A NEW BARYONIC EQUATION OF STATE AT SUB-NUCLEAR DENSITIES FOR CORE-COLLAPSE SIMULATIONS

The free energy of the nucleons that are not contained in nuclei is based on RMF in this paper (Shen et al. 1998a). The choice is not essential, however, and one can replace it by any other theory or model. The important thing to ensure the continuous transition to uniform nuclear matter is to use the same theory or model for the nucleons outside nuclei at sub-nuclear densities and uniform nuclear matter at supra-nuclear densities. The details of the RMF model employed in this paper can be found in the original paper (Shen et al. 1998a). In the following, we summarize only the essential ingredients.

In RMF nuclear interactions are described by the exchange of mesons. We employed the following Lagrangian:

$$\begin{eqnarray} {\cal L}_{{\rm RMF}} & = & \bar{\psi }[i\gamma _{\mu }\partial ^{\mu } -M -g_{\sigma }\sigma -g_{\omega }\gamma _{\mu }\omega ^{\mu } -g_{\rho }\gamma _{\mu }\tau _a\rho ^{a\mu }]\psi \nonumber\\ \nonumber && +\frac{1}{2}\partial _{\mu }\sigma \partial ^{\mu }\sigma -\frac{1}{2}m^2_{\sigma }\sigma ^2-\frac{1}{3}g_{2}\sigma ^{3} -\frac{1}{4}g_{3}\sigma ^{4} \\ \nonumber && -\frac{1}{4}W_{\mu \nu }W^{\mu \nu } +\frac{1}{2}m^2_{\omega }\omega _{\mu }\omega ^{\mu } +\frac{1}{4}c_{3}(\omega _{\mu }\omega ^{\mu })^2 \\ && -\frac{1}{4}R^a_{\mu \nu }R^{a\mu \nu } +\frac{1}{2}m^2_{\rho }\rho ^a_{\mu }\rho ^{a\mu }. \end{eqnarray} \tag{ 1 }$$

In the above expression, the notation is the same as in Shen et al. (1998a): ψ, σ, ω, and ρ denote nucleons (proton and neutron), scalar-isoscalar meson, vector-isoscalar meson, and vector-isovector meson, respectively, and W_μν = ∂^μω^ν − ∂^νω^μ and R^a_μν = ∂^μρ^aν − ∂^νρ^aμ + g_ρ^abcρ^bμρ^cν. The nucleon–meson interactions are given by the Yukawa couplings and the isoscalar mesons (σ and ω) interact with themselves, which are expressed as the cubic and quartic terms. In the mean field theory, the mesons are assumed to be classical and replaced by their ensemble averages whereas the Dirac equation for nucleons is quantized and the free energy is evaluated based on the energy spectrum of nucleons obtained this way. M is the mass of nucleons and assumed to be 938 MeV. We use the TM1 parameter set as in Shen et al. (1998a), in which the masses of mesons, m_σ, m_ω, m_ρ, and the coupling constants, g_σ, g_ω, g_ρ, g₂, g₃, c₃, are determined so that not only the saturation of uniform nuclear matter but also the properties of finite nuclei could be best reproduced (Sugahara & Toki 1994).

It is interesting to study how the resulting EOS is affected by the choice of RMF parameter or, more generally, by the choice of EOS for uniform nuclear matter. If Hempel & Schaffner-Bielich (2010), in which several RMF parameterizations were compared, is any guide, differences in the RMF parameterization will be reflected in thermodynamic quantities near the saturation density. Because RMF is employed also for the bulk energy of nuclei in our EOS as explained later in Section 2.2.3, differences in symmetry energy originated from different parameterizations may be reflected in the neutron-richness of nuclei, for which we have no available experimental mass data and the empirical mass formula is employed. The abundance of nuclei with experimental mass data, on the other hand, will not be changed very much. It should also be noted that RMF in general tends to overestimate the neutron matter energy at sub-nuclear densities (Sumiyoshi et al. 1995; Akmal et al. 1998; Steiner et al. 2005; Gandolfi et al. 2010; Hebeler & Schwenk 2010). It is stressed that one of the features of our EOS is that it is easy to adopt different EOSs at supra-nuclear densities as long as they are available. The above issues will be discussed in more detail in the forthcoming paper.

In this paper, the free energy density of uniform nuclear matter, f^RMF(n_B, T, Y_p) with n_B, T, and Y_p being the baryon number density, temperature, and proton fraction, respectively, is used mainly at sub-nuclear densities. In doing so, one important thing to be taken into account is the fact that the free nucleons cannot exist in the volume occupied by nuclei. The volume, V', that can accommodate the nucleons is hence given by

$$\begin{equation} V^{\prime } =V-\sum _{i} N_i V_i^N, \end{equation} \tag{ 2 }$$

where the index i specifies a nucleus, V is the total volume, N_i is the particle number of nucleus i, and V^N_i is the volume of nucleus i, which is expressed by the mass number, A_i, and number density, n_i, of the same nucleus as well as the nuclear saturation density, n_si, as follows:

$$\begin{equation} V_i^N= \frac{A_i}{n_{si}}, \end{equation} \tag{ 3 }$$

where we assume that the nuclei are homogeneous, having the nuclear saturation density, n_si. The reason why n_si has the subscript i will be made clear later (see Section 2.2). Then the densities of free protons and neutrons in the volume V', which are denoted by n'_p and n'_n, respectively, are given by

$$\begin{equation} n^{\prime }_{p/n}=\frac{N_{p/n}}{{V^{\prime }}}= \ \frac{n_{p/n}}{\eta }, \end{equation} \tag{ 4 }$$

where N_p/n and n_p/n are the particle numbers and the number densities, respectively, in the total volume V of nucleons and η is expressed as η = V'/V = 1 − ∑_iA_in_i/n_si. The free energy density for the nucleons in V' should be evaluated at Y_p = n'_p/(n'_p + n'_n) and n_B = (n'_p + n'_n) and is given for the entire volume as

$$\begin{equation} f_{p,n} = \eta \, f^{\prime }_{p,n}, \end{equation} \tag{ 5 }$$

with f'_{p, n} = f^RMF(n'_p, n'_n, T).

Free energy of nuclei is the most crucial part of our model free energy and is described in detail in the following. The free energy of nuclei consists of two parts, one concerning a translational energy and the other part roughly a nuclear mass. The modeling of the latter is based on a semi-empirical mass formula and takes into account the Coulomb, surface, and bulk energies. We explain each contribution to the mass formula one by one and then move on to the treatment of the translational energy.

Before proceeding, however, we state here the definition of the nuclear saturation density, n_si(Z_i/A_i, T), more precisely, since it will appear frequently in the following sections. It is in principle the baryon number density at which the free energy per baryon, F^RMF(T, n_B, Y_p) with Y_p = Z_i/A_i, by RMF takes its minimum value or, put another way, the uniform nuclear matter with the same proton fraction as the nucleus has the lowest energy in RMF. Note that the proton fraction of each nuclei is different from that for the whole system and that our nuclear saturation density is actually different from nucleus to nucleus and hence has the subscript that indicates which nucleus is referred to. There are two exceptions to this definition, though. The first one is the case in which the saturation density given above is smaller than the number density of the whole system, n_B. This happens if n_B reaches the saturation density of some nucleus, n_si(Z_i/A_i, T), before the whole system becomes uniform. This is indeed possible because the saturation density tends to be smaller for more neutron-rich nuclei. In such a case, the nucleus will be overcompressed and we replace the original saturation density with the number density of the whole system, n_B. The second case occurs at high temperatures, for which the free energy, F^RMF(T, n_B, Z_i/A_i), has no minimum because the entropy contribution, the TS term with S being entropy, overwhelms the internal energy. In this case, the nuclear saturation density, n_si, is set to be the density where the matter becomes uniform in the H. Shen EOS.

2.2.1. Coulomb Energy

To calculate the Coulomb energy density in the free energy for nuclei, we introduce a Wigner–Seitz cell (W-S cell) for each species of nuclei and impose charge neutrality in it. It is assumed that a nucleus is centered in the W-S cell, the volume of which is denoted as V_i and that the cell also contains nucleons as a vapor outside the nucleus as well as electrons, which are assumed to be uniform in the entire cell. The charge neutrality in the cell can be expressed as

$$\begin{equation} V_i n_e =Z_i +\big(V_i-V_i^N\big)n^{\prime }_p, \end{equation} \tag{ 6 }$$

which can be solved for V_i and gives

$$\begin{equation} V_i = \frac{Z_i - n^{\prime }_p V_i^N}{n_e-n^{\prime }_p}, \end{equation} \tag{ 7 }$$

where V^N_i is the nuclear volume in the cell and n_e denotes the number density of electrons. The vapor volume and nuclear volume fraction for each nucleus are given by V^B_i = V_i − V^N_i and u_i = V^N_i/V_i, respectively.

As the density approaches the nuclear saturation density, the nuclear pasta phase is encountered, in which various shapes of nuclei exist. As mentioned already, this phase has to be somehow treated in order to obtain the smooth transition to uniform nuclear matter. It is well beyond the scope of our project to treat all known pastas accurately, however. We hence resort to a rather crude approximation, in which the bubble phase is explicitly considered. It is known that bubble nuclei, in which the vapor nucleons are surrounded by the nucleus, appear at the end of the pasta phase. We assume there that a spherical vapor is located at the center of a W-S cell surrounded by a shell-shaped nucleus. In this paper, we assume further that each nucleus enters the nuclear pasta phase individually when the volume fraction, u_i, reaches 0.3 and that the bubble shape is realized when it exceeds 0.7 (Watanabe et al. 2005). The intermediate states (0.3 < u_i < 0.7) are simply interpolated from the normal and bubble states. This criterion is admittedly rather arbitrary, but we have tried another range of values, 0.4 < u_i < 0.6, and confirmed that the results are hardly affected. We hence believe that our EOS is not very sensitive to the criterion.

The evaluation of the Coulomb energy in the W-S cell is straightforward and given by the following expression:

$$\begin{equation} E_i^C= \int _{\rm cell} \frac{q(r) \, dq(r)}{r}, \end{equation} \tag{ 8 }$$

where the cell is assumed to be spherically symmetric and q(r) is the sum of electron and proton charges up to the radius r. Assuming further that the protons are uniformly distributed in the nucleus and the vapor separately, we obtain the Coulomb energies for the normal and bubble phases from the above integration as follows:

$$\begin{eqnarray} && E_i^C =\nonumber\\ &&\quad\left\lbrace \begin{array}{@{}ll}{\frac{3}{5}\left(\frac{3}{4 \pi }\right)^{-1/3} \frac{e^2}{n_{si}^2} \left(\frac{Z_i - n^{\prime }_p V_i^N}{A_i}\right)^2 {V_i^N}^{5/3} D(u_i)} & (u_i\le 0.3), \\\ms {\frac{3}{5}\left(\frac{3}{4 \pi }\right)^{-1/3} \frac{e^2}{n_{si}^2} \left(\frac{Z_i - n^{\prime }_p V_i^N}{A_i}\right)^2 {V_i^B}^{5/3} D(1-u_i)} & (u_i\ge 0.7), \end{array} \right. \nonumber\\ \end{eqnarray} \tag{ 9 }$$

with $D(u_i)=1-\frac{3}{2}u_i^{1/3}+\frac{1}{2}u_i$, where e is the elementary charge. Note that the two energies coincide with each other at u_i = 0.5, since the nuclear volume, V^N_i, is equal to the vapor volume, V^B_i, there although the value is not used and the interpolation is done from the values at u_i = 0.3 and u_i = 0.7 to crudely take into account the existence of other pastas. We employ cubic polynomials of u_i for interpolation. The four coefficients of the polynomial are determined by the condition that the Coulomb energy is continuous and smooth as a function of u_i at u_i = 0.3 and u_i = 0.7.

2.2.2. Surface Energy

In the semi-empirical mass formula we employ in this paper, the surface energies of nuclei are given by the product of the nuclear surface area and the surface tension, which is approximated by the following analytic expression:

$$\begin{equation} \sigma _i=\sigma _0 - \frac{A_i^{2/3} }{4 \pi r_{N_i}^2} [S_s(1- 2(Z_i/A_i)^2 ], \end{equation} \tag{ 10 }$$

where $r_{N_i} =(3/4 \pi V_i^N)^{1/3}$ is the radius of nucleus i and σ₀ denotes the surface tension for symmetric nuclei. The second term on the right-hand side is added to accommodate highly neutron-rich nuclei, for which no experimental data are available but a proper consideration of the surface symmetry energy is almost mandatory. The values of the constants, σ₀ = 1.15 MeV fm⁻³ and S_s = 45.8 MeV, are adopted from the paper by Lattimer & Swesty (1991). Note that this expression gives negative values if the nuclei become extremely neutron-rich. We hence assume in this paper that only those neutron-rich nuclei with a surface tension higher than a prescribed lower limit, σ = 0.272 MeV fm⁻³, can exist. This value is obtained from the mass data of the most neutron-rich nuclei with (Z, A) = (1, 7) (Audi et al. 2003) and the subtraction of the bulk and Coulomb energies given by our mass formula.

Noting that the nuclear surface should be replaced by the bubble surface in the bubble phase, we obtain the following expressions for the surface energy:

$$\begin{eqnarray} &&E_i^S=\left\lbrace \begin{array}{@{}l}4 \pi {r^2_{Ni}} \sigma _i \left(1-{\frac{n^{\prime }_p+n^{\prime }_n}{n_{si}}} \right)^2 = 4\pi \left({\frac{3}{4 \pi }} V_i^N \right)^{2/3} \\ \qquad\qquad\qquad\quad\;\sigma _i \left(1-{\frac{n^{\prime }_p+n^{\prime }_n}{n_{si}}} \right)^2 \quad (u_i\le 0.3), \\ 4 \pi {r^2_{Bi}} \, \sigma _i \left(1-{\frac{n^{\prime }_p+n^{\prime }_n}{n_{si}}} \right)^2 =4\pi \left({\frac{3}{4 \pi }} V_i^B \right)^{2/3} \\ \qquad\qquad\qquad\quad\;\sigma _i \left(1-{\frac{n^{\prime }_p+n^{\prime }_n}{n_{si}}} \right)^2 \quad (u_i\ge 0.7), \end{array} \right.\nonumber\\ \end{eqnarray} \tag{ 11 }$$

where r_Bi = (3/4πV^B_i)^1/3 is the radius of bubble i. The last factor, (1 − (n'_p + n'_n)/n_si)², in these expressions is introduced to take into account the effect that the surface energy is reduced as the density contrast decreases between the nucleus and the nucleon vapor. The specific functional form is inspired by the surface energy employed in the Thomas–Fermi approximation in the H. Shen EOS, |∇(n_n + n_p)|² (Equation (35) of Shen et al. 1998a). As shown later, this prescription is necessary to ensure the continuous transition to uniform nuclear matter, since the transition occurs not only by the coalescence of nuclei but also by the increase in the density of nucleon vapor. For the other pastas (0.3 < u_i < 0.7), we employ the same cubic interpolation as for the Coulomb energy (Section 2.2.1).

2.2.3. Bulk Energy

The bulk energy of nuclei should be different at high densities, ρ_B ≳ 10¹² g cm⁻³, from the one at lower densities, since nuclei cannot be regarded as isolated objects and effects of other nuclei, nucleons, and electrons cannot be ignored (Vautherin 1994). In this paper, we will take this into account phenomenologically as follows.

As the density increases, the nuclear bulk energy, E^B_i, should approach the value for uniform nuclear matter with the same proton fraction. We obtain the latter from RMF as

$$\begin{equation} E_i^{B}(T)= A_i F^{{\rm RMF}}(n_{si},T,Z_i/A_i), \end{equation} \tag{ 12 }$$

where F^RMF(n_B, T, Y_p) is the free energy per baryon by RMF. Note that the proton fraction in this expression is not the one for the whole system but the one for each nucleus; the density is taken to be the saturation density defined earlier (Section 2.2) for each nucleus. This implies that nuclei have their own densities that are different from each other. Since the saturation density tends to be lower for neutron-rich nuclei, the density of the whole system becomes equal to the one for some neutron-rich nucleus one after another as it increases. In such a case the density of the nucleus that becomes uniform in its W-S cell is set to be the system density as mentioned earlier. These prescriptions ensure the continuous transition to uniform nuclear matter.

At low densities, on the other hand, experimental mass data are used whenever available (Audi et al. 2003). This is indeed important in reproducing the well-known results of the ordinary NSE. As will be shown later, the nuclear shell and even–odd effects manifest themselves in the abundance of nuclei and should be taken into account whenever possible. These experimental mass data are accommodated in our mass formula by adjusting the bulk energy, which is expressed as

$$\begin{equation} E_i^{B}=M_i^{\rm {data}}-\big[E_i^{C}\big]_{{\rm vacuum}}-\big[E_i^{S}\big]_{{\rm vacuum}}, \end{equation} \tag{ 13 }$$

where subtracted from the experimental mass, M^data_i, are the Coulomb and surface energies [E^{C, S}_i]_vacuum of the nucleus isolated in vacuum, that is, the energies calculated from our mass formula for n_e = n'_p = n'_n = 0. This ensures that the experimental mass is reproduced exactly in the low-density limit. At finite densities, on the other hand, the mass is given by

$$\begin{eqnarray} M_i & = & E_i^{B} + E_i^{S} + E_i^{C}= M_i^{\rm {data}}+E_i^{S}(n^{\prime }_n,n^{\prime }_p,n_e) \nonumber \\ &&+ E_i^{C}(n^{\prime }_p,n_e) -\big[E_i^{C}\big]_{{\rm vacuum}}-\big[E_i^{S}\big]_{{\rm vacuum}}. \end{eqnarray} \tag{ 14 }$$

Note that the bulk energy in our mass formula actually includes other energies such as the bulk symmetry, shell, and pair energies.

For nuclei with no available experimental data, we employ the bulk energy obtained from RMF even at low densities as explained above. Because these nuclei, which are either very heavy Z_i ≳ 100 or highly neutron-rich Z_i/A_i ≲ 0.35, are abundant normally at high densities ($\rho _B \gtrsim 10^{12} \,\rm {g\, \rm cm^{-3}}$) alone, the above prescription is justified. The only exception is the combination of very low temperatures (T ≲ 1 MeV) and low proton fractions (Y_p ≲ 0.3). Such cases are not encountered in the simulations of core-collapse supernovae, however, and will not be a practical problem. These low- and high-density values (Equations (13) and (12)) are linearly interpolated between $\rho _B = 10^{12} \,\rm {g\, \rm cm^{-3}}$ and the nuclear saturation density, $\rho _B \sim 10^{14.2} \,\rm {g\, \rm cm^{-3}}$. The linear interpolation makes the free energy not smooth and the pressure discontinuous at the boundaries of the interpolation region. In practice, however, the variation of the bulk energy is quite minor compared with those of the Coulomb and translational energies and the discontinuities of the pressure are negligible. Moreover, nuclei that have experimental mass data and hence need the interpolation are not abundant near the high-density end, a fact which makes the discontinuities even less important there. We hence believe that the above prescription is sufficient.

2.2.4. Translational Energy

The translational energy of nucleus i in our model free energy is based on that for the ideal Boltzmann gas and given by

$$\begin{equation} E_i^{t}=k_B T \left\lbrace \log \left(\frac{n_i}{g_i(T) n_{Qi}}\right)- 1 \right\rbrace \left(1-\frac{n_B}{n_s}\right), \end{equation} \tag{ 15 }$$

where k_B is the Boltzmann constant and n_Qi  =  (M_ik_BT/2πℏ²)^3/2. Since nuclei have internal degrees of freedom, the contribution from excited states of nuclei to the free energy cannot be ignored at high temperatures. In the above formula, the effect is encapsulated in g_i(T), which is approximated by Fai & Randrup (1982) as

$$\begin{equation} g_{i}(T)=g_{i}^0 +\frac{c_1}{A_i^{5/3}}\int _0^\infty dE e^{-E/T}\exp (\sqrt{2 a(A_i) E}), \end{equation} \tag{ 16 }$$

where g⁰_i denotes the contribution of the ground state (the spin degrees of freedom and normally much smaller than g_i(T)) and is set to be g⁰_i = 1 for even nuclei and g⁰_i = 3 for odd ones in this paper for simplicity. Other coefficients in the contribution from the excited states are given as a(A_i) = (A_i/8)(1 − c₂A^−1/3_i) MeV⁻¹, c₁ = 0.2 MeV⁻¹, and c₂ = 0.8. In order to estimate the sensitivity of results to g_i(T), we multiplied the original g_i(T) rather arbitrarily by a factor of 0.5 or 2 at Y_p = 0.3 and T = 5 MeV and compared the results. We found small differences at ρ ∼ 10¹³ g cm⁻³. For example, the free energy is decreased or increased by 4% at most; the entropy changes by −1% and +3% whereas the mass fraction of all nuclei varies by −10% and +8% from the original values.

The last factor on the right-hand side of Equation (15) takes account of the excluded-volume effect: each nucleus can move in the space that is not occupied by other nuclei and free nucleons. The factor reduces the translational energy at high densities and is important to ensure the continuous transition to uniform nuclear matter as mentioned earlier. In reality, however, the nuclear translational energy is supposed to be suppressed by the formation of Coulomb lattice when the so-called plasma factor, which is defined as $\Gamma \equiv (\bar{Z} e)^2/(\bar{r} k_B T)$ with the average proton number $\bar{Z}$ and distance between nuclei $\bar{r}$, reaches a critical value ∼171 (Slattery et al. 1998); the factor introduced may be regarded as a very crude phenomenological approximation to this situation. The present form of the factor, (1 − n_B/n_s) = (V − V_baryon)/V, actually gives a linear suppression in terms of the occupied volume V_baryon = ∑_iA_iN_i/n_s. Here we always employ the nuclear saturation density for symmetric nuclei $n_s=\left[ n_{si}(Z_i/A_i,T) \right] _{Z_i/A_i=0.5}$ for numerical convenience, which will be justified, provided the approximation itself is very crude. Also note that the translational energy is a minor contribution except at low densities, where the factor is almost unity.

The abundances of nuclei as a function of ρ_B, T, and Y_p are obtained by minimizing the model free energy derived so far with respect to the number densities of nuclei and nucleons under the constraints,

$$\begin{eqnarray} n_p+n_n+\sum _i{A_i n_i} & = & n_B=\rho _B/m_B, \nonumber \\ n_p+\sum _i{Z_i n_i} & = & n_e=Y_p n_B, \end{eqnarray} \tag{ 17 }$$

just as in the ordinary NSE calculations. Note that n_p/n is the number density of proton/neutron in the total volume V of the system as defined in Section 2.1. By introducing Lagrange multipliers, α and β, for these constraints, the minimization condition is given by

$$\begin{eqnarray} &&\frac{\partial }{\partial {n_j}} \Big\lbrace f - \alpha \Big(n_p+n_n+\sum _i{A_i n_i} - n_B\Big) \nonumber\\ &&\quad- \beta \Big(n_p+\sum _i{Z_i n_i}-n_e \Big) \Big\rbrace =0, \end{eqnarray} \tag{ 18 }$$

where f is the free energy density,

$$\begin{eqnarray} f &=& \eta f_{p,n}^{{\rm RMF}}(n^{\prime }_p,n^{\prime }_n)+\sum _i n_i \big\lbrace E_i^{t}+ E_{i}^{B}(T,n_{s}, Z_i / A_i)\nonumber\\ &&+E_i^{S}(n^{\prime }_n,n^{\prime }_p)+ E_i^{C}(n^{\prime }_p) \big\rbrace, \end{eqnarray} \tag{ 19 }$$

and the index j runs over all nuclear species and nucleons. Taking j = p, n, we find as usual that the Lagrange multipliers, α and β, are related to the chemical potentials of proton (μ_p = ∂f/∂n_p) and neutron (μ_n = ∂f/∂n_n) as

$$\begin{eqnarray} \alpha & = & \mu _n, \end{eqnarray} \tag{ 20 }$$

$$\begin{eqnarray} \beta & = & \mu _p -\mu _n. \end{eqnarray} \tag{ 21 }$$

Differentiation with respect to the number densities of nuclei, on the other hand, gives the usual relations between the chemical potentials of nuclei μ_i and those of protons and neutrons:

$$\begin{equation} \mu _i=Z_i \mu _p+(A_i-Z_i)\mu _n. \end{equation} \tag{ 22 }$$

Differentiating the free energy for nuclei with respect to the number densities of nuclei, n_i, and employing Equation (22), we can express n_i as:

$$\begin{equation} n_i = g_i \, n_{Qi} \exp \left(\frac{Z_i \mu _p+(A_i -Z_i) \mu _n - M_i}{k_B T (1-n_B/n_s)} \right). \end{equation} \tag{ 23 }$$

Note that the only difference from the counterpart in the ordinary NSE calculations,

$$\begin{equation} n_i = g_i\, n_{Qi} \exp \left(\frac{Z_i \mu _p+(A_i -Z_i) \mu _n - M_i}{k_B T} \right), \end{equation} \tag{ 24 }$$

seems to be the factor (1 − n_B/n_s) in the denominator in the exponential.

This is not true, however, and the calculations for our model are much more involved than for ordinary NSE. In the latter case, all we have to do is to solve the two conservation equations, Equation (17), for μ_p and μ_n, since the number densities of all nuclei as well as nucleons are expressed by them alone (e.g., Equation (24)). In our case, on the other hand, the masses of nuclei, M_i, depend on the number densities of nucleons in the vapor, n'_p and n'_n, which are in turn related with μ_p and μ_n as follows:

$$\begin{equation} \mu _{p/n} =\frac{\partial f}{ \partial n_{p/n}}= \mu ^{\prime {\rm RMF}}_{p/n}(n^{\prime }_p,n^{\prime }_n) + \sum _i n_i {\frac{\partial M_i(n^{\prime }_p,n^{\prime }_n)}{ \partial n_{p/n}}}, \end{equation} \tag{ 25 }$$

where μ'^RMF_p/n are the chemical potentials of nucleons in the vapor of volume V', which are obtained from RMF (see Section 2.1), and the second term, which originates from the dependence of the mass of nuclei on the nucleon densities in the vapor and hence is summed over all nuclear species, is the source of complication in the solution. As a result, we have to solve not only the two conservation equations, Equation (17), but also the relations between μ_p/n and n'_p/n, Equation (25), for the four variables, μ_p, μ_n, n'_p, and n'_n, which then give the number density of nuclei, n_i, by Equation (23).

After minimization, we obtain the free energy density together with the abundance of various nuclei and free nucleons as a function of ρ_B, T, and Y_p. As mentioned repeatedly, the Coulomb-, surface-, and translational-energy contributions to the free energy of nuclei are going to zero as the density increases and only the nuclear bulk-energy and free-nucleon contributions remain just prior to the transition to uniform nuclear matter. Since both of them are entirely based on RMF at this point, the free energy density obtained by the sum of them coincides with the one for uniform nuclear matter obtained by RMF. At low densities, on the other hand, the excluded-volume effect is negligible and the masses of nuclei agree with those of the experimental data. In addition, the RMF free energy for the vapor nucleons reduces to the one for the ideal Boltzmann gas of nucleons, since the interactions between nucleons become negligible. As a result, the free energy density obtained with the present formulation reproduces the well-known free energy of the ideal gas composed of nucleons and nuclei, the masses of which are given by the experimental data.

Once the free energy is obtained, other physical quantities are derived by its partial differentiations. The baryonic pressure, for example, is obtained by differentiation with respect to the baryonic density as follows:

$$\begin{eqnarray} p_B &=&n_B \left[\partial {f}/\partial {n_B}\right]_{T,Ye}-f \end{eqnarray} \tag{ 26 }$$

$$\begin{eqnarray} &=& p_{p,n}^{{\rm RMF}}+ \sum _i \big(p_i^{th} + p_i^{{\rm ex}} + p_i^{{\rm bulk}} + p_i^{{\rm surf}} + p_i^{{\rm Coul}}\big), \end{eqnarray} \tag{ 27 }$$

where p^RMF_{p, n} is the contribution from the nucleons in the vapor; p^bulk_i, p^surf_i, and p^Coul_i originate from the bulk, surface, and Coulomb energies of nuclei in the free energy, respectively; both p^th_i = n_ik_BT(1 − n_B/n_s) and p^ex_i = n_ik_BT(n_B/n_s)(log (n_i/n_Qi) − 1) come from the translational energy of nuclei in the free energy, with the former being the ordinary pressure of nuclei as a Boltzmann gas and the latter arising from the excluded-volume effect approximated by the factor (1 − n_B/n_s). They are minor at high densities. Also note that p^bulk_i and p^surf_i are normally negligible. The Coulomb-energy contribution, p^Coul_i, is negative owing to the attractive electrostatic interactions among uniformly distributed electrons and protons inside nuclei and is particularly important for ρ_B ≳ 10¹¹ g cm⁻³ and Y_p ≳ 0.3. If nucleus i is in the ordinary droplet phase, it is given as

$$\begin{eqnarray} p_i^{{\rm Coul}} &=& n_i n_B \frac{3}{5}\left(\frac{3}{4 \pi }\right)^{-1/3} \frac{e^2}{n_{si}^2} \left(\frac{Z_i - n^{\prime }_p V_i^N}{A_i}\right)^2 {V_i^N}^{5/3} \nonumber \\ && \times \frac{1}{2} \big(- u_i^{1/3}+u_i\big) \frac{(\eta n_e -n^{\prime }_p)(1-u_i)}{(n_e-n^{\prime }_p) \eta n_B}, \end{eqnarray} \tag{ 28 }$$

whereas in the bubble phase it is expressed as

$$\begin{eqnarray} p_i^{{\rm Coul}} &=& n_i n_B \frac{3}{5}\left(\frac{3}{4 \pi }\right)^{-1/3} \frac{e^2}{n_{si}^2} \left(\frac{Z_i - n^{\prime }_p V_i^N}{A_i}\right)^2 \nonumber \\ && \times \left\lbrace {V_i^B}^{5/3} \frac{1}{2} \big(- u_i^{1/3}+u_i\big) \frac{(\eta n_e -n^{\prime }_p)(1-u_i)}{(n_e-n^{\prime }_p) \eta n_B}\right. \nonumber \\ &&\left.+ \frac{5}{3} {V_i^B}^{2/3} D(1-u) \frac{\partial V_i^B}{ \partial n_B} \right\rbrace. \end{eqnarray} \tag{ 29 }$$

The entropy per baryon is calculated from the following expression:

$$\begin{eqnarray} &&\hspace{-14pc}s = -\frac{[\partial {f}/\partial {T}]_{\rho _B,Ye}}{n_B} \end{eqnarray} \tag{ 30 }$$

$$\begin{eqnarray} &=&\eta s^{{\rm RMF}}_{p,n} + \sum _i \frac{n_i k_B}{n_B} \Biggl [ \left\lbrace \frac{5}{2}- \log \left(\frac{n_i}{g_i(T) n_{Qi}} \right) \right\rbrace (1-n_B/n_s) \nonumber \\ & & + (1-n_B/n_s) \, T \, g_i^{\prime }/g_i - \partial {M_i}/\partial {T} \Biggl ], \end{eqnarray} \tag{ 31 }$$

where $g^{\prime }_i(T)= [\partial {g_i(T)}/\partial {T}]_{\rho _B,Ye}$. The first term in Equation (31) is the entropy of uniform nuclear matter and the second one is the entropy of nuclei as a Boltzmann gas with the excluded-volume effect, which is negligible at low densities. The third term accounts for the thermal excitations of nuclei with (1 − n_B/n_s)Tg'_i/g_i being non-negligible at high temperatures as shown later. The last term originates from the fact that the bulk energy of nuclei is temperature-dependent in our formulation and given as follows:

$$\begin{equation} \frac{\partial {M_i}}{\partial {T}} = \frac{\partial {E_i^{B}}}{\partial {T}} = - A_i s_i^{{\rm RMF}} (T,n_{si}, Z_i / A_i). \end{equation} \tag{ 32 }$$

The contribution of this term is normally negligible except near the nuclear saturation density.

The free energy thus obtained is continuous at any density but not smooth at the saturation density. In fact, the pressure, or the density derivative of the free energy, is not continuous at the saturation density. This is most easily understood from Equation (27). The contribution from the excluded-volume effect in the translational energy, p^ex_i, in the equation does not vanish at the saturation density. The continuity of pressure would have been obtained if we had employed a suppression factor (the last term on the right-hand side of Equation (15)) with a higher power. We do not think this is necessary, however, since the discontinuity is practically negligible compared with the dominant electron pressure in core-collapse simulations. The derivatives of free energy with respect to T and Y_p are continuous at any density as observed in Equation (31) for entropy.

The mass fractions of nuclei are shown in the (N, Z) plane with N being the neutron number for ρ_B = 10¹¹ g cm⁻³, T = 1 MeV, and Y_p = 0.3 in Figure 1. It is clear that nuclei are abundant in the vicinities of the neutron magic numbers (N = 28, 50, and 80). In fact, the abundance peak arises around (N, Z) ∼ (50, 30) in our EOS in this case. For the same density, temperature, and Y_p, on the other hand, the H. Shen EOS predicts that the representative nucleus has (N, Z) = (57.2, 33.2). The difference originates mainly from the fact that the H. Shen EOS does not include the shell effect in its Thomas–Fermi approximation. At higher densities, heavier nuclei become abundant, which is evident in Figure 2, which shows the mass fractions of nuclei for ρ_B = 10^13.5 g cm⁻³, T = 1 MeV, and Y_p = 0.3. It should be noted that nuclei with Z ≳ 100 are quite abundant and, in fact, are dominant in this case. Hence the neglect of these nuclei in Hempel & Schaffner-Bielich (2010) cannot be justified at these high densities. It should also be mentioned that the nuclear distribution is much smoother in this case than it is for lower density. This is because the shell effect is reduced in our EOS at high densities, where the nuclear bulk energy is interpolated from the one derived experimentally and the one obtained theoretically for uniform nuclear matter, with the latter being dominant at this high density. H. Shen's EOS predicts that the most abundant nucleus is the one with (N, Z) = (203.8, 89.4), which is still different from the prediction of our EOS, (N, Z) = (243, 110). These differences may have an important ramification for the electron capture in the collapsing core (Hix et al. 2003).

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In order to expedite comparison with the existing EOSs with SNA, we show in Figure 3 the mass fractions of light (Z_i ⩽ 5) and heavy (Z_i ⩾ 6) nuclei together with free nucleons in the vapor for T = 1 MeV. Irrespective of the values of Y_p selected here (Y_p = 0.5, 0.3, 0.1), the free nucleons dominate other constituents up to ρ_B ∼ 10⁷ g cm⁻³ at this temperature. At larger densities (10⁷ g cm⁻³ ≲ ρ_B ≲ 10¹⁰ g cm⁻³), some of free nucleons are taken into symmetric light nuclei such as deuterons and alpha particles. For densities larger than ρ_B ∼ 10¹⁰ g cm⁻³, light nuclei disappear and heavy nuclei are populated dominantly. Near the nuclear saturation density, the matter is mostly made of nuclei in the bubble or uniform states. In the case of Y_p = 0.5, the free protons coexist with symmetric light nuclei and neutron-rich heavy nuclei at ρ_B ∼ 10¹⁰ g cm⁻³ and only symmetric heavy nuclei survive at larger densities. For Y_p = 0.3, 0.1, on the other hand, most of the free protons are contained in the nuclei and free neutrons coexist with heavy nuclei. In the case of Y_p = 0.1, in particular, there is a density regime near the nuclear saturation density, where very neutron-rich light nuclei such as the one with (Z, A) = (1, 10) exist in the pasta or uniform states as we mention again later. Figure 4 shows the matter compositions at a bit higher temperature, T = 5 MeV. It is evident for all the three values of Y_p that free nucleons are dominant up to higher densities, ρ_B ∼ 10¹² g cm⁻³. It is mentioned incidentally that the mass fractions of light nuclei in our EOS are larger in general than those of alpha particles in the H. Shen EOS. This is partly because the H. Shen EOS has only alpha particles as light nuclei whereas ours takes into account other nuclei such as deuterons. The difference somewhat affects the thermodynamical properties. For example, the entropy per baryon in our EOS is larger than that in the H. Shen EOS at ρ_B ∼ 10¹² g cm⁻³ and T ≳ 5 MeV owing to larger degrees of freedom in the light nuclei in the former as shown in Section 3.2.

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The average mass number, $\bar{A}$, of heavy (Z_i ⩾ 6) nuclei is displayed in Figure 5. For comparison, not only our own results but those of Hempel's and the mass number of the representative nuclei, A_rep, for the H. Shen EOS are also shown. It is found that for T = 1 MeV (the left column of the figure), the former two give average mass numbers that are systematically smaller than those of the H. Shen EOS at ρ_B ≲ 10¹² g cm⁻³, around which the supernova core becomes optically thick and neutrinos are trapped and equilibrated thermally and chemically with the matter. It is also clear that the average mass number grows stepwise for our and Hempel's EOSs whereas it is monotonic for H. Shen's EOS. This is simply due to the shell effect: nuclei are abundant in the vicinity of the neutron magic numbers (N = 28, 50, 80). At higher densities, ρ_B ≳ 10¹² g cm⁻³, the effect is suppressed in our EOS, as mentioned already, because the nuclear bulk energy is interpolated from the experimentally derived value and the one obtained theoretically by RMF, which does not include the shell effect. Also note that at these densities there appear very heavy and/or neutron-rich nuclei, for which no experimental mass data are available; we employ RMF for the estimation of their bulk energies (see Section 2.2.3). Heavy nuclei continue to grow until the nuclear saturation density in our EOS, whereas the growth stops at some density and nuclei disappear before the saturation density in Hempel's EOS (see the insets of the figure), which is at odds with the conventional idea. This point will be mentioned shortly again.

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The discernible difference between our results and those of Hempel at ρ_B ∼ 10¹² g cm⁻³ for Y_p = 0.1, 0.3 is attributed to the difference in the masses of neutron-rich nuclei, which we calculate by RMF whereas Hempel & Schaffner-Bielich (2010) obtained this from the mass table by Geng et al. (2005). The dip in the mass number at $\rho _B \sim 10^{13} \,\rm {g\, \rm cm^{-3}}$ for Y_p = 0.1 in our EOS is artificial, as it is produced by neglecting the shell effect in our mass formula. As nuclei become very neutron-rich, a rather rapid shift of abundant nuclei from those with experimental data to those without occurs, resulting in the observed change of the mass number. The mass number does not increase until higher densities are reached. As shown in the right column of Figure 5 for T = 5 MeV, for example, the mass number starts to increase at ∼10¹³ g cm⁻³. This is because the increase of entropy by the translational degrees of freedom of free nucleons is effective to minimize the free energy at this high temperature, whereas the decrease of internal energy by the binding energy of nuclei is more advantageous at lower temperatures. Another interesting feature at this temperature is the decrease of the mass number near the nuclear saturation density for Y_p = 0.3, 0.5 even in our EOS. There are two reasons for this: one is the formation of bubble nuclei and the reduction of surface energy as a result; the other reason is that the less heavy nuclei become, the greater entropy they have due to their larger translational degrees of freedom and they lower the free energy. It should be noted that unlike in the Hempel EOS, nuclei are not dissociated completely in our EOS.

The rate of coherent scatterings of neutrinos on nuclei is proportional not to the mass number of nuclei but to its square as long as the wavelength of neutrino is much larger than the nuclear size (Thompson et al. 2003). As mentioned already, in SNA the rate is evaluated by the mass number squared for the representative nucleus, the validity of which needs confirmation. In the top panels of Figure 6, we compare the average mass number squared, $\overline{A^2 }$, with the square of the average mass number, $\bar{A}^2$, and the mass number squared of the representative nuclei in the H. Shen EOS. In the middle and bottom panels, the standard deviations, $\sigma _A=\sqrt{\vphantom{A^A}\smash{\hbox{${\overline{A^2}-\bar{A}^2}$}}}$, and the dispersion normalized by the average mass number squared, $\sigma _A^2/\overline{A^2 }$, are also shown. It is intriguing to find that the average mass number squared is almost indistinguishable from the square of the average mass number. The reason for this small dispersion is that heavy nuclei are rather concentrated in the vicinities of the neutron magic numbers. In this sense, SNA is not too bad of an approximation in evaluating the rate of coherent scatterings of neutrinos on nuclei. It should be noted, however, that the mass numbers of the representative nuclei in the H. Shen EOS are systematically larger than the average mass numbers obtained in our EOS. Incidentally, at very high densities (ρ_B ≳ 10¹⁴ g cm⁻³), most of the nuclei are in the pasta phase.

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We demonstrate here that the transition to uniform nuclear matter occurs in one of the following two ways depending on the proton fraction of the entire system, Y_p. The first case corresponds to relatively large proton fractions, Y_p ≳ 0.3. At ρ_B ≳ 10¹³ g cm⁻³, most of the nuclei in abundance have a similar proton fraction Z_i/A_i, which is close to Y_p as shown in Figure 2. As the density rises up to the nuclear saturation density, their W-S cells get smaller and these abundant nuclei enter the pasta phase almost simultaneously. In the bubble phase, the volumes occupied by the nuclei in their W-S cells become larger and finally the nuclei coalesce with each other. The sequence is schematically illustrated for the nucleus with (Z, A) = (6, 20) in Figure 7. Note that the density in the vapor, or the bubble, remains lower than that in the nuclei up to the saturation density in this case.

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The second case occurs for more neutron-rich systems, Y_p ≲ 0.3. The proton fractions, Z_i/A_i, of nuclei having large mass fractions are not necessarily close to the Y_p value of the entire system and take various values at ρ_B ≳ 10¹³ g cm⁻³. This is due to the fact that more symmetric nuclei have greater binding energies. Hence, the nuclei are less neutron-rich than the vapor in general. As a result of this diversity of the proton fractions of the nuclei in abundance, these nuclei reach the pasta phase at different densities. Put another way, nuclei of different shapes coexist with substantial mass fractions at a particular density above ρ_B ∼ 10¹³ g cm⁻³. For example, at Y_p = 0.1, ρ_B = 10^13.9 g cm⁻³, and T = 1 MeV, a very heavy nucleus with (Z, A) = (48, 228) still remains in the normal phase whereas a very neutron-rich light nucleus with (Z, A) = (1, 10) as well as a neutron-rich slightly heavy nucleus with (Z, A) = (6, 49) are already in the bubble phase. This light nucleus already becomes uniform in the W-S cell at ρ_B = 10^14.1 g cm⁻³ whereas the others are still non-uniform. This actually corresponds to the formation of a large nucleus with no bubbles inside, which coexists with other bubble nuclei. As the density increases further beyond the nuclear saturation density up to the point where the entire system becomes uniform, the density of this big nucleus also increases, which we referred to as overcompression in Section 2.2. The reason for the occurrence of such overcompression is that the binding energies of nuclei that are less neutron-rich than the whole system are greater in general and the non-uniform matter is still energetically favored at the nuclear saturation density for the whole system. As shown in Figure 8, the transition to uniform nuclear matter for the very neutron-rich light nuclei occurs by coalescence with each other just as in the first case mentioned above. On the other hand, the transition for the nuclei with (Z, A) = (6, 49) proceeds not by coalescence but by the increase in the density of the vapor. Although it is not shown in the figure, it is interesting that the very heavy nucleus with (Z, A) = (48, 228) does not reach the bubble phase and is in the intermediate pasta phase just prior to the transition to uniform nuclear matter.

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Once the minimization is completed, the baryonic contribution to the free energy, to which our discussions are limited in this paper, is obtained as a function of ρ_B, T,  and Y_p. All the thermodynamical quantities are then derived from it through thermodynamical relations. It turns out that most of them are not very different from the ones in the Shen and Hempel EOSs: for the temperatures, T = 1, 5, 10 MeV considered in this paper, the three EOSs are essentially identical at low densities, ρ_B ≲ 10⁸ g cm⁻³, since the matter is simply composed of free nucleons as an ideal Boltzmann gas in all the EOSs. At higher densities, 10⁸ g cm⁻³ ≲ ρ_B ≲ 10¹³ g cm⁻³, a small difference appears, since some of the free nucleons are taken into nuclei, the treatment of which is different among the EOSs. At still greater densities, ρ_B ≳ 10¹³ g cm⁻³, the difference is more apparent. In the following we will look at the behavior in more detail.

Figure 9 shows the free energies per baryon as a function of density for the three combinations of temperature and proton fraction: T = 1 MeV, Y_p = 0.5; T = 5 MeV, Y_p = 0.3; and T = 10 MeV, Y_p = 0.1. In all cases, the free energies among the three EOSs are close to one another. Note that our EOS coincides with the H. Shen EOS at the highest density, as it should. This is also true of the Hempel EOS although how uniform nuclear matter is reached is different between the two EOSs as described in the previous section. Looking more closely, we find for T = 1 MeV and Y_p = 0.5 that Hempel's free energy is larger than those of our and H. Shen's EOSs at $\rho _B \sim 10^{14} \,\rm {g\, \rm cm^{-3}}$ and the hump is followed by a steep decline. The difference originates from the fact that the Hempel EOS includes only nuclei with Z ≲ 100 and that, unlike in our treatment, only the Coulomb energy is manipulated by using the W-S cell to account for the existence of other nuclei. Their prescription leads to the total dissociation of heavy nuclei into free nucleons before reaching the nuclear saturation density. The difference between our EOS and the H. Shen EOS, on the other hand, comes from the fact that the latter includes only single representative heavy nuclei. In the case of T = 5 MeV and Y_p = 0.3, our and Hempel's free energies are smaller than that of H. Shen's EOS at 10¹² g cm⁻³ ≲ ρ_B ≲ 10¹³ g cm⁻³. Under this condition light nuclei (deuterons in particular) are abundant as shown in Figure 4. The H. Shen EOS takes into account alpha particles alone as light nuclei. The greater degrees of freedom in the light nuclei in our EOS, especially the substantial population of deuteron, make the difference observed. The deviation of Hempel's free energy from ours at ρ_B ≳ 10¹³ g cm⁻³ occurs for the same reason as in the case of T = 1 MeV and Y_p = 0.5. For T = 10 MeV and Y_p = 0.1, our free energy is smaller than those of the other two EOSs at ρ_B ≳ 10¹³ g cm⁻³. The reason is that the existence of the pasta phase is taken into account in our EOS. In fact, under this condition our EOS predicts the coexistence of various neutron-rich nuclei in the pasta phase or in the uniform phase, which corresponds to a large nucleus with no bubbles inside and free neutrons (see Section 3.1 and Figure 8). This is in sharp contrast to the matter composed of normal nuclei and free neutrons considered in the other two EOSs.

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Pressure is shown as a function of density for three combinations of temperature and proton fraction in Figure 10. The three EOSs agree with one another at low densities, ρ_B ≲ 10¹² g cm⁻³. The difference appears, however, in the density regime where the pressure becomes negative for the H. Shen and Hempel EOSs. The baryonic pressure can be negative when the Coulomb-energy contribution, which is negative owing to the attractive nature of the Coulomb interactions, dominates over the other positive contributions. Note that the positive leptonic pressure is much larger than the baryonic one normally and the total pressure never becomes negative. For T = 5 MeV and Y_p = 0.3, our EOS gives positive pressures at ρ_B ∼ 10¹⁴ g cm⁻³, where the other EOSs predict negative values. This is due to the existence of the pasta phase, which our EOS takes into account but the others do not. In fact, the absolute values of the Coulomb-energy contribution are reduced when the change of the charge distribution in the W-S cell is considered in the pasta phase. As a result the pressure in our EOS remains positive. It is also noted that the contribution from the excluded-volume effect, ∑_ip^ex_i, gives a non-negligible positive contribution to the pressure. At the lower temperature, T = 1 MeV, even our EOS gives negative baryonic pressures and the main difference with the other EOSs lies in the lowest value, ρ_B ∼ 10¹² g cm⁻³, of the densities that give negative pressures. It is found that our EOS predicts a slightly higher density than the others. This is because our EOS gives larger mass fractions of free neutrons, which give a positive contribution to the pressure. The difference with the H. Shen EOS is a consequence of the shell effect, which is naturally included in the experimental mass data but ignored in the Thomas–Fermi approximation adopted in the H. Shen EOS. The difference with Hempel's result, on the other hand, comes from the difference in the theoretical estimate of the masses of neutron-rich nuclei with no experimental data. In the case of T = 10 MeV and Y_p = 0.5, our and Hempel's pressures are larger than that of H. Shen's EOS at ρ_B ≳ 10^13.5 g cm⁻³. This is because H. Shen's EOS neglects the contribution from the translational motions of nuclei, which is not negligible at this high temperature. It is also noted that at high densities, ρ_B ∼ 10¹³ g cm⁻³, the contribution from the excluded-volume effect is not negligible, either, the treatment of which is different between our and Hempel's EOSs (see Section 2.2.4).

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The entropy per baryon is displayed as a function of density for three combinations of temperature and proton fraction in Figure 11. For high temperatures, T = 5, 10 MeV, our results are larger than those of H. Shen's EOS at ρ_B ∼ 10¹² g cm⁻³ owing to a larger population of light nuclei such as deuteron as well as the inclusion of the entropy from excited states of nuclei, g_i(T) in Equation (16). The difference at ρ_B ∼ 10¹³ g cm⁻³ originates from the existence of heavier nuclei in our EOS than in the others. Because of the larger degrees of freedom, the entropy is actually larger in our EOS. At lower temperatures, T = 1 MeV, the entropy per baryon in our EOS is almost equal to those in the other EOSs.

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