Analytical relations between nuclear symmetry energy and single-nucleon potentials in isospin asymmetric nuclear matter

Nuclear Physics A 865 (2011) 1–16 www.elsevier.com/locate/nuclphysa

Analytical relations between nuclear symmetry energy and single-nucleon potentials in isospin asymmetric nuclear matter Chang Xu a,b , Bao-An Li a,∗ , Lie-Wen Chen a,c , Che Ming Ko d a Department of Physics and Astronomy, Texas A&M University-Commerce, Commerce, TX 75429-3011, USA b Department of Physics, Nanjing University, Nanjing 210008, China c Department of Physics, Shanghai Jiao Tong University, Shanghai 200240, China d Cyclotron Institute and Department of Physics and Astronomy, Texas A&M University, College Station,

TX 77843-3366, USA Received 27 April 2010; received in revised form 29 March 2011; accepted 27 June 2011 Available online 1 July 2011

Abstract Using the Hugenholtz–Van Hove theorem, we derive general expressions for the quadratic and quartic symmetry energies in terms of the isoscalar and isovector parts of single-nucleon potentials in isospin asymmetric nuclear matter. These expressions are useful for gaining deeper insights into the microscopic origins of the uncertainties in our knowledge on nuclear symmetry energies especially at supra-saturation densities. As examples, the formalism is applied to two model single-nucleon potentials that are widely used in transport model simulations of heavy-ion reactions. © 2011 Elsevier B.V. All rights reserved. Keywords: Symmetry energy; Nuclear potential; Heavy-ion collision; Transport model

1. Introduction One of the central issues currently under intense investigation in both nuclear physics and astrophysics is the Equation of State (EOS) of neutron-rich nuclear matter [1–3]. For cold nuclear matter of isospin asymmetry δ = (ρn − ρp )/(ρn + ρp ) at density ρ, the energy per nucleon * Corresponding author.

E-mail address: [email protected] (B.-A. Li). 0375-9474/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysa.2011.06.027

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E(ρ, δ) can be expressed as an even series  of δ that respects the charge symmetry of strong interactions, namely, E(ρ, δ) = E0 (ρ, 0) + i=2,4,6,... Esym,i (ρ)δ i where Esym,i (ρ) is the so-called symmetry energy of the ith order [3] and E0 (ρ, 0) is the EOS of symmetric nuclear matter. The quadratic term Esym,2 (ρ) is most important and its value at normal nuclear matter density ρ0 is known to be around 30 MeV from analyzing nuclear masses within liquid-drop models. Essentially, all microscopic many-body calculations have indicated that the higher-order terms are usually negligible around ρ0 , leading to the so-called empirical parabolic law of EOS even for δ approaching unity for pure neutron matter. The Esym,2 (ρ) is then generally regarded as the symmetry energy. For instance, the value of the quartic term has been estimated to be less than 1 MeV at ρ0 [4,5]. However, the presence of higher-order terms at supra-saturation densities can significantly modify the proton fraction in neutron stars at β-equilibrium and thus the cooling mechanism of proto-neutron stars [6,7]. It was also found that a tiny quartic term can cause a big change in the calculated core-crust transition density in neutron stars [8,9]. Therefore, precise evaluations of the quartic symmetry energy in neutron-rich matter are useful. Although much information about the EOS of symmetric nuclear matter E0 (ρ, 0) has been accumulated over the past four decades, our knowledge about the density dependence of Esym,i (ρ) is unfortunately still very poor. It has been generally recognized that the Esym,i (ρ), especially the quadratic and quartic terms, is critical for understanding not only the structure of rare isotopes and the reaction mechanism of heavy-ion collisions, but also many interesting issues in astrophysics [9–23]. Therefore, to determine the Esym,i (ρ) in neutron-rich matter has recently become a major goal in both nuclear physics and astrophysics. While significant progress has been made recently in constraining the Esym,2 (ρ) especially around and below the saturation density, see, e.g., [18–21], much more work needs to be done to constrain more tightly the Esym,i (ρ) at supra-saturation densities where model predictions are rather diverse [26–37]. As dedicated experiments using advanced new detectors have now been planned to investigate the high density behavior of Esym,2 (ρ) at many radioactive beam facilities around the world, it has become an urgent task to investigate theoretically more deeply the fundamental origin of the extremely uncertain high density behavior of Esym,2 (ρ). It is also of great interest to evaluate possible corrections due to the Esym,4 (ρ) term to the equation of state of asymmetric nuclear matter. Among existing proposals to extract information about the Esym,i (ρ) using terrestrial nuclear laboratory experiments, transport model simulations have shown that many observables in heavyion reactions are particularly useful for studying the Esym,i (ρ) in a broad density range [3,10,12, 14,15]. In these studies, the EOS enters the reaction dynamics and affects the final observables through the single-nucleon potential Un/p (ρ, δ, k) where k is the nucleon momentum. Except in situations where statistical equilibrium is established and thus many observables are directly related to the binding energy E(ρ, δ) after correcting for finite-size effects, what is being directly probed in heavy-ion reactions is the single-nucleon potential Un/p (ρ, δ, k). Moreover, the latter is also an input in shell model studies of nuclear structure. What is the direct relationship between the symmetry energy and the isoscalar and isovector parts of the single-nucleon potential Un/p (ρ, δ, k)? This is the central question we will address in this work. First of all, why is this question physically meaningful and important? To answer this question, we first notice that for all purposes in both nuclear physics and astrophysics, it is important to known how the density dependence of the nuclear symmetry energy is related to the underlying isospin dependence of the strong interaction, see, e.g., detailed discussions by Steiner et al. in Ref. [2]. This relation then allows us to not only better understand why the symmetry energy is still very uncertain but also to connect potentially with the QCD theory of nuclear strong interaction. Usually, one starts from a model energy density functional constrained by empirical

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properties of nuclear matter. The single-nucleon potential is then obtained by a functional derivative of the energy density with respect to the distribution function. The symmetry energy can also be easily obtained if the energy density functional is known. Thus, the single-nucleon potential and the symmetry energies are intrinsically correlated as they are all derived from the same energy density functional. However, it is important to stress that the single-nucleon potential is the one directly connected to the experimental observables. This is achieved, for example, by comparing shell model calculations or transport model simulations with the experimental data. In principle, to extract information about the symmetry energy from experiments, one can simply parameterize the single nucleon-potential and adjust its parameters within known constraints to fit the data. Therefore, being able to know directly the corresponding symmetry energy is very useful. Of course, the same quadratic and quartic symmetry energies should be obtained if one would follow the normal route to construct the energy density functional first. Nevertheless, for studying the density dependence of the nuclear symmetry energy and applying it in solving astrophysical problems, the direct relationship between the symmetry energy and the single-nucleon potential without first going through the procedure to construct the energy density functional is advantageous. For example, from nucleon–nucleus scattering and (p, n) charge exchange experiments one can directly extract from the data both the isoscalar and isovector nucleon optical potentials at normal density. One can then easily calculate the symmetry energy and its density slope at normal density directly from the optical potentials without having to first construct the nuclear density functional as demonstrated recently in Ref. [38]. Moreover, to find the relationship between the symmetry energy and the isoscalar and isovector single-particle potentials is actually a major goal of the current efforts in developing nuclear energy density functionals. To be more specific, we quote here two of the major questions identified at the 2005 Nuclear Density Functional Theory (DFT) program at the Institute of Nuclear Theory in Seattle [39]: (1) what is the form of the nuclear energy density functional and (2) what are the constraints of the nuclear energy density functional? In fact, they are still questions under intense investigation today [40]. In deliberating the second question, it was stated that [39] “Aside for the intrinsic limitations of the functionals, they also are limited by the insufficient constraints from data employed to determine the parameters. The density and gradient dependences of the isovector terms are poorly known, both for the ordinary densities and the pairing fields. To make a progress, a consorted effort will be required to study new functionals when applied to finite nuclei and infinite or semi-infinite nuclear matter. Another goal is to understand connections between the symmetry energy and isoscalar and isovector mean fields, and in particular the influence of effective mass and pair correlations on symmetry energy versus the isospin. Such understanding will allow us to better determine isospin corrections to nuclear mean fields and energy density functionals”. Our work reported here is a response to the call to help solve the questions quoted above. Interestingly, the relation between the symmetry energy and the isoscalar and isovector mean fields has already been derived first by Brueckner, Dabrowski and Haensel [41,42] using Kmatrices within the Brueckner theory in the 1960s. They showed that if one expands Un/p (ρ, δ, k) to the leading order in δ as in the well-known Lane potential [43], i.e., Un/p (ρ, δ, k) ≈ U0 (ρ, k) ± Usym,1 (ρ, k)δ

(1)

where U0 (ρ, k) and Usym,1 (ρ, k) are, respectively, the nucleon isoscalar and isovector (symmetry) potentials, the quadratic symmetry energy is then [41,42]  1 1 ∂U0  1 kF + Usym,1 (ρ, kF ) (2) Esym,2 (ρ) = t (kF ) + 3 6 ∂k  2 kF

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where t (kF ) is the nucleon kinetic energy at the Fermi momentum kF = (3π 2 ρ/2)1/3 in symmetric nuclear matter of density ρ. The above equation indicates that the symmetry energy Esym,2 (ρ) depends only on the single-particle kinetic and potential energies at the Fermi momentum kF . This is not surprising since the microscopic origin of the symmetry energy is the difference in the Fermi surfaces of neutrons and protons in the asymmetric nuclear matter. The first term kin = 1 t (k ) = h¯ 2 ( 3π 2 ) 23 ρ 23 is the trivial kinetic contribution due to the different Fermi moEsym F 3 6m 2 0 menta of neutrons and protons in the asymmetric nuclear matter; the second term 16 ∂U ∂k |kF kF is due to the momentum dependence of the isoscalar potential and also the fact that neutrons and protons have different Fermi momenta in the asymmetric nuclear matter; while the term 1 2 Usym (ρ, kF ) is due to the explicit isospin dependence of the nuclear strong interaction. For the isoscalar potential U0 (ρ, k), reliable information about its density and momentum dependence has already been obtained from high energy heavy-ion collisions, see, e.g., Ref. [13], albeit there is still some room for further improvements, particularly at high momenta/densities. On the contrary, the isovector potential Usym,1 (ρ, k) is still not very well determined especially at high densities and momenta. In fact, it has been identified as the key quantity responsible for the uncertain high density behavior of the symmetry energy as stressed in Ref. [3]. In the present work, we first derive Eq. (2) using the Hugenholtz–Van Hove (HVH) theorem [45]. Compared to the derivation by Brueckner, Dabrowski and Haensel [41,42], our derivation based on the HVH theorem is more straightforward. Another alternative derivation that is more tedious but educational is given in Appendix A. We then derive an expression for the quartic symmetry energy Esym,4 (ρ) in terms of the single-nucleon potential by keeping higher-order terms in the expansion of both the EOS and the single-nucleon potential. Applying the HVH formalism to two model single-nucleon potentials, namely, the Bombaci–Gale–Bertsch–Das Gupta (BGBD) potential [17] and a modified Gogny Momentum-Dependent-Interaction (MDI) [26,44], which are among the most widely used ones in studying isospin physics based on transport model simulations of heavy-ion reactions [3,14,15], we examine the relative contributions from the kinetic and various potential terms to Esym,2 (ρ) and Esym,4 (ρ). We put the emphasis on identifying those terms that dominate the high density behaviors of Esym,2 (ρ). Finally, we evaluate the relative importance of the Esym,4 (ρ) term by studying the Esym,4 (ρ)/Esym,2 (ρ) ratio as a function of density. The paper is organized as follows. In Section 2, based on the HVH theorem we derive expressions for the symmetry energy terms Esym,2 (ρ) and Esym,4 (ρ) in terms of the single-nucleon isoscalar and isovector potentials. Numerical results and discussions for both the BGBD and MDI interactions are given in Sections 3.1 and 3.2, respectively. A summary is given in Section 4. Finally, in Appendix A general expressions of the symmetry energies Esym,2 (ρ) and Esym,4 (ρ) are derived starting from the total energy density of the system. 2. Relation between the symmetry energy and the single-nucleon potential The Hugenholtz–Van Hove theorem [45] is a fundamental relation among the Fermi energy EF , the average energy per particle E and the pressure of the system P at the absolute temperature of zero. For a one-component system, in terms of the energy density ξ = ρE, the general HVH theorem can be written as [45,46] EF =

d(ρE) dE dξ = =E+ρ = E + P /ρ. dρ dρ dρ

(3)

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The above relation has been strictly proven to be valid for any interacting self-bound infinite Fermi system. It does not depend upon the precise nature of the interaction. In the special case of nuclear matter at saturation density where the pressure P vanishes, the average energy per nucleon becomes equal to the Fermi energy, i.e., EF = E. It is worthwhile to stress that the general HVH theorem of Eq. (3) is valid at any arbitrary density as long as the temperature remains zero [45,46]. In fact, a successful theory for nuclear matter is required not only to describe satisfactorily all saturation properties of nuclear matter but also to fulfill the general HVH theorem at any density. In the following, we use the general HVH theorem to derive the relation between the nuclear symmetry energy and the single-nucleon potential. According to the HVH theorem, the chemical potentials of neutrons and protons in isospin asymmetric nuclear matter of energy density ξ(ρ, δ) = ρE(ρ, δ) are, respectively [45,46],     ∂ξ , t kFn + Un ρ, δ, kFn = ∂ρn   p ∂ξ p , t kF + Up ρ, δ, kF = ∂ρp

(4) (5)

where t (k) = h¯ k 2 /2m is the kinetic energy and Un/p is the neutron/proton single-particle potenp tial. The Fermi momenta of neutrons and protons are kFn = kF (1 + δ)1/3 and kF = kF (1 − δ)1/3 , respectively. Subtracting Eq. (5) from Eq. (4) gives [41,42]  p        n ∂ξ ∂ξ p  t kF − t kF + Un ρ, δ, kFn − Up ρ, δ, kF = − . ∂ρn ∂ρp

(6)

The nucleon single-particle potentials can be expanded as a power series of δ while respecting the charge symmetry of nuclear interactions under the exchange of neutrons and protons,  Uτ (ρ, δ, k) = U0 (ρ, k) + Usym,i (ρ, k)(τ δ)i i=1,2,3,...

= U0 (ρ, k) + Usym,1 (ρ, k)(τ δ) + Usym,2 (k)(τ δ)2 + · · ·

(7)

where τ = 1 (−1) for neutrons (protons). If one neglects the higher-order terms (δ 2 , δ 3 , . . . ), Eq. (7) reduces to the Lane potential in Eq. (1). Expanding both the kinetic and potential energies around the Fermi momentum kF , the left side of Eq. (6) can be further written as  p        n p  t kF − t kF + Un ρ, δ, kFn − Up ρ, δ, kF   1 ∂ i [t (k) + U0 (ρ, k)]   ki =  F i! ∂k i i=1,2,3,...



×



i F (j )δ



i

  Usym,l (ρ, kF ) δ l − (−δ)l +



i F (j )δ j

δl −

j =1,2,3,...

j

F (j )(−δ)

j =1,2,3,...

l=1,2,3,...

×



−

j

j =1,2,3,...

+

kF





 1 ∂ i Usym,l (ρ, k)  i  kF i! ∂k i kF l=1,2,3,... i=1,2,3,... i F (j )(−δ)j (−δ)l 



j =1,2,3,...

 2 ∂[t (k) + U0 (ρ, k)]  =  kF + 2Usym,1 (ρ, kF ) δ + · · · , 3 ∂k

kF

(8)

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where we have introduced the function F (j ) = j1! [ 13 ( 13 − 1) · · · ( 13 − j + 1)]. For the right side of Eq. (6), expanding in powers of δ gives  ∂ξ ∂ξ 2 ∂ξ = − = 2iEsym,i (ρ)δ i−1 ∂ρn ∂ρp ρ ∂δ i=2,4,6,...

= 4Esym,2 (ρ)δ + 8Esym,4 (ρ)δ 3 + 12Esym,6 (ρ)δ 5 + · · · .

(9)

Comparing the coefficient of each δ i term in Eq. (8) with that in Eq. (9) then gives the symmetry energy of any order. For instance, the quadratic term  1 ∂[t (k) + U0 (ρ, k)]  1 Esym,2 (ρ) = kF + Usym,1 (ρ, kF )  6 ∂k 2 kF  1 ∂U0  1 1 kF + Usym,1 (ρ, kF ) (10) = t (kF ) + 3 6 ∂k  2 kF

is identical to that in Eq. (2), while the quartic term can be written as  5 ∂[t (k) + U0 (ρ, k)]  Esym,4 (ρ) =  kF 324 ∂k kF   1 ∂ 2 [t (k) + U0 (ρ, k)]  2 1 ∂ 3 [t (k) + U0 (ρ, k)]  3 −  kF + 648  kF 108 ∂k 2 ∂k 3 kF kF   1 ∂Usym,1 (ρ, k)  1 ∂ 2 Usym,1 (ρ, k)  2 −  kF + 72  kF 36 ∂k ∂k 2 kF k  F 1 ∂Usym,2 (ρ, k)  1 +  kF + 4 Usym,3 (ρ, kF ) . 12 ∂k

(11)

kF

Starting from the total energy density of the system that can be considered as the integral form of the HVH theorem, the same expressions for the Esym,2 (ρ) and Esym,4 (ρ) are derived in Appendix A. 3. Applications and discussions As shown in the previous section, the symmetry energies can be explicitly separated into the kinetic energy term T and the potential terms U0 and Usym,i at the Fermi momentum kF . To evaluate their relative contributions to the symmetry energies Esym,2 (ρ) and Esym,4 (ρ), we consider in this section two typical single-nucleon potentials that have been widely used in transport model simulations of heavy-ion reactions. One of them is the Bombaci–Gale–Bertsch–Das Gupta (BGBD) potential [17] that has been used extensively by the Catania group, see, e.g., Refs. [14, 15] for reviews. The other one is the modified Gogny potential (MDI), see, e.g., Ref. [3] for a review. We notice that new versions of both the BGBD [24] and MDI [25] interactions were recently proposed. For the purpose of the present study, however, it is sufficient to consider only the original BGBD and MDI interactions. 3.1. The Bombaci–Gale–Bertsch–Das Gupta potential As a first example, we use the phenomenological potential of Bombaci–Gale–Bertsch– Das Gupta [17]

C. Xu et al. / Nuclear Physics A 865 (2011) 1–16



B 1 2 + x3 uσ δ 2 Uτ (u, δ, k) = Au + Buσ − (σ − 1) 3 σ +1 2  



4 B 1 2 1 ± − A + x0 u − + x3 uσ δ 3 2 3σ +1 2 4 1 (3C − 4z1 )Iτ + (C + 2z1 )Iτ  + 5ρ0 2 

C − 8z1 δ u · g(k), + C +τ 5

7

(12)

where u = ρ/ρ0 is the reduced density and τ = 1 (−1) for neutrons (protons). In the above,

we have Iτ = [2/(2π)3 ] d 3 k fτ (k)g(k) with g(k) = 1/[1 + ( Λk )2 ] being a momentum regulator and fτ (k) being the phase space distribution function. The parameter Λ has the value Λ = 1.5kF0 , where kF0 is the nucleon Fermi wave number in symmetric nuclear matter at ρ0 . With A = −144 MeV, B = 203.3 MeV, C = −75 MeV and σ = 7/6, the BGBD potential reproduces all ground state properties including an incompressibility K0 = 210 MeV for symmetric nuclear matter [17]. The three parameters x0 , x3 and z1 can be adjusted to give different symmetry energy Esym,2 (ρ) and the neutron–proton effective mass splitting m∗n − m∗p [17,48–50]. For example, the parameter set z1 = −36.75 MeV, x0 = −1.477 and x3 = −1.01 leads to m∗n > m∗p while the one with z1 = 50 MeV, x0 = 1.589 and x3 = −0.195 leads to m∗n < m∗p at all non-zero densities and isospin asymmetries. On expanding the BGBD potential in δ, it is easy to obtain the coefficients of the first four terms: U0 (ρ, k), Usym,1 (ρ, k), Usym,2 (ρ, k) and Usym,3 (ρ, k). Thus, the second-order symmetry energy Esym,2 (ρ) is given by  1 1 ∂U0  1 kF + Usym,1 (ρ, kF ) Esym,2 (ρ) = t (kF ) +  3 6 ∂k kF 2

2 2/3 2 3π h¯ C k2 = ρ 2/3 − u F2 g(kF )2 6m 2 3 Λ  



2 B 1 1 1 + x0 u − + x3 uσ + − A 3 2 3σ +1 2 C − 8z1 + u · g(kF ), (13) 5 and the fourth-order symmetry energy Esym,4 (ρ) is Esym,4 (ρ) =



2 2  5kF2 3π ρ 3 h¯ 2 C k 2 10kF4 − u F2 + + 1 g(kF )4 162m 2 81 Λ Λ4 Λ2

 C − 8z1 kF2 5kF2 u 2 + + 1 g(kF )3 . 135 Λ Λ2

(14)

In Fig. 1 we compare Esym,2 (ρ) and its three components in the two cases of m∗n > m∗p and m∗n < m∗p . It is seen that the kinetic and isoscalar contributions are the same in both cases. However, they have significantly different isovector potentials Usym,1 , leading thus to different Esym,2 (ρ) especially at supra-saturation densities. Various contributions to the fourth-order symmetry energies Esym,4 (ρ) in the two cases are compared in Fig. 2. Similar to Esym,2 (ρ), the contributions of the T and U0 terms to Esym,4 (ρ)

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Fig. 1. The kinetic energy part (T ), the isoscalar potential part (U0 ) and the isovector potential part (Usym,1 ) of the symmetry energy Esym,2 from the BGBD potential with m∗n > m∗p (left) and for m∗n < m∗p (right).

Fig. 2. The kinetic and various potential contributions to the fourth-order symmetry energy Esym,4 with the BGBD potential for m∗n > m∗p (left) and for m∗n < m∗p (right).

are positive and they are the same in both cases. Interestingly, the Usym,1 term also plays the most important role in determining the high-density behavior of Esym,4 (ρ). It is positive in the case of m∗n > m∗p but negative in the case of m∗n < m∗p , resulting in very different behaviors of Esym,4 (ρ) at supra-saturation densities. Moreover, it is interesting to note that Esym,4 (ρ) receives no contribution from the Usym,2 term. This is not surprising because the Usym,2 term in the BGBD interaction is momentum independent and its contribution to Esym,4 (ρ) is actu∂U

(k)

sym,2 1 |kF kF = 0. On the contrary, the Usym,3 term still contributes to Esym,4 (ρ) via ally 12 ∂k 1 4 Usym,3 (kF ) although it is momentum independent too. In the two cases considered here, the contributions from the Usym,3 term also have opposite sign. To compare the fourth-order term Esym,4 (ρ) with the second-order term Esym,2 (ρ) more clearly, we show in Fig. 3 (left panel) their ratio Esym,4 (ρ)/Esym,2 (ρ) as a function of the reduced density ρ/ρ0 . Obviously, the relative value of Esym,4 (ρ) is generally small. However, it

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Fig. 3. Left panel: the ratio of Esym,4 over Esym,2 with the BGBD potential for m∗n > m∗p (case 1) and for m∗n < m∗p (case 2). Right panel: the ratio of Esym,4 over Esym,2 obtained from the MDI interaction as a function of reduced density ρ/ρ0 for x = 1, 0, and −1.

can reach up to about ±10% at high densities for both cases of m∗n > m∗p and m∗n < m∗p . It may thus lead to an appreciable modification in the proton fraction and therefore the properties of neutron stars at β-equilibrium. 3.2. A modified Gogny Momentum-Dependent-Interaction In this subsection, we discuss the symmetry energy obtained from the MDI interaction [26], which is derived from the Hartree–Fock approximation using a modified Gogny effective interaction [44] ρτ  ρτ + Al (x) ρ0 ρ0

σ   ρ B ρ σ −1 1 − xδ 2 − 4τ x +B δρτ  ρ0 σ + 1 ρ0σ  fτ (r , p ) 2Cτ,τ + d 3 p ρ0 1 + (p − p )2 /Λ2  fτ  (r , p ) 2Cτ,τ  + . d 3 p ρ0 1 + (p − p )2 /Λ2

Uτ (ρ, δ, p)  = Au (x)

(15)

In the above, τ = 1 (−1) for neutrons (protons) and τ = τ  ; σ = 4/3 is the density-dependence parameter; fτ (r , p)  is the phase space distribution function at coordinate r and momentum p.  The parameters B, Cτ,τ , Cτ,τ  and Λ are obtained by fitting the nuclear matter saturation properties [26]. The momentum dependence of the symmetry potential stems from the different interaction strength parameters Cτ,τ  and Cτ,τ for a nucleon of isospin τ interacting, respectively, with unlike and like nucleons in the background fields. More specifically, Cunlike = −103.4 MeV while Clike = −11.7 MeV. The quantities Au (x) = −95.98 − x σ2B +1 and 2B Al (x) = −120.57 + x σ +1 are parameters. The parameters B and σ in the MDI single-particle

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potential are related to the t0 and α in the Gogny effective interaction via t0 = 83 σ B+1 ρ1σ and 0 σ = α + 1 [44]. The parameter x is related to the spin(isospin)-dependence parameter x0 via x = (1 + 2x0 )/3 [51]. By expanding the single-nucleon potential in δ and using Eq. (10), the second-order symmetry energy Esym,2 (ρ) is  1 1 ∂U0  1 kF + Usym,1 (ρ, kF ) Esym,2 (ρ) = t (kF ) +  3 6 ∂k kF 2

 2/3 h¯ 2 3π 2 = ρ 2/3 6m 2  2 

4pF + Λ2 (Cτ,τ + Cτ,τ  ) πΛ2 Λ2 4p ln + − 2p + F F 3ρ0 pF h3 Λ2 (Al − Au ) ρ B ρσ −x 4 ρ0 σ + 1 ρ0σ 

2 4pF + Λ2 (Cτ,τ − Cτ,τ  ) πΛ2 , + 2p ln F 3ρ0 h3 Λ2 +

(16)

and according to Eq. (11) the fourth-order symmetry energy Esym,4 (ρ) is Esym,4 (ρ) =

2 2 3π ρ 3 h¯ 2 162m 2

2 4pf2 + Λ2 4(7Λ4 pf4 + 42Λ2 pf6 + 40pf8 Cτ,τ 4π 2 2 2 − 5 Λ 7Λ pf ln − 3 ρ0 ρ h3 Λ2 (4pf2 + Λ2 )2 −



 8pf6  4pf2 + Λ2 Cτ,τ  4π 2 2  2 2 4 4 7Λ . ln Λ p + 16p − 28p − f f f 35 ρ0 ρ h3 Λ2 Λ2 (17)

As one expects, the above expressions are identical to those derived directly from the MDI energy density functional using [25]  1 ∂ 2 E(ρ, δ)  Esym,2 (ρ) = , 2! ∂δ 2 δ=0  1 ∂ 4 E(ρ, δ)  Esym,4 (ρ) = . 4! ∂δ 4 δ=0

(18)

In Fig. 4, we show the kinetic (T ), isoscalar (U0 ) and isovector (Usym,1 ) potential contributions to Esym,2 for the three different spin (isospin)-dependence parameter x = 1, 0, and −1. We notice that the kinetic (T ) and the isoscalar potential (U0 ) contributions are the same for the three different x values. As pointed out in Ref. [26], it is the isovector potential Usym,1 that is causing the different density dependence of Esym,2 . For instance, with x = 1 the Usym,1 term decreases very quickly with increasing density and thus results in a super-soft symmetry energy at suprasaturation densities. On the contrary, the symmetry energy Esym,2 at supra-saturation densities is

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Fig. 4. The kinetic energy part (T ), the isoscalar potential part (U0 ) and the isovector potential part (Usym,1 ) of the symmetry energy Esym,2 from the MDI interaction with x = 1, 0 and −1.

Fig. 5. The kinetic energy and potential contributions to the fourth-order symmetry energy Esym,4 from the MDI interaction.

very stiff for both x = 0 and x = −1 as the contribution of the Usym,1 term becomes very positive with smaller values of x. Generally, the behavior of Esym,2 from the MDI interaction is mainly determined by the different choices of the spin (isospin)-dependence parameter x. Unlike the Esym,2 , only the kinetic energy and the momentum-dependent part of the interactions contribute to the Esym,4 for the MDI parametrization. Thus, the behavior of Esym,4 from the MDI interaction is independent of the parameter x. Shown in Fig. 5 are the various contributions to the fourth-order symmetry energy Esym,4 . Comparing these with the results obtained using the BGBD in Fig. 2, we find that the T and U0 terms from these two interactions are almost identical. However, there exists some differences for other terms. For the MDI interaction, the Usym,2 term is negative and becomes very important for determining Esym,4 . One the contrary, the contributions from the Usym,1 and Usym,3 terms are positive and they are relatively small as compared to the Usym,2 . It is also seen from Fig. 5 that the sum of all these contributions, i.e., Esym,4 , is positive, which is similar to that from the BGBD interaction for the case of m∗n > m∗p . To compare more directly the Esym,4 with

12

C. Xu et al. / Nuclear Physics A 865 (2011) 1–16

Esym,2 of the MDI parametrization, their ratio Esym,4 /Esym,2 is plotted in Fig. 3 (right panel) as a function of reduced density for x = 1, 0, and −1. It is seen that with x = 1 there is a sharp break in the curve around 3ρ0 . This is because the second-order symmetry energy Esym,2 changes from positive to negative around 3ρ0 in this case. However, this is not the case for both x = −1 and x = 0 where Esym,2 remains positive at all densities. For both the BGBD and MDI interactions, the generally small values of Esym,4 up to several times the normal density clearly show that the parabolic approximation of the EOS is well justified for most purposes. However, the Esym,4 especially with the BGBD interaction may still play an important role in some astrophysical processes. For instance, special cares have to be taken in evaluating the core-crust transition density where the energy curvatures are involved. Although the values of the Esym,4 term are normally small, its contribution to the curvatures is important and could significantly affect the value of the core-crust transition density especially with stiff symmetry energies [8,9]. Moreover, it may also affect the proton fraction which determines whether the direct Urca process can occur in neutron stars (see Ref. [6] and references therein). It is well known that the β equilibrium condition of neutron stars requires that μe = μn − μp ,

(19)

where μe , μn , and μp are the chemical potentials of electron, neutron, and proton, respectively, if muons and other charged constituents are ignored (ρe = ρp ). Considering the charge neutrality, 1

1

the μe can then be expressed as μe = h¯ c(3π 2 ρp ) 3 = h¯ c[3π 2 ρ(1 − δ)/2] 3 [47]. Thus the isospin asymmetry δ or proton fraction (x = ρp /ρ = (1 − δ)/2) in neutron stars is determined by 1  h¯ c 3π 2 ρ(1 − δ)/2 3 = 4Esym,2 (ρ)δ + 8Esym,4 (ρ)δ 3 .

(20)

Although the relative contribution of Esym,4 to the EOS is small, however, it could be important for determining the composition of neutron stars as the Esym,4 gains a factor of 2 compared to that from the Esym,2 in Eq. (20). For example, the proton fraction estimated from Eq. (20) can change from x = 4% to x = 3% at the saturation density by adding a fourth-order term Esym,4 = 2 MeV. At abnormal densities, depending on the density dependence of the Esym,2 the proton fraction can also be easily altered by adding a relatively small fourth-order term. Thus, the quartic term Esym,4 may play an appreciable role in determining the composition and cooling of neutron stars. 4. Summary In summary, using the Hugenholtz–Van Hove theorem we have derived general expressions for the quadratic and quartic symmetry energies in terms of the isoscalar and isovector parts of the single-particle potentials in isospin asymmetric nuclear matter. These expressions allows us to connect directly the symmetry energies with the underlying isospin dependence of strong interactions. Since the single-particle potentials are direct inputs in both shell and transport models, the derived expressions facilitate the extraction of symmetry energies from experimental data. They thus help constrain the corresponding energy density functionals. As two examples, the BGBD and the MDI potentials are used in deriving the corresponding symmetry energies. For both interactions, the isovector potential is responsible for the uncertain high density behavior of the quadratic symmetry energy. The analytical formulas for the nuclear symmetry energies in terms of the isoscalar and isovector potentials are expected to be useful for extracting reliable information about the EOS of neutron-rich nuclear matter from experimental data. Also, the magnitude

C. Xu et al. / Nuclear Physics A 865 (2011) 1–16

13

of the quartic symmetry energy is found to be generally small compared to the quadratic term. However, it could be important for determining the compositions and the core-crust transition densities of neutron stars. Acknowledgements We thank S.A. Coon, J. Dabrowski and H.S. Köhler for discussions and communications on some of the issues studied here. This work is supported in part by the US National Science Foundation grants PHY-0757839 and PHY-0758115, the National Aeronautics and Space Administration under grant NNX11AC41G issued through the Science Mission Directorate, the Research Corporation under grant No. 7123, the Welch Foundation under grant No. A-1358, the Texas Coordinating Board of Higher Education grant No. 003565-0004-2007, the National Natural Science Foundation of China grants 10735010, 10775068, 10805026, and 10975097, Shanghai Rising-Star Program under grant No. 11QH1401100, the National Basic Research Program of China (973 Program) under Contract Nos. 2007CB815004 and 2010CB833000. Appendix A We present here another way of deriving the expressions for Esym,2 (ρ) and Esym,4 (ρ). For a one-component Fermi system, one could determine its total energy starting with empty space, adding particles until the desired density is reached [52]. Each newly added particle would contribute an energy k(ρ x )2 /2m + U (ρ x , k(ρ x )), where k is the Fermi momentum corresponding to the density ρ x of particles already added to the system. As stressed by Bertsch and Das Gupta, the single-particle potential U (ρ x , k(ρ x )) is not the same as the potential energy per particle as one might at first guess [52]. The total energy density of the asymmetric nuclear matter written in coordinate space is (ρn ,ρp )

ξ(ρn , ρp ) =

   k(ρnx )2 + Un ρnx , ρpx , k ρnx dρnx 2m

(0,0)



+

k(ρpx )2 2m

  x x  x  x + Up ρn , ρp , k ρp dρp ,

(21)

where somewhat different notations from the main text for the single-particle potential are used for preciseness and convenience. It should be noted that the integral in Eq. (21) is independent of path as ensured by the HVH theorem (the full differential). By defining the varying total density ρnx + ρpx = ρ x and asymmetry δ ∗ = (ρn ,ρp )

ξ(ρn , ρp ) =

the energy density can be expressed as

   k(ρnx )2 dρnx + Un ρ x , δ ∗ , k ρnx 2m

(0,0)

+

ρnx −ρpx ρnx +ρpx ,

k(ρpx )2 2m

    + Up ρ x , δ ∗ , k ρpx dρpx .

(22)

For the kinetic energy, it is more convenient to work in the momentum space where the kinetic n,p energy per nucleon can be expanded around δ = 0 with kF = kF (1 ± δ)1/3 [53,54]

14

C. Xu et al. / Nuclear Physics A 865 (2011) 1–16 p

n

kF T =α

kF t (k)k 2 dk + α

0

t (k)k 2 dk 0

kF = 2α

 1 ∂t (k)  t (k)k dk + kF δ 2 6 ∂k kF 2

0

+

   5 ∂t (k)  1 ∂ 2 t (k)  2 1 ∂ 3 t (k)  3 4 δ + ···, k − k + k F 324 ∂k kF 108 ∂k 2 kF F 648 ∂k 3 kF F

where α = 3/(2kF3 ). Thus, the kinetic contribution to the Esym,2 (ρ) is  1 ∂t (k)  1 kin kF = t (kF ), Esym,2 (ρ) =  6 ∂k kF 3

(23)

(24)

and its contribution to Esym,4 (ρ) is kin Esym,4 (ρ) =

=

   5 ∂t (k)  1 ∂ 2 t (k)  2 1 ∂ 3 t (k)  3 k − k + k F 324 ∂k kF 108 ∂k 2 kF F 648 ∂k 3 kF F 1 t (kF ). 81

(25)

The single particle potential can be further expanded as a series of δ ∗        i Uτ ρ x , δ ∗ , k = U 0 ρ x , k + Usym,i ρ x , k τ δ ∗ ,

(26)

i=1,2,3,...

where τ = 1 (−1) for neutron (proton). Thus the potential energy density can be expanded as (ρn ,ρp )

  x x  x  x   x x  x  x  Un ρn , ρp , k ρn dρn + Up ρn , ρp , k ρp dρp

U= (0,0)

(ρn ,ρp )

=

   i Usym,i ρ x , k ρnx δ ∗ dρnx

i=1,2,3,...

(0,0)





   U0 ρ x , k ρnx +



  + U0 ρ , k ρpx + x



  x  x  ∗ i x dρp Usym,i ρ , k ρp −δ

i=1,2,3,... (ρn ,ρp )

= (0,0)

  x  x  ∂U0   U0 ρ , k ρ + ∂k 

 x  δ∗  x  x  ∗ + Usym,1 ρ , k ρ δ + · · · dρnx k ρ 3 k(ρ x )

    x  x  ∗   ∂U0   x  δ∗ x  − U ρ δ + U0 ρ x , k ρ x − k ρ , k ρ + · · · dρ sym,1 p . (27) ∂k k(ρ x ) 3 Note that the potential energy density can also be expressed as

C. Xu et al. / Nuclear Physics A 865 (2011) 1–16

15

U = Epot (ρn , ρp )ρ (ρn ,ρp )

    d Epot ρnx , ρpx ρ x

= (0,0)

(ρn ,ρp )

     pot   2 d E0 ρ x + Esym,2 ρ x δ ∗ + · · · ρ x

= (0,0)

x x 2  x x pot  x  (ρn − ρp ) d E0 ρ ρ + Esym,2 ρ + ··· ρnx + ρpx

(ρn ,ρp )

= (0,0)

(ρn ,ρp )

=

  ∂E0 pot   E0 ρ x + x ρ x + 2Esym,2 ρ x δ ∗ + · · · dρnx ∂ρ

(0,0)

   ∂E0 pot   + E0 ρ x + x ρ x − 2Esym,2 ρ x δ ∗ + · · · dρpx . ∂ρ

(28)

By comparing the right sides of above two equations, it is easy to obtain the second-order symmetry energy from the potential energy part at any density  1 ∂U0  1 pot Esym,2 = kF + Usym,1 (ρ, kF ). (29) 6 ∂k kF 2 Similarly, contributions from other Usym,i terms in δ can be obtained. By combining all coefficients of the δ 2 and δ 4 terms in both the kinetic and potential parts given above, the Esym,2 (ρ) and Esym,4 (ρ) obtained here are exactly the same as those obtained using directly the HVH theorem. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]

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