Jie Liu?Chao Gao?Niu Wan ?Chang Xu

Abstract Based on the Hugenholtz–Van Hove theorem,six basic quantities of the EoS in isospin asymmetric nuclear matter areexpressed in termsof thenucleon kinetic energy t(k),the isospin symmetric and asymmetric parts of the single-nucleon potentials U0（ρ，k）and Usym，i（ρ，k）.The six basic quantities include the quadratic symmetry energy Esym，2（ρ）,the quartic symmetry energy Esym，4（ρ）,their corresponding density slopes L2（ρ）and L4（ρ）,and the incompressibility coefficients K2（ρ）and K4（ρ）.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 densityρ0 are explicitly given.The contributions to these quantities from t(k),U0（ρ，k）,and Usym，i（ρ，k）are also analyzed at the normal nuclear densityρ0.It is clearly shown that the first-order asymmetric term Usym，1（ρ，k）(also known as the symmetry potential in the Lane potential)plays a vital role in determining the density dependence of the quadratic symmetry energy Esym，2（ρ）.It is also shown that the contributions from the high-order asymmetric partsof the single-nucleon potentials(Usym，i（ρ，k）with i＞ 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 ρsat（δ）,the corresponding quadratic incompressibility coefficient is found to have a simple empirical relation K sat，2=K2（ρ0）-4.14L2（ρ0）.

Keywords Equation of state.Symmetry energy.HVH theorem.Single-nucleon potential

1 Introduction

In the presentwork,we perform a systematic analysisof six basic quantities in the EoS based on the Hugenholtz–Van Hove(HVH)theorem [38],namely Esym，2（ρ）,Esym，4（ρ）,L2（ρ）,L4（ρ）,K2（ρ）,and K4（ρ）.Among them,the properties of Esym，2（ρ）,Esym，4（ρ）,and their slopes L2（ρ）and L4（ρ）were re-analyzed[39–43].The analytical expressionsof 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 interactionmodels:transportmodels 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 dynam ics(LQMD)model[51–53],and the self-consistentmean-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 densityρ0are calculated using two types of BGBD interactions:the MDIinteractions 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 nuclearmatter,E0（ρ）,K0（ρ）,and J0（ρ）are also analyzed in detail.Itshould 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 canonicalneutron stars,and the calculations of J0（ρ）in the present work m ight be helpful in further determining the properties of neutron stars.In particular,the contributions from the high-order terms of the single-nucleon potential Usym，3（ρ，k）and Usym，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 nuclearmatter 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.

2 Decom position of basic quantities of EoS in term s of global optical potential com ponents

2.1 Basic quantities in the Equation of State of asymm etric nuclear m atter

Fig.1 (Color online)The schematic diagram of basic quantities of the EoS in both isospin symmetric and asymmetric nuclearmatter,including E0（ρ）,Esym，2（ρ）,Esym，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 EFand the energy per nucleon E,the general Hugenholtz–Van Hove(HVH)theorem can be written as[38]

where τ=1 is for the neutron and τ=-1 for the proton,and U0（ρ，k）and Usym，i（ρ，k）are the symmetric and asymmetric parts,respectively.In particular,U0（ρ，k）and Usym，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:

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

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）

Obviously,E0（ρ0）=t（kF）+U0（ρ0，kF）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 δ2and δ4,namely

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）,Esym，2（ρ）,and Esym，4（ρ）are given as

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

2.3 The exact saturation densityρsat as a function of isospin asymmetry

For isospin asymmetric nuclear matter,the saturation density is different from that of the symmetric nuclear matterρ0.The former is defined as the exact saturation density and can be also written as a function of the isospin asymmetryδ[77]

At the exact saturation density ρsat（δ）,the energy per nucleon of asymmetric nuclear matter is given by

The corresponding incompressibility coefficient of the EoS is

3 Results and discussions

We performed a systematic analysis of the basic quantitiesin the EoSof both symmetric and asymmetric nuclear matter at the saturation densityρ0by 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 valuesof thebasic quantitiesin EoSare 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）,Esym，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 Esym，4（ρ0）value is almost the same as that in Ref.[77].To further estimate the error bars of these basic quantities,all thecalculated valuesin Table1 areplotted 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].

Table 1 Thesaturation density ρ0(fm-3)and basic quantities E0（ρ0）,K0（ρ0）,J0（ρ0）,Esym，2（ρ0）,Esym，4（ρ0）,L2（ρ0）,L4（ρ0）,K2（ρ0）,and K4（ρ0）for totally 25 interaction setsin four kindsof 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.[44–50,62–75]

Fig.2 (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.3 (Color online)Values of Esym，2（ρ0）,L2（ρ0）,K2（ρ0）,Esym，4（ρ0）,L4（ρ0）,and K4（ρ0）for asymmetric nuclear matter within 25 parameter sets of four kinds of interaction

In Fig.4,weshow themagnitudesof theseparated terms E0（ρ）,Esym，2（ρ）δ2,Esym，4（ρ）δ4as well as the total one E（ρ，δ）at two different densities(ρ0and 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）isdominant.The contribution of Esym，2（ρ）δ2increases with an increase in isospin asymmetryδ.It is also shown that the contribution from Esym，4（ρ）δ4is 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 atρ0,while Esym，2（ρ）δ2plays a more important role in the EoS,especially atδ2=0.5.It should also be noted that Esym，4（ρ）contributes only at a very high density and large isospin asymmetry.The magnitude of Esym，4（ρ） can significantly affect the calculation of the proton fraction in neutron stars atβ-equilibrium[14,41].

Fig.4 (Color online)The magnitudes of E0（ρ）,Esym，2（ρ）δ2,and Esym，4（ρ）δ4 in the EoSat two differentρ values and three differentδ2 values.The NRAPR Skyrme interaction is applied

We further expand E0（ρ）,Esym，2（ρ）,and Esym，4（ρ）as a series ofχwith their corresponding slopes and incompressibility coefficients.In Fig.5,we depict the contributions from each term at different densities 0.5ρ0,2ρ0and 3ρ0.As can be observed in Fig.5,the first-order terms(E0（ρ0）),(Esym，2（ρ0）),and(Esym，4（ρ0）)contribute largely at all densities.andterms become increasingly important with increasing density.For Esym，2（ρ）and Esym，4（ρ）at 3ρ0,the contributions from the slopes(and)and the incompressibility coefficients(and)are much larger than those at 0.5ρ0and 2ρ0.In particular,thE,,andtermsat 3ρ0can be asimportant asthefirstorder terms.Thus,high-order terms should be considered when analyzing the properties of nuclear matter systems at high densities,such as neutron stars.

Fig.5 (Color online)The magnitude of each order in E0（ρ）,Esym，2（ρ）and Esym，4（ρ）expressed by E0（ρ0）,K0（ρ0）and J0（ρ0）,Esym，2（ρ0）,L2（ρ0）and K2（ρ0）,and Esym，4（ρ0）,L4（ρ0）and K4（ρ0）,respectively.The NRAPRSkyrmeinteraction was applied

More interestingly,the basic quantities at the saturation density aredecomposed into the kinetic energy t(k)and the symmetric and asymmetric parts of the single-nucleon potential U0（ρ，k）and Usym，i（ρ，k）.As shown in Fig.6,the contributions from different terms t(k),U0（ρ，k）and Usym，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 Usym，1（ρ，k）,Usym，2（ρ，k）,Usym，3（ρ，k）,and Usym，4（ρ，k）cannot be neglected.It is clearly shown that the first-order term Usym，1（ρ，k）contributes to all six basic quantities.The second-order term Usym，2（ρ，k）does not contribute to Esym，2（ρ0）,but to its corresponding slope L2（ρ0）and the incompressibility coefficient K2（ρ0）.In principle,the Usym，2（ρ，k）term should also contribute to the fourth-order terms Esym，4（ρ0）,L4（ρ0）,and K4（ρ0）,but for the Skyrme interaction,Usym，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 Usym，3（ρ，k） and Usym，4（ρ，k）to the basic quantities.In Fig.7,we show the density-dependence of U0（ρ，kF）, Usym，1（ρ，kF）,Usym，2（ρ，kF）,Usym，3（ρ，kF）and Usym，4（ρ，kF）at the Fermi momentum kF= （3π2ρ/2）1/3by using the NRAPR Skyrme interaction.It can be clearly seen in Fig.7 that the magnitudes of U0（ρ，kF）and Usym，1（ρ，kF）are generally very large,while the ones of Usym，2（ρ，kF）,Usym，3（ρ，kF）and Usym，4（ρ，kF）are very small but increase with the increasing density.Our resultsindicate that the Usym，3（ρ，k）and Usym，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.6 (Color online)The single-nucleon potential decomposition of E0（ρ0）,K0（ρ0）,J0（ρ0）,Esym，2（ρ0）,L2（ρ0）,K2（ρ0）,Esym，4（ρ0）,L4（ρ0）,and K4（ρ0）.The NRAPR Skyrme interaction is applied

Fig.7 (Color online) The density-dependence of U0（ρ，kF）,Usym，1（ρ，kF）,Usym，2（ρ，kF）,Usym，3（ρ，kF）,and Usym，4（ρ，kF）.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 Kasy，2and Ksat，2from other studies are listed in thelast row of Table2.It isshown that thesecondorder coefficient ρsat，2,one of the most important isospindependent partsofρsat（δ）,hasa negative value in all cases,and the fourth-order coefficient ρsat，4also 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 asymmetryδ(see graph(a)of Fig.8).For the BGBD interaction(Case-2),the calculated value of ρsat，4is positive and relatively large.According to therelationship in Eq.(11),thiswould lead to a higher saturation density of asymmetric nuclear matter than that of symmetric nuclear matter with isospin asymmetryδclose to unity.For asymmetric nuclear matter at ρsat（δ）,the corresponding Esat，4values are rather diverse and are considered to be important for the proton fraction in neutron stars.

Table 2 The calculated values of expansion coefficientsρ0(fm-3), ρsat，2(fm-3),ρsat，4(fm-3),the quartic symmetry energy Esat，4(MeV),the quadratic incompressibility coefficient K sat，2(MeV),and its two main components K asy，2(MeV)and J0（ρ0）/K0（ρ0）.In the last three rows,the averaged values and constraints in previous studies are shown

Fig.8 (Color online)The isospin-dependence of the exact saturation density ρsat（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)

With the averaged results L2（ρ0）=57.0 MeV and K2（ρ0）=-123.6 MeV,the calculated value Ksat，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 Ksat，2for asymmetric nuclear matter.

4 Summary

Based on the Hugenholtz–Van Hove theorem,the general expressions for the six basic quantities of EoS are expanded in termsof the kinetic energy t(k),thesymmetric and asymmetric parts of the global optical potential U0（ρ，k）and Usym，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 densityρ0.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（kF）and U0（ρ，kF）play vital roles in determining the EoSof both symmetric and asymmetric nuclear matter.For asymmetric nuclear matter,Usym，1（ρ，k） contributes to all the quantities,whereas Usym，2（ρ，k）does not contribute to Esym，2（ρ0）,but contributesto the second-order terms L2（ρ0）and K2（ρ0）as well as the fourth-order terms Esym，4（ρ0）,L4（ρ0）,and K4（ρ0）.In addition,the contribution from Usym，3（ρ，k）cannot be neglected for Esym，4（ρ0）,L4（ρ0）,and K4（ρ0）.Usym，4（ρ，k）should also be included in the calculations for L4（ρ0）and K4（ρ0）.In addition,the quadratic incompressibility coefficient at ρsat（δ）is found to have a simple empirical relation Ksat，2=K2（ρ0）-4.14L2（ρ0）based on the present analysis.

Author ContributionsAll authors contributed to the study conception and design.Material preparation,data collection and analysis were performed by Jie Liu,Chao Gao,Niu Wan and Chang Xu.The first draft of the manuscript was written by Jie Liu and all authors commented on previous versions of the manuscript.All authors read and approved the final manuscript.