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Basic quantities of the Equation of State in isospin asymmetric nuclear matter

Jie Liu1, Chao Gao1, Niu Wan2, and Chang Xu1,111E-mail address: [email protected] 1School of Physics, Nanjing University, Nanjing 210093, China
2School of Physics and Optoelectronics, South China University of Technology, Guangzhou 510641, China
Abstract

Based on the Hugenholtz-Van Hove theorem, six basic quantities of the EoS in isospin asymmetric nuclear matter are expressed in terms of the nucleon kinetic energy t​(k)t(k), the isospin symmetric and asymmetric parts of the single-nucleon potentials U_​0​(ρ,k)U_{\_}0(\rho,k) and U_​sym,i​(ρ,k)U_{\_}{\text{\text{sym,i}}}(\rho,k). The six basic quantities include the quadratic symmetry energy E_​sym,2​(ρ)E_{\_}{\text{sym,2}}(\rho), the quartic symmetry energy E_​sym,4​(ρ)E_{\_}{\text{sym,4}}(\rho), their corresponding density slopes L_​2​(ρ)L_{\_}2(\rho) and L_​4​(ρ)L_{\_}4(\rho), and the incompressibility coefficients K_​2​(ρ)K_{\_}2(\rho) and K_​4​(ρ)K_{\_}4(\rho). By using four types of well-known effective nucleon-nucleon interaction models, namely the BGBD, MDI, Skyrme, and Gogny forces, the density- and isospin-dependent properties of these basic quantities are systematically calculated and their values at the saturation density ρ_​0\rho_{\_}0 are explicitly given. The contributions to these quantities from t​(k)t(k), U_​0​(ρ,k)U_{\_}0(\rho,k), and U_​sym,i​(ρ,k)U_{\_}{\text{sym,i}}(\rho,k) are also analyzed at the normal nuclear density ρ_​0\rho_{\_}0. It is clearly shown that the first-order asymmetric term U_​sym,1​(ρ,k)U_{\_}{\text{sym,1}}(\rho,k) (also known as the symmetry potential in Lane potential) plays a vital role in determining the density dependence of the quadratic symmetry energy E_​sym,2​(ρ)E_{\_}{\text{sym,2}}(\rho). It is also shown that the contributions from high-order asymmetric parts of the single-nucleon potentials (U_​sym,i​(ρ,k)U_{\_}{\text{sym,i}}(\rho,k) with i>1i>1) cannot be neglected in the calculations of the other five basic quantities. Moreover, by analyzing the properties of asymmetric nuclear matter at the exact saturation density ρ_​sat​(δ)\rho_{\_}{\text{sat}}(\delta), the corresponding quadratic incompressibility coefficient is found to have a simple empirical relation K_​sat,2=K_​2​(ρ_​0)−4.14​L_​2​(ρ_​0)K_{\_}{\text{sat,2}}=K_{\_}{2}(\rho_{\_}0)-4.14L_{\_}2(\rho_{\_}0).

pacs:
21.30.Fe, 21.65.Cd, 21.65.Ef, 21.65.Mn

I 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 PDan02 ; JML04 ; MBal16 ; CJJ21 . 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​(ρ,δ)=E_​0​(ρ)+i_=2,4,⋯∑​E_​sym,i​(ρ)​δiE(\rho,\delta)=E_{\_}0(\rho)+\overset{\sum}{{}_{\_}{i=2,4,\cdots}}{E_{\_}{\text{sym,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 ii-th order symmetry energy E_​sym,i​(ρ)=1i!​∂iE​(ρ,δ)∂δi∣_​δ=0E_{\_}{\text{sym,i}}(\rho)=\frac{1}{i!}\frac{\partial^{i}E(\rho,\delta)}{\partial\delta^{i}}\mid_{\_}{\delta=0}. In recent years, the EoS of nuclear matter has been extensively studied by (I) microscopic and phenomenological many-body approaches MB03 ; HY20 ; XLR21 ; JXU21 ; (II) the observables from heavy-ion reactions GC14 ; LWC14 ; JMDONG13 ; JXu09 ; GFWei20 ; OL09 ; (III) the astrophysical observations LiBA19 ; NBZhang-NST-18 ; YXU19 . For symmetric nuclear matter, the saturation density is constrained in a relatively narrow region ρ_​0=0.145∼0.180\rho_{\_}0=0.145\sim 0.180 fm-3 and the corresponding energy per nucleon E_​0​(ρ_​0)E_{\_}0(\rho_{\_}0) is approximately −16-16 MeV LiBA08 . The incompressibility coefficient K_​0​(ρ_​0)K_{\_}0(\rho_{\_}0) has a generally accepted value of 240±20240\pm 20 MeV constrained by theoretical approaches and the giant monopole resonance data DHY99 ; JPB80 ; SS06 . In addition, the skewness J_​0​(ρ_​0)J_{\_}0(\rho_{\_}0) was recently found to have significant effects on the structures of neutron stars, but its value was scattered widely from −800-800 MeV to 400400 MeV BJCai17 ; NBZ18 ; WJX19 . For asymmetric nuclear matter, the value of the quadratic symmetry energy E_​sym,2​(ρ_​0)E_{\_}{\text{sym,2}}(\rho_{\_}0) is constrained to be 31.7±3.231.7\pm 3.2 MeV LiBA13 ; MOer17 . However, its density slope and the incompressibility coefficient remain uncertain, i.e. L_​2​(ρ_​0)=58.7±28.1L_{\_}2(\rho_{\_}0)=58.7\pm 28.1 MeV LiBA13 ; MOer17 and K_​sat,2=−550±100K_{\_}{\text{sat,2}}=-550\pm 100 UGar07 ; TLi07 ; MLQ88 . 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 ZGXIAO09 ; LWC07 ; CXu12CPL ; NBZhang17 . The quartic symmetry energy E_​sym,4​(ρ_​0)E_{\_}{\text{sym,4}}(\rho_{\_}0) is predicted to be less than 11 MeV JP17 ; ZWL18 ; JMDONG18 . In contrast to the quadratic one, few studies have been conducted on the quartic density slope L_​4​(ρ_​0)L_{\_}4(\rho_{\_}0) and the corresponding incompressibility coefficient K_​4​(ρ_​0)K_{\_}4(\rho_{\_}0) CGB17 .

In the present work, we perform a systematic analysis of six basic quantities in the EoS based on the Hugenholtz-Van Hove (HVH) theorem NMH58 , namely the E_​sym,2​(ρ)E_{\_}{\text{sym,2}}(\rho), E_​sym,4​(ρ)E_{\_}{\text{sym,4}}(\rho), L_​2​(ρ)L_{\_}2(\rho), L_​4​(ρ)L_{\_}4(\rho), K_​2​(ρ)K_{\_}2(\rho), and K_​4​(ρ)K_{\_}4(\rho). Among them, the properties of E_​sym,2​(ρ)E_{\_}{\text{sym,2}}(\rho), E_​sym,4​(ρ)E_{\_}{\text{sym,4}}(\rho), and their slopes L_​2​(ρ)L_{\_}2(\rho) and L_​4​(ρ)L_{\_}4(\rho) were re-analyzed CXu11 ; NW17 ; CXu14 ; NW16 ; MJ21 . The analytical expressions of the incompressibility coefficients K_​2​(ρ)K_{\_}2(\rho) and K_​4​(ρ)K_{\_}4(\rho) 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 CG87 ; IB91 ; CBD03 ; JR04 , the isospin- and momentum-dependent MDI interaction CBD03 ; LiBA04a ; LiBA04b ; LWC05c , and the Lanzhou quantum molecular dynamics (LQMD) model ZQFENG11 ; ZQFENG12 ; FZHang20 , the self-consistent mean-field approach including the zero-range momentum-dependent Skyrme interaction THRS59 ; DV72 ; YZWANG19 , the finite-range Gogny interaction DMB67 ; JD75 ; DG77 and the relativistic mean-field model BOGUTA77 ; FO13 , etc.. The values of these quantities at the saturation density ρ_​0\rho_{\_}0 are calculated using two types of BGBD interactions, the MDI interactions with x=−1x=-1, 0 and 11, 1616 sets of the Skyrme interactions MD12 ; GSkI ; KDE0v1 ; LNS ; MSL0 ; AWS05 ; SKRA ; SkT1 ; Skxs20 ; SQMC650 ; SV-sym32 , and 44 sets of the Gogny interactions JFB91 ; FC08 ; SG09 . 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, E_​0​(ρ)E_{\_}{0}(\rho), K_​0​(ρ)K_{\_}0(\rho), and J_​0​(ρ)J_{\_}0(\rho) are also analyzed in detail. It should be emphasized that the skewness J_​0​(ρ_​0)J_{\_}0(\rho_{\_}0) 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 on J_​0​(ρ)J_{\_}0(\rho) 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)U_{\_}{\text{sym,3}}(\rho,k) and U_​sym,4​(ρ,k)U_{\_}{\text{sym,4}}(\rho,k) to these basic quantities are evaluated in detail.

The paper is organized as follows. In Sec. II, 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 Sec. III, the calculated results by using four different effective interaction models are given. Finally, a summary is presented in Sec. IV.

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

II.1 Basic quantities in the Equation of State of asymmetric nuclear matter

Refer to caption
Figure 1: (Color online) The schematic diagram of basic quantities of the EoS in both isospin symmetric and asymmetric nuclear matter, including E_​0​(ρ)E_{\_}0(\rho), E_​sym,2​(ρ)E_{\_}{\text{sym,2}}(\rho), E_​sym,4​(ρ)E_{\_}{\text{sym,4}}(\rho), K_​0​(ρ_​0)K_{\_}0(\rho_{\_}0), J_​0​(ρ_​0)J_{\_}0(\rho_{\_}0), L_​2​(ρ_​0)L_{\_}2(\rho_{\_}0), K_​2​(ρ_​0)K_{\_}2(\rho_{\_}0), L_​4​(ρ_​0)L_{\_}4(\rho_{\_}0), and K_​4​(ρ_​0)K_{\_}4(\rho_{\_}0).

For isospin asymmetric nuclear matter, the EoS can be expanded as a series of isospin asymmetry δ=(ρ_​n−ρ_​p)/ρ\delta=(\rho_{\_}\text{n}-\rho_{\_}\text{p})/\rho. If the high-order terms are neglected, it can be expressed as E​(ρ,δ)=E_​0​(ρ)+E_​sym,2​(ρ)​δ2+E_​sym,4​(ρ)​δ4E(\rho,\delta)=E_{\_}0(\rho)+E_{\_}{\text{sym,2}}(\rho)\delta^{2}+E_{\_}{\text{sym,4}}(\rho)\delta^{4} (see Fig. 1). Each term can be further expanded around the saturation density of symmetric nuclear matter ρ_​0\rho_{\_}0 as a series of a dimensionless variable χ=ρ−ρ_​03​ρ_​0\chi=\frac{\rho-\rho_{\_}0}{3\rho_{\_}0}, which characterizes the deviations of the nuclear density ρ\rho from ρ_​0\rho_{\_}0. The density slope and incompressibility coefficient of the ii-th order symmetry energy are defined as L_​i​(ρ)=3​ρ​∂E_​sym,i​(ρ)∂ρL_{\_}i(\rho)=3\rho\frac{\partial E_{\_}{\text{sym,i}}(\rho)}{\partial\rho} and K_​i​(ρ)=9​ρ2​∂2E_​sym,i​(ρ)∂ρ2K_{\_}i(\rho)=9\rho^{2}\frac{\partial^{2}E_{\_}{\text{sym,i}}(\rho)}{\partial\rho^{2}}, respectively. The skewness of the EoS for symmetric nuclear matter is given by J_​0​(ρ)=27​ρ3​∂3E_​0​(ρ)∂ρ3J_{\_}0(\rho)=27\rho^{3}\frac{\partial^{3}E_{\_}{0}(\rho)}{\partial\rho^{3}}.

II.2 The Hugenholtz-Van Hove (HVH) theorem and decomposition of basic quantities of asymmetric nuclear matter

Relating the Fermi energy E_​FE_{\_}F and the energy per nucleon EE, the general Hugenholtz-Van Hove (HVH) theorem can be written as NMH58

E_​F=d​ξd​ρ=E+ρ​d​Ed​ρ=E+Pρ,\displaystyle E_{\_}{F}=\frac{d\xi}{d\rho}=E+\rho\frac{dE}{d\rho}=E+\frac{P}{\rho}, (1)

where ξ=ρ​E\xi=\rho E and P=ρ2​∂E∂ρP=\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 CXu11

t​(k_​Fn)+U_​n​(ρ,δ,k_​Fn)=∂ξ∂ρ_​n,\displaystyle t(k_{\_}{\text{F}}^{\text{n}})+U_{\_}\text{n}(\rho,\delta,k_{\_}{\text{F}}^{\text{n}})=\frac{\partial\xi}{\partial\rho_{\_}\text{n}}, (2a)
t​(k_​Fp)+U_​p​(ρ,δ,k_​Fp)=∂ξ∂ρ_​p,\displaystyle t(k_{\_}{\text{F}}^{\text{p}})+U_{\_}\text{p}(\rho,\delta,k_{\_}{\text{F}}^{\text{p}})=\frac{\partial\xi}{\partial\rho_{\_}\text{p}}, (2b)

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

U_​n/p​(ρ,δ,k)=\displaystyle U_{\_}{\text{\text{n/p}}}(\rho,\delta,k)= U_​0​(ρ,k)+U_​sym,1​(ρ,k)​τ​δ+U_​sym,2​(ρ,k)​(τ​δ)2\displaystyle U_{\_}0(\rho,k)+U_{\_}{\text{sym,1}}(\rho,k)\tau\delta+U_{\_}{\text{sym,2}}(\rho,k)(\tau\delta)^{2}
+U_​sym,3​(ρ,k)​(τ​δ)3+U_​sym,4​(ρ,k)​(τ​δ)4,\displaystyle+U_{\_}{\text{sym,3}}(\rho,k)(\tau\delta)^{3}+U_{\_}{\text{sym,4}}(\rho,k)(\tau\delta)^{4}, (3)

where τ=1\tau=1 is for the neutron and τ=−1\tau=-1 for the proton, and U_​0​(ρ,k)U_{\_}0(\rho,k) and U_​sym,i​(ρ,k)U_{\_}{\text{sym,i}}(\rho,k) are the symmetric and asymmetric parts, respectively. In particular, U_​0​(ρ,k)U_{\_}0(\rho,k) and U_​sym,1​(ρ,k)U_{\_}{\text{sym,1}}(\rho,k) are called isoscalar and isovector (symmetry) potentials in the popular Lane potential AM62 .

By subtracting Eq. (2b) from Eq. (2a), one can obtain

[t​(k_​Fn)−t​(k_​Fp)]+[U_​n​(ρ,δ,k_​Fn)−U_​p​(ρ,δ,k_​Fp)]=∂ξ∂ρ_​n−∂ξ∂ρ_​p.\displaystyle[t(k_{\_}{\text{F}}^{\text{n}})-t(k_{\_}{\text{F}}^{\text{p}})]+[U_{\_}\text{n}(\rho,\delta,k_{\_}{\text{F}}^{\text{n}})-U_{\_}\text{p}(\rho,\delta,k_{\_}{\text{F}}^{\text{p}})]=\frac{\partial\xi}{\partial\rho_{\_}\text{n}}-\frac{\partial\xi}{\partial\rho_{\_}\text{p}}. (4)

Expressing both sides of Eq. (4) in terms of δ\delta and comparing the coefficients of δ\delta and δ3\delta^{3}, we can obtain the general expressions of the quadratic and quartic symmetry energies as

E_​sym,2​(ρ)=\displaystyle E_{\_}{\text{sym,2}}(\rho)= 16​∂[t​(k)+U_​0​(ρ,k)]∂k|_​k_​F​k_​F+12​U_​sym,1​(ρ,k_​F),\displaystyle\frac{1}{6}\frac{\partial[t(k)+U_{\_}0(\rho,k)]}{\partial k}|_{\_}{k_{\_}{\text{F}}}k_{\_}{\text{F}}+\frac{1}{2}U_{\_}{\text{sym,1}}(\rho,k_{\_}{\text{F}}), (5a)
E_​sym,4​(ρ)=\displaystyle E_{\_}{\text{sym,4}}(\rho)= 5324​∂[t​(k)+U_​0​(ρ,k)]∂k|_​k_​F​k_​F−1108​∂2[t​(k)+U_​0​(ρ,k)]∂k2|_​k_​F​k_​F2\displaystyle\frac{5}{324}\frac{\partial[t(k)+U_{\_}0(\rho,k)]}{\partial k}|_{\_}{k_{\_}{\text{F}}}k_{\_}{\text{F}}-\frac{1}{108}\frac{\partial^{2}[t(k)+U_{\_}0(\rho,k)]}{\partial k^{2}}|_{\_}{k_{\_}{\text{F}}}k_{\_}{\text{F}}^{2}
+1648​∂3[t​(k)+U_​0​(ρ,k)]∂k3|_​k_​F​k_​F3\displaystyle+\frac{1}{648}\frac{\partial^{3}[t(k)+U_{\_}0(\rho,k)]}{\partial k^{3}}|_{\_}{k_{\_}{\text{F}}}k_{\_}{\text{F}}^{3}
−136​∂U_​sym,1​(ρ,k)∂k|_​k_​F​k_​F+172​∂2U_​sym,1​(ρ,k)∂k2|_​k_​F​k_​F2\displaystyle-\frac{1}{36}\frac{\partial U_{\_}{\text{sym,1}}(\rho,k)}{\partial k}|_{\_}{k_{\_}{\text{F}}}k_{\_}{\text{F}}+\frac{1}{72}\frac{\partial^{2}U_{\_}{\text{sym,1}}(\rho,k)}{\partial k^{2}}|_{\_}{k_{\_}{\text{F}}}k_{\_}{\text{F}}^{2}
+112​∂U_​sym,2​(ρ,k)∂k|_​k_​F​k_​F+14​U_​sym,3​(ρ,k_​F).\displaystyle+\frac{1}{12}\frac{\partial U_{\_}{\text{sym,2}}(\rho,k)}{\partial k}|_{\_}{k_{\_}{\text{F}}}k_{\_}{\text{F}}+\frac{1}{4}U_{\_}{\text{sym,3}}(\rho,k_{\_}{\text{F}}). (5b)

By adding Eqs. (2a) to (2b), expanding both sides of this summation in terms of δ\delta, and comparing the coefficients of δ0\delta^{0}, we can obtain an important relationship between E_​0​(ρ)E_{\_}0(\rho) and its density slope L_​0​(ρ)L_{\_}0(\rho)

E_​0​(ρ)+ρ​∂E_​0​(ρ)∂ρ=t​(k_​F)+U_​0​(ρ,k_​F),\displaystyle E_{\_}0(\rho)+\rho\frac{\partial E_{\_}0(\rho)}{\partial\rho}=t(k_{\_}{\text{F}})+U_{\_}0(\rho,k_{\_}{\text{F}}), (6)

where L_​0​(ρ)L_{\_}0(\rho) is defined as 3​ρ​∂E_​0​(ρ)∂ρ3\rho\frac{\partial E_{\_}0(\rho)}{\partial\rho} and can be rewritten as

L_​0​(ρ)=3​[t​(k_​F)+U_​0​(ρ,k_​F)]−3​E_​0​(ρ).\displaystyle L_{\_}0(\rho)=3[t(k_{\_}{\text{F}})+U_{\_}0(\rho,k_{\_}{\text{F}})]-3E_{\_}0(\rho). (7)

Obviously, E_​0​(ρ_​0)=t​(k_​F)+U_​0​(ρ_​0,k_​F)E_{\_}0(\rho_{\_}0)=t(k_{\_}{\text{F}})+U_{\_}0(\rho_{\_}0,k_{\_}{\text{F}}) and E_​0​(ρ)E_{\_}0(\rho) can be calculated from the energy density of symmetric nuclear matter ξ​(ρ,δ=0)\xi(\rho,\delta=0). Simultaneously, the general expressions of the density slopes L_​2​(ρ)L_{\_}2(\rho) and L_​4​(ρ)L_{\_}4(\rho) can also be given by comparing the coefficients of δ2\delta^{2} and δ4\delta^{4}, namely,

L_​2​(ρ)=\displaystyle L_{\_}2(\rho)= 16​∂[t​(k)+U_​0​(ρ,k)]∂k|_​k_​F​k_​F+16​∂2[t​(k)+U_​0​(ρ,k)]∂k2|_​k_​F​k_​F2\displaystyle\frac{1}{6}\frac{\partial[t(k)+U_{\_}0(\rho,k)]}{\partial k}|_{\_}{k_{\_}{\text{F}}}k_{\_}{\text{F}}+\frac{1}{6}\frac{\partial^{2}[t(k)+U_{\_}0(\rho,k)]}{\partial k^{2}}|_{\_}{k_{\_}{\text{F}}}k_{\_}{\text{F}}^{2}
+∂U_​sym,1​(ρ,k)∂k|_​k_​F​k_​F+32​U_​sym,1​(ρ,k_​F)+3​U_​sym,2​(ρ,k_​F),\displaystyle+\frac{\partial U_{\_}{\text{sym,1}}(\rho,k)}{\partial k}|_{\_}{k_{\_}{\text{F}}}k_{\_}{\text{F}}+\frac{3}{2}U_{\_}{\text{sym,1}}(\rho,k_{\_}{\text{F}})+3U_{\_}{\text{sym,2}}(\rho,k_{\_}{\text{F}}), (8a)
L_​4​(ρ)=\displaystyle L_{\_}4(\rho)= 5324​∂[t​(k)+U_​0​(ρ,k)]∂k|_​k_​F​k_​F−1324​∂2[t​(k)+U_​0​(ρ,k)]∂k2|_​k_​F​k_​F2\displaystyle\frac{5}{324}\frac{\partial[t(k)+U_{\_}0(\rho,k)]}{\partial k}|_{\_}{k_{\_}{\text{F}}}k_{\_}{\text{F}}-\frac{1}{324}\frac{\partial^{2}[t(k)+U_{\_}0(\rho,k)]}{\partial k^{2}}|_{\_}{k_{\_}{\text{F}}}k_{\_}{\text{F}}^{2}
−1216​∂3[t​(k)+U_​0​(ρ,k)]∂k3|_​k_​F​k_​F3+1648​∂4[t​(k)+U_​0​(ρ,k)]∂k4|_​k_​F​k_​F4\displaystyle-\frac{1}{216}\frac{\partial^{3}[t(k)+U_{\_}0(\rho,k)]}{\partial k^{3}}|_{\_}{k_{\_}{\text{F}}}k_{\_}{\text{F}}^{3}+\frac{1}{648}\frac{\partial^{4}[t(k)+U_{\_}0(\rho,k)]}{\partial k^{4}}|_{\_}{k_{\_}{\text{F}}}k_{\_}{\text{F}}^{4}
−7108​∂U_​sym,1​(ρ,k)∂k|_​k_​F​k_​F+172​∂2U_​sym,1​(ρ,k)∂k2|_​k_​F​k_​F2+154​∂3U_​sym,1​(ρ,k)∂k3|_​k_​F​k_​F3\displaystyle-\frac{7}{108}\frac{\partial U_{\_}{\text{sym,1}}(\rho,k)}{\partial k}|_{\_}{k_{\_}{\text{F}}}k_{\_}{\text{F}}+\frac{1}{72}\frac{\partial^{2}U_{\_}{\text{sym,1}}(\rho,k)}{\partial k^{2}}|_{\_}{k_{\_}{\text{F}}}k_{\_}{\text{F}}^{2}+\frac{1}{54}\frac{\partial^{3}U_{\_}{\text{sym,1}}(\rho,k)}{\partial k^{3}}|_{\_}{k_{\_}{\text{F}}}k_{\_}{\text{F}}^{3}
+512​∂U_​sym,2​(ρ,k)∂k|_​k_​F​k_​F+16​∂2U_​sym,2​(ρ,k)∂k2|_​k_​F​k_​F2\displaystyle+\frac{5}{12}\frac{\partial U_{\_}{\text{sym,2}}(\rho,k)}{\partial k}|_{\_}{k_{\_}{\text{F}}}k_{\_}{\text{F}}+\frac{1}{6}\frac{\partial^{2}U_{\_}{\text{sym,2}}(\rho,k)}{\partial k^{2}}|_{\_}{k_{\_}{\text{F}}}k_{\_}{\text{F}}^{2}
+∂U_​sym,3​(ρ,k)∂k|_​k_​F​k_​F+94​U_​sym,3​(ρ,k_​F)+3​U_​sym,4​(ρ,k_​F).\displaystyle+\frac{\partial U_{\_}{\text{sym,3}}(\rho,k)}{\partial k}|_{\_}{k_{\_}{\text{F}}}k_{\_}{\text{F}}+\frac{9}{4}U_{\_}{\text{sym,3}}(\rho,k_{\_}{\text{F}})+3U_{\_}{\text{sym,4}}(\rho,k_{\_}{\text{F}}). (8b)

Taking the derivative of the summation of Eqs. (2a) and (2b) with respect to ρ\rho and comparing the coefficients, the incompressibility coefficients of E_​0​(ρ)E_{\_}{0}(\rho), E_​sym,2​(ρ)E_{\_}{\text{sym,2}}(\rho), and E_​sym,4​(ρ)E_{\_}{\text{sym,4}}(\rho) are given as

K_​0​(ρ)=\displaystyle K_{\_}0(\rho)= 9​ρ​∂[t​(k_​F)+U_​0​(ρ,k_​F)]∂ρ−18​[t​(k_​F)+U_​0​(ρ,k_​F)]+18​E_​0​(ρ),\displaystyle 9\rho\frac{\partial[t(k_{\_}{\text{F}})+U_{\_}0(\rho,k_{\_}{\text{F}})]}{\partial\rho}-18[t(k_{\_}{\text{F}})+U_{\_}0(\rho,k_{\_}{\text{F}})]+18E_{\_}0(\rho), (9a)
K_​2​(ρ)=\displaystyle K_{\_}2(\rho)= −13​∂[t​(k)+U_​0​(ρ,k)]∂k|_​k_​F​k_​F+13​∂2[t​(k)+U_​0​(ρ,k)]∂k2|_​k_​F​k_​F2\displaystyle-\frac{1}{3}\frac{\partial[t(k)+U_{\_}0(\rho,k)]}{\partial k}|_{\_}{k_{\_}{\text{F}}}k_{\_}{\text{F}}+\frac{1}{3}\frac{\partial^{2}[t(k)+U_{\_}0(\rho,k)]}{\partial k^{2}}|_{\_}{k_{\_}{\text{F}}}k_{\_}{\text{F}}^{2}
−k_​F​ρ​∂∂ρ​∂[t​(k)+U_​0​(ρ,k)]∂k|_​k_​F+12​k_​F2​ρ​∂∂ρ​∂2[t​(k)+U_​0​(ρ,k)]∂k2|_​k_​F\displaystyle-k_{\_}{\text{F}}\rho\frac{\partial}{\partial\rho}\frac{\partial[t(k)+U_{\_}0(\rho,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,k)]}{\partial k^{2}}|_{\_}{k_{\_}{\text{F}}}
+∂U_​sym,1​(ρ,k)∂k|_​k_​F​k_​F+3​k_​F​ρ​∂∂ρ​∂U_​sym,1​(ρ,k)∂k|_​k_​F+9​ρ​∂U_​sym,2​(ρ,k_​F)∂ρ,\displaystyle+\frac{\partial U_{\_}{\text{sym,1}}(\rho,k)}{\partial k}|_{\_}{k_{\_}{\text{F}}}k_{\_}{\text{F}}+3k_{\_}{\text{F}}\rho\frac{\partial}{\partial\rho}\frac{\partial U_{\_}{\text{sym,1}}(\rho,k)}{\partial k}|_{\_}{k_{\_}{\text{F}}}+9\rho\frac{\partial U_{\_}{\text{sym,2}}(\rho,k_{\_}{\text{F}})}{\partial\rho}, (9b)
K_​4​(ρ)=\displaystyle K_{\_}4(\rho)= −5162​∂[t​(k)+U_​0​(ρ,k)]∂k|_​k_​F​k_​F+23162​∂2[t​(k)+U_​0​(ρ,k)]∂k2|_​k_​F​k_​F2\displaystyle-\frac{5}{162}\frac{\partial[t(k)+U_{\_}0(\rho,k)]}{\partial k}|_{\_}{k_{\_}{\text{F}}}k_{\_}{\text{F}}+\frac{23}{162}\frac{\partial^{2}[t(k)+U_{\_}0(\rho,k)]}{\partial k^{2}}|_{\_}{k_{\_}{\text{F}}}k_{\_}{\text{F}}^{2}
−112​∂3[t​(k)+U_​0​(ρ,k)]∂k3|_​k_​F​k_​F3+5324​∂4[t​(k)+U_​0​(ρ,k)]∂k4|_​k_​F​k_​F4\displaystyle-\frac{1}{12}\frac{\partial^{3}[t(k)+U_{\_}0(\rho,k)]}{\partial k^{3}}|_{\_}{k_{\_}{\text{F}}}k_{\_}{\text{F}}^{3}+\frac{5}{324}\frac{\partial^{4}[t(k)+U_{\_}0(\rho,k)]}{\partial k^{4}}|_{\_}{k_{\_}{\text{F}}}k_{\_}{\text{F}}^{4}
−1027​k_​F​ρ​∂∂ρ​∂[t​(k)+U_​0​(ρ,k)]∂k|_​k_​F+1354​k_​F2​ρ​∂∂ρ​∂2[t​(k)+U_​0​(ρ,k)]∂k2|_​k_​F\displaystyle-\frac{10}{27}k_{\_}{\text{F}}\rho\frac{\partial}{\partial\rho}\frac{\partial[t(k)+U_{\_}0(\rho,k)]}{\partial k}|_{\_}{k_{\_}{\text{F}}}+\frac{13}{54}k_{\_}{\text{F}}^{2}\rho\frac{\partial}{\partial\rho}\frac{\partial^{2}[t(k)+U_{\_}0(\rho,k)]}{\partial k^{2}}|_{\_}{k_{\_}{\text{F}}}
−118​k_​F3​ρ​∂∂ρ​∂3[t​(k)+U_​0​(ρ,k)]∂k3|_​k_​F+1216​k_​F4​ρ​∂∂ρ​∂4[t​(k)+U_​0​(ρ,k)]∂k4|_​k_​F\displaystyle-\frac{1}{18}k_{\_}{\text{F}}^{3}\rho\frac{\partial}{\partial\rho}\frac{\partial^{3}[t(k)+U_{\_}0(\rho,k)]}{\partial k^{3}}|_{\_}{k_{\_}{\text{F}}}+\frac{1}{216}k_{\_}{\text{F}}^{4}\rho\frac{\partial}{\partial\rho}\frac{\partial^{4}[t(k)+U_{\_}0(\rho,k)]}{\partial k^{4}}|_{\_}{k_{\_}{\text{F}}}
−1154​∂U_​sym,1​(ρ,k)∂k|_​k_​F​k_​F−536​∂2U_​sym,1​(ρ,k)∂k2|_​k_​F​k_​F2+16​∂3U_​sym,1​(ρ,k)∂k3|_​k_​F​k_​F3\displaystyle-\frac{11}{54}\frac{\partial U_{\_}{\text{sym,1}}(\rho,k)}{\partial k}|_{\_}{k_{\_}{\text{F}}}k_{\_}{\text{F}}-\frac{5}{36}\frac{\partial^{2}U_{\_}{\text{sym,1}}(\rho,k)}{\partial k^{2}}|_{\_}{k_{\_}{\text{F}}}k_{\_}{\text{F}}^{2}+\frac{1}{6}\frac{\partial^{3}U_{\_}{\text{sym,1}}(\rho,k)}{\partial k^{3}}|_{\_}{k_{\_}{\text{F}}}k_{\_}{\text{F}}^{3}
+59​k_​F​ρ​∂∂ρ​∂U_​sym,1​(ρ,k)∂k|_​k_​F−13​k_​F2​ρ​∂∂ρ​∂2U_​sym,1​(ρ,k)∂k2|_​k_​F+118​k_​F3​ρ​∂∂ρ​∂3U_​sym,1​(ρ,k)∂k3|_​k_​F\displaystyle+\frac{5}{9}k_{\_}{\text{F}}\rho\frac{\partial}{\partial\rho}\frac{\partial U_{\_}{\text{sym,1}}(\rho,k)}{\partial k}|_{\_}{k_{\_}{\text{F}}}-\frac{1}{3}k_{\_}{\text{F}}^{2}\rho\frac{\partial}{\partial\rho}\frac{\partial^{2}U_{\_}{\text{sym,1}}(\rho,k)}{\partial k^{2}}|_{\_}{k_{\_}{\text{F}}}+\frac{1}{18}k_{\_}{\text{F}}^{3}\rho\frac{\partial}{\partial\rho}\frac{\partial^{3}U_{\_}{\text{sym,1}}(\rho,k)}{\partial k^{3}}|_{\_}{k_{\_}{\text{F}}}
+136​∂U_​sym,2​(ρ,k)∂k|_​k_​F​k_​F+43​∂2U_​sym,2​(ρ,k)∂k2|_​k_​F​k_​F2\displaystyle+\frac{13}{6}\frac{\partial U_{\_}{\text{sym,2}}(\rho,k)}{\partial k}|_{\_}{k_{\_}{\text{F}}}k_{\_}{\text{F}}+\frac{4}{3}\frac{\partial^{2}U_{\_}{\text{sym,2}}(\rho,k)}{\partial k^{2}}|_{\_}{k_{\_}{\text{F}}}k_{\_}{\text{F}}^{2}
−k_​F​ρ​∂∂ρ​∂U_​sym,2​(ρ,k)∂k|_​k_​F+12​k_​F2​ρ​∂∂ρ​∂2U_​sym,2​(ρ,k)∂k2|_​k_​F\displaystyle-k_{\_}{\text{F}}\rho\frac{\partial}{\partial\rho}\frac{\partial U_{\_}{\text{sym,2}}(\rho,k)}{\partial k}|_{\_}{k_{\_}{\text{F}}}+\frac{1}{2}k_{\_}{\text{F}}^{2}\rho\frac{\partial}{\partial\rho}\frac{\partial^{2}U_{\_}{\text{sym,2}}(\rho,k)}{\partial k^{2}}|_{\_}{k_{\_}{\text{F}}}
+272​U_​sym,3​(ρ,k_​F)+7​∂U_​sym,3​(ρ,k)∂k|_​k_​F​k_​F+3​k_​F​ρ​∂∂ρ​∂U_​sym,3​(ρ,k)∂k|_​k_​F\displaystyle+\frac{27}{2}U_{\_}{\text{sym,3}}(\rho,k_{\_}{\text{F}})+7\frac{\partial U_{\_}{\text{sym,3}}(\rho,k)}{\partial k}|_{\_}{k_{\_}{\text{F}}}k_{\_}{\text{F}}+3k_{\_}{\text{F}}\rho\frac{\partial}{\partial\rho}\frac{\partial U_{\_}{\text{sym,3}}(\rho,k)}{\partial k}|_{\_}{k_{\_}{\text{F}}}
+18​U_​sym,4​(ρ,k_​F)+9​ρ​∂U_​sym,4​(ρ,k_​F)∂ρ.\displaystyle+18U_{\_}{\text{sym,4}}(\rho,k_{\_}{\text{F}})+9\rho\frac{\partial U_{\_}{\text{sym,4}}(\rho,k_{\_}{\text{F}})}{\partial\rho}. (9c)

Similarly, taking the second derivative of Eq. (6) gives the skewness of E_​0​(ρ)E_{\_}0(\rho) as follows

J_​0​(ρ)=27​ρ2​∂2[t​(k_​F)+U_​0​(ρ,k_​F)]∂ρ2−81​ρ​∂[t​(k_​F)+U_​0​(ρ,k_​F)]∂ρ+162​[t​(k_​F)+U_​0​(ρ,k_​F)]−162​E_​0​(ρ).\displaystyle J_{\_}0(\rho)=27\rho^{2}\frac{\partial^{2}[t(k_{\_}{\text{F}})+U_{\_}0(\rho,k_{\_}{\text{F}})]}{\partial\rho^{2}}-81\rho\frac{\partial[t(k_{\_}{\text{F}})+U_{\_}0(\rho,k_{\_}{\text{F}})]}{\partial\rho}+162[t(k_{\_}{\text{F}})+U_{\_}0(\rho,k_{\_}{\text{F}})]-162E_{\_}0(\rho). (10)

II.3 The exact saturation density ρ_​sat\rho_{\_}{\text{sat}} as a function of isospin asymmetry

For isospin asymmetric nuclear matter, the saturation density is different from that of the symmetric nuclear matter ρ_​0\rho_{\_}0. The former is defined as the exact saturation density and can be also written as a function of the isospin asymmetry δ\delta LWC09

ρ_​sat​(δ)=ρ_​0+ρ_​sat,2​δ2+ρ_​sat,4​δ4+O​(δ6).\displaystyle\rho_{\_}{\text{sat}}(\delta)=\rho_{\_}{0}+\rho_{\_}{\text{sat,2}}\delta^{2}+\rho_{\_}{\text{sat,4}}\delta^{4}+O(\delta^{6}). (11)

For symmetric nuclear matter with δ=0\delta=0, the ρ_​sat​(δ)\rho_{\_}{\text{sat}}(\delta) is reduced to ρ_​0\rho_{\_}0. According to the property of the saturation point ∂E​(ρ,δ)∂ρ|_​ρ_​sat​(δ)=0\frac{\partial E(\rho,\delta)}{\partial\rho}|_{\_}{\rho_{\_}{\text{sat}}(\delta)}=0 and expanding the EoS in terms of χ\chi, the exact saturation density can be expressed as

ρ_​sat​(δ)=ρ_​0−3​L_​2​(ρ_​0)K_​0​(ρ_​0)​ρ_​0⋅δ2+[3​K_​2​(ρ_​0)​L_​2​(ρ_​0)K_​0​(ρ_​0)2−3​L_​4​(ρ_​0)K_​0​(ρ_​0)−3​J_​0​(ρ_​0)​L_​22​(ρ_​0)2​K_​0​(ρ_​0)3]​ρ_​0⋅δ4.\displaystyle\rho_{\_}{\text{sat}}(\delta)=\rho_{\_}0-\frac{3L_{\_}2(\rho_{\_}0)}{K_{\_}0(\rho_{\_}0)}\rho_{\_}0\cdot\delta^{2}+[\frac{3K_{\_}{2}(\rho_{\_}0)L_{\_}2(\rho_{\_}0)}{K_{\_}0(\rho_{\_}0)^{2}}-\frac{3L_{\_}4(\rho_{\_}0)}{K_{\_}0(\rho_{\_}0)}-\frac{3J_{\_}0(\rho_{\_}0)L_{\_}2^{2}(\rho_{\_}0)}{2K_{\_}0(\rho_{\_}0)^{3}}]\rho_{\_}0\cdot\delta^{4}. (12)

At the exact saturation density ρ_​sat​(δ)\rho_{\_}{\text{sat}}(\delta), the energy per nucleon of asymmetric nuclear matter is given by

E_​sat​(δ)=E​(ρ_​sat​(δ),δ)=\displaystyle E_{\_}{\text{sat}}(\delta)=E(\rho_{\_}{\text{sat}}(\delta),\delta)= E_​0​(ρ_​0)+E_​sym,2​(ρ_​0)​δ2+[E_​sym,4​(ρ_​0)−L_​22​(ρ_​0)2​K_​0​(ρ_​0)]​δ4\displaystyle E_{\_}0(\rho_{\_}0)+E_{\_}{\text{sym,2}}(\rho_{\_}0)\delta^{2}+[E_{\_}{\text{sym,4}}(\rho_{\_}0)-\frac{L_{\_}2^{2}(\rho_{\_}0)}{2K_{\_}0(\rho_{\_}0)}]\delta^{4}
=\displaystyle= E_​s​a​t,0+E_​sat,2​δ2+E_​sat,4​δ4.\displaystyle E_{\_}{sat,0}+E_{\_}{\text{sat,2}}\delta^{2}+E_{\_}{\text{sat,4}}\delta^{4}. (13)

The corresponding incompressibility coefficient of the EoS is

K_​sat​(δ)=\displaystyle K_{\_}{\text{sat}}(\delta)= 9​ρ_2​sat​(δ)​∂2E​(ρ,δ)∂2ρ|_​ρ_​s​a​t​(δ)\displaystyle 9\rho^{2}_{\_}{\text{sat}}(\delta)\frac{\partial^{2}E(\rho,\delta)}{\partial^{2}\rho}|_{\_}{\rho_{\_}{sat(\delta)}}
=\displaystyle= K_​0​(ρ_​0)+[K_​2​(ρ_​0)−6​L_​2​(ρ_​0)−J_​0​(ρ_​0)K_​0​(ρ_​0)​L_​2​(ρ_​0)]​δ2+O​[δ4]\displaystyle K_{\_}0(\rho_{\_}0)+[K_{\_}{2}(\rho_{\_}0)-6L_{\_}2(\rho_{\_}0)-\frac{J_{\_}0(\rho_{\_}0)}{K_{\_}0(\rho_{\_}0)}L_{\_}2(\rho_{\_}0)]\delta^{2}+O[\delta^{4}]
=\displaystyle= K_​s​a​t,0+K_​sat,2​δ2+O​[δ4].\displaystyle K_{\_}{sat,0}+K_{\_}{\text{sat,2}}\delta^{2}+O[\delta^{4}]. (14)

It is clearly shown that the quartic symmetry energy at the exact saturation density is E_​sat,4=E_​sym,4​(ρ_​0)−L_​22​(ρ_​0)2​K_​0​(ρ_​0)E_{\_}{\text{sat,4}}=E_{\_}{\text{sym,4}}(\rho_{\_}0)-\frac{L_{\_}2^{2}(\rho_{\_}0)}{2K_{\_}0(\rho_{\_}0)}, and the quadratic incompressibility coefficient is

K_​sat,2=K_​2​(ρ_​0)−6​L_​2​(ρ_​0)−J_​0​(ρ_​0)K_​0​(ρ_​0)​L_​2​(ρ_​0).\displaystyle K_{\_}{\text{sat,2}}=K_{\_}{2}(\rho_{\_}0)-6L_{\_}2(\rho_{\_}0)-\frac{J_{\_}0(\rho_{\_}0)}{K_{\_}0(\rho_{\_}0)}L_{\_}2(\rho_{\_}0). (15)

In previous studies LiBA08 ; MLQ88 , the K_​sat,2K_{\_}{\text{sat,2}} is approximated as K_​sat,2→K_​asy,2=K_​2​(ρ_​0)−6​L_​2​(ρ_​0)K_{\_}{\text{sat,2}}\rightarrow K_{\_}{\text{asy,2}}=K_{\_}{2}(\rho_{\_}0)-6L_{\_}2(\rho_{\_}0) by neglecting the −J_​0​(ρ_​0)K_​0​(ρ_​0)​L_​2​(ρ_​0)-\frac{J_{\_}0(\rho_{\_}0)}{K_{\_}0(\rho_{\_}0)}L_{\_}2(\rho_{\_}0) term for simplicity. We will discuss its effect on K_​sat,2K_{\_}{\text{sat,2}} in the following section.

III 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 ρ_​0\rho_{\_}0 by using 2525 interaction parameter sets, which include two BGBD interactions with different neutron-proton effective masses CG87 ; IB91 ; CBD03 ; JR04 , the MDI interaction with x=−1x=-1, 0, and 11 CBD03 ; LiBA04a ; LiBA04b ; LWC05c , the 1616 Skyrme interactions MD12 ; GSkI ; KDE0v1 ; LNS ; MSL0 ; AWS05 ; SKRA ; SkT1 ; Skxs20 ; SQMC650 ; SV-sym32 , and four Gogny interactions JFB91 ; FC08 ; SG09 . 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 total 2525 interaction parameter sets are summarized in Table 1. The average values of the basic quantities in the 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 E_​0​(ρ_​0)E_{\_}{0}(\rho_{\_}{0}), K_​0​(ρ_​0)K_{\_}0(\rho_{\_}{0}), E_​sym,2​(ρ_​0)E_{\_}{\text{sym,2}}(\rho_{\_}{0}), and L_​2​(ρ_​0)L_{\_}{2}(\rho_{\_}{0}) are consistent with the constraints extracted from both theoretical calculations and experimental data LiBA08 ; SS06 ; LiBA13 ; MOer17 . Interestingly, the averaged E_​sym,4​(ρ_​0)E_{\_}{\text{sym,4}}(\rho_{\_}{0}) value is almost the same as the constraint in Ref. LWC09 . 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 E_​0​(ρ_​0)E_{\_}{0}(\rho_{\_}{0}) and K_​0​(ρ_​0)K_{\_}{0}(\rho_{\_}{0}) are well constrained in a narrow range and the corresponding error bars are small. The error bar of skewness J_​0​(ρ_​0)=−411.3±37.0J_{\_}0(\rho_{\_}0)=-411.3\pm 37.0 MeV is relatively large, especially for the Gogny interactions. It is also noted that the skewness, together with K_​2​(ρ_​0)K_{\_}{2}(\rho_{\_}0), has recently received much attention in the calculations of the neutron stars’ maximum mass and the radius of canonical neutron stars LiBA19 ; NBZ18 ; WJX19 . The error bars of the high-order terms L_​4​(ρ_​0)L_{\_}{4}(\rho_{\_}0), K_​2​(ρ_​0)K_{\_}{2}(\rho_{\_}0) and K_​4​(ρ_​0)K_{\_}{4}(\rho_{\_}0) are also given, i.e. L_​4​(ρ_​0)=1.42±2.14L_{\_}{4}(\rho_{\_}0)=1.42\pm 2.14 MeV, K_​2​(ρ_​0)=−123.6±83.8K_{\_}{2}(\rho_{\_}0)=-123.6\pm 83.8 MeV, and K_​4​(ρ_​0)=−1.25±5.89K_{\_}{4}(\rho_{\_}0)=-1.25\pm 5.89 MeV. In addition, for the MDI interaction, the L_​2​(ρ_​0)L_{\_}{2}(\rho_{\_}0) and K_​2​(ρ_​0)K_{\_}{2}(\rho_{\_}0) values with different spin(isospin)-dependent parameter xx are scattered over a wide range. This is because the different choices of parameter xx are to simulate very different density dependences of the symmetry energies at high densities CBD03 ; LiBA04a ; LiBA04b .

Table 1: The saturation density ρ_​0\rho_{\_}{0} (fm-3) and basic quantities E_​0​(ρ_​0)E_{\_}{0}(\rho_{\_}{0}), K_​0​(ρ_​0)K_{\_}0(\rho_{\_}{0}), J_​0​(ρ_​0)J_{\_}{0}(\rho_{\_}{0}), E_​sym,2​(ρ_​0)E_{\_}{\text{sym,2}}(\rho_{\_}{0}), E_​sym,4​(ρ_​0)E_{\_}{\text{sym,4}}(\rho_{\_}{0}), L_​2​(ρ_​0)L_{\_}{2}(\rho_{\_}{0}), L_​4​(ρ_​0)L_{\_}4(\rho_{\_}{0}), K_​2​(ρ_​0)K_{\_}{2}(\rho_{\_}{0}), and K_​4​(ρ_​0)K_{\_}{4}(\rho_{\_}{0}) for totally 2525 interaction sets in four kinds of interactions. The units of these quantities were MeV. In the last three rows, the averaged values and constraints in previous studies are shown. All interactions were taken from Ref. CG87 ; IB91 ; CBD03 ; JR04 ; LiBA04a ; LiBA04b ; LWC05c ; MD12 ; GSkI ; KDE0v1 ; LNS ; MSL0 ; AWS05 ; SKRA ; SkT1 ; Skxs20 ; SQMC650 ; SV-sym32 ; JFB91 ; FC08 ; SG09 .
Force ρ_​0\rho_{\_}0 E_​0​(ρ_​0)E_{\_}0(\rho_{\_}0) K_​0​(ρ_​0)K_{\_}{0}(\rho_{\_}0) J_​0​(ρ_​0)J_{\_}{0}(\rho_{\_}0) E_​sym,2​(ρ_​0)E_{\_}{\text{sym,2}}(\rho_{\_}0) L_​2​(ρ_​0)L_{\_}{2}(\rho_{\_}0) K_​2​(ρ_​0)K_{\_}{2}(\rho_{\_}0) E_​sym,4​(ρ_​0)E_{\_}{\text{sym,4}}(\rho_{\_}0) L_​4​(ρ_​0)L_{\_}{4}(\rho_{\_}0) K_​4​(ρ_​0)K_{\_}{4}(\rho_{\_}0)
BGBD
Case-1 0.160 -15.8 215.9 -447.5 32.9 87.9 -32.7 1.72 6.82 7.14
Case-2 0.160 -15.8 215.9 -447.5 33.0 121.8 101.0 -0.73 -4.26 7.14
MDI
x=1x=1 0.160 -16.1 212.4 -447.3 30.5 14.7 -264.0 0.62 0.53 -4.83
x=0x=0 0.160 -16.1 212.4 -447.3 30.5 60.2 -81.7 0.62 0.53 -4.83
x=−1x=-1 0.160 -16.1 212.4 -447.3 30.5 105.8 100.6 0.62 0.53 -4.83
Skyrme
GSKI 0.159 -16.0 230.3 -405.7 32.0 63.5 -95.3 0.38 0.56 -1.61
GSKII 0.159 -16.1 234.1 -400.2 30.5 48.6 -158.3 0.92 3.26 3.80
KDE0v1 0.165 -16.2 228.4 -386.3 34.6 54.7 -127.4 0.46 0.92 -0.94
LNS 0.175 -15.3 211.5 -384.0 33.5 61.5 -127.7 0.82 2.67 2.44
MSL0 0.160 -16.0 230.0 -380.3 30.0 60.0 -99.3 0.81 2.70 2.66
NRAPR 0.161 -15.9 226.6 -364.1 32.8 59.7 -123.7 0.96 3.41 4.09
Ska25s20 0.161 -16.1 221.5 -415.0 34.2 65.1 -118.2 0.46 0.93 0.88
Ska35s20 0.158 -16.1 240.3 -378.6 33.5 64.4 -120.9 0.45 0.90 -0.90
SKRA 0.159 -15.8 216.1 -377.2 31.3 53.0 -138.8 0.95 3.39 4.07
SkT1 0.161 -16.0 236.1 -383.5 32.0 56.2 -134.8 0.46 0.91 -0.91
SkT2 0.161 -15.9 235.7 -382.6 32.0 56.2 -134.7 0.46 0.91 -0.91
SkT3 0.161 -15.9 235.7 -382.7 31.5 55.3 -132.1 0.46 0.91 -0.91
Skxs20 0.162 -15.8 202.4 -426.5 35.5 67.1 -122.5 0.53 1.27 -0.22
SQMC650 0.172 -15.6 218.2 -376.9 33.7 52.9 -173.2 1.05 3.82 4.77
SQMC700 0.171 -15.5 220.7 -369.9 33.5 59.1 -140.8 0.97 3.44 4.03
SV-sym32 0.159 -15.9 232.8 -378.3 31.9 57.0 -148.2 0.89 3.11 3.50
Gogny
D1 0.166 -16.4 227.2 -446.9 30.7 18.6 -273.6 0.76 1.75 -1.78
D1S 0.163 -16.0 201.8 -508.4 31.1 22.5 -241.0 0.44 -0.51 -7.56
D1N 0.161 -16.0 224.5 -430.9 29.6 33.6 -168.2 0.21 -1.95 -11.80
D1M 0.165 -16.0 226.2 -466.9 28.6 24.8 -133.3 0.69 -1.05 -20.81
Average 0.162 -15.94 222.8 -411.3 32.0 57.0 -123.6 0.64 1.42 -1.25
Constraint -16 240 31.7 58.7 0.62
Ref. LiBA08 SS06 LiBA13 ; MOer17 LiBA13 ; MOer17 LWC09
Refer to caption
Figure 2: (Color online) Values of basic quantities E_​0​(ρ_​0)E_{\_}0(\rho_{\_}0), K_​0​(ρ_​0)K_{\_}0(\rho_{\_}0), and J_​0​(ρ_​0)J_{\_}0(\rho_{\_}0) for symmetric nuclear matter at 2525 parameter sets of the BGBD, MDI, Skyrme, and Gogny interactions. The solid and dashed lines represent the average values and their deviations, respectively.
Refer to caption
Figure 3: (Color online) Values of E_​sym,2​(ρ_​0)E_{\_}{\text{sym,2}}(\rho_{\_}0), L_​2​(ρ_​0)L_{\_}2(\rho_{\_}0), K_​2​(ρ_​0)K_{\_}2(\rho_{\_}0), E_​sym,4​(ρ_​0)E_{\_}{\text{sym,4}}(\rho_{\_}0), L_​4​(ρ_​0)L_{\_}4(\rho_{\_}0), and K_​4​(ρ_​0)K_{\_}4(\rho_{\_}0) for asymmetric nuclear matter within 2525 parameter sets of four kinds of interactions.
Refer to caption
Figure 4: (Color online) The magnitudes of E_​0​(ρ)E_{\_}{0}(\rho), E_​sym,2​(ρ)​δ2E_{\_}{\text{sym,2}}(\rho)\delta^{2}, and E_​sym,4​(ρ)​δ4E_{\_}{\text{sym,4}}(\rho)\delta^{4} in the EoS at two different ρ\rho values and three different δ2\delta^{2} values. The NRAPR Skyrme interaction was applied.
Refer to caption
Figure 5: (Color online) The magnitude of each order in E_​0​(ρ)E_{\_}{0}(\rho), E_​sym,2​(ρ)E_{\_}{\text{sym,2}}(\rho), and E_​sym,4​(ρ)E_{\_}{\text{sym,4}}(\rho) expressed by E_​0​(ρ_​0)E_{\_}{0}(\rho_{\_}0), K_​0​(ρ_​0)K_{\_}{0}(\rho_{\_}0) and J_​0​(ρ_​0)J_{\_}{0}(\rho_{\_}0), E_​sym,2​(ρ_​0)E_{\_}{\text{sym,2}}(\rho_{\_}0), L_​2​(ρ_​0)L_{\_}{2}(\rho_{\_}0) and K_​2​(ρ_​0)K_{\_}{2}(\rho_{\_}0), and E_​sym,4​(ρ_​0)E_{\_}{\text{sym,4}}(\rho_{\_}0), L_​4​(ρ_​0)L_{\_}{4}(\rho_{\_}0) and K_​4​(ρ_​0)K_{\_}{4}(\rho_{\_}0), respectively. The NRAPR Skyrme interaction was applied.

In Fig. 4, we show the magnitudes of the separated terms E_​0​(ρ)E_{\_}{0}(\rho), E_​sym,2​(ρ)​δ2E_{\_}{\text{sym,2}}(\rho)\delta^{2}, E_​sym,4​(ρ)​δ4E_{\_}{\text{sym,4}}(\rho)\delta^{4} as well as the total one E​(ρ,δ)E(\rho,\delta) at two different densities (ρ_​0\rho_{\_}0 and 2​ρ_​02\rho_{\_}0) and three different isospin asymmetries (δ2\delta^{2}=0.10.1, 0.20.2 and 0.50.5) by taking the NRAPR Skyrme interaction as an example. At the saturation density ρ_​0\rho_{\_}0 (see graphs (a)-(c)), the contribution of E_​0​(ρ)E_{\_}{0}(\rho) to E​(ρ,δ)E(\rho,\delta) is dominant. The contribution of E_​sym,2​(ρ)​δ2E_{\_}{\text{sym,2}}(\rho)\delta^{2} increases with an increase in isospin asymmetry δ\delta. It is also shown that the contribution from E_​sym,4​(ρ)​δ4E_{\_}{\text{sym,4}}(\rho)\delta^{4} is small and comes into play at large isospin asymmetry with δ2=0.5\delta^{2}=0.5. At 2​ρ_​02\rho_{\_}0 (see graphs (d)-(f)), E_​0​(ρ)E_{\_}{0}(\rho) contribution is suppressed compared with that at ρ_​0\rho_{\_}0, while E_​sym,2​(ρ)​δ2E_{\_}{\text{sym,2}}(\rho)\delta^{2} plays a more important role in the EoS, especially at δ2=0.5\delta^{2}=0.5. It should also be noted that the E_​sym,4​(ρ)E_{\_}{\text{sym,4}}(\rho) contributes only at very high density and large isospin asymmetry. The magnitude of E_​sym,4​(ρ)E_{\_}{\text{sym,4}}(\rho) can significantly affect the calculation of the proton fraction in neutron stars at β\beta-equilibrium JXu09 ; CXu11 .

We further expand E_​0​(ρ)E_{\_}{0}(\rho), E_​sym,2​(ρ)E_{\_}{\text{sym,2}}(\rho), and E_​sym,4​(ρ)E_{\_}{\text{sym,4}}(\rho) as a series of χ\chi with their corresponding slopes and the incompressibility coefficients. In Fig. 5, we depict the contributions from each term at different densities 0.5​ρ_​00.5\rho_{\_}0, 2​ρ_​02\rho_{\_}0 and 3​ρ_​03\rho_{\_}0. As can be seen from Fig. 5, the first-order terms E_0​0E^{0}_{\_}0 (E_​0​(ρ_​0)E_{\_}0(\rho_{\_}0)), E_0​2E^{0}_{\_}2 (E_​sym,2​(ρ_​0)E_{\_}{\text{sym,2}}(\rho_{\_}0)), and E_0​4E^{0}_{\_}4 (E_​sym,4​(ρ_​0)E_{\_}{\text{sym,4}}(\rho_{\_}0)) contribute largely at all densities. E_​0KE_{\_}{0}^{\text{K}} and E_​0JE_{\_}{0}^{\text{J}} terms become increasingly important with increasing density. For E_​sym,2​(ρ)E_{\_}{\text{sym,2}}(\rho) and E_​sym,4​(ρ)E_{\_}{\text{sym,4}}(\rho) at 3​ρ_​03\rho_{\_}0, the contributions from their slopes (E_​2LE_{\_}2^{\text{L}} and E_​4LE_{\_}4^{\text{L}}) and the incompressibility coefficients (E_​2KE_{\_}2^{\text{K}} and E_​4KE_{\_}4^{\text{K}}) are much larger than those at 0.5​ρ_​00.5\rho_{\_}0 and 2​ρ_​02\rho_{\_}0. In particular, the E_​0JE_{\_}0^{\text{J}}, E_​2KE_{\_}2^{\text{K}}, and E_​4KE_{\_}4^{\text{K}} terms at 3​ρ_​03\rho_{\_}0 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 the neutron stars.

Refer to caption
Figure 6: (Color online) The single-nucleon potential decomposition of E_​0​(ρ_​0)E_{\_}{0}(\rho_{\_}0), K_​0​(ρ_​0)K_{\_}{0}(\rho_{\_}0), J_​0​(ρ_​0)J_{\_}{0}(\rho_{\_}0), E_​sym,2​(ρ_​0)E_{\_}{\text{sym,2}}(\rho_{\_}0), L_​2​(ρ_​0)L_{\_}{2}(\rho_{\_}0), K_​2​(ρ_​0)K_{\_}{2}(\rho_{\_}0), E_​sym,4​(ρ_​0)E_{\_}{\text{sym,4}}(\rho_{\_}0), L_​4​(ρ_​0)L_{\_}{4}(\rho_{\_}0), and K_​4​(ρ_​0)K_{\_}{4}(\rho_{\_}0). The NRAPR Skyrme interaction was applied.
Refer to caption
Figure 7: (Color online) The density-dependence of U_​0​(ρ,k_​F)U_{\_}{0}(\rho,k_{\_}{\text{F}}), U_​sym,1​(ρ,k_​F)U_{\_}{\text{sym,1}}(\rho,k_{\_}{\text{F}}), U_​sym,2​(ρ,k_​F)U_{\_}{\text{sym,2}}(\rho,k_{\_}{\text{F}}), U_​sym,3​(ρ,k_​F)U_{\_}{\text{sym,3}}(\rho,k_{\_}{\text{F}}), and U_​sym,4​(ρ,k_​F)U_{\_}{\text{sym,4}}(\rho,k_{\_}{\text{F}}). The NRAPR Skyrme interaction was applied.

More interestingly, the basic quantities at the saturation density are decomposed into the kinetic energy t​(k)t(k) and the symmetric and asymmetric parts of the single-nucleon potential U_​0​(ρ,k)U_{\_}0(\rho,k) and U_​sym,i​(ρ,k)U_{\_}{\text{sym,i}}(\rho,k). As shown in Fig. 6, the contributions from different terms t​(k)t(k), U_​0​(ρ,k)U_{\_}0(\rho,k) and U_​sym,i​(ρ,k)​(i=1,2,3,4)U_{\_}{\text{sym,i}}(\rho,k)(i=1,2,3,4) are denoted by superscripts of TT, U​0U0, U​1U1, U​2U2, U​3U3 and U​4U4, respectively. It is clear that the E_​0​(ρ_​0)E_{\_}0(\rho_{\_}0), K_​0​(ρ_​0)K_{\_}0(\rho_{\_}0), and J_​0​(ρ_​0)J_{\_}0(\rho_{\_}0) are completely determined by t​(k)t(k) and U_​0​(ρ,k)U_{\_}0(\rho,k). For other quantities, the contributions from the asymmetric parts U_​sym,1​(ρ,k)U_{\_}{\text{sym,1}}(\rho,k), U_​sym,2​(ρ,k)U_{\_}{\text{sym,2}}(\rho,k), U_​sym,3​(ρ,k)U_{\_}{\text{sym,3}}(\rho,k), and U_​sym,4​(ρ,k)U_{\_}{\text{sym,4}}(\rho,k) cannot be neglected. It is clearly shown that the first-order term U_​sym,1​(ρ,k)U_{\_}{\text{sym,1}}(\rho,k) contributes to all the six basic quantities. The second-order term U_​sym,2​(ρ,k)U_{\_}{\text{sym,2}}(\rho,k) does not contribute to E_​sym,2​(ρ_​0)E_{\_}{\text{sym,2}}(\rho_{\_}0), but to its corresponding slope L_​2​(ρ_​0)L_{\_}{2}(\rho_{\_}0) and the incompressibility coefficient K_​2​(ρ_​0)K_{\_}{2}(\rho_{\_}0). In principle, the U_​sym,2​(ρ,k)U_{\_}{\text{sym,2}}(\rho,k) term should also contribute to the fourth-order terms E_​sym,4​(ρ_​0)E_{\_}{\text{sym,4}}(\rho_{\_}0), L_​4​(ρ_​0)L_{\_}{4}(\rho_{\_}0), and K_​4​(ρ_​0)K_{\_}{4}(\rho_{\_}0), but for the Skyrme interaction, U_​sym,2​(ρ,k)U_{\_}{\text{sym,2}}(\rho,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)U_{\_}{\text{sym,3}}(\rho,k) and U_​sym,4​(ρ,k)U_{\_}{\text{sym,4}}(\rho,k) to the basic quantities. In Fig. 7, we show the density-dependence of U_​0​(ρ,k_​F)U_{\_}{0}(\rho,k_{\_}{\text{F}}), U_​sym,1​(ρ,k_​F)U_{\_}{\text{sym,1}}(\rho,k_{\_}{\text{F}}), U_​sym,2​(ρ,k_​F)U_{\_}{\text{sym,2}}(\rho,k_{\_}{\text{F}}), U_​sym,3​(ρ,k_​F)U_{\_}{\text{sym,3}}(\rho,k_{\_}{\text{F}}), and U_​sym,4​(ρ,k_​F)U_{\_}{\text{sym,4}}(\rho,k_{\_}{\text{F}}) at the Fermi momentum k_​F=(3​π2​ρ/2)1/3k_{\_}{\text{F}}=(3\pi^{2}\rho/2)^{1/3} by using the NRAPR Skyrme interaction. It can be clearly seen from Fig. 7 that the magnitudes of U_​0​(ρ,k_​F)U_{\_}{0}(\rho,k_{\_}{\text{F}}) and U_​sym,1​(ρ,k_​F)U_{\_}{\text{sym,1}}(\rho,k_{\_}{\text{F}}) are generally very large, while the ones of U_​sym,2​(ρ,k_​F)U_{\_}{\text{sym,2}}(\rho,k_{\_}{\text{F}}), U_​sym,3​(ρ,k_​F)U_{\_}{\text{sym,3}}(\rho,k_{\_}{\text{F}}) and U_​sym,4​(ρ,k_​F)U_{\_}{\text{sym,4}}(\rho,k_{\_}{\text{F}}) are very small but increase with the increasing density. Our results indicate that the U_​sym,3​(ρ,k)U_{\_}{\text{sym,3}}(\rho,k) and U_​sym,4​(ρ,k)U_{\_}{\text{sym,4}}(\rho,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.

Table 2: The calculated values of expansion coefficients ρ_​0\rho_{\_}0 (fm-3), ρ_​sat,2\rho_{\_}{\text{sat,2}} (fm-3), ρ_​sat,4\rho_{\_}{\text{sat,4}} (fm-3), the quartic symmetry energy E_​sat,4E_{\_}{\text{sat,4}} (MeV), the quadratic incompressibility coefficient K_​sat,2K_{\_}{\text{sat,2}} (MeV), and its two main components K_​asy,2K_{\_}{\text{asy,2}} (MeV) and J_​0​(ρ_​0)/K_​0​(ρ_​0)J_{\_}0(\rho_{\_}0)/K_{\_}0(\rho_{\_}0). In the last three rows, the averaged values and constraints in previous studies are shown.
Force ρ_​0\rho_{\_}0 ρ_​sat,2\rho_{\_}{\text{sat,2}} ρ_​sat,4\rho_{\_}{\text{sat,4}} E_​sat,4E_{\_}{\text{sat,4}} K_​asy,2K_{\_}{\text{asy,2}} K_​sat,2K_{\_}{\text{sat,2}} J_​0​(ρ_​0)/K_​0​(ρ_​0)J_{\_}0(\rho_{\_}0)/K_{\_}0(\rho_{\_}0)
BGBD
Case-1 0.160 -0.195 0.038 -16.17 -560.1 -377.9 -2.07
Case-2 0.160 -0.271 0.295 -35.11 -630.0 -377.5 -2.07
MDI
x=1x=1 0.160 -0.033 -0.040 0.11 -352.2 -321.2 -2.11
x=0x=0 0.160 -0.136 -0.013 -7.91 -442.9 -316.1 -2.11
x=−1x=-1 0.160 -0.239 0.237 -25.73 -534.2 -311.4 -2.11
Skyrme
GSKI 0.159 -0.131 -0.024 -8.36 -476.03 -364.23 -1.76
GSKII 0.159 -0.099 -0.056 -4.12 -450.04 -366.94 -1.71
KDE0v1 0.165 -0.119 -0.044 -6.09 -455.71 -363.13 -1.69
LNS 0.175 -0.153 -0.059 -8.12 -496.75 -385.10 -1.82
MSL0 0.160 -0.125 -0.033 -7.01 -459.33 -360.11 -1.65
NRAPR 0.161 -0.127 -0.050 -6.90 -481.82 -385.91 -1.61
Ska25s20 0.161 -0.142 -0.039 -9.11 -508.89 -386.89 -1.87
Ska35s20 0.158 -0.127 -0.039 -8.19 -507.47 -405.95 -1.58
SKRA 0.159 -0.117 -0.058 -5.55 -456.89 -364.36 -1.75
SkT1 0.161 -0.115 -0.045 -6.23 -471.90 -380.66 -1.62
SkT2 0.161 -0.115 -0.045 -6.23 -471.62 -380.45 -1.62
SkT3 0.161 -0.113 -0.044 -6.03 -463.93 -374.14 -1.62
Skxs20 0.162 -0.161 -0.044 -10.60 -525.16 -383.74 -2.11
SQMC650 0.172 -0.125 -0.082 -5.37 -490.78 -399.34 -1.73
SQMC700 0.171 -0.137 -0.065 -6.93 -495.14 -396.16 -1.68
SV-sym32 0.159 -0.117 -0.057 -6.10 -490.44 -397.74 -1.62
Gogny
D1 0.166 -0.041 -0.050 0.001 -385.2 -348.6 -1.97
D1S 0.163 -0.055 -0.056 -0.81 -376.0 -319.3 -2.52
D1N 0.161 -0.072 -0.039 -2.30 -369.8 -305.3 -1.92
D1M 0.165 -0.054 -0.023 -0.67 -282.1 -230.9 -2.06
Average 0.162 -0.125 -0.017 -7.98 -465.4 -360.1 -1.86
Constraint -500 -370 / -550
Ref. LWC07 LWC09 / UGar07 ; TLi07
Refer to caption
Figure 8: (Color online) The isospin-dependence of the exact saturation density ρ_​sat​(δ)\rho_{\_}{\text{sat}}(\delta) within 1414 typical interaction parameter sets in graph (a) and the comparisons between the error bars of the quadratic incompressibility coefficients K_​2​(ρ_​0)K_{\_}2(\rho_{\_}0), K_​asy,2K_{\_}{\text{asy,2}}, and K_​sat,2K_{\_}{\text{sat,2}} calculated by using 2525 interaction parameter sets in graph (b).

By analyzing the isospin-dependence of the saturation properties of asymmetric nuclear matter, a number of important quantities are calculated using 2525 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,2K_{\_}{\text{asy,2}} and K_​sat,2K_{\_}{\text{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\rho_{\_}{\text{sat,2}}, one of the most important isospin-dependent parts of ρ_​sat​(δ)\rho_{\_}{\text{sat}}(\delta), has a negative value in all cases, and the fourth-order coefficient ρ_​sat,4\rho_{\_}{\text{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\delta (see graph (a) of Fig. 8). For the BGBD interaction (Case-2), the calculated value of ρ_​sat,4\rho_{\_}{\text{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 the isospin asymmetry δ\delta close to unit. For asymmetric nuclear matter at ρ_​sat​(δ)\rho_{\_}{\text{sat}}(\delta), the corresponding E_​sat,4E_{\_}{\text{sat,4}} values are rather diverse, which is considered to be important for the proton fraction in neutron stars.

As shown in graph (b) of Fig. 8, the results of K_​2​(ρ_​0)K_{\_}2(\rho_{\_}0), K_​asy,2K_{\_}{\text{asy,2}}, and K_​sat,2K_{\_}{\text{sat,2}} are given and their values are constrained to be K_​2=−123.6±83.8K_{\_}{2}=-123.6\pm 83.8 MeV, K_​asy,2=−465.4±70.0K_{\_}{\text{asy,2}}=-465.4\pm 70.0 MeV, and K_​sat,2=−360.1±39.0K_{\_}{\text{sat,2}}=-360.1\pm 39.0 MeV, respectively. The averaged K_​asy,2K_{\_}{\text{asy,2}} value is close to the previous theoretical constraint −500±50-500\pm 50 MeV given in Ref. LWC07 if the error bar is considered. In Table II, there are two previous constraints for K_​sat,2K_{\_}{\text{sat,2}}. One is K_​sat,2=−370±120K_{\_}{\text{sat,2}}=-370\pm 120 MeV from a modified Skyrme-like (MSL) model LWC09 , and the other is −550±100-550\pm 100 MeV by analyzing the measured data of the isotopic dependence of the giant monopole resonance (GMR) in the even-A Sn isotopes UGar07 ; TLi07 . Compared with these previous studies, it is clear that the K_​asy,2K_{\_}{\text{asy,2}} and K_​sat,2K_{\_}{\text{sat,2}} values remain uncertain and require more data to further constrain their values. In addition, as mentioned before, the term −J_​0​(ρ_​0)K_​0​(ρ_​0)​L_​2​(ρ_​0)-\frac{J_{\_}0(\rho_{\_}0)}{K_{\_}0(\rho_{\_}0)}L_{\_}2(\rho_{\_}0) in Eq. (15) is typically ignored for simplicity. However, it 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 J_​0​(ρ_​0)/K_​0​(ρ_​0)J_{\_}0(\rho_{\_}0)/K_{\_}0(\rho_{\_}0) is constrained in the range of −1.86±0.23-1.86\pm 0.23. Finally we obtain a simple relation for K_​sat,2K_{\_}{\text{sat,2}}

K_​sat,2=K_​2​(ρ_​0)−4.14​L_​2​(ρ_​0).K_{\_}{\text{\text{sat,2}}}=K_{\_}2(\rho_{\_}0)-4.14L_{\_}2(\rho_{\_}0). (16)

With the averaged results L_​2​(ρ_​0)=57.0L_{\_}2(\rho_{\_}0)=57.0 MeV and K_​2​(ρ_​0)=−123.6K_{\_}2(\rho_{\_}0)=-123.6 MeV, the calculated value K_​sat,2=−359.6K_{\_}{\text{\text{sat,2}}}=-359.6 MeV is in good agreement with the average value of −360.1±39.0-360.1\pm 39.0 MeV from the 2525 interaction sets. This simple empirical relation could be useful for estimating the value of K_​sat,2K_{\_}{\text{\text{sat,2}}} for asymmetric nuclear matter.

IV 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)t(k), the symmetric and asymmetric parts of global optical potential U_​0​(ρ,k)U_{\_}0(\rho,k) and U_​sym,i​(ρ,k)U_{\_}{\text{sym,i}}(\rho,k). The analytical expressions of coefficients K_​2​(ρ)K_{\_}2(\rho) and K_​4​(ρ)K_{\_}4(\rho) are given for the fist time. By using 2525 kinds of interaction sets, the values of these quantities were systematically calculated at the saturation density ρ_​0\rho_{\_}0. It is emphasized that there are very few studies on quantities L_​4​(ρ_​0)L_{\_}4(\rho_{\_}0), K_​2​(ρ_​0)K_{\_}2(\rho_{\_}0) and K_​4​(ρ_​0)K_{\_}4(\rho_{\_}0) and their average values from a total of 2525 interaction sets are L_​4​(ρ_​0)=1.42±2.14L_{\_}4(\rho_{\_}0)=1.42\pm 2.14 MeV, K_​2​(ρ_​0)=−123.6±83.8K_{\_}2(\rho_{\_}0)=-123.6\pm 83.8 MeV, and K_​4​(ρ_​0)=−1.25±5.89K_{\_}4(\rho_{\_}0)=-1.25\pm 5.89 MeV, respectively. The averaged values of the other quantities weere 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 the saturation density. It is clearly shown that t​(k_​F)t(k_{\_}{\text{F}}) and U_​0​(ρ,k_​F)U_{\_}0(\rho,k_{\_}{\text{F}}) play vital roles in determining the EoS of both symmetric and asymmetric nuclear matter. For asymmetric nuclear matter, the U_​sym,1​(ρ,k)U_{\_}{\text{sym,1}}(\rho,k) contributes to all the quantities, whereas U_​sym,2​(ρ,k)U_{\_}{\text{sym,2}}(\rho,k) does not contribute to E_​sym,2​(ρ_​0)E_{\_}{\text{sym,2}}(\rho_{\_}0), but contributes to the second-order terms L_​2​(ρ_​0)L_{\_}{2}(\rho_{\_}0) and K_​2​(ρ_​0)K_{\_}{2}(\rho_{\_}0) as well as the fourth-order terms E_​sym,4​(ρ_​0)E_{\_}{\text{sym,4}}(\rho_{\_}0), L_​4​(ρ_​0)L_{\_}{4}(\rho_{\_}0), and K_​4​(ρ_​0)K_{\_}{4}(\rho_{\_}0). The contribution from U_​sym,3​(ρ,k)U_{\_}{\text{sym,3}}(\rho,k) cannot be neglected for E_​sym,4​(ρ_​0)E_{\_}{\text{sym,4}}(\rho_{\_}0), L_​4​(ρ_​0)L_{\_}{4}(\rho_{\_}0), and K_​4​(ρ_​0)K_{\_}{4}(\rho_{\_}0). U_​sym,4​(ρ,k)U_{\_}{\text{sym,4}}(\rho,k) should also be included in the calculations for L_​4​(ρ_​0)L_{\_}{4}(\rho_{\_}0) and K_​4​(ρ_​0)K_{\_}{4}(\rho_{\_}0). In addition, the quadratic incompressibility coefficient at ρ_​sat​(δ)\rho_{\_}{\text{sat}}(\delta) is found to have a simple empirical relation K_​sat,2=K_​2​(ρ_​0)−4.14​L_​2​(ρ_​0)K_{\_}{\text{sat,2}}=K_{\_}{2}(\rho_{\_}0)-4.14L_{\_}2(\rho_{\_}0) based on the present analysis.

ACKNOWLEDGEMENTS

This work is supported by the National Natural Science Foundation of China (Grant No. 11822503).

References