Thermal effects on nuclear symmetry energy with a momentum-dependent effective interaction
aa r X i v : . [ nu c l - t h ] J u l Thermal effects on nuclear symmetry energy with amomentum-dependent effective interaction
Ch.C. MoustakidisDepartment of Theoretical Physics, Aristotle University of Thessaloniki,54124 Thessaloniki, GreeceNovember 6, 2018
Abstract
The knowledge of the nuclear symmetry energy of hot neutron-rich matter is important for understandingthe dynamical evolution of massive stars and the supernova explosion mechanisms. In particular, the electroncapture rate on nuclei and/or free protons in presupernova explosions is especially sensitive to the symmetryenergy at finite temperature. In view of the above, in the present work we calculate the symmetry energy asa function of the temperature for various values of the baryon density, by applying a momentum-dependenteffective interaction. In addition to a previous work, the thermal effects are studied separately both in thekinetic part and the interaction part of the symmetry energy. We focus also on the calculations of the mean fieldpotential, employed extensively in heavy ion reaction research, both for nuclear and pure neutron matter. Theproton fraction and the electron chemical potential, which are crucial quantities for representing the thermalevolution of supernova and neutron stars, are calculated for various values of the temperature. Finally, weconstruct a temperature dependent equation of state of β -stable nuclear matter, the basic ingredient for theevaluation of the neutron star properties. The determination of the nuclear symmetry energy (SE) based on microscopic and/or phenomenological approachesis of great interest in nuclear physics as well as in nuclear astrophysics. For instance, it is important for the study ofthe structure and reactions of neutron-rich nuclei, the Type II supernova explosions, neutron-star mergers and thestability of neutron stars. In addition, the SE is the basic ingredient for the determination of the proton fractionand electron chemical potential. The above quantities determine the cooling rate and neutrino emission flux ofprotoneutron stars and the possibility of kaon condensation in dense matter [1, 2].Heavy-ion reactions are a unique means to produce in terrestrial laboratories hot neutron-rich matter similarto those existing in many astrophysical situations [3]. Although the behavior of the SE for densities below thesaturation point still remains unknown, significant progress has been made only most recently in constraining theSE at subnormal densities and around the normal density from the isospin diffusion data in heavy-ion collisions[4, 5]. This has led to a significantly more refined constraint on neutron-skin thickness of heavy nuclei [6, 7] andthe mass-radius correlation of neutron stars [8]. For densities above the saturation point the trend of the SE ismodel dependent and exhibits completely different behavior.Up to now the main part of the calculations concerning the density dependence of the SE is related with the coldnuclear matter ( T = 0). However, recently, there is an increasing interest for the study of the SE and the propertiesof neutron stars at finite temperature [3, 9, 10, 11, 12, 13, 14, 15]. The motivation of the present work is to clarifythe effects of finite temperature on SE and to find also the appropriate relations describing that effect. Especiallywe focus on the interaction part of the SE, where so far it has received little theoretical attention concerning itsdependence on the temperature.In order to investigate the thermal properties of the SE, we apply a momentum dependent effective interactionmodel. In that way, we are able to study simultaneously thermal effects not only on the kinetic part of the symmetryenergy but also on the interaction part. The present model has been introduced by Gale et al. [16, 17, 18, 19] inorder to examine the influence of momentum-dependent interactions on the momentum flow of heavy ion collisions.Over the years the model has been extensively applied in the study not only of the heavy ion collisions but also in1he properties of nuclear matter by a proper modification [20, 21, 22, 23]. A review analysis of the present modelis presented in Refs. [2, 18].In the present work we study the thermal properties of the nuclear symmetry energy by applying the abovephenomenological model focusing mainly on the temperature dependence of the kinetic and interaction part of theSE as well as the total SE. Though it is well known how the temperature affects the kinetic part of the symmetryenergy [3, 24, 25] the temperature dependence of the interaction part of the SE has so far received little theoreticalattention. In addition, we determine the temperature dependence of the proton fraction as well as of the electronchemical potential. Both of the above quantities are related with the thermal evaluation of the supernova and theproton-neutron stars. The single particle potential for the pure neutron matter and the symmetric nuclear matter,extensively applied in heavy ion collision research, is also estimated for various values of the temperature. Finally,we construct the equation of state (EOS) of β -stable matter which is the basic ingredient for calculations of theneutron star properties.The plan of the paper is as follows. In Sec. II the model and the relative formulae are discussed and analyzed.Results are reported and discussed in Sec. III, while the summary of the work is given in Sec. IV. The schematic potential model, used in the present work, is designed to reproduce the results of the more mi-croscopic calculations of both nuclear and neutron-rich matter at zero temperature and can be extended to finitetemperature [2]. The energy density of the asymmetric nuclear matter (ANM) is given by the relation ǫ ( n n , n p , T ) = ǫ nkin ( n n , T ) + ǫ pkin ( n p , T ) + V int ( n n , n p , T ) , (1)where n n ( n p ) is the neutron (proton) density and the total baryon density is n = n n + n p . The contribution ofthe kinetic parts are ǫ nkin ( n n , T ) + ǫ pkin ( n p , T ) = 2 Z d k (2 π ) ¯ h k m ( f n ( n n , k, T ) + f p ( n p , k, T )) , (2)where f τ , (for τ = n, p ) is the Fermi-Dirac distribution function with the form f τ ( n τ , k, T ) = (cid:20) (cid:18) e τ ( n, k, T ) − µ τ ( n, T ) T (cid:19)(cid:21) − . (3)The nucleon density n τ is evaluated from the following integral n τ = 2 Z d k (2 π ) f τ ( n τ , k, T ) = 2 Z d k (2 π ) (cid:20) (cid:18) e τ ( n, k, T ) − µ τ ( n, T ) T (cid:19)(cid:21) − . (4)In Eq. (3), e τ ( n, k, T ) is the single particle energy (SPE) and µ τ ( n, T ) stands for the chemical potential of eachspecies. The SPE has the form e τ ( n, k, T ) = ¯ h k m + U τ ( n, k, T ) , (5)where the single particle potential U τ ( n, k, T ), is obtained by differentiating V int i.e. U τ = ∂V int ( n n , n p , T ) /∂n τ .Including the effect of finite-range forces between nucleons, in order to avoid acausal behavior at high densities,the potential contribution is parameterized as follows [2] V int ( n n , n p , T ) = 13 An (cid:20) − ( 12 + x )(1 − x ) (cid:21) u + Bn (cid:2) − ( + x )(1 − x ) (cid:3) u σ +1 B ′ (cid:2) − ( + x )(1 − x ) (cid:3) u σ − + 25 u X i =1 , (cid:20) (2 C i + 4 Z i ) 2 Z d k (2 π ) g ( k, Λ i )( f n + f p )+ ( C i − Z i ) 2 Z d k (2 π ) g ( k, Λ i )( f n (1 − x ) + f p x ) (cid:21) , (6)2here x = n p /n is the proton fraction and u = n/n , with n denoting the equilibrium symmetric nuclear matterdensity n = 0 .
16 fm − . The constants A , B , σ , C , C and B ′ , which enter in the description of symmetric nuclearmatter and the additional parameters x , x , Z , and Z , used to determine the properties of asymmetric nuclearmatter, are treated as parameters constrained by empirical knowledge [2]. The function g ( k, Λ i ) suitably chosento simulate finite range effects is of the following form g ( k, Λ i ) = " (cid:18) k Λ i (cid:19) − , (7)where the finite range parameters are Λ = 1 . k F and Λ = 3 k F and k F is the Fermi momentum at the saturationpoint n .The entropy density s τ ( n, T ) required for the calculations of the total pressure and for the EOS, has the samefunctional form as that of a non interacting gas system, that is s τ ( n, T ) = − Z d k (2 π ) [ f τ ln f τ + (1 − f τ ) ln(1 − f τ )] . (8)The ratio entropy/baryon is given by S τ ( n, T ) = s τ ( n, T ) /n . The baryon pressure P b ( n, T ), needed to constructthe EOS, is given by P b ( n, T ) = T X τ = p,n s τ ( n, T ) + X τ = p,n n τ µ τ ( n, T ) − ǫ anm ( n, T ) . (9)Finally, the total energy density and pressure of charge neutral and chemically equilibrium nuclear matter are ǫ tot ( n, T ) = ǫ b ( n, T ) + X l = e − ,µ − ǫ l ( n, T ) , (10) P tot ( n, T ) = P b ( n, T ) + X l = e − ,µ − P l ( n, T ) . (11)The leptons (electrons and muons) originating from the condition of the β -stable matter are considered as non-interacting Fermi gases.The above analysis holds in general for the asymmetric nuclear matter. Below, in order to calculate the thermaleffect on the SE, we will focus our study on two cases, i.e. the symmetric nuclear matter (SNM) and the pureneutron matter (PNM). The energy density of SNM is given by Eqs. (1) and (6) by setting x = 1 /
2, that is [2] ǫ snm ( n, T ) = 2 Z d k (2 π ) ¯ h k m f n + 2 Z d k (2 π ) ¯ h k m f p + 12 An u + Bn u σ +1 B ′ u σ − + u X i =1 , C i Z d k (2 π ) g ( k, Λ i ) f n + u X i =1 , C i Z d k (2 π ) g ( k, Λ i ) f p . (12)In addition, the single particle potential U τsnm ( n, k, T ) in the case of SNM, defined from the relation U τsnm = ∂V snm /∂n τ , is easily calculated and given by U τsnm ( n, k, T ) = ˜ U τsnm ( n, T ) + u X i =1 , C i " (cid:18) k Λ i (cid:19) − . (13)It is obvious from Eq. (13) that U τsnm ( n, k, T ) is separated in two terms. The first one corresponds to the momentumindependent part, while the second one corresponds to the momentum dependent one. The term ˜ U τsnm ( n, T ) has3he following form ˜ U τsnm ( n, T ) = Au + Bu σ ( σ + 1 + 2 B ′ u σ − )(1 + B ′ u σ − ) + 2 n X i =1 , C i Z d k (2 π ) " (cid:18) k Λ i (cid:19) − f τ , τ = p, n. (14)At zero temperature ( T = 0), where θ ( k F τ − k ), the integrals in Eqs. (12) and (14) are calculated analytically(see Appendix A for more details). The energy density of PNM is given by Eqs. (1) and (6) by setting x = 0 and f p = 0, that is [2] ǫ pnm ( n, T ) = 2 Z d k (2 π ) ¯ h k m f n + 13 An (1 − x ) u + Bn (1 − x ) u σ +1 B ′ (1 − x ) u σ − + 25 u X i =1 , (3 C i − Z i ) 2 Z d k (2 π ) g ( k, Λ i ) f n . (15)The single particle potential U npnm ( n, k, T ) in the case of PNM is defined from the relation U npnm = ∂V pnm /∂n n iswritten as U npnm ( n, k, T ) = ˜ U npnm ( n, T ) + 25 u X i =1 , (3 C i − Z i ) " (cid:18) k Λ i (cid:19) − . (16)The momentum-independent part is˜ U npnm ( n, T ) = 23 A (1 − x ) u + B (1 − x ) u σ [1 + B ′ (1 − x ) u σ − ] (cid:18) ( σ + 1) + 43 B ′ (1 − x ) u σ − (cid:19) + 25 n X i =1 , (3 C i − Z i )2 Z d k (2 π ) " (cid:18) k Λ i (cid:19) − f n . (17)The integrals in Eqs. (15) and (17), similarly to the case of SNM, at T = 0 are calculated analytically (seeAppendix A for more details). The energy density of ANM at density n and temperature T , in a good approximation, is expressed as ǫ anm ( n, T, x ) = ǫ snm ( n, T, x = 1 /
2) + ǫ sym ( n, T, x ) , (18)where ǫ sym ( n, T, x ) = n (1 − x ) E totsym ( n, T ) = n (1 − x ) (cid:0) E kinsym ( n, T ) + E intsym ( n, T ) (cid:1) . (19)In Eq. (19) the nuclear symmetry energy E totsym ( n, T ) is separated in two parts corresponding to the kinetic con-tribution E kinsym ( n, T ) and the interaction contribution E intsym ( n, T ). In the present work we will concentrate on thesystematic study of the thermal properties of the above two quantities.From Eqs. (18) and (19) and setting x = 0 we obtain that the nuclear symmetry energy E totsym ( n, T ) is given by E totsym ( n, T ) = 1 n ( ǫ pnm ( n, T ) − ǫ snm ( n, T )) . (20)Thus, from Eqs. (12) and (15) and by a suitable choice of the parameters x , x , Z and Z , we can obtain differentforms for the density dependence of the symmetry energy E totsym ( n, T ). It is well known that the need to exploredifferent forms for E totsym ( n, T ) stems from the uncertain behavior at high density [2]. In the present work, since we4re interested mainly in the study of thermal effects on the SE, we choose a specific form of the SE enabling us toreproduce accurately the results of many other theoretical studies [26]. According to this choice the SE, at T = 0,is expressed as E totsym ( n, T = 0) = 13 u / | {z } Kinetic + 17 F ( u ) | {z } Interaction = 13 u / | {z } Kinetic + 17 u |{z} Interaction , (21)where the contributions of the kinetic and the interaction term are separated clearly. The parameters x , x , Z and Z are chosen in order that Eq. (20), for T = 0, to reproduce the results of Eq. (21). In addition, theparameters A , B , σ , C , C and B ′ are determined in order that E ( n = n ) − mc = −
16 MeV, n = 0 .
16 fm − ,and the incompressibility to be K = 240 MeV.The single particle potential U τanm ( n, k, T ), in the case of ANM defined from the relation U τanm = ∂V anm /∂n τ ,is written as U τanm ( n, k, T ) = U τsnm ( n, k, T ) + ∂V sym ∂n τ = U τsnm ( n, k, T ) + U τsym ( n, T, x ) , (22)where V sym ( n, T, x ) = (1 − x ) nE intsym ( n, T ) . (23)It is easy to find that the term U τsym ( n, T ), in the case of T = 0 and by applying expression (21), is given by (seealso ref. [27]) U τsym ( n, T, x ) = ± u (1 − x ) , (24)where + and − stand for neutrons and protons respectively. In the general case where thermal effects are includedin our calculations, the E intsym ( n, T ) takes the form E intsym ( n, T ) = au b , (25)where a and b are temperature dependent constants (see Eq. (41) on Sec. III). Thus, after some algebra, we get ina good approximation, the relation U τsym ( n, T, x ) ≃ ± au b (1 − x ) . (26)The above relation is needed for the calculation of the single particle energy e τ ( n, k, T ) in the β -stable matter andafterwards for the calculation of the Fermi-Dirac function f τ ( n, T ) which is the basic ingredient for the determina-tion of the entropy density s τ ( n, T ). The key quantity for the determination of the equation of state in β -stable matter is the proton fraction x , whichis a basic ingredient of Eq. (19). In β -stable matter the processes [28] n −→ p + e − + ¯ ν e , p + e − −→ n + ν e , (27)take place simultaneously. We assume that neutrinos generated in these reactions have left the system. This impliesthat ˆ µ = µ n − µ p = µ e , (28)where µ n , µ p and µ e are the chemical potentials of the neutron, proton and electron respectively. Given the totalenergy density ǫ ≡ ǫ ( n n , n p ), the neutron and proton chemical potentials can be defined as µ n = ∂ǫ∂n n | n p , µ p = ∂ǫ∂n p | n n . (29)Hence we can show that ˆ µ = µ n − µ p = − ∂ǫ/n∂x | n = − ∂E∂x | n . (30)In β equilibrium one has ∂E∂x = ∂∂x ( E b ( n, x ) + E e ( x )) = 0 , (31)5here E b ( n, x ) the energy per baryon and E e ( x ) the electron energy. The charge condition implies that n e = n p = nx or k F e = k F p . Combining relations (18), (19) and (30) we get µ e ( n, T ) = ˆ µ ( n, T ) = 4(1 − x ) E totsym ( n, T ) . (32)From Eq. (32) it is obvious that the proton fraction x is not only a function of the baryon density n but, in addition,depends on the temperature T i.e. x = x ( n, T ).For relativistic non-degenerate free electrons we have n e = xn = 2(2 π ) Z d k (cid:20) √ ¯ h k c + m e c − µ e ( n,T ) T (cid:21) . (33)Or, using Eq. (32) and performing the angular integration we get n e = xn = 1 π Z ∞ k dk (cid:20) √ ¯ h k c + m e c − − x ) E totsym ( n,T ) T (cid:21) . (34)Eq. (34) determines the equilibrium electron (proton) fraction x ( n, T ) since the density and momentum dependentsymmetry energy E totsym ( n, T ) is known. We focus our attention on the calculation of the E totsym ( n, T ) with the help of Eq. (20). Thus, one has to calculatefirst the energy densities in pure and in symmetric nuclear matter as a function of the density n and for fixed valuesof temperature T . As an example of the calculations procedure at finite temperature (the results for T = 0 areincluded in the Appendix A), we consider the case of pure neutron matter. The procedure is similar in the case ofsymmetric nuclear matter (see Ref. [2]).The outline of our approach is the following: For a fixed neutron density n n and temperature T , Eq. (4) maybe solved iteratively in order to calculate the variable η ( n ; T ) = µ τ ( n ; T ) − ˜ U ( n ; T ) T . (35)The knowledge of η ( n, T ) allows the last term in Eq. (17) to be evaluated, yielding ˜ U ( n ; T ) which may then beused to infer the chemical potential from µ τ ( n ; T ) = T η ( n ; T ) + ˜ U ( n ; T ) , (36)required as an input to the calculation of the single particle spectrum e τ ( n, k, T ) in Eq. (5). Using e τ ( n, k ; T ), theenergy density in Eq. (15) is evaluated. According to our calculation recipe, given in the previous subsection, we calculate the energy densities of PNMand SNM as functions of the density, for various values of the temperature T . As a second step, we calculate the E totsym ( n, T ) from Eq. (20). The knowledge of E totsym ( n, T ) is required for the evaluation of the proton fraction x from Eq. (34) as well as for the electron chemical potential µ e = ˆ µ from Eq. (32). Finally from Eqs. (9), (10) and(11) we construct the EOS of β -stable matter for various values of the temperature T . It is worth pointing outthat in the present work we do not include the muon case, since we restrict ourselves mainly on the temperaturedependent behavior of the SE. According to our plan, in future work we will extend the treatment to include alsothe muon case in order to study the detailed composition and the thermal properties of neutron-rich matter withapplications in neutron star structure and thermal evaluation.In Fig. 1 we check the validity of approximation (18). We plot the difference E ( n, T, x ) − E ( n, T, x = 1 /
2) asa function of (1 − x ) at temperature T = 0, T = 20 and T = 50 MeV for three baryon number fractions i.e.6 = 1, u = 2 and u = 3. It is seen that an almost linear relation holds between E ( n, T, x ) − E ( n, T, x = 1 /
2) and(1 − x ) , even closer to the case of pure neutron matter ( x = 0), indicating the validity of approximation (18).In Fig. 2 we indicate the behavior of the SE as a function of the temperature T for various fixed values of thebaryon density n . More precisely, in any case, we plot E totsym ( T ; n ), as well as E kinsym ( T ; n ) and E intsym ( T ; n ) as afunction of T for n = 0 . , . , . , . − . The most striking feature of the above analysis is a decrease of theSE (total, kinetic and interaction part) by increasing the temperature. This is consistent with the predictions ofmicroscopic and/or phenomenological theories [3, 13, 14]In order to illustrate further the dependence of the symmetry energy on the temperature and to find thequantitative characteristic on this dependence, the values of E sym ( T ; n ) for various values of the density n arederived with the least-squares fit method and found to take the general form E sym ( T ; n ) = A T /T ) c + B. (37)The values of the density dependent parameters A , B , T and c , for E totsym ( T ; n ), E kinsym ( T ; n ) and E intsym ( T ; n )for n = 0 . , . , . − are presented in Table 1. It is easy to find that in the case of low temperature limit( T /T ≪
1) all kinds of the symmetry energy decrease approximately according to E sym ( T ; n ) ∝ C − C T (where C and C density dependent constants). In the high density limit ( T /T ≫
1) the symmetry energy decreasesapproximately according to E sym ( T ; n ) ∝ C T − + C (where also C and C are density dependents constants).It is noted that the same behavior holds for E totsym ( T ; n ) as well as for E kinsym ( T ; n ) and E intsym ( T ; n ). This behavior iswell expected for the kinetic part of the symmetry energy (see also Ref. [3, 25]), where analytical calculations arepossible (see the prove in Appendix B). From the above study, it is concluded that there is a similar temperaturedependence both for the kinetic and the interaction part of the symmetry energy and consequently for the totalsymmetry energy, in the case of momentum dependent interaction. Recently, the temperature dependence of thekinetic and interaction part of the SE has been studied and illustrated in Ref. [14]. The results of the present workagree with those of Ref. [14] although different models have been employed to evaluate SE.In Fig. 3, we plot E totsym ( T ; n ) as a function of temperature for various low values of the baryon density. Inthe same figure we also include experimental data of the measured temperature dependent symmetry energy fromTexas A&M University (TAMU)[29] and the INDRA-ALADIN Collaboration at GSI [30]. The comparison thenallows to estimate the required density of the fragment-emitting of the experiments. As pointed out by Li et.al. [3]the experimentally observed evolution of the SE is mainly due to the change in density rather than temperature.Fig. 4 illustrates the behavior of the E totalsym ( n ; T ) (a), E kinsym ( n ; T ) (b), E intsym ( n ; T ) (c), as a function of the baryondensity n for various fixed values of the temperature T . The case T = 0 corresponds to the fundamental expressionof the present work i.e. E totsym ( u ; T = 0) = 13 u / + 17 u. (38)In any case, the trends of the various parts of the symmetry energy are similar. An increase in the temperatureleads just to a shift to lower values for the symmetry energy. It is worth pointing out that, the maximum decreaseof E totsym ( n ; T ), in the area under study (for T = 0 MeV up to T = 50 MeV), is between 40% (for n = 0 . − ) and4% (for n = 1fm − ). Correspondingly, the decrease of E kinsym ( n ; T ) is between 57% (for n = 0 . − ) and 5% (for n = 1 fm − ) and of the E intsym ( n ; T ) is between 22% (for n = 0 . − ) and 5% (for n = 1 fm − ). It is obvious thatthe thermal effects are more pronounced on the kinetic part than in the interaction part of the symmetry energyand in addition, more pronounced in lower values of the baryon density.The total symmetry energy E totsym ( u ; T ), for various values of the temperature T , was derived with the least-squares fit on the numerical results taken from Eq. (20) and has the form E totsym ( u ; T = 5) = 1 .
676 + 29 . u − . u + 0 . u − . u ,E totsym ( u ; T = 10) = − .
118 + 30 . u − . u + 0 . u − . u ,E sym ( u ; T = 20) = − .
910 + 29 . u − . u + 0 . u − . u ,E totsym ( u ; T = 50) = 0 .
099 + 18 . u + 2 . u − . u + 0 . u . (39)It is also useful to record some relations for E totsym ( u ; T ) derived by least-squares fit on the numerical results, inthe case where SE is parameterized in a way similar to that one holding for T = 0. In that case, the parametrizationis the following (the case E totsym ( u ; T = 0) is included also for comparison) E totsym ( u ; T = 0) = 13 u / + 17 u, totsym ( u ; T = 5) = E totsym ( u ; T = 0) − . u − . ,E totsym ( u ; T = 10) = E totsym ( u ; T = 0) − . u − . ,E sym ( u ; T = 20) = E totsym ( u ; T = 0) − . u − . ,E totsym ( u ; T = 50) = E totsym ( u ; T = 0) − . u − . . (40)From Eq. (40), the decrease of the SE as a result of increasing T , is evident.The interaction part of the symmetry energy E intsym ( u ; T ) for various values of the temperature T was derivedby a least-squares fit on the numerical results taken from Eqs. (19) and (20) and has the form E intsym ( u ; T = 5) = 17 . u . ,E intsym ( u ; T = 10) = 16 . u . ,E intsym ( u ; T = 20) = 16 . u . ,E intsym ( u ; T = 50) = 13 . u . . (41)Similarly, for the kinetic part of the symmetry energy E kinsym ( u ; T ) we obtain E kinsym ( u ; T = 5) = 12 . u . ,E kinsym ( u ; T = 10) = 12 . u . ,E kinsym ( u ; T = 20) = 11 . u . ,E kinsym ( u ; T = 50) = 8 . u . . (42)In Fig. 5 we plot the total energy per particle of the PNM (a) and of the SNM as a function of the density forvarious values of the temperature. In both cases it is concluded that the thermal effects become more pronouncedwhen T >
10 MeV and for baryon densities n < . − .Fig. 6 displays the single particle potential U pnm ( n, T, k ) of the PNM as a function of the momentum k forvarious values of the density n and temperature T . An increase of T leads to corresponding increase of the values ofthe U pnm ( n, T, k ), an effect, expected to be more pronounced for lower values of the baryon density ( n = 0 . − )compared to highest ( n = 0 . − ). The same trend holds also for the single particle potential U snm ( n, T, k ) ofthe SNM plotted in Fig. 7. Observing Figs. 6 and 7 one might expect that the change of T will affect slightly thenucleons with high momentum k . This could be seen by plotting the single particle energy e τ ( n, k, T ) (see Eq. (5))as a function of k . However, the above effect cannot be seen in the present work, where we plot just the singleparticle potential U τ ( n, k, T ) as a function of k .In Fig. 8 we display the single particle potential of neutron U n ( n, T, k ) (Fig. (a),(b)) and proton U p ( n, T, k )(Fig. (c),(d)), in β -stable matter, as a function of the momentum k for various values of the temperature T for n = 0 . n = 0 . − . The potential U τ ( n, T, k ) is evaluated according to Eq. (22. The most striking featureof Fig. 8 is the reduced thermal effect for high values of the baryon density, especially in the case of the neutronsingle particle potential. In the case of the proton, thermal effects are more pronounced.In Fig. 9(a) the proton fraction x is displayed, calculated from Eq. (34) as a function of n for various values of T . Thermal effects increase the value of x between 57% (for n = 0 . − ) and 2% (for n = 1 fm − ). This effectis directly related with the dependence of x on the symmetry energy. As discussed previously, the temperatureinfluences slightly the symmetry energy at high values of the density and consequently this is reflected in the valuesof x . It is stressed that x depends on T in two ways, as one can see from Eq. (34). That is, it depends directly on T due the Dirac-Fermi distribution and also depends on the symmetry energy which is also temperature dependent.In Fig. 9(b) we present the electron chemical potential µ e as a function of the density n for various T . Anincrease of T decreases µ e . The effect is more pronounced when T >
20 MeV. We mention that the rate of electroncapture on both free and bound protons depends in a very sensitive way on the difference ˆ µ = µ n − µ p = µ e between neutron and proton chemical potentials [9]. Larger values of ˆ µ = µ e inhibit the neutronization process,since it becomes more difficult to transform a proton into a neutron.Finally, in Fig. 10 we present the equation of state of beta stable matter constructed by applying the presentmomentum-dependent interaction model, for various values of the temperature T . It is obvious that the thermaleffects are enhanced when T >
20 MeV. The above EOS is very important for the calculation of the neutron starsproperties and also in combination with the calculated proton fraction and electron chemical potentials for thethermal evaluation of the neutron stars. 8
Summary
The knowledge of the nuclear symmetry energy of hot neutron-rich matter is important for understanding thedynamical evolution of massive stars and the supernova explosion mechanisms. In view of the above statement, weinvestigate, in the present work, the thermal effects on the nuclear symmetry energy. In order to perform the aboveinvestigation we apply a model with a momentum-dependent effective interaction. In that way, we are able to studythe thermal effect not only on the kinetic part of the symmetry energy but also on the interaction part which, inturn, due to a momentum dependence, is affected by the variation of the temperature. It is concluded that, ingeneral, by increasing T we obtain a decreasing SE. Our finding that both kinetic and interaction parts exhibit thesame trend both for low and high values of the temperature is an interesting result. Analytical relations, derivedby the method of least squares fit are given also for the above quantities. Temperature effects on the pure neutronmatter and also on symmetric nuclear matter are also investigated and presented. The single particle potential ofproton and neutron is of interest in heavy ions collisions experiments, is calculated also for pure neutron matter,symmetric nuclear matter and β -stable matter for various values of the baryon density and fixed values of T. Itis concluded that thermal effects are more pronounced for low values of the density n , where for high values of n the effects are almost negligible. Quantities, which are of great interest for the thermal evaluation of supernovaand neutron stars, i.e. the proton fraction x = x ( n, T ) and the electron chemical potential µ e = µ e ( n, T ), arecalculated and their temperature and density dependence is investigated. Thermal effects are larger for low valuesof the density and high values of T. Appendix A
The energy density of the SNM as well as of the PNM, at zero temperature are easily calculated from Eqs. (12)and (15) respectively by setting f τ = θ ( k F τ − k ) (where θ ( k F τ − k ) is the theta function and k F τ is the Fermimomentum of the nucleon τ ) and takes the following forms ǫ snm ( n, k ; T = 0) = 35 E F n u / + 12 An u + Bn u σ +1 B ′ u σ − + 3 n u X i =1 , C i (cid:18) Λ i k F (cid:19) u / i k F − tan − u / i k F ! , (43) ǫ pnm ( n, k ; T = 0) = 2 / E F n u / + 13 An (1 − x ) u + Bn (1 − x ) u σ +1 B ′ (1 − x ) u σ − + 35 n u X i =1 , (3 C i − Z i ) (cid:18) Λ i k F (cid:19) (2 u ) / i k F − tan − (2 u ) / i k F ! , (44)where E F = ¯ h k F / m is the Fermi energy of nuclear matter at the equilibrium density. Appendix B
In order to compare the numerical results obtained from the kinetic part of the symmetry energy E kinsym ( n, T ) withthose predicted from analytical calculations, we calculate E totalsym ( n, T ) in the low and in the hight temperature limitas follows Low temperature limit
The kinetic energy per nucleon E τkin ( n, T ) at low temperature ( T ≪ E F ) has the form [31, 32, 33] E τkin ( n, T ) = 35 E τF " π (cid:18) TE τF (cid:19) , (45)9here E τF = (¯ hk τF ) / m = ¯ h (3 π n τ ) / / m . Considering that δ = 1 − x = ( n n − n p ) / ( n n + n p ) after somealgebra we found that the E kin ( n, T, δ ) of a two-component Fermi gas has the form E kin ( n, δ, T ) = h E F i (cid:16) (1 + δ ) / + (1 − δ ) / (cid:17) + 310 1 h E F i (cid:16) π T (cid:17) (cid:16) (1 + δ ) / + (1 − δ ) / (cid:17) , (46)where h E F i = 3 / E F . Expanding expression (46) around the symmetric point δ = 0 or x = 1 / E kin ( n, T ) = h E F i + 320 π h E F i T + (1 − x ) (cid:18) h E F i − π h E F i T (cid:19)| {z } E kinsym ( n,T ) , (47)with the contribution of the symmetry energy written explicitly. It is obvious that in the low temperature limit E kinsym ( n, T ) behaves as E kinsym ( n, T ) ∝ C − C T . High temperature limit
The kinetic energy per nucleon E kin ( n, T, δ ) of a two-component Fermi gas at high temperature ( T ≫ E F ) isreplaced by a virial expansion in nλ where λ = q π ¯ h /mT is the quantum wavelength. So, E kin ( n, T ) is givenby the relation [32, 25] E kin ( n, δ, T ) = 32 T + 34 T X ν C ν (cid:18) λ n (cid:19) ν (cid:0) (1 − δ ) ν +1 + (1 + δ ) ν +1 (cid:1) . (48)Expanding expression (48) around the symmetric point δ = 0 or x = 1 / E kin ( n, T, δ ) = 32 T " X ν C ν (cid:18) λ n (cid:19) ν + (1 − x ) T X ν C ν (cid:18) λ n (cid:19) ν ν ( ν + 1)2 | {z } E kinsym ( n,T ) . (49)It is seen that in the high temperature limit E kinsym ( n, T ) behaves as E kinsym ( n, T ) ∝ C T − / + C T − + · · · . Acknowledgments
The author would like to thank Prof. S.E. Massen and Dr. C.P. Panos for useful comments on the manuscript andalso Prof. A.Z. Mekjian for valuable comments and correspondence. The work was supported by the PythagorasII Research project (80861) of EΠEAEK and the European Union.
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Quantum Theory of Many-Particle Systems (Dover Publications, Mineola, NewYork, 2003). 11 .0(cid:13) 0.2(cid:13) 0.4(cid:13) 0.6(cid:13) 0.8(cid:13) 1.0(cid:13)0(cid:13)20(cid:13)40(cid:13)60(cid:13)80(cid:13) (cid:13) (cid:13) E ( n , x )- E ( n , x = / ) (cid:13) (1-2x)(cid:13) T=0(cid:13) u=1(cid:13) u=2(cid:13) u=3(cid:13) (cid:13) (cid:13) E ( n , T , x )- E ( n , T , x = / ) (cid:13) (1-2x)(cid:13) T=20(cid:13) u=1(cid:13) u=2(cid:13) u=3(cid:13) (cid:13) (cid:13) E ( n , T , x )- E ( n , T , x = / ) (cid:13) (1-2x)(cid:13) T=50(cid:13) u=1(cid:13) u=2(cid:13) u=3(cid:13)
Figure 1: The difference E ( n, T, x ) − E ( n, T, x = 1 /
2) as a function of (1 − x ) at temperature T = 0, T = 20 and T = 50 MeV for three baryon number fractions u = 1, u = 2 and u = 3.Table 1: The values of the density dependent parameters A , B , T and c , for E totsym ( u ; T ), E kinsym ( u ; T ) and E intsym ( u ; T )for n = 0 . , . , . − . For more details see text.n=0.1 fm − n=0.3 fm − n=0.5 fm − Parameters E totsym E kinsym E intsym E totsym E kinsym E intsym E totsym E kinsym E intsym A B T c (cid:13) (cid:13) E (cid:13) sy m (cid:13) ( T ; n ) ( M e V ) (cid:13) T (MeV)(cid:13) n=0.1 fm(cid:13) -3(cid:13)
Kinetic(cid:13) Interaction(cid:13) Total(cid:13) (cid:13) (cid:13) E (cid:13) sy m (cid:13) ( T ; n ) ( M e V ) (cid:13) T (MeV)(cid:13) n=0.2 fm(cid:13) -3(cid:13)
Kinetic(cid:13) Interaction(cid:13) Total(cid:13) (cid:13) (cid:13) E (cid:13) sy m (cid:13) ( T ; n ) ( M e V ) (cid:13) T (MeV)(cid:13) n=0.3 fm(cid:13) -3(cid:13)
Kinetic(cid:13) Interaction(cid:13) Total(cid:13) (cid:13) (cid:13) E (cid:13) sy m (cid:13) ( T ; n ) ( M e V ) (cid:13) T (MeV)(cid:13) n=0.5 fm(cid:13) -3(cid:13)
Kinetic(cid:13) Interaction(cid:13) Total(cid:13)
Figure 2: Temperature dependence of the total nuclear symmetry energy and its interaction and kinetic energypart for various values of the baryon density n . 13 (cid:13) 2(cid:13) 4(cid:13) 6(cid:13) 8(cid:13) 10(cid:13)10(cid:13)15(cid:13)20(cid:13)25(cid:13) (cid:13) (cid:13) E (cid:13) sy m (cid:13) ( T ; n ) ( M e V ) (cid:13) T (MeV)(cid:13)
Figure 3: Temperature dependence of the symmetry energy for low values of the baryon density ( n =0 . , . , . , . , .
14 fm − ). The experimental data are from Ref. [29] (solid squares) and Ref. [30] (opensquares) are included for comparison. (cid:13) (cid:13) E (cid:13) sy m (cid:13) ( n ; T ) ( M e V ) (cid:13) n (fm(cid:13) -3(cid:13) )(cid:13) (a)(cid:13) T=0(cid:13) T=5(cid:13) T=10(cid:13) T=20(cid:13) T=50(cid:13) (cid:13) (cid:13) E (cid:13) k i n (cid:13) ( n ; T ) ( M e V ) (cid:13) n (fm(cid:13) -3(cid:13) )(cid:13) (b)(cid:13) T=0(cid:13) T=5(cid:13) T=10(cid:13) T=20(cid:13) T=50(cid:13) (cid:13) (cid:13) E (cid:13) i n t (cid:13) ( n ; T ) ( M e V ) (cid:13) n (fm(cid:13) -3(cid:13) )(cid:13) (c)(cid:13) T=0(cid:13) T=5(cid:13) T=10(cid:13) T=20(cid:13) T=50(cid:13) Figure 4: Density dependence of the total nuclear symmetry energy E totsym ( n, T ) as well its kinetic E kinsym ( n, T ) andinteraction E intsym ( n, T ) part for various values of the temperature T .14 .2(cid:13) 0.4(cid:13) 0.6(cid:13) 0.8(cid:13) 1.0(cid:13)0(cid:13)50(cid:13)100(cid:13)150(cid:13)200(cid:13)250(cid:13)300(cid:13)350(cid:13) (cid:13) (cid:13) E (cid:13) pn m (cid:13) ( n ; T ) ( M e V ) (cid:13) n (fm(cid:13) -3(cid:13) )(cid:13) (a)(cid:13) T=0(cid:13) T=5(cid:13) T=10(cid:13) T=20(cid:13) T=30(cid:13) T=50(cid:13) (cid:13) (cid:13) E (cid:13) s n m (cid:13) ( n ; T ) ( M e V ) (cid:13) n (fm(cid:13) -3(cid:13) )(cid:13) (b)(cid:13) T=0(cid:13) T=5(cid:13) T=10(cid:13) T=20(cid:13) T=30(cid:13) T=50(cid:13) Figure 5: (a) The energy per particle of pure neutron matter as a function of the baryon density for various valuesof the temperature T . (b) The energy per particle of symmetric nuclear matter as a function of the baryon densityfor various values of the temperature T . (cid:13) (cid:13) U (cid:13) pn m (cid:13) ( k ; n , T ) ( M e V ) (cid:13) k (fm(cid:13) -1(cid:13) )(cid:13) n=0.1 fm(cid:13) -3(cid:13) T=0(cid:13) T=10(cid:13) T=20(cid:13) T=50(cid:13) (cid:13) (cid:13) U (cid:13) pn m (cid:13) ( k ; n , T ) ( M e V ) (cid:13) k (fm(cid:13) -1(cid:13) )(cid:13) n=0.3 fm(cid:13) -3(cid:13) T=0(cid:13) T=10(cid:13) T=20(cid:13) T=50(cid:13) (cid:13) (cid:13) U (cid:13) pn m (cid:13) ( k ; n , T ) ( M e V ) (cid:13) k (fm(cid:13) -1(cid:13) )(cid:13) n=0.5 fm(cid:13) -3(cid:13) T=0(cid:13) T=10(cid:13) T=20(cid:13) T=50(cid:13)
Figure 6: The single particle potential of the pure neutron matter as a function of the momentum k for variousvalues of the temperature T and for n = 0 . , . . − respectively.15 (cid:13) 1(cid:13) 2(cid:13) 3(cid:13) 4(cid:13) 5(cid:13)-60(cid:13)-50(cid:13)-40(cid:13)-30(cid:13)-20(cid:13)-10(cid:13) (cid:13) (cid:13) U (cid:13) s n m (cid:13) ( k ; n , T ) ( M e V ) (cid:13) k (fm(cid:13) -1(cid:13) )(cid:13) n=0.1 fm(cid:13) -3(cid:13) T=0(cid:13) T=10(cid:13) T=20(cid:13) T=50(cid:13) (cid:13) (cid:13) U (cid:13) s n m (cid:13) ( k ; n , T ) ( M e V ) (cid:13) k (fm(cid:13) -1(cid:13) )(cid:13) n=0.3 fm(cid:13) -3(cid:13) T=0(cid:13) T=10(cid:13) T=20(cid:13) T=50(cid:13) (cid:13) (cid:13) U (cid:13) s n m (cid:13) ( k ; n , T ) ( M e V ) (cid:13) k (fm(cid:13) -1(cid:13) )(cid:13) n=0.5 fm(cid:13) -3(cid:13) T=0(cid:13) T=10(cid:13) T=20(cid:13) T=50(cid:13)
Figure 7: The single particle potential of the symmetric nuclear matter as a function of the momentum k for variousvalues of the temperature T and for n = 0 . , . . − respectively.16 (cid:13) 1(cid:13) 2(cid:13) 3(cid:13) 4(cid:13) 5(cid:13)-40(cid:13)-30(cid:13)-20(cid:13)-10(cid:13)0(cid:13) (cid:13) (cid:13) U (cid:13) n (cid:13) ( k ; n , T ) ( M e V ) (cid:13) k (fm(cid:13) -3(cid:13) )(cid:13) (a)(cid:13) n=0.1 fm(cid:13) -3(cid:13) T=0(cid:13) T=10(cid:13) T=20(cid:13) T=50(cid:13) (cid:13) (cid:13) U (cid:13) n (cid:13) ( k ; n , T ) ( M e V ) (cid:13) k (fm(cid:13) -3(cid:13) )(cid:13) (b)(cid:13) n=0.5 fm(cid:13) -3(cid:13) T=0(cid:13) T=10(cid:13) T=20(cid:13) T=50(cid:13) (cid:13) (cid:13) U (cid:13) p (cid:13) ( k ; n , T ) ( M e V ) (cid:13) k (fm(cid:13) -3(cid:13) )(cid:13) (c)(cid:13) n=0.1 fm(cid:13) -3(cid:13) T=0(cid:13) T=10(cid:13) T=20(cid:13) T=50(cid:13) (cid:13) (cid:13) U (cid:13) p (cid:13) ( k ; n , T ) ( M e V ) (cid:13) k (fm(cid:13) -3(cid:13) )(cid:13) (d)(cid:13) n=0.5 fm(cid:13) -3(cid:13) T=0(cid:13) T=10(cid:13) T=20(cid:13) T=50(cid:13)
Figure 8: The single particle potential of β -stable matter for neutron ((a)and (b)) and for proton ((c) and (d)) asa function of the momentum k for various values of the temperature T for n = 0 . . − .17 .2 0.4 0.6 0.8 1.00.000.050.100.150.20 x ( n ; T ) n (fm -3 ) T=0 T=10 T=20 T=50 µ e ( M e V ) n (fm -3 ) T=0 T=10 T=20 T=50
Figure 9: (a) The proton fraction x in β -stable matter as a function of the density n for various values of thetemperature T . The straight line corresponds to the case x = 11%. (b) The electron chemical potential µ e = ˆ µ = µ n − µ p as a function of the density n for various values of the temperature T .
300 400 500 600 700 800 900050100150200250300 P ( M e V /f m ) ε (Mev/ fm ) T=0 T=10 T=20 T=50