TABULATED EQUATION OF STATE FOR SUPERNOVA MATTER INCLUDING FULL NUCLEAR ENSEMBLE

For describing stellar matter, we adopt the grand canonical approximation where macroscopic states are characterized by temperature T and chemical potentials of the constituents. The chemical potentials for nuclear species (A, Z) are expressed as

$$\begin{equation} \begin{array}{ll}\mu _{AZ}=A\mu _B+Z\mu _Q. \end{array} \end{equation} \tag{ 1 }$$

For protons, μ_p = μ_B + μ_Q, and for neutrons, μ_n = μ_B. The chemical potentials μ_B and μ_Q are found from the conservation laws for baryon number B and electric charge Q in the normalization volume V:

$$\begin{eqnarray} &&\rho =\frac{B}{V}=\sum _{AZ}A\rho _{AZ}, \nonumber \\ &&\rho _Q=\frac{Q}{V}=\sum _{AZ}Z\rho _{AZ}-\rho _e=0. \end{eqnarray} \tag{ 2 }$$

Here, ρ_AZ is the average density of a nuclear species with mass A and charge Z (see below); $\rho _e=\rho _{e^-}-\rho _{e^+}$ is the net electron density. The second equation requires any macroscopic volume of the star (V) to be electrically neutral.

The lepton number conservation is a valid concept only if ν and $\tilde{\nu }$ are trapped in the matter within the neutrino-sphere (Prakash et al. 1997). If they escape freely from the star, the lepton number conservation is irrelevant and μ_L = 0. In this case, the two remaining chemical potentials are determined by Equations (2). In the β-equilibrated matter the electron chemical potential μ_e is found from the condition μ_e = μ_n − μ_p. However, β equilibrium may not be achieved in a fast explosive process. In this case, it is more appropriate to perform calculations for fixed values of electron fraction Y_e. Then, μ_e is determined from the given electron density ρ_e = Y_eρ.

The nuclear component of stellar matter is represented as a mixture of gases of different species (A, Z), including nuclei and nucleons. It is convenient to introduce the numbers of particles of different kind N_AZ in a normalization volume V. In SMSM, we use the grand canonical version of the SMM formulated in Botvina et al. (1985), and developed further in Botvina et al. (2002). The corresponding thermodynamic potential of the system can be expressed as

$$\begin{eqnarray} &&\Omega (T,\mu _B,\mu _Q, \lbrace N_{AZ}\rbrace,V)= F^{\rm tr}(T,\lbrace N_{AZ}\rbrace,V_f) \nonumber \\ &&\quad+\sum _{AZ}N_{AZ}F_{AZ}(T,\rho) -\mu _B\sum _{AZ}AN_{AZ}-\mu _Q\sum _{AZ}ZN_{AZ},\ \quad \end{eqnarray} \tag{ 3 }$$

where the first term accounts for the translational degrees of freedom of nuclear fragments and the second term is associated with their internal degrees of freedom. Assuming the Maxwell–Boltzmann statistics for all nuclear species, including nucleons, the translational free energy can be explicitly written as⁶

$$\begin{equation} F^{\rm tr}(T,\lbrace N_{AZ}\rbrace,V)=-T\sum _{AZ}N_{AZ}\left[\ln {\left(\frac{g_{AZ}V_fA^{3/2}}{N_{AZ}\lambda _T^3}\right)}+1\right], \end{equation} \tag{ 4 }$$

where g_AZ is the ground-state degeneracy factor for species (A, Z), λ_T = (2π/m_NT)^1/2 is the nucleon thermal wavelength, m_N ≈ 939 MeV is the average nucleon mass, and V_f is so-called free volume of the system, which accounts for the finite size of nuclear species and is only a fraction of the total volume V. We assume that all nuclei with A > 4 have normal nuclear density ρ₀ ≈ 0.15 fm⁻³ so that the proper volume of a nucleus with mass A is A/ρ₀. At the relatively low densities considered here, one can adopt the excluded volume approximation, V_f/V ≈ (1 − ρ/ρ₀). This approximation is commonly accepted in statistical models (see the extended discussion of this problem in Sagun et al. 2014). Certain information about the free volume in multifragmentation reactions has been extracted from analysis of experimental data (Scharenberg et al. 2001).

The internal excitations of nuclei play an important role in regulating their abundances since they significantly increase their entropy. Some authors (see, e.g., Ishizuka et al. 2003) limit the excitation spectrum by particle-stable levels known for low excited nuclei. Within the SMM, we follow quite different philosophy motivated by experimental investigations of nuclear disintegration reactions. Namely, we assume that excited states are populated according the internal temperature of nuclei, which is assumed to be the same as the temperature of the surrounding medium. In this case, not only particle-stable states but also particle-unstable states will contribute to the excitation energy and entropy. This assumption can be justified by the dynamical equilibrium between emission and absorption processes in the hot medium. Moreover, in the supernova environment, both the excited states and the binding energies of nuclei may be strongly affected by the surrounding matter. For this reason, we find it more appropriate to use an approach which can easily be generalized to include in-medium modifications. Namely, the internal free energy of species (A, Z) with A > 4 is parameterized in the spirit of the liquid drop model, which has been proven to be very successful in nuclear physics:

$$\begin{equation} F_{AZ}(T,\rho)=F_{AZ}^B+F_{AZ}^S+F_{AZ}^{\rm sym}+F_{AZ}^C\protect \protect. \end{equation} \tag{ 5 }$$

Here, the right-hand side contains, respectively, the bulk, the surface, the symmetry, and the Coulomb terms. The first three terms are taken in the standard form (Bondorf et al. 1995),

$$\begin{eqnarray} &&F_{AZ}^B(T)=\left(-w_0-\frac{T^2}{\varepsilon _0}\right)A, \end{eqnarray} \tag{ 6 }$$

$$\begin{eqnarray} &&F_{AZ}^S(T)=\beta _0\left(\frac{T_c^2-T^2}{T_c^2+T^2}\right)^{5/4}A^{2/3}\protect \protect, \end{eqnarray} \tag{ 7 }$$

$$\begin{eqnarray} &&F_{AZ}^{\rm sym}=\gamma \frac{(A-2Z)^2}{A}, \end{eqnarray} \tag{ 8 }$$

where w₀ = 16 MeV, ε₀ = 16 MeV, β₀ = 18 MeV, T_c = 18 MeV, and γ = 25 MeV are the model parameters which are extracted from nuclear phenomenology and provide a good description of multifragmentation data (Bondorf et al. 1995; Botvina et al. 1995; Scharenberg et al. 2001; D'Agostino et al. 1996, 1999; Bellaize et al. 2002; Avdeyev et al. 2002). However, these parameters, especially the symmetry coefficient γ, may be different in hot nuclei, and therefore they should be determined from corresponding experimental data (see the discussions in Botvina et al. 2006, 2002; Buyukcizmeci et al. 2008; Ogul et al. 2011).

In the electrically neutral environment, the nuclear Coulomb term should be modified to include the screening effect of electrons. This can be done, e.g., within the Wigner–Seitz approximation as was proposed in Lamb et al. (1981) and Lattimer et al. (1985). One should imagine that the whole system is divided into spherical cells each containing one nucleus. The radius of the cell is determined by the condition that it contains the same number of electrons as the number of protons in the nucleus. The interaction between the cells is neglected. Then, assuming a constant electron density, one obtains

$$\begin{eqnarray} F_{AZ}^C(\rho)&=&\frac{3}{5}c(\rho)\frac{(eZ)^2}{r_0A^{1/3}}\cdot \nonumber\\ c(\rho)&=&\left[1-\frac{3}{2}\left(\frac{\rho _e}{\rho _{0p}}\right)^{1/3} +\frac{1}{2}\left(\frac{\rho _e}{\rho _{0p}}\right)\right], \end{eqnarray} \tag{ 9 }$$

where r₀ = 1.17 fm, ρ_e = Y_eρ is the average electron density, and ρ_0p = (Z/A)ρ₀ is the proton density inside the nuclei. The screening function c(ρ) is 1 at ρ_e = 0 and 0 at ρ_e = ρ_0p. In fact, in this work, all presented results are obtained with the approximation ρ_e/ρ_0p ≈ ρ/ρ₀, as in Lattimer et al. (1985), which works well when neutrons are mostly bound in nuclei so that ρ_0p ≈ Y_eρ₀. Although this approximation gives satisfactory results in many cases, it may deviate at very high temperatures and very low baryon densities. However, under these conditions, the nuclei are getting smaller and the Coulomb interaction effects become less important. Generally, the reduction of the Coulomb energy due to electron screening favors the formation of heavy nuclei.

In the SMM nucleons and light nuclei (A ⩽ 4) are considered as structureless particles characterized only by exact masses, proper volumes, and spin degeneracy factors g_AZ (Bondorf et al. 1995): for nucleons, g_n = g_p = 2; for deuterons, g₂₁ = 3; for ³H, g₃₁ = 2; for ³He, g₃₂ = 2; for ⁴He, g_AZ = 1. Their Coulomb interaction is taken into account within the same Wigner–Seitz approximation. For all nuclear species with A > 4, we use g_AZ = 1, but include internal excitations.

Within the grand canonical approximation, the conservation law Equations (2) are fulfilled only for the mean densities of nuclear species ρ_AZ = 〈N_AZ〉/V. The mean fragment numbers 〈N_AZ〉 in volume V are found by minimizing the thermodynamic potential Equation (3) with respect to fragment multiplicities {N_AZ} at fixed T and V. In the lowest-order approximation, this gives

$$\begin{equation} \langle N_{AZ}\rangle =g_{AZ}\frac{V_f}{\lambda _T^3}A^{3/2}{\rm exp}\left[-\frac{1}{T}(F_{AZ}-\mu _{AZ})\right]\protect \protect, \end{equation} \tag{ 10 }$$

where μ_AZ is defined by Equation (1). The corrections to this expression become significant at higher densities and low temperatures when heavy nuclei are abundant.

The pressure associated with nuclear species can be calculated by differentiating Equation (3) with respect to V at fixed T and {N_AZ}, which gives

$$\begin{equation} P_{\rm nuc}=P_{\rm tr}+P_{C}\protect \protect, \end{equation} \tag{ 11 }$$

where first term comes from the translational motion of fragments and the second term from the density-dependent Coulomb interaction. Their explicit expressions are

$$\begin{equation} P_{\rm tr}=T\sum _{AZ}\rho _{AZ},\quad P_{C}=\rho \sum _{AZ}\rho _{AZ} \frac{\partial F_{AZ}^C}{\partial \rho }\protect \protect. \end{equation} \tag{ 12 }$$

As one can see from Equation (9), the Coulomb term gives a negative contribution to pressure (Baym et al. 1971).

In Figure 2, we present the translational pressure P_tr, Equation (12), caused by single nucleons and nuclei. The total nuclear pressure P_nuc is also shown in Figure 2 as a function of temperature. It is important that at T ⩾ 5 MeV, when matter nearly completely dissociates into nucleons and lightest clusters, P_C is close to zero. In this region, the total nuclear pressure coincides with the pure nuclear pressure. The Coulomb pressure becomes very important when heavy clusters dominate in the system. One can see in Figure 2 that at low temperatures and high density, the total nuclear pressure may be negative. In this case, the nuclear clusterization is favorable for the collapse. However, the positive pressure of the relativistic electron Fermi gas is considerably (more than an order of magnitude) larger. Therefore, the total pressure P_tot given by the sum of the total nuclear, electron, and photon pressures in such an environment will always be positive, and the condition of thermodynamical stability(∂P_tot/∂ρ ⩾ 0) will be fulfilled.

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Finally, the entropy of nuclear species is calculated by differentiating Equation (3) with respect to T at fixed V and {N_AZ}, which gives

$$\begin{equation} S_{\rm nuc}=\sum _{AZ}N_{AZ}\left[\ln {\left(\frac{g_{AZ}V_fA^{3/2}}{N_{AZ}\lambda _T^3}\right)}+\frac{5}{2}\right]-\sum _{AZ}N_{AZ}\frac{\partial F_{AZ}}{\partial T}. \end{equation} \tag{ 13 }$$

Here, the first term comes from the translational motion, and the second term is from internal excitations of fragments.

As follows from Equation (10), the fate of heavy nuclei depends strongly on the relationship between F_AZ and μ_AZ. In order to avoid an exponentially divergent contribution to the baryon density, at least in the thermodynamic limit (A → ∞), inequality F_AZ ⩾ μ_AZ must hold. The equality sign here corresponds to the situation when a large (infinite) nuclear fragment coexists with the gas of smaller clusters (Bugaev et al. 2001). When F_AZ > μ_AZ, only small clusters with a nearly exponential mass spectrum are present. However, there exists a region of thermodynamic quantities corresponding to F_AZ ≈ μ_AZ when the mass distribution of nuclear species is close to a power-law A^−τ with τ ≈ 2. This is a characteristic feature of the liquid-gas phase transition. The advantage of our approach is that we consider all of the fragments present in this transition region and, therefore, can study this phase of nuclear matter in full detail.

We assume that besides nuclear species, the supernova matter also contains electrons, positrons, and photons.⁷ At T, μ_e > m_e, the pressure of the relativistic electron–positron gas can be written as

$$\begin{eqnarray} &&P_{e}=\frac{g_e\mu _e^4}{24\pi ^2} \nonumber \\ &&\quad\cdot \left[1+2\left(\frac{\pi T}{\mu _e}\right)^2+ \frac{7}{15}\left(\frac{\pi T}{\mu _e}\right)^4-\frac{m_e^2}{\mu _e^2} \left(3+\left(\frac{\pi T}{\mu _e}\right)^2\right)\right],\qquad \end{eqnarray} \tag{ 14 }$$

where first-order corrections (${\sim} m_e^2$) due to the finite electron mass are included and g_e = 2 is the spin degeneracy factor for electrons. Due to the correction terms of order (m_e/πT)² and (m_e/μ_e)², these expressions can be used even at T, μ_e of the order of m_e. At these condition, the contributions of electrons to the pressure and entropy become negligible. The corresponding expressions for net number density ρ_e and entropy density s_e are obtained from standard thermodynamic relations, ρ_e = ∂P_e/∂μ_e and s_e = ∂P_e/∂T, which give

$$\begin{eqnarray} &&\rho _e=\frac{g_e\mu _e^3}{6\pi ^2}\left[1+\frac{1}{\mu _e^2}\left(\pi ^2T^2- \frac{3}{2}m_e^2\right)\right], \end{eqnarray} \tag{ 15 }$$

$$\begin{eqnarray} &&s_e= \frac{g_e T\mu _e^2}{6}\left[1+\frac{7}{15}\left(\frac{\pi T}{\mu _e}\right)^2- \frac{m_e^2}{2\mu _e^2}\right]. \end{eqnarray} \tag{ 16 }$$

The photons are always close to the thermal equilibrium, and they are treated as massless Bose gas with zero chemical potential. The corresponding density ρ_γ, energy density e_γ, pressure P_γ, and entropy density s_γ of photons are given by the standard formulae

$$\begin{equation} \rho _{\gamma }=\frac{g_{\gamma }\xi (3) T^3}{\pi ^2 }, \quad e_{\gamma }=\frac{g_{\gamma }\pi ^2 T^4}{30 }, \quad P_{\gamma }=\frac{e_{\gamma }}{3}, \quad s_{\gamma }=\frac{4 e_{\gamma }}{3 T}, \end{equation} \tag{ 17 }$$

where g_γ = 2. A possible participation in the statistical ensemble of $\nu _e, \tilde{\nu _{e}}$, μ, ν_μ, and $\tilde{\nu _{\rm \mu }}$ is not considered in this work, so the current SMSM tables do not include these particles. However, the computer SMSM code includes the options of β equilibrium, as well as the full lepton (including neutrino) conservation; therefore, such calculations are possible (see Botvina & Mishustin 2004, 2010).

The SMSM was realized with a generalization of the special computer code developed previously for the grand canonical calculations of the full ensemble of nuclear species, as performed in Botvina et al. (2002). All kinds of particles contribute to the free energy, pressure, and other thermodynamical characteristics of the system, and we sum up all these contributions. The densities of all particles are calculated self-consistently by taking into account the relations between their chemical potentials. In the solution of the coupled equations, we used the step-by-step approximation method and control the found potentials with the relative precision of 0.001. We have checked that it is sufficient for our purposes. In this procedure, we directly observe how the successive steps approach the correct values during the simulations. We take baryon number B = 1000 and perform calculations for all fragments with 1⩽A ⩽1000 and 0⩽Z ⩽ A in the ensemble. This restriction on the size of nuclear fragments is fully justified in our case since fragments with larger masses (A >1000) can be produced only at very high densities ρ ≳ 0.3ρ₀ (Lamb et al. 1981; Lattimer et al. 1985), which are appropriate for the regions deep inside the protoneutron star. Such nuclei, as well as the "pasta" phase at the high densities, are not considered here.

The SMSM EOS tables cover the following ranges of control parameters:

• 1.  
  Temperature: T = 0.2–25 MeV; for 35 T values.
• 2.  
  Electron fraction Y_e: 0.02–0.56; linear mesh of Y_e = 0.02, giving 28 Y_e values. It is equal to the total proton fraction X_p, due to charge neutrality.
• 3.  
  Baryon number density fraction ρ/ρ₀ = (10⁻⁸–0.32), giving 31 ρ/ρ₀ values.

In our calculations, we consider all nuclear species with 1 ⩽ A ⩽ 1000 and 0 ⩽ Z ⩽ A. These restrictions on fragment size are justified within the above-defined intervals of control parameters. The different T (35 points), ρ/ρ₀ (31 point), and Y_e (28 points) values are listed in Table 1. They sum up to 30,380 different grid points. We believe that physical conditions for the SMSM EOS are sufficiently well represented by this grid. Running time for calculations was approximately 7.8 days (Work station:4xDual core AMD 2.2 GHz processor). We have obtained the file SMSM-EOS-Tables.txt (datafile2.txt, size 6.8 MB).

We can certainly say that our model is not applicable at densities above 0.3ρ₀ and temperatures above 25 MeV. At higher densities, the matter will be rearranged into more complicated geometrical structures known as "pasta" phases (Lamb et al. 1981; Newton & Stone 2009). We are not considering them in our present work. The assumption of statistical equilibrium also requires that the temperature is high enough to allow nuclear transformations leading to this equilibrium. As is commonly believed, this could be a good approximation at T > 1 MeV. Nevertheless, for completeness, we extend our calculations to T = 0.2 MeV, when the statistical equilibrium could be problematic.

The information is stored in a format that is very similar to the tables of Shen et al. (1998) or Hempel & Schaffner-Bielich (2010) or Furusawa et al. (2011) so that it can easily be implemented in running codes. The SMSM EOS tables ("SMSM-EOS-Tables.txt") is the main file which is available in the online journal.

In the beginning of file, we write the general parameters of the liquid-drop description: w₀ = 16 MeV, β₀ = 18 MeV, T_c = 18 MeV, γ = 25 MeV, ε₀ = 16 MeV, r₀ = 1.17 fm. The tables are written in the order of increasing T. For each T, we present the results in the order of increasing Y_e and ρ/ρ₀ (see Table 1). The first three lines in the SMSM EOS tables are shown in Table 2 for guidance regarding its form and content. Each line of the table corresponds to one density grid point. It contains 21 different thermodynamic quantities which are defined as follows.

• 1.  
  T, temperature (MeV),
• 2.  
  Y_e, electron fraction: Y_e = ρ_e/ρ,
• 3.  
  ρ/ρ₀, density fraction,
• 4.  
  X_n, the mass fraction of free neutrons is given by X_n = ρ_n/ρ,
• 5.  
  X_p, the mass fraction of free protons: X_p = ρ_p/ρ,
• 6.  
  X_d, the mass fraction of deuterons (A = 2, Z = 1) is given by X_d = 2ρ_d/ρ,
• 7.  
  X_t, the mass fraction of tritons (A = 3, Z = 1) is given by X_t = 3ρ_t/ρ,
• 8.  
  $X_{{^3}{\rm He}}$, the mass fraction of helions (A = 3, Z = 2) is given by $X_{{^3}He} = 3 \rho _{{^3}{\rm He}}/\rho$,
• 9.  
  X_alpha, the mass fraction of alphas (A = 4, Z = 2) is given by X_alpha = 4ρ_alpha/ρ,
• 10.  
  X_heavy, the mass fraction of heavy nuclei (A > 4, Z > 2) is defined by

  $$\begin{equation*} X_{{\rm heavy}} = \frac{\sum _{A>4,Z>2}A\rho _{AZ}}{\rho }, \end{equation*}$$
• 11.  
  〈A_heavy〉, the average mass number of heavy nuclei (A > 4, Z > 2) is defined by

  $$\begin{equation*} \langle A_{{\rm heavy}}\rangle =\frac{\sum _{A>4,Z>2}A\rho _{AZ}}{\sum _{A>4, Z>2}\rho _{AZ}}, \end{equation*}$$

  〈A_heavy〉 is set to zero if X_heavy = 0.
• 12.  
  〈Z_heavy〉, the average charge number of heavy nuclei (A > 4, Z > 2) is defined by

  $$\begin{eqnarray*} &&\langle Z_{{\rm heavy}}\rangle =\frac{\sum _{A>4,Z>2}Z\rho _{AZ}}{\sum _{A>4,Z>2}\rho _{AZ}},\nonumber \end{eqnarray*}$$
• 13.  
  E_tot, total energy, can be written as the sum of nuclear E_nuc, electron E_e, and photon E_γ energy contributions (MeV nucleon⁻¹),

  $$\begin{eqnarray*} &&E_{\rm tot}=E_{\rm nuc}+E_{\rm e}+E_{\rm \gamma }.\nonumber \end{eqnarray*}$$
• 14.  
  E_nuc, the energy of nuclear species (MeV nucleon⁻¹), is calculated as F+TS, using Equations (4), (5), and (13).
• 15.  
  S_tot, total entropy, is the sum of nuclear S_nuc, electron S_e, and photon S_γ entropy contributions (1/nucleon),

  $$\begin{equation*} S_{\rm tot}=S_{\rm nuc}+S_{\rm e}+S_{\rm \gamma }.\nonumber \end{equation*}$$
• 16.  
  S_nuc, nuclear entropy (1/nucleon) is given by Equation (13).
• 17.  
  P_tot, total pressure, is the sum of the pressure of nuclear species P_nuc (Equation (11)), electrons P_e (Equation (14)), and photons P_γ (Equation (17); MeV fm⁻³),

  $$\begin{eqnarray*} &&P_{\rm tot}=P_{\rm nuc}+P_{e}+P_{\rm \gamma }.\nonumber \end{eqnarray*}$$
• 18.  
  P_nuc, total nuclear pressure, is the sum of translational pressure and Coulomb pressure (Equation (12); MeV fm⁻³),
• 19.  
  μ_e, chemical potential of electrons (MeV),
• 20.  
  μ_B, chemical potential of baryons (MeV),
• 21.  
  μ_Q, chemical potential of charged particles (MeV).

Table 2. SMSM EOS Tables

Column 1	Column 2	Column 3	Column 4	Column 5
T(MeV)	Ye	ρ/ρ0	Xn	Xp
0.20	0.02	0.100E-07	0.9447	0.3514E-38
0.20	0.02	0.178E-07	0.9445	0.7178E-39
0.20	0.02	0.316E-07	0.9442	0.1388E-39
Column 6	Column 7	Column 8	Column 9	Column 10
Xd	Xt	$X_{{^3}He}$	Xalpha	Xheavy
0.1248E-35	0.2067E-26	0	0.2871E-24	0.05527
0.4503E-36	0.1327E-26	0	0.6668E-25	0.05551
0.1609E-36	0.8426E-27	0	0.1519E-25	0.05582
Column 11	Column 12	Column 13	Column 14	Column 15
〈Aheavy〉	〈Zheavy〉	Etot	Enuc	Stot
		(MeV nucleon−1)	(MeV nucleon−1)	(1/nucleon)
89.66	33.44	0.08469	−0.1659	13.81
90.70	33.66	−0.02678	−0.1664	12.53
91.76	33.88	−0.09010	−0.1675	11.57
Column 16	Column 17	Column 18	Column 19	Column 20
Snuc	Ptot	Pnuc	μe	μB
(1/nucleon)	(MeV fm−3)	(MeV fm−3)	(MeV)	(MeV)
12.13	0.4077E-09	0.2816E-09	0.01716	−2.063
11.58	0.6277E-09	0.5011E-09	0.03048	−1.948
11.03	0.1019E-08	0.8906E-09	0.05394	−1.833
Column 21
μQ
(MeV)
−16.96
−17.28
−17.60

Only a portion of this table is shown here to demonstrate its form and content. A machine-readable version of the full table is available.

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The thermodynamic consistency of the SMSM EOS table is guarantied by using the grand canonical ensemble with the thermodynamical potential (Equation (3)). This approximation should be good for the large volumes of matter considered here. The numerical accuracy of our calculations is typically better than 1%. The mass conservation and electrical neutrality conditions are especially checked. For example, the mass fractions of the different particle species sum up to unity:

$$\begin{eqnarray*} &&\delta _X =1-X_n+X_p+X_d+X_t+X_{^3He}+X_{{\rm alpha}}+X_{{\rm heavy}}.\nonumber \end{eqnarray*}$$

δ_X is found around 0.1%.

In addition to standard output, i.e., the 21 different thermodynamic quantities listed above, such as the pressure, entropy, energy, fractions of light particles, and heavy nuclei (actually, in Shen et al.'s (1998) case, a single heavy nucleus), it is possible to determine the ensemble-averaged densities of all heavy nuclei, ρ_AZ. This is achieved by first finding the chemical potentials μ_B and μ_Q and then using Equation (10). This formula is valid for all (T, ρ, Y_e) values and can be applied for nuclei with all possible mass numbers and charges. We believe that the knowledge about full nuclear ensemble could be quite helpful for accurate calculations of weak reactions with neutrinos and electrons in supernova simulations.

The mass distributions contain important information about the composition of nuclear matter in the nuclear liquid-gas phase transition (coexistence region). The concept of statistical equilibrium assumes an intensive interaction between the fragments via specific microscopic processes, such as absorption and emission of neutrons, which provide equilibration (see, e.g., the discussion in Botvina & Mishustin 2010). In this situation, the nuclei may remain hot and have modified properties, such as binding energies, excited states, etc., which differ from those in cold, isolated nuclei. We note that the data file for mass distributions "SMSM-EOS-Mass-Distribution-Tables.txt" (datafile3.txt) is also stored for each calculation. This file is also available in a machine-readable form in the online journal and on the Web page of SMSM. The file size is 319 MB, it contains 30,380 calculations, and every line has 1003 columns. The first three columns show T, Y_e, and ρ/ρ₀, and columns 4–1003 show the number of mass yields per nucleon (relative yield) for A = 1–1000. Data are presented for 35 T, 28 Y_e, and 31 ρ/ρ₀ values with the same order as in Table 1. The first 3 lines and the first 10 columns in "SMSM-EOS-Mass-Distribution-Tables" are shown in Table 3.

Since we have 35(T) ×28(Y_e) = 980 values for 31 ρ/ρ₀ values, we grouped four density intervals from lower to higher as

• 1.  
  Density-1 = ρ/ρ₀ = 10⁻⁸–5.62.10⁻⁷ (eight values),
• 2.  
  Density-2 = ρ/ρ₀ = 10⁻⁶–5.62.10⁻⁵ (eight values),
• 3.  
  Density-3 = ρ/ρ₀ = 10⁻⁴–5.62.10⁻³ (eight values),
• 4.  
  Density-4 = ρ/ρ₀ = 10⁻²–3.17.10⁻¹ (seven values).

For these intervals, we have plotted 3920 figures as EPS files containing information about mass distributions for each grid point. We have presented example figures for T = 1, 3 MeV, Y_e = 0.2–0.4, and different densities in Figures 3–6. As seen in these figures, we also put X_n, X_p, X_alpha, X_heavy, 〈A_heavy〉, μ_B, and μ_Q values from tables with the same order of ρ/ρ₀ as the control parameters inside of figures. These figures are presented on the SMSM Web page with files

• 1.  
  SMSM-Massdist-Figs-Density-1.zip,
• 2.  
  SMSM-Massdist-Figs-Density-2.zip,
• 3.  
  SMSM-Massdist-Figs-Density-3.zip,
• 4.  
  SMSM-Massdist-Figs-Density-4.zip.

Each file contains 980 EPS figures. For example, inside SMSM-Massdist-Figs-Density-4.zip, one can find the mass distribution figure for T = 1 MeV, Y_e = 0.20, and ρ/ρ₀ = 10⁻²–3.17.10⁻¹ in T-1-Ye-020-DENSITY-4.eps, as shown in Figure 3.

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At very low temperatures, the SMSM predicts a Gaussian-like distribution for heavy nuclei as seen in Figures 3 and 4. In this case, an approximation of a single heavy nucleus adopted in the EOS by Lattimer & Swesty (1991) and Shen et al. (1998) may work reasonably well for calculations of thermodynamical characteristics of matter. However, at T ⩾ 1 MeV, the gap between the Gaussian peak and light clusters and nucleons is essentially filled by nuclei of intermediate masses leading to characteristic U-shaped distributions. The examples of such distributions are shown in Figures 5 and 6. They give a typical evolution of mass distributions of nuclei in the coexistence region of the liquid-gas phase transition. Previously, it was demonstrated for disintegration of finite nuclei (Bondorf et al. 1995). These distributions continuously transform with decreasing density as well as with increasing temperatures into exponential falloff of fragment yields as functions of A.

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Using SMSM EOS tables and Equation (10), it is also possible to obtain isotopic distributions of stellar matter. Examples of isotopic distributions for Z = 8 and Z = 26 are shown in Figure 7. As seen from the figure, the SMSM predicts Gaussian-type distributions, which is a consequence of the liquid-drop description of fragments. Isotopic yields help to understand the trends observed for summed quantities such as mass yields or mass fractions. They are also needed to calculate average 〈Z〉 values, which are listed in the tables. This information is important for realistic calculations of the weak reactions with electrons and neutrinos.

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Recently, in Buyukcizmeci et al. (2013), we have compared mass and isotope distributions of the SMSM (Botvina & Mishustin 2010) with predictions of two other models by Hempel & Schaffner-Bielich (2010) and Furusawa et al. (2011), under conditions expected during the collapse of massive stars and supernova explosions. Recently, we have found significant differences between mass distributions predicted by these three models, which use different assumptions on properties of hot fragments in a dense environment, especially at low electron fractions, low temperatures, and high densities. We note that the differences are also present in the behavior of nuclear pressure (see Figure 22 in Buyukcizmeci et al. 2013).

Indeed, the main assumption of the SMSM is that most interaction effects are included into the internal binding energy of the fragments. Apart from that, we also take into account the Coulomb energy and excluded volume effects. As one can see from the tables, the mass fraction of single nucleons is rather small (less than 20%) at baryon densities around 0.1ρ₀ and electron fractions around 0.2. Their actual density is therefore only 0.02ρ₀. The interaction effects are rather weak at such a low density (see, e.g., Talahmeh & Jaqaman 2013). At higher densities and lower electron fractions, the share of unbound nucleons becomes even larger and their interaction may become more important (Typel et al. 2010). But even at baryon densities around 0.3ρ₀ and temperatures T  ∼  1 MeV, most nucleons are bound in clusters, and most interaction effects come from the nuclear binding energy. Exactly in this domain of parameters, different models predict very different nuclear ensembles (see Buyukcizmeci et al. 2013). Certainly, improving the EOS calculations in this region is an important issue for future studies. Recently, Furusawa et al. (2013) have made an attempt to improve the description of nucleons and light nuclei in supernova matter.

An important special case when full β equilibrium is reached in stellar matter corresponds to the condition μ_e = μ_n − μ_p. It is expected to be fulfilled in certain situations, such as the slow collapse of a massive star, late stages of a supernova explosion, or in crusts of neutron stars. On the other hand, the β equilibrium can be a useful physical limit for theoretical estimates of the nuclear composition without full knowledge of weak reactions. The SMSM allows for this kind of calculations (Botvina & Mishustin 2004, 2010) and the corresponding tables of matter properties at baryon densities ρ = (10⁻⁸–0.32)ρ₀ and temperatures T = 0.2–25 MeV are under construction. At a given ρ and T, the electron fraction Y_e can be evaluated iteratively to fulfill the above-mentioned condition.

In Figure 8, we show the calculated electron fraction, the chemical potential of electrons μ_e, and the average mass number of heavy nuclei 〈A_heavy〉 (A > 4) under the β-equilibrium condition as a function of temperature for various densities. Figure 8 gives valuable information about the composition of stellar matter under the condition of β equilibrium. In Figure 8(a), one can see an interesting behavior, i.e., a non-monotonic change of Y_e with increasing temperature, which has a simple explanation. As follows from Figure 8(b), electrons have larger chemical potentials at higher densities (ρ = 0.1–0.3ρ₀; Figure 8(b)), where heavy nuclei (Figure 8(c)) can survive even at high temperatures T = 6–8 MeV. That is why the fraction of electrons Y_e increases slowly with temperature. On the other hand, at lower densities (ρ ⩽ 0.01ρ₀), the heavy nuclei can exist only at low temperatures, and they disintegrate into free nucleons and light clusters at temperatures T > 3–5 MeV. However, in Figure 8(a), one can see that Y_e exhibits a minimum at T ≈ 2 MeV for 10⁻⁵/10⁻⁴ρ₀ and T = 3/4 MeV for (10⁻³/10⁻²ρ₀. As follows from Figure 8(c), these temperatures correspond to the transition from heavy nuclei to free nucleons and light clusters. At higher densities (ρ = (0.3/0.1)ρ₀), when heavy nuclei are present in the ensemble even at higher temperatures, the electron fraction grows slowly with the temperature, reaching (10/12)% at T = 10 Mev. These results reveal an interesting correlation between the nuclear composition and electron fraction in β-equilibrated stellar matter.

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