Symmetry energy of hot nuclei in the relativistic Thomas-Fermi approximation
aa r X i v : . [ nu c l - t h ] N ov Symmetry energy of hot nuclei in the relativistic Thomas-Fermiapproximation
Z. W. Zhang, S. S. Bao, J. N. Hu, and H. Shen ∗ School of Physics, Nankai University, Tianjin 300071, China
Abstract
We develop a self-consistent description of hot nuclei within the relativistic Thomas–Fermi ap-proximation using the relativistic mean-field model for nuclear interactions. The temperaturedependence of the symmetry energy and other physical quantities of a nucleus are calculated byemploying the subtraction procedure in order to isolate the nucleus from the surrounding nucleongas. It is found that the symmetry energy coefficient of finite nuclei is significantly affected bythe Coulomb polarization effect. We also examine the dependence of the results on nuclear inter-actions and make a comparison between the results obtained from relativistic and nonrelativisticThomas-Fermi calculations.
PACS numbers: 21.10.Dr, 21.30.Fe, 21.65.EfKeywords: Symmetry energy, Thomas-Fermi approximation, Relativistic mean-field model ∗ Electronic address: [email protected] . INTRODUCTION The properties of hot nuclei, such as excitation energies, entropies, symmetry energies,and density distributions, are of great interest in both nuclear physics and astrophysics [1].Especially, the symmetry energy and its dependence on density and temperature play a cru-cial role in understanding various phenomena in heavy-ion collisions, supernova explosions,and neutron-star properties [2–4]. Hot nuclei formed in nucleus-nucleus collisions are thermo-dynamically unstable against the emission of nucleons. Theoretically, an external pressure isimposed on the nucleus to compensate for the tendency of nucleon emission. This pressureis assumed to be exerted by a surrounding gas representing evaporated nucleons, which is inequilibrium with the hot nucleus. In order to isolate the nucleus from the surrounding gas,a subtraction procedure was first proposed in the Hartree-Fock framework [5] and then usedin the Thomas-Fermi approach [6]. The subtraction procedure is based on the existence oftwo solutions to the equations of motion, one corresponding to the nuclear liquid phase inequilibrium with the surrounding gas (
N G ) and the other corresponding to the nucleon gasalone ( G ). The density profile of the nucleus ( N ) is then obtained by subtracting the gasdensity from that of the liquid-plus-gas phase. As a result, physical quantities of the isolatednucleus obtained using the subtraction procedure could be independent of the size of the boxin which the calculation is performed. This subtraction procedure has been widely used inthe nonrelativistic Thomas-Fermi approximation with Skyrme effective interactions [7–13].It is interesting and important to develop a relativistic Thomas-Fermi model for the descrip-tion of hot nuclei by employing the subtraction procedure and to investigate the temperaturedependence of the symmetry energy of finite nuclei.The relativistic Thomas-Fermi approximation has been developed and applied to studythe droplet formation [14, 15] and nuclear pasta phases [16–18] in asymmetric nuclear matterat subnuclear densities. This method is considered to be self-consistent in the treatmentof surface effects and nucleon distributions. The relativistic Thomas-Fermi approximationhas been used to describe finite nuclei [19, 20] and nonuniform nuclear matter for supernovasimulations [21]. In Refs. [19, 20], the caloric curve for finite nuclei was studied within therelativistic Thomas-Fermi approximation, and the results obtained were found to depend onthe input freeze-out volume, which is actually the box size for performing the calculation. Inthe present work, we aim to study the properties of hot nuclei within the relativistic Thomas–2ermi approximation by employing the subtraction procedure, so that the results obtainedcan be independent of the box size. For the nuclear interaction, we adopt the relativisticmean-field (RMF) model, which has been successfully used to study various phenomena innuclear physics [22–24]. In the RMF approach, nucleons interact via the exchange of scalarand vector mesons, and the parameters are generally fitted to nuclear matter saturationproperties or ground-state properties of finite nuclei. In the present calculations, we con-sider four different RMF parametrizations, NL3 [25], TM1 [26], FSU [27], and IUFSU [28],so that we can examine the dependence of results on the RMF parametrization. TheseRMF parametrizations are known to be successful in describing the ground state propertiesof finite nuclei including unstable ones. In this work, we intend to adopt these differentRMF parametrizations to study the properties of hot nuclei and investigate the temperaturedependence of the symmetry energy of finite nuclei within the relativistic Thomas-Fermiapproximation.Because of the increasing importance of the symmetry energy in nuclear physics andastrophysics, there have been numerous studies on the symmetry energy of nuclear matterbased on various many-body methods [2–4]. However, there are fewer calculations for thesymmetry energy of finite nuclei [29–35]. In Ref. [32], the symmetry energy of finite nucleiwas calculated based on a density-functional theory at both zero and finite temperatures.A parametrized Thomas-Fermi approach was used in Ref. [33] to estimate the density de-pendence of the symmetry energy from nuclear masses. In Ref. [35], the symmetry energycoefficients of finite nuclei were extracted in the framework of the Skyrme-Hartree-Fockapproach. The nonrelativistic Thomas-Fermi approximation with Skyrme effective interac-tions was employed for evaluating the symmetry energy of finite nuclei and its dependenceon temperature [10, 11]. In a recent work by Agrawal et al. [13], various definitions of thesymmetry energy coefficients of infinite nuclear matter and finite nuclei, as well as theirtemperature dependencies, have been extensively investigated by using Skyrme interactions,and it was found that the symmetry energy of nuclear matter has a weak dependence ontemperature, while that of finite nuclei shows a rapid decrease with increasing temperature.In the present work, we will use the relativistic Thomas-Fermi approximation with the RMFmodel for nuclear interactions to study the temperature dependence of the symmetry energyof finite nuclei. We will also make a comparison between the relativistic and nonrelativisticresults. 3his article is organized as follows. In Sec. II, we briefly describe the RMF model andthe relativistic Thomas-Fermi approximation by employing the subtraction procedure for thedescription of hot nuclei. In Sec. III, we present the numerical results of hot nuclei propertiesand the temperature dependence of the symmetry energy of finite nuclei. Section IV isdevoted to the conclusions. II. FORMALISM
In this section, we first give a brief description of the RMF model used for nuclear inter-actions. Then we develop a relativistic Thomas-Fermi model by employing the subtractionprocedure for the description of hot nuclei. The symmetry energy of finite nuclei can becalculated by a pair of nuclei that have the same mass number ( A ) but different numbers ofprotons ( Z ) and neutrons ( N ). A. Relativistic mean-field model
In the RMF approach, nucleons interact through the exchange of various mesons. Themesons considered are the isoscalar scalar and vector mesons ( σ and ω ) and isovector vectormeson ( ρ ). The nucleonic Lagrangian density reads L RMF = X i = p,n ¯ ψ i h iγ µ ∂ µ − M − g σ σ − g ω γ µ ω µ − g ρ γ µ τ a ρ aµ i ψ i + 12 ∂ µ σ∂ µ σ − m σ σ − g σ − g σ − W µν W µν + 12 m ω ω µ ω µ + 14 c ( ω µ ω µ ) − R aµν R aµν + 12 m ρ ρ aµ ρ aµ + Λ v (cid:0) g ω ω µ ω µ (cid:1) (cid:0) g ρ ρ aµ ρ aµ (cid:1) , (1)where W µν and R aµν are the antisymmetric field tensors for ω µ and ρ aµ , respectively. Inthe RMF approach, meson fields are treated as classical fields and the field operators arereplaced by their expectation values. For a static system, the nonvanishing expectationvalues are σ = h σ i , ω = h ω i , and ρ = h ρ i .For nonuniform nuclear matter at finite temperature, the local energy density derived4rom the Lagrangian density (1) is given by ε ( r ) = X i = p,n π Z ∞ dk k p k + M ∗ (cid:0) f ki + + f ki − (cid:1) + 12 ( ∇ σ ) + 12 m σ σ + 13 g σ + 14 g σ −
12 ( ∇ ω ) − m ω ω − c ω + g ω ω ( n p + n n ) −
12 ( ∇ ρ ) − m ρ ρ + g ρ ρ ( n p − n n ) − Λ v (cid:0) g ω ω (cid:1) (cid:0) g ρ ρ (cid:1) , (2)where M ∗ = M + g σ σ is the effective nucleon mass and n i is the number density of species i ( i = p or n ). The entropy density is given by s ( r ) = X i = p,n π Z ∞ dk k (cid:2) − f ki + ln f ki + − (cid:0) − f ki + (cid:1) ln (cid:0) − f ki + (cid:1) − f ki − ln f ki − − (cid:0) − f ki − (cid:1) ln (cid:0) − f ki − (cid:1)(cid:3) , (3)where f ki + and f ki − ( i = p or n ) are the occupation probabilities of the particle and antiparticleat momentum k , respectively. The number density of protons ( i = p ) or neutrons ( i = n )at position r is given by n i ( r ) = 1 π Z ∞ dk k (cid:0) f ki + − f ki − (cid:1) . (4)In the RMF model, the parameters are generally fitted to nuclear matter saturationproperties or ground-state properties of finite nuclei. In the present work, we consider fourdifferent RMF parametrizations, NL3 [25], TM1 [26], FSU [27], and IUFSU [28], so that wecan examine the dependence of results on the RMF parametrization. It is known that theseRMF parametrizations are successful in reproducing the ground-state properties of finitenuclei including unstable ones. The NL3 model includes nonlinear terms of the σ mesononly, while the TM1 model includes nonlinear terms for both σ and ω mesons. An additional ω - ρ coupling term is added in the FSU and IUFSU models; it plays an important role inmodifying the density dependence of the symmetry energy and affecting the neutron-starproperties [27, 28, 36–39]. The IUFSU parametrization was developed from FSU by reducingthe neutron-skin thickness of Pb and increasing the maximum neutron-star mass in theparameter fitting [28]. The TM1 model was successfully used to construct the equation ofstate for supernova simulations and neutron-star calculations [40, 41]. For completeness, wepresent the parameter sets and corresponding properties of nuclear matter and finite nuclei5or these RMF models in Table I. It is noticeable that the symmetry energy slope L of theRMF models ranges from a low value of 47 . . L on the results obtainedby using these different RMF models. B. Relativistic Thomas-Fermi approximation for hot nuclei
We use the relativistic Thomas-Fermi approximation to describe hot nuclei by employ-ing the subtraction procedure as described in Refs. [5, 6]. The subtraction procedure wasproposed in order to isolate the hot nucleus from the surrounding gas, so that the resultingproperties of the nucleus could be independent of the size of the box in which the calculationwas performed. At given temperature and chemical potentials, there exist two solutions tothe equations in the RMF model derived from the Lagrangian density (1), one correspond-ing to the nuclear liquid phase with the surrounding gas (
N G ) and the other correspondingto the nucleon gas alone ( G ). The density profile of the nucleus ( N ) is then obtained bysubtracting the gas density from that of the liquid-plus-gas phase. Without the inclusion ofthe Coulomb interaction in the Lagrangian density (1), the gas phase is just diluted uniformnuclear matter, while the liquid-plus-gas phase is an uncharged nucleus in equilibrium withsurrounding nucleon gas. The inclusion of the Coulomb interaction leads to a difficulty indescribing hot nuclei. Because the Coulomb repulsion increases with the box size, it drivesprotons to the border and finally results in a divergence. To overcome this difficulty, theauthor of Ref. [5] proposed calculating the Coulomb potential from the subtracted protondensity, which is the proton density of the isolated nucleus. This implies that protons inthe gas phase do not contribute to the Coulomb potential, but they can be influenced bythe Coulomb potential of the nucleus. This prescription for the inclusion of the Coulombinteraction is quite successful in describing hot nuclei, and, as a result, properties of the hotnucleus are independent of the box size.Using the relativistic Thomas-Fermi approximation with the subtraction procedure, westudy a hot nucleus based on the thermodynamic potential of the isolated nucleus, which isdefined by Ω = Ω NG − Ω G + E C , (5)where Ω NG and Ω G are the nucleonic thermodynamic potentials in the N G and G phases,6espectively. We employ the RMF model to calculate the thermodynamic potential Ω a ( a = N G or G ), which can be written asΩ a = E a − T S a − X i = p,n µ i N ai . (6)Here, the energy E a , entropy S a , and particle number N ai in the phase a are obtained by E a = Z ε a ( r ) d r, (7) S a = Z s a ( r ) d r, (8) N ai = Z n ai ( r ) d r, (9)where ε a ( r ), s a ( r ), and n ai ( r ) are the local energy density, entropy density, and particlenumber density in the RMF model given by Eqs. (2), (3), and (4), respectively. The Coulombenergy is calculated from the subtracted proton density as E C = Z (cid:20) e (cid:0) n NGp − n Gp (cid:1) A −
12 ( ∇ A ) (cid:21) d r, (10)where A is the electrostatic potential.The equilibrium state of the isolated nucleus can be obtained by minimization of thethermodynamic potential Ω defined in Eq. (5). The meson mean fields in the N G phasesatisfy the variational equation δ Ω δφ NG = 0 , φ NG = σ NG , ω NG , ρ NG , (11)which leads to the following equations of motion for meson mean fields in the N G phase: −∇ σ NG + m σ σ NG + g (cid:0) σ NG (cid:1) + g (cid:0) σ NG (cid:1) = − g σ (cid:0) n NGs,p + n NGs,n (cid:1) , (12a) −∇ ω NG + m ω ω NG + c (cid:0) ω NG (cid:1) + 2Λ v g ω g ρ ω NG (cid:0) ρ NG (cid:1) = g ω (cid:0) n NGp + n NGn (cid:1) , (12b) −∇ ρ NG + m ρ ρ NG + 2Λ v g ω g ρ (cid:0) ω NG (cid:1) ρ NG = g ρ (cid:0) n NGp − n NGn (cid:1) . (12c)The occupation probability f k,NGi + ( f k,NGi − ) of species i ( i = p or n ) can be derived from thevariational equation, δ Ω δf k,NGi ± = 0 , (13)which results in the Fermi-Dirac distribution of particles and antiparticles, f k,NGi ± = (cid:26) (cid:20)(cid:18)q k + ( M ∗ ,NG ) + g ω ω NG + g ρ τ ρ NG + e τ + 12 A ∓ µ i (cid:19) /T (cid:21)(cid:27) − . (14) G phase as −∇ σ G + m σ σ G + g (cid:0) σ G (cid:1) + g (cid:0) σ G (cid:1) = − g σ (cid:0) n Gs,p + n Gs,n (cid:1) , (15a) −∇ ω G + m ω ω G + c (cid:0) ω G (cid:1) + 2Λ v g ω g ρ ω G (cid:0) ρ G (cid:1) = g ω (cid:0) n Gp + n Gn (cid:1) , (15b) −∇ ρ G + m ρ ρ G + 2Λ v g ω g ρ (cid:0) ω G (cid:1) ρ G = g ρ (cid:0) n Gp − n Gn (cid:1) , (15c)and the occupation probability in the G phase as f k,Gi ± = (cid:26) (cid:20)(cid:18)q k + ( M ∗ ,G ) + g ω ω G + g ρ τ ρ G + e τ + 12 A ∓ µ i (cid:19) /T (cid:21)(cid:27) − . (16) In the equations for meson mean fields, n as,i and n ai denote, respectively, the scalar andnumber densities of species i ( i = p or n ) in the a ( a = N G or G ) phase. The numberdensity n ai is calculated from Eq. (4), while the scalar density n as,i is given by n as,i ( r ) = 1 π Z ∞ dk k M ∗ ,a q k + ( M ∗ ,a ) (cid:16) f k,ai + + f k,ai − (cid:17) . (17)By minimizing Ω with respect to the electrostatic potential A , we obtain the Poissonequation for A as − ∇ A = e (cid:0) n NGp − n Gp (cid:1) . (18)The inclusion of the Coulomb energy in Ω leads to a coupling between the two sets ofequations for the N G and G phases. Therefore, the coupled equations (12), (15), and (18)should be solved simultaneously at given temperature T and chemical potentials µ p and µ n .To solve Eqs. (12) and (15), we take the boundary conditions for meson mean fields in phase a ( a = N G or G ) as dφ a dr ( r = 0) = 0 , dφ a dr ( r = R ) = 0 , φ a = σ a , ω a , ρ a , (19)where r = 0 and r = R represent, respectively, the center and the edge of a spherical boxwith radius R . For the electrostatic potential A , the boundary conditions are taken as dA dr ( r = 0) = 0 , A ( r = R ) = eN p πR , (20)where N p is the proton number of the isolated nucleus given by Eq. (21) below, and e = p π/
137 is the electromagnetic coupling constant. The box radius R is generally taken tobe sufficiently large (about 15–20 fm) so that the results of the isolated nucleus could be8ndependent of the box size. In the present calculations, we set R = 20 fm, and we havechecked that the resulting properties of the nucleus remain unchanged when varying R from15 to 20 fm for T ≤ N p protons and N n neutrons at temperature T , the proton and neutronchemical potentials µ p and µ n can be determined from given N p and N n . Once the chemicalpotentials are known, the occupation probabilities and density distributions can be obtainedeasily. In practice, we solve self-consistently the coupled set of Eqs. (12), (15), and (18)under the constraints of given N p and N n . After getting the solutions for the N G and G phases, we can extract the properties of hot nuclei based on the subtraction procedure. Theproton and neutron numbers, N p and N n , are given by N i = N NGi − N Gi = Z n i ( r ) d r, i = p, n, (21)where n i ( r ) = n NGi ( r ) − n Gi ( r ) is the local density of the isolated nucleus, which decreasesto zero at large distances. Therefore, physical quantities of the isolated nucleus could beindependent of the size of the box in which the calculation is performed. The total energyof the hot nucleus is given by E = E NG − E G + E C , (22)where E NG and E G are the nucleonic energies without the Coulomb interaction in the N G and G phases, which are calculated from Eq. (7). The Coulomb energy E C is given byEq. (10). For a nucleus at temperature T , its excitation energy is defined as E ∗ ( T ) = E ( T ) − E ( T = 0) . (23)The entropy and other extensive quantities of the isolated nucleus can be calculated bysubtracting the contribution of the G phase from that of the N G phase.
C. Symmetry energy of finite nuclei
The symmetry energy is a key quantity in the study of exotic nuclei, heavy-ion collisions,and astrophysical phenomena [2–4]. For infinite nuclear matter, the symmetry energy isdefined by expanding the energy per particle, ǫ ( n, α ), in terms of the isospin asymmetryparameter, α = ( n n − n p ) /n , as ǫ ( n, α ) = ǫ ( n,
0) + a v sym ( n ) α + O ( α ) , (24)9here n = n n + n p is the nucleon number density. a v sym ( n ) is the symmetry energy coefficientof nuclear matter at density n , and its value at saturation density n is about 30–34 MeV [11,13, 42]. On the other hand, the symmetry energy of finite nuclei can be defined based onthe Bethe-Weizs¨acker mass formula, which gives the expression for the binding energy perparticle as EA = ǫ ( A, Z ) = ǫ vol + ǫ surf + ǫ sym + ǫ Coul + ǫ pair = − a vol + a surf A − / + a sym ( N − Z ) A + a Coul Z A / + ǫ pair , (25)where N and Z are the neutron and proton numbers, respectively, while A = N + Z isthe mass number. The volume, surface, symmetry, and Coulomb energies per particle aredenoted by ǫ vol , ǫ surf , ǫ sym , and ǫ Coul , respectively, while the last term ǫ pair represents thepairing correction. Here, a sym is the symmetry energy coefficient of finite nuclei, which isgenerally dependent on the mass number A . The symmetry energy of finite nuclei includesthe volume and surface terms, and they are related by [29] a sym ( A ) = a v sym (cid:0) a v sym /a s sym (cid:1) A − / , (26)where a v sym and a s sym are the volume and surface symmetry energy coefficients, respectively.In the limit of large A , a sym ( A ) can be expanded in power of A − / , which results in therelation [42] a sym ( A ) = a v sym + a ssym A − / , (27)with a ssym = − ( a v sym ) / ( a s sym ). The volume symmetry energy coefficient a v sym is identifiedas the symmetry energy coefficient of infinite nuclear matter at saturation density, and itsvalue is around 30–34 MeV as mentioned above. The ratio of volume and surface coefficients, a v sym /a s sym , was found to be in a range of ∼ . A at temperature T from a different method [10, 31]: a sym ( A, T ) = [ ǫ b ( A, X , T ) − ǫ b ( A, X , T )] / (cid:0) X − X (cid:1) , (28)where X and X are the neutron excesses of a pair of nuclei having the same mass number A but different proton number Z . For a nucleus with Z protons and N neutrons, the neutron10xcess is defined by X = ( N − Z ) /A . Here, ǫ b = ǫ − ǫ C is the energy per particle obtained bysubtracting the Coulomb part. The total energy per particle, ǫ = E/A , is calculated fromEq. (22), while the Coulomb energy per particle, ǫ C = E C /A , is obtained from Eq. (10).In practice, we choose a pair of even-even nuclei with the same A but different Z for thecalculation of a sym ( A, T ) used in Eq. (28). According to the definition given by Eq. (28),the resulting a sym ( A, T ) would also depend on the choice of nuclear pair, namely, the choiceof Z and Z . This dependence has been discussed in Ref. [10] by using the nonrelativisticThomas-Fermi approximation with Skyrme effective interactions. In the present study, wewill employ the relativistic Thomas-Fermi approximation with the RMF model for nuclearinteractions to extract the symmetry energy coefficient of finite nuclei from Eq. (28). We willalso evaluate the symmetry energy coefficient from uncharged nuclei; namely, the Coulombinteraction is switched off, so that the symmetry energy coefficient extracted would not behindered by the Coulomb interaction. In the next section, we will present and discuss ourresults and make a comparison with those obtained in the nonrelativistic approach. III. RESULTS AND DISCUSSION
In this section, we investigate properties of hot nuclei within the relativistic Thomas-Fermiapproximation by employing the subtraction procedure. For a nucleus with N p protons and N n neutrons at temperature T , we solve the coupled set of Eqs. (12), (15), and (18) with thechemical potentials µ p and µ n constrained by given N p and N n . After getting the solutionsfor the N G and G phases, we can obtain physical properties of the isolated nucleus ( N )using the subtraction procedure. In Figs. 1 and 2, we display the density distributions ofneutrons and protons for Fe and
Pb at T = 0, 4, and 8 MeV obtained by using theTM1 parametrization. From top to bottom, we show, respectively, the results of the nuclearliquid-plus-gas phase ( N G ), nucleon gas phase ( G ), and subtracted nuclear liquid phase( N ). It is clear that the densities of the isolated nucleus ( N ) vanish at large distances,and, as a result, physical quantities of the nucleus could be independent of the box size. Atzero temperature, the proton and neutron densities of the G phase are found to be exactlyzero, while their values are finite but very small at finite temperature. As temperatureincreases, the neutron and proton densities of the G phase increase significantly. Moreover,densities at the center of the nucleus are reduced and the nuclear surface becomes more11iffuse with increasing T . As can be seen in the middle panels of Figs. 1 and 2, the densitiesof the G phase are obviously polarized due to the inclusion of the Coulomb interaction. Thenucleon distributions at the center of a heavy nucleus such as Pb are significantly affectedby the repulsive Coulomb potential; namely, the nucleon densities (especially the protondensities) at the center of the nucleus are slightly lower than those at the surface region, asshown in the bottom panel of Fig. 2. These tendencies are consistent with those obtainedin nonrelativistic approaches [5, 6].We show in Figs. 3 and 4 the root-mean-square (rms) radii of neutrons and protons, R n and R p , and their difference known as the neutron-skin thickness, ∆ r np = R n − R p , as afunction of the temperature T for Fe and
Pb, respectively. It is well known that thereis a correlation between the neutron-skin thickness ∆ r np and the symmetry energy slope L [27, 28, 36, 37]. In order to study the temperature dependence of this correlation, wecalculate ∆ r np by using four different RMF parametrizations, namely, NL3, TM1, FSU,and IUFSU, which cover a wide range of L as listed in Table I. It is shown that a larger L favors a larger ∆ r np , which does not change much with increasing T . The values of ∆ r np for Pb at T = 0 are presented in Table I. On the other hand, both R n and R p are found toincrease significantly with increasing T . This is because the nucleon distributions becomemore diffuse at higher temperature, as shown in Figs. 1 and 2. The increase of R n and R p inTM1 is somewhat slower than that in the other three cases, which can be roughly explainedby a relatively small decrease in the saturation density of nuclear matter. With increasing T , the saturation density of nuclear matter obtained in TM1 deceases more slowly comparedto that obtained in other three cases, and, as a result, the size of a nuclear drop with thesame mass number A would increase more slowly with temperature, as shown by black-solidlines in Figs. 3 and 4. The temperature dependence of R n and R p obtained in this workis comparable to that shown in Ref. [6]. For the influence of L on R n and R p , it is shownthat R n of heavy nuclei is more sensitive to L than R p , which is related to the correlationbetween ∆ r np and L .The excitation energies of hot nuclei are calculated from Eq. (23) using the four differentRMF parametrizations listed in Table I. We plot in Fig. 5 the temperature T as a function ofthe excitation energy per particle, E ∗ /A (the so-called caloric curve), for four representativenuclei, Fe,
Sn,
Sm, and
Pb. It is shown that different RMF parametrizationsproduce very similar results in all cases. This is because all of these models have been fitted12o experimental masses of finite nuclei covering a wide range of atomic numbers. Therefore,binding energies obtained by these models are close to each other. As one can see in Fig. 5, E ∗ /A increases slowly at low temperature, while it rises more rapidly as T increases. It isknown that there exists a limiting temperature T lim for a hot nucleus, above which the nucleusbecomes unstable due to the Coulomb interaction [5, 6, 43, 44]. The limiting temperature T lim depends on the nucleus, and it generally decreases with increasing mass number A and with increasing charge-to-mass ratio Z/A [6]. It was found in the finite-temperatureHartree-Fock calculation [5] and the nonrelativistic Thomas-Fermi approximation [6] that T lim is about 8–10 MeV depending on the nucleus and effective interactions. In the presentstudy, we find that the instability of hot nuclei caused by the Coulomb interaction occursat T > T ∼ A at temperature T is calculated from Eq. (28) by a pair of nucleiwith proton numbers Z and Z . In practice, we choose the nuclear pair ( A , Z ) and( A , Z ) near the β -stability line with Z − Z = 2 for evaluating the symmetry energycoefficient a sym . According to the liquid-drop model, the symmetry energy coefficient offinite nuclei is dependent on the mass number A due to the combination of volume andsurface contributions [29, 42], but it would not be very sensitive to the choice of nuclearpair, namely, the choice of Z and Z . However, calculations done by De and Samaddar [10]show that a sym depends sensitively on the choice of the nuclear pair (see Figs. 6, 7, and 8of Ref. [10]). In order to test the dependence of a sym on the choice of Z with Z = Z − a sym by red-dashed lines as a function of Z for A = 56 and A = 208 at T = 0, 4, and 8 MeV using the TM1 and FSU parametrizations,respectively. It is shown that the dependence of a sym on Z becomes more pronounced forhigher temperature and close to isospin symmetry. For the case of T = 8 MeV and A = 56,there is a sharp drop in a sym at Z = 28, which is calculated from the nuclear pair (56,28) and (56, 26). This value even becomes negative for the FSU parametrization, as shownin the bottom-left panel of Fig. 7. Negative a sym extracted from the same nuclear pair at T = 8 MeV has also been presented in Fig. 6 of Ref. [10]. In principle, the symmetry energycoefficient could be extracted from uncharged nuclei in which the Coulomb interaction isswitched off [42]. This is because the symmetry energy is solely determined by nuclear13orces. It would be more appropriate to switch off the Coulomb interaction in extracting thesymmetry energy coefficient of finite nuclei, rather than to subtract the Coulomb energy aftercalculating charged nuclei as defined by Eq. (28). This is because the Coulomb interactioncan significantly polarize nucleon densities and affect nuclear surface properties [29, 35]. As aresult, the symmetry energy coefficient calculated from Eq. (28) is hindered by the Coulombpolarization. In Figs. 6 and 7, we compare the results of a sym extracted from unchargednuclei (without Coulomb) with those obtained from charged nuclei (with Coulomb). As onecan see, a sym without the Coulomb interaction becomes much less sensitive to the choiceof nuclear pair in comparison to that with the Coulomb interaction. We note that a lowersensitivity to the nuclear pair is expected according to the liquid-drop model. In Figs. 8and 9, we display and compare the temperature dependence of a sym obtained from chargednuclei (with Coulomb) and uncharged nuclei (without Coulomb) using the TM1 and FSUparametrizations, respectively. The calculations are performed for four pairs of nuclei with Z = Z −
2, while the representative nuclei ( A , Z ) are chosen to be Fe,
Sn,
Sm,and
Pb. It is shown that a sym with the Coulomb interaction decreases more rapidlythan that without the Coulomb interaction, which may be related to a more pronouncedCoulomb polarization effect at higher temperature. It has been pointed out in Ref. [10] thata fast drop of a sym even results in negative values of a sym , which is likely to arise from theCoulomb polarization of nucleon densities. In the present study, we find that a sym obtainedfrom uncharged nuclei is less sensitive to the choice of nuclear pair as expected and shows asmaller decrease than that with the Coulomb interaction. The relatively weak temperaturedependence of a sym obtained without the Coulomb interaction seems to be more consistentwith the behavior of infinite nuclear matter [45].In order to examine the impact of the RMF parametrization on the results obtained, wecalculate a sym from uncharged nuclei by using four different RMF parametrizations, NL3,TM1, FSU, and IUFSU. We show in Fig. 10 the results calculated from four pairs of nucleiwith Z = Z −
2, while the representative nuclei ( A , Z ) are again chosen to be Fe,
Sn,
Sm, and
Pb. One can see that there are no large differences among these RMFparametrizations, and all of them show a decreasing a sym with increasing temperature. Itis shown that the largest L of NL3 is associated with the smallest a sym at T = 0 forall nuclei considered. This is because the surface term for the model with a larger L cancontribute more to the symmetry energy; namely, the ratio of volume and surface coefficients,14 v sym /a s sym , appearing in Eq. (26) has a bigger value for a larger L . The correlation between a v sym /a s sym and L calculated from Skyrme interactions has been shown in Fig. 15 of Ref. [30].Here, we can calculate the ratio a v sym /a s sym at T = 0 for each RMF parametrization fromEq. (26) using the results of Pb. With a v sym given in Table I and a sym ( A = 208 , T = 0)calculated from the pair of uncharged nuclei (208, 82) and (208, 80), we obtain that theratio a v sym /a s sym ranges from 1 .
56 for IUFSU to 3 .
54 for NL3 (see Table I). The correlationbetween a v sym /a s sym and L obtained in the present study is consistent with that shown inFig. 15 of Ref. [30]. Due to the large value of a v sym /a s sym for NL3, we obtain a relativelysmall a sym ( A = 208 , T = 0) = 23 . a v sym = 37 . a sym ( A = 208 , T = 0) = 24 . a v sym = 31 . a sym ( A = 208 , T = 0) = 24 . . a sym ( A = 56)obtained in TM1 is almost identical to that obtained in FSU for T < a sym ( A, T = 0) for A = 56, 112 and 150 extracted from unchargednuclei agree well with those calculated from Eq. (26) using parameters given in Table I. Thisimplies that the mass dependence of the symmetry energy coefficient of finite nuclei givenby Eq. (26) is approximately valid in the relativistic Thomas-Fermi approximation. FromEq. (26), it is easy to understand the increase of a sym ( A, T = 0) by ∼ A = 56to A = 208 as shown in Fig. 10. At finite temperature, the symmetry energy coefficient a sym decreases smoothly with increasing T . It is found that a sym obtained in IUFSU fallsmore rapidly than others, which may be related to its lower value of L . It is interestingto compare our results with those of Ref. [10], which were obtained in the nonrelativisticThomas-Fermi approximation. We find that the temperature dependence of a sym shown inFig. 10 is much smaller than that shown in Fig. 5 of Ref. [10]. For instance, a sym for A = 56and A = 112 with the SBM effective interaction, as shown by red-dashed lines in Fig. 5of Ref. [10], decreases by ∼
10 MeV up to T = 8 MeV, whereas the corresponding value inour calculation without the Coulomb interaction is ∼ a sym with the Coulomb interaction decreases more rapidly thanthat without the Coulomb interaction, and a drop of ∼
10 MeV is achieved for A = 56 and15 = 112 with the FSU parametrization, which is very close to the value of Ref. [10] with theSBM interaction, as mentioned above. We note that saturation properties of nuclear matterare very similar between the SBM interaction and the FSU interaction. Therefore, thedifferent temperature dependence between our results shown in Fig. 10 and that presentedin Fig. 5 of Ref. [10] may be mainly attributed to different treatments of the Coulombinteraction. As discussed above, the symmetry energy coefficient of finite nuclei can besignificantly affected by the Coulomb polarization effect. It would be more appropriate toswitch off the Coulomb interaction in extracting the symmetry energy coefficient of finitenuclei, rather than to subtract the Coulomb energy after calculating charged nuclei. IV. CONCLUSIONS
We have developed a relativistic Thomas-Fermi approximation by employing the subtrac-tion procedure for the description of hot nuclei. The subtraction procedure is necessary forisolating the nucleus from the surrounding nucleon gas, so that the resulting properties ofthe nucleus could be independent of the size of the box in which the calculation is performed.For the nuclear interaction, we have adopted the RMF model and considered four successfulparametrizations, NL3, TM1, FSU, and IUFSU, which cover a wide range of the symmetryenergy slope L . By comparing the results with different RMF parametrizations, it is possibleto study the impact of the RMF parametrization on properties of hot nuclei. The correla-tion between the neutron-skin thickness ∆ r np and the symmetry energy slope L has beenconfirmed in the present calculation, and we have found that ∆ r np is almost independent oftemperature T . On the other hand, the rms radii of neutrons and protons, R n and R p , havebeen shown to increase significantly with increasing T , and nucleon distributions becomemore diffuse at higher temperature. The excitation energies of hot nuclei have been foundto be insensitive to the RMF parametrization. We have achieved very similar caloric curvesfor different RMF parametrizations.We have investigated the symmetry energy of finite nuclei within the relativistic Thomas-Fermi approximation. The symmetry energy coefficient a sym of finite nuclei, which is depen-dent on the mass number A and temperature T , has been extracted from a nuclear pair ( A , Z ) and ( A , Z ) near the β -stability line with Z − Z = 2. It has been found that a sym calculated from charged nuclei depends sensitively on the choice of nuclear pair, especially16t high temperature and close to isospin symmetry. This is because the Coulomb interac-tion can significantly polarize nucleon densities and affect nuclear surface properties. Onthe other hand, a sym calculated from uncharged nuclei becomes much less sensitive to thechoice of nuclear pair in comparison to that obtained from charged nuclei. Furthermore,the temperature dependence of a sym extracted from uncharged nuclei has been shown to bemuch smaller than that calculated from charged nuclei, which may be due to the Coulombpolarization effect becoming more pronounced at higher temperature. Therefore, we con-clude that the symmetry energy coefficient of finite nuclei can be significantly affected bythe Coulomb polarization effect. It would be more appropriate to switch off the Coulombinteraction in extracting the symmetry energy coefficient of finite nuclei, rather than to sub-tract the Coulomb energy after calculating charged nuclei, so that the resulting a sym wouldnot be hindered by the Coulomb polarization. We have studied the temperature depen-dence of a sym for several representative nuclei using different RMF parametrizations in orderto examine the impact of the RMF parametrization on the results obtained. It has beenfound that a sym extracted from uncharged nuclei decreases smoothly with increasing T , anda drop of ∼ T = 8 MeV depending on A and on the RMFparametrization used. The tendency of the temperature dependence of a sym is similar fordifferent RMF parametrizations. We have compared our results with those obtained in thenonrelativistic Thomas-Fermi approximation [10]. It has been shown that the temperaturedependence of a sym obtained in the present work from uncharged nuclei is much smallerthan that presented in Ref. [10], which may be mainly attributed to different treatmentsof the Coulomb interaction. We have evaluated the ratio of volume and surface symmetryenergy coefficients, a v sym /a s sym , at zero temperature. It has been found that a larger symme-try energy slope L of nuclear matter corresponds to a bigger value of a v sym /a s sym , which isconsistent with that obtained by Skyrme interactions [30]. When the temperature increasesfrom zero to a finite value, both a v sym and a s sym may change with the temperature. We planto investigate the temperature dependence of a v sym and a s sym in a further study.17 cknowledgment This work was supported in part by the National Natural Science Foundation of China(Grant No. 11375089). [1] S. Shlomo and V. M. Kolomietz, Rep. Prog. Phys. , 1 (2005).[2] V. Baran, M. Colonna, V. Greco, and M. Di Toro, Phys. Rep. , 335 (2005).[3] J. M. Lattimer and M. Prakash, Phys. Rep. , 109 (2007).[4] B. A. Li, L. W. Chen, and C. M. Ko, Phys. Rep. , 113 (2008).[5] P. Bonche, S. Levit, and D. Vautherin, Nucl. Phys. A , 278 (1984); , 265 (1985).[6] E. Suraud, Nucl. Phys. A , 109 (1987).[7] J. N. De, X. Vi˜nas, S. K. Patra, and M. Centelles, Phys. Rev. C , 057306 (2001).[8] T. Sil, J. N. De, S. K. Samaddar, X. Vi˜nas, M. Centelles, B. K. Agrawal, and S. K. Patra,Phys. Rev. C , 045803 (2002).[9] S. K. Samaddar, J. N. De, X. Vi˜nas, and M. Centelles, Phys. Rev. C , 054608 (2007).[10] J. N. De and S. K. Samaddar, Phys. Rev. C , 024310 (2012).[11] J. N. De, S. K. Samaddar, and B. K. Agrawal, Phys. Lett. B , 361 (2012).[12] B. K. Agrawal, D. Bandyopadhyay, J. N. De, and S. K. Samaddar, Phys. Rev. C , 044320(2014).[13] B. K. Agrawal, J. N. De, S. K. Samaddar, M. Centelles, and X. Vi˜nas, Eur. Phys. J. A ,19 (2014).[14] D. P. Menezes and C. Providˆencia, Nucl. Phys. A , 283 (1999).[15] D. P. Menezes and C. Providˆencia, Phys. Rev. C , 024313 (1999).[16] S. S. Avancini, D. P. Menezes, M. D. Alloy, J. R. Marinelli, M. M. W. Moraes, and C.Providˆencia, Phys. Rev. C , 015802 (2008).[17] S. S. Avancini, S. Chiacchiera, D. P. Menezes, and C. Providˆencia, Phys. Rev. C , 055807(2010); , 059904(E) (2012).[18] F. Grill, C. Providˆencia, and S. S. Avancini, Phys. Rev. C , 055808 (2012).[19] D. P. Menezes and C. Providˆencia, Phys. Rev. C , 044306 (2001).[20] T. Sil, B. K. Agrawal, J. N. De, and S. K. Samaddar, Phys. Rev. C , 054604 (2001).
21] Z. W. Zhang and H. Shen, Astrophys. J. , 185 (2014).[22] B. D. Serot and J. D. Walecka, Adv. Nucl. Phys. , 1 (1986).[23] Y. K. Gambhir, P. Ring, and A. Thimet, Ann. Phys. (N.Y.) , 132 (1990).[24] J. Meng, H. Toki, S. G. Zhou, S. Q. Zhang, W. H. Long, and L. S. Geng, Prog. Part. Nucl.Phys. , 470 (2006).[25] G. A. Lalazissis, J. K¨onig, and P. Ring, Phys. Rev. C , 540 (1997).[26] Y. Sugahara and H. Toki, Nucl. Phys. A , 557 (1994).[27] B. G. Todd-Rutel and J. Piekarewicz, Phys. Rev. Lett. , 122501 (2005).[28] F. J. Fattoyev, C. J. Horowitz, J. Piekarewicz, and G. Shen, Phys. Rev. C , 055803 (2010).[29] P. Danielewicz, Nucl. Phys. A , 233 (2003).[30] P. Danielewicz and J. Lee, Nucl. Phys. A , 36 (2009).[31] D. J. Dean, K. Langanke, and J. M. Sampaio, Phys. Rev. C , 045802 (2002).[32] S. J. Lee and A. Z. Mekjian, Phys. Rev. C , 064319 (2010).[33] K. Oyamatsu and K. Iida, Phys. Rev. C , 054302 (2010).[34] H. Mei, Y. Huang, J. M. Yao, and H. Chen, J. Phys. G , 015107 (2012).[35] J. Dong, W. Zuo, and J. Gu, Phys. Rev. C , 014303 (2013).[36] C. J. Horowitz and J. Piekarewicz, Phys. Rev. Lett. , 5647 (2001).[37] J. Carriere, C. J. Horowitz, and J. Piekarewicz, Astrophys. J. , 463 (2003).[38] R. Cavagnoli, D. P. Menezes, and C. Providˆencia, Phys. Rev. C , 065810 (2011).[39] C. Providˆencia and A. Rabhi, Phys. Rev. C , 055801 (2013).[40] H. Shen, H. Toki, K. Oyamatsu, and K. Sumiyoshi, Astrophys. J. Suppl. , 20 (2011).[41] H. Shen, Phys. Rev. C , 035802 (2002).[42] P.-G. Reinhard, M. Bender, W. Nazarewicz, and T. Vertse, Phys. Rev. C , 014309 (2006).[43] S. Levit and P. Bonche, Nucl. Phys. A , 426 (1985).[44] H. R. Jaqaman, Phys. Rev. C , 1677 (1989).[45] Ch. C. Moustakidis, Phys. Rev. C , 025805 (2007). ABLE I: Parameter sets for the four RMF parametrizations used in this work and correspondingproperties of nuclear matter and finite nuclei. The masses are all given in MeV. The quantities E , K , a v sym , and L are, respectively, the energy per particle, incompressibility coefficient, symmetryenergy coefficient, and symmetry energy slope of nuclear matter at saturation density n . Thelast two lines show the neutron-skin thickness of Pb, ∆ r np , and the ratio of volume and surfacesymmetry energy coefficients, a v sym /a s sym , extracted from the results of Pb at zero temperature.NL3 TM1 FSU IUFSU M m σ m ω m ρ g σ g ω g ρ g (fm − ) –10.4310 –7.2325 –4.2771 –8.4929 g –28.885 0.6183 49.8556 0.4877 c v n (fm − ) 0.148 0.145 0.148 0.155 E (MeV) –16.3 –16.3 –16.3 –16.4 K (MeV) 272 281 230 231 a v sym (MeV) 37.4 36.9 32.6 31.3 L (MeV) 118.2 110.8 60.5 47.2∆ r np (fm) 0.223 0.211 0.167 0.116 a v sym /a s sym .000.020.040.060.080.100.0000.0020.0040.0060.008 0 5 10 150.000.020.040.060.08 0 5 10 15 20 NGneutron NGprotonG den s i t y ( f m - ) G T=0 T=4 T=8 r (fm) N Fe N FIG. 1: (Color online) Density distributions of neutrons (left panels) and protons (right panels) for Fe at T = 0, 4, and 8 MeV obtained using the TM1 parametrization. The results of the nuclearliquid-plus-gas phase ( N G ), nucleon gas phase ( G ), and subtracted nuclear liquid phase ( N ) areshown in the top, middle, and bottom panels, respectively. .000.020.040.060.080.100.0000.0020.0040.0060.008 0 5 10 150.000.020.040.060.08 0 5 10 15 20 NGneutron NGprotonG den s i t y ( f m - ) G T=0 T=4 T=8 r (fm) N Pb N FIG. 2: (Color online) Same as Fig. 1, but for
Pb. .43.84.24.63.43.84.2 0 2 4 6 80.00.10.2 R n Fe R p R n , R p , D r np ( f m ) T (MeV) D r np NL3 TM1 FSU IUFSU
FIG. 3: (Color online) The rms radii of neutrons and protons, R n and R p , and the neutron-skinthickness, ∆ r np = R n − R p , as a function of the temperature T for Fe obtained using four differentRMF parametrizations. .66.06.46.85.45.86.2 0 2 4 6 80.00.10.20.3 R n Pb R p R n , R p , D r np ( f m ) T (MeV) D r np NL3 TM1 FSU IUFSU
FIG. 4: (Color online) Same as Fig. 3, but for
Pb.
048 0 2 4048 0 2 4 6 Fe Sn NL3 TM1 FSU IUFSU Sm T ( M e V ) Pb E * /A (MeV) FIG. 5: (Color online) Temperature T as a function of the excitation energy per particle, E ∗ /A (caloric curve), for Fe,
Sn,
Sm, and
Pb obtained using four different RMF parametriza-tions. A=208 T=4A=56 T=0A=56 T=8 A=208 T=0A=208 T=8 a sy m ( M e V ) without Coulomb with Coulomb Z A=56 T=4A=56 T=8
FIG. 6: (Color online) Symmetry energy coefficient a sym as a function of Z for A = 56 and A = 208 at T = 0, 4, and 8 MeV obtained using the TM1 parametrization. The nuclear pair usedfor calculating a sym is chosen to be ( A , Z ) and ( A , Z − -1001020 80 82 84 86 8801020 A=56 T=0A=56 T=8 A=208 T=0A=208 T=8 a sy m ( M e V ) without Coulomb with Coulomb Z A=208 T=4A=56 T=4A=56 T=8
FIG. 7: (Color online) Same as Fig. 6, but for the FSU parametrization. a sy m ( M e V ) Fe Sn Sm T (MeV) Pb without Coulomb with Coulomb FIG. 8: (Color online) Temperature dependence of a sym for Fe,
Sn,
Sm, and
Pb obtainedusing the TM1 parametrization. The results obtained from uncharged nuclei (without Coulomb)are compared with those calculated from charged nuclei by subtracting the Coulomb energy (withCoulomb). a sy m ( M e V ) Fe Sn Sm T (MeV) Pb without Coulomb with Coulomb FIG. 9: (Color online) Same as Fig. 8, but for the FSU parametrization. a sy m ( M e V ) T (MeV)
NL3 TM1 FSU IUFSU Fe Sn Pb Sm FIG. 10: (Color online) Temperature dependence of a sym calculated from uncharged nuclei for Fe,
Sn,
Sm, and