Introduction

Research on the isospin- and density-dependent properties of the equation of state (EoS) in isospin asymmetric nuclear matter is a longstanding issue in both nuclear physics and astrophysics[1-4]. With respect to the exchange symmetry between protons and neutrons, the EoS for asymmetric nuclear matter can be expressed as an even series of isospin asymmetry $E(\rho \mathrm{,}\delta )=E_0(\rho )+\stackrel{{\displaystyle \sum }}{{}_{i=\mathrm{2,4,}\cdots }}{E}_{\text{sym}\mathrm{,}i}(\rho ){\delta }^i$, in which the first term is the energy per nucleon in symmetric nuclear matter and the coefficients of the isospin-dependent terms are known as the i-th order symmetry energy $E_{\text{sym}\mathrm{,}i}(\rho )=\frac{1}{i!}\frac{{\partial }^{i}E(\rho \mathrm{,}\delta )}{\partial {\delta }^i}{|}_{\delta =0}$. In recent years, the EoS of nuclear matter has been extensively studied by (I) microscopic and phenomenological many-body approaches[5-8]; (II) the observables from heavy-ion reactions[9-14]; (III) the astrophysical observations[15-17]. For symmetric nuclear matter, the saturation density is constrained in a relatively narrow region ρ₀=0.145~0.180 fm^-3 and the corresponding energy per nucleon E₀(ρ₀) is approximately -16 MeV[18]. The incompressibility coefficient K₀(ρ₀) has a generally accepted value of 240±20 MeV constrained by both theoretical approaches and giant monopole resonance data[19-21]. In addition, the skewness J₀(ρ₀) was recently found to have significant effects on the structures of neutron stars, but its value is scattered widely from -800 MeV to 400 MeV[22-24]. For asymmetric nuclear matter, the value of the quadratic symmetry energy E_sym,2(ρ₀ is constrained to be 31.7±3.2 MeV[25, 26]. However, its density slope and incompressibility coefficient remain uncertain, that is, L₂(ρ₀)=58.7±28.1 MeV[25, 26] and K_sat,2=-550±100[27-29]. It should be emphasized that, at both sub-saturation and supra-saturation densities, the quadratic symmetry energy is not well constrained, especially at supra-saturation densities[30-33]. The quartic symmetry energy E_sym,4(ρ₀) is predicted to be less than 1 MeV[34-36]. In contrast to the quadratic ones, few studies have been conducted on the quartic density slope L₄(ρ₀) and the corresponding incompressibility coefficient K₄(ρ₀)[37].

In the present work, we perform a systematic analysis of six basic quantities in the EoS based on the Hugenholtz-Van Hove (HVH) theorem[38], namely, E_sym,2(ρ), E_sym,4(ρ), L₂(ρ), L₄(ρ), K₂(ρ), and K₄(ρ). Among them, the properties of E_sym,2(ρ), E_sym,4(ρ), and their slopes L₂(ρ) and L₄(ρ) were re-analyzed[39-43]. The analytical expressions of the incompressibility coefficients K₂(ρ) and K₄(ρ) in terms of single-nucleon potentials are given for the first time. In the literature, there are various effective interaction models: transport models such as the Bombaci-Gale-Bertsch-Das Gupta (BGBD) interaction[44-47], the isospin-and momentum-dependent MDI interaction[47-50], the Lanzhou quantum molecular dynamics (LQMD) model[51-53], and the self-consistent mean-field approach including the zero-range momentum-dependent Skyrme interaction[54-56], the finite-range Gogny interaction[57-59], and the relativistic mean-field model[60, 61]. The values of these quantities at the saturation density ρ₀ are calculated using two types of BGBD interactions: the MDI interactions with x=-1, 0 and 1, 16 sets of the Skyrme interactions[62-72], and 4 sets of Gogny interactions[73-75]. By taking the NRAPR Skyrme interaction as an example, we show the isospin- and density-dependent properties of the EoS for asymmetric nuclear matter explicitly. Meanwhile, for symmetric nuclear matter, E₀(ρ), K₀(ρ), and J₀(ρ) are also analyzed in detail. It should be emphasized that the skewness J₀(ρ₀) was recently found to be closely related to not only the maximum mass of neutron stars but also the radius of canonical neutron stars, and the calculations of J₀(ρ) in the present work might be helpful in further determining the properties of neutron stars. In particular, the contributions from the high-order terms of the single-nucleon potential U_sym,3(ρ,k) and U_sym,4(ρ,k) to these basic quantities are evaluated in detail.

The paper is organized as follows. In Sect. 2, based on the HVH theorem, we express the basic quantities of the EoS in terms of the nucleon kinetic energy and the symmetric and asymmetric parts of the single-nucleon potential. The isospin-dependent saturation properties of the asymmetric nuclear matter are also discussed. In Sect. 3, the calculated results by using four different effective interaction models are given. Finally, a summary is presented in Sect. 4.

Decomposition of basic quantities of EoS in terms of global optical potential components

2.1

Basic quantities in the Equation of State of asymmetric nuclear matter

For isospin asymmetric nuclear matter, the EoS can be expanded as a series of isospin asymmetry δ=(ρn-ρp)/ρ. If the high-order terms are neglected, the EoS can be expressed as E(ρ,δ) = E₀(ρ)+E_sym,2(ρ)δ²+E_sym,4(ρ)δ⁴ (see Fig. 1). Each term can be further expanded around the saturation density of symmetric nuclear matter ρ₀ as a series of dimensionless variables $\chi =\frac{\rho -{\rho }_0}{3{\rho }_0}$, which characterizes the deviations of the nuclear density ρ from ρ₀. The density slope and incompressibility coefficient of the i-th order symmetry energy are defined as $L_i(\rho )=3\rho \frac{\partial E_{\text{sym}\mathrm{,}i}(\rho )}{\partial \rho }$ and $K_i(\rho )=9{\rho }^2\frac{{\partial }^2{E}_{\text{sym}\mathrm{,}i}(\rho )}{\partial {\rho }^2}$, respectively. The skewness of the EoS for symmetric nuclear matter is given by $J_0(\rho )=27{\rho }^3\frac{{\partial }^3{E}_0(\rho )}{\partial {\rho }^3}$.

Fig. 1Full-screen View

(Color online) The schematic diagram of basic quantities of the EoS in both isospin symmetric and asymmetric nuclear matter, including E₀(ρ), E_sym,2(ρ), E_sym,4(ρ), K₀(ρ₀), J₀(ρ₀), L₂(ρ₀), K₂(ρ₀), L₄(ρ₀), and K₄(ρ₀).

2.2

The Hugenholtz-Van Hove (HVH) theorem and decomposition of basic quantities of asymmetric nuclear matter

Relating the Fermi energy E_F and the energy per nucleon E, the general Hugenholtz-Van Hove (HVH) theorem can be written as[38]

$E_{\text{F}}=\frac{\text{d}\xi }{\text{d}\rho }=E+\rho \frac{\text{d}E}{\text{d}\rho }=E+\frac{P}{\rho }\mathrm{,}$ (1)

where ξ=ρE and $P={\rho }^2\frac{\partial E}{\partial \rho }$ are the energy density and pressure of the fermion system at an absolute temperature of zero. Accordingly, the Fermi energies of neutrons and protons in asymmetric nuclear matter can be expressed as[41]:

$t(k_{\text{F}}^{\text{n}})+U_{\text{n}}(\rho \mathrm{,}\delta \mathrm{,}{k}_{\text{F}}^{\text{n}})=\frac{\partial \xi }{\partial {\rho }_{\text{n}}}\mathrm{,}$ (2a) $t(k_{\text{F}}^{\text{p}})+U_{\text{p}}(\rho \mathrm{,}\delta \mathrm{,}{k}_{\text{F}}^{\text{p}})=\frac{\partial \xi }{\partial {\rho }_{\text{p}}}\mathrm{,}$ (2b)

where $t(k_{\text{F}}^{\text{n/p}})$ and $U_{\text{n/p}}(\rho \mathrm{,}\delta \mathrm{,}{k}_{\text{F}}^{\text{n/p}})$ are the kinetic energy and the single-nucleon potential of the neutron/proton with the Fermi momentum $k_{\text{F}}^{\text{n/p}}=k_{\text{F}}{(1+\tau \delta )}^{1/3}$. Furthermore, U_n/p(ρ,δ,k) can be expanded by a series of isospin asymmetries δ as

$\begin{array}{l}{U}_{\text{n/p}}(\rho \mathrm{,}\delta \mathrm{,}k)=U_0(\rho \mathrm{,}k)+U_{\text{sym,1}}(\rho \mathrm{,}k)\tau \delta \\ +U_{\text{sym,2}}(\rho \mathrm{,}k)(\tau \delta {)}^2+U_{\text{sym,3}}(\rho \mathrm{,}k)(\tau \delta {)}^3\\ +U_{\text{sym,4}}(\rho \mathrm{,}k)(\tau \delta {)}^4\mathrm{,}\end{array}$ (3)

where τ=1 is for the neutron and τ=-1 for the proton, and U₀(ρ,k) and U_sym,i(ρ,k) are the symmetric and asymmetric parts, respectively. In particular, U₀(ρ,k) and U_sym,1(ρ,k) are called isoscalar and isovector (symmetry) potentials in the popular Lane potential[76].

By subtracting Eq. (2b) from Eq. (2a), we obtain:

$[t(k_{\text{F}}^{\text{n}})-t(k_{\text{F}}^{\text{p}})]+[U_{\text{n}}(\rho \mathrm{,}\delta \mathrm{,}{k}_{\text{F}}^{\text{n}})-U_{\text{p}}(\rho \mathrm{,}\delta \mathrm{,}{k}_{\text{F}}^{\text{p}})]=\frac{\partial \xi }{\partial {\rho }_{\text{n}}}-\frac{\partial \xi }{\partial {\rho }_{\text{p}}}\mathrm{.}$ (4)

Expressing both sides of Eq. (4) in terms of δ and comparing the coefficients of δ and δ³, we can obtain the general expressions of the quadratic and quartic symmetry energies as

$E_{\text{sym,2}}(\rho )=\frac{1}{6}{\frac{\partial [t(k)+U_0(\rho \mathrm{,}k)]}{\partial k}|}_{k_{\text{F}}}{k}_{\text{F}}+\frac{1}{2}{U}_{\text{sym,1}}(\rho \mathrm{,}{k}_{\text{F}})\mathrm{,}$ (5a) $\begin{array}{l}{E}_{\text{sym,4}}(\rho )=\frac{5}{324}{\frac{\partial [t(k)+U_0(\rho \mathrm{,}k)]}{\partial k}|}_{k_{\text{F}}}{k}_{\text{F}}-\frac{1}{108}{\frac{{\partial }^2[t(k)+U_0(\rho \mathrm{,}k)]}{\partial k^2}|}_{k_{\text{F}}}{k}_{\text{F}}^2+\frac{1}{648}{\frac{{\partial }^3[t(k)+U_0(\rho \mathrm{,}k)]}{\partial k^3}|}_{k_{\text{F}}}{k}_{\text{F}}^3\\ -\frac{1}{36}{\frac{\partial U_{\text{sym,1}}(\rho \mathrm{,}k)}{\partial k}|}_{k_{\text{F}}}{k}_{\text{F}}+\frac{1}{72}\frac{{\partial }^2{U}_{\text{sym,1}}(\rho \mathrm{,}k)}{\partial k^2}{|}_{k_{\text{F}}}{k}_{\text{F}}^2+\frac{1}{12}{\frac{\partial U_{\text{sym,2}}(\rho \mathrm{,}k)}{\partial k}|}_{k_{\text{F}}}{k}_{\text{F}}+\frac{1}{4}{U}_{\text{sym,3}}(\rho \mathrm{,}{k}_{\text{F}})\mathrm{.}\end{array}$ (5b)

By adding Eqs. (2a) to (2b), expanding both sides of this summation in terms of δ, and comparing the coefficients of δ⁰, we can obtain an important relationship between E₀(ρ) and its density slope L₀(ρ)

$E_0(\rho )+\rho \frac{\partial E_0(\rho )}{\partial \rho }=t(k_{\text{F}})+U_0(\rho \mathrm{,}{k}_{\text{F}})\mathrm{,}$ (6)

where L₀(ρ) is defined as $3\rho \frac{\partial E_0(\rho )}{\partial \rho }$ and can be rewritten as

$L_0(\rho )=3[t(k_{\text{F}})+U_0(\rho \mathrm{,}{k}_{\text{F}})]-3E_0(\rho )\mathrm{.}$ (7)

Obviously, E₀(ρ₀)=t(k_F)+U₀(ρ₀,k_F) and E₀(ρ) can be calculated from the energy density of the symmetric nuclear matter ξ(ρ,δ=0). Simultaneously, the general expressions of the density slopes L₂(ρ) and L₄(ρ) can also be given by comparing the coefficients of δ² and δ⁴, namely,

$\begin{array}{l}{L}_2(\rho )=\frac{1}{6}{\frac{\partial [t(k)+U_0(\rho \mathrm{,}k)]}{\partial k}|}_{k_{\text{F}}}{k}_{\text{F}}+\frac{1}{6}{\frac{{\partial }^2[t(k)+U_0(\rho \mathrm{,}k)]}{\partial k^2}|}_{k_{\text{F}}}{k}_{\text{F}}^2\\ +{\frac{\partial U_{\text{sym,1}}(\rho \mathrm{,}k)}{\partial k}|}_{k_{\text{F}}}{k}_{\text{F}}+\frac{3}{2}{U}_{\text{sym,1}}(\rho \mathrm{,}{k}_{\text{F}})+3U_{\text{sym,2}}(\rho \mathrm{,}{k}_{\text{F}})\mathrm{,}\end{array}$ (8a) (8b)

Taking the derivative of the summation of Eqs. (2a) and (2b) with respect to ρ and comparing the coefficients, the incompressibility coefficients of E₀(ρ), E_sym,2(ρ), and E_sym,4(ρ) are given as

$K_0(\rho )=9\rho \frac{\partial [t(k_{\text{F}})+U_0(\rho \mathrm{,}{k}_{\text{F}})]}{\partial \rho }-18[t(k_{\text{F}})+U_0(\rho \mathrm{,}{k}_{\text{F}})]+18E_0(\rho )\mathrm{,}$ (9a) $\begin{array}{l}{K}_2(\rho )=-\frac{1}{3}{\frac{\partial [t(k)+U_0(\rho \mathrm{,}k)]}{\partial k}|}_{k_{\text{F}}}{k}_{\text{F}}+\frac{1}{3}{\frac{{\partial }^2[t(k)+U_0(\rho \mathrm{,}k)]}{\partial k^2}|}_{k_{\text{F}}}{k}_{\text{F}}^2\\ -k_{\text{F}}\rho {\frac{\partial }{\partial \rho }\frac{\partial [t(k)+U_0(\rho \mathrm{,}k)]}{\partial k}|}_{k_{\text{F}}}+\frac{1}{2}{k}_{\text{F}}^2\rho \frac{\partial }{\partial \rho }{\frac{{\partial }^2[t(k)+U_0(\rho \mathrm{,}k)]}{\partial k^2}|}_{k_{\text{F}}}\\ +{\frac{\partial U_{\text{sym,1}}(\rho \mathrm{,}k)}{\partial k}|}_{k_{\text{F}}}{k}_{\text{F}}+3k_{\text{F}}\rho {\frac{\partial }{\partial \rho }\frac{\partial U_{\text{sym,1}}(\rho \mathrm{,}k)}{\partial k}|}_{k_{\text{F}}}+9\rho \frac{\partial U_{\text{sym,2}}(\rho \mathrm{,}{k}_{\text{F}})}{\partial \rho }\mathrm{,}\end{array}$ (9b) (9c)

Similarly, by taking the second derivative of Eq. (6) gives the skewness of E₀(ρ) as follows:

$J_0(\rho )=27{\rho }^2\frac{{\partial }^2[t(k_{\text{F}})+U_0(\rho \mathrm{,}{k}_{\text{F}})]}{\partial {\rho }^2}-81\rho \frac{\partial [t(k_{\text{F}})+U_0(\rho \mathrm{,}{k}_{\text{F}})]}{\partial \rho }+162[t(k_{\text{F}})+U_0(\rho \mathrm{,}{k}_{\text{F}})]-162E_0(\rho )\mathrm{.}$ (10)

Results and discussions

We performed a systematic analysis of the basic quantities in the EoS of both symmetric and asymmetric nuclear matter at the saturation density ρ₀ by using 25 interaction parameter sets, which include two BGBD interactions with different neutron-proton effective masses[44-47], the MDI interaction with x=-1, 0, and 1[47-50], 16 Skyrme interactions[62-72], and four Gogny interactions[73-75]. It is known that most of these interactions are fitted to the properties of finite nuclei, and the extrapolations to abnormal densities can be rather diverse. However, the comparison of a large number of results from different interactions could possibly provide useful information on the tendency of the density dependence of these basic quantities. Detailed numerical results from the total 25 interaction parameter sets are summarized in Table 1. The average values of the basic quantities in EoS are also given. For comparison, we also list the constraints summarized in other studies (see the last row of Table 1). As shown in Table 1, the calculated values of E₀(ρ₀), K₀(ρ₀), E_sym,2(ρ₀), and L₂(ρ₀) are consistent with the constraints extracted from both theoretical calculations and experimental data[18, 21, 25, 26]. Interestingly, the averaged E_sym,4(ρ₀) value is almost the same as that in Ref.[77]. To further estimate the error bars of these basic quantities, all the calculated values in Table 1 are plotted in Figs. 2 and 3. It is seen from Fig. 2 that the data points of E₀(ρ₀) and K₀(ρ₀) are well constrained in a narrow range and the corresponding error bars are small. The error bar of skewness J₀(ρ₀)=-411.3±37.0 MeV is relatively large, especially for Gogny interactions. It is also noted that the skewness, together with K₂(ρ₀), has recently received much attention in the calculation of the maximum mass of neutron stars and the radius of canonical neutron stars[15, 22, 23]. The error bars of the high-order terms L₄(ρ₀), K₂(ρ₀), and K₄(ρ₀) are also given, that is, L₄(ρ₀)=1.42±2.14 MeV, K₂(ρ₀)=-123.6±83.8 MeV, and K₄(ρ₀)=-1.25±5.89 MeV. In addition, for the MDI interaction, the L₂(ρ₀) and K₂(ρ₀) values with different spin(isospin)-dependent parameter x are scattered over a wide range. This is because the different choices of parameter x are to simulate very different density dependences of the symmetry energies at high densities[47-49].

Fig. 2Full-screen View

(Color online) Values of basic quantities E₀(ρ₀), K₀(ρ₀), and J₀(ρ₀) for symmetric nuclear matter at 25 parameter sets of the BGBD, MDI, Skyrme, and Gogny interactions. The solid and dashed lines represent the average values and their deviations, respectively.

Fig. 3Full-screen View

(Color online) Values of E_sym,2(ρ₀, L₂(ρ₀), K₂(ρ₀), E_sym,4(ρ₀), L₄(ρ₀), and K₄(ρ₀) for asymmetric nuclear matter within 25 parameter sets of four kinds of interaction.

In Fig. 4, we show the magnitudes of the separated terms E₀(ρ), E_sym,2(ρ)δ², E_sym,4(ρ)δ⁴ as well as the total one E(ρ,δ) at two different densities (ρ₀ and 2ρ₀) and three different isospin asymmetries (δ²=0.1, 0.2 and 0.5) by taking the NRAPR Skyrme interaction as an example. At the saturation density ρ₀ (see graphs (a)–(c)), the contribution of E₀(ρ) to E(ρ,δ) is dominant. The contribution of E_sym,2(ρ)δ² increases with an increase in isospin asymmetry δ. It is also shown that the contribution from E_sym,4(ρ)δ⁴ is small and comes into play at large isospin asymmetry with δ²=0.5. At 2ρ₀ (see graphs (d)–(f)), the E₀(ρ) contribution is suppressed compared with that at ρ₀, while E_sym,2(ρ)δ² plays a more important role in the EoS, especially at δ²=0.5. It should also be noted that E_sym,4(ρ) contributes only at a very high density and large isospin asymmetry. The magnitude of E_sym,4(ρ) can significantly affect the calculation of the proton fraction in neutron stars at β-equilibrium[14, 41].

Fig. 4Full-screen View

(Color online) The magnitudes of E₀(ρ), E_sym,2(ρ)δ², and E_sym,4(ρ)δ⁴ in the EoS at two different ρ values and three different δ² values. The NRAPR Skyrme interaction is applied.

We further expand E₀(ρ), E_sym,2(ρ), and E_sym,4(ρ) as a series of χ with their corresponding slopes and incompressibility coefficients. In Fig. 5, we depict the contributions from each term at different densities 0.5ρ_0, 2ρ₀ and 3ρ₀. As can be observed in Fig. 5, the first-order terms $E_0^0$ (E₀(ρ₀)), $E_2^0$ (E_sym,2(ρ₀), and $E_4^0$ (E_sym,4(ρ₀)) contribute largely at all densities. $E_0^{\text{K}}$ and $E_0^{\text{J}}$ terms become increasingly important with increasing density. For E_sym,2(ρ) and E_sym,4(ρ) at 3ρ₀, the contributions from the slopes ($E_2^{\text{L}}$ and $E_4^{\text{L}}$) and the incompressibility coefficients ($E_2^{\text{K}}$ and $E_4^{\text{K}}$) are much larger than those at 0.5ρ_0 and 2ρ₀. In particular, the $E_0^{\text{J}}$, $E_2^{\text{K}}$ , and $E_4^{\text{K}}$ terms at 3ρ₀ can be as important as the first-order terms. Thus, high-order terms should be considered when analyzing the properties of nuclear matter systems at high densities, such as neutron stars.

Fig. 5Full-screen View

(Color online) The magnitude of each order in E₀(ρ), E_sym,2(ρ) and E_sym,4(ρ) expressed by E₀(ρ₀), K₀(ρ₀) and J₀(ρ₀), E_sym,2(ρ₀, L₂(ρ₀) and K₂(ρ₀), and E_sym,4(ρ₀), L₄(ρ₀) and K₄(ρ₀), respectively. The NRAPR Skyrme interaction was applied.

More interestingly, the basic quantities at the saturation density are decomposed into the kinetic energy t(k) and the symmetric and asymmetric parts of the single-nucleon potential U₀(ρ,k) and U_sym,i(ρ,k). As shown in Fig. 6, the contributions from different terms t(k), U₀(ρ,k) and U_sym,i(ρ,k) (i=1,2,3,4) are denoted by superscripts of T, U0, U1, U2, U3 and U4, respectively. It is clear that E₀(ρ₀), K₀(ρ₀), and J₀(ρ₀) are completely determined by t(k) and U₀(ρ,k). For other quantities, the contributions from the asymmetric parts U_sym,1(ρ,k), U_sym,2(ρ,k), U_sym,3(ρ,k), and U_sym,4(ρ,k) cannot be neglected. It is clearly shown that the first-order term U_sym,1(ρ,k) contributes to all six basic quantities. The second-order term U_sym,2(ρ,k) does not contribute to E_sym,2(ρ₀, but to its corresponding slope L₂(ρ₀) and the incompressibility coefficient K₂(ρ₀). In principle, the U_sym,2(ρ,k) term should also contribute to the fourth-order terms E_sym,4(ρ₀), L₄(ρ₀), and K₄(ρ₀), but for the Skyrme interaction, U_sym,2(ρ,k) is not momentum-dependent and does not contribute. In addition, there are very few studies on the contributions of high-order terms U_sym,3(ρ,k) and U_sym,4(ρ,k) to the basic quantities. In Fig. 7, we show the density-dependence of U₀(ρ,k_F), U_sym,1(ρ,k_F), U_sym,2(ρ,k_F), U_sym,3(ρ,k_F) and U_sym,4(ρ,k_F) at the Fermi momentum k_F=(3π²ρ/2)^1/3 by using the NRAPR Skyrme interaction. It can be clearly seen in Fig. 7 that the magnitudes of U₀(ρ,k_F) and U_sym,1(ρ,k_F) are generally very large, while the ones of U_sym,2(ρ,k_F), U_sym,3(ρ,k_F) and U_sym,4(ρ,k_F) are very small but increase with the increasing density. Our results indicate that the U_sym,3(ρ,k) and U_sym,4(ρ,k) contributions should be taken into account for the fourth-order terms to understand the properties of asymmetric nuclear matter, especially for the cases with very large isospin asymmetries and high densities.

Fig. 6Full-screen View

(Color online) The single-nucleon potential decomposition of E₀(ρ₀), K₀(ρ₀), J₀(ρ₀), E_sym,2(ρ₀, L₂(ρ₀), K₂(ρ₀), E_sym,4(ρ₀), L₄(ρ₀), and K₄(ρ₀). The NRAPR Skyrme interaction is applied.

Fig. 7Full-screen View

(Color online) The density-dependence of U₀(ρ,k_F), U_sym,1(ρ,k_F), U_sym,2(ρ,k_F), U_sym,3(ρ,k_F), and U_sym,4(ρ,k_F). The NRAPR Skyrme interaction was applied.

By analyzing the isospin dependence of the saturation properties of asymmetric nuclear matter, a number of important quantities are calculated using 25 interaction parameter sets, and their numerical results as well as their averaged values are also listed in Table 2. For comparison, the constraints of K_asy,2 and K_sat,2 from other studies are listed in the last row of Table 2. It is shown that the second-order coefficient ρ_sat,2, one of the most important isospin-dependent parts of ρ_sat(δ), has a negative value in all cases, and the fourth-order coefficient ρ_sat,4 also has a negative value for the Skyrme and Gogny interactions. This means that in most cases, the saturation density of asymmetric nuclear matter is lower than that of symmetric nuclear matter, especially at larger isospin asymmetry δ (see graph (a) of Fig. 8). For the BGBD interaction (Case-2), the calculated value of ρ_sat,4 is positive and relatively large. According to the relationship in Eq. (11), this would lead to a higher saturation density of asymmetric nuclear matter than that of symmetric nuclear matter with isospin asymmetry δ close to unity. For asymmetric nuclear matter at ρ_sat(δ), the corresponding E_sat,4 values are rather diverse and are considered to be important for the proton fraction in neutron stars.

Fig. 8Full-screen View

(Color online) The isospin-dependence of the exact saturation density ρ_sat(δ) within 14 typical interaction parameter sets in graph (a) and the comparisons between the error bars of the quadratic incompressibility coefficients K₂(ρ₀), K_asy,2, and K_sat,2 calculated by using 25 interaction parameter sets in graph (b).

As shown in graph (b) of Fig. 8, the results of K₂(ρ₀), K_asy,2, and K_sat,2 are given and their values are constrained to be K₂=-123.6±83.8 MeV, K_asy,2=-465.4±70.0 MeV, and K_sat,2=-360.1±39.0 MeV, respectively. The averaged K_asy,2 value is close to the previous theoretical constraint of -500±50 MeV given in Ref.[31] if the error bar is considered. In Table 2, there are two previous constraints for K_sat,2. One is K_sat,2=-370±120 MeV from a modified Skyrme-like (MSL) model[77], and the other is -550±100 MeV by analyzing the measured data of the isotopic dependence of the giant monopole resonance (GMR) in the even-A Sn isotopes[27, 28]. Compared with these previous studies, it is clear that the K_asy,2 and K_sat,2 values remain uncertain and require more data to further constrain their values. In addition, as mentioned before, the term $-\frac{J_0({\rho }_0)}{K_0({\rho }_0)}{L}_2({\rho }_0)$ in Eq. (15) is typically ignored for simplicity. However, this is clearly shown in Fig. 8(b) that the contribution of this term is non-negligible. In the present work, we include the contribution of this high-order term, and the ratio J₀(ρ₀)/K₀(ρ₀) is constrained in the range of -1.86±0.23. Finally we obtain a simple relation for K_sat,2

With the averaged results L₂(ρ₀)=57.0 MeV and K₂(ρ₀)=-123.6 MeV, the calculated value K_sat,2=-359.6 MeV is in good agreement with the average value of -360.1±39.0 MeV from the 25 interaction sets. This simple empirical relation could be useful for estimating the value of K_sat,2 for asymmetric nuclear matter.

SUMMARY

Based on the Hugenholtz-Van Hove theorem, the general expressions for the six basic quantities of EoS are expanded in terms of the kinetic energy t(k), the symmetric and asymmetric parts of the global optical potential U₀(ρ,k) and U_sym,i(ρ,k). The analytical expressions of the coefficients K₂(ρ) and K₄(ρ) are given for the first time. By using 25 types of interaction sets, the values of these quantities were systematically calculated at the saturation density ρ₀. It is emphasized that there are very few studies on quantities L₄(ρ₀), K₂(ρ₀), and K₄(ρ₀) and their average values from a total of 25 interaction sets are L₄(ρ₀)=1.42±2.14 MeV, K₂(ρ₀)=-123.6±83.8 MeV, and K₄(ρ₀)=-1.25±5.89 MeV, respectively. The averaged values of the other quantities were consistent with those of previous studies. Furthermore, the different contributions of the kinetic term, the isoscalar and isovector potentials to these basic quantities were systematically analyzed at saturation density. It is clearly shown that t(k_F) and U₀(ρ,k_F) play vital roles in determining the EoS of both symmetric and asymmetric nuclear matter. For asymmetric nuclear matter, U_sym,1(ρ,k) contributes to all the quantities, whereas U_sym,2(ρ,k) does not contribute to E_sym,2(ρ₀), but contributes to the second-order terms L₂(ρ₀) and K₂(ρ₀) as well as the fourth-order terms E_sym,4(ρ₀), L₄(ρ₀), and K₄(ρ₀). In addition, the contribution from U_sym,3(ρ,k) cannot be neglected for E_sym,4(ρ₀), L₄(ρ₀), and K₄(ρ₀). U_sym,4(ρ,k) should also be included in the calculations for L₄(ρ₀) and K₄(ρ₀). In addition, the quadratic incompressibility coefficient at ρ_sat(δ) is found to have a simple empirical relation K_sat,2=K₂(ρ₀)-4.14 L₂(ρ₀) based on the present analysis.