Nuclear condensation and symmetry energy of dilute nuclear matter: an S-matrix approach
aa r X i v : . [ nu c l - t h ] D ec Nuclear condensation and symmetry energy of dilute nuclear matter:an S -matrix approach J.N. De ∗ and S.K. Samaddar † Theory Division, Saha Institute of Nuclear Physics, 1/AF Bidhannagar, Kolkata 700064.
Based on the general analysis of the grand canonical partition function in the S -matrix framework,the calculated results on symmetry energy, free energy and entropy of dilute warm nuclear matterare presented. At a given temperature and density, the symmetry energy or symmetry free energy ofthe clusterized nuclear matter in the S -matrix formulation deviates, particularly at low temperatureand relatively higher density, in a subtle way, from the linear dependence on the square of theisospin asymmetry parameter X = ( ρ n − ρ p ) / ( ρ n + ρ p ), contrary to those obtained for homogeneousnucleonic matter. The symmetry coefficients, in conventional definition, can then be even negative.The symmetry entropy similarly shows a very different behavior. PACS numbers: 12.40.Ee, 21.65.Mn, 21.65.Ef, 24.10.PaKeywords: symmetry energy, symmetry entropy, nuclear matter, statistical mechanics, S -matrix I. INTRODUCTION
Understanding properties of symmetry energy for lowdensity nuclear matter is of much topical interest in bothastrophysical and laboratory context. The supernovasimulation dynamics has a sensitive dependence on thesymmetry energy [1]; a higher symmetry energy, for ex-ample, leads to a lower electron ( e − )-capture rate in thesupernova collapse phase that may result in a strongerexplosive shock. The variations in the e − -capture ratealso produces changes in the neutrino luminosities thatare potentially observable. The isotopic abundance ofrelatively heavier elements in explosive nucleosynthesis isfurther directly correlated to the symmetry energy. Theneutron skin thickness of heavy nuclei has also a directdependence on the density variation of the symmetry en-ergy [2, 3]. At subnormal densities (0.2 ≤ ρ/ρ ≤ ρ is the saturation density of nuclear matter), the sym-metry coefficients that are a measure of the symmetryenergy per baryon ( e sym ) have recently been estimatedfrom analysis of data related to isotopic distributions [4],isospin diffusion [5, 6], and isoscaling [7, 8] in heavy ionreactions. The different analyzes give somewhat differ-ent results. In the density domain mentioned, the sym-metry coefficient C E (= e sym /X ) varies with density as C E ( ρ ) ∼ C E ( ρ )( ρ/ρ ) γ with γ lying in a broad range[8, 9], between 0.55 − ∗ Electronic address: [email protected] † Electronic address: [email protected] nuclear matter has been calculated over a wide densityrange in many guises; many-body theories using variousnucleon-nucleon interactions or interaction Lagrangians[9] have lead to varying results [10]. There have alsobeen some recent investigations on the symmetry freeenergy of hot nuclear matter [11] and also of finite nuclei[12]. These calculations differ in details, but the generalqualitative behavior of the symmetry coefficients withdensity do not deviate much from the experimental trend.A similar power law variation is exhibited; the exponent γ lies mostly within the broad limits as extracted fromexperimental analyzes.The above calculations have been done in the mean-field (MF) approximation; the system is taken to be ho-mogeneous nucleonic matter. For dilute nuclear matter,however, the system minimizes its total energy or free en-ergy by forming clusters [13, 14]. A detailed knowledgeof the composition of nuclear matter is then needed toappreciate how the symmetry energies are affected whenmatter gets clusterized. This has a direct role in a bet-ter understanding of neutrino-driven energy transfer ininhomogeneous supernova matter [15]. Using the virialexpansion technique, clusterization in dilute nuclear mat-ter and its import in the evaluation of the symmetrycoefficients has been investigated recently by Horowitzand Schwenk [16] where they considered the matter tobe composed of n, p, α . The investigations with the samecluster species have been followed further [17] to connectexperimentally the temperature and density-dependententropic contributions to the symmetry free energy co-efficient. The resulting symmetry coefficients are foundto be considerably larger than the corresponding onesobtained from MF calculations. This is an importantresult, it shows the strong role of clusterization and nat-urally calls for a realization of symmetry coefficients ifall possible permissible clusters are incorporated in thecalculation. The present article is an attempt in this di-rection. II. ELEMENTS OF THEORY
The grand canonical partition function in the S -matrixformalism of statistical mechanics proposed by Dashen etal [18] sets the logical framework of our calculation. Thedetails of the formalism, as applied to nuclear matteris given in Ref. [19]; for the sake of completeness, theessentials are presented in the following.The partition function Z for the two-component nu-clear matter composed of the elementary species neutronsand protons is written as Z = T r e − β ( H − µ p ˆ N p − µ n ˆ N n ) , (1)where β is the inverse of temperature T of the system, H the total Hamiltonian, ˆ N p,n the number operators forprotons and neutrons, and µ p,n are the correspondingchemical potentials. The partition function can be de-composed as Z = ∞ X Z,N =0 ζ Zp ζ Nn T r
Z,N e − βH , (2)where the fugacities are given by ζ p = e βµ p , ζ n = e βµ n .The trace T r
Z,N is taken over states of Z protons and N neutrons. For small ζ p and ζ n , the quantity ln Z can beexpanded in a virial seriesln Z = X Z,N ′ D Z,N ζ Zp ζ Nn . (3)The prime on Σ indicates that the term with Z = N =0 isexcluded. Evaluation of the virial coefficients D Z,N givesthe partition function and thence the thermodynamic be-havior of the system.Following the temperature-Green’s-function method,it was shown in Ref. [18] that all the dynamical informa-tion concerning the microscopic interaction in the grandpotential of the system can be collected in two types ofterms so that the partition function is written asln Z = ln Z (0) part + ln Z scat . (4)The first term corresponds to contributions from stablesingle particle states of clusters of different sizes (neu-trons and protons included) formed in the infinite systembehaving like an ideal quantum gas [the superscript (0)indicates this behavior] and the second term correspondsto contribution from multiparticle scattering states, re-spectively. The particle piece can further be split in con-tributions from ground states and excited states of thebound nucleon clusters, so thatln Z (0) part = ln Z (0) gr + ln Z (0) ex . (5) The first term in Eq. (5) is a sum of ideal gas terms, onefor each of the ground states of all the possible speciesof mass A with Z protons and N neutrons that can beformed in the system,ln Z (0) gr = ∓ V X Z,N g Z d p (2 π ) × ln (cid:16) ∓ ζ Z,N e − β ( p / Am ) (cid:17) . (6)Here m is the nucleon mass, p is the momentum of thenucleus, and ζ Z,N the ’effective fugacity’ given by ζ Z,N = e β ( µ Z,N + B Z,N ) . B Z,N is the binding energy of the nucleusand µ Z,N its chemical potential. From the condition ofchemical equilibrium among the different species, µ Z,N = Zµ p + N µ n .The ∓ sign in Eq. (6) corresponds to nuclei with A evenor odd, obeying Bose or Fermi statistics, V is the volumeof the system and g is the spin degeneracy. The sumincludes the original elementary species, namely the neu-trons and protons. The Coulomb interaction is assumedabsent in nuclear matter, ideally the sum in Eq. (6) wouldinvolve infinite terms as the maximum cluster mass A canthen even be infinite. However, for applications to realphysical systems such as neutron star matter, Coulombeffect is to be included in the binding energies of nuclei.In that case, the sum is finite, conditioned by the sta-bility of nuclei with inclusion of Coulomb in the bindingenergies. Calculations in the S -matrix formalism withinclusion of Coulomb in the fragment binding energieswould henceforth be referred to as SMF, those withoutCoulomb would be called SNC. Eq. (6) can be readilyexpanded in a virial series asln Z (0) gr = V X Z,N gλ ( Am ) ζ Z,N ± ζ Z,N / + · · · ! , (7)in powers of effective fugacities provided | ζ Z,N | < λ ( Am ) = p π/ ( AmT ) is the thermal wavelengthof a cluster of mass Am . We work in natural units ~ = c = 1. A nucleus in a particular excited state istaken as a distinctly different species and can be treatedin the same footing as the ground state. The density ofstates is quite high in relatively heavy nuclei, increasesnearly exponentially with the square root of excitationenergy E and so the contribution of the excited states ofa single nucleus is written as an integral over E of theideal gas term weighted with the level density ω ( A, E ),ln Z (0) ex = ∓ V X Z,N ′ g Z E s E dE ω ( A, E ) × Z d p (2 π ) ln (cid:16) ∓ ζ Z,N e − β ( p / Am + E ) (cid:17) . (8)For the level density, we take the expression [20] ω ( A, E ) = √ π a / e √ aE E / . (9)The level density parameter a is taken as A/ − ,its empirical value.In Eq. (8), the prime on Σ denotes exclusion of thelight nuclei ( A ≤
8) from the sum. For these nuclei,we take only the ground states; their degeneracy factor g is taken from experiments. For other nuclei, for bothground and excited states, g is taken as 1 or 2, accordingas they are bosonic or fermionic. The lower limit E isdictated by the location of the first excited state. Theupper limit E s is the separation energy. We take E =2MeV and E s =8 MeV.The scattering piece ln Z scat of Eq. (4) can be formally,but explicitly written [19] for nuclear matter:ln Z scat = V X Z t ,N t e βµ Zt,Nt λ ( A t m ) X σ e βB Zt,Nt,σ × Z ∞ dǫ e − βǫ πi T r Z t ,N t ,σ (cid:18) A S − ( ǫ ) ∂∂ǫ S ( ǫ ) (cid:19) c . (10)Here, the double sum refers to the sum over all possiblescattering channels, each having its chemical potential µ and formed by taking any number of particles from anyof the stable species (proton, neutron and nuclei in theirground and excited states) and the trace is over all planewave states for each of these channels. S is the scatteringoperator and A the boson symmetrization or fermion an-tisymmetrization operator. The subscript c denotes onlythe connected parts in the diagrammatics of the expres-sion in the parenthesis. A channel in the set has a totalnumber of Z t protons and N t neutrons ( A t = N t + Z t ); σ denotes all other labels required to fix a channel withinthis set. B Z t ,N t ,σ is the sum of the individual bindingenergies of all the particles in the channel and ǫ is thetotal kinetic energy in the c.m. frame of the scatteringpartners. Examination of Eq. (10) shows that channelswith larger binding energies are more important, becauseof the factor e βB Zt,Nt,σ . Furthermore, two-particle chan-nels are expected to dominate over multiparticle channelswith the same Z t and N t from binding energy consider-ation. We therefore consider only two-particle scatteringchannels. It is convenient to divide the channels into lightones, consisting of low mass particles ( A <
8, say) andheavy ones ( A ≥ Z scat = ln Z light + ln Z heavy , (11)as the sum of contributions from the light and heavychannels.Experimentally it is known that the scattering of rela-tively heavier nuclei is dominated by a multitude of res-onances near the threshold. The S -matrix elements arethen approximated by resonances. Following [21, 22],each of these resonances are treated like an ideal gasterm and then ln Z heavy can be written in the same formof ln Z (0) ex , assuming their level density to be the sameas those of the excited states. The integration over E inEq. (8) now extends from E s to E r , the limit of resonancedomination. The damping of the integral in Eq. (10) due to the presence of the Boltzmann factor assures contri-bution only from low energies; we therefore take E r ≃ N N, N t, N He , N α ( N refersto nucleon) and αα are considered for evaluation of Z light . Inclusion of other light particle scattering chan-nels may have some influence on the present results. Inintermediate energy heavy ion collisions, for example,deuteron usually has considerable multiplicity and thecontribution to ln Z light from deuteron scattering chan-nels is worth further exploration. With the choice of thescattering channels as mentioned above, we then writeln Z light = ln Z NN +ln Z Nt +ln Z NHe +ln Z Nα +ln Z αα . (12)We explicitly write the contribution from the N N chan-nel. If we consider only elastic two-body scattering, thetrace in Eq. (10) becomes a sum over the derivative ofthe phase shifts of the appropriate partial waves. It givesformulas of the same form as derived by Beth and Uh-lenbeck [23] for the second virial coefficient. It is givenas ln Z NN = Vλ (2 m ) { ( ζ p + ζ n )∆ I =1 NN + ζ p ζ n ( − I =1 NN + ∆ I =0 NN ) } . (13)Here I is the isospin index and∆ INN = 1 πT Z ∞ dǫe − βǫ X S,L,J (2 J + 1) δ NN S +1 L J ( ǫ ) . (14)The quantity δ NN S +1 L J ( ǫ ) refers to the NN phase shift inthe LSJ channel. The contributing partial waves are de-termined by I through the requirement of antisymmetryon the total wave function of the N N system. The otherterms in Eq. (12) have nearly similar forms [19]. The ∆’sfor the
N N, N α and αα channels are evaluated in [16]and those for N He and N t are available in [24]. Thiscompletes the evaluation of partition function for nuclearmatter. It is then straightforward to get the relevant ob-servables like the pressure P , the number density of the i -th species ρ i , free energy per baryon f or the entropyper baryon s from the relations, P = T ln Z V , ρ i = ζ i (cid:18) ∂∂ζ i ln Z V (cid:19) V,T ,f = 1 ρ X i µ i ρ i − P ! , s = 1 ρ (cid:18) ∂P∂T (cid:19) µ i , (15)with the baryon density ρ = P i A i ρ i . III. RESULTS AND DISCUSSIONS
We have calculated the nuclear equation of state(EOS), fragment distributions, the symmetry free en-ergy, symmetry entropy and the symmetry coefficients(symmetry energy coefficient C E and the symmetry freeenergy coefficient C F ) for dilute nuclear matter. Thecalculations have been done in the SMF and the SNCmodel where all possible nuclear species are consideredand compared, in order to explore the role of heavyspecies, with calculations [16, 24] where only the lightspecies ( n, p, d, t, He and α ) are taken. The latter modelis subsequently referred to as the light species model(LSM). The three models (SMF, SNC and LSM) wouldbe collectively referred to as the condensation models. Tohighlight the importance of clusterization on the physicalobservables, the MF results are also presented.In practice, an asymptotic wave function may not havea precise meaning at relatively high density; it would thenbe difficult to have a meaningful expression of the parti-tion function in terms of the S -matrix elements. We,therefore, restrict our calculations to low density nu-clear matter and have considered up to a baryon density ρ =0.01 fm − .In Fig. 1, the EOS ( P − ρ ) is displayed for symmetricnuclear matter ( ρ n = ρ p ) at temperatures T =2, 4, and8 MeV. The dotted lines refer to calculations in the MFmodel with the SkM ∗ interaction. At low temperatures,it is seen that the system enters the unphysical region( dP/dρ <
0) in the density range considered. This can beavoided by applying Maxwell’s construction. In the SMF,because of the many-body correlations (condensation),this unphysical behavior does not arise. Since Coulombinteraction is absent in nuclear matter, to compare re-sults from the mean-field, a set of S -matrix calculationshave been done with Coulomb switched off in the nuclearbinding energies (SNC). The SNC calculations are repre-sented in the figure as dot-dashed lines. With isothermalcompression, at lower temperatures, the pressure levelsoff at very low densities as shown by the dot-dash lines inFig .1(a) and 1(b), signaling a behavior like a first-orderphase transition. At the higher temperature 8 MeV, thesaid transition starts at a density beyond 0.01 fm − , it isnot seen in the figure. The sum in Eq. (3) for the SNCcalculation runs up to infinity in principle; in practice,one takes a finite sum for calculational facilitation. Wehave taken the maximum mass A max =1000. The resultsare found to be not very sensitive to further increase of A max [25]. The binding energies of these nuclei are ob-tained using a simple liquid-drop mass formula [26] withCoulomb switched off.To help comparison with physical systems such as neu-tron star matter, we have also considered phenomenologi-cal binding energies (Coulomb included) [27] of the nucleithat limits the number of terms in the sum to ∼ A max = 339 and Z max =136. The EOS in the SMF are shown by the full lines.It is seen that with Coulomb in the binding energies, thesignature of the first-order phase transition is washed outwith monotonic increase in pressure on isothermal com-pression. The results in the LSM model are shown bythe dashed lines. At a given density, in the SMF, thefragment multiplicity is comparatively lesser as more nucleons get bound in larger clusters; the pressure is alsothen lesser compared to the LSM model; the deviationsare significant, particularly at higher density and at lowertemperature.The composition of matter at different densities andtemperatures for symmetric nuclear matter are shown inFig. 2 through the charge multiplicities. The left, mid-dle and right panels correspond to temperatures T =2, 4,and 8 MeV, respectively, at three different baryon den-sities ρ = 0.0001, 0.001 and 0.01 fm − . The full linesdisplay the results from the SMF, the dashed lines arethe ones from the LSM model and the dot-dashed linescorrespond to the ones from the SNC model. Examina-tion of the results brings out a few important findings:i) At very low densities, the multiplicities up to Z =2are practically the same in the SMF and LSM models.In the SMF, heavier fragments may be formed, but thatis insignificant. ii) With increase in density, heavy frag-ment formation can no longer be neglected. Increase intemperature hinders the heavy cluster formation. Themultiplicity distributions display a saw-toothed nature.This is odd-even effect due to inclusion of pairing in thephenomenological binding energies of the nuclear clus-ters; this effect is diluted with increase in temperature.iii) At relatively low temperatures and higher densities,in the SNC model, the matter consists of only nucleonsand very heavy nuclei; the matter resembles liquid-likealong with a negligible fraction of nucleonic gas. Thesefeatures are observed at all the three densities at T =2MeV and at ρ =0.01 fm − for T =4 MeV. These resultsare not shown in the figure. With increasing temperatureand decreasing density, the liquid-like structures disap-pear. These features appear from a delicate dependenceof the chemical potential on the density and temperatureand the dependence of fragment binding energy on in-creasing fragment mass which tends to saturate at ∼ ρ (fm −3 ) P ( M e V f m − ) −0.00200.0020.004 (a)(b)(c) X=0.0T=2X=0.0T=4X=0.0T=8
FIG. 1: (Color online) The EOS for symmetric nuclear matterat T = 2, 4, and 8 MeV. Calculations are shown for modelsin mean-field (dotted cyan), LSM (dashed black), SNC (dot-dashed magenta), and SMF (full line, blue) Z −2 −4 −2 −4 F r a g m e n t d e n s i t y (f m − ) −2 −4 (a) T=2 ρ=0.0001 (b) T=2 ρ =0.001 (c) T=2 ρ =0.01 (d) T=4 ρ =0.0001 (e) T=4 ρ =0.001 (f) T=4 ρ =0.01 (g) T=8 ρ =0.0001 (h)T=8 ρ =0.001 (i) T=8 ρ =0.01 FIG. 2: The charge distributions at different temperaturesand densities as shown are compared in the models of LSM(dashed line), SNC (dot-dashed) and SMF (full line).
MeV per nucleon.The evolution of symmetry free energy per baryon f sym as a function of asymmetry X at different temperaturesand densities is displayed in Fig. 3. The symmetry freeenergy for a given density and temperature is defined as f sym = f ( X ) − f (0) + 1 ρ X i [ ρ i ( X ) − ρ i (0)] B ic , (16)where B ic is the Coulomb contribution to the binding en-ergy of the i -th fragment species. The symmetry energy X f sy m ( M e V ) (a)T=2 (b)T=4(c)T=8 (d)T=2 FIG. 3: ( Color on-line) The symmetry free energy shown asa function of X at different temperatures. The dashed, dot-dashed and full lines correspond to ρ =0.0001, 0.001 and 0.01fm − , respectively. The results are calculated in the modelsof the mean-field (thin cyan), LSM (thick black), SNC (thickmagenta), and the SMF (thin blue). X −0.1−0.0500.05 s sy m (a) (b)(c) T=4T=4 (d)T=8 T=8 FIG. 4: (Color online) The symmetry entropy s sym as afunction of X at different temperatures and densities. Thenotations are the same as described in the caption to Fig. 3. e sym can be defined likewise. The dashed, dot-dashedand the full lines correspond to calculations at densi-ties of ρ =0.0001, 0.001 and 0.01 fm − , respectively. InFigs. 3(a), 3(b) and 3(c), the symmetry free energies pernucleon in the LSM (thick black lines) and the SMF (thinblue lines) are compared at the three densities at differ-ent temperatures. As seen in Fig. 3(a) at T =2 MeV, atthe lowest density, the two calculations yield nearly thesame results; with increasing density, the difference in thetwo model predictions shows up prominently because ofthe formation of larger clusters in the SMF. This differ-ence washes out gradually with increasing temperatureand the results are not discernible as seen in Fig. 3(b) ρ (fm −3 ) C E ( M e V ) ρ (fm −3 ) C F ( M e V ) (a)T=4 (b)T=4(d)T=8(c)T=8 FIG. 5: (Color online) The symmetry coefficients C E and C F as a function of density at different temperatures in themodels of mean-field (dotted cyan), LSM (dashed black), SNC(dot-dashed magenta) and SMF (full lines, blue). T (MeV) C E ( M e V ) T (MeV) C F ( M e V ) (a) ρ =0.0001 (b) ρ =0.001 (c) ρ =0.01 (d) ρ =0.0001 (e) ρ =0.001 (f) ρ =0.01 FIG. 6: The symmetry coefficients C E and C F displayed asa function of temperature at densities ρ =0.0001, 0.001, and0.01 fm − . The notations are the same as described in thecaption to Fig. 5. and Fig. 3(c). In Fig. 3(d), the symmetry free ener-gies at T =2 MeV for the three densities from the SNC(thick magenta lines) and MF (thin cyan lines) modelsare compared. In the liquid-drop mass formula, the sym-metry energy is taken to be linear in X . This is seen tobe nearly true also for symmetry free energy of nuclearmatter in a density region around the saturation density[11] in the MF model; we find the same for dilute nuclearmatter as is shown in Fig. 3(d). The symmetry energiescan then be written as e sym = C E X , f sym = C F X , (17)where C E and C F are the symmetry energy and sym-metry free energy coefficients. Clusterization affects thislinearity at low temperatures; as the temperature is in-creased, the linearity tends to be restored as seen fromFig. 3(b) and Fig. 3(c). The symmetry energy has a sim-ilar behavior and is not shown here.One interesting result that is borne out of the SMF cal-culation is that at a given T and ρ , asymmetric nuclearmatter may become more stable than symmetric matter[ f ( X ) < f (0)]. This may result in negative f sym as isshown in Fig. 3(a). This happens at relatively higherdensities. We find it to be due to the absence of isospin-conjugate (mirror) nuclei because of Coulomb interac-tion. Restricting the sum in Eq. (3) to only mirror nucleiremoves this negativity. In the SNC model, clusters al-ways occur in isospin-conjugate pairs, f sym is then alwayspositive as seen from Fig. 3(d).The symmetry entropy per baryon, defined as s sym = s ( X ) − s (0) is presented as a function of asym-metry for different densities and temperatures in Fig. 4.The notations used are the same as those used for Fig. 3.Comparison between the MF and the SNC models at thethree densities is displayed in Fig. 4(b) and Fig. 4(d). In the MF model, the symmetry entropy decreases withasymmetry. This is akin to the entropy of mixing . Theentropy per nucleon of an ideal two-component nucleongas is given, from Gibbs-Duhem relation, by s ( X ) = 52 − ρ n ρ ln ζ n − ρ p ρ ln ζ p . (18)Using the fact that for low density nucleonic matter ρ n,p ≃ λ ζ n,p , (19)the symmetry entropy can be shown to behave as s sym ( X ) = − X (cid:18) X + .... (cid:19) , (20)which for low values of asymmetry decreases linearly with X . This is independent of both temperature and den-sity and is nearly manifested for the low density nuclearmatter we have considered. As can be seen from the com-parison with the SNC calculations, clusterization changesthis behavior; in the density and temperature domainwhere clusterization becomes important, the symmetryentropy is larger compared to that in the MF model andcan even be positive. In Fig. 4(a) and Fig. 4(c), com-parison is made between the results from LSM and SMF.At high temperatures and low densities, the difference inthe results from these models is not discernible; however,at relatively high density and low temperature, s sym inthe LSM is appreciably larger than that in the SMF asdisplayed in Fig. 4(a) for ρ =0.01 fm − and T = 4 MeV.This is attributed to the relatively rapid growth of mul-tiplicity (mostly of neutrons at the cost of p , He and α )in LSM model compared to that in SMF.The conventional definition of the symmetry coeffi-cients C E and C F as given by Eq. (17) holds good when e sym and f sym are linear or nearly linear in X . In Fig. 5,we display these symmetry coefficients as a function ofdensity for temperatures when the symmetry energies arenearly linear in X in the different models. In the den-sity region investigated, in the MF model, the symmetrycoefficients increase linearly with density in contrast toa power-law dependence ∼ ( ρ/ρ ) γ with γ ∼ C E in Fig. 5(a), which at higherdensities saturates to ∼
28 MeV, the value for normal nu-clear matter as given in the liquid-drop mass formula weuse [26]. The symmetry coefficient C E in the LSM andin SMF are close at high temperature T = 8 MeV, butdiffer at T = 4 MeV where clusterization is more impor-tant. This difference is filled up for symmetry free energycoefficient C F due to the symmetry free entropy [shownin Fig. 4(a)] resulting in practically the same C F in boththe models.The temperature dependence of the symmetry coeffi-cients C E and C F at a few densities in all the modelsconsidered are displayed in the left and right panels ofFig. 6, respectively. In the mean-field model, shown bythe dotted lines, C E is linear and approximately constantfor a given density as seen earlier [28]. At very low den-sity ρ =0.0001 fm − , it is nearly zero and not discerniblein the figure. In the condensation models a pronouncedincrease in the values of C E is observed, particularly atlow temperatures. In the SNC model this value is closeto that for normal nuclear matter as expected. At largertemperatures, values of C E obtained in the condensationmodels approach those calculated from MF.The symmetry free energy coefficient C F , by ourchoice, can be written as C F = C E − T s sym X . (21)For dilute nuclear matter, in the MF model, C E is linearin T at a particular density and s sym , as stated earlier, isnegative and proportional to X . A linear increase of C F with temperature is therefore expected in the mean-fieldmodel which is realized as displayed by the dotted linesin the right panels of Fig. 6. At higher temperatures, re-sults for C F in all the models tend to merge, particularlyat low density because of the hindrance to form clusters.At lower temperatures, C F is significantly higher withclusterization compared to that in the MF. In Eq.(21),the first term on the right hand side decreases with tem-perature whereas the second term increases with conden-sation; this interplay causes a minimum in C F which ismore pronounced at lower densities. IV. CONCLUDING REMARKS
The role of condensation on some properties of dilutenuclear matter, namely, the nuclear EOS, the symme-try free energy, entropy and the symmetry coefficientshas been addressed in this article in the S -matrix frame-work. This approach has the advantage that the relevant observables can be directly connected to the experimen-tally measured quantities like the nuclear binding ener-gies and the scattering phase-shifts. This approach con-tains no interaction potential parameters and hence, theresults are mostly model-independent.Except at very low densities, the nuclear EOS in the S -matrix approach differs appreciably from the one ob-tained in the MF model. The MF model, supplementedwith Maxwell’s construction displays a liquid-gas phasetransition on isothermal compression, in the S -matrixapproach, the system responds to the compression by amarked growth of clusters out of the dilute nucleonic gas.The growth is hindered with isochoric heating.One of the very remarkable features of the results onsymmetry energies (both e sym and f sym ) in SMF is thatthe symmetry energies are generally nonlinear in X , con-trary to the results in the MF model. The symmetryenergies may even be negative in the SMF at low tem-peratures, high densities and at small X . The symmetryentropy similarly displays a subtle behavior with cluster-ization. In the conventional definition, at a particulartemperature and density, the symmetry coefficients arethen no longer independent of the asymmetry parameterand may be negative. In the region where the symme-try energies are practically linear in X , the symmetryenergy and free energy coefficients in the S -matrix ap-proach are found to be appreciably larger compared tothose obtained in the MF model. The nuclear EOS andthe symmetry energies and coefficients are thus seen tohave a significant dependence on the many-nucleon cor-relations or cluster formation in the low density nuclearmatter. Inclusion of these effects are thus warranted forthe study of physical phenomena like supernova dynam-ics that depend sensitively on the symmetry energy andits temperature and density dependence. Acknowledgments
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