Properties of asymmetric nuclear matter in different approaches
P. Gögelein, E.N.E. van Dalen, Kh. Gad, Kh. S. A. Hassaneen, H. Müther
aa r X i v : . [ nu c l - t h ] S e p Properties of asymmetric nuclear matter in different approaches
P. G¨ogelein, E.N.E. van Dalen, Kh. Gad, Kh. S. A. Hassaneen, and H. M¨uther Institut f¨ur Theoretische Physik, Universit¨at T¨ubingen, D-72076 T¨ubingen, Germany Physics Department, Faculty of Science, Sohag University, Sohag, Egypt
Properties of asymmetric nuclear matter are derived from various many-body approaches. Thisincludes phenomenological ones like the Skyrme Hartree-Fock and relativistic mean field approaches,which are adjusted to fit properties of nuclei, as well as more microscopic attempts like the Brueckner-Hartree-Fock approximation, a self-consistent Greens function method and the so-called V lowk ap-proach, which are based on realistic nucleon-nucleon interactions which reproduce the nucleon-nucleon phase shifts. These microscopic approaches are supplemented by a density-dependent con-tact interaction to achieve the empirical saturation property of symmetric nuclear matter. The pre-dictions of all these approaches are discussed for nuclear matter at high densities in β -equilibrium.Special attention is paid to behavior of the isovector component of the effective mass in neutron-richmatter. PACS numbers: 21.60.Jz, 21.65.-f, 26.60.-c, 97.60.JdKeywords: Nuclear equation of state, isospin dependence, symmetry energy, effective mass.
I. INTRODUCTION
The equation of state for nuclear matter under exotic conditions is one of the main topics in modern nuclear physics.This interest in the properties of nuclear matter at high densities and large proton-neutron asymmetries is partlymotivated by the fact that this information is required in theoretical models of compact objects in astrophysics likeneutron stars or the simulation of supernovae. However, the study of nuclear systems with large isospin asymmetriesare also the subject the forthcoming radioactive-ion-beam facilities such as FAIR at GSI or SPIRAL2 at GANIL.Different ways have been developed to obtain predictions for the properties of nuclear systems under exotic condi-tions. One way is to start from phenomenological models which successfully describe the properties of stable nuclei. Avery popular approach along this line is the use of an effective density dependent Skyrme-type interaction[1, 2, 3, 4, 5].Modern Skyrme parameterizations have been developed, which were constrained in their fitting procedures to obtainresults for neutron-rich nuclear matter compatible to those of microscopic calculations. Here we mention the Skyrme-Lyon (SLy) forces[6], which have been used in studies of the neutron-star crust[7, 8].Also the relativistic mean-field approximation has very successfully been used to describe the properties of stablenuclei[9]. Attempts have been made to derive a set of density-dependent meson-nucleon coupling constants of densitydependent relativistic mean field (DDRMF) calculations from microscopic Dirac Brueckner Hartree Fock calculationswith a readjustment in order to reproduce the bulk properties of stable nuclei and the saturation point of symmetricnuclear matter[10].The parameters of such Skyrme Hartree Fock and DDRMF calculations have been fitted to the data of stable nucleiand the predictive power of these simple phenomenological approaches could be rather limited. This would mean thatthe predictions for nuclear systems with exotic values for density and proton-neutron asymmetries may not be veryreliable.Therefore we will also consider some so-called microscopic approaches, which start from models of the nucleon-nucleon (NN) interaction, which are adjusted to describe the experimental phase shifts of NN scattering at energiesbelow the pion threshold[11, 12]. The traditional models of such realistic NN interactions like e.g. the charge-dependent Bonn potential CDBONN[11] or the Argonne potential V18[12] contain rather strong short range compo-nents, which make it inevitable to employ non-perturbative approximation schemes for the solution of the many-bodyproblem for the nuclear hamiltonian based on such interactions[13]. Such non-perturbative approximations includethe Brueckner hole-line expansion with the Brueckner Hartree Fock (BHF) approximation, the self-consistent evalua-tion of Green’s function using the T-matrix approximation[14, 15, 16, 17, 18, 19, 20, 21] (SCGF) and also variationalapproaches using correlated basis functions [22, 23, 24].If such microscopic calculations would reproduce the properties of nuclear systems under normal conditions, onecould argue that a scheme which reproduce the data of two nucleons in the vacuum (NN phase shifts) as well as nucleardata at normal densities should also provide reliable results for nuclear systems at densities beyond the saturationdensity of nuclear matter. Unfortunately, however, such microscopic calculations fail to reproduce the saturation pointof symmetric nuclear matter or the bulk properties of finite nuclei with good precision[13]. For example in nuclearmatter such calculations yield results for the saturation point, which are located on the so-called Coester band[25],i.e. they either yield too little binding energy or a saturation density well above the empirical value of ρ = 0.16 fm − .In recent years large progress has been made developing tools for essentially exact calculations for nuclear systemwith mass number up to about A = 12[26]. These calculations demonstrate that using realistic models for the NNinteraction in a non-relativistic hamiltonian precise results for nuclear systems are obtained if and only if these realistictwo-body interactions are supplemented by a three-body force.Therefore we will consider in the present study the results of BHF and SCGF calculations, adding a simple density-dependent contact interaction which is adjusted to describe the saturation point of symmetric nuclear matter.During the last years a different calculation scheme has evolved, which is also based on the NN scattering data buttries to decouple the low- and high- momentum components in the nuclear hamiltonian using renormalization groupmethods[27, 28, 29, 30]. The interaction resulting for the low-momentum regime of nuclear structure, which we willrefer to as V lowk is rather soft, which implies that non-perturbative tools of many-body theory can be applied[31]. Alsoit is very attractive, that V lowk turns out to be essentially model-independent, if the cut-off for the low-momentumregime is chosen appropriately.However, evaluating the properties of nuclear matter by using V lowk in a Hartree-Fock or BHF calculation, onedoes not obtain a saturation point[28, 32], the binding energy per nucleon increases with increasing density andnuclear systems tend to collapse to high densities. This will be compensated by adding a density-dependent contactinteraction, which is adjusted in the same fashion as for the BHF and SCGF cases discussed above.Therefore we will apply two different schemes: two phenomenological methods (Skyrme Hartree Fock and relativisticmean field DDRMF) employing parameterizations, which tend to reproduce some features of microscopic calculationsand three microscopic approaches (BHF, SCGF and Hartree-Fock with V lowk ), which are supplemented by a simplecontact interaction to reproduce the empirical saturation properties of symmetric nuclear matter. For all these methodswe study bulk features of nuclear matter at large densities and proton neutron asymmetries. A comparison of theresulting values for nuclear compressibility, symmetry energy, proton fraction in β equilibrium and effective masses,which characterize the density of states around the Fermi energy, should provide information about the uncertaintiesin the extrapolation of nuclear properties in exotic regions of density and asymmetry. Also we expect some hintsabout the reliability and problems of the individual approaches.After this introduction the section two shall briefly outline the different approximation schemes which we are goingto consider. The results are discussed in section three and the main conclusions are summarized in the final section. II. DIFFERENT MANY-BODY APPROACHESA. Skyrme Hartree-Fock
A very popular many-body approach in nuclear physics is the Skyrme-Hartree-Fock approach which can be founde.g. in [1, 2, 3, 4]. The Skyrme interaction leads to an energy functional E = Z H ( r ) d r , (1)where H is the Hamiltonian density in the Hartree-Fock approximation. In case of infinite asymmetric nuclear matterthe Hamiltonian density writes [2, 6] H = H K + H eff + H + H (2)where H K is the kinetic energy term, H eff an effective mass term, H a zero range term and H a density dependentterm. These terms are given by H K = ~ m τ, H eff = (cid:2) t (2 + x ) + t (2 + x ) (cid:3) τ ρ + (cid:2) t (2 x + 1) − t (2 x + 1) (cid:3)(cid:2) τ p ρ p + τ n ρ n (cid:3) , H = t (cid:2) (2 + x ) ρ − (2 x + 1)( ρ p + ρ n ) (cid:3) , H = t ρ α (cid:2) (2 + x ) ρ − (2 x + 1)( ρ p + ρ n ) (cid:3) , (3)where the coefficients t i , x i , W , and α are the parameters of a generalized Skyrme force. In the present study weapply the commonly used parameterization SLy4.The densities ρ and τ are defined in terms of the corresponding densities for protons and neutrons ρ = ρ p + ρ n and τ = τ p + τ n . If we identify the isospin label ( i = p, n ), the matter densities for protons and neutrons are given by ρ i = 2 Z k Fi d k (2 π ) = 13 π k F i , (4)where k F i denotes the corresponding Fermi momentum and the spin degeneracy gives a factor of 2. We obtain thekinetic energy density by τ i = 2 Z k Fi d k (2 π ) k = 15 π (3 π ρ i ) / . (5)The Hamiltonian density leads to the energy per nucleon E/A = H /ρ and the single particle energy may be writtenas a function of momentum k ε i ( k ) = k m ∗ i + V i (6)with an effective mass m ∗ i and a density dependent Potential V i . B. Brueckner-Hartree-Fock
Starting from realistic NN interactions, we have to use more advanced many-body approximations like theBrueckner-Hartree-Fock (BHF) approach, which have the capability to account for the effects of correlations, whichare due to the strong tensor and short-range components of such realistic NN interactions. In the BHF approximationthis is achieved by evaluating the so-called G -matrix, which corresponds to the in-medium scattering matrix. Theself-energy of a nucleon with isospin i , momentum k and energy ω in asymmetric nuclear matter is defined in theBHF approximation by [13, 20] Σ BHFi = X j Z d q h kq | G (Ω) | kq i ij n j ( q ) . (7)In this equation n j ( q ) refers to the occupation probability of a free Fermi gas of protons ( j = p ) and neutrons( j = n ) like in the mean-field or Hartree-Fock approach. This means that for asymmetric matter with a total density ρ = ρ p + ρ n this probability is defined by n j ( q ) = ( | q | ≤ k F j , | q | > k F j , (8)with Fermi momenta for protons ( k F p ) and neutrons ( k F n ).The antisymmetrized G matrix elements in eq. (7) are obtained from a given NN interaction by solving the Bethe-Goldstone equation h kq | G (Ω) | kq i ij = h kq | V | kq i ij + Z d p d p h kq | V | p p i ij × Q ( p i, p j )Ω − ( ε p ,i + ε p ,j ) + iη × h p p | G (Ω) | kq i ij . (9)The single-particle energies ε pi of the intermediate states should be the corresponding BHF single-particle energieswhich are defined in terms of the real part of the BHF self-energy of eq. (7) by ε ki = k m + Re (cid:2) Σ BHFi ( k , ω = ε ki ) (cid:3) , (10)with a starting energy parameter Ω = ω + ε qj in the Bethe-Goldstone equation (9).The Pauli operator Q ( p i, p j ) restricts the intermediate states to particle states with momenta p , p , which areabove the corresponding Fermi momentum. However, the single-particle spectrum is often parameterized in the formof an effective mass ε ki ≈ k m ∗ i + U i , (11)so that a so-called angle-averaged propagator can be defined, which reduces the Bethe-Goldstone equation to anintegral equation in one dimension. The exact Pauli operator has been treated in [33]. C. Relativistic models
An interesting extension of the BHF approach is the Dirac-Brueckner-Hartree-Fock (DBHF) approach which ac-counts for relativistic effects as well as correlation effects as described in terms of the G matrix. In the DBHFapproximation one evaluates the self-energy Σ i ( k ) in a way very similar to the BHF approximation keeping track ofthe Lorentz structure. Attempts have been made to fit a density dependent Relativistic Mean Field model (DDRMF)to the results of such DBHF calculations [10, 34]. Here we employ such a DDRMF model which has been obtained byfitting density dependent coupling constants for the meson-nucleon vertices to reproduce the self-energy of the DBHFcalculations of van Dalen et al. [35], which was based on the Bonn A potential.Both approaches, the DDRMF as well as the DBHF, start from a Lagrangian, which includes baryons, mesons andthe interaction L = L B + L M + L int . (12)However, they differ in the included mesons and the coupling operators [10, 35, 36, 37]. In the DDRMF model the σ , δ , ω , and ρ mesons are included in scalar and vector couplings, respectively. The variation of the Lagrangian leads toa Dirac equation for the nucleons [ γ · k ∗ i + m ∗ Di ] u ( k, s, i ) = γ E ∗ i u ( k, s, i ) , (13)where u ( k, s, i ) denotes the plane wave Dirac spinor with momentum k , spin s and isospin iu ( k, s, i ) = s E ∗ i + m ∗ Di E ∗ i σ · k ∗ i E ∗ i + m ∗ Di ! χ / ( s ) χ / ( i ) . (14)The starred quantities contain the different components of the nucleon self-energy: the scalar, time-like vector andspace-like contributions m ∗ Di = M + Σ S,i ( k, k F ) E ∗ i = E i ( k ) − Σ ,i ( k, k F ) k ∗ i = k i + ˆ k i Σ V,i ( k, k F ) , (15)where ˆ k i is the unit vector along the momentum k i of the nucleon. The general form of the nucleon self-energy ininfinite matter is obtained by evaluation of the meson exchange in spin saturated nuclear matterΣ i ( k, k F ) = Σ S,i ( k, k F ) + γ Σ ,i ( k, k F ) + γ · ˆ k i Σ V,i ( k, k F ) . (16)In the fitting process special attention has to be payed to the rearrangement contribution to the time-like vectorself-energy [38], the relativistic operator structure [39], and the proper renormalization due to the space-like vectorcontribution to the self-energy [10].The single particle energy in the DDRMF model is obtained from the Dirac equation (13) E i ( k ) = q k ∗ i + m ∗ Di + Σ ,i , (17)and the energy-momentum tensor leads to the energy density in asymmetric nuclear matter E = h T i = 1 π X i = p,n Z k F,i k dk q k ∗ i + m ∗ Di + 12 X i = p,n (Σ S,i ρ si + Σ ,i ρ i ) , with the scalar densities ρ si and the baryon densities ρ i ρ si = 1 π Z k F,i k dk m ∗ Di E ∗ i ( k ) ,ρ i = 1 π Z k F,i k dk. (18) D. Self-consistent Green’s function
One of the drawbacks of the BHF approximation is the fact that it does not provide results for the equation of state,which are consistent from the point of view of thermodynamics. As an example we mention that BHF results do notfulfill e.g. the Hugenholtz van Hove theorem. This is due to the fact that the BHF approximation does not considerthe propagation of particle and hole states on equal footing. An extension of the BHF approximation, which obeysthis symmetry is the self-consistent Green’s function (SCGF) method using the so-called T -matrix approximation.During the last years techniques have been developed, which allow to evaluate the solution of the SCGF equationsfor microscopic NN interactions[15, 16, 17, 18, 19]. Those calculation demonstrate that for the case of realisticNN interactions, the contribution of particle-particle ladders dominates the contribution of corresponding hole-holepropagation terms. This justifies the use of the BHF approximation and a procedure, which goes beyond BHF andaccounts for hole-hole terms in a perturbative way[40, 41]. This leads to a modification of the self-energy in the BHFapproximation by adding a hole-hole term of the form∆Σ h pi ( k, ω ) = X j Z ∞ k Fj d p Z k Fi d h Z k Fj d h × h kp | G (Ω) | h h i ij ω + ε pj − ε h i − ε h j − iη . (19)The quasi-particle energy for the extended self-energy can be defined as ε qpki = k m + Re[Σ BHFi ( k, ω = ε qpki ) + ∆Σ h pτ ( k, ω = ε qpki )] , (20)Accordingly, the Fermi energy is obtained evaluating this definition at the Fermi momentum k = k F i for protons andneutrons, respectively, ε F i = ε qpk F i . (21)The spectral functions for hole and particle strength, S hi ( k, ω ) and S pi ( k, ω ), are obtained from the real and imaginarypart of the self-energy Σ = Σ BHF + ∆Σ h p S h ( p ) i ( k, ω ) = ± π Im Σ i ( k, ω )[ ω − k / m − Re Σ i ( k, ω )] + [Im Σ i ( k, ω )] , (22)where the plus and minus sign on the left-hand side of this equation refers to the case of hole ( h , ω < ε F i ) andparticle states ( p , ω > ε F i ), respectively. The hole strength represents the probability that a nucleon with isospin i ,momentum k , and energy ω can be removed from the ground state of the nuclear system with the removal energy ω ,whereas the particle strength denotes the probability that such a nucleon can be added to the ground state of thesystem with A nucleons resulting in a state of the A + 1 particle system which has an energy of ω relative to theground state of the A particle system. Hence the occupation probability is obtained by integrating the hole part ofthe spectral function n i ( k ) = Z ε Fi −∞ dω S hi ( k, ω ) . (23)Note that this yields values for the occupation probability, which ranges between values of 0 and 1 for all momenta k , leading to a partial depletion of the hole-states in the Fermi gas model ( k < k F ) and partial occupations for stateswith momenta h > k F . A similar integral yields the mean energy for the distribution of the hole and particle strength,respectively h ε hi ( k ) i = R ε Fi −∞ dω ω S hi ( k, ω ) n i ( k ) , (24) h ε pi ( k ) i = R ∞ ε Fi dω ω S pi ( k, ω )1 − n i ( k ) . (25)Our self-consistent Green’s function calculation is defined by identifying the single particle energy in the Bethe-Goldstone equation as well as in the 2 h p correction term in eq. (19) by ε kτ = ( h ε hτ ( k ) i for k < k F τ , h ε pτ ( k ) i for k > k F τ . (26)This definition leads to a single particle Greens function, which is defined for each momentum k by just one poleat ω = ε kτ . Hence in the calculation of the self-energy the mean value of the spectral distribution is considered.However, the modified occupation of nucleons obtained by the spectral functions are not included in the calculationof the self-energies. The total energy per nucleon is evaluated by EA = P i R d k R ε Fi −∞ dω S hi ( k, ω )( k / m + ω ) / P i R d k n i ( k ) . (27) E. Renormalization of the NN interaction
It is very reasonable to assume that the long-range or low-momentum part of the NN interaction is fairly welldescribed in terms of meson-exchange, while different (quark) degrees of freedom are getting important to describethe short-rang or high-momentum components of the NN interaction. Therefore it is quite attractive to disentanglethese low-momentum and high-momentum components from each other. This means that one tries to define a modelspace, which accounts for the low-momentum degrees of freedom and renormalizes the effective hamiltonian for thislow-momentum regime to account for the effects of the high-momentum parts, which are integrated out.This concept of a model space and effective operators appropriately renormalized for this model space has a longhistory in approaches to the nuclear many-body physics. As an example we mention the effort to evaluate effectiveoperators to be used in Hamiltonian diagonalization calculations of finite nuclei. For a review on this topic see e.g. [42].The concept of a model space for the study of infinite nuclear matter was used e.g. by Kuo et al.[43, 44, 45].During the last years this concept has received a lot of attention and led to the definition of the so-called V lowk interaction. One way to determine this interaction is to follow the unitary-model-operator approach (UMOA)[46].We follow the usual notation and define a projection operator P , which projects onto the model space of two-nucleonwave functions with relative momenta k smaller than a chosen cut-off Λ. The operator projecting on the complementof this subspace is identified by Q and these operators satisfy the usual relations like P + Q = 1, P = P , Q = Q , and P Q = 0 = QP . It is the aim of the Unitary Model Operator Approach (UMOA) to define a unitary transformation U in such a way, that the transformed Hamiltonian does not couple the P and Q space, i.e. QU − HU P = 0 . (28)The technique to determine this unitary transformation is nicely been described by Fujii et al.[47] (see also [31]). Itleads to an effective hamiltonian H eff = h + V eff , which contains the term of the kinetic energy h and an effectiveinteraction V eff given by V eff = V lowk = U − ( h + V ) U − h . (29)Diagonalising this effective hamiltonian in the low-momentum model-space, one obtains eigenvalues which are identicalto the diagonalisation of the original hamiltonian h + V in the complete space. This means solving the LipmannSchwinger equation for NN scattering using this V lowk with an cut-off Λ yields the same phase shifts as obtainedfor the realistic interaction V without a cutoff. One finds that the resulting V lowk is essentially independent of theunderlying realistic interaction V , if is fitted to the experimental phase shifts and if the cut-off Λ is chosen aroundΛ = 2 fm − , which means that the model space includes scattering up to the pion threshold. In that sense V lowk isunique and, as it reproduces the NN scattering phase shifts it can also be regarded as a realistic interaction like e.g.the CDBONN or Argonne V18 interactions. density ρ [fm -3 ] e n e r gy / nu c l e on [ M e V ] SLy4DDRMFBHFBHF + ctSCGFSCGF + ctV lowk V lowk + ct FIG. 1: (Color online) Comparison of binding energy per nucleon of symmetric nuclear matter as obtained fromSkyrme SLy4, DDRMF, BHF, SCGF, and Vlowk. Results of approaches based on realistic NN interactions are alsocompared with an additional contact interaction of the form displayed in eq.30Since, however, the high-momentum or short-range components have been integrated out by means of the unitarytransformation of eq.(28), the V lowk does not induce any short-range correlations into the nuclear wave function. Thisleads to the nice feature that mean-field calculations using V lowk lead to reasonable results and corrections of many-body theories beyond mean field are weak[31]. On the other hand, however, it is this lack of short-range correlationeffects, which are modified in the medium, which prevents the emergence of a saturation point in calculations ofsymmetric nuclear matter[28, 32].In order to to achieve saturation in nuclear matter one has to add three-body interaction terms or a density-dependent two-nucleon interaction. This is not very astonishing as it is known that a renormalization of a two-bodyoperator leads to many-body terms[48, 49]. Therefore it is quite natural to supplement the effective interaction V lowk by a simple contact interaction, which we have chosen following the notation of the Skyrme interaction to be of theform ∆ H = 12 t ρ + 112 t ρ α , (30)where ρ is the matter density, t , t and α are parameters. For a fixed value of α (typically α =0.5) we have fitted t and t in such a way that a Hartree-Fock calculation using V lowk plus the contact term of eq.(30) yields the empiricalsaturation point for symmetric nuclear matter.The same parameterization of a contact term has been used to evaluate corrections to the self-energy of BHF andSCGF in such a way that also these calculations reproduce the saturation of symmetric nuclear matter.Note that this contact term is an isoscalar term and does not influence the symmetry energy, proton fractions in β -equilibrium, and effective masses. III. RESULTS AND DISCUSSION
All results of calculations, which refer to realistic NN interactions, have been obtained using the CDBONN[11]interaction. This includes all BHF and SCGF calculations. Also the evaluation of V lowk has been based on theproton-neutron part of CDBONN. Using a cut-off parameter Λ = 2 fm − , these results do not significantly depend onthe underlying interaction. The Skyrme Hartree-Fock calculations have been done using the parameterization SLy4and for the relativistic mean-field calculation the parameterization for DDRMF in [10] has been used.First let us turn to the binding energy of symmetric nuclear matter, which are displayed in Fig. 1. Compared toother realistic NN interactions the CDBONN potential, which we have chosen here is a rather soft NN interactionwith a weak tensor force. This is indicated by the results for the saturation point of symmetric nuclear matter as SLy4 DDRMF BHF BHF(+ct) SCGF SCGF(+ct) V lowk + ct ρ [fm − ] 0.160 0.178 0.374 0.161 0.212 0.160 0.160 E/A ( ρ ) [MeV] -15.97 -16.25 -23.97 -16.01 -11.47 -16.06 -16.0 K [MeV] 230 337 286 214 203 270 258 a S ( ρ ) [MeV] 32.0 32.1 51.4 31.9 34.0 28.3 21.7TABLE I: Properties of symmetric nuclear matter are compared for Skyrme SLy4, DDRMF, BHF, SCGF, and V lowk . Theresults, which are listed in the columns labeled with +ct are obtained employing the additional contact interaction of eq.(30)with parameters as listed in table II. The quantities listed include the saturation density ρ , the binding energy at saturation E/A , the compressibility modulus K and the symmetry energy at saturation density a S ( ρ ).BHF SCGF V lowk t [MeV fm ] -153 -311 -438.1 t [MeV fm α ] 2720 3670 6248TABLE II: Parameters t and t defining the contact interaction of eq.(30)as obtained for the fit to the saturation point ρ = 0 . fm − and E/A = − . α = 0 . obtained in the BHF approximation (the minimum of the dashed black line in Fig. 1 and data in table I ). Thesaturation density is larger than twice the empirical value and the calculated energy is well below, which means thatthe CDBONN result is located in the large binding energy high density part of the Coester band[13].The hole-hole contributions to the nucleon self-energy, which are included in the SCGF approach yield a repulsivecontribution the energy per nucleon, which increases with increasing nuclear density (dashed-dotted green line inFig. 1). This shifts the saturation point to a lower density and binding energy per nucleon so that also the saturationpoint obtained with SCGF is within the Coester band.The Hartree-Fock calculation using V lowk does not lead to a minimum in the energy versus density plot (see solidred line in Fig. 1), as we have already mentioned before[28, 32].In order to reproduce the empirical saturation point of symmetric nuclear we have added an isoscalar interactionterm as defined in eq.(30) choosing a value for α =0.5 and fitting the parameters t and t . The results for thesefitting parameters are listed in table II and the corresponding energy versus density curves are displayed in Fig. 1using the same line shape and color with and without adding the contact term but add an additional symbol to thelines displaying the results with inclusion of the contact term. For all three cases the fit yields an attractive two-bodycontact interaction and a repulsive t term.The results for the calculated saturation points in table I are supplemented by the corresponding values for thenuclear compressibility modulus K = 9 ρ ∂ ( E/A ) ∂ρ (cid:12)(cid:12)(cid:12)(cid:12) ρ = ρ (31)This nuclear compressibility, which is calculated at the saturation density ρ , together with the increase of energy atlarge density displayed in Fig. 1 characterize the stiffness of the EoS of symmetric nuclear matter. Comparing thedifferent approaches we find that the relativistic features included in the DDRMF approach lead to the stiffer EoSaround the saturation density as well as at higher densities. The SCGF and the V lowk calculations yield rather similarresults after the contact terms are included, which are a little bit softer than the DDRMF results and characterizedby a compression modulus of 270 MeV and 258 MeV for SCGF and V lowk , respectively. At higher densities the resultsare also very close to those obtained for the Skyrme Hartree-Fock using SLy4. Note, however, that SLy4 yields arather low value for K as compared to the SCGF and V lowk calculations. The softest EoS for symmetric matteramong those approaches which fit the empirical saturation point is provided by the BHF approximation.It is obvious that these results for the EoS for BHF, SCGF and V lowk are rather sensible to the choice of theexponent α in the t term of the contact interaction. A larger value of α would lead to stiffer EoS. We have pickedthe value α = 0.5 to obtain results for the EoS, which are similar to those resulting from the empirical approaches.Note, that the choice of α does not have any effect on the results referring to proton neutron asymmetries, which isthe main focus of this study.Table II also displays results for the symmetry energy a S ( ρ ) = ∂ ( E/A ) ∂I (cid:12)(cid:12)(cid:12)(cid:12) ρ , I = N − ZA = 1 − Y p , (32) density ρ [fm -3 ] s y mm e t r y e n e r gy [ M e V ] SLy4DDRMFBHFSCGFV lowk
FIG. 2: (Color online) Comparison of the symmetry energy a S ( ρ ) as a function of density ρ as obtained fromSkyrme SLy4, DDRMF, BHF, SCGF, and V lowk approaches.evaluated for each approach at the corresponding saturation density ρ . The two phenomenological approaches SLy4and DDRMF yield results which are in the range of the experimental value of 32 ± a S ( ρ ) since these values arecalculated at the corresponding saturation densities, which are larger than the empirical saturation density.The symmetry energy calculated in the SCGF approach is slightly smaller than the one obtained from the BHFapproximation. This is valid for all densities under consideration (see Fig. 2). This difference can easily be explained:As we already mentioned above, the contribution of the hole-hole terms is repulsive, which leads to larger energies forSCGF as compared to BHF for all densities in symmetric nuclear matter (Fig. 1) as well as in pure neutron matter(see Fig. 3). Since, however, the contribution of ladder diagrams is larger in the proton-neutron interaction (due tothe strong tensor terms in the S − D partial wave) than in the neutron-neutron interaction, this repulsive effectis stronger in symmetric nuclear matter than in neutron enriched matter. Therefore the symmetry energy calculatedin SCGF is slightly smaller if the hole-hole terms are included in SCGF[50].The symmetry energy at saturation density obtained from V lowk plus contact term is only about two third of theexperimental value (see table I) and the value is significantly below the other two microscopic approaches also at higherdensities. This can be understood from the following considerations: The Hartree-Fock calculations using V lowk donot account for the attractive contributions due to the NN ladder terms involving NN states with relative momentabelow the cut-off Λ. This missing attraction is compensated by the fit of the contact interaction to the empiricalsaturation point of symmetric matter. While the contact interaction is chosen to be identical for proton-neutron andneutron-neutron interaction, the ladder terms are more attractive for the isospin T = 0 partial waves (see above), i.e.the proton-neutron interaction. This leads to a significant underestimate for the symmetry energy at all densities.Note, however, that this failure of V lowk should disappear if V lowk would be employed in an appropriate many-bodycalculation beyond the mean field approximation.The symmetry energy rises as a function of density for all approaches considered. Note, however, that the twophenomenological approaches Skyrme Hartree-Fock using SLy4 and DDRMF provide rather different predictionsat high densities although the symmetry energy at normal density is identical. The relativistic approach predictssymmetry energies for high densities, which are well above all those derived from the microscopic calculations, whilethe Skyrme interaction yields a symmetry energy which is even below the V lowk estimate at densities above four timessaturation density.Rather similar features also observed, when we inspect the properties of nuclear matter in β -equilibrium, neutralizingthe charge of the protons by electrons, displayed in Fig. 4. The upper panel of this figure displays the proton abundance Y p = Z/A , which are to some extent related to the symmetry energy: large symmetry energy should correspond tolarge proton abundances. So the largest proton abundances are predicted within the DDRMF approach. Already ata density around 0.4 fm − Y p exceeds the about 10%, which implies that the direct URCA process could be enabled,which should be reflected in a fast cooling of a neutron star.0 density ρ [fm -3 ] e n e r gy / nu c l e on E / A [ M e V ] SLy4DDRMFBHF + ctSCGF + ctV lowk + ct
FIG. 3: (Color online) Energy per nucleon of pure neutron matter as a function of density as obtained from SkyrmeSLy4, DDRMF, BHF, SCGF, and V lowk approaches. p r o t on a bund a n ce [ % ] density ρ [fm -3 ] e n e r gy / nu c l e on [ M e V ] Skyrme SLy4DDRMFBHF + ctSCGF + ctV lowk + ct
FIG. 4: (Color online) Results for a system of infinite matter consisting of protons, neutrons and electrons in β -equilibrium. The upper panel show the proton abundances and the lower panel displays the energy per nucleon asa function of density using the various approximation schemes discussed in the text.1 density ρ [fm -3 ] e ff ec ti v e m a ss m * [ M ] SLy4BHFV lowk
Neutrons: lineProtons: line with symbol
FIG. 5: (Color online) Effective masses for protons (lines with symbols) and neutrons (lines without symbols) asobtained for nuclear matter in β -equilibrium using Skyrme HF (SLy4), BHF, and V lowk approaches.The other extreme case is the prediction derived from SLy4. In this approach the proton abundance does not exceeda value of 6%.The V lowk and SCGF approaches lead to similar proton abundances at large densities. This demonstrates that theevaluation of the proton abundance in β -equilibrium cannot directly be deduced from the symmetry energy, sincethe former observable is derived from proton and neutron energies at large asymmetries ( Z << N ), whereas thesymmetry energy is calculated from the second derivative at N = Z (see eq.(32). The BHF approach shows slightlylower values for Y p at high density, but the results are still in the same range as SCGF and V lowk .At low densities the Skyrme HF approach yields large proton fractions as compared to the results of the othercalculations. Large proton fractions at low densities tend to enhance density inhomogeneities and thus favor theexistence of a large variety of pasta structures. Therefore the Skyrme HF (Sly4) and the DDRMF approach, whichhave been explored in detail in [8, 10], should favor the formation of pasta structures as compared to the microscopicapproaches.Comparing the energies of matter in β -equilibrium derived from the various approaches as a function of density(Fig. 4,lower panel) we find the same trends as in the case of pure neutron matter displayed in Fig. 3. The absolutevalues are lower in the case β -equilibrium (about 75% of the energies for neutron matter). Furthermore the relativedifferences between the various approaches are smaller. While in the case of neutron matter the energy differencesbetween the various predictions are as large as 50% of the mean value, the corresponding number for matter in β -equilibrium is only around 25%. The approximation schemes leading to large energies for neutron matter alsoshow large symmetry energies, which result into relatively large proton abundances and smaller energies for matterin β -equilibrium.The equation of state of nuclear matter in β -equilibrium is the main input to predict mass and radii of neutronstars. A stiffer equation of state supports a larger maximum mass and a lower central density. In addition a thickercrust is found for the stiffer equation of state[51].Another important information for the evaluation of dynamical features of matter in neutron stars is the densityof states, which can be characterized by an effective mass. The term effective mass is used in various connections inmany-body physics. This includes the effective Dirac mass of the relativistic mean field approach m ∗ D (see eq.(15)),as well as effective masses, which express the non-locality of the self-energy in space and time, which corresponds toa momentum and energy dependence. The density of states, however, is related to the single-particle spectrum closeto the Fermi energy, which in the case of nuclear matter can be parameterized in terms of an effective mass by theexpression ε ( k ) = k m ∗ + U . (33)Such effective masses for protons and neutrons determined for nuclear matter in β -equilibrium are displayed in Fig. 5as a function of density considering non-relativistic approximation schemes.2It is a general feature of all approaches considered that the effective masses for protons as well as neutrons decreasewith increasing density. However, there is a striking difference between the phenomenological Skyrme approximationand the BHF and V lowk approach, which are based on realistic NN interactions: The effective mass for protons issmaller than the corresponding one for neutrons in neutron rich matter for the calculations using realistic interactions,while it is opposite applying the Skyrme parameterization. In fact, if we define the effective masses for protons m ∗ p and neutrons m ∗ n in terms of isoscalar m ∗ S and isovector masses m ∗ V by1 m ∗ n = 1 m ∗ S + I (cid:18) m ∗ S − m ∗ V (cid:19) m ∗ p = 1 m ∗ S − I (cid:18) m ∗ S − m ∗ V (cid:19) with I = N − ZA , (34)it turns out most of the Skyrme parameterizations yield an effective isovector mass m ∗ V , which is even larger thanthe bare nucleon mass M [5], which implies that it is larger than the effective isoscalar mass m ∗ S . This means thatthe effective mass for neutrons is smaller than the corresponding one for the protons in neutron rich matter ( I > κ of the Thomas-Reiche-Kuhn sum-rule[4, 52]. Attemptshave been made to distinguish between effective masses, which describe the energies around the Fermi energy, andthose characterizing the bulk spectrum by introducing a term, which leads to a surface peaking of the effective massterm in finite nuclei[53].Non-relativistic descriptions of nuclear matter, which are based on realistic interactions yield an effective isovectormass m ∗ V which is smaller than the corresponding effective isoscalar mass, which leads to a larger effective massfor neutrons than for protons in neutron-rich matter (see Fig. 5). In order to analyze this finding we inspect thedependence of the nucleon self-energy in the BHF approximation Σ BHFi , defined in eq.(7), as a function of energy ω and momentum k of the nucleon considered. Following the discussion of Mahaux and Sartor[54] one can define theeffective k -mass m k ( k ) M = (cid:20) Mk ∂ Σ( k, ω ) ∂k (cid:21) − (35)and the effective E -mass m E ( ω ) M = (cid:20) − ∂ Σ( k, ω ) ∂ω (cid:21) . (36)The effective mass can then be calculated from the effective k -mass and the effective E -mass by m ∗ ( k ) M = m ∗ k ( k ) M m ∗ E ( ω = ε ( k )) M . (37)Results for the effective k -mass and E − mass as obtained from BHF calculations for asymmetric nuclear matter ata density ρ = 0.17 fm − and a proton abundance Y π of 25 % ( I = 0 .
5) are displayed in Fig. 6. We notice that theeffective k -mass for the protons is significantly below the corresponding value for the neutrons at all momenta. Sincethe k -masses tend to increase as a function of the nucleon momentum k , the difference in the Fermi momenta forprotons and neutrons enhance the difference m ∗ k,n ( k F n ) − m ∗ k,p ( k F p ).The effective k -mass describes the non-locality of the BHF self-energy. This non-locality and thereby also thesefeatures of the effective k -mass are rather independent on the realistic interaction used. Furthermore it turns outthat the values for the k -mass are essentially identical if one derives them from the nucleon BHF self-energy usingthe G -matrix or from the bare interaction V or from V lowk [41]. This non-locality of the self-energy is dominated byFock-exchange contribution originating from π -exchange. In neutron-rich matter this contribution leads to a strongerdepletion for the proton mass than for the neutron mass[20, 55].The effective E -mass, representing the non-locality of the self-energy in time, yield values larger than M for momentaaround k F . Also in this case the deviation of m ∗ from M is larger for protons than for neutrons. The effective E -massoriginates from the energy-dependence of the G -matrix and is due to the admixture of 2-particle 1-hole configurationsto the single-particle states. One may also say that the effective E -mass is due to correlations beyond mean field. Itaccounts to the coupling of vibrational modes.Anyway, the enhancement of the effective mass m ∗ , which is due to the effective E -mass in eq.(37) is not strongenough to compensate the effects of the k -mass. Therefore the final effective mass is below the bare mass M and theeffective mass for neutrons remains larger than the corresponding one for protons.3 k [fm -1 ] e ff ec ti v e m a ss m * [ M ] k-mass, protonsk-mass, neutronsE-mass, protonsE-mass, neutrons FIG. 6: (Color online) Effective k -mass m ∗ k ( k ) (solid lines) and effective E -mass m ∗ E ( k ) (dashed lines) for neutronsand protons (lines with symbol) as obtained from the BHF calculations for asymmetric nuclear matter at thedensity ρ = 0 .
17 fm − and a proton abundance of 25 %. The Fermi momenta for protons and neutrons are indicatedby vertical dotted linesThe effects of the E -mass are weaker for V lowk than for the G -matrix. This is obvious since V lowk only accounts forladder diagrams included with particle-particle states above the cut-off, whereas the G -matrix includes all particle-particle states above the Fermi-momenta. This explains the lower effective masses obtained for V lowk than for theBHF approximation (see Fig. 5). Results on effective masses obtained from SCGF are rather similar to the BHFresults, therefore we do not discuss them here explicitly.We want to add that the results for the effective E -mass can be rather different in finite nuclei than in nuclearmatter. As it has already been mentioned above, this E -mass is due to the admixture of vibrational modes, which arequite different in finite nuclei as compared to nuclear matter. This may explain the differences in the predictions forthe effective isovector mass resulting from Skyrme interactions, which are based on fits of properties for finite nuclei,and those originating from realistic interactions.Finally we are going to address the results for effective masses as they originate from relativistic mean-field approx-imations. Here we have to distinguish between the values for the Dirac mass and the effective mass parameterizingthe single-particle spectrum according to eq.(33). The Dirac mass m ∗ D is defined in terms of the scalar part of theself-energy (see eq.(15)). In the case of the relativistic mean-field approximation the self-energies do not depend onenergy or momentum and the scalar self-energies for protons and neutrons are solely due to the direct contributions ofthe scalar mesons σ and δ . The Dirac masses obtained from DDRMF in nuclear matter at β -equilibrium are displayedin Fig. 7. We see that the Dirac masses decrease with density but show larger values for the protons than for theneutrons.In order to compare these results for the Dirac mass with the corresponding non-relativistic effective mass, we haveto compare the expressions for the single-particle energies eq.(17) and eq.(33) and adjust the parameters in such away that the expressions yield identical results and slopes as a function of k at the Fermi momentum. This leads to m ∗ i = q k F i + ( m ∗ Di ) , (38)where the label i refers to the case of proton and neutron. Results for these non-relativistic masses are displayedin Fig. 7 by solid lines. We find that the enhancement of the Dirac mass by the corresponding Fermi momenta ineq.(38) is significant for the neutrons in particular. This leads to the effect that at high densities the non-relativisticeffective mass for neutrons gets larger than the corresponding mass for protons, a behavior which is opposite to theone observed for the Dirac masses.It is worth noting that this feature, the difference of the Dirac masses m ∗ Dn − m ∗ Dp is negative while the difference ofthe corresponding non-relativistic masses is positive has been observed before within the framework of Dirac Brueckner4 density ρ [fm -3 ] e ff ec ti v e m a ss m * [ M ] nonrel.Dirac Neutrons: lineProtons: line with symbol
FIG. 7: (Color online) Effective masses originating from the DDRMF calculations of nuclear matter in β -equilibrium.The Dirac masses for protons and neutrons are represented by the dashed lines, while solid lines are used to identifythe effective masses according to eq.(33). The results for protons are shown in terms of lines with symbols.Hartree Fock (DBHF) even at small densities[35, 39, 56, 57]. The parameterization of the DDRMF approach has beenmade to reproduce the bulk properties of the DBHF of [35]. However, this adjustment cannot account for details likethe non-locality of the self-energies in DBHF. Therefore it does not reproduce such details as the effective isovectormass with good accuracy. IV. CONCLUSION
Various approaches to the nuclear many-body problem have been investigated to explore their predictions for nuclearmatter at high density and large proton - neutron asymmetries. Two of these approaches, the Skyrme Hartree-Fockand the Density Dependent Relativistic Mean Field approach are predominantly of phenomenological origin. Theirparameters have been adjusted to reproduce data of finite nuclei. However, the parameters have been selected in sucha way that also bulk properties of asymmetric nuclear matter derived from microscopic calculations are reproduced.The other three approaches are based on realistic NN interactions, which fit the NN scattering phase shifts. In theseapproximation schemes (Brueckner Hartree Fock BHF, Self-consistent Greens Function SCGF and Hartree Fock usinga renormalized interaction V lowk ) a isoscalar contact interaction has been added to reproduce the empirical saturationpoint of symmetric nuclear matter.These various approximation schemes lead to rather similar predictions for the energy per nucleon of symmetric andasymmetric nuclear matter at high densities. In detail one finds that the relativistic DDRMF leads to a rather stiffEquation of State (EoS) for symmetric matter while the BHF approach leads to a relatively soft EoS, a feature whichis compensated within the microscopic framework by the repulsive features of the hole-hole ladders included in SCGF.The phenomenological approximation schemes DDRMF (Skyrme Hartree Fock) over (under) estimate the symmetryenergy at high densities as compared to the microscopic approaches. The lack of long range (low energy) correlationeffects in V lowk leads to a symmetry energy which is too small already at normal density. These features are alsoreflected in the study of nuclear matter in the β -equilibrium and lead to moderate differences in the predictions forproton abundances and EoS.More significant differences are observed when we inspect details like the effective masses, in particular the isovectoreffective mass. In neutron-rich matter the microscopic approaches predict a positive difference between neutron andproton effective masses. This feature can be related to the non-locality of the self-energy induced by one-pion exchangeterm and is expressed in terms of an effective k -mass. This feature may partly be compensated by the effects ofvibrational excitation modes on the nucleon mean fields. The effects of such low-energy excitations might lead todifferent results in nuclear matter and finite nuclei. This could be study e.g. in many-body calculations employing V lowk , which account for the effects of vibrational modes explicitly.We also discuss the differences between effective Dirac masses and corresponding non-relativistic masses in neutron-5rich matter. While the difference between the neutron and proton Dirac masses is negative, the differences of thecorresponding non-relativistic masses tend to get positive. Further studies of the non-localities in space and timeof various components in the Dirac self-energy would be useful to explore the connections to the non-relativisticmicroscopic approaches more in detail. V. ACKNOWLEDGEMENTS
This work has been supported by the Deutsche Forschungsgemeinschaft DFG (Mu 705/5-1). [1] T.H.R. Skyrme, Nucl. Phys. , 615 (1959).[2] D. Vautherin and D.M. Brink, Phys. Rev. C , 626, (1972).[3] P. Bonche and D. Vautherin, Nucl Phys. A , 496 (1981).[4] P. Ring and P. Schuck, The Nuclear Many–Body Problem , (Springer, Berlin Heidelberg New York, 1980).[5] J.R. Stone and P.-G. Reinhard, Prog. Part. Nucl. Phys. , 587 (2007).[6] E. Chabanat, P. Bonche, P. Haensel, J. Meyer, and R. Schaeffer, Nucl. Phys. A , 710, (1997) and Nucl. Phys. A635 ,231, (1998).[7] F. Douchin, P. Haensel, and J. Meyer, Nucl. Phys.
A665 , 419 (2000).[8] P. G¨ogelein and H. M¨uther, Phys. Rev. C , 024312 (2007).[9] D. Vretenar, A. Afanasjev, G.A. Lalazissis, and P. Ring, Phys. Rept. , 101 (2005).[10] P. G¨ogelein, E.N.E. van Dalen, C. Fuchs, and H. M¨uther, Phys. Rev C , 025802 (2008).[11] R. Machleidt, F. Sammarruca, and Y. Song, Phys. Rev. C , R1483 (1996).[12] R.B. Wiringa, V.G.J. Stoks, and R. Schiavilla, Phys. Rev. C , 38(1995).[13] H. M¨uther and A. Polls, Prog. Part. Nucl. Phys. , 243 (2000).[14] W. H. Dickhoff and C. Barbieri, Prog. Part. Nucl. Phys. , 377 (2004).[15] Y. Dewulf, W. H. Dickhoff, D. Van Neck, E. R. Stoddard and M. Waroquier, Phys. Rev. Lett. , 152501 (2003).[16] P. Bo˙zek, Phys. Rev. C , 2619 (1999).[17] P. Bo˙zek, Phys. Rev. C , 054306 (2002).[18] T. Frick and H. M¨uther, Phys. Rev. C , 034310 (2003).[19] T. Frick, H. M¨uther, A. Rios, A. Polls, and A. Ramos, Phys. Rev. C , 014313 (2005).[20] Kh.S.A. Hassaneen and H. M¨uther, Phys. Rev. C , 054308 (2004).[21] Kh. Gad and Kh.S.A. Hassaneen, Nucl. Phys. A , 67 (2007).[22] A. Akmal and V.R. Pandharipande, Phys. Rev. C , 2261 (1997).[23] S. Fantoni and A. Fabrocini, in Microscopic Quantum Many-Body Theories and Their Applications , edited by J. Navarroand A. Polls (Springer, New York, 1998).[24] S. Fantoni, in
Introduction to Modern Methods of Quantum Many-Body Theory and Their Applications , edited by A.Fabrocini, S. Fantoni, and E. Krotscheck, Series on Advances in Quantum Many-Body Theory, Vol. 7, (World Scientific,Singapore 2002).[25] F. Coester, S. Cohen, B.D. Day, and C.M. Vincent, Phys. Rev. C , 769 (1970).[26] S.C. Pieper, R.B. Wiringa, and J. Carlson, Phys. Rev. C , 054325 (2004).[27] S.K. Bogner, T.T.S. Kuo, and A. Schwenk, Phys. Rept. , 1 (2003).[28] S.K. Bogner, A. Schwenk, R.J. Furnstahl, and A. Nogga, Nucl. Phys. A763 , 59 (2005).[29] S.K. Bogner, T.T.S. Kuo, and A. Schwenk, Phys. Rept. , 1 (2003).[30] H. Hergert and R. Roth, Phys. Rev. C , 051001 (2007).[31] P. Bo˙zek, D.J. Dean, and H. M¨uther, Phys. Rev. C , 014303 (2006).[32] J. Kuckei, F. Montani, H. M¨uther, and A. Sedrakian, Nucl. Phys. A 723 , 32 (2003).[33] E. Schiller, H. M¨uther, and P. Czerski, Phys. Rev. C , 2934 (1999); E. Schiller, H. M¨uther, and P. Czerski, Phys. Rev.C , 059901 (1999).[34] R. Fritz and H. M¨uther, Phys. Rev. C , 633 (1994); R. Fritz, Ph-D thesis, T¨ubingen (1994).[35] E.N.E. van Dalen, C. Fuchs, and A. Faessler, Eur. Phys. J. A , 29 (2007).[36] B.D. Serot and J.D. Walecka, Adv. Nucl. Phys. , 1 (1986).[37] R. Brockmann and R. Machleidt, Phys. Rev. C , 1965 (1990).[38] C. Fuchs, H. Lenske, and H.H. Wolter, Phys. Rev. C , 3043 (1995).[39] E. Schiller and H. M¨uther, Eur. Phys. J. A11 , 15 (2001).[40] P.Grange, J. Cugnon, and A. Lejeune, Nucl. Phys. A , 365 1987.[41] T. Frick, Kh. Gad, H. M¨uther, and P. Czerski, Phys. Rev. C , 034321 (2002).[42] D.J. Dean, T. Engeland, M. Hjorth-Jensen, M. Kartamychev, and E. Osnes, Prog. Part. Nucl. Phys. , 419 (2004).[43] Z.Y. Ma and T.T.S. Kuo, Phys. Lett. , 137 (1983).[44] H.Q. Song and T.T.S. Kuo, Phys. Rev. C , 2883 (1991). [45] T.T.S. Kuo and Y. Tzeng, Int. J. of Mod. Phys. E , 523 (1994).[46] K. Suzuki and R. Okamoto, Prog. Theor. Phys. ,1045 (1994).[47] S. Fujii, R. Okamoto, and K. Suzuki, Phys. Rev. C , 034328 (2004).[48] A. Polls, H. M¨uther, A. Faessler, T.T.S. Kuo, and E. Osnes, Nucl. Phys. A 401 , 124 (1983).[49] H. M¨uther, A. Polls, and T.T.S. Kuo, Nucl. Phys.
A 435 , 548 (1985).[50] A.E.L. Dieperink, Y. Dewulf, D. Van Neck, M. Waroquier, and V. Rodin, Phys. Rev. C , 064307 (2003).[51] L. Engvik, G. Bao, M. Hjorth-Jensen, E. Osnes, E. Oestaard, Astrophysical Journal , 794 (1996).[52] M. Bender, P.-H. Heenen, and P.-G. Reinhard, Rev. Mod. Phys. , 121 (2003).[53] M. Farine, J.M. Pearson, and F. Tondeur, Nucl. Phys. A 696 , 396 (2001).[54] C. Mahaux and R. Sartor,
Advances in Nuclear Physics , edited by J. W. Negele and E. Vogt (Plenum, New York, 1991),Vol. 20.[55] W. Zuo, L.G. Cao, B.A. Li, U. Lombardo, and C.W. Shen, Phys. Rev. C , 014005 (2005).[56] F. Sammarruca, W. Barredo, and P. Krastev, Phys. Rev. C , 064306 (2005).[57] E.N.E. van Dalen, C. Fuchs, and A. Faessler, Phys. Rev. Lett. , 022302 (2005); Phys. Rev. C72