Equilibrium nuclear ensembles taking into account vaporization of hot nuclei in dense stellar matter

Equilibrium nuclear ensembles taking into account vaporization of hot nuclei in dense stellar matter


We investigate the high-temperature effect on the nuclear matter that consists of mixture of nucleons and all nuclei in the dense and hot stellar environment. The individual nuclei are described within the compressible liquid-drop model that is based on Skyrme interactions for bulk energies and that takes into account modifications of the surface and Coulomb energies at finite temperatures and densities. The free-energy density is minimized with respect to the individual equilibrium densities of all heavy nuclei and the nuclear composition. We find that their optimized equilibrium densities become smaller and smaller at high temperatures because of the increase of thermal contributions to bulk free energies and the reduction of surface energies. The neutron-rich nuclei become unstable and disappear one after another at some temperatures. The calculations are performed for two sets of model parameters leading to different values of the slope parameter in the nuclear symmetry energy. It is found that the larger slope parameter reduces the equilibrium densities and the melting temperatures. We also compare the new model with some other approaches and find that the mass fractions of heavy nuclei in the previous calculations that omit vaporization are underestimated at  MeV and overestimated at  MeV. The further sophistication of calculations of nuclear vaporization and of light clusters would be required to construct the equation of state for explosive astrophysical phenomena.


I Introduction

Hot and dense matter can be realized both in terrestrial experiments of heavy ion collision and explosive astrophysical phenomena such as core collapse supernovae and mergers of compact stars (see e.g., Refs Botvina and Mishustin (2004); Janka (2012); Kotake et al. (2012); Burrows (2013); Foglizzo et al. (2015); Shibata and Taniguchi (2011); Faber and Rasio (2012)). Even after various endeavors of the previous decades, we have little information about the nuclear equations of state (EOSs) both at sub and supra nuclear densities and, up to now, there is no consensus about its theoretical description in the nuclear community Oertel et al. (2017). This is partially because it is difficult to reproduce the density, temperature and proton-neutron ratio dependences of nuclear matter arbitrarily in terrestrial laboratories, although several experimental efforts are underway. Furthermore, it cannot be calculated from first-principles due to enormous complexity.

The EOS provides information on nuclear composition in addition to thermodynamical quantities. The former plays an important role to alter the charge fraction of stellar matter through weak interactions Raduta et al. (2016, 2017); Furusawa et al. (2017a) and, as a result, affects the dynamics and synthesis of heavy elements in these events Hix et al. (2003); Lentz et al. (2012); Wanajo et al. (2014); Sekiguchi et al. (2015). The latter is directly linked to the structure of compact stars and the dynamics of their formation and merger.

In the literature, the single nucleus approximation (SNA) or the multi nucleus approximation (MNA) is used to construct huge EOS data covering a wide range of density, temperature and proton fraction for stellar matter Buyukcizmeci et al. (2013); Shen et al. (2011). In the former, the ensemble of heavy nuclei is represented by a single nucleus Lattimer and Swesty (1991); Shen et al. (1998a, b, 2011); Togashi et al. (2017), which is calculated by the compressible liquid drop model or Thomas Fermi model. In such calculations, the optimizations for nuclear structure, such as compression, are taken into consideration, whereas the approach is not able to provide a realistic nuclear composition. Furthermore, the mass number and mass fraction of the representative nucleus are considerably deviated from the actual values obtained for full nuclear ensemble Furusawa and Mishustin (2017). In the latter type of EOSs such as SMSM Botvina and Mishustin (2004, 2010); Buyukcizmeci et al. (2014), HS Hempel and Schaffner-Bielich (2010); Steiner et al. (2013) and FYSS Furusawa et al. (2011); Furusawa et al. (2013a, 2017b, 2017c) EOSs, the nuclear composition including all nuclear species is optimized for each set of thermodynamical condition. On the other hand, the MNA EOSs use very rough approximations, such as an incompressible-liquid drop model for nuclear binding energies at high densities and temperatures, to reduce the numerical cost for calculations of the full statistical ensemble. The fully self-consistent calculation with the multi-nucleus composition including in-medium effects on nuclear structure can hardly be performed due to the complexity Gulminelli and Raduta (2015); Furusawa and Mishustin (2017).

The purpose of this series of papers is to investigate what happens in the full nuclear ensemble as the density or temperature increases. In our previous paper Furusawa and Mishustin (2017), we took into account self-consistently the compression or decompression of heavy nuclei in the multi-nucleus description, i.e., the changes of equilibrium densities of individual nuclei embedded in a dense stellar environment. We then found that heavy nuclei in the ensemble are either decompressed or compressed depending on whether nucleons are dripped. Furthermore, we demonstrated that the EOS with MNA shows better agreement with the self-consistent calculation of the mass fraction and average mass number of heavy nuclei compared with the EOS with SNA below about times the nuclear saturation density and low temperatures.

In this paper, we investigate the self-consistent multi-nucleus system at higher temperatures up to 15 MeV. In the previous paper, we discussed the results below MeV, because the bulk energy was estimated by the simple parametric formula of the bulk energies of heavy nuclei, which is valid only at low temperatures and around the saturation density. In this work, we replace it by a more reliable calculation based on a Skyrme-type mean-field interaction (see Refs. Khodel et al. (1987); Smirnov et al. (1988); Oyamatsu (1993); Fayans et al. (2000); Oyamatsu and Iida (2003)) and the exact expression for thermal excitation energies. The improvement allows us to discuss the vaporization of nuclear species at the temperatures, at which the equilibrium densities can not be obtained and the nuclei disappear one after another. Such self-consistent calculations of nuclear abundance and dissolution have not been reported so far in the framework of the multi-nucleus description, whereas nuclear structures at high temperatures are investigated mainly in SNA (see, e.g., Newton and Stone (2009)). Gulminelli and Oertel Gulminelli and Raduta (2015) investigated the equilibrium between nucleons and nuclei as well as nuclear composition in the multi nucleus description, although the optimization of nuclear equilibrium density is not taken into consideration.

This work is organized as follows: In Sec. II, we formulate a model of EOS, which is used to find the nuclear abundance and to take into account the possibility of nuclear vaporization. The results at some typical astrophysical conditions for two different parameter sets for bulk nuclear matter are discussed in Sec. III. In addition, we provide systematical comparisons with other models for free energies of heavy and light nuclei in Sec. IV. Section V is devoted to the conclusion.

Ii Model

The basis of our EOS model is the same as the previous one. For details, we refer the readers to Furusawa and Mishustin Furusawa and Mishustin (2017). The EOS as a function of baryon density , temperature , and total proton fraction is obtained by optimizing the nuclear composition as well as the nuclear equilibration, or minimizing the free energy density with respect to the number densities of all particles and equilibrium densities of heavy nuclei. The free energy density is expressed as


where and are the free energy densities of free protons and neutrons and , and are the number densities, translational free energies and mass free energies of light clusters and heavy nuclei . For light clusters with the atomic number or the neutron number , we include only the nuclide with the available experimental mass data, while all heavy nuclei with and are taken into consideration as long as they are stable. The maximum asymmetry of heavy nuclei, , is around 0.8, which corresponds to extremely neutron-rich nuclei such as . The nuclear interaction of bulk nuclear matter is explicitly taken into account in and . A compressible liquid drop model is adopted for , which is described by the sum of bulk, Coulomb and surface free energies as . For , experimental mass data with Coulomb shifts are utilized. The interactions among different species are represented by the excluded volume effects in and . Below we explain each term of the model free energy.

We improve our previous model by replacing the parametric expression of the nuclear bulk energies by the more realistic calculation based on Skyrme type interactions, which provides the free energy per baryon of uniform nuclear matter, , as a function of local values of , charge fraction , and . The charge fraction means the ratio of total proton number to total baryon number. The bulk free energies of heavy nuclei are represented as


where is the equilibrium density of heavy nucleus to be optimized and is the mass number of nucleus . The bulk free energy consists of interaction and kinetic terms as follows:


The interaction term is calculated based on Oyamatsu et al. Oyamatsu and Iida (2003) as


where and are energy densities for symmetric and pure-neutron matter, respectively, and 1.58632 fm. The kinetic term is derived from the Fermi integrals of nucleons Lattimer and Swesty (1991):


where and and are proton and neutron masses, respectively. The chemical potentials, , are derived from the number densities of protons, , and neutrons, , and is defined as


In the previous model, the bulk energy per baryon was estimated by the simplified formula:


with  MeV Lattimer and Swesty (1991); Bondorf et al. (1995). The parameters for bulk properties at zero temperature, , , , and , characterize the EOS for uniform nuclear matter and are useful to compare them with the canonical values obtained from terrestrial experiments and observations of compact stars Oertel et al. (2017). In the new model, the five coefficients, , , , and , in Eq. (4) are determined to reproduce those bulk properties in Eq. (9) at and Oyamatsu and Iida (2003):


The last equation derives from the fact that pressure becomes zero at the saturation density. The bulk parameters are taken from the same reference Oyamatsu and Iida (2003) and corresponding coefficients given by Eqs. (10-14) are summarized in the Tab. 1. The nuclear bulk energies are shown in Fig. 1. The left panel indicates that the bulk energies for symmetric nuclear matter () of the two parameter sets are identical, since the difference in the values of is not influential at all. On the other hand, the free energies for asymmetric nuclear matter () rise more rapidly for the parameter set B with the larger value of as shown in the right panel.

The Coulomb energy of heavy nuclei is expressed in the same way as in the previous model:


where is the number density of electrons and the local number densities of free nucleons are defined as with their number densities in the whole volume, , and the excluded volume effect, . They are expressed as and with the total volume , the nuclear volume and the numbers of free protons, , and free neutrons, , in the same way as in the HS and FYSS EOSs. The individual nuclear volumes are estimated as and . Equation (15) is obtained within the Wigner-Seitz approximation assuming that whole system is divided into electrically-neutral spherical cells each containing one nucleus and corresponding numbers of free nucleons and electrons, see Refs. Furusawa et al. (2011); Ebel et al. (2018) for details. The cell volume, volume fraction and its function are expressed as , and and is the elementary charge.

The surface energy of heavy nuclei is calculated as


where is the radius of the nucleus. We employ the following surface tension Agrawal et al. (2014) : , where MeV/fm and 12.1 MeV. The critical temperature is determined to be consistent with the free energy density of bulk nuclear matter described by Eq. (3). It is defined as the temperature at which simultaneously and , where the bulk pressure is defined as . In the previous model, we just assumed MeV for all nuclei. Figure 2 displays the critical temperatures above which the bulk nuclear matter can not be bound. The neutron-rich matter has smaller critical temperatures due to the symmetry energies. The extremely neutron-rich matter with has no local minimum point in the bulk pressure. It can be confirmed that the larger value of for the parameter set B causes slightly lower .

The masses of light clusters, , are set to be their experimental values, , plus the Coulomb energy shifts, which are defined as by using the same formula for the Coulomb energies of heavy nuclei, Eq. (15). Note that the equilibrium densities of light clusters are assumed to be constant and equal to the saturation density of symmetric nuclear matter, . In the previous paper, we represented the light clusters by particles only, without the Coulomb correction, for the fair comparison with SNA EOSs. Effective repulsive interactions among light clusters and other baryons are introduced by using excluded-volume effects, as explained in the next paragraph. Explicit nuclear interactions for light clusters, however, are not considered in this model for simplicity, since there are many ambiguities in the model for the free energy of light clusters and since the main focus of this work is the calculation of heavy nuclei. We discuss some other approaches for light clusters briefly in Section IV.

The translational free energies of light and heavy nuclei in Eq. (1) are expressed as


where are the spin degeneracy factors of the ground state, which are set to unity for heavy nuclei and to the experimental values for light clusters. We consider contributions of excited states of heavy nuclei are included in the temperature-dependent mass free energy, , in the same way as in SMSM and FYSS EOSs, whereas some MNA EOSs such as HS EOS employ the partition functions counting the excited state (see Refs. Buyukcizmeci et al. (2013); Furusawa et al. (2013a) about the difference between the two approaches). Excited states for light clusters are also neglected here for simplicity (see Section IV). The excluded volume correction is employed with in the same way as in the SMSM and HS EOSs. Note that differs from the excluded volume effect for free nucleons, , in which only nuclei contribute to the exclusion Hempel and Schaffner-Bielich (2010).

The free energy densities of free nucleons dripped from nuclei are based on the same calculations for the bulk energies of heavy nuclei with the excluded volume effect. It is expressed as


The free energy per nucleons, , is estimated by Eq (3), and includes kinetic and interaction terms. In the previous model, it was estimated by the translational energies of point-like particles based on the ideal-gas approximation with the excluded volume.

To optimize the total free-energy densities, we solve four variables, the chemical potentials and local number densities of nucleons, , to satisfy four equations, the relations between the chemical potentials and number densities of nucleons as well as the baryon- and charge-number conservations. This process to solve the full nuclear ensemble is almost the same as the optimization in the MNA EOSs. In addition, we find the equilibrium densities of all nuclei, , by minimizing their mass free energies, , at every step of the above process. This equilibration is usually included in the optimization for SNA EOSs, in which the equilibrium density of the representative nucleus is adjusted.

Iii Result of self-consistent model

We discuss the temperature dependence of the results of the EOSs with the bulk parameter sets B and E for = 0.1  and 0.3  and 0.2 and 0.4, typical values in the supernova core, where heavy nuclei prevail even at high temperatures. Figure 3 displays the mass fraction and equilibrium densities, , in the plane for the model with the parameter set E at , and , 3, 5 and 10 MeV. Generally speaking, the abundant nuclei are determined by the balance among the chemical potentials, binding energies and entropies. As temperature increases, the difference in entropies becomes more important than that in binding energies. At extremely-high temperature, the neutron-rich nuclei, however, can not survive due to the vaporizations even with large chemical potentials and entropies. For instance, the nuclei of and are populated at and 5 MeV but disappear at 10 MeV, albeit relatively close to the abundance peak . Note that the nuclear equilibration depends not only on bulk properties but also on Coulomb and surface energies. Those nuclei with are diminished at  MeV although the temperature is lower than the critical temperature, MeV, of the bulk matter with . This is in part due to the reductions of the surface tensions by the factor, . The vaporizations of neutron-rich nuclei can be confirmed in Fig. 4, in which mass fractions and equilibrium densities of nuclei with are displayed for the models with parameter sets B and E. The neutron-rich nuclei have smaller equilibrium densities than symmetric nuclei and are cut off one by one at high temperatures. We found that the EOS with the parameter set B, which has the larger value of , gives smaller equilibrium densities.

Figure 5 displays the equilibrium densities of the averaged values, , and of the specific nucleus with , Ca, as functions of temperature. We find that the equilibrium densities of individual nuclei decrease as temperature increases. This is because of the increase of kinetic term in bulk energy, , as shown in Fig. 1 and the decrease of surface energy. On the other hand, the average equilibrium density increases around  MeV, since the lighter nuclei with take the place of the heavy nuclei with as will be shown in Fig. 7. The nuclei with small mass numbers generally have small Coulomb energies per baryon and large equilibrium densities. This inconsistency in the temperature dependence between the average and individual equilibrium densities is similar to that in the density dependence observed in the previous work. We showed that the individual nuclei are compressed when few nucleons are dripped, while the average equilibrium density always decreases along with the density rise because of the change in dominant nuclear species Furusawa and Mishustin (2017). At temperatures higher than  MeV, the dominant nuclear species are not greatly changed and the average equilibrium density is reduced according to their decompression. The nuclei with low , however, start to be vaporized one by one above  MeV as shown in Fig. 3. As a result, the equilibrium densities increase a little for when the dominant neutron-rich nuclei vaporize, which have smaller equilibrium densities than symmetric nuclei in general. For instance, the nuclei with such as Ca are diminished around MeV and and, then, the average equilibrium densities change non-smoothly.

Figure 6 provides the mass fractions of dripped nucleons, light clusters ( or ) and heavy nuclei ( and ) as functions of temperature. At high temperatures, the mass fraction of heavy nuclei is reduced, since the particles with larger mass numbers give smaller entropies per baryon. The differences in those values between parameter sets B and E for are smaller than in the case of , since their bulk energies around are not greatly different as shown in Fig. 1. As for , the mass fraction of heavy nuclei in the parameter set B is smaller than in the other model due to the larger value of , while the free neutron fraction is larger. The average mass and atomic numbers for heavy nuclei are also reduced at high temperatures to increase the entropy per baryon as shown in Fig. 7. The smaller equilibrium density for the model with the parameter set B provides larger surface energies and smaller Coulomb energies of nuclei and, as a result, the larger average mass and proton numbers.

Iv Comparison with other models

We also perform some calculations to compare different models for free energy of heavy nuclei (Models I-IV). The models are listed on Table 2. Model I is the self-consistent model of our new calculation, whereas incompressible-liquid drop models are utilized in the other models, in which the equilibrium densities are set to constant values at zero density and zero temperature as . In Model III, the critical temperature for Eq. (16) is assumed to be MeV, while the charge fraction dependence of , which is shown in Fig. 2, is taken into account in Model II. The former may be regarded as a surrogate for SMSM and FYSS EOSs and the latter may be close to a recent MNA EOS, SRO EOS da Silva Schneider et al. (2017), in which the dependence of is introduced in surface tensions.

Recently Pais and Typel Pais and Typel (2016) also construct an MNA EOS with an assumption of the dissolutions of heavy nuclei around MeV. In Model IV, we introduce the same nuclear dissolution factor, , as


where is the chemical potential of heavy nucleus . The flashing temperature, 11.26430 MeV, is determined by the local instability of symmetric nuclear matter calculated by the DD2 relativistic mean field model Typel et al. (2010). The value taken from the reference Pais and Typel (2016) is not consistent with the flashing temperature of our bulk matter calculation. It is between 12 and 16 MeV as shown in the left panel of Fig. 1, where the local minimum of bulk energy around disappears. In the mass evaluation for Model IV, the temperature dependence of surface tension is ignored or to avoid the double count of the finite temperature effect. For all four models, we utilize the same formulations for free energies of the nucleons with the parameter set B and light clusters to focus only on modeling heavy nuclei.

Figures 8 displays the mass fraction of heavy nuclei for the four models. Below MeV, the incompressible models, Models II-IV, are not so deviated from the self-consistent model, Model I. Their mass fractions, however, are underestimated a little up to  MeV, since, in Model I, the adjustment of equilibrium densities lowers the mass free energies Furusawa and Mishustin (2017). On the other hand, the incompressible heavy nuclei in Models II and III keep on prevailing even above  MeV and their fractions are significantly overestimated. The dependence of in Model II reduces the surface energies and increases the mass fraction of neutron-rich nuclei compared with those of Model III. Such neutron-rich nuclei, however, can not survive above , and, thus, the mass fraction of the former becomes smaller than that of the latter at  MeV, and . The reduction of heavy nuclei in Model IV is quantitatively similar to that in Model I and, thus, the dissolution factor in Model IV is likely to reproduce the nuclear vaporization to some extent even with the incompressible liquid drop model. The nuclear dissolution temperatures, MeV, and the flashing temperature in Model IV are lower than the vaporization temperatures in Model I, MeV. These temperatures are, respectively, related to the local instability of symmetric nuclear matter and the global instabilities of compressible liquid nuclei with various asymmetry dependences. Both the dissolution factor of Eq. (19) in Model IV and the estimate of surface tensions in Model I, however, are simple and not greatly reliable. Thus, quantitative justifications about the melting temperature would be difficult in this comparison.

Finally we compare some theoretical approaches for light clusters. In Models V and VI, the incompressible liquid drop model with MeV for heavy nuclei is utilized as in Models III, but we replace the light cluster calculation by other approaches. In Model V, a quantum approach is incorporated to evaluate Pauli and Self energy shifts of light clusters Röpke (2009); Typel et al. (2010) as in the similar way as in FYSS EOS. The Pauli energy shift, , is calculated by Eq. (72) in Typel et al. 2010 Typel et al. (2010). The self energy shift, , is evaluated based on interaction energies, [Eq. (4)], for dripped nucleons as


The excluded volume effects in Eq. (17) are ignored with , since Pauli exclusion is incorporated in the mass evaluation (see ref. Hempel et al. (2011) about the detailed comparison between the two approaches). In the other three models, the masses are evaluated as as explained in Section II. Model VI includes the temperature-dependent partition function for light clusters as


where is the chemical potential of light cluster . This partition function Fái and Randrup (1982) is adopted for all nuclei in HS EOS with the maximum energy modified. For the other models, we assume that . The properties of these models are summarized in Table 2.

Figure 9 displays the mass fractions of deuteron, , triton plus helion, , and alpha particle, , for Models I, III, V and VI. Model I gives the rapid increase in mass fraction of light clusters at the vaporization point of heavy nuclei, whereas Model III shows the smooth slope around 14 MeV. The vaporization temperatures of heavy nuclei in the new model may be checked in the heavy-ion reactions at intermediate energies by measuring the excitations of light clusters. The Pauli energy shifts in Model V reduce the mass fractions of light clusters compared with those in Model III for the most part. Especially for neutron-rich conditions , their mass fractions are suppressed because of strong Pauli-exclusion among neutrons. Only at and for deuteron, the self-energy shift increases the mass fraction a little. For Model VI, the increase in degree of freedom at high temperature increases the mass fractions of light clusters. The model for light clusters also affects the mass fraction of heavy nuclei, and vice versa. Actually the mass fraction of heavy nuclei in Models V (VI) tends to be more (less) than that in Model III even with the same free energy model for heavy nuclei. On the other hand, Fig. 9 shows that Model I gives less (more) light clusters than Model III at low (high) temperatures according to the change of mass fraction of heavy nuclei by the self-consistent optimization shown in Fig. 8, even if they employ the same model for light clusters.

V Conclusion

The new self-consistent calculation of nuclear abundances and nuclear equilibrium densities allows us to investigate how the full ensemble of nuclei behaves under the hot and dense stellar environment. We find that the nuclear equilibrium densities decrease and finally the neutron-rich nuclei are dissolved to nucleons one after another as temperature increases. The reduction of surface energies at high temperatures and the increase of kinetic contributions in bulk energies enhance the vaporization, and finally lead to disappearance of nuclei even at the temperatures below the critical temperatures for bulk matter. The larger value of the slope parameter, , provides the smaller equilibrium densities and lower critical temperatures due to the steeper slope of bulk energies for neutron-rich matter. The total mass fractions are reduced and the average mass and atomic numbers are increased owing to the small equilibrium densities in the model with the larger value of .

These modifications of equilibrium densities and vanishing of hot nuclei are not included in supernova EOSs considering the full nuclear ensemble beacuse of the simplifications adopted for nuclear binding energies. In the ordinary multi-nucleus EOSs, heavy nuclei are assumed to prevail even at MeV. We also find that the mass fractions in such incompressible liquid drop models are underestimated below about MeV or overestimated at higher temperatures due to the neglect of the decompression or vaporization, respectively. We also confirm that the dissolution factor Pais and Typel (2016) may be able to reproduce the disappearance of heavy nuclei qualitatively even with the incompressible liquid drop model. In addition, the mass fractions of light clusters are revealed to be sensitive to their mass evaluations, partition functions and treatment of excluded volume effects as well as the free energy model for heavy nuclei.

Our self-consistent calculations including the optimization for both nuclear abundances and structures may not be suitable for constructing data tables covering a wide rage of conditions , because of the huge computational resources required. Tentatively, feasible improvements for the previous multi-nucleus EOSs, in which the critical temperature was set to be 18 MeV Buyukcizmeci et al. (2014); Furusawa et al. (2017b, c) or Hempel and Schaffner-Bielich (2010); Steiner et al. (2013) for all nuclei, would be the consideration of the dissolution factor Pais and Typel (2016) or the iso-spin dependence of da Silva Schneider et al. (2017).

In our model, the surface tension and its temperature and density dependences are estimated simply to calculate the whole ensemble of nuclei. More sophisticated models such as Thomas Fermi calculations would improve results quantitatively although their self-consistent calculations may be more difficult. The free energy of light clusters should be also improved as discussed in the last part of Section IV and may influence in a essential way on the supernova dynamics and proto-neutron-star cooling Sumiyoshi and Röpke (2008); Typel et al. (2010); Furusawa et al. (2013b). More detailed analysis about the light clusters Typel et al. (2010); Hempel et al. (2011); Typel and Pais (2017); Avancini et al. (2017) may be required to discuss the properties of hot nuclear matter in addition to the descriptions of uniform nuclear matter and heavy nuclei.

S.F. was supported by Japan Society for the Promotion of Science Postdoctoral Fellowships for Research Abroad. I.M. acknowledges partial support from the Helmholz International Center for FAIR (Germany). This work was supported by RIKEN iTHES and iTHEMS Projects. A part of the numerical calculations were carried out on PC cluster at Center for Computational Astrophysics, National Astronomical Observatory of Japan.
parameter set B E
[fm] 0.15969 0.15979
[MeV] -16.184 -16.145
[MeV] 230 230
[MeV] 33.550 31.002
[MeV] 73.214 42.498
[MeV fm] -467.51 -466.32
[MeV fm] 2206.0 2192.9
[fm] 341.77 340.02
[MeV fm] -251.42 -206.48
[MeV fm] 1138.8 662.27
Table 1: The parameters of bulk nuclear matter Oyamatsu and Iida (2003, 2007).
light clusters
Model I optimized no excluded volume,
Model II constant no excluded volume,
Model III constant 18 MeV no excluded volume,
Model IV constant yes excluded volume,
Model V constant 18 MeV no quantum approach,
Model VI constant 18 MeV no excluded volume,
Table 2: Model list for Sec. IV. The columns of , , , and light clusters denote respectively whether the nuclear equilibrium density is solved self-consistently, the critical temperature for surface tension in Eq. (16), whether the dissolution factor of Eq. (20) is introduced, and the model free energy density of light clusters. All the models adopt the parameter set B for bulk nuclear matter.
Figure 1: Free energies per baryon of symmetric (left panel, ) and asymmetric (right panel, ) uniform nuclear matter for parameter sets B (blue solid lines) and E (red dashed lines) at 0 (thin lines), 4, 8, 12, and 16 MeV (medium lines) and (thick lines). The critical temperatures are 18.1 MeV for both parameter sets at and are 5.7 and 5.5 MeV for the parameter sets B and E, respectively, at as shown in Fig. 2.
Figure 2: The critical temperatures for bulk nuclear matter, above which the bulk pressure has no local minimum, as a function of charge fraction for parameter sets B (blue solid lines) and E (red dashed lines).
Figure 3: Mass fraction, , (left column) and equilibrium density, , (right column) in plane for the parameter set E at  , , and  1, 3, 5, and 10 MeV (from top to bottom).
Figure 4: Mass fraction, , (left column) and equilibrium density, , (right column) for the nuclei with as functions of charge fraction, , for parameter sets B (blue solid lines) and E (red dashed lines) at 1 MeV (thin lines), 5 MeV (medium lines) and 10 MeV(thick lines) and at , , , and from top to bottom.
Figure 5: Equilibrium density of Ca (thin lines) and average one over all the heavy nuclei, , (thick lines) as functions of for parameter sets B (blue solid lines) and E (red dashed lines) at (top row) and (bottom row) and 0.2 (left column) and 0.4 (right column).
Figure 6: Mass fractions of heavy nuclei (thick lines), light clusters (medium lines) and free nucleons (thin lines) as functions of for parameter sets B (blue solid lines) and E (red dashed lines) at (top row) and (bottom row) and 0.2 (left column) and 0.4 (right column).
Figure 7: Average mass and proton numbers (thick and thin lines) of heavy nuclei as functions of in the self-consistent model for parameter sets B (blue solid lines) and E (red dashed lines) at (top row) and (bottom row) and 0.2 (left column) and 0.4 (right column).
Figure 8: Comparison of mass fractions of heavy nuclei among different nuclear models listed on Table 2, the self-consistent calculation with the compressible-liquid drop model (Model I, red solid lines), the incompressible-liquid drop models with the same of surface tensions as in Model I (Model II, blue dashed lines) and with MeV for the all nuclei (Model III, green dashed-dotted lines) and the incompressible one with and the nuclear dissolution factor Pais and Typel (2016), Eq. (20), (Model IV, magenta dashed double-dotted lines) at (top row) and (bottom row) and 0.2 (left column) and 0.4 (right column). All the four models are calculated with the parameter set B.
Figure 9: Mass fractions of deuteron (top row), triton and helion (middle row) and alpha particle (bottom row) for Model I (red solid lines), Model III (green dashed-dotted lines), and Model V (orange dashed lines) with a quantum approach to light clusters and Model VI (black dashed double-dotted lines) with a level-density approach to light clusters at and 0.2 (left column) and 0.4 (right column). The three models other than Model I employ the same incompressible liquid drop models for the heavy nuclei.


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