Asymmetric nuclear matter : a variational approach
aa r X i v : . [ nu c l - t h ] M a y Asymmetric Nuclear Matter : A variational Approach
S. Sarangi
ICFAI Institute of Science & Technology, Bhubaneswar-751010, India
P. K. Panda
Indian Association for the Cultivation of Sciences, Jadavpur, Kolkata-700 032, India
S. K. Sahu
Physics Department, Banki College, Banki-754008, Cuttack, India
L.Maharana
Physics Department, Utkal University, Bhubaneswar-751004, India ∗ We discuss here a self-consistent method to calculate the properties of the cold asymmetric nuclearmatter. In this model, the nuclear matter is dressed with s -wave pion pairs and the nucleon-nucleon (N-N) interaction is mediated by these pion pairs, ω and ρ mesons. The parameters ofthese interactions are calculated self-consistently to obtain the saturation properties like equilibriumbinding energy, pressure, compressibility and symmetry energy. The computed equation of state isthen used in the Tolman- Oppenheimer-Volkoff (TOV) equation to study the mass and radius of aneutron star in the pure neutron matter limit. PACS numbers: 21.65.+f,21.30.Fe, 24.10.Cn,26.60.+c
I. INTRODUCTION
The search for an appropriate nuclear equation of state has been an area of considerable research interest becauseof its wide and far reaching relevance in heavy ion collision experiments and nuclear astrophysics. In particular, thestudies in two obvious limits, namely, the symmetric nuclear matter (SNM) and the pure neutron matter (PNM) havehelped constrain several properties of nuclear matter such as binding energy per nucleon, compressibility modulus,symmetry energy and its density dependence at nuclear saturation density ρ [1, 2, 3] to varying degrees of success. Oflate, the avaliability of flow data from heavy ion collision experiments and phenomenological data from observation ofcompact stars have renewed the efforts to further constrain these properties and to explore their density and isospincontent (asymmetry) variation behaviours [4, 5, 6, 7].One of the fundamental concerns in the construction of nuclear equation of state is the parametrization of thenucleon-nucleon (N-N) interaction. Different approaches have been developed to address this problem. These methodscan be broadly classified into three general types [8], namely, the ab initio methods, the effective field theory approachesand calculations based on phenomenological density functionals. The ab initio methods include the Brueckner-Hartree-Fock (BHF) [9, 10, 11] approach, the (relativistic) Dirac-Brueckner-Hartree-Fock (DBHF) [12, 13, 14, 15, 16]calculations, the Green Function Monte-Carlo (GFMC) [17, 18, 19] method using the basic N-N interactions givenby boson exchange potentials. The other approach of this type, also known as the variational approach, is pioneeredby the Argonne Group [20, 21]. This method is also based on basic two-body (N-N) interactions in a non-relativisticformalism with relativistic effects introduced successively at later stages. The effective field theory (EFT) approachesare based on density functional theories [22, 23] like chiral perturbation theory [24, 25]. These calculations involve afew density dependent model parameters evaluated iteratively. The third type of approach, namely, the calculationsbased on phenomenological density functionals include models with effective density dependent interactions such asGogny or Skyrme forces [26] and the relativistic mean field (RMF) models [27, 28, 29, 30]. The parameters of thesemodels are evaluated by carefully fitting the bulk properties of nuclear matter and properties of closed shell nuclei toexperimental values. Our work presented here belongs to this class of approaches in the non-relativistic approximation.The RMF models represent the N-N interactions through the coupling of nucleons with isoscalar scalar σ mesons,isoscalar vector ω mesons, isovector vector ρ mesons and the photon quanta besides the self- and cross-interactionsamong these mesons [29]. Nuclear equations of state have also been constructed using the quark meson coupling model ∗ Electronic address: [email protected] (QMC) [31] where baryons are described as systems of non-overlapping MIT bags which interact through effectivescalar and vector mean fields, very much in the same way as in the RMF model. The QMC model has also beenapplied to study the asymmetric nuclear matter at finite temperature [32].It has been shown earlier [33, 34], that the medium and long range attraction effect simulted by the σ mesons in RMFtheory can also be produced by the s -wave pion pairs. This “dressing” of nucleons by pion pairs has also been appliedto study the properties of deuteron[35] and He [36]. On this basis, we start with a nonrelativistic Hamiltonian densitywith πN interaction. The ω − repulsion and the isospin asymmetry part of the NN interaction are parametrized bytwo additional terms representing the coupling of nucleons with the ω and the ρ mesons respectively. The parametersof these interactions are then evaluated self-consistently by using the saturation properties like binding energy pernucleon, pressure, compressibility and the symmetry energy. The equation of state (EOS) of asymmetric nuclearmatter is subsequently calculated and compared with the results of other independent approaches available in currentliterature. The EOS of pure neutron matter is also used to calculate the mass and radius of a neutron star. Weorganize the paper as follows: In Section II, we present the theoretical formalism of the asymmetric nuclear matteras outlined above. The results are presented and discussed in Section III. Finally, in the last section the concludingremarks are drawn indicating the future outlook of the model. II. FORMALISM
We start with the effective pion nucleon Hamiltonian H ( x ) = H N ( x ) + H int ( x ) + H M ( x ) , (1)where the free nucleon part H N ( x ) is given by H N ( x ) = ψ † ( x ) ε x ψ ( x ) , (2)the free meson part H M ( x ) is defined as H M ( x ) = 12 (cid:2) ˙ ϕ i + ( ∇ ϕ i ) · ( ∇ ϕ i ) + m ϕ i (cid:3) , (3)and the πN interaction [33] is provided by H int ( x ) = ψ † ( x ) (cid:20) − iG ǫ x σ · p ϕ + G ǫ x ϕ (cid:21) ψ ( x ) . (4)In equations (2) and (4), ψ represents the non-relativistic two component spin-isospin quartet nucleon field. Thesingle particle nucleon energy operator ǫ x is given by ǫ x = ( M − ∇ x ) / with nucleon mass M and the pion-nucleoncoupling constant G . The isospin triplet pion fields of mass m are represented by ϕ .We expand the pion field operator ϕ i ( x ) in terms of the creation and annihilation operators of off-mass shell pionssatisfying equal time algebra as ϕ i ( x ) = 1 √ ω x ( a i ( x ) † + a i ( x )) , ˙ ϕ i ( x ) = i r ω x a i ( x ) † − a i ( x )) , (5)with energy ω x = ( m − ∇ x ) / in the perturbative basis. We continue to use the perturbative basis, but note thatsince we take an arbitrary number of pions in the unitary transformation U in equation (7) as given later, the resultswould be nonperturbative. The expectation value of the first term of H int ( x ) in eq. (4) vanishes and the pion pairof the second term provides the isoscalar scalar interaction of nucleons thereby simulating the effects of σ -mesons. Apion-pair creation operator given as B † = 12 Z f ( k ) a i ( k ) † a i ( − k ) † d k , (6)is then constructed in momentum space with the ansatz function f ( k ) to be determined later.We then define the unitary transformation U as U = e ( B † − B ) , (7)and note that U , operating on vacuum, creates an arbitrarily large number of scalar isospin singlet pairs of pions.The “pion dressing” of nuclear matter is then introduced through the state | f i = U | vac i = e ( B † − B ) | vac i , (8)where U constitutes a Bogoliubov transformtion given by U † a i ( k ) U = (cosh f ( k )) a i ( k ) + (sinh f ( k )) a i ( − k ) † , (9)We then proceed to calculate the energy expectation values. We consider N nucleons occupying a spherical volumeof radius R such that the density ρ = N/ ( πR ) remains constant as ( N, R ) → ∞ and we ignore the surface effects.We describe the system with a density operator ˆ ρ N such that its matrix elements are given by [33] ρ αβ ( x , y ) = T r [ˆ ρ N ψ β ( y ) † ψ α ( x )] , (10)and T r [ˆ ρ N ˆ N ] = Z ρ αα ( x , x ) d x = N = ρV. (11)We obtain the free nucleon energy density h f = h f | T r [ˆ ρ N H N ( x )] | f i = X τ = n,p γk τf π M + 310 k τ f M ! . (12)In the above equation, the spin degeneracy factor γ = 2, the index τ runs over the isospin degrees of freedom n and p and k τf represents the Fermi momenta of the nucleons. For asymmetric nuclear matter, we define the neutron andproton densities ρ n and ρ p respectively over the same spherical volume such that the nucleon density ρ = ρ n + ρ p .The Fermi momenta k τf are related to neutron and proton densities by the relation k τf = (6 π ρ τ /γ ) . We also definethe asymmetry parameter y = ( ρ n − ρ p ) /ρ . It can be easily seen that ρ τ = ρ (1 ± y ) for τ = n, p respectively.Using the operator expansion of equation (5), the free pion part of the Hamiltonian as given in equation (3) can bewritten as H M ( x ) = a i ( x ) † ω x a i ( x ) . (13)The free pion kinetic energy density is given by h k = h f |H M ( x ) | f i = 3(2 π ) Z d k ω ( k ) sinh f ( k ) , (14)where ω ( k ) = √ k + m . Using ǫ x ≃ M in the nonrelativistic limit, the interaction energy density h int can be writtenfrom equation (4) as h int = h f | T r [ˆ ρ N H int ( x )] | f i ≃ G ρ M h f | : ϕ i ( x ) ϕ i ( x ) : | f i . (15)Using the equations (7), (8) and (9), we have from equation (15) h int = G ρ M (cid:18) π ) Z d k ω ( k ) (cid:18) sinh 2 f ( k )2 + sinh f ( k ) (cid:19)(cid:19) . (16)The pion field dependent energy density terms add up to give h m (= h k + h int ) which is to be optimized with respectto the ansatz function f ( k ) for its evaluation. However, this ansatz function yields a divergent value for h m . Thishappens because we have taken the pions to be point like and have assumed that they can approach as near each otheras they like, which is physically inaccurate. Therefore, we introduce a phenomenological repulsion energy betweenthe pions of a pair given by h Rm = 3 a (2 π ) Z (sinh f ( k )) e R π k d k , (17)where the two parameters a and R π correspond to the strength and length scale, repectively, of the repulsion andare to be determined self-consistently later. Thus the pion field dependent term of the total energy density becomes h m = h k + h int + h Rm . Then the optimization of h m with respect to f ( k ) yieldstanh 2 f ( k ) = − G ρ M · ω ( k ) + G ρ M + aω ( k ) e R π k . (18)The expectation value of the pion field dependent parts of the total Hamiltonian density of eqn. (1) alongwith themodification introduced by the phenomenological term h Rm becomes h m = −
32 1(2 π ) (cid:16) G M (cid:17) ρ h ρ n I n + ρ p I p i (19). with the integrals I τ ( τ = n, p ) given by I τ = Z k τf πk dkω h ω + ae R π k ) / ( ω + ae R π k + G ρMω ) / + ( ω + ae R π k ) + G ρ Mω i (20)and ω = ω ( k ).We now introduce the energy of ω repulsion by the simple form h ω = λ ω ρ , (21)where the parameter λ ω corresponds to the strength of the interaction at constant density and is to be evaluatedlater. We note that equation (21) can arise from a Hamiltonian density given in terms of a local potential v R ( x ) as H R ( x ) = ψ ( x ) † ψ ( x ) Z v R ( x − y ) ψ ( y ) † ψ ( y ) d y , (22)where, when density is constant, we in fact have λ ω = Z v R ( x ) d x . The isospin dependent interaction is mediated by the isovector vector ρ mesons. We represent the contribution dueto this interaction, in a manner similar to the ω -meson energy, by the term h ρ = λ ρ ρ (23)where ρ = ( ρ n − ρ p ) and the strength parameter λ ρ is to be determined as described later. Thus we finally writedown the binding energy per nucleon E B of the cold asymmetric nuclear matter: E B = ερ − M, (24)where ε = ( h f + h m + h ω + h ρ ) is the energy density. The expression for ε contains the four model parameters a , R π , λ ω and λ ρ as introduced above. These parameters are then determined self-consistently through the saturationproperties of nuclear matter. The pressure P , compressibility modulus K and the symmetry energy E sym are givenby the standard relations: P = ρ ∂ ( ε/ρ ) ∂ρ (25) K = 9 ρ ∂ ( ε/ρ ) ∂ρ (26) E sym = (cid:18) ∂ ( ε/ρ ) ∂y (cid:19) y =0 . (27)The effective mass M ∗ is given by M ∗ = M + V s with V s = ( h int + h Rm ) /ρ. III. RESULTS AND DISCUSSION
We now discuss the results obtained in our calculations and compare with those available in literature. The fourparameters of the model are fixed by self-consistently solving eqs. (24) through (27) for the respective properties ofnuclear matter at saturation density ρ = 0.15 fm − . While pressure P vanishes at saturation density for symmetricnuclear matter (SNM), the values of binding energy per nucleon and symmetry energy are chosen to be −
16 MeVand 31 MeV respectively. In the numerical calculations, we have used the nucleon mass M = 940 MeV, the mesonmasses m = 140 MeV, m ω = 783 MeV and m ρ = 770 MeV and the π − N coupling constant G / π = 14 .
6. Inorder to ascertain the dependence of compressibility modulus on the parameter values, we vary the K value over arange 210 MeV to 280 MeV for the symmetric nuclear matter ( y = 0) and evaluate the parameters. It may be notedthat this is the range of the compressibility value which is under discussion in the current literature. For K values inthe range 210 MeV to 250 MeV, the program does not converge. The solutions begin to converge for compressibilitymodulus K around 258 MeV. We choose the value K = 260 MeV for our calculations. In Table I we present the fourfree parameters of the model for ready reference. TABLE I: Parameters of the model obtained by solving the equations (24)- (27) self consistently at saturation density. a R π λ ω λ ρ (MeV) (fm) (fm ) (fm )16.98 1.42 3.10 0.65 / -20020406080100120140160180 E B ( M e V ) y = 0.0 (SNM)y = 0.5y = 1.0 (PNM) FIG. 1: The binding energy per nucleon E B as a function of relative nucleon density ρ/ρ calculated for different values of theasymmetry parameter y . The values y = 0.0 and 1.0 correspond to symmetric nuclear matter (SNM) and pure neutron matter(PNM) respectively. For this set of parameter values the effective mass of nucleons at saturation density is found to be M ∗ /M = 0 .
81. Inthe Fig. 1, we present the binding energy per nucleon E B calculated for different values of the asymmetry parameter y as a function of the relative nuclear density ρ/ρ . The values y = 0.0 and 1.0 correspond to SNM and PNMrespectively. As expected, the binding energy per nucleon E B of SNM initially decreases with increase in density,reaches a minimum at ρ = ρ and then increases. In case of PNM, the binding energy increases monotonically withincreasing density in consistence with its well known behaviour. In Fig. 2(a), we compare the E B of SNM as a functionof the nucleon density with a few representative results in the literature, namely, the Walecka model [27] (long-shortdashed curve), the DBHF calculations of Li et al. with Bonn A potential (short-dashed curve) (data for both themodels are taken from [13]) and the variational A18 + δ v + UIX* (corrected) model of Akmal at al. (APR) [21](long-dashed curve). While the Walecka and Bonn A models are relativistic, the variational model is nonrelativisticwith relativistic effects and three body correlations introduced successively. Our model produces an EOS softer thanthat of Walecka and Bonn A, but stiffer than the variational calculation results of the Argonne group. It is well-knownthat the Walecka model yields a very high compressibility K . However, its improvised versions developed later withself- and cross-couplings of the meson fields have been able to bring down the compressibility modulus in the ball parkof 230 ±
10 MeV [7]. Our model yields nuclear matter saturation properties correctly alongwith the compressibility of K = 260 MeV which is resonably close to the empirical data. In Fig. 2(b), we plot E B as a function of the relativenucleon density for PNM. Similar to the SNM case, our EOS is softer than that of Walecka and Bonn A models, butstiffer than the variational model. We use this EOS to calculate the mass and radius of a neutron star of PNM asdiscussed later.The density dependence of pressure of SNM and PNM are calculated using the eqn. 25. These results are plotted(solid blue curves) in Figs. 3(a) and (b). Recently, Danielewicz et al. [4] have deduced the empirical bounds on theEOS in the density range of 2 < ρ/ρ < . FIG. 2: (a)The binding energy per nucleon E B as a function of relative nucleon density ρ/ρ for SNM. The results of presentwork (P.W.) are compared with the results of DBHF calculations with Bonn A potential [13], the variational calculations ofthe Argonne group [21] and the Walecka model [27]. The data for the Bonn A and Walecka model curves are taken from [13].(b) Same as Fig-(2a), but for PNM. The potentials per nucleon in our model can be defined from the meson dependent energy terms of eqs. (19), (21)and (23). Contribution to potential from the scalar part of the meson interaction is due to the pion condensates andis given by V s = ( h int + h Rm ) /ρ as defined earlier. The contribution by vector mesons has two components, namely,due to the ω and the ρ mesons and is given by V v = V ω + V ρ = ( h ω + h ρ ) /ρ . In the Figs. 4 (a) and (b), we plot V s and V v as functions of relative density ρ/ρ calculated for PNM (Fig. 4(a)) and for SNM (Fig. 4(b)) respectively. Themagnitudes of the potentials calculated by our model are weaker compared to those produced by DBHF calculationswith Bonn A interaction [13] as shown in both the panels of Fig. 4. In Fig. 4(a), we show the contributions to therepulsive vector potential due to ω mesons (short-dashed curve), ρ mesons (long-dashed curve) and their combinedcontribution (long-short-dashed curve). The contribution due to ρ mesons rises linearly at a slow rate and has a low etal.et al.*et al.** FIG. 3: (a) The pressure as a function of relative nucleon density for SNM as generated by the present work (P.W.) (solid bluecurve). The color-filled region in green corresponds to the bounds deduced from experimental flow data and simulations studiesby Danielewicz et al. [4]. The data for the curves corresponding to RMF(NL3) calculations and the variational calculations ofAkmal et al. (APR) are taken from [4].(b) Pressure as a function of relative nucleon density for PNM. The shaded region and thecolor-filled region in green correspond to the bounds deduced by Danielewicz et al. using the “stiff” and “soft” parametrizationsof Prakash et al. [37]. Our EOS is consistent with these bounds in the cases of both SNM and PNM. contribution at saturation density. This indicates that major contribution to the short-range repulsion part of nuclearforce is from ω meson interaction.Knowledge of density dependence of symmetry energy is expected to play a key role in understanding the structureand properties of neutron-rich nuclei and neutron stars at densities above and below the saturation density. Thereforethis problem has been receiving considerable attention of late. Several theoretical and experimental investigationsaddressing this problem have been reported ([3, 8, 39] and references therein). While the results of independentstudies show reasonable consistency at sub-saturation densities ρ ≤ ρ , they are at wide variance with each other atsupra-saturation densities ρ > ρ . This wide variation has given rise to the so-called classification of “soft” and “stiff”dependence of symmetry energy on density [38, 39].Fig. 5 shows a representation of the spectrum of such results alongwith the results of the present work (solid bluecurve). While the Gogny and Skyrme forces (dark rib-dotted and dotted curves respectively with data taken from[8, 39]) produce “soft” dependence on one end, the NL3 force (dot-dashed curve with data taken from [8]) produces avery “stiff” dependence on the other end. The analysis of experimental and simulation studies of intermediate energyheavy-ion reactions as reported by Shetty et al. [39] (red triangles and long-short-dashed red curve repectively), resultsof DBHF calculations of Li et al. and Huber et al. [13, 29, 40] (rib-dashed and magenta ribbed curve), variationalmodel [3, 21] (short-dashed curve), RMF calculations with nonlinear Walecka model including ρ mesons by Liu etal. [30] (long-dashed green curve) as shown in Fig. 5 suggest “stiff” dependence with various degrees of stiffness.The experimental results (represented by the red triangles with data taken from Shetty et al. [39]) are derived fromthe isoscaling parameter α which, in turn, is obtained from relative isotopic yields due to multifragmentation ofexcited nuclei produced by bombarding beams of Fe and Ni on Fe and Ni targets. Shetty et al. have shownthat the results of multifragmentation simulation studies carried out with Antisymmetrized Molecular Dynamics(AMD) model using Gogny-AS interaction and Statistical Multifragmentation Model (SMM) are consistent with theabove-mentioned experimental results and suggest (as shown by the red long-short-dashed curve) a moderately stiffdependence of the symmetry energy on density. Our results (represented by the solid blue curve) calculated usingeqn. (27) are consistent with these results at subsaturation densities but are stiffer at supra-saturation densities. More (b) SNM -600-400-2000200400600800 P o t e n ti a l s ( M e V ) Bonn A (Scalar) [13]Bonn A (Vector) [13]P. W. (V s )P. W. (V )P. W. (V )P. W. (V +V ) (a) PNM FIG. 4: (a) The potentials V s , V ω and V ρ (as defined in the text) in PNM as calculated by our model are compared withthe Bonn A results of Li et al. [13]. The contributions made by the ω -meson (short-dashed curve) and ρ -meson (long-dashedcurve) mediated interactions are distinctly shown for comparison. (b) The potentials in SNM. Because of isospin symmetry, V ρ (see text for definition) vanishes. Both the scalar (solid curve) and vector (short-dashed curve) potentials produced by ourcalculations are weaker in magnitude compared to those of Bonn A calculations. observational or experimental information is required to be built into our model to further constrain the symmetryenergy at higher densities. In Fig.5, the curve due to Huber et al. [40] (with data taken from [29]) correspond totheir DBHF ‘HD’ model calculations which involves only the σ , ω and ρ mesons. Similarly the long-dashed greencurve due to Liu et al. [30] is from the basic non-linear Walecka model with σ , ω and ρ mesons. Our formalism is theclosest to these two models with the exception that in our model the effect of σ mesons is simulated by the π mesoncondensates. It is also noteworthy that our results are consistent with these results for densities upto 2 ρ .The wide variation of density dependence of symmetry energy at supra-saturation densities has given rise to theneed of constraining it. As discussed by Shetty et al [39], a general functional form E sym = E sym ( ρ/ρ ) γ hasemerged. Studies by various groups have produced the fits with E sym ∼ −
33 MeV and γ ∼ . − .
05. A similarparametrization of the E sym produced by our EOS with E sym = 31 MeV yields the exponent parameter γ = 0.85.We next use the equation of state for PNM derived by our model in the Tolman-Oppenheimer-Volkoff (TOV)equation to calculate the mass and radius of a PNM neutron star. The mass and radius of the star are found to be2 . M ⊙ and 11.7 km respectively. IV. CONCLUSION
In this work we have presented a quantum mechanical nonperturbative formalism to study cold asymmetric nuclearmatter using a variational method. The system is assumed to be a collection of nucleons interacting via exchange of π pairs, ω and ρ mesons. The equation of state (EOS) for different values of asymmetry parameter is derived from thedynamics of the interacting system in a self-consistent manner. This formalism yields results similar to those of the ab initio DBHF models, variational models and the RMF models without invoking the σ mesons. The compressibilitymodulus and effective mass are found to be K = 260 MeV and M ∗ /M = 0.81 respectively. The symmetry energycalculated from the EOS suggests a moderately “stiff” dependence at supra-saturation densities and corroborates therecent arguments of Shetty et al. [39]. A parametrization of the density dependence of symmetry energy of the form E s y m ( M e V ) Multifragmentation Expt. [39]P.W.AMD (Gogny-AS) [39]AMD (Gogny) [39]DBHF (Bonn A) [29]DBHF ( ) [29,40]APR [3,21]RMF (NL ) [30]Skyrme [8]NL3 [8]
FIG. 5: Symmetry energy E sym calculated from the EOS (as in Eq. 27) (P.W.) (solid blue line) is plotted as a function ofdensity along with results of other groups. The data for experimental points and the results of the antisymmetrized moleculardynamics (AMD) simulations with Gogny-AS and Gogny interactions are taken from Shetty et al [39], DBHF (Bonn A) resultsare taken from [29], RMF (NL ρ ) data are from [30], the variational model of Akmal et al. (APR) [21] results are from [3], DBHF( σωρ ) model of Huber et al. [40] data are from [29], the Skyrme amd NL3 results are from [8]. Our result shows consistencywith those of other groups and corroborates the moderately “stiff” dependence of E sym as advocated by Shetty et al. [39]. E sym = E sym ( ρ/ρ ) γ with the symmetry energy E sym at saturation density being 31 MeV produces γ = 0 .
85. TheEOS of pure neutron matter (PNM) derived by the formalism yields the mass and radius of a PNM neutron star tobe 2 . M ⊙ and 11.7 km respectively. V. ACKNOWLEDGEMENTS
P.K.P would like to acknowledge Julian Schwinger foundation for financial support. P.K.P wishes to thank ProfessorF.B. Malik and Professor Virulh Sa-yakanit for inviting the CMT31 workshop. The authors are also thankful toProfessor S.P. Misra for many useful discussions.
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