Light and heavy clusters in warm stellar matter

Light and heavy clusters in warm stellar matter 1 corresp first-author Corresponding author, pais.lena@uc.pt Corresponding author, pais.lena@uc.pt pais.lena@uc.pt 2 1 3 4 CFisUC, Department of Physics, University of Coimbra, 3004-516 Coimbra, Portugal. CFisUC, Department of Physics, University of Coimbra, 3004-516 Coimbra, Portugal. LPC (CNRS/ENSICAEN/Université de Caen Normandie), UMR6534, 14050 Caen cédex, France. LPC (CNRS/ENSICAEN/Université de Caen Normandie), UMR6534, 14050 Caen cédex, France. Institut für Physik, Universität Rostock, D-18051 Rostock, Germany. Institut für Physik, Universität Rostock, D-18051 Rostock, Germany. National Research Nuclear University (MEPhI), 115409 Moscow, Russia. National Research Nuclear University (MEPhI), 115409 Moscow, Russia. Light and heavy clusters are calculated for warm stellar matter in the framework of relativistic mean-field models, in the single-nucleus approximation. The clusters abundances are determined from the minimization of the free energy. In-medium effects of light cluster properties are included by introducing an explicit binding energy shift analytically calculated in the Thomas-Fermi approximation, and the coupling constants are fixed by imposing that the virial limit at low density is recovered. The resulting light cluster abundances come out to be in reasonable agreement with constraints at higher density coming from heavy ion collision data. Some comparisons with microscopic calculations are also shown. nuclear clusters warm stellar matter in-medium effects 1 Introduction 1 At densities below the nuclear saturation density and not too high temperatures ( <math id="M1"><mrow><mi>T</mi><mo>≲</mo><mn>20</mn></mrow></math> 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 1 - 6 6 1-6 ], and heavy (pasta phases) [ 7 7 - 14 14 7-14 ] nuclei can be expected. Besides core-collapse supernova matter [ 15 15 ], also neutron stars [ 16 16 , 17 17 ], and neutron star mergers [ 18 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 19 - 21 21 19-21 ]. This work follows the one in Ref. [ 22 22 ], where light clusters are calculated in the relativistic mean field (RMF) framework [ 23 23 ]. Both in-medium mass shifts and in-medium modification of the cluster couplings are d iscussed. 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 24 ]. At very low densities, we use the model-independent constraint, the virial EoS (VEoS) [ 1 1 , 25 25 , 26 26 ], to fix the cluster-meson couplings so that the VEoS particle fractions obtained in Ref. [ 26 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 27 , 28 28 ], so that we employ the Thomas-Fermi formulation of Ref. [ 29 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 g sj = x s A j g s is proposed, where g s is the coupling constant with the nucleons ( n , p ), A j the cluster mass number, and x s is a universal cluster coupling fraction, with an associated uncertainty. Besides the four standard charged light particles, 4 He, 3 H, 3 He, and 2 H, as the density increases, heavier clusters can also form, like 5 H, 7 H. 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 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 1 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 22 ]. The total binding energy of each cluster is defined as <math display="block" id="M2"><mrow><msub><mi>B</mi><mi>j</mi></msub><mo>=</mo><msub><mi>A</mi><mi>j</mi></msub><msup><mi>m</mi><mo>*</mo></msup><mo>−</mo><msubsup><mi>M</mi><mi>j</mi><mo>*</mo></msubsup><mtext> </mtext><mn>,</mn></mrow></math> with m * the nucleon effective mass, and <math id="M3"><mrow><msubsup><mi>M</mi><mi>j</mi><mo>*</mo></msubsup></mrow></math> the effective mass of each cluster, given by <math display="block" id="M4"><mrow><msubsup><mi>M</mi><mi>j</mi><mo>*</mo></msubsup><mo>=</mo><msub><mi>A</mi><mi>j</mi></msub><mi>m</mi><mo>−</mo><msub><mi>g</mi><mrow><mi>s</mi><mi>j</mi></mrow></msub><msub><mi>ϕ</mi><mn>0</mn></msub><mo>−</mo><mrow><mo>(</mo><mrow><msubsup><mi>B</mi><mi>j</mi><mn>0</mn></msubsup><mo>+</mo><mi>δ</mi><msub><mi>B</mi><mi>j</mi></msub></mrow><mo>)</mo></mrow><mn>,</mn></mrow></math> where <math id="M5"><mrow><msubsup><mi>B</mi><mi>j</mi><mn>0</mn></msubsup></mrow></math> is the cluster binding energy in the vacuum, and δ B j is defined as [ 22 22 ] <math display="block" id="M6"><mrow><mi>δ</mi><msub><mi>B</mi><mi>j</mi></msub><mo>=</mo><mfrac><mrow><msub><mi>Z</mi><mi>j</mi></msub></mrow><mrow><msub><mi>ρ</mi><mn>0</mn></msub></mrow></mfrac><mrow><mo>(</mo><mrow><msubsup><mi>ϵ</mi><mi>p</mi><mo>*</mo></msubsup><mo>−</mo><mi>m</mi><msubsup><mi>ρ</mi><mi>p</mi><mo>*</mo></msubsup></mrow><mo>)</mo></mrow><mo>+</mo><mfrac><mrow><msub><mi>N</mi><mi>j</mi></msub></mrow><mrow><msub><mi>ρ</mi><mn>0</mn></msub></mrow></mfrac><mrow><mo>(</mo><mrow><msubsup><mi>ϵ</mi><mi>n</mi><mo>*</mo></msubsup><mo>−</mo><mi>m</mi><msubsup><mi>ρ</mi><mi>n</mi><mo>*</mo></msubsup></mrow><mo>)</mo></mrow><mtext> </mtext><mn>,</mn></mrow></math> The binding energy shift, δ B j , takes in-medium effects into account, and needs to be determined. It is the energetic counterpart of the classical excluded-volume mechanism. Since <math id="M7"><mrow><msubsup><mi>ϵ</mi><mi>j</mi><mo>*</mo></msubsup></mrow></math> and <math id="M8"><mrow><msubsup><mi>ρ</mi><mi>j</mi><mo>*</mo></msubsup></mrow></math> are the energy density and density of the gas in the lowest states, defined as <math display="block" id="M9"><mrow><msubsup><mi>ϵ</mi><mi>j</mi><mo>*</mo></msubsup><mo>=</mo><mfrac><mn>1</mn><mrow><msup><mi>π</mi><mn>2</mn></msup></mrow></mfrac><mstyle displaystyle="true"><mrow><msubsup><mo>∫</mo><mn>0</mn><mrow><msub><mi>p</mi><mrow><msub><mi>F</mi><mi>j</mi></msub></mrow></msub><mo stretchy="false">(</mo><mtext>g</mtext><mi>a</mi><mi>s</mi><mo stretchy="false">)</mo></mrow></msubsup></mrow></mstyle><msup><mi>p</mi><mn>2</mn></msup><msub><mi>e</mi><mi>j</mi></msub><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mi>f</mi><mrow><mi>j</mi><mo>+</mo></mrow></msub><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">)</mo><mo>+</mo><msub><mi>f</mi><mrow><mi>j</mi><mo>−</mo></mrow></msub><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mtext>d</mtext><mi>p</mi></mrow></math> <math display="block" id="M10"><mrow><msubsup><mi>ρ</mi><mi>j</mi><mo>*</mo></msubsup><mo>=</mo><mfrac><mn>1</mn><mrow><msup><mi>π</mi><mn>2</mn></msup></mrow></mfrac><mstyle displaystyle="true"><mrow><msubsup><mo>∫</mo><mn>0</mn><mrow><msub><mi>p</mi><mrow><msub><mi>F</mi><mi>j</mi></msub></mrow></msub><mo stretchy="false">(</mo><mtext>g</mtext><mi>a</mi><mi>s</mi><mo stretchy="false">)</mo></mrow></msubsup></mrow></mstyle><msup><mi>p</mi><mn>2</mn></msup><mo stretchy="false">(</mo><msub><mi>f</mi><mrow><mi>j</mi><mo>+</mo></mrow></msub><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">)</mo><mo>+</mo><msub><mi>f</mi><mrow><mi>j</mi><mo>−</mo></mrow></msub><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mtext>d</mtext><mi>p</mi><mtext> </mtext><mn>,</mn></mrow></math> 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: <math display="block" id="M11"><mrow><msub><mi>f</mi><mrow><mi>j</mi><mo>±</mo></mrow></msub><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>1</mn><mrow><mi>exp</mi><mo stretchy="false">[</mo><mo stretchy="false">(</mo><msub><mi>e</mi><mi>j</mi></msub><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">)</mo><mo>∓</mo><msub><mi>ν</mi><mi>j</mi></msub><mo stretchy="false">)</mo><mo>/</mo><mi>T</mi><mo stretchy="false">]</mo><mo>+</mo><mn>1</mn></mrow></mfrac><mn>,</mn></mrow></math> with <math id="M12"><mrow><msub><mi>e</mi><mi>j</mi></msub><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">)</mo><mo>=</mo><msqrt><mrow><msubsup><mi>p</mi><mi>j</mi><mn>2</mn></msubsup><mo>+</mo><msup><mi>m</mi><mrow><mo>*</mo><mn>2</mn></mrow></msup></mrow></msqrt></mrow></math> , and ν j = μ j - g v V 0 - g ρ τ j b 0 /2, j = n , p . The other quantity that considers in-medium effects is the scalar cluster-meson coupling, g sj = x sj A j g s , which is determined from experimental constraints. We fix x sj so that in the low-density limit the Virial EoS is reproduced. We obtained [ 22 22 ] x sj =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 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 <math display="block" id="M13"><mrow><mi>F</mi><mo>=</mo><mi>f</mi><msup><mi>F</mi><mi>I</mi></msup><mo>+</mo><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>f</mi><mo stretchy="false">)</mo><msup><mi>F</mi><mrow><mi>I</mi><mi>I</mi></mrow></msup><mo>+</mo><msub><mi>F</mi><mtext>e</mtext></msub><mo>+</mo><msub><mi>ε</mi><mrow><mtext>surf</mtext></mrow></msub><mo>+</mo><msub><mi>ε</mi><mrow><mtext>Coul</mtext></mrow></msub><mtext>,</mtext></mrow></math> This minimization is done with respect to four variables: the size of the geometric configuration, r d , which gives, just like in the CP case, the condition ε surf = 2 ε Coul , the baryonic density in the high-density phase, ρ I , the proton density in the high-density phase, <math id="M14"><mrow><msubsup><mi>ρ</mi><mi>p</mi><mi>I</mi></msubsup></mrow></math> , and the volume fraction of the high-density phase, f , defined as <math display="block" id="M15"><mrow><mi>f</mi><mo>=</mo><mfrac><mrow><mi>ρ</mi><mo>−</mo><msup><mi>ρ</mi><mrow><mi>I</mi><mi>I</mi></mrow></msup></mrow><mrow><msup><mi>ρ</mi><mi>I</mi></msup><mo>−</mo><msup><mi>ρ</mi><mrow><mi>I</mi><mi>I</mi></mrow></msup></mrow></mfrac><mtext> </mtext><mn>.</mn></mrow></math> The equilibrium conditions then become <math display="block" id="M16"><mtable columnalign="left"><mtr><mtd><msup><mi>P</mi><mi>I</mi></msup><mo>=</mo><msup><mi>P</mi><mrow><mi>I</mi><mi>I</mi></mrow></msup><mo>−</mo><msub><mi>ε</mi><mrow><mtext>surf</mtext></mrow></msub><mrow><mo>(</mo><mrow><mfrac><mn>1</mn><mrow><mn>2</mn><mi>α</mi></mrow></mfrac><mo>+</mo><mfrac><mn>1</mn><mrow><mn>2</mn><mi>Φ</mi></mrow></mfrac><mfrac><mrow><mo>∂</mo><mi>Φ</mi></mrow><mrow><mo>∂</mo><mi>f</mi></mrow></mfrac><mo>−</mo><mfrac><mrow><msubsup><mi>ρ</mi><mi>p</mi><mrow><mi>I</mi><mi>I</mi></mrow></msubsup></mrow><mrow><mi>f</mi><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>f</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msubsup><mi>ρ</mi><mi>p</mi><mi>I</mi></msubsup><mo>−</mo><msubsup><mi>ρ</mi><mi>p</mi><mrow><mi>I</mi><mi>I</mi></mrow></msubsup><mo stretchy="false">)</mo></mrow></mfrac></mrow><mo>)</mo></mrow><mn>,</mn></mtd></mtr><mtr><mtd><msubsup><mi>μ</mi><mi>n</mi><mi>I</mi></msubsup><mo>=</mo><msubsup><mi>μ</mi><mi>n</mi><mrow><mi>I</mi><mi>I</mi></mrow></msubsup><mtext> </mtext><mn>,</mn></mtd></mtr><mtr><mtd><msubsup><mi>μ</mi><mi>p</mi><mi>I</mi></msubsup><mo>=</mo><msubsup><mi>μ</mi><mi>p</mi><mrow><mi>I</mi><mi>I</mi></mrow></msubsup><mo>−</mo><mfrac><mrow><msub><mi>ε</mi><mrow><mtext>surf</mtext></mrow></msub></mrow><mrow><mi>f</mi><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>f</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msubsup><mi>ρ</mi><mi>p</mi><mi>I</mi></msubsup><mo>−</mo><msubsup><mi>ρ</mi><mi>p</mi><mrow><mi>I</mi><mi>I</mi></mrow></msubsup><mo stretchy="false">)</mo></mrow></mfrac><mtext> </mtext><mn>,</mn></mtd></mtr></mtable></math> 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 30 ] <math display="block" id="M17"><mrow><mi>Φ</mi><mo>=</mo><mrow><mo>{</mo><mrow><mtable><mtr><mtd><mrow/></mtd><mtd><mrow><mrow><mo>(</mo><mrow><mfrac><mrow><mn>2</mn><mo>−</mo><mi>D</mi><msup><mi>α</mi><mrow><mn>1</mn><mo>−</mo><mn>2</mn><mo>/</mo><mi>D</mi></mrow></msup></mrow><mrow><mi>D</mi><mo>−</mo><mn>2</mn></mrow></mfrac><mo>+</mo><mi>α</mi></mrow><mo>)</mo></mrow><mfrac><mn>1</mn><mrow><mi>D</mi><mo>+</mo><mn>2</mn></mrow></mfrac><mtext> </mtext><mn>,</mn><mtext> </mtext><mi>D</mi><mo>=</mo><mn>1,3</mn></mrow></mtd></mtr><mtr><mtd><mrow/></mtd><mtd><mrow><mfrac><mrow><mi>α</mi><mo>−</mo><mn>1</mn><mo>−</mo><mi>ln</mi><mi>α</mi></mrow><mrow><mi>D</mi><mo>+</mo><mn>2</mn></mrow></mfrac><mtext> </mtext><mn>,</mn><mtext> </mtext><mi>D</mi><mo>=</mo><mn>2</mn><mtext> </mtext><mn>.</mn></mrow></mtd></mtr></mtable></mrow></mrow></mrow></math> 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: <math display="block" id="M18"><mtable columnalign="left"><mtr><mtd><msubsup><mi>μ</mi><mrow><msub><mi>A</mi><mrow><mtext>cl</mtext></mrow></msub></mrow><mi>I</mi></msubsup><mo>=</mo><mi>N</mi><msubsup><mi>μ</mi><mi>n</mi><mi>I</mi></msubsup><mo>−</mo><mi>Z</mi><msubsup><mi>μ</mi><mi>p</mi><mi>I</mi></msubsup><mn>,</mn></mtd></mtr><mtr><mtd><msubsup><mi>μ</mi><mrow><msub><mi>A</mi><mrow><mtext>cl</mtext></mrow></msub></mrow><mrow><mi>I</mi><mi>I</mi></mrow></msubsup><mo>=</mo><mi>N</mi><msubsup><mi>μ</mi><mi>n</mi><mrow><mi>I</mi><mi>I</mi></mrow></msubsup><mo>−</mo><mi>Z</mi><msubsup><mi>μ</mi><mi>p</mi><mrow><mi>I</mi><mi>I</mi></mrow></msubsup><mtext> </mtext><mn>,2</mn><mo>≤</mo><msub><mi>A</mi><mrow><mtext>cl</mtext></mrow></msub><mo>≤</mo><mn>12</mn><mtext> </mtext><mn>,</mn></mtd></mtr></mtable></math> and charge neutrality must also be imposed: <math display="block" id="M19"><mrow><msub><mi>ρ</mi><mtext>e</mtext></msub><mo>=</mo><msub><mi>Y</mi><mi>p</mi></msub><mi>ρ</mi><mo>=</mo><mi>f</mi><msubsup><mi>ρ</mi><mtext>c</mtext><mi>I</mi></msubsup><mo>+</mo><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>f</mi><mo stretchy="false">)</mo><msubsup><mi>ρ</mi><mtext>c</mtext><mrow><mi>I</mi><mi>I</mi></mrow></msubsup><mtext> </mtext><mn>,</mn></mrow></math> 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 1 In the following, we show some of the results obtained in this work, at finite fixed temperatures and for fixed proton fractions y p 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, x s , from the Virial equation of state (VEoS). Then we investigate the effect of introducing the binding energy shift δ B j , 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 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 x s : Virial EoS 2 The cluster-meson couplings are obtained from the best fit of the RMF cluster mass fractions, defined as X j = A j n j /n , to the VEoS data, taking the FSU parametrization [ 32 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 δB j of Eq. (3) is completely negligible, and does not affect the particle fractions (see also Figure 2 Figure 2 below), therefore it was put to zero for this calculation. Only the g sj parameters are optimized, <math display="block" id="M20"><mrow><msub><mi>g</mi><mrow><mi>s</mi><mi>j</mi></mrow></msub><mo>=</mo><msub><mi>x</mi><mrow><mi>s</mi><mi>j</mi></mrow></msub><msub><mi>A</mi><mi>j</mi></msub><msub><mi>g</mi><mi>s</mi></msub><mn>,</mn></mrow></math> while the vector couplings are set to <math display="block" id="M21"><mrow><msub><mi>g</mi><mrow><mi>v</mi><mi>j</mi></mrow></msub><mo>=</mo><msub><mi>A</mi><mi>j</mi></msub><msub><mi>g</mi><mi>v</mi></msub><mn>.</mn></mrow></math> Reasonable values for g sj are (0.85 ± 0.05) A j g s , see Fig. 1 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 <math id="M22"><mrow><mi>ρ</mi><msubsup><mi>λ</mi><mi>n</mi><mn>3</mn></msubsup><mo>=</mo><mn>1</mn><mo>/</mo><mn>10</mn></mrow></math> [ 26 26 ], ρ being the baryon density and λ n 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 33 ]: its contribution is essential if the binding energy per nucleon is small when compared with T . Effect of the binding energy shift δ B j 2 Let us now discuss the effect of introducing a non-zero binding energy shift δ B j , Eq. (3). In Fig. 2 Fig. 2 , we compare the binding energy of the α -clusters, obtained taking δB α , as defined by Eq. (3), with the binding energy <math display="block" id="M24"><mrow><msub><mi>B</mi><mi>j</mi></msub><mo>=</mo><msubsup><mi>B</mi><mi>j</mi><mn>0</mn></msubsup><mo>+</mo><mi>δ</mi><msubsup><mi>B</mi><mi>j</mi><mrow><mtext>QS</mtext></mrow></msubsup></mrow></math> obtained from quantum statistical (QS) calculations, with <math id="M25"><mrow><mi>δ</mi><msubsup><mi>B</mi><mi>j</mi><mrow><mtext>Q</mtext><mi>S</mi></mrow></msubsup><mo stretchy="false">(</mo><mi>P</mi><mo>;</mo><msub><mi>ρ</mi><mi>n</mi></msub><mn>,</mn><msub><mi>ρ</mi><mi>p</mi></msub><mn>,</mn><mi>T</mi><mo stretchy="false">)</mo></mrow></math> taking different center-of-mass momenta P =0, 1, 2 fm -1 , according to Ref. [ 33 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 34 ] <math display="block" id="M26"><mrow><mi>δ</mi><msubsup><mi>B</mi><mi>α</mi><mrow><mtext>Pauli</mtext></mrow></msubsup><mo stretchy="false">(</mo><mi>P</mi><mo>=</mo><mn>0</mn><mo>;</mo><msub><mi>ρ</mi><mi>n</mi></msub><mn>,</mn><msub><mi>ρ</mi><mi>p</mi></msub><mn>,</mn><mi>T</mi><mo stretchy="false">)</mo><mo>=</mo><mo>−</mo><mfrac><mrow><mn>164371</mn><mtext> </mtext><mtext> </mtext><mi>ρ</mi></mrow><mrow><msup><mrow><mo stretchy="false">(</mo><mi>T</mi><mo>+</mo><mn>10.67</mn><mo stretchy="false">)</mo></mrow><mrow><mn>3</mn><mo>/</mo><mn>2</mn></mrow></msup></mrow></mfrac></mrow></math> (in units of MeV, with T in units of MeV, and ρ in units of fm -3 ). Lastly, Fig. 2 Fig. 2 also shows a calculation from Typel et al. [ 35 35 ], where, in order to suppress cluster formation at higher densities, they introduced an empirical quadratic form given by <math display="block" id="M27"><mtable columnalign="left"><mtr><mtd><mi>δ</mi><msubsup><mi>B</mi><mi>α</mi><mrow><mtext>Typel</mtext></mrow></msubsup><mo stretchy="false">(</mo><mi>T</mi><mo stretchy="false">)</mo><mo>=</mo><mi>δ</mi><msubsup><mi>B</mi><mi>α</mi><mrow><mtext>Pauli</mtext></mrow></msubsup><mo stretchy="false">(</mo><mi>P</mi><mo>=</mo><mn>0</mn><mo>;</mo><msub><mi>ρ</mi><mi>n</mi></msub><mn>,</mn><msub><mi>ρ</mi><mi>p</mi></msub><mn>,</mn><mi>T</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mo>×</mo><mrow><mo>[</mo><mrow><mn>1</mn><mo>−</mo><mfrac><mrow><mi>δ</mi><msubsup><mi>B</mi><mi>α</mi><mrow><mtext>Pauli</mtext></mrow></msubsup><mo stretchy="false">(</mo><mi>P</mi><mo>=</mo><mn>0</mn><mo>;</mo><msub><mi>ρ</mi><mi>n</mi></msub><mn>,</mn><msub><mi>ρ</mi><mi>p</mi></msub><mn>,</mn><mi>T</mi><mo stretchy="false">)</mo></mrow><mrow><mn>2</mn><msubsup><mi>B</mi><mi>α</mi><mn>0</mn></msubsup></mrow></mfrac></mrow><mo>]</mo></mrow><mn>.</mn></mtd></mtr></mtable></math> We can see from Fig. 2 Fig. 2 that the additional binding energy shift δ B j 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 x sj and temperature T on the binding energy shift. From Fig. 3 Fig. 3 we conclude that the larger x sj the slower - δ B j increases, and also that a larger temperature determines a softer behavior, with - δ B j taking larger values at the lower densities and smaller ones close to the dissolution density. However, we should stress that Figs. 2 Figs. 2 and 3 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 B j on the global cluster distributions 2 We show in the present subsection that the clusters are dissolved below the nuclear saturation density, ρ 0 . In Fig. 4 Fig. 4 we show the clusters mass fractions for matter with a fixed proton fraction of y p =0.41, T =5 MeV, and x sj =0.85. We observe that if we neglect the δB j 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 ( X j < 10 -4 ) of light clusters are within the density range 0.04fm -3 < ρ < 0.06fm -3 . Equilibrium constants 2 In the high-density limit, a constraint was proposed in Ref. [ 31 31 ]. These chemical equilibrium constants, K c [ j ], calculated with data from heavy-ion collisions, are defined as <math display="block" id="M28"><mrow><msub><mi>K</mi><mi>c</mi></msub><mo stretchy="false">[</mo><mi>j</mi><mo stretchy="false">]</mo><mo>=</mo><mfrac><mrow><msub><mi>ρ</mi><mi>j</mi></msub></mrow><mrow><msubsup><mi>ρ</mi><mi>n</mi><mrow><msub><mi>N</mi><mi>j</mi></msub></mrow></msubsup><msubsup><mi>ρ</mi><mi>p</mi><mrow><msub><mi>Z</mi><mi>j</mi></msub></mrow></msubsup></mrow></mfrac></mrow></math> where ρ j is the number density of cluster j , with neutron number N j and proton number Z j , and ρ p , ρ n 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 31 , 36 36 ]. In Fig. 5 Fig. 5 , we show the chemical equilibrium constants for all the light clusters considered, taking the range of the couplings to be g sj =(0.85±0.05) A j g s , and we compare with the experimental results of Ref. [ 31 31 ]. We can see that taking the coupling fractions x sj =0.85±0.05 essentially describes the experimental equilibrium constants. We have checked that x s =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 g sj = (0.85 ± 0.05) A j g s . For the deuteron, the experimental data seem to be described by the upper limit x s =0.9. Possibly a more detailed approach would allow for a different coupling g sj for each cluster. Influence on the pasta structure 2 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 B j , 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., 4 He, 2 H, 3 H, and 3 He, we are also taking all the bound clusters with A ≤ 12. In Fig. 6 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 x s =0.85, and, in all calculations, x v =1, with g s i = x s g s A i and g v i = x v g v A i . 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 1 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 B j . The fraction x sj =0.85 ± 0.05 reproduces both virial limit and K c 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. 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C 91 , 045805 ( 2015 ). doi: 10.1103/PhysRevC.91.045805 http://doi.org/10.1103/PhysRevC.91.045805 Constraining supernova equations of state with equilibrium constants from heavy-ion collisions 10.1007/s41365-018-0518-6 This work was partly supported by the FCT (Portugal) Project No. UID/FIS/04564/2016, and by former NewCompStar, COST Action MP1304. H.P. is supported by FCT (Portugal) under Project No. SFRH/BPD/95566/2013. She is very thankful to the Organizers of IWND 2018 for the opportunity to present this work, as well as the financial support received. 2018-09-04 2018-11-08 2018-11-08 2018-11-26 2018-12-01 2021-12-20 © China Science Publishing & Media Ltd. (Science Press), Shanghai Institute of Applied Physics, the Chinese Academy of Sciences, ChineseNuclear Society and Springer Nature Singapore Pte Ltd. 2018 This is a post-peer-review, pre-copyedit version of an article published in Nuclear Science and Techniques. The final authenticated version is available online at: http://dx.doi.org/10.1007/s41365-018-0518-6 http://dx.doi.org/10.1007/s41365-018-0518-6 . 2018 Nuclear Science and Techniques 12 29