1 Introduction
At densities below the nuclear saturation density and not too high temperatures (
This work follows the one in Ref. [22], where light clusters are calculated in the relativistic mean field (RMF) framework [23]. Both in-medium mass shifts and in-medium modification of the cluster couplings are discussed. We also perform a new calculation for the heavy cluster within the compressible liquid drop (CLD) model, including light clusters. The results shown in this work will be further explored in a more detailed article, now in preparation [24].
At very low densities, we use the model-independent constraint, the virial EoS (VEoS) [1, 25, 26], to fix the cluster-meson couplings so that the VEoS particle fractions obtained in Ref. [26] are well reproduced. This constraint only depends on the experimentally determined binding energies and scattering phase shifts, and provides the correct zero density limit for the equation of state at finite temperature.
In the high-density regime, the cluster dissolution mechanism is quite well described by the geometrical excluded volume mechanism [27, 28], so that we employ the Thomas-Fermi formulation of Ref. [29] to evaluate the associated cluster mass shift, and we obtain a simple analytical formula for the effective mass shift. To reproduce empirical data, an in-medium modified coupling of cluster j with the scalar meson σ of the form gsj=xsAjgs is proposed, where gs is the coupling constant with the nucleons (n, p), Aj the cluster mass number, and xs is a universal cluster coupling fraction, with an associated uncertainty.
Besides the four standard charged light particles, 4He, 3H, 3He, and 2H, as the density increases, heavier clusters can also form, like 5H, 7H. Eventually these clusters become very heavy, and the pasta phases appear. In this work, we are also interested in exploring the effect of such pasta structures within a CLD calculation [30] that includes light clusters with different sizes, since we want to understand if these heavier light clusters, which are usually ignored in pasta calculations, should also be included in calculations for stellar matter. The CLD calculation is based on the coexistence phase (CP) approximation, where the Gibbs equilibrium conditions are imposed to get the lowest free-energy state, with the difference that in the CLD method, the surface and Coulomb terms are added to the free energy before the minimisation is performed.
2 Theoretical description
We consider a system of protons and neutrons that interact via the exchange of mesons: the scalar σ, the vector ω, and the isovector ρ. Light clusters, deuteron (d), triton (t), helion (h) and α, are taken into account as new degrees of freedom. Electrons must also be included since we are dealing with stellar matter. The Lagrangian density, based on the non-linear Walecka model, is given in Ref. [22].
The total binding energy of each cluster is defined as
with m* the nucleon effective mass, and
where
The binding energy shift, δ Bj, takes in-medium effects into account, and needs to be determined. It is the energetic counterpart of the classical excluded-volume mechanism. Since
we avoid double counting because the energy states occupied by the gas are excluded. In the above expressions, fj±(p) are the Fermi distribution functions for the particles and anti-particles:
with
The other quantity that considers in-medium effects is the scalar cluster-meson coupling, gsj=xsjAjgs, which is determined from experimental constraints. We fix xsj so that in the low-density limit the Virial EoS is reproduced. We obtained [22] xsj=0.85± 0.05 as good universal scalar cluster-meson coupling, that not only reproduces reasonably well the Virial EoS but also reproduces well data coming from heavy-ion collisions in the high density limit.
In the compressible liquid drop model (CLD) [30], matter is divided in two main regions: a high density phase (I), where the heavy cluster forms, and the low density phase (II), where a background nucleon gas exists and where the light clusters can form.
We obtain the equilibrium conditions of the system from the minimization of the total free energy, including the surface and Coulomb terms. The free energy density is given by
This minimization is done with respect to four variables: the size of the geometric configuration, rd, which gives, just like in the CP case, the condition εsurf= 2εCoul, the baryonic density in the high-density phase, ρI, the proton density in the high-density phase,
The equilibrium conditions then become
with α=f for droplets, rods and slabs, α=1-f for tubes and bubbles. The expression for Φ depends on the dimension, D and volume fraction, f, of the heavy clusters, and is given by Ref. [30]
For each phase, the light clusters, which we extend to A=12, are in chemical equilibrium, with the chemical potential of each cluster defined as:
and charge neutrality must also be imposed:
with ρe the electron density and ρc the charge density. Equations (9), (11) and (12) need to be solved self-consistently for the low-energy state to be found.
3 Results and discussion
In the following, we show some of the results obtained in this work, at finite fixed temperatures and for fixed proton fractions yp which describes the ratio of the total proton density to the baryon density. We start by explaining how we determined the cluster-meson coupling fraction, xs, from the Virial equation of state (VEoS). Then we investigate the effect of introducing the binding energy shift δ Bj, and its consequence on the clusters distributions, and we also calculate the equilibrium constants, comparing our results with data coming from heavy-ion collisions [31]. Finally, a calculation with heavy cluster from a compressible liquid drop (CLD) approximation is done, where we also include light clusters with a nucleon number, A, up to 12.
Determination of xs: Virial EoS
The cluster-meson couplings are obtained from the best fit of the RMF cluster mass fractions, defined as Xj = Ajnj/n, to the VEoS data, taking the FSU parametrization [32] model. This model has been chosen because it describes adequately the properties of nuclear matter at saturation and subsaturation densities. The fit is done choosing a sufficiently low density close to the cluster onset, where the virial EoS is still valid, and, at the same time, the interaction already has non-negligible effects. We have considered densities between 10-6 fm-3 and 10-4 fm-3, a range of densities where we expect the VEoS to be a good approximation. In this low-density domain, the binding energy shift δBj of Eq. (3) is completely negligible, and does not affect the particle fractions (see also Figure 2 below), therefore it was put to zero for this calculation.
Only the gsj parameters are optimized,
while the vector couplings are set to
Reasonable values for gsj are (0.85 ± 0.05)Ajgs, see Fig. 1, where the colored bands show the range of particle fractions covered by this interval at low densities, for T=4 and 10 MeV. The solid vertical black lines, defined by
Effect of the binding energy shift δ Bj
Let us now discuss the effect of introducing a non-zero binding energy shift δ Bj, Eq. (3). In Fig. 2, we compare the binding energy of the α-clusters, obtained taking δBα, as defined by Eq. (3), with the binding energy
obtained from quantum statistical (QS) calculations, with
Also shown in this Figure is a QS calculation from a perturbative approach where the Pauli blocking shift of α particles with center-of-mass momentum (wave number) P = 0 was obtained at the lowest order of density ρ [34]
(in units of MeV, with T in units of MeV, and ρ in units of fm-3). Lastly, Fig. 2 also shows a calculation from Typel et al. [35], where, in order to suppress cluster formation at higher densities, they introduced an empirical quadratic form given by
We can see from Fig. 2 that the additional binding energy shift δ Bj given by Eq. (3) is completely negligible in the domain of validity of the VEoS, which means that the cluster couplings do not depend on this term. Even for higher densities and still in the range where the total binding energy of the clusters is positive, this extra correction is small but will rise fast as the density increases, as it can be seen in the next Figure.
It is also interesting to discuss the effect of the coupling xsj and temperature T on the binding energy shift. From Fig. 3 we conclude that the larger xsj the slower -δ Bj increases, and also that a larger temperature determines a softer behavior, with -δ Bj taking larger values at the lower densities and smaller ones close to the dissolution density.
However, we should stress that Figs. 2 and 3 do not give a complete picture of the in-medium effects and cluster dissolution mechanism, because the mass shift strongly modifies the equations of motion for the meson fields. The particle fractions are thus affected in a highly complex way because of the self-consistency of the approach, which additionally induces temperature effects, as we will see next.
Effect of δ Bj on the global cluster distributions
We show in the present subsection that the clusters are dissolved below the nuclear saturation density, ρ0. In Fig. 4 we show the clusters mass fractions for matter with a fixed proton fraction of yp=0.41, T=5 MeV, and xsj=0.85. We observe that if we neglect the δBj term, the clusters do not dissolve, which is precisely the role of this extra term in the binding energy. For the temperatures and proton fractions presented, the typical values for the dissolution (Xj < 10-4) of light clusters are within the density range 0.04fm-3 < ρ < 0.06fm-3.
Equilibrium constants
In the high-density limit, a constraint was proposed in Ref. [31]. These chemical equilibrium constants, Kc[j], calculated with data from heavy-ion collisions, are defined as
where ρj is the number density of cluster j, with neutron number Nj and proton number Zj, and ρp, ρn are, respectively, the number densities of free protons and neutrons. For this calculation we fix the proton fraction to 0.41 as was done in [31, 36].
In Fig. 5, we show the chemical equilibrium constants for all the light clusters considered, taking the range of the couplings to be gsj=(0.85±0.05)Ajgs, and we compare with the experimental results of Ref. [31]. We can see that taking the coupling fractions xsj=0.85±0.05 essentially describes the experimental equilibrium constants. We have checked that xs=0.95 would be too large.
This experimental data seem to put extra constraints, that together with VEoS, suggest that a good universal coupling for all clusters is gsj = (0.85 ± 0.05)Ajgs. For the deuteron, the experimental data seem to be described by the upper limit xs=0.9. Possibly a more detailed approach would allow for a different coupling gsj for each cluster.
Influence on the pasta structure
Until now, we have considered homogeneous matter (HM) with light clusters. We next test how is the fraction of heavy clusters (pasta) affected with the inclusion of the light clusters. For that, we consider a CLD calculation with light clusters, where the inclusion of the binding energy shift term of the light clusters, δ Bj, is also considered. The heavy cluster is always calculated in the droplet configuration.
Another important remark is that, besides considering the 4 usual light clusters, i.e., 4He, 2H, 3H, and 3He, we are also taking all the bound clusters with A ≤ 12.
In Fig. 6, we show for a fixed proton fraction of 0.2 and T=5,7 MeV, the total mass fraction of clusters, light and heavy, in both a CLD and a HM calculation. For the cluster-meson couplings, we are taking xs=0.85, and, in all calculations, xv=1, with gsi=xsgsAi and gvi=xvgvAi. Looking at the abundance of light clusters, we see that it is higher in the HM calculation, because the CLD also considers the heavy cluster. However, the melting of these clusters in the CLD case occurs at a higher density. If we consider the heavy cluster abundancy, we see that it decreases when the calculation includes light clusters, and its onset occurs at a higher density as compared to the CLD case. The background of free nucleons is also higher in this case.
4 Summary
In summary, a simple parametrisation of in-medium effects acting on light clusters was proposed in a RMF framework. The interactions of the clusters with the medium was described by a modification of the σ-meson coupling constant. The cluster dissolution was obtained by the density-dependent extra term on the binding energy, δ Bj. The fraction xsj=0.85 ± 0.05 reproduces both virial limit and Kc from HIC. The inclusion in the CLD (heavy cluster) calculation of a larger number of degrees of freedom through light clusters not only reduces the size of the heavy cluster but also increases the fraction of free nucleons in the background gas. Overall, we find that the influence of light, loosely bound clusters, beside 2H, 3H, 3He, and 4He, is not negligible, and they should be explicitly included in the EoS for core-collapse supernova simulations and neutron star mergers.
Cluster formation and the virial equation of state of low-density nuclear matter
. Nucl. Phys. A 776, 55-79 (2006). doi: 10.1016/j.nuclphysa.2006.05.009;Composition and thermodynamics of nuclear matter with light clusters
. Phys. Rev. C 81, 015803 (2010). doi: 10.1103/PhysRevC.81.015803;Light clusters in nuclear matter and the “pasta” phase
. Phys. Rev. C 85, 035806 (2012). doi: 10.1103/PhysRevC.85.035806;Statistical description of complex nuclear phases in supernovae and proto-neutron stars
. Phys. Rev. C 82, 065801 (2010). doi: 10.1103/PhysRevC.82.065801;A statistical model for a complete supernova equation of state
. Nucl. Phys. A 837, 210-254 (2010). doi: 10.1016/j.nuclphysa.2010.02.010;Description of light clusters in relativistic nuclear models
. Phys. Rev. C 85, 055811 (2012). doi: 10.1103/PhysRevC.85.055811Structure of matter below nuclear saturation density
. Phys. Rev. Lett. 50, 2066 (1983). doi: 10.1103/PhysRevLett.50.2066;Dynamical response of the nuclear “pasta” in neutron star crusts
. Phys. Rev. C 72, 035801 (2005). doi: 10.1103/PhysRevC.72.035801;Nuclear “pasta” structures and the charge screening effect
. Phys. Rev. C 72, 015802 (2005). doi: 10.1103/PhysRevC.72.015802;Simulation of transitions between “Pasta” phases in dense matter
. Phys. Rev. Lett. 94, 031101 (2005). doi: 10.1103/PhysRevLett.94.031101;Erratum: Phase diagram of nuclear “pasta” and its uncertainties in supernova cores
[Phys. Rev. C 77, 035806 (2008)].Erratum: Phase diagram of nuclear “pasta” and its uncertainties in supernova cores
Phys. Rev. C 81, 049902 (2010). doi: 10.1103/PhysRevC.81.049902;Exploring the nuclear pasta phase in core-collapse supernova matter
. Phys. Rev. Lett. 109, 151101 (2012). doi: 10.1103/PhysRevLett.109.151101;Nuclear “waffles”
. Phys. Rev. C 90, 055805 (2014). doi: 10.1103/PhysRevC.90.055805;Equation of state and thickness of the inner crust of neutron stars
. Phys. Rev. C 90, 045803 (2014). doi: 10.1103/PhysRevC.90.045803Equations of state for supernovae and compact stars
. Rev. Mod. Phys. 89, 015007 (2017). doi: 10.1103/RevModPhys.89.015007GW170814: A three-detector observation of gravitational waves from a binary black hole coalescence
. Phys. Rev. Lett. 119, 141101(2017). doi: 10.1103/PhysRevLett.119.141101Influence of light nuclei on neutrino-driven supernova outflows
. Phys. Rev. C 78, 015806 (2008). doi: 10.1103/PhysRevC.78.015806;The influence of inelastic neutrino reactions with light nuclei on the standing accretion shock instability in core-collapse supernovae
. Astrophys. J. 774, 78 (2013). doi: 10.1088/0004-637X/774/1/78;Supernova equations of state including full nuclear ensemble with in-medium effects
. Nucl. Phys. A 957, 188-207 (2017). doi: 10.1016/j.nuclphysa.2016.09.002Light clusters in warm stellar matter: Explicit mass shifts and universal cluster-meson couplings
. Phys. Rev. C 97, 045805 (2018). doi: 10.1103/PhysRevC.97.045805Relativistic mean-field hadronic models under nuclear matter constraints
. Phys. Rev. C 90, 055203 (2014). doi: 10.1103/PhysRevC.90.055203The virial equation of state of low-density neutron matter
.Phys. Lett. B 638, 153-159 (2006). doi: 10.1016/j.physletb.2006.05.055;Constraining mean-field models of the nuclear matter equation of state at low densities
. Nucl. Phys. A 887, 42-76 (2012). doi: 10.1016/j.nuclphysa.2012.05.006Light clusters in nuclear matter: Excluded volume versus quantum many-body approaches
. Phys. Rev. C 84, 055804 (2011). doi: 10.1103/PhysRevC.84.055804Unified treatment of subsaturation stellar matter at zero and finite temperature
. Phys. Rev. C 92, 055803 (2015). doi: 10.1103/PhysRevC.92.055803Light clusters, pasta phases, and phase transitions in core-collapse supernova matter
. Phys. Rev. C 91, 055801 (2015). doi: 10.1103/PhysRevC.91.055801Laboratory tests of low density astrophysical nuclear equations of state
. Phys. Rev. Lett. 108, 172701 (2012). doi: 10.1103/PhysRevLett.108.172701Neutron-rich nuclei and neutron stars: A new accurately calibrated interaction for the study of neutron-rich matter
. Phys. Rev. Lett. 95, 122501 (2005). doi: 10.1103/PhysRevLett.95.122501Nuclear matter equation of state including two-, three-, and four-nucleon correlations
. Phys. Rev. C 92, 054001 (2015). doi: 10.1103/PhysRevC.92.054001Light nuclei quasiparticle energy shifts in hot and dense nuclear matter
. Phys. Rev. C 79, 014002 (2009). doi: 10.1103/PhysRevC.79.014002Composition and thermodynamics of nuclear matter with light clusters
. Phys. Rev. C 81, 015803 (2010). doi: 10.1103/PhysRevC.81.015803Constraining supernova equations of state with equilibrium constants from heavy-ion collisions
. Phys. Rev. C 91, 045805 (2015). doi: 10.1103/PhysRevC.91.045805