Critical properties of calibrated relativistic mean-field models for the transition to warm, non-homogeneous nuclear matter
Olfa Boukari, Helena Pais, Sofija Antić, Constança Providência
aa r X i v : . [ nu c l - t h ] J u l Critical properties for warm non-homogeneous stellar matter from calibrated models
Olfa Boukari , Helena Pais , Sofija Anti´c , and Constan¸ca Providˆencia Laboratoire de Physique de la Mati`ere Condens´ee,Facult´e des Sciences de Tunis, Campus Universitaire, Le Belv´ed`ere-1060, Tunisia. CFisUC, Department of Physics, University of Coimbra, 3004-516 Coimbra, Portugal. CSSM and CoEPP, School of Physical Sciences,University of Adelaide, Adelaide SA 5005, Australia.
The extension to warm and asymmetric clusterized nuclear matter is developed for a set of wellcalibrated equations of state. It is shown that even though different equations of state are constrainedby the same experimental, theoretical and observational data, and the properties of symmetricnuclear matter are consistent within the models, the properties of very asymmetric nuclear matter,such as the one found inside of neutron stars, differ a lot for various models. Some models predictlarger transition densities to homogeneous matter for beta-equilibrated matter than for symmetricnuclear matter.
I. INTRODUCTION
Core-collapse supernovae (CCSN) and neutron star(NS) mergers are two astrophysical explosive eventswhere matter can reach temperatures above ∼
50 MeV.In CCSN matter, β − equilibrium is not immediatelyreached, and a fixed proton fraction in the range of0 < Y p < . T = 0 MeV.These nuclear RMF models are listed as follows: 1)SFHo and SFHx [11], constructed to fulfill constraintsfrom effective chiral field theory calculations for neutronmatter [12], from nuclear experiments on matter nearand below the saturation density, and from neutron ra-dius measurements. In Ref. [11], the authors have shownthat the two EOS, among the models they tested, wouldlead to the most compact protoneutron stars in the firstmilliseconds after the bounce; 2) FSU2R and FSU2H pro-posed in [13, 14], calibrated to well settled properties ofnuclear matter, and neutron star observations. FSU2Hallows for the appearance of hyperonic degrees of free-dom, while still predicting two solar-mass neutron stars;3) TM1 [15], and TM1e [16], recently published, and rep-resents an extended version of the TM1 model, where theauthors include an ω − ρ meson coupling term in order tomodel the density dependence of the symmetry energy.This leads to a lower slope of the symmetry energy ascompared to the original model, and a decrease in theneutron star radii; 4) DD2 [17] and DDME2 [18], whichsatisfy the same well established nuclear matter proper-ties [19, 20], and allow the appearance of hyperons intwo-solar mass neutron stars; 5) finally, D1 and D2 [21],closely related to DD2. The difference between D1 andD2 is in the inclusion of an additional energy dependencein the nucleon’s scalar and vector self-energies for the D2model, which was introduced in order to satisfy the op-tical potential constraint at high nuclear densities.The main conclusion of the present work is that whilecalibrated models behave in a very similar way at zerotemperature and symmetric matter, large differenceswere identified for both the critical temperatures anddensities of β -equilibrated matter in very asymmetricmatter. In some models, like SFHo and SFHx, the onsetof homogeneous matter in β -equilibrated matter occursat similar or larger densities, than the ones found forsymmetric nuclear matter. This will have consequenceson the predictions of CCSN or NS merger simulations.The structure of the paper is the following: in SectionII, the general formalism of RMF models and spinodalcalculation are briefly introduced, Section III discussesand compares the results on critical points, transitiondensities, and distilation effect between different models,and, finally, in Section IV, a few conclusions are drawn. II. THE FORMALISM
A brief summary of the RMF formalism is given inthe first part of the section, while the thermodynami-cal spinodal calculation and respective critical points areaddressed in the second subsection.
A. Field Theoretical Models with RMF Lagrangian
In our set of RMF models, the nucleons, with mass M ,interact with the scalar-isoscalar meson field σ with mass m σ , the vector-isoscalar meson field ω µ with mass m ω ,and the vector-isovector meson field ρ µ with mass m ρ .The Lagrangian density is given by: L = X i = p,n L i + L σ + L ω + L ρ + L σωρ , (1)where the nucleon Lagrangian reads L i = ¯ ψ i [ γ µ iD µ − M ∗ ] ψ i , (2)with iD µ = i∂ µ − g ω ω µ − g ρ τ · ρ µ . (3)The Dirac effective mass is given by M ∗ = M − g σ σ . (4)In the above equations, g σ , g ω and g ρ are the meson-nucleon couplings, and τ are the SU(2) isospin matrices.The mesonic Lagrangians are: L σ = + 12 (cid:18) ∂ µ φ∂ µ σ − m σ σ − κσ − λσ (cid:19) , L ω = −
14 Ω µν Ω µν + 12 m ω ω µ ω µ + ζ ζg ω ( ω µ ω µ ) , L ρ = − B µν · B µν + 12 m ρ ρ µ · ρ µ + ξ g ρ ( ρ µ ρ µ ) , (5) where Ω µν = ∂ µ ω ν − ∂ ν ω µ , B µν = ∂ µ ρ ν − ∂ ν ρ µ − g ρ ( ρ µ × ρ ν ), and κ , λ , ζ , and ξ are coupling constants.The mesonic Lagrangian is supplemented with the fol-lowing non-linear term that mixes the σ, ω , and ρ mesons[11]: L σωρ = g ρ f ( σ, ω µ ω µ ) ρ µ · ρ µ . (6)For the SFHo and SFHx models, f is given by f ( σ, ω µ ω µ ) = X i =1 a i σ i + X j =1 b j ( ω µ ω µ ) j , (7)while for the FSU2R, FSU2H, TM1 and TM1e models,this function f reduces to f ( ω µ ω µ ) = Λ v g v ω µ ω µ . (8)For these four models, the coupling constant of the non-linear term ξ is absent.For the density-dependent models, DD2, DDME2, andD1, the isoscalar couplings of the mesons i to the baryonsare written in the following way g i ( n B ) = g i ( n ) a i b i ( x + d i ) c i ( x + d i ) , (9)and the isovector ones are given by g i ( n B ) = g i ( n ) exp [ − a i ( x − . (10)Here, n is the symmetric nuclear saturation density, and x = n B /n . For the D2 model, there are additionalterms in the vector density because of the energy de-pendent self-energies, meaning that n B and n ω are nolonger equal. For all density-dependent models, the cou-pling constants k , λ , ξ , and ζ are zero, together with the f function.The energy density E is given by: E N L = X i = p,n E i + 12 m σ σ − m ω ω − m ρ ρ + κ σ + λ σ − ζ
24 ( g ω ω ) − ξ
24 ( g ρ ρ ) − g ρ ρ f, (11)for the non-linear (NL) models, which includes severalnon-linear mesonic terms, and by E DD = X i = p,n E i + 12 m σ σ − m ω ω − m ρ ρ − Σ R n B , (12)for the density-dependent (DD) models. Σ R is therearrangement term that appears only in the density-dependent models (see Refs. [17, 18, 21]), and is givenby Σ R = ∂g ω ∂n B ω n B + ∂g ρ ∂n B ρ ( ρ p − ρ n ) / − ∂g σ ∂n B σρ s . (13)In Eqs. (11) and (12), the single-particle energies E i aregiven by E i = 1 π Z dp p ǫ ∗ i ( f i + + f i − ) , (14)the nucleon number density is ρ i = 1 π Z dp p ( f i + − f i − ) , (15)the scalar density is ρ is = 1 π Z dp p M ∗ ǫ ∗ i ( f i + + f i − ) , (16)the distribution functions are defined as f i ± = 11 + exp [( ǫ ∗ i ∓ ν i ) /T ] , (17)with ǫ ∗ i = p p + M ∗ , and the nucleons effective chemi-cal potential as ν i = µ i − g v V − g ρ t i b − Σ R , (18)where t i is the third component of the isospin operator,and the rearrangement term is included only for the DDmodels. The entropy density S is calculated from S = − X i = n,p Z d p π [ f i + ln f i + + (1 − f i + ) ln (1 − f i + )+ ( f i + ↔ f i − )] . (19)The free energy density F is then obtained from the ther-modynamic relation F = E − T S . (20) B. Stability Conditions
In the present study, we determine the region of in-stability of nuclear matter constituted by protons andneutrons by calculating the spinodal surface in the( ρ p , ρ n , T ) space. Stability conditions for asymmetricmatter impose that the curvature matrix of the free en-ergy density [22] C ij = (cid:18) ∂ F ∂ρ i ∂ρ j (cid:19) T , (21)or, equivalently, C = ∂µ n ∂ρ n ∂µ n ∂ρ p ∂µ p ∂ρ n ∂µ p ∂ρ p ! , (22)is positive. The stability conditions impose Tr( C ) > C ) >
0, which is equivalent to the requirementthat the two eigenvalues λ ± = 12 (cid:16) Tr( C ) ± p Tr( C ) − C ) (cid:17) (23)are positive. The largest eigenvalue, λ + , is always posi-tive, and the instability region is delimited by the surface λ − = 0. Interesting information is given by the associ-ated eigenvectors δρ ± , defined as δρ ± p δρ ± n = λ ± − ∂µ n ∂ρ n ∂µ n ∂ρ p . In particular, the eigenvector associated with the eigen-value that defines the spinodal surface determines theinstability direction, i.e. the direction along which thefree energy decreases.The critical points for different temperatures T , whichare important for the definition of conditions under whichthe system is expected to clusterize, are also going to becalculated. These points satisfy simultaneously [22, 23]Det( C ) = 0 (24)Det( M ) = 0 , (25)with M = C C ∂ |C| ∂ρ p ∂ |C| ∂ρ n ! . (26)The thermodynamical spinodals and respective criticalpoints are going to be calculated for a series of the intro-duced RMF models in the next section. III. RESULTS AND DISCUSSION
In this Section, we start by elaborating in more de-tail on the models we use. For each of them, we calcu-late the thermodynamic instability regions, the criticalpoints, the transition densities, and the isospin distilla-tion effect for a given temperature. To conclude, a dis-cussion of the results will be presented.
A. Models
In the present study we consider a set of RMF mod-els calibrated to properties of nuclei and nuclear mat-ter. These models fall into two different types: onewith density-dependent couplings, DD2, DDME2, D1,and D2, which we designate by DD models, and the otherwith non-linear couplings, SFHo, SFHx, FSU2R, FSU2H,TM1, and TM1e, which we designate by NL models.In Table I, some symmetric nuclear matter propertiescalculated at saturation density are given for the all themodels that we explore.Concerning the NL models, SFHo and SFHx includeseveral non-linear terms of higher order. They were con-structed in such a way that they both satisfy constraintscoming from nuclear masses, giant monopole resonances,and binding energies and charge radii of
Pb and Zr.Besides, they satisfy the 2- M ⊙ constraint [24], and thepressure of neutron matter is always positive and increas-ing. TABLE I. The symmetric nuclear matter properties at satu-ration density for the models under study: the nuclear satu-ration density ρ , the binding energy per particle B/A , theincompressibility K , the symmetry energy E sym , the slope ofthe symmetry energy L , and the nucleon effective mass M ∗ .All quantities are in MeV, except for ρ that is given in fm − ,and the effective nucleon mass is normalized to the nucleonmass.Model ρ B/A K E sym
L M ∗ /M SFHo 0.158 16.19 245 31.6 47 0.76SFHx 0.163 16.16 239 28.7 23 0.71FSU2R 0.15 16.28 238 30.7 47 0.59FSU2H 0.15 16.28 238 30.5 45 0.59TM1 0.145 16.3 281 36.9 111 0.63TM1e 0.145 16.3 281 31.4 40 0.63DDME2 0.152 16.14 251 32.3 51 0.57DD2 0.149 16.02 243 31.7 58 0.56D1 0.15 16.0 240 32.0 60 0.56D2 0.146 16.0 240 32.0 60 0.56 ρ/ρ E sy m ( M e V ) DD2DDME2SFHoSFHxFSU2RFSU2HD1D2TM1TM1e
FIG. 1. The symmetry energy as a function of the density forthe models under consideration.
FSU2H and FSU2R were calibrated in order to repro-duce the properties of finite nuclei, constraints from kaonproduction and collective flow in HIC, and to predictneutron matter pressures consistent with effective chiralforces. Both models reproduce 2 M ⊙ stars, have a sym-metry energy and its slope at saturation consistent withcurrent laboratory predictions, and their neutron skinthickness is compatible with several experiments, bothfor Pb and for Ca, as from measurements of theelectric dipole polarizability of nuclei.TM1e accurately describes finite nuclei, gives twosolar-mass neutron stars and radii compatible with thelatest astrophysical observations by NICER [25]. Itsslope of the symmetry energy is also consistent with as-trophysical observations and terrestrial nuclear experi-ments [26–28], while TM1 fails these constraints.With respect to the density-dependent models, D1 andD2 are close to DD2, which was fitted to properties of ρ n (fm -3 )00.020.040.060.080.1 ρ p (f m - ) SFHo
Symetric mattercritical points β equilibrium T=0 MeV β equilibrium T=10 MeVT=0 MeV FIG. 2. (Color online) The spinodal regions on the ( ρ n , ρ p )plane for the SFHo model at T = 0 , , ,
12 and 14 MeV.Also shown are the β -equilibrium EoS at T = 0 (green solid)and 10 MeV (green dashed), the critical points line (blackdashed), and the symmetric matter line (blue solid). nuclei and reproduces 2 M ⊙ stars. D2 includes an energydependence, that was fitted to the optical potentials [29].This model does not reach the 2-solar-mass constraintsince the EoS becomes very soft when the optical po-tential constraint is satisfied. DDME2 was adjusted toreproduce the properties of symmetric and asymmetricnuclear matter, binding energies, charge radii, and neu-tron radii of spherical nuclei.The density dependence of the symmetry energy forthese models is plotted in Fig. 1. Some conclusionsmay be drawn: SFHo is the model that presents a softersymmetry energy above ≈ . ρ and, even below thisdensity, it is only SFHx that is slightly softer. WhileDDME2, DD2, SFHo and SFHx are quite similar below0 . ρ , FSU2R, FSU2H and TM1e are clearly stiffer inthis range of densities. TM1 has an almost linear be-havior with density, presenting the smallest values below ≈ . − , and the largest above that value. In fact,above ≈ . − , all models have a similar behavior ex-cept TM1 that is much stiffer, and SFHx that is quitesoft. We will discuss how these behaviors reflect them-selves on the instability regions. B. Spinodal sections and critical points
In Fig. 2, we show the spinodal sections obtainedwith the SFHo model at different temperatures, imposing λ − = 0, defined in Eq. (23). The larger the temperature,the smaller the section, which will eventually be reducedto a point at the critical temperature, that correspondsto the critical end point (CEP), and occurs for symmetricmatter. For SFHo, the CEP occurs at T = 16 .
14 MeVand ρ = 0 .
051 fm − . It is interesting to notice that the T = 0 spinodal is convex at the ρ p = ρ n point. Many of TABLE II. The transition density ρ t , the correspondent pro-ton fraction Y p t , and the density of symmetric matter ob-tained at T = 0 MeV for some of the models considered inthis work.Model ρ t (fm − ) Y p t ρ sym (fm − )SFHx 0.122 0.041 0.103SFHo 0.105 0.047 0.101FSU2R 0.087 0.045 0.095FSU2H 0.092 0.046 0.095TM1e 0.094 0.050 0.094TM1 0.047 0.025 0.070DD2 0.081 0.034 0.095D1 0.082 0.032 0.102DDME2 0.087 0.039 0.099 the models previously studied are concave at this point,see for instance [30] for a discussion. In Ref. [30], onlythe model SIII [31] shows a quite abnormal behavior.A consequence of this behavior is the prediction thathighly asymmetric matter is still non-homogeneous atdensities close, or even above, the transition density fromnon-homogeneous to homogeneous matter of symmetricmatter, designated in the following as ρ sym . However,one would expect that the contribution of the repulsivesymmetry term to the binding energy of nuclear matterwould move the transition density to lower densities, asthe proton-neutron asymmetry increases.In the same Figure, the EoS for β -equilibrium mattercalculated at two different temperatures, T = 0 and 10MeV, is also represented. The crust-core transition den-sity at a given temperature may be estimated from theintersection of the EoS with the spinodal at that sametemperature. In Refs. [8, 32], it was shown that this is agood estimation although slightly larger than the valuesobtained within a Thomas-Fermi or a dynamical spinodalcalculations. For the two temperatures shown, we con-clude that: i) The T = 0 MeV EoS intercepts the T = 0spinodal at ρ t = 0 .
105 fm − , indicating that the crust ofa neutron star described by this model extends until ap-proximately this density. The line y p = 0 . ρ sym = 0 .
101 fm − , a density slightly smallerthan ρ t ; ii) the T = 10 MeV EoS does not intercept therespective spinodal, and this indicates that β -equilibriummatter at this temperature is homogeneous.The line of critical points is also displayed in the figure.At a given temperature, these are the two points in thespinodal section that have maximum pressure, and wherethe direction of the instability is parallel to the tangent tothe spinodal. This means that the pressure above P max belongs to the homogeneous matter phase.In Table II, the transition density of both β -equilibrium matter ρ t , and of symmetric matter, ρ sym ,are given, together with the proton fraction at the β -equilibrium transition for each model. For β -equilibriummatter the transition occurs for Y p ≪ .
5. All models ρ p ( f m - ) Critical points ρ p ( f m - )
000 0.04 0.08 0.12 ρ n (fm -3 ) ρ p ( f m - ) ρ n (fm -3 ) ρ p ( f m - ) T=0 MeV61012 FIG. 3. (Color online) The spinodal sections on the ( ρ n , ρ p )plane for SFHx (top left), SFHo (top right), FSU2H (bottomleft) and FSU2R (bottom right) at T = 0 , , ,
12, and 14MeV. The SFHx model is the only one that presents an un-stable region at T = 15 MeV. The critical points line is givenby the black dashed line. ρ p (f m - ) ρ n (fm -3 )00.020.040.060.080.10.12 ρ p (f m - ) T=0 MeV FIG. 4. (Color online) The spinodal sections on the ( ρ n , ρ p )plane for TM1 (top) and TM1e (bottom), at T = 0 , , , have ρ sym > ρ t , except TM1e, SFHo and SFHx, the lastmodel having an extreme transition density of ≈ . − . For TM1e both densities are equal. SFHo andSFHx are also the models that predict larger crust-coretransition densities.The spinodal sections obtained at different tempera-tures for the NL models we consider in this study areplotted in Figs. 3 and 4. SFHo and SFHx present a ρ p ( f m - ) ρ p ( f m - ) ρ n (fm -3 ) ρ p ( f m - ) ρ n (fm -3 ) ρ p ( f m - ) DD2
T=0MeV6101213
DDME2 D1 D2 FIG. 5. (Color online) The spinodal sections on the ( ρ n , ρ p )plane for DD2(top left), DDME2 (top right),D1 (bottom left)and D2 (bottom right) at T = 0 , ,
10, and 12 MeV. Thesmallest unstable regions shown are for T = 13 MeV (DD2and DDME2), 12.2 MeV (D1), and 14 MeV (D2). convex curvature at the transition density of symmet-ric matter. This seems to point to some problem in themodel. They also have a bigger instability region as com-pared to the other models. Comparing TM1 and TM1e,it is clearly seen that the ones with a smaller slope L at saturation have spinodal sections that extend to moreasymmetric matter, right up to almost the CEP, whichoccurs for symmetric nuclear matter. This implies thatin warm stellar matter in beta-equilibrium, as the onefound in neutron star mergers, finite clusters will appearat larger temperatures and proton asymmetries, havingdirect implications in processes like neutrino cross sec-tions.On the other hand, the spinodals for DD models, whichare plotted in Fig. 5, show a behavior closer to the onepresented by TM1, although having a much smaller slope L : the spinodal sections are smaller, do not extend to soasymmetric nuclear matter and they are all concave at y p = 0 . T, ρ c , y pc ) are plotted.At T = 0 MeV, the models SFHx, FSU2R, FSU2H andTM1e have a proton fraction at the critical point equalto zero or very close to zero. All other models have asimilar proton fraction of the order of 0.028-0.039. At T = 6 MeV, SFHx, FSU2R, FSU2H, and even TM1e,still present a critical proton fraction of the order of 0.01or below (for SFHx it is still zero), while for all the othermodels, it grows up to ≈ . − . TABLE III. The critical densities ρ c and the correspondentproton fractions Y p c for different temperatures for the modelsconsidered in this work.Model T (MeV) ρ c (fm − ) Y p c SFHx 0 0.1010 0.0SFHo 0.1015 0.0283FSU2R 0.0827 0.0037FSU2H 0.0876 0.0022TM1 0.0774 0.0496TM1e 0.0902 0.0041D2 0.0775 0.0296D1 0.0840 0.0390DD2 0.0796 0.0302DDME2 0.0839 0.0274SFHx 6 0.1015 0.0SFHo 0.0886 0.0850FSU2R 0.0673 0.0083FSU2H 0.0728 0.0063TM1e 0.0778 0.0154D2 0.0679 0.0809D1 0.0775 0.1110DD2 0.0702 0.0855DDME2 0.0750 0.0882SFHx 10 0.1019 0.0056SFHo 0.0746 0.1395FSU2R 0.0633 0.0304FSU2H 0.0676 0.0251TM1 0.0601 0.1594TM1e 0.0708 0.0339D2 0.0569 0.1412D1 0.0661 0.2181DD2 0.0578 0.1523DDME2 0.0612 0.1707SFHx 14 0.0920 0.09SFHo 0.0583 0.2509FSU2R 0.0490 0.2686FSU2H 0.0477 0.2607TM1e 0.0619 0.1244D2 0.0463 0.4167D1 - -DD2 - -DDME2 - - keeping a critical proton fraction equal to zero for
T < ≈ . − for T <
12 MeV. The models FSU2H and FSU2R also showa critical proton fraction very close to zero for
T < p T c ( M e V ) FSU2HFSU2RSFHxSFHoD1D2DD2DDME2 ρ (fm -3 ) T c ( M e V ) FIG. 6. (Color online) The critical proton fraction (top) andthe critical density (bottom) as a function of the temperature T for some of the models considered in this work. core-collapse supernova matter or neutron star mergerssince the non-homogeneous matter will extend to largerdensities and larger temperatures. The models SFHx,FSU2R, FSU2H and TM1e predict clusterization of quiteasymmetric matter for quite high temperatures. This willaffect the evolution of asymmetric stellar matter as foundin neutron star mergers, or core-collapse supernova mat-ter after the neutrino trapped stage.The CEP properties, i.e. the critical temperatures andcorresponding critical densities and pressures, are givenfor each model in Table IV. At the critical temperature,matter is symmetric. The largest critical temperature, ofthe order of 16 MeV, is obtained for SFHx and SFHo. D1presents the smallest critical temperature of the order of12 MeV.In [33], the authors made a compilation of experimen-tal determinations of the critical temperature of sym-metric nuclear matter. The measurements were per-formed within multifragmentation reactions or fission,and the critical temperature values fluctuate between15 and 23 MeV. However, some of the estimations areobtained with large uncertainties. The analysis withsmaller uncertainties [34] determined a critical temper-ature of 16.6 ± .
86 MeV, considering the limiting tem-perature values obtained in five different mass regions[35], where the authors obtained a temperature above15 MeV, using both multifragmentation and fission pro-cesses. In Ref. [36], the authors used results from six
TABLE IV. The critical temperatures, and their correspon-dent critical densities and pressures at the CEP, for the mod-els considered in this work. The proton fraction is equal to0.5.Model T c (MeV) ρ c (fm − ) P c (MeV.fm − )SFHx 15.81 0.052 0.242SFHo 16.14 0.051 0.249FSU2R 14.19 0.045 0.186FSU2H 14.16 0.044 0.183TM1 15.62 0.049 0.239TM1e 15.61 0.049 0.239DD2 13.73 0.046 0.178DDME2 13.12 0.045 0.156D1 12.22 0.058 0.187D2 14.14 0.046 0.193 different sets of experimental data, both involving com-pound nuclei or multifragmention, and the critical tem-perature of 17.9 ± . ± .
01 fm − , and 0.31 ± .
07 MeV/fm ,respectively. They used Fisher’s droplet model, that wasmodified to account for several effects, such as Coulomb,finite size or angular momentum effects.Regarding the models we consider in this study, crit-ical temperatures above 15 MeV are obtained for TM1,TM1e, SFHo, and SFHx. DD models have generally acritical temperature of the order of 14 MeV, or below,and FSU2R and FSU2H have a critical temperature justabove 14 MeV. Concerning the critical density, all mod-els have a density ρ c & .
044 fm − , but only the mod-els SFHx, SFHo, TM1, TM1e, and D1 predict a density & .
05 fm − , as determined in Ref. [36]. SFHx, SFHo,TM1 and TM1e are the models that predict a criticalpressure within the range obtained in Ref. [36].In [33], the authors have determined the CEP of sev-eral RMF models, and, from all the models tested, onlythe DD models and the models named Z271 predicted acritical temperature above 15 MeV, and the critical pres-sure and density within the range proposed in [36]. Weshould, however, refer that the models Z271 predict amaximum stellar mass below 1.7 M ⊙ , as shown in [37]. C. Transition densities
In the following, we discuss the transition densitiesfrom non-homogeneous to homogeneous matter underdifferent proton fraction conditions.In Fig. 7, we show the transition densities as afunction of the temperature for two different cases: i) β − equilibrium; ii) a fixed proton fraction of 0.3, a frac-tion that is representative in core-collapse supernovamatter. Inside the represented region, matter is, in prin-ciple, non-homogeneous. This is only an estimation of ρ t (f m - ) DD2DDME2FSU2RFSU2HSFHxSFHo β -eq ρ t ( f m - ) y p =0.3 FIG. 7. (Color online) The transition density, ρ t , as a functionof the temperature for β − equilibrium (top) and fixed protonfraction (bottom) matter for some of the models consideredin this work. the instability region, since we are not taking into ac-count finite size effects.For y p = 0 .
3, all models coincide at low densities andtemperatures below 10 MeV. At the upper limit, the tran-sition densities take the values 0 . ± .
01 fm − at T = 0,and up to T ≈
10 MeV, they decrease ∼ .
02 fm − .There exists experimental data that constrain matterwith this kind of asymmetry, and they show that thetemperature does not affect much the properties of nu-clear matter below 10 MeV. A larger discrepancy is foundfor temperatures above 10 MeV. The critical temperaturefor this matter asymmetry varies between 12 and almost16 MeV, with SFHo and SFHx models giving the largesttemperatures, and DD2 and DDME2 the lowest ones. β − equilibrium matter has a much smaller proton frac-tion, and there are no experimental data that can con-strain the EoS of this kind of matter. Let us, however,recall that all the models satisfy constraints coming fromchiral effective field theory calculations for neutron mat-ter. For β − equilibrium matter, we verify that the in-stability region estimated by the models considered varya lot. SFHx predicts a T = 0 transition density abovethe one obtained for y p = 0 .
3, and a critical tempera-ture ≈
14 MeV. Although with more reasonable transi-tion densities at low temperatures, FSU2H and FSU2Ralso predict very large critical temperatures, ≈
12 MeV.All the other models predict a critical temperature ofthe order of 3 MeV, but show a large dispersion on the transition density, with SFHo going above 0.1 fm − . InRef. [38], the authors have discussed the influence of thedensity dependence of symmetry energy on the supernovaevolution considering the models TM1 and TM1e. Theyconcluded that there are only minor effects around thecore bounce and in the first milliseconds considering theevolution of stars with masses of the order of 12-15 M ⊙ ,precisely because the proton fractions are still not too farfrom symmetric matter at this stage, and the predictionsfrom both models do not differ much. However, moredrastic differences between TM1 and TM1e were foundat a later stage, with TM1e giving rise to larger neutrinoemissions and a slower decay of the neutrino luminosities.As referred before, the thermodynamic calculation ofthe instability regions only allows an estimation of theregion where non-homogeneous matter is expected. Fi-nite size effects due to the finite range of nuclear forceand Coulomb interaction effects will affect the extensionof the region of instability, as discussed in [8]. The au-thors showed that the transition density obtained froma dynamical spinodal approach would predict transitiondensities that are ≈ . − lower and proton fractions10% smaller, which are good lower limit estimations, ascompared to a thermodynamical spinodal calculation. AThomas-Fermi calculation of the non-homogeneous mat-ter may give slightly larger transition densities, as shownin [32].In Fig. 8, we show the transition densities between thedifferent nuclear pasta phases, together with the transi-tion density to homogeneous matter, for five of the mod-els under consideration. These densities were calculatedfrom a Thomas-Fermi approximation at T = 0 MeV and β − equilibrium matter [39]. As expected, the crust-coretransitions obtained in these calculations are lower thanthe ones estimated from our thermodynamical approach,by not more than 0.01 fm − . It is interesting to noticethat while DD2 and DDME2 predict a large extensionof the spherical clusters in the inner crust, a shorter ex-tension of the rod phase, and no slab phase, or a verynarrow one, the models FSU2H, FSU2R, and TM1e pre-dict similar extensions of the droplet-like, rod-like andslab-like pasta structures. These different geometries willcertainly affect the transport properties of the neutronstar inner crust. D. Distillation effect
Transport properties are also affected by the protoncontent of the gas phase, when matter clusterizes. Inthe following, we analyse how the system tends to sep-arate into two phases, and the isospin content of each.We designate by isospin distillation effect the tendencyof matter to separate into a low-density phase, the gasphase, that is more neutron rich, i.e. with low protonfraction, and a high-density phase, the clusters, with aproton fraction closer to the one of symmetric matter,i.e. with high proton fraction.
DropletsRodsSlabsTubesBubblesHomMat 0.02 0.04 0.06 0.08 0.1T=0 MeV, β -eqTF calculation ρ (fm -3 ) DD2DDME2FSU2RFSU2HTM1e FIG. 8. (Color online) The different pasta structures from aThomas-Fermi calculation for cold β − equilibrium matter forsome of the models under consideration. δ ρ p - / δ ρ n - FSU2HFSU2R
SFHoSFHx δ ρ p - / δ ρ n - DD2DDME2
D1D2 ρ/ρ δ ρ p - / δ ρ n - TM1 ρ/ρ TM1eT=6MeV T=12MeV T=0MeVy p = 0.05y p =0.3 FIG. 9. (Color online) The fluctuations δρ − p /δρ − n at T =0 ,
6, and 12 MeV as function of the density, with Y p = 0 . .
05 (thin lines), for FSU2R and FSU2H(top left), SFHo and SFHx (top right), DD2 and DDME2(middle left), D1 and D2 (middle right), TM1 (bottom left)and TM1e (bottom right).
In Fig. 9, we show the isospin distillation effect forall models, by plotting the ratio of the proton to theneutron density fluctuations inside the instability region.The higher the ratios, the higher the distillation effect,because the clusterized phase becomes proton richer. Asexpected, the lower the temperature, the higher the dis-tillation effect. The SFHo and SFHx models predict thelargest distillation effects, meaning that, within thesetwo models, the gas phase has the lowest proton frac-tion. This is a reflection of their small symmetry energy.FSU2H and FSU2R present a very similar behavior to theone of DD2 and DDME2 at the lowest densities, but theirdensity ratios decrease faster with the density, and fordensities around half the saturation density, these mod-els have the smallest distillation effect. Finally, D1 andD2 do not differ much from DD2 model.
IV. CONCLUSIONS
In the present work, we have studied the extension ofthe non-homogeneous phase of warm and asymmetric nu-clear matter, considering several recently-proposed cali-brated RMF models. At T = 0 MeV, these models havebeen constrained by nuclear properties, ab-initio theo-retical calculations for neutron matter, and neutron starobservations. No constaint was imposed at finite tem-perature. The thermodynamical spinodal sections in the( ρ p , ρ n ) plane for several temperatures and the criticalpoints have been calculated.The main conclusions are: i) for symmetric nuclearmatter, the transition density to homogeneous matterspreads over a range narrower than 0.01 fm − , 0 . <ρ sym < . y p = 0 .
3, the transition density to homogeneous mat-ter obtained from the models considered is compatiblewithin ≈ .
02 fm − , for temperatures below 8 MeV;iii) above T = 8 MeV, the models differ much more,and the critical temperatures vary in a range of 4 MeV,12 . < T c < . β -equilibrated stellar matter, dif-fer a lot, both on the transition density, and on the criti-cal temperature above which β -equilibrated matter is notclusterized. SFHo and SFHx models predict transitiondensities from clusterized matter to homogeneous matterfor β -equilibrated matter larger than that for symmetricmatter. This behavior is somehow strange since it wouldbe expected that the extension of the instability region ofasymmetric matter would be smaller than the one of sym-metric nuclear matter due to the symmetry energy contri-bution, which is a repulsive contribution. Concerning thecritical temperature of β -equilibrated matter, the modelsSFHo, SFHx, FSU2R and FSU2H predict a temperaturethat is just . et al. [38] have shown, by using two models,TM1 and TM1e [16], that are only different in the isospinchannel, that a softer symmetry energy is responsible fora more drastic evolution of the protoneutron star withlarger neutrino emissions, giving rise to higher neutrinoluminosities and average energies. We may, therefore,expect stronger effects with SFHo and SFHx models.0 ACKNOWLEDGMENTS
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