PHYSICAL REVIEW C 82, 065801 (2010)
Statistical description of complex nuclear phases in supernovae and proto-neutron stars
Ad. R. Raduta1 and F. Gulminelli21NIPNE, Bucharest-Magurele, POB-MG6, Romania
2LPC (IN2P3-CNRS/Ensicaen et Universite), F-14076 Caen cedex, France(Received 10 September 2010; revised manuscript received 22 October 2010; published 9 December 2010)
We develop a phenomenological statistical model for dilute star matter at finite temperature, in which freenucleons are treated within a mean-field approximation and nuclei are considered to form a loosely interactingcluster gas. Its domain of applicability, that is, baryonic densities ranging from about > 108 g/cm3 to normalnuclear density, temperatures between 1 and 20 MeV, and proton fractions between 0.5 and 0, makes it suitablefor the description of baryonic matter produced in supernovae explosions and proto-neutron stars. The firstfinding is that, contrary to the common belief, the crust-core transition is not first order, and for all subsaturationdensities matter can be viewed as a continuous fluid mixture between free nucleons and massive nuclei. Asa consequence, the equations of state and the associated observables do not present any discontinuity over thewhole thermodynamic range. We further investigate the nuclear matter composition over a wide range of densitiesand temperatures. At high density and temperature our model accounts for a much larger mass fraction boundin medium nuclei with respect to traditional approaches as Lattimer-Swesty, with sizable consequences on thethermodynamic quantities. The equations of state agree well with the presently used EOS only at low temperaturesand in the homogeneous matter phase, while important differences are present in the crust-core transition region.The correlation among the composition of baryonic matter and neutrino opacity is finally discussed, and we showthat the two problems can be effectively decoupled.
DOI: 10.1103/PhysRevC.82.065801 PACS number(s): 21.65.Mn, 24.10.Pa, 26.50.+x, 26.60.c
Nuclear matter is not only a theoretical idealization provid-ing benchmark studies of the effective nuclear interaction, butit is also believed to constitute the major baryonic componentof massive objects in the universe, as exploding supernovaecores and neutron stars. The structure and properties ofthese astrophysical objects at baryonic densities exceedingnormal nuclear matter density is still highly speculative .Conversely, for subsaturation densities, it is well establishedthat matter is mainly composed of neutrons, protons, electrons,positrons, and photons in thermal and typically also chemicalequilibrium [9,10]. Depending on the thermodynamic condi-tion, neutrinos, and antineutrinos can also participate to theequilibrium.
Such composite matter is subject to the contrasting cou-plings of the electromagnetic and the strong interaction.Because of the electron screening, the two couplings act oncomparable length scales giving rise to the phenomenon offrustration , well-known in condensed matter physics. Because of this, a specific phase diagram, differingfrom the one of nuclear matter and including inhomogeneouscomponents, is expected in stellar matter .
Many theoretical studies exist at zero temperature. In a coldneutron star, going from the dilute crust to the dense core, atransition is known to occur from a solid phase constituted offinite nuclei on a Wigner lattice immersed in a backgroundof delocalized electrons and neutrons through intermediateinhomogeneous phases composed of nonspherical nuclei(pasta phases) to a liquid phase composed of uniform neutrons,protons, and electrons .
At finite temperature the matter structure and propertiesare not as well settled. The most popular phenomenological
approaches are the Lattimer-Swesty  (LS) and the Shen equation of state, recently updated in Ref. . In thesestandard treatments currently used in most supernovae codes,the dilute stellar matter at finite temperature is described inthe baryonic sector as a statistical equilibrium among protons,neutrons, s, and a single heavy nucleus. The transition tohomogeneous matter in the neutron star core is supposed tobe first order in these modelizations and obtained througha Maxwell construction in the total density at fixed protonfraction.
It is clear that such single nucleus approximation (SNA) ishighly schematic and improvements are possible. Concerningintegrated quantities as thermodynamic functions and equa-tions of state, such variables may be largely insensitive tothe detailed matter composition , though we show in thisarticle that this is not always the case. However, it is alsoknown that the composition at relatively high density close tosaturation, together with the pressure and symmetry energy,governs the electron capture rate, which in turn determinesthe proton fraction at bounce and the size of the homologouscore, a key quantity to fix the strength of the shock-wave andthe output of the supernovae explosion . Moreoverthe composition may also affect the nucleosynthesis of heavyelements, which is still poorly understood , as well asthe neutrino scattering through the core after bounce [37,38]and the cooling rate of neutron stars [39,40]. For these reasons,in recent years, many efforts have been done to improve thesimplistic representation of stellar matter given by the SNAapproach.
The different modelizations which consider a possibledistribution of all different nuclear species inside dilute stellarmatter are known under the generic name of nuclear statistical
0556-2813/2010/82(6)/065801(28) 065801-1 2010 The American Physical Society
AD. R. RADUTA AND F. GULMINELLI PHYSICAL REVIEW C 82, 065801 (2010)
equilibrium (NSE) . The basic idea behind thesemodels is the Fisher conjecture that strong interactions indilute matter may be entirely exhausted by clusterization. In these approaches stellar matter in the baryonicsector is then viewed as a noninteracting ideal gas of allpossible nuclear species in thermal equilibrium. The resultis that thermodynamic quantities like entropies and pressureappear very similar to the ones calculated with standardapproaches, while noticeable differences are seen in the mattercomposition. In particular, an important contribution of lightand intermediate mass fragments is seen at high temperature,which is neglected in standard SNA approaches.
The strongest limitation of NSE-based approaches is thatthey completely neglect in-medium effects, which are knownto be very important in nuclear matter. Since the onlynuclear interactions are given by the cluster self-energies, thehomogeneous matter composing the neutron star core cannotbe modelized, nor can it be the phenomenon of neutron drip inthe inner crust, well described by mean-field models . As aconsequence, these models cannot be applied at densities closeto saturation and the crust-core transition cannot be described.
To overcome this problem, different microscopic as well as phenomenological  approaches have beendeveloped in the very recent past. In this article we wouldlike to introduce a phenomenological model that treats thenuclei component within an improved NSE, while it describesthe unbound protons and neutrons in the finite temperatureHartree-Fock approximation.
The plan of the article is as follows. The first part ofthe article is devoted to the description of the model. Theclusterized component, the homogeneous component, theproperties of the mixture, and the lepton sector are describedin successive sections together with their thermodynamicproperties. Particular attention is devoted to the modelizationof the crust-core transition. We show that the inclusion ofexcluded volume is sufficient to describe the transition fromthe clusterized crust to the homogeneous core and that thistransition is continuous. Different generic as well as specificarguments are given against the possibility of a first-ordertransition. The second part of the article gives some resultsrelevant for the star matter phenomenology. The first sectionshows observables following constant chemical potentialpaths, in order to connect the observables with the properties ofthe phase diagram. Then the behavior of the different quantitiesfor constant proton fractions is displayed in order to comparewith more standard treatments of supernova matter. Finally,the last section addresses the problem of neutrino trappingand the interplay between the matter opacity to neutrinos andmatter composition. Conclusions and outlooks conclude thearticle.
II. THE MODEL
The model aims to describe the thermal and chemicalproperties of nuclear matter present in supernovae and(proto-)neutron stars at densities ranging from the normalnuclear density 0 to 1060, temperatures between 0 and20 MeV, and proton concentration between 0.5 and 0. Inthis regime, the star matter typically consists of a mixture
of nucleons, light and heavy nuclear clusters, neutrinos (ifwe consider the thermodynamic stage where neutrinos aretrapped), photons, and a charge-neutralizing background ofelectrons and positrons.
As there is no interaction among electrons, neutrinos, pho-tons, and nuclear matter, the different systems may be treatedseparately and their contributions to the global thermodynamicpotential and equations of state added up.
In the grand-canonical ensemble this reads,
G(, n, p, e, V )
= G(bar)(, n, p, V ) +G(lep)(, e, V )+G( )(, V ), (1)
where the grand-canonical potential,
G(, n, p, e, V ) = lnZgc(, n, p, e, V )= S[,n,p], (2)
is the Legendre transformation of the entropy S with respect tothe fixed intensive variables. In the previous equations V is anarbitrary macroscopic volume and Zgc is the grand-canonicalpartition sum.
The observables conjugated to the ones fixed by thereservoir and geometry can be immediately calculated aspartial derivatives of G. Thus, the total energy density is
e = 1V
= e(bar) + e(lep) + e( ), (3)the different particle densities are
i = 1V
where i = n, p, e, and, finally, the total pressure is
p = GV
= p(bar) + p(lep) + p( ). (5)
A. The baryon sector
The light and heavy nuclei are assumed to form a gas ofloosely interacting clusters which coexist in the Wigner-Seitzcell with a homogeneous background of delocalized nucleons.To avoid exceeding the normal nuclear density and naturallyallow for homogeneous-unhomogeneous matter transition,nuclei and nucleons are forbidden to occupy the same volume.In the following we start describing the modelization of thesetwo components separately, and we turn successively to theproperties of the mixture obtained when the two are supposedto be simultaneously present in the Wigner-Seitz cell.
1. The homogeneous nuclear matter component
Mean-field models constitute a natural choice for approach-ing interacting particle systems. By introducing a mean-fieldpotential, the physical problem is reduced to the simplifiedversion of a system of noninteracting particles. Effectivenucleon-nucleon interactions allow one to express the system
STATISTICAL DESCRIPTION OF COMPLEX NUCLEAR . . . PHYSICAL REVIEW C 82, 065801 (2010)
average energy as a simple single-particle density functionaland to cast the nuclear matter statistics in a way which isformally very similar to an ideal Fermi gas .
The mean-field energy density of an infinite homogeneoussystem e(HM) = H 0/V is a functional of the particle densitiesq and kinetic densities q for neutrons (q = n) or protons(q = p). At finite temperature, the mean-field approximationconsists in expressing the grand-canonical partition functionof the interacting particle system as the sum of the grand-canonical partition function of the corresponding independent-particle system associated to the mean-field single-particleenergies, with the temperature weighted difference betweenthe average single-particle energy (W 0 = lnZ0) and themean-field energy
lnZ (HM) lnZ (HM)0 + (W 0 H 0). (6)The one-body partition sum is defined as
Z (HM)0 = Tr[e(W0nNnpNp)] = Zn0Zp0 (7)and can be expressed as a function of the neutron and protonkinetic energy density
[1 + e
dp = h2
q = 2
q = 2
In these equations the effective chemical potential qincludes the self-energies according to q = q q e(HM),mn,p = (eHM/n,p)1/2 are the neutron (proton) effectivemass, and the factor 2 comes from the spin degeneracy.
Equation (9) establishes a self-consistent relation betweenthe density of q particles q and their chemical potential q .
Introducing the single particles energies iq = p2i
2mq+ q e(HM),
the above densities can be written as regular Fermi integralsby shifting the chemical potential according to q = q q e
HM. The Fermi-Dirac distribution indeed reads:
nq(p) = 11 + exp[(p2/2mq q)]
Equations (11) and (9) define a self-consistent problemsince mq depends on the densities. For each couple (
a unique solution (n, p) is found by iteratively solving theself-consistency between n,p and mn,p. Then Eq. (10) is usedto calculate n,p.
At the thermodynamic limit the system volume V divergestogether with the particle numbers Nn, Np, and thethermodynamics is completely defined as a function of the twoparticle densities (n, p) or, equivalently, the two chemicalpotentials (n,p).
With Skyrme-based interactions, the energy density ofhomogeneous, spin-saturated matter with no Coulomb effects
TABLE I. SKM* force parameters .
t0 (MeV fm3) 2645t1 (MeV fm5) 410t2 (MeV fm5) 135t3 (MeV fm3+3 ) 15 595x0 0.09x1 0.0x2 0.0x3 0.0 1/6
is written as:
e(HM) = h2
2m(n + p)
+ t0(x0 + 2)(n + p)2/4 t0(2x0 + 1)(2n + 2p
+ t3(x3 + 2)(n + p)+2/24 t3(2x3 + 1) (n + p)
(2n + 2p
)/24 + [t1(x1 + 2)
+ t2(x2 + 2)](n + p)(n + p)/8 + [t2(2x2 + 1) t1(2x1 + 1)](nn + pp)/8, (12)
where t0, t1, t2, t3, x0, x1, x2, x3, and are Skyrme parameters.Several Skyrme potentials have been developed over the
years for describing the properties of both infinite nuclearmatter and atomic nuclei and address the associated thermo-dynamics. It was thus in particular shown that nuclear mattermanifests liquid-gas like first-order phase transitions up to acritical temperature .
In order to make direct quantitative comparisons betweenour model and the one of Lattimer and Swesty , throughthis article nucleon-nucleon interactions are accounted foraccording to the SKM* parametrization  if not explicitelymentioned otherwise. Table I summarizes the force parameterswhile the main properties of nuclear matter are summarized inTable II.
The thermodynamic properties of the system are beststudied introducing the constrained entropy:
s(HM)c = s(HM) eHM, (13)where the entropy density is nothing but the Legendretransform of the mean-field partition sum:
s(HM) = lnZ (HM)/V + [e(HM) nn pp]. (...