & 96.09 & 0.9547 & 0.9502 & 0.9507 & 0.9518 & 0.9528 & 0.9610 & 0.9471 \\\\\nSkI2 & 104.33 & 0.9609 & 0.9618 & 0.9631 & 0.9643 & 0.9654 & 0.9736 & 0.9587 \\\\\nSkI5 & 129.33 & 0.9562 & 0.9591 & 0.9614 & 0.9633 & 0.9648 & 0.9754 & 0.9549 \\\\ \\hline\n\\end{tabular}\n\\caption{Values of the core-crust transition asymmetry $\\delta_t$ for Skyrme interactions \ncalculated within the thermodynamical approach. The results have been computed with the exact expression of the EoS (Exact), the parabolic approximation (PA), \nor the approximations of the full EoS with Eq.~(\\ref{eq:EOSexpgeneral}) up to second ($\\delta^2$), fourth ($\\delta^4$), sixth ($\\delta^6$) \neighth ($\\delta^8$) and tenth ($\\delta^{10}$) order. The value for each interaction of the slope parameter of the symmetry energy $L$ is also included in units of MeV.}\n\\end{table}\n\n\n\\begin{table}[htb]\n\\begin{tabular}{ccccccccc}\n\\hline\n\\multicolumn{9}{c}{THERMODYNAMICAL METHOD} \\\\ \\hline\nForce & $L$ & \\begin{tabular}[c]{@{}c@{}}$\\rho_t$ \\\\ ($\\delta^2$)\\end{tabular} & \n\\begin{tabular}[c]{@{}c@{}}$\\rho_t$\\\\ ($\\delta^4$)\\end{tabular} & \\begin{tabular}[c]{@{}c@{}}$\\rho_t$\\\\ \n($\\delta^6$)\\end{tabular} & \\begin{tabular}[c]{@{}c@{}}$\\rho_t$\\\\ ($\\delta^8$)\\end{tabular} & \n\\begin{tabular}[c]{@{}c@{}}$\\rho_t$\\\\ ($\\delta^{10}$)\\end{tabular} & \\begin{tabular}[c]{@{}c@{}}$\\rho_t$\\\\\n(Exact)\\end{tabular} & \\begin{tabular}[c]{@{}c@{}}$\\rho_t$\\\\ (PA)\\end{tabular} \\\\ \\hline\\hline\nMSk7 & 9.41 & 0.1291 & 0.1276 & 0.1270 & 0.1266 & 0.1263 & 0.1251 & 0.1273 \\\\\nSIII & 9.91 & 0.1225 & 0.1202 & 0.1196 & 0.1192 & 0.1190 & 0.1181 & 0.1186 \\\\\nSkP & 19.68 & 0.1204 & 0.1170 & 0.1156 & 0.1147 & 0.1140 & 0.1116 & 0.1153 \\\\\nHFB-27 & 28.50 & 0.1074 & 0.1055 & 0.1046 & 0.1039 & 0.1034 & 0.1013 & 0.1057 \\\\\nSKX & 33.19 & 0.1076 & 0.1058 & 0.1047 & 0.1040 & 0.1034 & 0.1015 & 0.1061 \\\\\nHFB-17 & 36.29 & 0.1019 & 0.1002 & 0.0991 & 0.0983 & 0.0977 & 0.0951 & 0.1011 \\\\\nSGII & 37.63 & 0.0976 & 0.0951 & 0.0934 & 0.0921 & 0.0911 & 0.0857 & 0.0963 \\\\\nUNEDF1 & 40.01 & 0.1004 & 0.0978 & 0.0961 & 0.0949 & 0.0940 & 0.0896 & 0.0992 \\\\\nSk$\\chi$500 & 40.74 & 0.1005 & 0.0995 & 0.0987 & 0.0981 & 0.0977 & 0.0956 & 0.1009 \\\\\nSk$\\chi$450 & 42.06 & 0.0969 & 0.0954 & 0.0944 & 0.0937 & 0.0931 & 0.0909 & 0.0965 \\\\\nUNEDF0 & 45.08 & 0.1021 & 0.0994 & 0.0976 & 0.0963 & 0.0953 & 0.0911 & 0.1010 \\\\\nSkM* & 45.78 & 0.0980 & 0.0952 & 0.0934 & 0.0920 & 0.0910 & 0.0861 & 0.0967 \\\\\nSLy4 & 45.96 & 0.0945 & 0.0931 & 0.0921 & 0.0914 & 0.0908 & 0.0886 & 0.0942 \\\\\nSLy7 & 47.22 & 0.0931 & 0.0917 & 0.0907 & 0.0897 & 0.0894 & 0.0872 & 0.0928 \\\\\nSLy5 & 48.27 & 0.0941 & 0.0926 & 0.0916 & 0.0908 & 0.0902 & 0.0877 & 0.0938 \\\\\nSk$\\chi$414 & 51.92 & 0.1006 & 0.0989 & 0.0978 & 0.0970 & 0.0964 & 0.0939 & 0.1004 \\\\\nMSka & 57.17 & 0.0990 & 0.0967 & 0.0951 & 0.0940 & 0.0932 & 0.0892 & 0.0984 \\\\\nMSL0 & 60.00 & 0.0949 & 0.0916 & 0.0893 & 0.0877 & 0.0864 & 0.0795 & 0.0942 \\\\\nSIV & 63.50 & 0.0984 & 0.0954 & 0.0934 & 0.0919 & 0.0908 & 0.0858 & 0.0975 \\\\\nSkMP & 70.31 & 0.0915 & 0.0874 & 0.0846 & 0.0826 & 0.0810 & 0.0714 & 0.0908 \\\\\nSKa & 74.62 & 0.0940 & 0.0904 & 0.0880 & 0.0862 & 0.0849 & 0.0785 & 0.0933 \\\\\nR$_\\sigma$ & 85.69 & 0.0948 & 0.0890 & 0.0852 & 0.0825 & 0.0805 & 0.0657 & 0.0943 \\\\\nG$_\\sigma$ & 94.01 & 0.0961 & 0.0893 & 0.0851 & 0.0820 & 0.0797 & 0.0620 & 0.0957 \\\\\nSV & 96.09 & 0.0954 & 0.0898 & 0.0862 & 0.0835 & 0.0815 & 0.0702 & 0.0952 \\\\\nSkI2 & 104.33 & 0.0903 & 0.0851 & 0.0817 & 0.0791 & 0.0772 & 0.0632 & 0.0898 \\\\\nSkI5 & 129.33 & 0.0901 & 0.0846 & 0.0807 & 0.0778 & 0.0755 & 0.0595 & 0.0893 \\\\ \\hline\n\\end{tabular}\n\\caption{Values of the core-crust transition density $\\rho_t$ (fm$^{-3}$) for Skyrme interactions \ncalculated within the thermodynamical approach. The results have been computed with the exact expression of the EoS (Exact), the parabolic approximation (PA), \nor the approximations of the full EoS with Eq.~(\\ref{eq:EOSexpgeneral}) up to second ($\\delta^2$), fourth ($\\delta^4$), sixth ($\\delta^6$) \neighth ($\\delta^8$) and tenth ($\\delta^{10}$) order. The value for each interaction of the slope parameter of the symmetry energy $L$ is also included in units of MeV.}\n\\end{table}\n\n\n\\begin{table}[htb]\n\\begin{tabular}{ccccccccc}\n\\hline\n\\multicolumn{9}{c}{THERMODYNAMICAL METHOD} \\\\ \\hline\nForce & $L$ & \\begin{tabular}[c]{@{}c@{}}P$_t$\\\\ ($\\delta^2$)\\end{tabular} & \n\\begin{tabular}[c]{@{}c@{}}P$_t$\\\\ ($\\delta^4$)\\end{tabular} & \\begin{tabular}[c]{@{}c@{}}P$_t$\\\\\n($\\delta^6$)\\end{tabular} & \\begin{tabular}[c]{@{}c@{}}P$_t$\\\\ ($\\delta^8$)\\end{tabular} & \\begin{tabular}[c]{@{}c@{}}P$_t$\\\\\n($\\delta^{10}$)\\end{tabular} & \\begin{tabular}[c]{@{}c@{}}P$_t$\\\\ (Exact)\\end{tabular} \n& \\begin{tabular}[c]{@{}c@{}}P$_t$\\\\ (PA)\\end{tabular} \\\\ \\hline\\hline\nMSk7 & 9.41 & 0.4270 & 0.4404 & 0.4424 & 0.4398 & 0.4391 & 0.4366 & 0.4563 \\\\\nSIII & 9.91 & 0.3864 & 0.4376 & 0.4438 & 0.4424 & 0.4418 & 0.4386 & 0.4703 \\\\\nSkP & 19.68 & 0.6854 & 0.7106 & 0.7048 & 0.6958 & 0.6898 & 0.6681 & 0.7309 \\\\\nHFB-27 & 28.50 & 0.5702 & 0.5833 & 0.5766 & 0.5679 & 0.5620 & 0.5398 & 0.6119 \\\\\nSKX & 33.19 & 0.6708 & 0.6847 & 0.6748 & 0.6640 & 0.6568 & 0.6318 & 0.7198 \\\\\nHFB-17 & 36.29 & 0.5649 & 0.5640 & 0.5531 & 0.5419 & 0.5343 & 0.5039 & 0.5957 \\\\\nSGII & 37.63 & 0.5118 & 0.5128 & 0.4953 & 0.4781 & 0.4657 & 0.4016 & 0.5545 \\\\\nUNEDF1 & 40.01 & 0.6413 & 0.6335 & 0.6126 & 0.5939 & 0.5806 & 0.5212 & 0.6785 \\\\\nSk$\\chi$500 & 40.74 & 0.4018 & 0.3888 & 0.3807 & 0.3728 & 0.3681 & 0.3481 & 0.4099 \\\\\nSk$\\chi$450 & 42.06 & 0.5440 & 0.5384 & 0.5270 & 0.5162 & 0.5090 & 0.4812 & 0.5680 \\\\\nUNEDF0 & 45.08 & 0.7409 & 0.7277 & 0.7017 & 0.6797 & 0.6641 & 0.5991 & 0.7820 \\\\\nSkM* & 45.78 & 0.6567 & 0.6469 & 0.6226 & 0.6012 & 0.5857 & 0.5173 & 0.6982 \\\\\nSLy4 & 45.96 & 0.5274 & 0.5170 & 0.5061 & 0.4958 & 0.4888 & 0.4623 & 0.5459 \\\\\nSLy7 & 47.22 & 0.5212 & 0.5099 & 0.4987 & 0.4883 & 0.4812 & 0.4540 & 0.5388 \\\\\nSLy5 & 48.27 & 0.5361 & 0.5237 & 0.5114 & 0.5000 & 0.4923 & 0.4620 & 0.5546 \\\\\nSk$\\chi$414 & 51.92 & 0.5909 & 0.5771 & 0.5624 & 0.5495 & 0.5410 & 0.5078 & 0.6112 \\\\\nMSka & 57.17 & 0.6741 & 0.6600 & 0.6342 & 0.6122 & 0.5965 & 0.5284 & 0.7229 \\\\\nMSL0 & 60.00 & 0.6774 & 0.6408 & 0.6031 & 0.5729 & 0.5510 & 0.4412 & 0.7145 \\\\\nSIV & 63.50 & 0.7695 & 0.7569 & 0.7195 & 0.6880 & 0.6651 & 0.5682 & 0.8568 \\\\\nSkMP & 70.31 & 0.6904 & 0.6339 & 0.5827 & 0.5432 & 0.5144 & 0.3590 & 0.7344 \\\\\nSKa & 74.62 & 0.8102 & 0.7631 & 0.7139 & 0.6758 & 0.6483 & 0.5259 & 0.8671 \\\\\nR$_\\sigma$ & 85.69 & 0.8977 & 0.7663 & 0.6769 & 0.6133 & 0.5676 & 0.3024 & 0.9385 \\\\\nG$_\\sigma$ & 94.01 & 1.0270 & 0.8463 & 0.7323 & 0.6535 & 0.5974 & 0.2686 & 1.0671 \\\\\nSV & 96.09 & 0.9172 & 0.8098 & 0.7181 & 0.6523 & 0.6054 & 0.3779 & 1.0363 \\\\\nSkI2 & 104.33 & 0.8850 & 0.7410 & 0.6511 & 0.5877 & 0.5423 & 0.2845 & 0.8885 \\\\\nSkI5 & 129.33 & 1.0561 & 0.8589 & 0.7367 & 0.6522 & 0.5921 & 0.2651 & 1.0319 \\\\ \\hline\n\\end{tabular}\n\\caption{Values of the core-crust transition pressure $P_t$ (MeV fm$^{-3}$) for Skyrme interactions \ncalculated within the thermodynamical approach. The results have been computed with the exact expression of the EoS (Exact), the parabolic approximation (PA), \nor the approximations of the full EoS with Eq.~(\\ref{eq:EOSexpgeneral}) up to second ($\\delta^2$), fourth ($\\delta^4$), sixth ($\\delta^6$) \neighth ($\\delta^8$) and tenth ($\\delta^{10}$) order. The value for each interaction of the slope parameter of the symmetry energy $L$ is also included in units of MeV.}\n\\end{table}\n\n\n\n\n\\begin{table}[htb]\n\\centering\n\\resizebox{\\columnwidth}{!}{\n \\begin{tabular}{ccccccccccc}\n \\hline\n \\multicolumn{11}{c}{THERMODYNAMICAL METHOD}  \\\\    \\hline\nForce                       & D1     & D1S    & D1M    & D1N     & D250   & D260   & D280    & D300  &D1M$^*$&D1M$^{**}$ \\\\ \\hline\\hline\n$L$                         &18.36   & 22.43  & 24.83  & 33.58   & 24.90  &  17.57 & 46.53   & 25.84  & 43.18 & 33.91\\\\\\hline\n$\\delta_t^{\\delta^2}$       & 0.9215 & 0.9199 & 0.9366 & 0.9373  & 0.9167 & 0.9227 & 0.9202  & 0.9190 &0.9386 &0.9375\\\\\n$\\delta_t^{\\delta^4}$       & 0.9148 & 0.9148 & 0.9290 & 0.9336  & 0.9119 & 0.9136 & 0.9127  & 0.9128 &0.9315  & 0.9301 \\\\\n$\\delta_t^{\\delta^6}$       & 0.9127 & 0.9129 & 0.9265 & 0.9321  & 0.9101 & 0.9112 & 0.9110  & 0.9110 &  0.9292&0.9278\\\\\n$\\delta_t^{\\mathrm{exact}}$ & 0.9106 & 0.9111 & 0.9241 & 0.9310  & 0.9086 & 0.9092 & 0.9110  & 0.9096 & 0.9275 &0.9257\\\\\n$\\delta_t^\\mathrm{PA}$      & 0.9152 & 0.9142 & 0.9296 & 0.9327  & 0.9111 & 0.9153 & 0.9136  & 0.9134 & 0.9316 &0.9305 \\\\ \\hline\n\n$\\rho_t^{\\delta^2}$         & 0.1243 & 0.1141 & 0.1061 & 0.1008  & 0.1156 & 0.1228 & 0.1001  & 0.1161 & 0.0974 &0.1000\\\\\n$\\rho_t^{\\delta^4}$         & 0.1222 & 0.1129 & 0.1061 & 0.0996  & 0.1143 & 0.1198 & 0.0984  & 0.1145 &  0.095&0.0997\\\\\n$\\rho_t^{\\delta^6}$         & 0.1211 & 0.1117 & 0.1053 & 0.0984  & 0.1131 & 0.1188 & 0.0973  & 0.1136 & 0.0949 &0.0989\\\\\n$\\rho_t^{\\mathrm{exact}}$   & 0.1176 & 0.1077 & 0.1027 & 0.0942 & 0.1097 & 0.1159 & 0.0938 & 0.1109 & 0.0909 &0.0960\\\\\n$\\rho_t^\\mathrm{PA}$               & 0.1222 & 0.1160 & 0.1078 & 0.1027  & 0.1168 & 0.1171 & 0.0986  & 0.1142& 0.0940 & 0.1019\\\\ \\hline\n\n$P_t^{\\delta^2}$            & 0.6279 & 0.6316 & 0.3326 & 0.4882  & 0.7034 & 0.5892 & 0.6984  & 0.6776 & 0.3528 &0.3464 \\\\\n$P_t^{\\delta^4}$            & 0.6479 & 0.6239 & 0.3531 & 0.4676  & 0.6908 & 0.6483 & 0.7170  & 0.6998 & 0.3605 &0.3599\\\\\n$P_t^{\\delta^6}$            & 0.6452 & 0.6156 & 0.3554 & 0.4582  & 0.6811 & 0.6509 & 0.7053  & 0.6955 & 0.3575 &0.359\\\\\n$P_t^{\\mathrm{exact}}$      & 0.6184 & 0.5817 & 0.3390 & 0.4164  & 0.6464 & 0.6272 & 0.6493  & 0.6647 &0.3301  &0.3368\\\\\n$P_t^\\mathrm{PA}$                  & 0.6853 & 0.6725 & 0.3986 & 0.5173  & 0.7368 & 0.6809 & 0.7668  & 0.7356 & 0.4125 &0.4085\\\\\\hline\n\\end{tabular\n}\n\\caption{Values of the core-crust transition density $\\rho_t$ (in fm$^{-3}$) for Gogny forces\ncalculated within the thermodynamical approach and using exact expression of the EoS ($\\rho_t^\\mathrm{exact}$), the parabolic approximation ($\\rho_t^{PA}$), \nor the approximations of the full EoS with Eq.~(\\ref{eq:EOSexpgeneral}) up to second ($\\rho_t^{\\delta^2}$), fourth ($\\rho_t^{\\delta^4}$) and sixth ($\\rho_t^{\\delta^6}$) order.\nThe table includes the corresponding values of the transition pressure $P_t$ (in MeV fm$^{-3}$) and isospin asymmetry~$\\delta_t$.}\n\\label{Table-transition}\n\\end{table}\n\n\n\n\\begin{table}[htb]\n\\begin{tabular}{ccccccccc}\n\\hline\n\\multicolumn{9}{c}{DYNAMICAL METHOD} \\\\ \\hline\nForce & $L$& \\begin{tabular}[c]{@{}c@{}}$\\delta_t$ \\\\ ($\\delta^2$)\\end{tabular} & \\begin{tabular}[c]{@{}c@{}}$\\delta_t$\\\\ \n($\\delta^4$)\\end{tabular} & \\begin{tabular}[c]{@{}c@{}}$\\delta_t$\\\\ ($\\delta^6$)\\end{tabular} & \\begin{tabular}[c]{@{}c@{}}$\\delta_t$\\\\ \n($\\delta^8$)\\end{tabular} & \\begin{tabular}[c]{@{}c@{}}$\\delta_t$\\\\ ($\\delta^{10}$)\\end{tabular} & \\begin{tabular}[c]{@{}c@{}}$\\delta_t$\\\\ \n(Exact)\\end{tabular} & \\begin{tabular}[c]{@{}c@{}}$\\delta_t$\\\\ (PA)\\end{tabular} \\\\ \\hline \\hline\nMSk7 & 9.41 & 0.9312 & 0.9272 & 0.9259 & 0.9253 & 0.9250 & 0.9243 & 0.9276 \\\\\nSIII & 9.91 & 0.9272 & 0.9201 & 0.9180 & 0.9171 & 0.9167 & 0.9159 & 0.9212 \\\\\nSkP & 19.68 & 0.9274 & 0.9214 & 0.9199 & 0.9194 & 0.9191 & 0.9192 & 0.9228 \\\\\nHFB-27 & 28.50 & 0.9327 & 0.9278 & 0.9264 & 0.9259 & 0.9257 & 0.9258 & 0.9285 \\\\\nSKX & 33.19 & 0.9276 & 0.9219 & 0.9206 & 0.9202 & 0.9200 & 0.9203 & 0.9228 \\\\\nHFB-17 & 36.29 & 0.9392 & 0.9354 & 0.9344 & 0.9341 & 0.9339 & 0.9344 & 0.9357 \\\\\nSGII & 37.63 & 0.9587 & 0.9547 & 0.9538 & 0.9535 & 0.9535 & 0.9549 & 0.9553 \\\\\nUNEDF1 & 40.01 & 0.9498 & 0.9461 & 0.9455 & 0.9454 & 0.9455 & 0.9473 & 0.9463 \\\\\nSk$\\chi$500 & 40.74 & 0.9473 & 0.9452 & 0.9445 & 0.9442 & 0.9440 & 0.9440 & 0.9450 \\\\\nSk$\\chi$450 & 42.06 & 0.9381 & 0.9347 & 0.9338 & 0.9336 & 0.9335 & 0.9339 & 0.9348 \\\\\nUNEDF0 & 45.08 & 0.9453 & 0.9413 & 0.9407 & 0.9407 & 0.9409 & 0.9429 & 0.9414 \\\\\nSkM* & 45.78 & 0.9490 & 0.9449 & 0.9442 & 0.9441 & 0.9443 & 0.9462 & 0.9453 \\\\\nSLy4 & 45.96 & 0.9346 & 0.9318 & 0.9309 & 0.9306 & 0.9305 & 0.9308 & 0.9315 \\\\\nSLy7 & 47.22 & 0.9351 & 0.9323 & 0.9315 & 0.9312 & 0.9311 & 0.9315 & 0.9321 \\\\\nSLy5 & 48.27 & 0.9370 & 0.9342 & 0.9333 & 0.9331 & 0.9330 & 0.9335 & 0.9339 \\\\\nSk$\\chi$414 & 51.92 & 0.9394 & 0.9365 & 0.9358 & 0.9356 & 0.9355 & 0.9361 & 0.9364 \\\\\nMSka & 57.17 & 0.9491 & 0.9449 & 0.9442 & 0.9440 & 0.9441 & 0.9458 & 0.9450 \\\\\nMSL0 & 60.00 & 0.9592 & 0.9563 & 0.9560 & 0.9562 & 0.9565 & 0.9598 & 0.9559 \\\\\nSIV & 63.50 & 0.9490 & 0.9431 & 0.9422 & 0.9422 & 0.9424 & 0.9454 & 0.9433 \\\\\nSkMP & 70.31 & 0.9659 & 0.9635 & 0.9634 & 0.9638 & 0.9643 & 0.9691 & 0.9628 \\\\\nSKa & 74.62 & 0.9517 & 0.9480 & 0.9478 & 0.9482 & 0.9487 & 0.9530 & 0.9473 \\\\\nR$_\\sigma$ & 85.69 & 0.9699 & 0.9689 & 0.9696 & 0.9704 & 0.9712 & 0.9786 & 0.9670 \\\\\nG$_\\sigma$ & 94.01 & 0.9705 & 0.9701 & 0.9712 & 0.9723 & 0.9733 & 0.9822 & 0.9676 \\\\\nSV & 96.09 & 0.9645 & 0.9605 & 0.9607 & 0.9614 & 0.9622 & 0.9695 & 0.9590 \\\\\nSkI2 & 104.33 & 0.9694 & 0.9697 & 0.9706 & 0.9715 & 0.9723 & 0.9794 & 0.9675 \\\\\nSkI5 & 129.33 & 0.9682 & 0.9697 & 0.9711 & 0.9723 & 0.9732 & 0.9815 & 0.9668 \\\\ \\hline\n\\end{tabular}\n\\caption{Values of the core-crust transition asymmetry $\\delta_t$ for Skyrme interactions \ncalculated within the dynamical approach. The results have been computed with the exact expression of the EoS (Exact), the parabolic approximation (PA), \nor the approximations of the full EoS with Eq.~(\\ref{eq:EOSexpgeneral}) up to second ($\\delta^2$), fourth ($\\delta^4$), sixth ($\\delta^6$) \neighth ($\\delta^8$) and tenth ($\\delta^{10}$) order. The value for each interaction of the slope parameter of the symmetry energy $L$ is also included in units of MeV.}\n\\end{table}\n\n\\begin{table}[htb]\n\\begin{tabular}{ccccccccc}\n\\hline\n\\multicolumn{9}{c}{DYNAMICAL METHOD} \\\\ \\hline\nForce & $L$  & \\begin{tabular}[c]{@{}c@{}}$\\rho_t$ \\\\ ($\\delta^2$)\\end{tabular} & \n\\begin{tabular}[c]{@{}c@{}}$\\rho_t$\\\\ ($\\delta^4$)\\end{tabular} & \\begin{tabular}[c]{@{}c@{}}$\\rho_t$\\\\\n($\\delta^6$)\\end{tabular} & \\begin{tabular}[c]{@{}c@{}}$\\rho_t$\\\\ ($\\delta^8$)\\end{tabular} & \\begin{tabular}[c]{@{}c@{}}$\\rho_t$\\\\\n($\\delta^{10}$)\\end{tabular} & \\begin{tabular}[c]{@{}c@{}}$\\rho_t$\\\\ (Exact)\\end{tabular} & \\begin{tabular}[c]{@{}c@{}}$\\rho_t$\\\\ (PA)\\end{tabular} \\\\ \\hline\\hline\nMSk7 & 9.41 & 0.1184 & 0.1170 & 0.1163 & 0.1159 & 0.1157 & 0.1145 & 0.1166 \\\\\nSIII & 9.91 & 0.1163 & 0.1140 & 0.1134 & 0.1131 & 0.1128 & 0.1120 & 0.1125 \\\\\nSkP & 19.68 & 0.1065 & 0.1034 & 0.1022 & 0.1013 & 0.1007 & 0.0983 & 0.1021 \\\\\nHFB-27 & 28.50 & 0.0978 & 0.0961 & 0.0952 & 0.0945 & 0.0940 & 0.0919 & 0.0961 \\\\\nSKX & 33.19 & 0.1002 & 0.0985 & 0.0975 & 0.0968 & 0.0962 & 0.0943 & 0.0986 \\\\\nHFB-17 & 36.29 & 0.0922 & 0.0907 & 0.0897 & 0.0889 & 0.0883 & 0.0858 & 0.0914 \\\\\nSGII & 37.63 & 0.0881 & 0.0858 & 0.0843 & 0.0832 & 0.0823 & 0.0771 & 0.0867 \\\\\nUNEDF1 & 40.01 & 0.0891 & 0.0866 & 0.0850 & 0.0838 & 0.0828 & 0.0780 & 0.0881 \\\\\nSk$\\chi$500 & 40.74 & 0.0932 & 0.0923 & 0.0916 & 0.0910 & 0.0906 & 0.0886 & 0.0935 \\\\\nSk$\\chi$450 & 42.06 & 0.0889 & 0.0875 & 0.0865 & 0.0858 & 0.0853 & 0.0831 & 0.0883 \\\\\nUNEDF0 & 45.08 & 0.0910 & 0.0884 & 0.0867 & 0.0855 & 0.0845 & 0.0800 & 0.0899 \\\\\nSkM* & 45.78 & 0.0867 & 0.0843 & 0.0827 & 0.0814 & 0.0805 & 0.0756 & 0.0854 \\\\\nSLy4 & 45.96 & 0.0851 & 0.0838 & 0.0830 & 0.0823 & 0.0818 & 0.0797 & 0.0847 \\\\\nSLy7 & 47.22 & 0.0840 & 0.0828 & 0.0820 & 0.0813 & 0.0808 & 0.0786 & 0.0838 \\\\\nSLy5 & 48.27 & 0.0845 & 0.0833 & 0.0823 & 0.0816 & 0.0811 & 0.0788 & 0.0842 \\\\\nSk$\\chi$414 & 51.92 & 0.0922 & 0.0907 & 0.0897 & 0.0889 & 0.0884 & 0.0859 & 0.0919 \\\\\nMSka & 57.17 & 0.0913 & 0.0893 & 0.0879 & 0.0868 & 0.0860 & 0.0821 & 0.0906 \\\\\nMSL0 & 60.00 & 0.0839 & 0.0811 & 0.0792 & 0.0777 & 0.0766 & 0.0697 & 0.0832 \\\\\nSIV & 63.50 & 0.0883 & 0.0857 & 0.0840 & 0.0826 & 0.0816 & 0.0764 & 0.0872 \\\\\nSkMP & 70.31 & 0.0796 & 0.0763 & 0.0740 & 0.0723 & 0.0709 & 0.0615 & 0.0789 \\\\\nSKa & 74.62 & 0.0828 & 0.0798 & 0.0777 & 0.0762 & 0.0750 & 0.0685 & 0.0820 \\\\\nR$_\\sigma$ & 85.69 & 0.0833 & 0.0785 & 0.0754 & 0.0731 & 0.0713 & 0.0571 & 0.0828 \\\\\nG$_\\sigma$ & 94.01 & 0.0838 & 0.0782 & 0.0747 & 0.0721 & 0.0700 & 0.0532 & 0.0833 \\\\\nSV & 96.09 & 0.0817 & 0.0774 & 0.0745 & 0.0724 & 0.0708 & 0.0598 & 0.0811 \\\\\nSkI2 & 104.33 & 0.0777 & 0.0735 & 0.0706 & 0.0684 & 0.0668 & 0.0535 & 0.0773 \\\\\nSkI5 & 129.33 & 0.0757 & 0.0713 & 0.0682 & 0.0659 & 0.0641 & 0.0497 & 0.0752 \\\\ \\hline\n\\end{tabular}\n\\caption{Values of the core-crust transition density $\\rho_t$ (fm$^{-3}$) for Skyrme interactions \ncalculated within the dynamical approach. The results have been computed with the exact expression of the EoS (Exact), the parabolic approximation (PA), \nor the approximations of the full EoS with Eq.~(\\ref{eq:EOSexpgeneral}) up to second ($\\delta^2$), fourth ($\\delta^4$), sixth ($\\delta^6$) \neighth ($\\delta^8$) and tenth ($\\delta^{10}$) order. The value for each interaction of the slope parameter of the symmetry energy $L$ is also included in units of MeV.}\n\\end{table}\n\n\n\\begin{table}[htb]\n\\begin{tabular}{ccccccccc}\n\\hline\n\\multicolumn{9}{c}{DYNAMICAL METHOD} \\\\ \\hline\nForce & $L$ & \\begin{tabular}[c]{@{}c@{}}P$_t$\\\\ ($\\delta^2$)\\end{tabular} & \n\\begin{tabular}[c]{@{}c@{}}P$_t$\\\\ ($\\delta^4$)\\end{tabular} & \\begin{tabular}[c]{@{}c@{}}P$_t$\\\\ \n($\\delta^6$)\\end{tabular} & \\begin{tabular}[c]{@{}c@{}}P$_t$\\\\ ($\\delta^8$)\\end{tabular} & \\begin{tabular}[c]{@{}c@{}}P$_t$\\\\ \n($\\delta^{10}$)\\end{tabular} & \\begin{tabular}[c]{@{}c@{}}P$_t$\\\\ (Exact)\\end{tabular} &\n\\begin{tabular}[c]{@{}c@{}}P$_t$\\\\ (PA)\\end{tabular} \\\\ \\hline \\hline\nMSk7 & 9.41 & 0.3753 & 0.3886 & 0.3898 & 0.3873 & 0.3866 & 0.3840 & 0.4024 \\\\\nSIII & 9.91 & 0.3515 & 0.3950 & 0.4001 & 0.3986 & 0.3981 & 0.3953 & 0.4237 \\\\\nSkP & 19.68 & 0.5645 & 0.5802 & 0.5744 & 0.5665 & 0.5611 & 0.5411 & 0.5959 \\\\\nHFB-27 & 28.50 & 0.4664 & 0.4769 & 0.4717 & 0.4645 & 0.4598 & 0.4410 & 0.4992 \\\\\nSKX & 33.19 & 0.5737 & 0.5854 & 0.5772 & 0.5680 & 0.5618 & 0.5396 & 0.6138 \\\\\nHFB-17 & 36.29 & 0.4465 & 0.4469 & 0.4387 & 0.4299 & 0.4240 & 0.3988 & 0.4712 \\\\\nSGII & 37.63 & 0.3925 & 0.3955 & 0.3831 & 0.3703 & 0.3610 & 0.3095 & 0.4252 \\\\\nUNEDF1 & 40.01 & 0.4845 & 0.4774 & 0.4603 & 0.4448 & 0.4338 & 0.3803 & 0.5139 \\\\\nSk$\\chi$500 & 40.74 & 0.3237 & 0.3153 & 0.3097 & 0.3036 & 0.3001 & 0.2848 & 0.3320 \\\\\nSk$\\chi$450 & 42.06 & 0.4397 & 0.4368 & 0.4282 & 0.4197 & 0.4140 & 0.3910 & 0.4597 \\\\\nUNEDF0 & 45.08 & 0.5651 & 0.5549 & 0.5343 & 0.5165 & 0.5037 & 0.4459 & 0.5966 \\\\\nSkM* & 45.78 & 0.4914 & 0.4860 & 0.4685 & 0.4526 & 0.4409 & 0.3844 & 0.5214 \\\\\nSLy4 & 45.96 & 0.4092 & 0.4033 & 0.3958 & 0.3882 & 0.3831 & 0.3625 & 0.4245 \\\\\nSLy7 & 47.22 & 0.4068 & 0.4001 & 0.3922 & 0.3845 & 0.3792 & 0.3580 & 0.4215 \\\\\nSLy5 & 48.27 & 0.4124 & 0.4053 & 0.3968 & 0.3885 & 0.3829 & 0.3595 & 0.4277 \\\\\nSk$\\chi$414 & 51.92 & 0.4731 & 0.4644 & 0.4535 & 0.4435 & 0.4368 & 0.4095 & 0.4906 \\\\\nMSka & 57.17 & 0.5304 & 0.5233 & 0.5042 & 0.4872 & 0.4750 & 0.4182 & 0.5701 \\\\\nMSL0 & 60.00 & 0.4834 & 0.4614 & 0.4357 & 0.4142 & 0.3985 & 0.3117 & 0.5112 \\\\\nSIV & 63.50 & 0.5609 & 0.5587 & 0.5335 & 0.5109 & 0.4941 & 0.4151 & 0.6247 \\\\\nSkMP & 70.31 & 0.4592 & 0.4283 & 0.3961 & 0.3700 & 0.3508 & 0.2353 & 0.4903 \\\\\nSKa & 74.62 & 0.5700 & 0.5434 & 0.5104 & 0.4837 & 0.4641 & 0.3661 & 0.6103 \\\\\nR$_\\sigma$ & 85.69 & 0.6025 & 0.5221 & 0.4634 & 0.4203 & 0.3892 & 0.1925 & 0.6334 \\\\\nG$_\\sigma$ & 94.01 & 0.6715 & 0.5613 & 0.4876 & 0.4353 & 0.3979 & 0.1611 & 0.7020 \\\\\nSV & 96.09 & 0.5487 & 0.5044 & 0.4536 & 0.4145 & 0.3860 & 0.2279 & 0.6238 \\\\\nSkI2 & 104.33 & 0.5462 & 0.4616 & 0.4071 & 0.3676 & 0.3391 & 0.1623 & 0.5536 \\\\\nSkI5 & 129.33 & 0.5978 & 0.4891 & 0.4224 & 0.3752 & 0.3416 & 0.1403 & 0.5917 \\\\ \\hline\n\\end{tabular}\n\\caption{Values of the core-crust transition pressure $P_t$ (MeV fm$^{-3}$) for Skyrme interactions \ncalculated within the dynamical approach. The results have been computed with the exact expression of the EoS (Exact), the parabolic approximation (PA), \nor the approximations of the full EoS with Eq.~(\\ref{eq:EOSexpgeneral}) up to second ($\\delta^2$), fourth ($\\delta^4$), sixth ($\\delta^6$) \neighth ($\\delta^8$) and tenth ($\\delta^{10}$) order. The value for each interaction of the slope parameter of the symmetry energy $L$ is also included in units of MeV.}\n\\end{table}\n\n\n\\begin{table}[htb]\n\\centering\n\\begin{tabular}{llcccc}\n\\hline\n\\multicolumn{6}{c}{DYNAMICAL METHOD} \\\\ \\hline\n\\multicolumn{2}{c}{Force} & $L$ & $\\delta_t$ & $\\rho_t$ & $P_t$ \\\\ \\hline\\hline\n\\multicolumn{1}{c|}{\\multirow{10}{*}{Gogny}} & D1 & 18.36 & 0.9137 & 0.1045 & 0.5070 \\\\\n\\multicolumn{1}{c|}{} & D1S & 22.43 & 0.9145 & 0.0951 & 0.4723 \\\\\n\\multicolumn{1}{c|}{} & D1M & 24.83 & 0.9257 & 0.0949 & 0.2839 \\\\\n\\multicolumn{1}{c|}{} & D1N & 33.58 & 0.9345 & 0.0847 & 0.3280 \\\\\n\\multicolumn{1}{c|}{} & D250 & 24.90 & 0.9121 & 0.0987 & 0.5382 \\\\\n\\multicolumn{1}{c|}{} & D260 & 17.57 & 0.9126 & 0.1044 & 0.5188 \\\\\n\\multicolumn{1}{c|}{} & D280 & 46.53 & 0.9181 & 0.0841 & 0.5046 \\\\\n\\multicolumn{1}{c|}{} & D300 & 25.84 & 0.9135 & 0.1013 & 0.5547 \\\\\n\\multicolumn{1}{c|}{} & D1M* & 43.18 & 0.9300 & 0.0838 & 0.2702 \\\\\n\\multicolumn{1}{c|}{} & D1M** & 33.91 & 0.9279 & 0.0886 & 0.2786 \\\\ \\hline\n\\multicolumn{1}{c|}{\\multirow{15}{*}{MDI}} & $x=-1.4$ & 123.98 & 0.9967 & 0.0331 & -0.0299 \\\\\n\\multicolumn{1}{c|}{} & $x=-1.2$ & 114.86 & 0.9950 & 0.0344 & -0.0144 \\\\\n\\multicolumn{1}{c|}{} & $x=-1$ & 105.75 & 0.9925 & 0.0364 & 0.0045 \\\\\n\\multicolumn{1}{c|}{} & $x=-0.8$ & 96.63 & 0.9889 & 0.0393 & 0.0286 \\\\\n\\multicolumn{1}{c|}{} & $x=-0.6$ & 87.51 & 0.9841 & 0.0430 & 0.0600 \\\\\n\\multicolumn{1}{c|}{} & $x=-0.4$ & 78.40 & 0.9778 & 0.0476 & 0.0991 \\\\\n\\multicolumn{1}{c|}{} & $x=-0.2$ & 69.28 & 0.9701 & 0.0526 & 0.1445 \\\\\n\\multicolumn{1}{c|}{} & $x=0$ & 60.17 & 0.9612 & 0.0579 & 0.1936 \\\\\n\\multicolumn{1}{c|}{} & $x=0.2$ & 51.05 & 0.9515 & 0.0636 & 0.0467 \\\\\n\\multicolumn{1}{c|}{} & $x=0.4$ & 41.94 & 0.9411 & 0.0697 & 0.0480 \\\\\n\\multicolumn{1}{c|}{} & $x=0.6$ & 32.82 & 0.9303 & 0.0768 & 0.0483 \\\\\n\\multicolumn{1}{c|}{} & $x=0.8$ & 23.70 & 0.9194 & 0.0855 & 0.0472 \\\\\n\\multicolumn{1}{c|}{} & $x=1$ & 14.59 & 0.9089 & 0.0980 & 0.0442 \\\\\n\\multicolumn{1}{c|}{} & $x=1.1$ & 10.03 & 0.9044 & 0.1082 & 0.0410 \\\\\n\\multicolumn{1}{c|}{} & $x=1.15$ & 7.75 & 0.9019 & 0.1160 & 0.0382 \\\\ \\hline\n\\end{tabular}\n\\caption{Values of the core-crust transition asymmetry $\\delta_t$, transition density $\\rho_t$ (fm$^{-3}$) and transition pressure $P_t$ (MeV fm$^{-3}$) for a set of \nGogny and MDI interactions \ncalculated within the dynamical approach. The results have been computed with the exact expression of the EoS.\nThe value for each interaction of the slope parameter of the symmetry energy $L$ is also included in units of MeV.}\n\\end{table}\n\n\n\\begin{table}[htb]\n\\centering\n\\begin{tabular}{lccc}\n\\hline\n\\multicolumn{4}{c}{DYNAMICAL METHOD} \\\\ \\hline\nForce & $\\delta_t$ & $\\rho_t$ & $P_t$ \\\\ \\hline\\hline\nSEI $L=34$ MeV & 0.9132 & 0.0923 & 0.5508 \\\\\nSEI $L=39$ MeV & 0.9183 & 0.0889 & 0.5327 \\\\\nSEI $L=45$ MeV& 0.9251 & 0.0848 & 0.5056 \\\\\nSEI $L=51$ MeV& 0.9326 & 0.0805 & 0.4719 \\\\\nSEI $L=58$ MeV& 0.9404 & 0.0763 & 0.4319 \\\\\nSEI $L=65$ MeV& 0.9477 & 0.0724 & 0.3902 \\\\\nSEI $L=67$ MeV& 0.9505 & 0.0709 & 0.3732 \\\\\nSEI $L=71$ MeV& 0.9543 & 0.0688 & 0.3484 \\\\\nSEI $L=75$ MeV& 0.9586 & 0.0664 & 0.3187 \\\\\nSEI $L=77$ MeV& 0.9611 & 0.0649 & 0.3007 \\\\\nSEI $L=82$ MeV& 0.9661 & 0.0619 & 0.2632 \\\\\nSEI $L=86$ MeV& 0.9705 & 0.0591 & 0.2276 \\\\\nSEI $L=89$ MeV& 0.9727 & 0.0575 & 0.2085 \\\\\nSEI $L=92$ MeV& 0.9760 & 0.0552 & 0.1802 \\\\\nSEI $L=96$ MeV& 0.9789 & 0.0530 & 0.1548 \\\\\nSEI $L=100$ MeV& 0.9824 & 0.0501 & 0.1223 \\\\\nSEI $L=105$ MeV& 0.9857 & 0.0471 & 0.0909 \\\\\nSEI $L=111$ MeV& 0.9895 & 0.0434 & 0.0549 \\\\\nSEI $L=115$ MeV& 0.9913 & 0.0415 & 0.0373 \\\\ \\hline\n\\end{tabular}\n\\caption{Values of the core-crust transition asymmetry $\\delta_t$, transition density $\\rho_t$ (fm$^{-3}$) and transition pressure $P_t$ (MeV fm$^{-3}$) for a set of \nSEI interactions of $\\gamma=1\/2$ and different slope of the symmetry energy\ncalculated within the dynamical approach. The results have been computed with the exact expression of the EoS.\nThe value for each interaction of the slope parameter of the symmetry energy $L$ is also included in units of MeV.}\n\\end{table}\n\n\\chapter{Extended Thomas-Fermi approximation with finite-range forces}\\label{app_vdyn}\nIn this Appendix we find the contribution to the surface part of the curvature matrix in the dynamical method (Chapter~\\ref{chapter4}) to find the core-crust transition. \nThe contribution coming from the exchange and kinetic parts of the interaction are found using a density matrix (DM) expansion in the \nExtended-Thomas-Fermi (ETF) approximation. To find the contributions coming from the direct part of the interaction we perform \nan expansion of the direct energy in terms of the gradients of the nuclear derivatives, which \nin momentum space can be summed at all orders. \n\n\nThe total energy density provided by a finite-range density-dependent effective nucleon-nucleon \ninteraction can be decomposed as \n\\begin{equation}\n \\mathcal{H} = \\mathcal{H}_{kin} +  \\mathcal{H}_{zr} + \\mathcal{H}_{dir} + \\mathcal{H}_{exch}+ \n \\mathcal{H}_{Coul}+\\mathcal{H}_{LS},\n \\label{eqendens}\n\\end{equation} \nwhere $\\mathcal{H}_{kin}$, $\\mathcal{H}_{zr}$, $\\mathcal{H}_{dir}$, $\\mathcal{H}_{exch}$, \n$\\mathcal{H}_{Coul}$ and $\\mathcal{H}_{LS}$ are the\n kinetic, zero-range, finite-range direct, finite-range exchange, Coulomb and spin-orbit contributions.\nThe finite-range part of the interaction can be written in a general way as \n\\begin{equation}\\label{eqVfin}\nV({\\bf r},{\\bf r'}) = \\sum_i \\left(W_i + B_i P_{\\sigma} - \nH_i P_{\\tau} -+\nM_i P_{\\sigma} P_{\\tau}\\right) v_i({\\bf r} , {\\bf r'}),\\\\\n\\end{equation}\nwhere $P^{\\sigma}$ and $P^{\\tau}$ are the spin and isospin exchange operators and $v_i({\\bf r} , {\\bf r'})$ \nare the form factors of the force. \nFor Gaussian-type interactions the form factor is\\footnote{Notice that the notation \n of the range in the Gaussian form factor for Gogny interactions has changed from $\\mu$ in Chapter~\\ref{chapter1} to $\\alpha$ to not confuse it with the \n corresponding range parameter of the Yukawa form factors found in the SEI and MDI interactions.}\n\\begin{equation}\n v_i({\\bf r} , {\\bf r'})=e^{-|{\\bf r} - {\\bf r'}|^2\/\\alpha_i^2} ,\n\\end{equation}\n while for a Yukawa force one has\n \\begin{equation}\n  v_i({\\bf r} , {\\bf r'})=\\frac{e^{-\\mu_i |{\\bf r} - {\\bf r'}|}}{\\mu_i |{\\bf r} - {\\bf r'}|}.\n \\end{equation}\n Let us remind that the form factor of the SEI interaction, which we have used to present all of \n the expressions of Yukawa type functionals, has an additional parameter $\\mu_i$ in the denominator \n compared with the usual expression of the MDI interaction in the literature.\n Hence, the Yukawa form factor for MDI interactions in the literature is usually written as\n \\begin{equation}\n  v_i({\\bf r} , {\\bf r'})=\\frac{e^{-\\mu_i |{\\bf r}   - {\\bf r'}|}}{|{\\bf r} - {\\bf r'}|}.\n \\end{equation}\n It is important to take into account this fact when using our expressions for the MDI interactions. \n \nThe finite-range term provides the direct ($\\mathcal{H}_{dir}$) and exchange ($\\mathcal{H}_{exch}$) \ncontributions to the total energy density in Eq.~(\\ref{eqendens}).\nThe HF energy due to the finite-range part of the interaction, neglecting zero-range, Coulomb and spin-orbit contributions, reads \n\\begin{eqnarray}\nE_{HF} &=& \\sum_q \\int d{\\bf r} \\left[\\frac{\\hbar^2}{2m}\\tau({\\bf r})\n+ {\\cal H}_{dir} + {\\cal H}_{exch} \\right]_q= \\sum_q \\int d{\\bf r}\\left[\\frac{\\hbar^2}{2m}\\tau({\\bf r})\\right. \\nonumber\\\\\n&+& \\left.\\frac{1}{2}\\rho({\\bf r})V^{H}({\\bf r})\n+ \\frac{1}{2}\\int d{\\bf r'} V^{F}({\\bf r},{\\bf r'})\\rho \\left({\\bf r} ,{\\bf r'}\\right)\n\\right]_q,\n\\label{eq2}\n\\end{eqnarray}\nwhere the subscript $q$ refers to each kind of nucleon. In this equation $\\rho({\\bf r})$ and $\\tau({\\bf r})$ \nare the particle and kinetic energy densities, respectively, and $\\rho \\left({\\bf r} ,{\\bf r'}\\right)$ \nis the one-body density matrix.\nThe direct ($V^H$) and exchange ($V^F$) contributions to the HF potential due to the finite-range interaction (\\ref{eqVfin}) are \ngiven by \n\\begin{equation}\\label{eq3}\nV_q^H ({\\bf r}) = \\sum_i \\int d{\\bf r'} v_i ({\\bf r} , {\\bf r'}) \\left[ D_{L, dir}^i \\rho_q ({\\bf r'}) + D_{U,dir}^i \\rho_{q'} ({\\bf r'})\\right]\n\\end{equation}\nand\n\\begin{eqnarray}\nV_q^F({\\bf r}, {\\bf r'})&=& - \\sum_i v_i ({\\bf r} , {\\bf r'}) \\left[ D_{L, exch}^i \\rho_q ({\\bf r} ,{\\bf r'})\n+ D_{U,exch}^i \\rho_{q'} ({\\bf r},{\\bf r'}) \\right],\n\\label{eq4}\n\\end{eqnarray}\nrespectively. In Eqs.~(\\ref{eq3}) and (\\ref{eq4}) the coefficients $D_{L, dir}^i$, $D_{U,dir}^i$, $D_{L, exch}^i$ and $D_{U,exch}^i$ \n are the usual contributions of the spin\nand isospin strengths for the like and unlike nucleons in the direct and exchange potentials:\n\\begin{eqnarray}\n&&D_{L,dir}^i = W_i + \\frac{B_i}{2} - H_i - \\frac{M_i}{2} \\nonumber \\\\\n&&D_{U,dir}^i= W_i + \\frac{B_i}{2}\\nonumber\\\\\n&&D_{L,exch}^i=M_i + \\frac{H_i}{2} - B_i - \\frac{W_i}{2}\\nonumber\\\\\n&&D_{U,exch}^i=M_i + \\frac{H_i}{2}.\n\\end{eqnarray}\n\nThe non-local one-body DM $ \\displaystyle\\rho({\\bf r},{\\bf r'})=\\sum_k \\varphi_k^*({\\bf r})\\varphi_k ({\\bf r'})$\nplays an essential role in Hartree-Fock (HF) calculations using effective finite-range forces,\nwhere its full knowledge is needed. \nThe HFB theory with finite-range forces is well\nestablished from a theoretical point of view \\cite{decharge80,nakada03} and calculations in\nfinite nuclei are feasible nowadays with a reasonable computing time. However, these calculations \nare still complicated and usually require specific codes, as for example the one provided by Ref.~\\cite{robledo02}. \nTherefore, approximate methods based on the expansion of the DM in terms of local densities and their gradients usually allow one to\nreduce the non-local energy density to a local form.\n\nThe simplest approximation to the DM is to replace locally its quantal value by its expression \nin nuclear matter, i.e., the so-called Slater or Thomas-Fermi (TF) approximation. A more elaborated\ntreatment was developed by Negele and Vautherin \\cite{negele72a, negele72b}, which expanded the DM into a\nbulk term (Slater) plus a corrective contribution that takes into account the finite-size effects.\nCampi and Bouyssy \\cite{campi78a,campi78b} proposed another approximation consisting of a Slater term alone but \nwith an effective Fermi momentum, which partially resummates the finite-size corrective terms.\nLater on, Soubbotin and Vi\\~nas developed the Extended Thomas-Fermi (ETF) approximation \nof the one-body DM in the case of a non-local single-particle Hamiltonian \\cite{soubbotin00}. \nThe ETF DM includes, on top of the Slater part, corrections of $\\hbar^2$ order, which are\nexpressed through second-order derivatives of the neutron and proton densities. \nIn the same Ref.~\\cite{soubbotin00} the similarities and differences\nwith previous DM expansions \\cite{negele72a, negele72b,campi78a,campi78b} are discussed in detail.\nThe ETF approximation to the HF energy for non-local potentials\nconsists of replacing the quantal density matrix by the semiclassical one that contains, in\naddition to the bulk (Slater) term, corrective terms depending on the second-order derivatives\nof the proton and neutron densities, which account for contributions up to $\\hbar^2$-order. \nThe angle-averaged semiclassical ETF density matrix, derived in Refs.~\\cite{centelles98,gridnev98,soubbotin00}, \nfor each kind of nucleon reads \n\\begin{eqnarray}\n{\\tilde \\rho}({\\bf R},s) &=&=\\rho_U ({\\bf R},s) +\\rho_2 ({\\bf R},s) =  \\frac{3j_1(k_F s)}{k_f s}\\rho \\nonumber\\\\\n&+& \\frac{s^2}{216}\\left\\{\\left[\\left(9 - 2k_F\\frac{f_k}{f}\n- 2k_F^2\\frac{f_{kk}}{f} +  k_F^2\\frac{f_k^2}{f}\\right)\\frac{j_1(k_F s)}{k_F s} -4 j_0(k_F s)\\right]\n\\frac{({\\bf \\nabla}\\rho)^2}{\\rho}\\right. \\nonumber \\\\\n&-&\\left.\\left[\\left(18 +  6k_F\\frac{f_k}{f}\\right) \\frac{j_1(k_F s)}{k_F s} - 3j_0(k_F s)\\right]\\Delta \\rho \n-\\left[18\\rho \\frac{\\Delta f}{f} \\right.\\right.\\nonumber \\\\\n&+& \\left.\\left. \\left(18 - 6k_F \\frac{f_k}{f}\\right)\\frac{{\\bf \\nabla}\\rho \\cdot {\\bf \\nabla} f}{f}\n+12 k_F \\frac{{\\bf \\nabla}\\rho\\cdot {\\bf \\nabla} f_k}{f} - 9\\rho \\frac{({\\bf \\nabla} f)^2}{f} \\right]\n\\frac{j_1(k_F s)}{k_F s} \\right\\}, \\nonumber \\\\\n\\label{eqA1}\n\\end{eqnarray}\nwhere $\\rho_U$ is the uniform Slater bulk term, $\\rho_2$ contains the ETF gradient corrections of order $\\hbar^2$ \nto the DM, and ${\\bf R}=({\\bf r}+{\\bf r'})\/2$ and ${\\bf s}={\\bf r}-{\\bf r'}$ are the center of mass and relative \ncoordinates of the two nucleons, respectively.\nMoreover, in the r.h.s. of~(\\ref{eqA1}), $\\rho=\\rho ({\\bf R})$ is the local density, $k_F=(3 \\pi^2 \\rho({\\bf R}))^{1\/3}$ is the corresponding Fermi momentum for each type of nucleon, and $j_1(k_F s)$\nis the $l=1$ spherical Bessel function. In Eq.~(\\ref{eqA1}), $f= \\left.f({\\bf R},k)\\right|_{k=k_F}$ is the inverse of the position and momentum\ndependent effective mass, defined for each type of nucleon as\n\\begin{equation}\nf({\\bf R},k) = \\frac{m}{m^{*}({\\bf R},k)}= 1 + \\frac{m}{\\hbar^2k}\\frac{\\partial V^{F}_{0}({\\bf R},k)}{\\partial k}.\n\\label{eqA2}\n\\end{equation} \nThe value of $f$ in Eq.~(\\ref{eqA1}) is computed at $k=k_{F}$ (see below), and $f_{k}= \\left.f_{k}({\\bf R},k)\\right|_{k=k_F}$ and \n$f_{kk}= \\left.f_{kk}({\\bf R},k)\\right|_{k=k_F}$ denote its first and second\nderivatives with respect to $k$.\nIn Eq.~(\\ref{eqA2}), $V^{F}_{0}({\\bf R},k)$ is the Wigner transform of the exchange potential \n(\\ref{eq4}) given by\n\\begin{eqnarray}\\label{eqA3} \nV^{F}_0({\\bf R},k) = \\int d{\\bf s} V^{F}_0({\\bf R},s)e^{-i{\\bf k}\\cdot{\\bf s}} \n= \\int d{\\bf s} V_0^{F} ({\\bf R},s) j_0 (k,s),\n\\end{eqnarray}\nAt TF level, computed with\nEq.~(\\ref{eq4}) using the Slater DM (\\ref{eqA1}), it can be rewritten as\n\\begin{eqnarray}\nV^{F}_{0}({\\bf R},k) &=& - \\sum_i \\left[ D_{L,exch}^i \\int d{\\bf s}v(s)\\frac{3j_1(k_{Fq} s)}{k_{Fq} s}\\rho_q j_0(k s)\\right.\\nonumber\\\\\n&+& \\left.D_{U,exch}^i \\int d{\\bf s}v(s)\\frac{3j_1(k_{Fq'} s)}{k_{Fq'} s}\\rho_{q'} j_0(k s)\\right].\n\\label{eqA9}\n  \\end{eqnarray}\nNotice that due to the structure of the exchange potential (\\ref{eqA9}), the space dependence of the effective mass for each \nkind of nucleon is through the Fermi momenta of both type of nucleon, neutrons and protons,\n i.e., $ f_q=f_q(k,k_{Fn}({\\bf R}),k_{Fp}({\\bf R}))$ ($q=n,p$).\nWhen the inverse \neffective mass and its derivatives with respect to $k$ are used in (\\ref{eqA1}), an additional space\ndependence arises from the replacement of the momentum $k$ by the local Fermi momentum\n$k_F({\\bf R})$.\n \nUsing the DM (\\ref{eqA1}), the explicit form of the semiclassical kinetic energy at the ETF level for either neutrons or protons \ncan be written as:\n\\begin{eqnarray}\n\\tau_{ETF}({\\bf R}) &=& \\left.\\left(\\frac{1}{4}\\Delta_R - \\Delta_s\\right) {\\tilde \\rho}({\\bf R},s)\\right|_{s=0} =\n\\tau_{0} + \\tau_{2},\n\\label{eqA4}\n\\end{eqnarray}\nwhich consists of the well-known TF term\n\\begin{equation}\n \\tau_{0}=\\frac{3}{5}k_{F}^2\\rho,\n\\end{equation}\nplus the $\\hbar^2$ contribution\n\\begin{eqnarray}\n\\tau_{2}({\\bf R}) &=& \n\\frac{1}{36}\\frac{({\\bf \\nabla} \\rho)^2}{\\rho} \\left[ 1 + \\frac{2}{3}k_F \\frac{f_k}{f} +\n\\frac{2}{3}k_F^2 \\frac{f_{kk}}{f}- \\frac{1}{3}k_F^2 \\frac{f_k^2}{f^2} \\right] + \n\\frac{1}{12}\\Delta \\rho \\left[4 + \\frac{2}{3}k_F \\frac{f_k}{f} \\right]\\nonumber \\\\\n&+&  \\frac{1}{6}\\rho \\frac{\\Delta f}{f}+ \\frac{1}{6}\\frac{{\\bf \\nabla}\\rho \\cdot {\\bf \\nabla}f}{f}\n\\left[1 - \\frac{1}{3}k_F \\frac{f_k}{f}\\right] + \\frac{1}{9}\\frac{{\\bf \\nabla}\\rho \\cdot {\\bf \\nabla}f_k}{f}\n- \\frac{1}{12}\\rho \\frac{({\\bf \\nabla}f)^2}{f^2}.\n\\label{eqA41}\n\\end{eqnarray}\nThis $\\hbar^2$ contribution reduces to the standard $\\hbar^2$\nexpression for local forces \\cite{skms} if the effective mass depends only on the \nposition and not on the momentum.\n\nCollecting all pieces together, the HF energy~(\\ref{eq2}) at ETF\nlevel calculated using the DM given by Eq.~(\\ref{eqA1}) reads\n\\begin{eqnarray}\nE_{HF}^{ETF} &=& \\sum_{q} \\int d{\\bf R}\\left[\\frac{\\hbar^2}{2m}\\frac{3}{5}(3\\pi^2)^{2\/3}\\rho^{5\/3}\\right. \\nonumber\\\\\n&+& \\left.\\frac{1}{2}\\rho({\\bf R})V^{H}({\\bf R}) + \\frac{1}{2} \\int d{\\bf s}  \\rho_U ({\\bf R}, {\\bf s}) V_0^F ({\\bf R}, {\\bf s}) \n+ \\frac{\\hbar^2 \\tau_2({\\bf R})}{2m} + {\\cal H}_{exch,2}({\\bf R})\\right]_q,\\nonumber\\\\\n\\label{eq5}\n\\end{eqnarray}\nwhere ${\\cal H}_{exch,2}({\\bf R})$ \nis the $\\hbar^2$ contribution to the exchange energy. The contributions from the exchange and kinetic energies \nto the surface term of the curvature matrix~(\\ref{eq12}) in Chapter~\\ref{chapter4}\nwill come from the $\\tau_2$ and ${\\cal H}_{exch,2}({\\bf R})$, respectively.\n\nThe exchange energy density, which is local within the ETF approximation, is obtained from the exchange potential (\\ref{eqA3}) and\nis given by\n\\begin{eqnarray}\n{\\cal H}_{exch}({\\bf R}) = \n\\frac{1}{2 }\\int d{\\bf s} \\rho_0({\\bf R},s) V_0^F ({\\bf R},s) +\n\\int d{\\bf s} \\rho_2({\\bf R},s) V^{F}_{0}({\\bf R},s),\n\\label{eqA5}\n\\end{eqnarray}\nfrom where, and after some algebra explained in detail in Ref.~\\cite{soubbotin00}, one can recast the $\\hbar^2$ \ncontribution to the exchange energy for each kind of nucleon as\n\\begin{equation}\n{\\cal H}_{exch,2}({\\bf R}) = \n \\frac{\\hbar^2}{2m} \\left[(f-1)\\left(\\tau_{ETF}-\\frac{3}{5}k_F^2 \\rho - \\frac{1}{4}\\Delta \\rho \\right) \n+ k_F f_k \\left(\\frac{1}{27}\\frac{({\\bf \\nabla} \\rho)^2}{\\rho} - \\frac{1}{36}\\Delta \\rho\\right)\\right].\n\\label{eqA6}\n\\end{equation}\nNotice that $\\tau_{ETF}-\\frac{3}{5}k_{F}^2\\rho=\\tau_{2}$ and that \n$\\Delta \\rho$ vanishes under integral sign if spherical symmetry in coordinate space is assumed. \n\nFrom Eq.~(\\ref{eq5}) we see that the energy in the ETF approximation for finite-range forces consists of a \npure TF part, which depends only on the local densities of each type of particles, plus additional $\\hbar^2$ \ncorrections coming from the $\\hbar$-expansion of the kinetic and exchange energy densities, which depend \non the local Fermi momentum $k_F$ and on second-order derivatives of the nuclear density:\n\\begin{eqnarray}\n\\int d{\\bf R}\\left[{\\cal H}_{kin}({\\bf R}) +  {\\cal H}_{exch,2}({\\bf R})\\right] &=& \n\\frac{\\hbar^2}{2m}\\int d{\\bf R}\\left\\{\\tau_{0} +  \n \\left[f\\tau_{2} - \\frac{1}{4}f \\Delta \\rho \\right.\\right.\\nonumber\\\\\n&+& \\left.\\left. k_F f_k \\left(\\frac{1}{27}\n\\frac{({\\bf \\nabla} \\rho)^2}{\\rho} - \\frac{1}{36}\\Delta \\rho\\right)\\right]\\right\\}.\n\\label{eqA7}\n\\end{eqnarray}\nThe full $\\hbar^2$ contribution to the total energy corresponding to the finite-range central interaction \n(\\ref{eqVfin}), given by Eq.~(\\ref{eqA7}) for each kind of nucleon, can be written, after partial integration, as:\n\\begin{eqnarray}\n&&\\int d{\\bf R}\\bigg[\\frac{\\hbar^2}{2m}\\tau_2({\\bf R}) + {\\cal H}_{exch,2}({\\bf R})\\bigg]_q =\\\\\n&&\\int d{\\bf R}\\bigg[B_{nn}(\\rho_{n},\\rho_{p})\\big({\\bf \\nabla}\\rho_n\\big)^2\n+ B_{pp}(\\rho_{n},\\rho_{p})\\big({\\bf \\nabla}\\rho_p\\big)^2\n+ 2 B_{np}(\\rho_{n},\\rho_{p}){\\bf \\nabla}\\rho_n\\cdot{\\bf \\nabla}\\rho_p\\bigg],\\nonumber\n\\label{eqA10}\n\\end{eqnarray}\nwhere the like coefficients of the gradients of the densities are\n\\begin{eqnarray}\nB_{nn}(\\rho_n,\\rho_p) &=& \\frac{\\hbar^2}{2m} \\frac{1}{108}\\left\\{\\left[3 f_n + k_{Fn}(2f_{nk} - 3 f_{nk_{Fn}})\n+  k^2_{Fn}(5f_{nkk} + 3 f_{nkk_{Fn}})  \\right. \\right.\n\\nonumber \\\\\n&-& \\left. \\left. k^2_{Fn}\\frac{(2f_{nk} +  f_{nk_{Fn}})^2}{f_n}\\right]\\frac{1}{\\rho_n}\n- \\frac{\\rho_p}{\\rho_n^2}k^2_{Fn}\\frac{f^2_{pk_{Fn}}}{f_p}\\right\\}\n\\label{eqA11}\n\\end{eqnarray}\nand a similar expression for $B_{pp}(\\rho_n,\\rho_p)$ obtained by exchanging $n$ by $p$ in Eq.~(\\ref{eqA11}). The \nunlike coefficient  $B_{np}(\\rho_n,\\rho_p)$ of Eq.~(\\ref{eqA10}) reads:\n\\begin{eqnarray}\nB_{np}(\\rho_n,\\rho_p)&=& B_{pn}(\\rho_p,\\rho_n)= - \\frac{\\hbar^2}{2m} \\frac{1}{316}\\left\\{\\left[ \\vphantom{ \\frac{2 k_{Fn} k_{Fp}(2f_{nk} + f_{nk_{Fn}})f_{nk_{Fp}}}{f_n}} 3 k_{Fp}f_{nk_{Fp}}\n- 3 k_{Fn} k_{Fp}f_{nkk_{Fp}} \\right.\\right.\n\\nonumber\\\\\n&+&  \\left.\\left.\\frac{2 k_{Fn} k_{Fp}(2f_{nk} + f_{nk_{Fn}})f_{nk_{Fp}}}{f_n}\\right]\n\\frac{1}{\\rho_p}\\right.\n\\nonumber \\\\ \n&+& \\left.\\left[3 k_{Fn}f_{pk_{Fn}} - 3 k_{Fp} k_{Fn}f_{pkk_{Fn}} +  \n\\frac{2 k_{Fp} k_{Fn}(2f_{pk} + f_{pk_{Fp}})f_{pk_{Fn}}}{f_p}\\right]\\frac{1}{\\rho_n}\\right\\}.\\nonumber\\\\\n\\label{eqA12}\n\\end{eqnarray}\nAs stated before, all derivatives of the neutron (proton) inverse effective mass $f_q (k, k_{Fq}, k_{Fq'})$\nwith respect to the momentum $k$, $f_{qk}( k_{Fq}, k_{Fq'})$, are \nevaluated at the neutron (proton) Fermi momentum $k_{Fq}$, i.e.\n$f_{qk}  (k_{Fq}, k_{Fq'}) =\\left.\\frac{\\partial f_q (k, k_{Fq}, k_{Fq'})}{\\partial k}\\right|_{k=k_{Fq}}$,\n$f_{qkk}  (k_{Fq}, k_{Fq'}) =\\left.\\frac{\\partial^2 f_q (k, k_{Fq}, k_{Fq'})}{\\partial k^2}\\right|_{k=k_{Fq}}$,\n$f_{qkk_{Fq'}}  (k_{Fq}, k_{Fq'}) =\\left.\\frac{\\partial^2 f_q (k, k_{Fq}, k_{Fq'})}{\\partial k \\partial k_{Fq'}}\\right|_{k=k_{Fq}}$, etc.\n\n\nWe proceed to derive the direct term in Eq.~(\\ref{eq9}) in Section~\\ref{Theory_dyn} of Chapter~\\ref{chapter4} due to the fluctuating density.\nLet us first obtain the gradient expansion of the direct energy coming from the finite-range part of the force. For the sake of \nsimplicity, we consider a single Wigner term. The result for the case including spin and isospin exchange operators can be obtained analogously.\nIn the case of a Wigner term we have \n\\begin{equation}\\label{eq:Edir}\n E_{dir} = \\frac{1}{2}\\int d{\\bf R} d{\\bf s} \\rho ({\\bf R}) \\rho({\\bf R}-{\\bf s}) v ({\\bf s}).\n\\end{equation}\nFollowing the procedure of \nRef.~\\cite{durand93}, a central finite-range interaction can be expanded in a series of distributions as follows:  \n\\begin{equation}\nv({s}) = \\sum_{n=0}^{\\infty} c_{2n}\\nabla^{2n} \\delta ({\\bf s}),\n\\label{eqB2}\n\\end{equation}\nwhere the coefficients $c_{2n}$ are chosen in such a way \nthat the expansion (\\ref{eqB2}) gives the same moments of the interaction $v$($s$). \nThis implies that \\cite{durand93}\n\\begin{equation}\nc_{2n} = \\frac{1}{(2n+1)!} \\int d{\\bf s} s^{2n} v({s}),\n\\label{eqB3}\n\\end{equation}\nwhich allows one to determine the values of the coefficients $c_{2n}$ for any value of $n$.\nUsing this expansion, the direct energy (\\ref{eq:Edir}) can be written as \n\\begin{eqnarray}\nE_{dir}= \\sum_{n=0}^{\\infty} \\frac{c_{2n}}{2} \\int d{\\bf R} d{\\bf s} \\rho({\\bf R}) \n\\rho({\\bf R}- {\\bf s}) \\nabla^{2n} \\delta({\\bf s})\n=\\sum_{n=0}^{\\infty}  \\frac{c_{2n}}{2} \\int d{\\bf R} \\rho({\\bf R})\\nabla^{2n} \\rho({\\bf R}).\n\\label{eqB4}\n\\end{eqnarray}      \nExpanding now the density $\\rho({\\bf R})$ in its uniform and varying contributions,\nthe direct energy due to the fluctuating part of the density\nbecomes:\n\\begin{equation}\n\\delta E_{dir} =  \\sum_{n=0}^{\\infty} \\frac{c_{2n}}{2} \n\\int d{\\bf R} \\delta \\rho({\\bf R})\\nabla^{2n} \\delta \\rho({\\bf R}).\n\\label{eqB5}\n\\end{equation}\nNotice that the splitting of the density (\\ref{eq6}) also provides a contribution to the\nnon-fluctuating energy $E_0({\\rho_U})$ through the constant density $\\rho_U$.\nLinear terms in $\\delta \\rho ({\\bf R})$ do not contribute \nto the direct energy by the reasons discussed previously.\nProceeding as explained in the following lines, to transform integrals in coordinate space into\nintegrals in momentum space (see Eqs.~(\\ref{eq8}) and (\\ref{eqB1})) after some algebra the fluctuating correction to \nthe direct energy can be written as follows:\n\\begin{equation}\n\\delta E_{dir} = \\frac{1}{2} \\int \\frac{d{\\bf k}}{(2\\pi)^3} \\delta n({\\bf k})\\delta  n^*({\\bf k}) \n{\\cal F}(k),\n\\label{eqB6}\n\\end{equation}\nwhere \n\\begin{equation}\n {\\cal F}(k)= \\sum_{n=0}^{\\infty} c_{2n}k^{2n}\n\\end{equation}\nis a series encoding the \nresponse of the direct energy to the perturbation induced by the varying density.\nThis series is the Taylor expansion of the Fourier transform of the form factor $v(s)$. In the case of \nGaussian ($e^{-s^2\/\\alpha^2}$) or Yukawian ($e^{-\\mu s}\/\\mu s$) form factors one obtains, respectively, \n\\begin{equation}\\label{Fgauss}\n {\\cal F}(k)=\\pi^{3\/2} \\alpha^3 e^{-\\alpha^2 k^2\/4}\n\\end{equation}\nand\n\\begin{equation}\\label{Fyuk}\n {\\cal F}(k)=\\frac{4\\pi}{\\mu(\\mu^2 + k^2)}.\n\\end{equation}\n\nFor the nuclear direct potential the first \n term ($c_0$) of the series for ${\\cal F}(k)$, which can be written as \n${\\cal F}(0)$, corresponds to the bulk contribution associated to the fluctuating density \n$\\delta \\rho({\\bf R})$, i.e. it also contributes to $\\mu_n$ and $\\mu_p$ in Eq.~(\\ref{eq10}). \nThen, the fluctuating correction to the direct energy in Eq.~(\\ref{eq10}) is given by   \n\\begin{eqnarray}\n \\delta E_{dir} &=& \\frac{1}{2} \\int \\frac{d{\\bf k}}{(2\\pi)^3} \\sum_i\\left[D_{L,dir}^i\\left(\\delta n_n({\\bf k})\\delta n^*_n({\\bf k})\n + \\delta n_p({\\bf k})\\delta n^*_p({\\bf k})\\right) \\right. \\nonumber \\\\\n&+& \\left. D_{U,dir}^i\\left(\\delta n_n({\\bf k})\\delta n^*_p({\\bf k})\n+\\delta n_p({\\bf k})\\delta n^*_n({\\bf k})\\right)\n\\right]({\\cal F}_i(k)- {\\cal F}_i(0)) .\n\\end{eqnarray}\nLet us also point out that \nif the series ${\\cal F}(k)$ is cut at first order, i.e. taking only the $n=1$ term of the series, $c_2$, one recovers \nthe typical $k^2$ dependence corresponding to square gradient terms in the energy density functional. \nIf this expansion up to quadratic terms in $k$ is used, the dynamical potential can be written \nas Eq.~(\\ref{eq14a}), where the coefficient $\\beta (\\rho)$ reads\n\\begin{eqnarray}\n\\beta (\\rho) &=& \\left[\\sum_i D^i_{L,dir}c^i_2 + 2B_{pp} \\left(\\rho_n, \\rho_p\\right)\\right] + \\frac{\\left(\\frac{\\partial \\mu_p}{\\partial \\rho_n}\\right)^2}\n{\\left(\\frac{\\partial \\mu_n}{\\partial \\rho_n}\\right)^2}\\left[ \\sum_i D^i_{L,dir}c^i_2 + 2B_{nn}\\left(\\rho_n, \\rho_p\\right)\\right] \\nonumber\\\\\n&-&2 \\frac{\\frac{\\partial \\mu_p}{\\partial \\rho_n}}\n{\\frac{\\partial \\mu_n}{\\partial \\rho_n}}\\left[ \\sum_i D^i_{U,dir}c^i_2 + 2B_{np}\\left(\\rho_n, \\rho_p\\right)\\right],\n\\label{eqB8}\n\\end{eqnarray}\nand the $B_{qq}\\left(\\rho_n, \\rho_p\\right)$ and $B_{qq'}\\left(\\rho_n, \\rho_p\\right)$ functions have been given in \nEqs.~(\\ref{eqA11}) and (\\ref{eqA12}), corresponding to the $\\hbar^2$ contributions coming from the expansion of the energy\ndensity functional. Moreover, for Gaussian form factors one has \n\\begin{equation}\n c_2^i = -\\frac{\\pi^{3\/2} \\alpha_i^5}{4},\n \\end{equation}\nwhereas for Yukawian form factors one has \n\\begin{equation}\n c_2^i= -\\frac{4 \\pi}{\\mu_i^5}.\n\\end{equation}\n\n\\fancyhead[RE, LO]{Chapter C}\n\\end{appendices}\n\\newpage\n{\\small\n\\fancyhead[RE, LO]{Bibliography}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{sec:intro}\n\nComputer Aided Detection (CAD) is a technology that interprets and explains digital images. In the field of medicine, CAD systems are usually developed for extracting useful information from medical images (e.g. Radiography, MRI, Tomography, Ultrasound, PET, SPECT etc.) and providing a second opinion to doctors. Study shows that a well-designed CAD system can increase medical doctors' performance and, therefore, reduce incorrect actions and diagnosis time \\cite{bib1}. Several CAD systems have been implemented commercially for detecting breast cancer, lung cancer, colon cancer, Alzheimers disease, among others.\n\nIn this paper, we are proposing a CAD system that is expected to detect lesions from oral CT images in mandibular region. According to the report of National Cancer Institute, oral cancer accounts for 2.5\\% of all cancers in the United States \\cite{bib2}. Research shows, if oral cancer is detected in early stages, the death rate can be reduced to 10\\% - 20\\% while later stages lead to 40\\% - 65\\% mortality. \n\nOral CT scan is commonly used for treatment planning of orthodontic issues, temporomandibular joint disorder diagnosis, correct placement of dental implants, evaluating the jaw, sinuses, nerve canals and nasal cavity; detecting, assessing and treating jaw tumors and many more. Interpretation of dental CT images are challenging because image modalities are often poor due to noise, contrast is low and artifacts are present; topology is complicated; teeth orientation is arbitrary and lacks clear lines of separation between normal and abnormal regions [4]. Moreover, the inspection of the CT scan requires dedicated training and dentists' time. Furthermore, the diagnosis may vary from dentist to dentist and experience plays a vital role in correct judgment and conclusion of diagnosis. Besides, some early lesions may not be visible clearly to the human eye. These issues and their probable solutions for a better diagnostic environment are the primary motivation of this work.\n\nOral CT images contain both maxilla and mandible of oral anatomy. Lesions in the mandibular region only were studied in this work, and a method to detect its abnormalities was proposed. The framework consists of two algorithms, one is for detection of Close border (CB) lesion (Type I) and the other is for detection of Open border (OB) lesion (Type II). Image processing involved in these two algorithms were implemented in 2D CT slice images and final decision and lesion marking was done by analyzing the results in a 3D CT volume. This paper reports the methodology and evaluates the results, to validate the goal of a CAD system.\n\n\n\\section{Related works}\\label{sec:RW}\n\nComputerized dental treatment systems and clinical decision support systems have seen success in recent years. Many diverse CAD systems were developed for diagnosing different oral diseases. Based on most researched cases, we have broadly classified the research areas into three different categories: caries diagnosis, bone density diagnosis and lesion\/bone defect diagnosis. Following sections describe the categories in short. \n \n\\subsection{Caries diagnosis} \nComputer aided caries diagnosis technology is perhaps the most studied category in the dental radiograph analysis field. The Logicon System (Carestream Dental LLC, Atlanta, GA) is a well-known technology for caries detection \\cite{bib3}. The output of the software is in graphical form, which shows whether the area in question is a sound tooth, or is decalcified or carious. Tracy et al. (2011) \\cite{bib1} studied the performance the of the CAD system where twelve blinded dentists reviewed 17 radiographs. The group concluded that, by using the CAD support, diagnosis of caries increased from 30\\% to 69\\%. \nLikewise, Firestone et al. (1998) \\cite{bib5} investigated the effect of a knowledge-based decision support system (CariesFinder, CF) on the diagnostic performance and therapeutic decisions. The study involved 102 approximal surface radiographic images and 16 general practitioners to identify the presence of caries and whether restoration was required. Their study showed that when the dental practitioners were equipped with a decision support system, their ability to diagnose dental caries correctly increased significantly. \nSimilarly, Olsen et al. (2009) \\cite{bib6} proposed a computerized diagnosis system that aimed to give feedback about the presence and extent of caries on the surface of teeth. Their method gave both qualitative and quantitative opinion to dental practitioners, by using digital images and a graphical user interface. \n\n\\subsection{Bone density diagnosis}\nKavitha et al. (2012) \\cite{bib7} proposed a CAD system that measures the cortical width of the mandible continuously to identify women with low bone mineral density (BMD) from dental panoramic images. The algorithm was developed using support vector machine classifier where images of 60 women were used for system training and 40 were used in testing. Results showed that the system is promising for identifying low skeletal BMD. \nMuramatsu et al. (2013) \\cite{bib8} also proposed a similar work for measuring mandibular cortical width with a 2.8 mm threshold. The algorithm showed 90\\% sensitivity and 90\\% specificity. \nReddy et al. (2011) \\cite{bib9} developed a CAD method to differentiate various metastatic lesions present in the human jawbones from Dental CT images. They developed a method to find most discriminative texture features from a region of interest, and compared support vector machine (SVM) and neural network classifier for classification among different bone groups. They have achieved an overall classification accuracy of 95\\%, and concluded that artificial neural networks and SVM are useful for classification of bone tumors. \n\n\\subsection{Lesion\/bone defect diagnosis} \nShuo Li et al. (2007) \\cite{bib10} developed a semi-automatic lesion detection framework for periapical dental X-rays using level set method. The algorithm was designed to locate two types of lesions: periapical lesion (PL) and bifurcation lesion (BL). Support vector machine was used for segmentation purpose. The algorithm automatically locates PL and BL with a severity level marked on it. Stelt et al. (1991) \\cite{bib11} also developed a similar algorithm to detect periodontal bone defects. They have used image processing techniques to find the lesions that were artificially introduced into the radiographs. Their system was designed to decrease interobserver variability and time-dependent variability. \n\nThe work in this research was focused on lesion\/bone defect diagnosis on CT images. The proposed method is a fully automatic scheme to highlight suspicious regions regardless of the type of a disease. In the following sections, materials used for this work and methodology of the algorithm are discussed, and finally the results are evaluated.\n\n\n\\section{ Materials}\n\nTotal oral CT scans of 52 patients were investigated, where 22 patients had abnormalities and 30 patients had normal appearances. The CT scans were obtained using a 495-slice spiral CT scanner (Vatech, PaX - i3D) with tube voltage of 90 KV, current of 10 mA and slice thickness of 0.2mm. The 2D slice data was reconstructed with a 512 x 512 matrix. A dentist verified normal and abnormal aspects of the images. These images were provided by Vatech Co., Ltd, South Korea, and were supplied in 16-bit RAW format.\n\n\\section{ Methodology }\n\nThis section elaborates the method of the CAD system (Figure \\ref{fig:Algorithm}). The framework consists of two algorithms for detection of two types of lesions: Close border lesions (Type I) and Open border lesions (Type II) (Figure \\ref{fig:2type4}). This classification is not done on a medical basis but is used for research purpose only. In addition, the goal of this CAD scheme was not to detect a specific syndrome, rather any kind of abnormality regardless of the diseases type. Therefore, the classification of lesions considered here served as a generalized grouping of all type of abnormalities that could be found in dental CT images. A Close border lesion is defined as one that has a well-defined boundary around the lesion, and a Open border problem is one that has a broken boundary line around the lesion. Methods for detecting these two types of lesions are described in the later sections.\n\n\n\\begin{figure}[tbp]\n\\centering\n\\includegraphics[width=0.5\\textwidth]{2type4.PNG} \n\\caption{Illustration of two types of abnormality (a) Close border lesion, (b) Open border lesion\n}\\label{fig:2type4}  \n\\end{figure}\n\n\\begin{figure}[tbp]\n\\centering\n\\includegraphics[width=0.8\\textwidth]{Algorithm.PNG} \n\\caption{Architecture of the proposed method\n}\\label{fig:Algorithm} \n\\end{figure}\n\n\n\\subsection{Preprocessing}\n\nTo enhance the contrast of the image, histogram normalization was carried out according to equation \\ref{eq:preprocessing}:\n\n\\begin{equation}\\label{eq:preprocessing}\nI_{out(x,y)}=  255 \\bigg( \\frac{I_{in(x,y)}-I_{in,min}}   {I_{in,max}-I_{in,min}}\\bigg)\n\\end{equation}\n\n\n\\subsection{Detection of Close border (CB) Lesion (Type I) }\n\nA CB lesion in dental CT images can be understood as a low-intensity region surrounded by a relatively high-intensity boundary. These regions can be circular, elliptical or any irregular shape. The problematic regions were usually neither very large nor very small in size, and had a unique texture and statistical properties. These regions were visually homogenous and could be identified in several slices for a particular patient. CB lesion detection algorithm was designed in three classic stages: initial lesion candidate detection, feature extraction and classification. These steps are described in sequential sections.\n\n\\subsubsection{Initial lesion candidate detection }\n\nThe purpose of initial lesion candidate detection is to detect all the possible areas that could be a potential lesion. Potential CB lesions can be considered as blobs in a digital CT image. Blob can be realized as regions in images that differ in properties compared to its surrounding area. The sensitivity of this blob detection process was high so the chances of missing out a possible lesion candidate were low in this stage.\n\n\\paragraph{Binarization.} \\label{Bin}\nThe binary image was computed from original gray scale image using Maximum Entropy threshold method \\cite{bib12}, which is an automatic threshold process. Calculating appropriate threshold value for binarization is vital for the performance of detection because inaccurate threshold could eliminate possible lesion candidates. This particular method was chosen due to its binarization accuracy and implementation simplicity. \n\n\\paragraph{Denoising.}\nTo eliminate noises from the binary image, morphological closing operation  \\cite{bib13} was carried out. Structuring element for the operation was chosen in such a way that it would fill holes of size up to 5mm diameter. Holes larger than 5mm were considered as a possible lesion.\n\n\\paragraph{Blob detection.}\n\nFinally, all possible lesion candidates were filtered out from denoised image by applying blob detection \\cite{bib14} method, which was developed by Haralick et al. (1992). All blobs larger than 5mm in size were detected. In addition, the anatomy present in the bottom 1\/3 of the image area was less likely to represent a lesion, therefore, detected blobs in this area were rejected. Figure \\ref{fig:ILCD} illustrates the initial lesion candidate detection process.\n\n\n\\begin{figure}[tbp]\n\\centering\n\\includegraphics[width=.8\\textwidth]{ILCD.PNG}   \n\\caption{Illustration of Initial Lesion Candidate Detection process\n}\\label{fig:ILCD}\n\\end{figure}\n\n\\subsubsection{Feature extraction}\nThe reason of feature extraction is to acquire sufficient information about an image. Once all the initial lesion candidates were detected, a bounding rectangle around each blob were calculated (Figure \\ref{fig:ROI_extraction}(a)). The rectangles were then superimposed over the intensity image in the same position relative to the detected blobs (Figure \\ref{fig:ROI_extraction}(b)). Then image sections inside each rectangle were cropped out for feature extraction and considered as Region of Interest (ROI). \n\nThree types of features were extracted from each ROI. These features are commonly known as First order statistics, Second order statistics and Image moments. \n\nFirst Order Statistics are the standard statistical measures of gray level values of a ROI. Mean, standard deviation, skewness and kurtosis were calculated from each ROI.\n\nSecond order statistics, computed from Grey-Level Co-occurrence Matrices (GLCM) \\cite{bib15}, are a good measure of image texture. This feature set was developed by Haralick et al. (1973) and is used in many real world applications for its reasonable performance in discriminating different image textures. Four second order statistics features were calculated namely: contrast, homogeneity, energy and entropy. For the computation of GLCM, only horizontal position of neighboring pixels were considered. \n\nImage moments provide a special kind of weighted average of image pixels intensities. This measure usually provide attractive information such as geometry information of objects in an image. Hu (1962) \\cite{bib16} developed seven image moments which are independent of position, size, orientation and parallel projection. These seven moments were calculated for extracting geometry information from ROIs. \n\nTherefore, a total of fifteen (First order statistics: 4, Second order statistics: 4, moments: 7) feature vectors were calculated from each ROI.\n\n\\begin{figure}[tbp]\n\\centering\n\\includegraphics[width=.8\\textwidth]{ROI_extraction.PNG} \n\\caption{Illustration of ROI extraction process. (a) Bounding rectangles calculated around each blob, (b) Bounding rectangles were superimposed over the intensity image to crop out the ROIs.}\\label{fig:ROI_extraction}  \n\\end{figure}\n\n\\subsubsection{Classification}\n\nArtificial Neural Network (ANN) was used for classification of data. Specifically, Multilayer Perceptron (MLP) neural network was employed for classification between two groups: Lesion and Normal. The network was trained with backpropagation learning \\cite{bib17}. Sample ROIs were used to train the network, where both normal and abnormal examples are present (Figure \\ref{fig:training}). Data was collected manually by using blob detection and feature extraction algorithms. The neural network had a configuration of fifteen input layer nodes for fifteen feature vectors, ten hidden layer nodes, and two output layer nodes. The trained network was then used for unseen data classification.\n\nFinally, a lesion area was marked on the intensity image if the ROI was classified as a lesion class.\n\n\\begin{figure}[tbp]\n\\centering\n\\includegraphics[width=.9\\textwidth]{training.PNG}  \n\\caption{Illustrations of training samples (a) Samples of lesion ROI, (b) Samples of normal ROI.}\\label{fig:training} \n\\end{figure}\n\n\\subsection{Detection of Open border (OB) lesion (Type II)}\n\nOpen border problems cannot be solved by blob detection method, as lesions had no distinctive closed border separating it from the background. However, this problem was solved by using morphological image processing operations. The method is schematically shown in Figure \\ref{fig:BDmethod}: \n\n\\begin{enumerate}\n\\item [Step 1]\n\nA grayscale CT slice. (Figure \\ref{fig:BDmethod}(1)). \n\n\\item [Step 2]\n\nBinary image of the grayscale CT slice (Figure \\ref{fig:BDmethod}(2)). See section \\ref{Bin} for binarization.\n\\item [Step 3]\n\nMorphological closing operation was carried out on the binary image to remove noise and artifacts from it. Structuring element for this closing operation was designed in such a way that it can remove holes or openings smaller than 5mm diameter. Moreover, any anatomy present in the bottom 1\/3 of the image was eliminated from further consideration. (Figure \\ref{fig:BDmethod}(3)). \n\n\\item [Step 4]\n\nMorphological closing operation was carried out once more to fill large sized holes or openings (i.e. closing with larger disk shaped structuring element). The design of the structuring element was such that it can fill holes up to 30mm of diameter (Figure \\ref{fig:BDmethod}(4)). \n\n\\item [Step 5]\n\nThe output image from step 3 was then subtracted from the image of step 4. The resulting image contained any architectural defect present in the intensity image. In this stage, blobs were detected that were larger than 5mm diameter and smaller than 30mm diameter. If any blob found within the specified range, then it was considered as a probable lesion (Figure \\ref{fig:BDmethod}(5)). \n\n\\item [Step 6]\n\nFinally, the detected blob centroid was shown on the intensity image (Figure \\ref{fig:BDmethod}(6)).\n\\end{enumerate}\n\n\\begin{figure}[tbp]\n\\centering\n\\includegraphics[width=1.0\\textwidth]{BDmethod.PNG}   \n\\caption{Illustration of Open border lesion detection process. (1) Grayscale CT slice, (2) Binary image, (3) Noise reduction, (4) Large hole filling, (5) Subtraction result of image 3 from image 4., (6) Lesion area marking on the intensity image. \n}\\label{fig:BDmethod}\n\\end{figure}\n\n\n\\section{Results and Discussions}\n\n\nIn this section, results of the two algorithms are discussed separately. In the first section, results of CB lesion detection algorithm have been evaluated, and results of OB lesion detection algorithm will be evaluated in the later section. \n\nIn our CT dataset, 22 patients contained oral lesions where 10 patients had CB lesions only, 8 patients had  OB lesions only and 4 patients had both types of lesions. We had also considered 30 patients who did not have any lesions (normal cases). Thus, 52 CT cases composed the dataset to evaluate the lesion detection framework.\n\nFree Response Receiver Operating Characteristics (FROC) curves were used for evaluating the CAD system. Sensitivity and false positive per patient were the parameters used for FROC curves where, \n\n\n\\begin{equation}\nSensitivity=  \\frac{TPs}{TPs+FNs}\n\\end{equation}\n\n\n\n\\begin{equation}\nFalse\\; positive\\; per\\; patient = \\frac{FPs}{FPs+TNs}\n\\end{equation}\n\n\n\nWhere TP: True Positive, FP: False Positive, TN: True Negative and FN: False Negative. \n\n\\subsection{Evaluation of Close border lesion detection algorithm}\n\nTo validate the proposed CB lesion detection method, we divided the CB patient cases into two sets- one for training and the other for testing. The training set contained 7 abnormal and 15 normal patients, and the testing set also contained 7 abnormal and 15 normal patients. Results of the algorithm are presented in Figure \\ref{fig:CBL}. The testing set yielded a maximum of 71\\% sensitivity with 0.31 false positives per patient. The FROC curve shows results at different operating points where the algorithm can be functioned. However, when all the 14 CB cases and all 30 normal cases were used for training, and the same dataset was used for testing, maximum 92\\% sensitivity was achieved with 0.32 false positives per patient.\n\nThe algorithm performance was observed to be lower for those lesions that do not have a well-defined boundary around it. It is still a challenging task to detect lesions with unclear appearances. Moreover,  lesion size also had an impact to the performance of the algorithm. Our method intends to achieve high detection accuracy on oral lesions sized between 7.5mm to 20mm diameter. Detection accuracy decreased for lesions larger than 25mm or smaller than 7mm diameter. \n\nFor training, only 7 abnormal and 15 normal example cases were used, which was a very small quantity of data to produce reliable results. However, the algorithm still performed in an acceptable manner by indicating 71\\% sensitivity. Upon availability of additional example cases, the algorithm is expected to predict results with better accuracy and could be used in clinical context.\n\n\\begin{figure}[tbp]\n\\centering\n\\includegraphics[width=0.7\\textwidth]{CBL.PNG}   \n\\caption{Comparison of FROC curves in Close border lesion detection method. The solid curve indicates the results of  the scheme when entire Close border dataset was used for both training and testing (14 abnormal and 30 normal patients). The dotted curve indicates the results when the testing set was not used for training.\n}\\label{fig:CBL} \n\\end{figure}\n\n\\subsection{Evaluation of Open border lesion detection algorithm}\n\nThe OB algorithm yielded better results than the CB algorithm. It achieved 85\\% sensitivity without any false positives on the OB dataset of 12 abnormal and 30 normal patients. Moreover, the algorithm was able to achieve 100\\% sensitivity with just 0.13 false positives per patient (See figure \\ref{fig:bd})\n\nThe system was designed to detect lesions of diameter between 10mm - 25mm. For the detection of larger or smaller sized lesions, more example data and research is needed.\n\nIn this method, no feature extraction or classification technique was used. Instead, a rule based process was implemented. The method was validated by dentists' opinion about healthy or lesion regions. Figure \\ref{fig:bd} shows the performance of the method at different operating points.\n\n\\begin{figure}[tbp]\n\\centering\n\\includegraphics[width=0.7\\textwidth]{bd.PNG}  \n\\caption{FROC Curve of Open border detection method\n}\\label{fig:bd}\n\\end{figure}\n\n\nBoth of the systems were implemented on C++ language. It took 7 - 10 seconds for executing the algorithms together for a single patient. The computer used for the testing had core i7, 4.00 GHz processor and 16 GB of RAM.\n\nOral CAD algorithms are relatively new and emerging technology in the field of medical image analysis. Most of the work done in the past were for serving a specific purpose, as discussed in the related works section. The design of our CAD scheme is a more generalized procedure of lesion detection framework and it provides a different justification as well. Moreover, the image database used in this study is entirely different than others. Therefore, for comparison, the authors could not find any similar CAD scheme either in the literature or in the industry and our method can be considered as a novel approach of solving a new set of problems.\n\n\n\\section{Conclusion}\n\\label{concl}\n\nAn approach to detect oral lesions in mandible region on CT images have been proposed in this work. Two types of lesions, Close border lesions and Open border lesions, were identified. They covered most of the problems that could be found on oral CT images. Results suggest that this method is capable of detecting CB lesions with 71\\% sensitivity at 0.31 false positives per patient and OB lesions with 100\\% sensitivity at 0.13 false positives per patient.\n\n\tSome of the limitations of the method include that: it cannot detect lesions brighter than its surroundings, lack of a well-defined boundary around a lesion decreases the detection accuracy and too large or too small lesions achieved less detection sensitivity. However, availability of more example images are expected to improve the results and overcome the limitations.\n    \n\tThe work done in this research focused on the mandibular region of the oral anatomy. Therefore, future research possibilities in this CAD framework includes: detection of lesions in maxilla, detection of brighter lesions and improve the detection accuracy for lesions with unclear appearances.\n\n\n\n\\acknowledgments\n\nThe authors are thankful to  Vatech (Value Added Technology CO., LTD), South Korea, for providing the CT images and sponsoring the project.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nWeak gravitational lensing is a unique tool for probing the mass distribution of the universe and for constraining dark matter halo properties of galaxies and clusters. In contrast to alternative mass estimate methods (employing e.g. X-ray temperatures, radial velocities, or mass-to-light ratios), weak lensing does not rely on any assumptions about virial equilibrium and is sensitive to all mass along the line of sight, making no distinction between luminous and dark matter.\n\nOver the past decade, an enormous international effort has been invested in improving the reliability of weak lensing analysis \\citep{step1, step2, great08, great10}, seeking to remove biases and systematic effects that limit the accuracy of the method. By far most of the work has been focused on measuring the shear signal, the coherent stretching and distortion of distant galaxy shapes by a foreground lensing mass, but recently the magnification signal has begun to attract attention as well \\citep{Scranton05, Hildebrandt09b, Hildebrandt11, LHJM10, Umetsu11, Huff11}.  \n\nWeak lensing magnification is, to first order, a measure of the convergence of a lensing mass. It can be detected through the stretching of solid angle on the sky, which leads to the amplification of source flux, since lensing conserves surface brightness (i.e. photons are neither created nor destroyed in purely lensing processes). In general, two different approaches can be taken to measure magnification. The method we employ here involves observing the effects on source number densities; an interesting alternative method is being explored by \\citet{Schmidt12}, which makes use of source size and flux information, and employs the same COSMOS X-ray groups used in this study.\n\nMagnification affects the source number densities in two ways, and the one that dominates is determined by the intrinsic magnitude number counts of the sources in question. Simply put, the brightest sources, which usually have steep number counts, will exhibit an {\\it increase} in number density when lensed, as the amplification allows more objects to be detected, while the number density of the faintest sources, having relatively shallow number counts, will {\\it decrease} \\citep{Narayan89}.\n\nCompared to shear measurements, magnification exhibits a slightly lower signal-to-noise ratio (S\/N), the reason it has been largely ignored until recently. However, what magnification lacks in signal strength, it makes up for in terms of its ability to be applied to lenses at higher redshift and to poorly resolved sources \\citep{Waerbeke10}. Since shear studies require measurements of galaxy shapes, in order for a source to be used it must necessarily be well resolved. This is in stark contrast to magnification studies using source number densities, which have no such requirement for the sources to be resolved at all! In principle, only source magnitudes, redshifts, and positions relative to a lens must be known. This simple fact makes it possible to extend weak-lensing magnification analyses to a much higher redshift than possible for shear, and allows a much higher source density to be included in the analysis.  See \\citet{Waerbeke10}, \\citet{RozoSchmidt10}, and \\citet{Umetsu11} for more detailed discussions of the benefits of combining magnification with shear in gravitational lensing studies. \n\nIn Section \\ref{theory}, we review the equations describing the effects of weak-lensing magnification on source number densities. Section \\ref{data} gives the properties of the X-ray groups and Lyman break galaxies (LBGs) that are used in this study. Then Section \\ref{results} describes the steps of our analysis, and results of the composite-halo model fitting. We summarize the results in Section \\ref{summary}, and compare with weak-lensing shear measurements that have previously been made on populations of galaxy groups. We use the WMAP7 $\\Lambda$CDM cosmological parameters $H_0 =$ 71 km s$^{-1}$ Mpc$^{-1}$ and $\\Omega_{\\Lambda} = 0.734$ \\citep{WMAP7}, and set $\\Omega_M = 1 - \\Omega_{\\Lambda}$.\n\n\\section{Theory}\n\\label{theory}\nThe amplification matrix $\\cal{A}$ maps the image deformation from the source to observer frame, and describes the first order effects of gravitational lensing:  \n\\begin{equation}\n\\cal{A} = \\left( \\begin{array}{cc}\n{1-\\kappa-\\gamma_1} & {-\\gamma_2} \\\\\n{-\\gamma_2} & {1-\\kappa+\\gamma_1} \\\\\n\\end{array} \\right).\n\\end{equation}\nIt is a function of the convergence $\\kappa$, and the shear $\\gamma$, which define the isotropic and anisotropic focusing of light rays, respectively. The magnification factor $\\mu$ is the inverse determinant of this matrix, so that\n\\begin{equation}\n\\mu = \\frac{1}{\\mathrm{det} \\cal{A}} = \n\\frac{1}{(1-\\kappa)^2 - \\left|\\gamma\\right|^2}\n\\end{equation}\n\\citep{BartelmannSchneider01}.  \n\nThe cumulative number counts of distant {\\it unlensed} sources $N_0$ are related to the observed {\\it lensed} number counts $N$, up to some flux $f$, by the equation\n\\begin{equation}\nN (>f) = \\frac{1}{\\mu} N_0 \\left( > \\frac{f}{\\mu} \\right).\n\\end{equation}\nHere the two distinct effects of weak-lensing magnification, on source number counts, are made explicit. The prefactor of $1 \/ \\mu$ is the dilution of source density, as the observed solid angle on the sky is stretched by a foreground massive lens. The modification to the flux $f \/ \\mu$ inside the argument of $N_0$ represents the effect of source amplification by a lens, such that one is able to detect intrinsically fainter objects due to gravitational lensing.\n\nSwitching from working in fluxes to magnitudes $m$, the differential number count relationship was demonstrated by \\citet{Narayan89} to be\n\\begin{equation}\nn(m)dm = \\mu ^{\\alpha -1} n_0 (m)dm,\n\\end{equation}\nwhere $\\alpha$ is defined according to\n\\begin{equation}\n\\alpha \\equiv \\alpha(m) = 2.5 \\frac{d}{dm} \\log n_0(m).\n\\end{equation}\nThus, distant source galaxies, lensed by an intervening concentration of mass, may have their observed number counts increased {\\it or} decreased depending on the sign of the quantity $(\\alpha -1)$. Sources for which $(\\alpha -1) > 0$ will appear to be correlated on the sky with a lens position, while sources with $(\\alpha -1) < 0$ will be anti-correlated, as a dearth of objects will be observed in the vicinity of a lens. The number density of galaxies for which the intrinsic number count slope gives $(\\alpha -1) \\approx 0$ will essentially be unaffected by lensing magnification, as the dilution and amplification effects will cancel, and no correlation signal will be observed for these objects \\citep{Scranton05}.\n\n\\section{Data}\n\\label{data}\n\\subsection{Lenses}\nThe lenses in this study consist of X-ray-selected galaxy groups in the COSMOS field. See \\citet{Leauthaud10} for the detailed properties of these groups. From the full sample of 206 groups investigated in the aforementioned study, we use the shear-calibrated mass estimates to construct the most massive subsample of groups for this magnification study. Here masses are characterized by the parameter $M_{200}$, the total mass interior to a sphere of radius $R_{200}$, within which the average density is 200 times the critical. \n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[scale=0.7]{m200_zphot_histograms_44x.eps}\n\n\\caption{Masses and photometric redshifts of the groups in this study. We select the most massive groups in our sample, $M_{200} \/ M_\\odot \\ge 3.56 \\times 10^{13}$. Using only the cleanest groups (characterized by having $\\ge$ 4 members, well-defined centroids, and no flags on possible mergers or projection effects), and applying appropriate masking, we are left with a sample of $44$ groups for this lensing magnification analysis.}\n\\label{hists}\n\\end{center}\n\\end{figure*}\n\nAny groups that have less than four member galaxies, that appear to be undergoing mergers, that have uncertain centroids, or that raise concerns about projection effects, are excluded from the analysis. These restrictions follow from the group catalog requirement \\texttt{FLAG\\_INCLUDE=1}, discussed in \\citet{George11}. The remaining $44$ most massive groups have shear determined masses in the range $ 3.56 \\times 10^{13} \\le M_{200} \/ M_\\odot \\le 1.70 \\times 10^{14} $, and we employ stacking to increase the S\/N of the magnification measurement. The redshift range of the groups is $ 0.32 \\le z \\le 0.98 $. Figure \\ref{hists} displays these lens properties.\n\nChoosing an optimal lens centroid about which to construct angular bins is an area of ongoing research, and common choices include the brightest central galaxy or the X-ray emission peak. If the location of the dark matter density peak were known a priori, then it would obviously be the ideal choice, but instead we must rely upon some combination of observables to approximate this position. In this paper, we define lensing mass centers by the location of the group galaxy with the highest stellar mass (MMGG$_\\mathrm{scale}$) lying within a distance $ (R_s + \\sigma_x) $ of the X-ray center, where $ R_s $ is the group scale radius and $ \\sigma_x $ is the uncertainty in the X-ray center position \\citep{George11}.  In order to be very confident about the locations of group centers, we exclude groups for which this galaxy is not the most massive member of the group. This choice of centroid has been shown to accurately trace the centers of halos in this sample by optimizing the shear signal on small scales \\citep{George12}.\n\n\n\\subsection{Sources}\nBackground sources are LBGs, a type of high-redshift star-forming galaxy that has been used successfully in previous magnification studies \\citep[see][]{Hildebrandt09b, Hildebrandt11}. These LBGs were selected using the typical three-color dropout technique. For the $U$-, $G$-, and $R$-dropouts the selections described in \\citet{Hildebrandt09a} were used, however the COSMOS Subaru $g^+$ and $r^+$ data were used instead of the CFHT-LS $g^*$ and $r^*$ data (see \\citet{Capak07} for the filter definitions). For the $B$-dropouts the selection from \\citet{Ouchi04} was used.  \n\nThe appeal of using LBGs for magnification is rooted in the fact that their luminosity functions (LFs) have been extensively studied and their redshift distributions are fairly narrow and accurate. After all quality cuts and image masking, we are left with 45,132 LBGs in total. The four distinct sets are comprised of 12,980 $U$-, 22,520 $G$-, 4870 $B$-, and 4762 $R$-dropouts, located at redshifts of $\\sim$3.1, 3.8, 4.0, and 4.8, respectively.\n\nWe first test our data selection by cross-correlating the foreground groups with LBGs separated into discrete magnitude bins. Here we use the basic \\citet{LandySzalay93} estimator, \n\\begin{equation}\nw(\\theta)=\\frac{\\text{D}_1 \\text{D}_2 - \\text{D}_1 \\text{R} - \\text{D}_2 \\text{R} + \\text{RR}}{\\text{RR}},\n\\end{equation}\nto simply compute cross-correlations between groups and background sources. D$_1$ and D$_2$ represent the data sets of lenses and sources, and R are the {\\it random objects} from a mock catalog we create, containing points uniformly distributed throughout the COSMOS survey area. Each product of terms is the number of pairs of those objects found to lie within some angular bin, normalized by the total number of pairs found at all angular separations. This cross-correlation estimator has been shown to be both robust and unbiased \\citep{Kerscher00}.  \n\nIn any lensing study, care must be taken to ensure that regions of an image containing artifacts such as saturated pixels, satellite tracks, or other spurious effects are masked out of the investigation. We consistently apply the same masks to the group, source, and random catalogs, prior to the correlation analysis. Using a large number (584,586) of objects in this random catalog serves to reduce shot noise.\n\nWe expect that the faintest (brightest) magnitude bins should yield a negative (positive) cross-correlation with the group centers, and this is exactly what we find. Figure \\ref{MagBinned} displays this anticipated result, where we simply use a number count weighted average to combine the signal of the distinct LBG samples. As discussed in \\citet{Hildebrandt09b}, this negative correlation is one of the strongest verifications that no redshift overlap exists between lens and source populations, for no viable reason other than lensing magnification can be given for such a signal to exist. Redshift overlap between samples must be avoided in magnification studies, as positive cross-correlations due to physical clustering would overwhelm any lensing-induced signal.\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[scale=0.9]{magbinnedLBGs_multiNFWfit.eps}\n\\caption{Angular cross-correlation of the X-ray groups with Lyman break galaxies, the latter separated into three magnitude-selected samples. The bright sample contains $U$, $G$, $B$, and $R$-dropouts in the magnitude ranges $23<r<25$, $23.5<i<25$,  $23.5<i<25$,  and $24<z<25.5$, respectively. Similarly, the medium ranges are $25<r<25.5$, $25<i<26$,  $25<i<26$,  $25.5<z<26$. The faint ranges are $r>25.5$, $i>26$,  $i>26$,  $z>26$. These magnitude ranges are selected to contain LBGs for which $(\\alpha-1)>0$, $\\approx 0$, and $<0$. The measured correlations for each LBG sample are simply averaged here (weighting by the number counts) in order to more clearly display this diagnostic check. The dashed curves are calculated from the composite-NFW fit, using weighting by the appropriate $\\langle \\alpha -1 \\rangle$ factor, which is given in each panel. The negative correlation observed for the faintest sample is a good indication that no redshift overlap exists between foreground lenses and background sources.}\n\\label{MagBinned}\n\\end{center}\n\\end{figure*}\n\n\\section{Analysis and Results}\n\\label{results}\n\\subsection{Measuring $\\alpha(m)$}\nAlong with the mass of the lens itself, the slope of the source number counts as a function of magnitude, parameterized by the quantity $\\alpha \\equiv \\alpha(m)$, controls the amplitude and sign of the expected magnification signal. To interpret the correlations that we measure, and to implement an optimally weighted procedure, we must determine the value of this quantity for every source galaxy that we intend to use in the measurement. Fortunately, LBGs have been extensively studied and many measurements of their LFs have been published.  \n\nFor the $U$-, $G$-, and $R$-dropouts, we use the recent measurements by \\citet{vanderBurg10}. For the $B$-dropouts we use the results of \\citet{Sawicki06}. These two sets of measurements both involved fitting a Schechter function \\citep{Schechter76} to their galaxy number counts, and their best-fit parameters that we use here are displayed in Table 1. The Schechter Function is given by\n\\begin{equation}\n\\Phi(M)=0.4\\ln(10)\\Phi^\\ast10^{0.4(\\alpha_{\\text{LF}}+1)(M^\\ast-M)} \\exp [-10^{0.4(M^\\ast-M)}],\n\\end{equation}\nwhere $\\Phi^\\ast$, $M^\\ast$, and $\\alpha_{\\text{LF}}$ are the normalization, characteristic magnitude, and faint-end slope of the LF. Note that the $\\alpha(m)$ which we want to calculate is not the same as the constant parameter $\\alpha_{\\text{LF}}$, but approaches it in the limit of very faint magnitudes.\n\nSolving this equation for $\\alpha(m)$, we obtain\n\\begin{equation}\n\\alpha(m) = 2.5 \\frac{d}{dm} \\log n_0(m)= 2.5 \\frac{d}{dM} \\log \\Phi(M)\\]\n\\[ = 10^{0.4(M^\\ast-M)}-\\alpha_{\\text{LF}}-1.\n\\end{equation}\nWe convert the observed apparent magnitudes $m$ of the LBGs to absolute magnitudes $M$ via the relationship $M = m - $DM$ + 2.5 \\log (1+z)$, where DM and $z$ are the distance modulus and redshift of the galaxy in question. Since we select apparent magnitudes in the $r$, $i$, and $z$ bands for the $U$-, $G$- and $B$-, and $R$-dropouts, we probe very similar restframe wavelengths and the $K$-correction between the samples is negligible. Thus we ignore it here. Using the LF parameters in Table \\ref{LFtable}, combined with the conversion to absolute magnitudes, we then obtain a measure of $\\alpha(m)$ for every LBG in the sample.\n\nIt is important to assess how uncertainties in the quantity $\\alpha(m)$ can affect the interpretation of the magnification measurement. Since we will rely on the quantity $(\\alpha-1)$ as a weight factor in this analysis, using a wrong $\\alpha$ could potentially lead to a bias in the mass measurement. For very faint objects the observed magnitudes become less certain due to shot noise. We propagate these magnitude errors through Equation 8 to obtain an uncertainty on $\\alpha(m)$, which is used to find a magnitude-based cut on the sources. We find that cutting $\\sim$10\\% of the very faintest sources, largely $R$-dropouts, gives us a good balance between removing the most uncertain $\\alpha$ values, but still retaining a significant number of sources for the analysis.\n\nHere we consider two possible sources of systematic error, which are combined in quadrature to yield the total systematic error, reported in Section 4.3. The first source is uncertainty on the LF parameters (see Table 1), which includes the effects of cosmic variance. We repeat the composite-halo fit, detailed in Section 4.3, for the range of permitted values of $M^\\ast$ and $\\alpha_{\\text{LF}}$, finding a maximum variation in the mass measurement of up to $\\sim$40\\%. Second, we consider the possibility for a small photometric offset to exist between the various surveys used in this work. Assuming a maximum offset of $\\pm$0.05 magnitudes between surveys, we vary all observed source magnitudes uniformly by offsets in the range $-0.5 \\leq \\delta m \\leq 0.5$, and find the maximum effect on the mass measurement to be $\\sim$15\\%.\n\n\\subsection{Optimally Weighted Cross-correlation}\nWe implement a modified version of the \\citet{LandySzalay93} estimator for the angular cross-correlation function, in which pair counts are weighted by their expectations from the differential source number counts as a function of magnitude. This weighted correlation function has been shown to optimally boost the magnification signal \\citep{Menard03}:    \n\\begin{equation}\nw(\\theta)_{\\text{optimal}}=\\frac{\\text{S}^{\\alpha-1} \\text{L} - \\text{S}^{\\alpha-1} \\text{R} - \\langle \\alpha-1 \\rangle \\text{LR}}{\\text{RR}} + \\langle \\alpha-1 \\rangle .\n\\end{equation}\nOptimal-weighting was first implemented by \\citet{Scranton05} and, apart from notation, this equation is identical to the estimator used in \\citet{Hildebrandt09b}.  As with the original basic estimator, each term represents the number of pairs of objects found in a given angular $\\theta$ bin, normalized by the total number of pairs at all angular separations.  \n\nS stands for the {\\it sources}, or background lensed galaxies, L are the {\\it lenses}, or X-ray groups, and once again R are the {\\it random objects}. The superscript $(\\alpha-1)$ on the S indicates that pair counts involving sources are to be weighted by this factor. After removing masked objects from the catalogs, and satisfying the above selection criteria, we are left with 39,710 LBG sources, $44$ X-ray group lenses, and 584,586 random objects for the analysis.\n\n\\begin{table}\n  \\begin{center}\n \n   \n  \n\n   \n   \\caption{Luminosity Function (Schechter) Parameters from External LBG Measurements. $^a$ LF parameters from \\protect \\citet{vanderBurg10}. $^b$ LF parameters from \\protect \\citet{Sawicki06}.}\n \\label{LFtable}\n    \\begin{tabular}{lcccl}\n      \\hline \\hline\n      LBG Sample & $M^*$ & $\\alpha_{\\text{LF}}$ & Number \\\\ \\hline\n      $U$ ($z \\sim$ 3.1)$^a$ & $-20.84^{+0.15}_{-0.13}$ & $-1.60^{+0.14}_{-0.11}$ & 12,980 \\\\\n      $G$ ($z \\sim$ 3.8)$^a$ & $-20.84^{+0.09}_{-0.09}$ & $-1.56^{+0.08}_{-0.08}$ & 22,520 \\\\\n      $B$ ($z \\sim$ 4.0)$^b$ & $-21.00^{+0.40}_{-0.46}$ & $-1.26^{+0.40}_{-0.36}$ & 4,870 \\\\ \n      $R$ ($z \\sim$ 4.8)$^a$ & $-20.94^{+0.10}_{-0.11}$ & $-1.65^{+0.09}_{-0.08}$ & 4,762 \\\\\n      \\hline\n    \\end{tabular}\n  \\end{center}\n\\end{table}\n\n\nThe brightest source galaxies, which are observationally found to lie in the steepest part of the LF, are expected to be positively correlated with the group centers, have the largest value of $(\\alpha-1)$, and so receive a relatively large weight in this correlation study. In contrast, the faintest background galaxies are expected to be anti-correlated, on average, with the group positions, because the effects of magnification dilution should be greater than the amplification of flux can compensate for, and these galaxies thus receive a negative weight. Sources for which $(\\alpha-1) \\approx 0$ ought to have the effects of dilution and amplification cancel out overall, and receive very little to no weight in this analysis \\citep{Scranton05}. \n\nThe optimally weighted correlation function is given in Figure 3, and shows the measured radial profiles for this stack of massive galaxy groups. Error bars are $1 \\sigma$ uncertainties, obtained by jackknife resampling of the source population. To do this, we create 50 jackknife samples of data, each with a different $1\/50$ of sources removed from it. Then we measure the optimal correlation function for each, and from these estimate the covariance matrix through\n\\begin{equation}\nC(\\theta_1, \\theta_2)= \\left( \\frac{N}{N-1} \\right)^2 \\times \\sum_{j=1}^N [w_j(\\theta_1)-\\bar{w}(\\theta_1)] \\times [w_j(\\theta_2)-\\bar{w}(\\theta_2)],\n\\end{equation}\nwhere the index $j$ runs over the $N=50$ jackknife measurements.\n\n\\subsection{Halo Mass Profiles}\nMeasuring the magnification-induced effects on source number counts behind massive lenses allows one to estimate properties of the lens, such as the mass profile. In this paper, we use a composite-halo approach which allows us to fit for the full range of both group masses and redshifts. The horizontal axes in Figures \\ref{MagBinned} and \\ref{multihalo} are therefore actual transverse distances obtained by taking account of the angular diameter size at each unique group redshift. We incorporate both the singular isothermal sphere (SIS) and the Navarro-Frenk-White (NFW; \\citep{nfw97}) density profiles into the composite-halo modeling.\n\n\n\n\nThe magnification contrast is $\\delta\\mu(\\theta) \\equiv \\mu(\\theta) -1$, and for an SIS halo it is simply given by \n\\begin{equation}\n\\delta \\mu_{\\text{SIS}}(\\theta)=\\frac{\\theta_{\\text{E}}}{\\theta-\\theta_{\\text{E}}},\n\\end{equation}\nwhere $\\theta_{\\text{E}}=4\\pi(\\frac{\\sigma_v}{c})^2\\frac{D_{ls}}{D_s}$ is the Einstein radius of the lens, and $D_{ls}$ and $D_s$ are angular diameter distances between lens and source, and observer and source, respectively. The velocity dispersion of the lens, $\\sigma_v$, can be expressed in terms of the mass and critical energy density of the universe at lens redshift $z$:\n\\begin{equation}\n\\sigma_v=\\left[ \\frac{\\pi}{6}200\\rho_{\\text{crit}}(z)M_{200}^2 G^3 \\right]^\\frac{1}{6} .\n\\end{equation}\n\nFor the NFW halo, the magnification contrast takes a slightly more complicated form. From Equation 2, we have\n\\begin{equation}\n\\delta\\mu_{\\text{NFW}}(\\theta)=\\left[ (1-\\kappa_{\\text{NFW}})^2 - \\left|\\gamma_{\\text{NFW}}\\right|^2 \\right]^{-1} -1.\n\\end{equation}\nWe use the analytical NFW expressions for $\\kappa$ and $\\gamma$ derived in \\citet{WrightBrainerd00} to evaluate $\\delta\\mu_{\\text{NFW}}$ for every lens-source pair in the study. The two NFW fit parameters are the scale radius $r_{\\text{s}}$ and the concentration $c$, which together determine the mass\n\\begin{equation}\nM_{200}=\\frac{4\\pi}{3}(200)\\rho_{\\text{crit}}(z)c^3r_{\\text{s}}^3 .\n\\end{equation}\nAs we do not find this magnification measurement precise enough to provide meaningful two-parameter constraints, we use the mass-concentration relation given in \\citet{Munoz11} to reduce this to a single-parameter fit.\n\nWe perform a composite-halo fit for both lens models, similar to the multi-SIS used in \\citet{Hildebrandt11}. This allows us to fit for a range of masses and redshifts, thereby avoiding any biases that would be introduced by simply fitting to a stacked average lens profile. The optimally weighted correlation function is related to the magnification contrast through\n\\begin{equation}\nw(a)_{\\text{optimal}}= \\frac{1}{N_{l}}\\sum_{i=1}^{N_{l}} \\langle (\\alpha -1)^2 \\rangle_i \\delta \\mu(z_i, aM_{\\text{shear},i}),\n\\end{equation}\nwhere $i$ runs over all lenses. Here the fit parameter $a$ characterizes the scaling relation between the $M_{200}$ previously measured from the shear, and the best fit $M_{200}$ from magnification, so that $a \\equiv M_{\\text{magnification}}\/M_{\\text{shear}}$. \n\nWe use the generalized minimum-$\\chi^2$ method to fit the composite-halo profiles to the magnification measurements (see Figure \\ref{multihalo}), using the full unbiased inverse covariance matrix, according to the prescription in \\citet{Hartlap07}. The $\\chi^2\/$dof is 1.5 for the composite-SIS and 0.8 for the composite-NFW ($\\chi_{\\text{SIS}}^2=7.5$, $\\chi_{\\text{NFW}}^2=4.2$, dof=5 in both cases). For the composite-SIS, we obtain a best-fit value of $a=1.2 \\pm 0.4 \\pm 0.4^{\\text{sys}}$, and with the composite-NFW we get $a=1.8 \\pm 0.5 \\pm 0.4^{\\text{sys}}$. These results indicate consistency with the previous shear mass measurements, albeit with large uncertainties.\n\n\n\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[scale=0.75]{wopt_CompositeHaloFits_44x_LFam1_aveErr0.7magcut.eps}\n\\caption{Composite-halo fits to the optimally weighted correlation function, using the LBG background source sample. The significance of the magnification detection is 4.9$\\sigma$. The dashed line is the composite-SIS and the solid line is the composite-NFW.  We find the best-fit relative scaling relations for each to be $a= M_{\\text{mag}}\/M_{\\text{shear}}= 1.2 \\pm 0.4 \\pm 0.4^{\\text{sys}}$ (SIS) and $a= 1.8 \\pm 0.5 \\pm 0.4^{\\text{sys}}$ (NFW). The dotted line shows the prediction from the shear measured values of $M_{200}$ (A. Leauthaud 2011, private communication).}\n\\label{multihalo}\n\\end{center}\n\\end{figure*}\n\n\n\\section{Summary and Conclusions}\n\\label{summary}\nWe report a 4.9$\\sigma$ detection of weak-lensing magnification from a population of X-ray-selected galaxy groups. This is the first magnification measurement using source number densities successfully performed on the group scale. \\citet{Schmidt12} have recently explored the magnification of these groups using source sizes and fluxes. For comparison, the shear detection significance is 11$\\sigma$ on the same selection of $44$ groups (A. Leauthaud 2011, private communication).\\footnote[1]{The significance quoted for the shear does not take into account the full covariance matrix, as we have done for the magnification measurement. Therefore this shear significance might be a bit optimistic.}\n\nTo improve S\/N in this measurement, we stack the lenses, consisting of $44$ massive X-ray-detected galaxy groups in the COSMOS 1.64 deg$^2$ field. We measure an optimally weighted cross-correlation between the X-ray groups and high-redshift LBGs, with $1 \\sigma$ error bars determined from jackknife resampling of the sources. Performing composite-halo fits to this optimally weighted signal yields a measurement of the relative scaling between shear- and magnification-derived masses. Our magnification measurement yields a mass $M_{\\text{mag}}=aM_{\\text{shear}}$ where the best-fit parameter $a= 1.2 \\pm 0.4 \\pm 0.4^{\\text{sys}}$ (SIS), and $a= 1.8 \\pm 0.5 \\pm 0.4^{\\text{sys}}$ (NFW), demonstrating a rough consistency with the shear measurement.\n\nAs discussed in Section 4.1, a central issue is the importance of having correct $(\\alpha-1)$ measures for every source galaxy, to ensure that the {\\it optimal weighting} truly is optimal. We perform a thorough error analysis that includes measurement uncertainties from the full covariance matrix, and investigate possible sources of systematic errors from both photometry and externally calibrated LF parameters.\n\nWe claim that LBGs are a preferred source sample when it comes to performing lensing magnification analyses using source number counts. A few reasons for the superiority of the LBG sample include more reliable redshift determinations, as well as greater lensing efficiencies and generally much higher values of the quantity $(\\alpha-1)$. The single most significant reason to choose LBGs for this type of analysis, however, is for the ease of obtaining a reliable measure of $\\alpha(m)$. Previous deep measurements of LBG LFs allow us to perform calculations yielding the optimal weight factor $(\\alpha-1)$, as well as its associated uncertainty.\n\nAlthough the S\/N of shear is superior to magnification in general, the latter probes the surface mass density of the lens directly, while the shear measures the differential mass density. Thus the combination of these two independent measurements is desirable, and breaks the lens mass-sheet degeneracy. In fact, \\citet{RozoSchmidt10} demonstrated that joining magnification into shear analyses, independent of survey details, can improve statistical precision by up to 40\\%-50\\%. Magnification using source number densities is also far less sensitive to the effects of atmospheric seeing than either shear or magnification using source sizes. Both of these methods require quality source images which, for very high redshift sources, can currently only be obtained from space-based data.\n\nImproving the overall weak-lensing-derived constraints on cosmological and astrophysical parameters is not the only benefit to incorporating magnification into our analyses, however. Measurements of magnification are sensitive to completely different systematics than shear, and therefore uniquely positioned to help improve calibration of these residual effects on shear measurements. For example, magnification (using number counts) is not at all sensitive to the possible intrinsic alignment of source galaxies, since it does not use any shape information. Magnification can also be used as a simultaneous probe of intergalactic dust extinction, a small but measurable effect through its wavelength dependence \\citep{Menard10}, and as a direct way to measure galaxy bias \\citep{Waerbeke10}.\n\nAs one proceeds to investigate dark matter structures at increasingly high redshift, it becomes more and more important to include the magnification component of the signal. This is a direct consequence of the fact that higher redshift lenses necessitate more distant sources, which are in turn much harder to measure shapes for. Proceeding exclusively with shear necessarily means that a high fraction of detected sources are being eliminated from the lensing analysis, and information is therefore being lost, simply because we lack the capabilities to robustly determine their shapes. With photometric redshifts available, the possibility to do magnification studies on our shear catalogs really comes along free of charge. Upcoming projects will survey the entire extragalactic sky, and the inclusion of magnification will be a necessary component of any robust weak lensing study.\n\n\\vspace{10.pt}\n\nThe authors thank Fabian Schmidt and Martha Milkeraitis for useful discussions related to this work. J.F. was supported by JPL grant number 1394704, and is now supported by NSERC and CIfAR. H.H. is supported by the Marie Curie IOF 252760 and by a CITA National Fellowship. This work was performed in part at JPL, run by Caltech under a contract for NASA. This work was supported by World Premier International Research Center Initiative (WPI Initiative), MEXT, Japan. This work is based in part on data collected at Subaru Telescope, which is operated by the National Astronomical Observatory of Japan, and on observations made with the NASA\/ESA {\\it Hubble Space Telescope}. This research has made use of the NASA\/IPAC Infrared Science Archive, which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration. This work is also based on observations obtained with MegaPrime\/MegaCam, a joint project of CFHT and CEA\/DAPNIA, at the Canada-France-Hawaii Telescope (CFHT) which is operated by the National Research Council (NRC) of Canada, the Institute National des Sciences de l'Univers of the Centre National de la Recherche Scientifique of France, and the University of Hawaii.\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}