1 Introduction

At densities below the nuclear saturation density and not too high temperatures ($T\lesssim 20$ MeV), core-collapse supernova matter is unstable with respect to density fluctuations such that inhomogeneous structures develop and clusters can appear. Light (deuterons, tritons, helions, α-particles) [1-6], and heavy (pasta phases) [7-14] nuclei can be expected. Besides core-collapse supernova matter [15], also neutron stars [16, 17], and neutron star mergers [18] are systems where light and heavy clusters may appear. These structures may have consequences in cooling of the object as they may change the neutrino mean free path [19-21].

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 $M_j^{*}$ the effective mass of each cluster, given by

where $B_j^0$ is the cluster binding energy in the vacuum, and δ Bj is defined as [22]

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 ${ϵ}_j^{*}$ and ${\rho }_j^{*}$ are the energy density and density of the gas in the lowest states, defined as

we avoid double counting because the energy states occupied by the gas are excluded. In the above expressions, f_j±(p) are the Fermi distribution functions for the particles and anti-particles:

with $e_j(p)=\sqrt{p_j^2+m^{*2}}$, and νj=μj-gvV₀-gρτjb₀/2, j=n,p.

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 ε_surf= 2ε_Coul, the baryonic density in the high-density phase, ρI, the proton density in the high-density phase, ${\rho }_p^I$, and the volume fraction of the high-density phase, f, defined as

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 ρ_e the electron density and ρ_c 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.

Fig. 2.Full-screen View

(Color online) Binding energy of α for the RMF-FSU calculation (this work), T=4 MeV, and yp=0.5 obtained with Eq. (2). For comparison, results neglecting the binding energy shift (3) (δBj=0), as well as the empirical form Eq. (17) from Typel et al. [35], and results obtained from a recent QS approach [33] for different center-of-mass momenta, P (in units of fm^-1), and a QS calculation of a perturbative approach [34] (Pauli), Eq. (16), are also shown.

Only the gsj parameters are optimized,

$g_{sj}=x_{sj}{A}_j{g}_s\mathrm{,}$ (13)

while the vector couplings are set to

$g_{vj}=A_j{g}_v\mathrm{.}$ (14)

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 $\rho {\lambda }_n^3=1/10$ [26], ρ being the baryon density and λn the nucleon thermal wavelength, represent the upper limit of the region of validity of the VEoS. Looking at this figure, we see that the only cluster fraction that is not consistent with the VEoS result is the deuteron. This can be explained by the fact that this cluster is the less bound cluster we are considering, and mean-field models usually fail to describe such systems. This is an effect of the continuum contribution of the deuteron channel, as was pointed out in Ref. [33]: its contribution is essential if the binding energy per nucleon is small when compared with T.

Fig. 1.Full-screen View

(Color online) Fraction of deuteron, Xd, triton, Xt, helion, Xh, and α, Xα, as a function of the density for FSU, T=4 MeV (top) and 10 MeV (bottom), with proton fraction yp=0.5, taking δB=0, xsj=0.85± 0.05, (variation indicated by the spreading of the bands), and comparing with results of the Virial EoS from [26]. Solid vertical black lines are given by $\rho {\lambda }_n^3=1/10$.

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

$B_j=B_j^0+\delta B_j^{\text{QS}}$ (15)

obtained from quantum statistical (QS) calculations, with $\delta B_j^{\text{Q}S}(P;{\rho }_n\mathrm{,}{\rho }_p\mathrm{,}T)$ taking different center-of-mass momenta P=0, 1, 2 fm^-1, according to Ref. [33].

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]

$\delta B_{\alpha }^{\text{Pauli}}(P=0;{\rho }_n\mathrm{,}{\rho }_p\mathrm{,}T)=-\frac{164371\rho }{{(T+10.67)}^{3/2}}$ (16)

(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

$\begin{array}{l}\delta B_{\alpha }^{\text{Typel}}(T)=\delta B_{\alpha }^{\text{Pauli}}(P=0;{\rho }_n\mathrm{,}{\rho }_p\mathrm{,}T)\\ \times \left[1-\frac{\delta B_{\alpha }^{\text{Pauli}}(P=0;{\rho }_n\mathrm{,}{\rho }_p\mathrm{,}T)}{2B_{\alpha }^0}\right]\mathrm{.}\end{array}$ (17)

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.

Fig. 3.Full-screen View

(Color online) Binding energy shift of α, δBα, as given by Eq.(3) for the RMF-FSU calculation, yp=0.5, T=5 MeV (top), and T=10 MeV (bottom), for xs=0.8,0.85,0.9.

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.

Fig. 4.Full-screen View

(Color online) Fraction of α, Xα (red), deuteron, Xd (blue), triton, Xt (light blue), and helion, Xh (green), as a function of the density for FSU, T=5 MeV, and yp=0.41, with (thick) and without (thin) δ Bj, for xsj=0.85, keeping gvj=Ajgv.

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

$K_c[j]=\frac{{\rho }_j}{{\rho }_n^{N_j}{\rho }_p^{Z_j}}$ (18)

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.

Fig. 5.Full-screen View

(Color online) Chemical equilibrium constants of α (a), helion (b), deuteron (c), and triton (d) for FSU, and yp=0.41, and the universal gsj fitting with gsj=(0.85±0.05)Ajgs, (red dotted lines). The experimental results of Qin et al [31] (yellow region) are also shown.

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.

Fig. 6.Full-screen View

(Color online) Total mass fraction of free particles (black), and light clusters (blue), for a CLD with (thick) and without (thin) light clusters, and HM (dashed) calculations for FSU, yp=0.2, and xs=0.85, xv=1, with T=5 MeV (top) and T=7 MeV (bottom). The heavy cluster mass fraction (red) from a CLD calculation is also shown. All the results shown take δBj ≠ 0. In both calculations we are taking Acl up to 12.

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.