aa r X i v : . [ nu c l - t h ] A p r Warm alpha-nucleon matter
S. K. Samaddar ∗ and J. N. De † Saha Institute of Nuclear Physics, 1/AF Bidhannagar, Kolkata , India
The properties of warm dilute alpha-nucleon matter are studied in a variational approach in theThomas-Fermi approximation starting from an effective two-body nucleon-nucleon interaction. Theequation of state, symmetry energy, incompressibility of the said matter as well as the α fractionare in consonance with those evaluated from the virial approach that sets a bench-mark for suchcalculations at low densities. PACS numbers: 21.65.Cd, 21.65.Ef, 21.65.Mn, 24.10.PaKeywords: nuclear matter, alpha-clusters, Thomas-Fermi approach, S -matrix I. INTRODUCTION
Cold nuclear matter at subsaturation density as α mat-ter has been subjected to a critical study for some time[1, 2]. The aim is to understand the α -clustering nearthe surface of heavy nuclei or the putative dilute alpha-condensate in light 4n nuclei. In astrophysical context,in following the evolution of the core-collapse supernovae,these studies have been extended to the case of warm nu-clear matter [3]. The homogeneous low-density nuclearmatter stabilizes as a mixture of nucleons and nucleon-clusters. It has lower free energy compared to that fornucleonic matter. The cluster composition is tempera-ture and density dependent, with increasing temperatureor decreasing density, the population of heavier clusterstends to diminish leading to a mixture of nucleons andlight clusters like d, t, He and α [4, 5]. The properties ofthe clusterized matter undergo a major change, e.g. , theincompressibility of clusterized nuclear matter is quitesmaller compared to that for homogeneous nucleon mat-ter [6]. This directly influences the collapse and bouncephase of the supernova matter. The symmetry energies ofnuclear matter are also affected significantly when mat-ter gets clusterized [7, 8]. This has an important role ina better understanding of neutrino-driven energy trans-fer in supernova matter [9]. The symmetry energy alsoinfluences the cluster composition in the crust of neutronstars and is thus instrumental in shaping the details oftheir mass, cooling and structure [10].The equation of state (EOS) of warm dilute nuclearmatter with only light clusters upto α has recently beeninvestigated in the virial approach [7, 11]; inclusion ofheavier clusters has also been made in the S -matrix (SM)framework [12]. These methods relate the calculationsdirectly to the experimental observables like the bind-ing energies and the phase-shifts and thus, as such, aremodel-independent. They are usually taken as bench-mark calculations in the domain of low-density and high-temperature; they are understood to exhaust all the dy- ∗ Electronic address: [email protected] † Electronic address: [email protected] namical information concerning the strong interactions inthe medium. For an interacting quantum gas, the virialexpansion, however, virtually ends at the second order.Formulation of higher order virial coefficients are very in-volved even at the formal level [13], making it difficult toestimate the domain of validity of the virial series trun-cated at the second order. It may further be noted thatthe density should be dilute enough so that the conceptof asymptotic wave functions as inherent in the virial ex-pansion should be meaningful.An alternate avenue could be to bypass the virial ex-pansion altogether and take recourse to nucleation in theframework of the mean-field model with a suitably choseneffective two-nucleon interaction that inherently takes aninclusive account of the scattering effects. Unlike thevirial ( S -matrix) approach which has direct contact withthe experimental data, this method has indirect contactbut it can be applied to relatively higher densities. Withincreasing density, a large number of different fragmentspecies would, however, be formed that makes the nu-merical calculation very lengthy. Before attempting anyfull-blown calculation, it may then be worthwhile as afirst step, to take only α -clustering in the nuclear matterand to examine whether the model works in the low-density region where the bench-mark calculations exist.The present work aims towards that end.For the study of the so-mentioned α -nucleon ( α N) mat-ter, we have chosen the Thomas-Fermi prescription forthe mean-field model and the finite range, momentumand density dependent modified Seyler-Blanchard (SBM)effective interaction [14]. The properties we explore in-clude the EOS of the α N matter, its symmetry energy, in-compressibility, α concentration etc. In Sec. II, the the-oretical framework for the mean-field and the S -matrixapproach is presented. Sec. III contains the results anddiscussions. Concluding remarks are given in Sec. IV. II. THEORETICAL FRAMEWORK
Given an effective two-nucleon interaction, the prop-erties of the α N matter can be evaluated by exploitingthe occupation functions of the n, p and alphas obtainedfrom minimization of the thermodynamic potential of the
TABLE I: The parameters of the effective interaction (inMeV fm units) C l C u a b d κ system. In Sec. II A, some details of the effective interac-tion used are given. In Sec II B, theoretical formulationfor obtaining the occupation functions from the Thomas-Fermi (TF) approximation is presented. In Sec. II C,expressions for various observables explored are given.In Sec. II D, a brief outline of the S -matrix approach ismade. A. The effective interaction
The form of the SBM effective interaction v is v ( r, p, ρ ) = C l,u [ v ( r, p ) + v ( r, ρ )] ,v = − (1 − p b ) f ( r , r ) ,v = d [ ρ ( r ) + ρ ( r )] κ f ( r , r ) , (1)with f ( r , r ) = e −| r − r | /a | r − r /a . (2)The subscripts l and u to the interaction strength C re-fer to like-pair (nn or pp) and unlike-pair (np) interac-tions, respectively. The range of the effective interactionis given by a, b is the measure of the strength of themomentum dependence of the interaction. The relativeseparation of the interacting nucleons is r = r − r andthe relative momentum is p = p − p ; d and κ are thetwo parameters governing the strength of the density de-pendence and ρ ( r ) and ρ ( r ) are the nucleon densitiesat the sites of the two interacting nucleons. The poten-tial parameters are given in Table I; the details for thedetermination of these parameters are given in [14].The incompressibility K ∞ of symmetric nucleon matteris mostly governed by the parameter κ ; for the potentialset we have chosen, the value of K ∞ =238 MeV.The equation of state of symmetric nuclear mattercalculated with the SBM interaction is seen to agreeextremely well [15] with that obtained in a variationalapproach by Friedman and Pandharipande [4] with v +TNI interaction. This interaction also reproducesquite well the binding energies, rms charge radii, chargedistributions and the giant monopole resonance energiesfor a host of even-even nuclei ranging from O to veryheavy systems [14]. Interactions of this type has beenused with great success by Myers and Swiatecki [16] inthe context of nuclear mass formula.
B. The occupation functions
The self-consistent occupation probabilities of nucleonsand alphas in α N matter at temperature T are obtainedin the TF approximation by minimizing the thermody-namic potential of the systemΩ = E − T S − X τ µ τ N τ − µ α N α . (3)Here τ represents the isospin index (n,p). The quantities E, S, µ τ , µ α , N τ and N α are the total internal energy, en-tropy, nucleon chemical potentials, α chemical potential,free nucleon numbers and the number of α particles, re-spectively, in the system. Chemical equilibrium in thesystem ensures µ α = 2( µ n + µ p ) . (4)The total energy of the α N matter in TF approximationis E = X τ (Z d r d p p m τ ˜ n τ ( p ) + 12 Z d r d p d r d p × [ v ( | r − r | , | p − p | ) + v ( | r − r | , ρ )][ C l ˜ n τ ( p )+ C u ˜ n − τ ( p )]˜ n τ ( p ) + 12 ( C l + C u ) Z d r d p d r d p × ˜ n τ ( p )˜ n α ( p ) Z V α d r Z d p α i ˜ n αi ( p α i )[ v ( R ′ , | p − ( p α i + p ) | ) + v ( R ′ , ( X τ ′ ρ τ ′ + ρ αi ))] ) + Z d r d p p m α ˜ n α ( p )+( C l + C u ) Z d r d p d r d p ˜ n α ( p )˜ n α ( p ) × Z V α d r d r ′ Z d p α i d p α ′ i ˜ n αi ( p α i )˜ n α ′ i ( p α ′ i ) × [ v ( R ′ , | ( p + p α i ) − ( p + p α ′ i | ) + v ( R ′ , ρ αi )] − N α B α (5)In Eq. (5), m τ and m α are the nucleon and α masses,the first and fourth terms correspond to the kinetic en-ergy of the nucleons and alphas, the second and the fifthterms refer to the interaction energy among nucleons andamong alphas, respectively and the third term is the in-teraction energy between nucleons and alphas. The var-ious space coordinates occurring in the third and fifthterms are shown in Figs. 1 and 2, respectively.These terms are evaluated in the single-folding anddouble-folding models. The last term is the bindingenergy contribution from the α particles. Here ˜ n τ = h n τ , ˜ n α = h n α where n τ and n α are the occupationprobabilities for nucleons and alphas, respectively. Sim-ilarly, ˜ n αi ( p α i ) = h n αi ( p α i ) represents the occupationprobability of the constituent nucleons in the α particle OR r R R α r r AB P
FIG. 1: Space coordinates shown for nucleon (located at A )and alpha (with center at B ) configuration. The origin of thecoordinate system is at O and P is any arbitrary point withinalpha. r rRRr r R α OP P AA
FIG. 2: Space coordinates shown for alpha-alpha configura-tion with O as the origin of the coordinate system. P and P ′ are arbitrary points within the alphas with A and A ′ as theircenters. and p α i is their intrinsic momentum inside the α . Thespace coordinates do not enter in the occupation func-tions ˜ n τ and ˜ n α as the system is infinite. For simplicity,the α particles are taken to be uniform nuclear dropswith a sharp surface and hence the space coordinates donot also occur in ˜ n αi . The notation R V α refers to config-uration integral over the volume of α . The integral over˜ p α i is over the Fermi sphere of the nucleon momenta in-side the α particles. Since alphas are difficult to excite(the first excited state in α is ∼
20 MeV), they are takento be in their ground states. All the other integrals areover the entire configuration or momentum space unlessotherwise specified. It then follows that Z ˜ n τ ( p ) d p = N τ /V = ρ τ , Z ˜ n α ( p ) d p = N α /V = ρ α , Z ˜ n αi ( p ) d p = 4 /V α = ρ αi , (6)where V is the volume of the α N system and V α = πR α ,with R α as the sharp-surface radius of the α drop takento be 2.16 fm obtained from experimental rms charge-radius of α ; ρ αi is the density of the constituent nucleonsof the α particles. The total baryon density ρ b is givenby ρ b = ρ + 4 ρ α where ρ = P τ ρ τ is the density of thefree nucleons and ρ α is the α -particle density.The total entropy of the α N system is S = X τ S τ + S α , (7)where in the Landau quasi-particle approximation, S τ = 2 h Z (cid:2) n τ ( p ) ln n τ ( p )+(1 − n τ ( p )) ln(1 − n τ ( p )) (cid:3) d r d p , (8)and S α = 1 h Z (cid:2) n α ( p ) ln n α ( p ) − (1 + n α ( p )) ln(1 + n α ( p )) (cid:3) d r d p . (9)Minimization of Ω with respect to n τ and n α , remember-ing that δn τ ( p ) and δn α ( p ) are separately arbitrary overthe whole phase space, at the end yields p m τ + Z d r d p n v ( | r − r | , | p − p | )+ v ( | r − r | , ρ ) o [ C l ˜ n τ ( p ) + C u ˜ n − τ ( p )]+ κd (2 ρ ) κ − X τ ′ Z d p ′ d r d p × [ C l ˜ n τ ′ ( p ) + C u ˜ n − τ ′ ( p )]˜ n τ ′ ( p ′ ) f ( r , r )+ 12 ( C l + C u ) Z d r d p ˜ n α ( p ) × Z d r d p α i ˜ n αi ( p α i ) (cid:8) v ( R ′ , | p − ( p α i + p ) | )+ v ( R ′ , ( ρ + ρ αi )) (cid:9) + 14 ( C l + C u ) κd ( ρ + ρ αi ) κ − × X τ ′ Z d p ′ d p ˜ n τ ′ ( p ′ )˜ n α ( p ) ρ αi × Z d r Z V α d r e −| R ′ | /a | R ′ | /a + T (cid:2) ln n τ ( p ) − ln(1 − n τ ( p )) (cid:3) − µ τ = 0 , (10)and p m α + 2( C l + C u ) Z d r d p ˜ n α ( p ) × Z d r d p α i d r ′ d p α ′ i ˜ n αi ( p α i )˜ n α ′ i ( p α ′ i ) × n v ( R ′ , | ( p + p α i ) − ( p + p α ′ i ) | ) + v ( R ′ , ρ αi ) o + 12 ( C l + C u ) X τ Z d p d p α i ˜ n τ ( p )˜ n αi ( p α i ) × Z d r Z V α d r n v ( R ′ , | p − ( p + p α i ) | )+ v ( R ′ , ρ + ρ αi ) o + T (cid:2) ln n α ( p ) − ln(1 + n α ( p )) (cid:3) − ( µ α + B α ) = 0 . (11)Without any loss of generality, r can be set equal tozero in Eqs. (10) and (11). The single-particle occupa-tion functions n τ ( p ) and n α ( p ) for nucleons and alphasare determined from Eqs. (10) and (11), respectively.Eq. (10), after some algebraic manipulations can be writ-ten as p m τ + V τ + p V τ + V τ + T (cid:2) ln n τ ( p ) − ln(1 − n τ ( p )) (cid:3) − µ τ = 0 . (12)The momentum-dependent nucleon single-particle poten-tial V τ ( p ) is given by V τ ( p ) = V τ + p V τ , (13)where V τ is the momentum-independent part. Eq. (12)leads to n τ ( p ) = (cid:20) exp (cid:26)(cid:18) p m ∗ τ + V τ + V τ − µ τ (cid:19) /T (cid:27)(cid:21) − , (14)where m ∗ τ is the nucleon effective mass, m ∗ τ = (cid:20) m τ + 2 V τ (cid:21) − , (15)and V τ is the rearrangement potential coming from thedensity dependence of the interaction. Similarly Eq. (11)can be written as p m α + V α + p V α + T (cid:2) ln n α ( p ) − ln(1 + n α ( p )) (cid:3) − ( µ α + B α ) = 0 , (16)which yields n α ( p ) = (cid:20) exp (cid:18)(cid:26) p m ∗ α + V α − ( µ α + B α ) (cid:27) /T (cid:19) − (cid:21) − (17)where m ∗ α = (cid:20) m α + 2 V α (cid:21) − , (18)is the α effective mass. V α is the momentum-independentpart of the α -single particle potential V α (= V α + p V α ) in the system. The nucleon and α masses are renormalizeddue to the momentum dependence in the interaction.The expressions for V τ can be arrived at as, V τ = − πa (cid:8) − d (2 ρ ) κ (cid:9) ( C l ρ τ + C u ρ − τ )+ 16 π a b h (cid:20) C l (2 m ∗ τ T ) / J / ( η τ ) + C u (2 m ∗− τ T ) / × J / ( η − τ ) (cid:21) + 14 I ( C l + C u ) ρ α ρ iα (cid:20) < p α > + < ( p αi ) >b + d ( ρ + ρ αi ) κ − (cid:21) . (19)The first two terms come from the interaction betweenfree nucleons, the last term originates from the presenceof alphas. In Eq. (19), I is the six-dimensional integral(see Fig. 1) I = Z V α d r Z d R e −| r + R | /a | r + R | /a . (20)This integral can be evaluated analytically. The quantity < p α > is the mean squared value of the α momentumin α N matter and < ( p αi ) > is the mean squared valueof the constituent nucleon momentum inside the α . Thevalue of < p α > is < p α > = (2 m ∗ α T ) B / ( η α ) /B / ( η α ) ≃ m ∗ α T, (21)and < ( p αi ) > ≃
35 ( P αF ) (22)where P αF is the value of the zero-temperature nucleonFermi momentum inside α , taken to be 220.5 MeV/c,consistent with the α sharp surface radius. The J k ( η )and B k ( η ) are the Fermi and Bose integrals, J k ( η ) = Z ∞ x k dxe ( x − η ) + 1 , (23)and B k ( η ) = Z ∞ x k dxe ( x − η ) − , (24)with η τ = ( µ τ − V τ − V τ ) /T,η α = ( µ α + B α − V α ) /T. (25)The expressions for V τ , V τ , V α and V α are given as V τ = 4 πa b [ C l ρ τ + C u ρ − τ ] + 14 I ( C l + C u ) ρ α ρ iα b , (26) V τ = 4 πa κd (2 ρ ) κ − X τ ′ [ C l ρ τ ′ + C u ρ − τ ′ ] ρ τ ′ + 14 I ( C l + C u ) κd ( ρ + ρ αi ) κ − ρ αi ρ α ρ, (27) V α = 14 ( C l + C u ) ρ αi ( ρ αi ρ α I α (cid:2) d (2 ρ αi ) κ −
1+ 3 m ∗ α T + ( P αF ) b (cid:3) + I (cid:2) ρ (cid:8) d ( ρ + ρ αi ) κ −
1+ 35 ( P αF ) b (cid:9) + X τ π (2 m ∗ τ T ) / J / ( η τ ) h b (cid:3)) , (28)and V α = 14 ( C l + C u ) ρ αi (cid:8) ρ αi ρ α I α + Iρ (cid:9) /b . (29)In both V τ and V τ , the last term stems from the α -N interaction. The effective nucleon mass in pure nu-cleonic matter thus gets modified due to clusterization.The integral I α occurring in Eqs. (28) and (29) is anine-dimensional integral (see Fig. 2), I α = Z V α d r Z V α d r ′ Z d R e −| R + r − r ′ | /a | R + r − r ′ | /a , (30)which can be evaluated numerically. If the alphas donot interpenetrate, the integral over R excludes the α volumes. C. Expressions for observables in TFapproximation i) Energy per baryon :
The energy per baryon e b of the α N matter can be calculated from Eq. (5). It canbe split into the following form, e b = e NN + e αN + e αα . (31)Here e NN comes from the kinetic energy of the free nu-cleons and the interactions among them, e αN arises fromthe interaction among the free nucleons and the alphasand e αα stems from the kinetic energy of the alphas andthe interaction among themselves. The expressions forthem are e NN = 1 ρ b X τ ρ τ (cid:2) T J / ( η τ ) /J / ( η τ ) { − m ∗ τ V τ } + 12 V τ (cid:3) , (32) e αN = 14 ρ b ( C l + C u ) Iρ α ρ iα hn m ∗ α T + 3 / P αF ) b − d ( ρ + ρ αi ) κ o ρ + 1 b X τ (cid:0) πh (2 m ∗ τ T ) / J / ( η τ ) (cid:1)i , (33) and e αα = 1 ρ b h πm α h (2 m ∗ α T ) / B / ( η α )+ 14 ( C l + C u ) I α ρ α ( ρ αi ) n d (2 ρ αi ) κ −
1+ 6 m ∗ α Tb + 65 ( P αF ) b oi . (34)In the above equations, as stated earlier, ρ b (= ρ + 4 ρ α )corresponds to the total baryon density, ρ and ρ α arethe free nucleon and α densities, respectively, in the α Nsystem. ii) Entropy per baryon:
The entropy per baryon s b can be evaluated using Eqs. (8) and (9). It is additiveand can be written as s b = s N + s α , (35)where s N and s α are the contributions to entropy fromfree nucleons and alphas respectively. Their expressionsreduce to s N = 1 ρ b X τ ρ τ h J / ( η τ ) /J / ( η τ ) − η τ i , (36)and s α = ρ α ρ b h B / ( η α ) /B / ( η α ) − η α i . (37) iii) Pressure of α N matter:
Once the energy andentropy of the composite system are known, the pressurecan be calculated from the Gibbs-Duhem thermodynamicidentity, P = X τ ρ τ µ τ + ρ α µ α − f b ρ b , (38)where f b is the free energy per baryon, f b = e b − T s b . iv) Incompressibility and the symmetry coeffi-cients: The incompressibility K can be computed fromthe derivative of pressure K = 9 dPdρ . (39)The symmetry free energy and symmetry energy coeffi-cients C F and C E are calculated from C F = 12 (cid:16) ∂ f b ∂X (cid:17) X =0 , (40) C E = 12 (cid:16) ∂ e b ∂X (cid:17) X =0 , (41)where X is the neutron-proton asymmetry of the α N sys-tem. It is given as X = ( ρ nb − ρ pb ) /ρ b , where ρ nb and ρ pb are the total neutron and proton density, respectively. D. The S -matrix approach The relevant key elements of the S -matrix framework[17] as applied in the context of dilute nuclear matter[8, 12] are outlined in brief below.The grand partition function of an interacting infinitesystem of neutrons and protons can be written as Z = ∞ X Z,N =0 ( ζ p ) Z ( ζ n ) N Tr Z,N e − βH . (42)where ζ p = e βµ p and ζ n = e βµ n are the elementary fu-gacities with β = 1 /T and µ ’s are the nucleonic chemicalpotentials. Here H is the total Hamiltonian of the sys-tem and the trace T r
Z,N is taken over states of Z protonsand N neutrons. The partition function can be split intotwo types of terms [17]ln Z = ln Z (0) part + ln Z scat . (43)The first term on the right hand side corresponds to con-tributions from stable single-particle states of clustersof different sizes including free nucleons formed in thesystem; the second term refers to all possible scatteringstates. The superscript (0) indicates that the clusters be-have as an ideal quantum gas. In general, ln Z (0) part con-tains contributions from the ground states as well as theparticle-stable excited states of all the clusters. The scat-tering term ln Z scat may be written as a sum of scatteringcontributions from a set of channels, each set having to-tal proton number Z t and neutron number N t . Since ourinterest in the present work is focused on α N matter, inln Z (0) part , we include only the nucleons and the groundstate of α ; similarly in ln Z scat , only the scattering chan-nels N N, αN and αα are considered, so thatln Z scat = ln Z NN + ln Z αN + ln Z αα . (44)Each of the terms in Eq. (44) can be expanded in therespective virial coefficients. Expansion upto the second-order coefficients are only considered. They are writtenas energy integrals of the relevant phase-shifts [6, 7]. Thepartition function can then be written explicitly asln Z = V n λ N [ ζ n + ζ p + b nn ζ n + b pp ζ p + 12 b np ζ n ζ p +8 ζ α + 8 b αα ζ α + 8 b αn ζ α ( ζ n + ζ p )] o , (45)where λ N = h √ πmT is the nucleon thermal wavelength, ζ α = e β ( µ α + B α ) , B α being the binding energy of α and µ α = 2( µ n + µ p ). The b nn , b np , etc., are the temperaturedependent virial coefficients [7, 12]. The value of thevirial coefficient b np has been adjusted so as to excludethe resonance formation of deuteron from n-p scatteringto be consistent with our choice of the α N matter.The knowledge of the partition function allows all therelevant observables to be calculated. The pressure is given by P = T ln Z /V . (46)The number density ρ i is calculated from ρ i = ζ i (cid:18) ∂∂ζ i ln Z V (cid:19) V,T , (47)where i stands for n,p, or α . Once the pressure, densitiesand chemical potentials are known, the free energy can beobtained from the Gibbs-Duhem relation. The entropyper baryon is calculated from s b = 1 ρ b (cid:16) ∂P∂T (cid:17) µ , (48)which yields the energy per baryon as e b = f b + T s b . Theexplicit expression for the entropy per baryon is s b = 1 ρ b ( PT − X i ρ i ln ζ i + Tλ N (cid:2) ζ n ζ p b ′ np + ( ζ n + ζ p ) b ′ nn +8 ζ α b ′ αα + 8 ζ α ( ζ n + ζ p ) b ′ αn (cid:3)) . (49)The prime on the virial coefficients denotes their temper-ature derivatives. III. RESULTS AND DISCUSSIONS
In the mean-field framework, the momentumand density-dependent finite-range modified Seyler-Blanchard force as scripted in Eqs. (1) and (2) has beenchosen as the effective two-nucleon interaction in ourcalculations. To start with, we take baryon matter ata given density ρ b at a temperature T with an isospinasymmetry X . The unknowns are the free nucleondensities ρ n , ρ p and the α concentration in the matter.The three constraints are the conservation of the totalbaryon number, the total isospin and the condition ofchemical equilibrium between the nucleons and alphas.Starting from a guess value for the α concentration,the unknowns are determined iteratively using theNewton-Raphson method. For our calculations, themasses of neutron and proton are taken to be the same,for α binding energy, the experimental value of 28.3MeV is used. For the evaluation of the αα -potential,the α -particles are assumed to be nuclear droplets withsharp boundary and that they do not interpenetrate.The calculations are done upto a baryon density ρ b =0.01 fm − . To show the effect of temperature ondifferent properties of the dilute matter, results are re-ported for temperatures T = 3, 5, and 10 MeV. In Fig. 3,the baryon fraction in α , Y α =4 ρ α /ρ b (hereafter referredto as α fraction) in α N matter as a function of density α N (TF) α N (SM) ρ b (fm -3 ) Y α T=3T=5T=10T=3T=5T=10X=0.0X=0.2(a)(b)
FIG. 3: (color online) The α fraction Y α = 4 ρ α /ρ b shown asa function of baryon density ρ b in TF and SM approaches at T =3, 5 and 10 MeV for symmetric matter ( X =0.0) and asym-metric matter ( X =0.2) in panels (a) and (b), respectively. α N (TF) α N (SM) X Y α ρ b =0.001 ρ b =0.01 T=3T=5T=3T=5T=10(a)(b)
FIG. 4: (color online) The α fraction Y α displayed as a func-tion of asymmetry X at baryon density ρ b =0.001 (upperpanel) and at 0.01 fm − (lower panel) at T = 3, 5 and 10MeV in TF and SM approaches. -15-10-50-25-20-15-10 F / A ( M e V ) N (TF) α N (TF) α N (SM) ρ b (fm -3 ) -70-60-50-40-30 T=3
T=5T=10 (a)(b)(c)
FIG. 5: (color online) Free energy per baryon
F/A shown asa function of ρ b at T =3, 5 and 10 MeV in the TF frameworkfor homogeneous nucleonic matter (blue lines) and α N matter(black lines). The red lines represent results from the SMapproach. at the three temperatures mentioned are shown for sym-metric ( X =0) and asymmetric ( X =0.2) nuclear matterin panels (a) and (b), respectively. The black lines corre-spond to results obtained in the TF approximation [ α N(TF)], the red lines refer to those in the SM approach[ α N (SM)] with consideration of only n, p and α as theconstituents of the baryonic matter. At low temperaturesand higher densities, it is seen that alphas are the majorconstituents of the matter, with increasing temperature,the free nucleon fraction increases at the cost of α den-sity. At moderate asymmetry X =0.2, the α populationis somewhat lower compared to that for symmetric nu-clear matter. In the temperature and density domainthat we explore, the results from both the SM and TFapproach are found to be quite close. The asymmetrydependence of α fraction Y α is displayed in Fig. 4 at tworepresentative densities ρ b =0.001 and 0.01 fm − at thethree temperatures. With increasing asymmetry, the α concentration decreases, the decrease is more prominentat lower temperature. At the lower density (Fig. 4(a)),results for T =10 MeV are not shown as Y α is close tozero.In Fig. 5, the free energy per baryon for the homoge-neous nucleonic matter (denoted by N(TF)) and the α Nmatter in the TF approximation are presented in panels(a), (b), and (c) at T = 3, 5, and 10 MeV, respectively.The calculations presented refer to symmetric nuclearmatter. The blue and black lines represent results forN(TF) and α N(TF). It is clearly seen that the cluster-ized matter has lower free energy compared to homoge- ρ b (fm -3 ) P ( M e V f m - ) N (TF) α N (TF) α N (SM)
T=3T=5T=10 (a)(b)(c)
FIG. 6: (color online) Pressure P as a function of ρ b . Thenotations are the same as in Fig. 5. neous nucleonic matter. This is more prominent at lowertemperatures, higher temperature tends to melt away theclusters. For comparison, results from the S -matrix ap-proach are also presented. They are shown by the redlines, nearly indistinguishable from those from α N(TF).Fig. 6 displays the pressure of the baryonic matter. Atlower temperatures ( T =3 and 5 MeV), the nucleonic mat-ter shows the rise and fall of the pressure with densityleading to unphysical region. For α N matter, however, nosuch unphysical region is observed in the density regionwe have studied. Both the TF and the SM approachesyield nearly the same value of pressure. At high temper-ature the α concentration becomes very less, the pressurein all the three approaches are then nearly the same inthis density region.In Fig. 7, the effective masses of nucleon and α areshown as a function of density at the temperatures men-tioned. The nucleon effective mass is calculated for bothnucleonic matter (blue line) and α N matter (full blackline) in the TF approximation. The nucleon effectivemass at a given ρ b in homogeneous nucleonic matter isalways lower compared to that in clusterized matter. Itis independent of temperature. In α N matter it nomi-nally decreases with temperature. At high temperature,the nucleon effective masses calculated in the homoge-neous and clusterized matter are nearly degenerate, withlowering of temperature, the degeneracy is lifted due tothe increase in the α concentration. The effective α mass is shown by the dashed black lines. With increas-ing temperature, the medium effect on the α mass getsstrikingly enhanced. This is due to the interplay of thetemperature-dependent contributions from the αα inter-actions and α N interactions corresponding to the first m N */m N (N)m N */m N (N α ) m * / m ρ b (fm -3 ) (a)(b)(c) ---- m α */m α (N α ) T=3T=5T=10
FIG. 7: (color online) The nucleon (full black lines) and α (dashed black lines) effective masses shown as a function of ρ b at T =3, 5 and 10 MeV in the TF framework for α N matter.The blue lines refer to the corresponding nucleon effectivemasses for homogeneous nucleonic matter. and the second term within the braces in Eq. (29).The incompressibility of the baryonic matter as a func-tion of density is displayed in Fig. 8 at the three temper-atures. At very low density and higher temperature, thematter is mostly nucleonic in all the three approaches,so the incompressibility K is ∼ T ; this one sees at thelower densities considered at T =10 MeV in panel (c) ofthis figure. Even at this very high temperature, however,the nucleonic interactions have their role as the densityincreases; this results in the reduction of the incompress-ibility from the ideal gas value. At the lower tempera-tures (panels (a) and (b)), clusterization softens the mat-ter towards compression compared to homogeneous mat-ter (shown in the lower density region); increasing den-sity, however, pushes the homogeneous matter towardsthe unphysical region leading to negative incompressibil-ity.The symmetry energy coefficients C E and C F of thebaryonic matter as a function of density are displayed inthe left and right panels, respectively, of Fig. 9 at thethree temperatures studied. The blue lines refer to cal-culations for the homogeneous matter, the black and redlines represent results for α N(TF) and α N(SM). Cluster-ized matter displays a marked increase in the symmetrycoefficients noticed already earlier [6, 7]. The two ap-proaches to clusterization lead to the same values of thesymmetry coefficients at lower densities, with increase indensity the difference widens, more so at lower tempera-tures.The results presented so far in the α N(TF) approach
N (TF) α N (TF) α N (SM) K ( M e V ) ρ b (fm -3 ) (a)(b)(c)T=3T=5 T=10
FIG. 8: (color online) The incompressibility K for baryonicmatter shown as a function of ρ b at T =3, 5 and 10 MeV. Thenotations are the same as in Fig. 5. have been calculated with the assumption that the alphasdo not overlap, they are mutually impenetrable sphericaldrops. This assumption relies on the fact that the alphasare very tightly bound and very hard to excite. To ex-plore the effect of overlap in alphas, we consider a possi-bility of penetration with at best a 5 % overlap in volume(the value of I α in Eq. (30) then changes accordingly).Calculations have been repeated with this changed condi-tion. The so-calculated free energy per baryon, pressureand the α fraction Y α in the baryonic matter are pre-sented in panels (a), (b) and (c), respectively, of Fig. 10at T =3 MeV (the dot-dashed black lines) and comparedwith those calculated with the no-overlap condition (thefull black lines) and also those from the α N(SM) ap-proach (the red lines). There is no significant change inthe free energy or in α fraction, but the pressure changesperceptibly, particularly at higher density. The goodagreement between the no-overlap α N(TF) calculationswith those from the bench-mark α N(SM) shows the via-bility of the approximation of the impenetrability of thealphas.
IV. CONCLUDING REMARKS
Clusterization in warm dilute nuclear matter has beentreated earlier in the virial approach or in the S -matrixframework. These are model-independent parameter-freecalculations. As explained in the introduction, thesemethods may have limitations at relatively high densi-ties and low temperatures. An alternate avenue for deal- C E , F ( M e V ) ρ b (fm -3 ) N (TF) α N (TF) α N (SM)
T=3T=5T=10C E T=3T=5T=10 C F (a)(b)(c) (d)(e)(f) FIG. 9: (color online) The symmetry energy C E (left panels)and symmetry free energy coefficients C F (right panels) shownas a function of ρ b at temperatures T =3, 5 and 10 MeV. Thenotations are the same as in Fig. 5. -12-10 F / A ( M e V ) P ( M e V f m - ) ρ b (fm -3 ) Y α No overlap (TF) 5% overlap (TF)(SM) (a)(b)(c) T=3 α N matter
FIG. 10: (color online) The free energy per particle, pressureand α fraction shown as a function of ρ b at T =3 MeV in panels(a), (b) and (c), respectively, for α drops with no overlap (fullblack lines) and with at best 5% overlap (dashed-dot blacklines) in the TF approximation. The same observables arealso shown in the SM approach (red lines). α as the constituentsof the matter at low densities and see how the resultscompare with those from the model-independent virialapproach.We have chosen the SBM interaction that nicely re-produces the bulk properties of nuclear matter and offinite nuclei. We have calculated the α fraction, free en-ergy, pressure, incompressibility and the symmetry coef-ficients of this α N matter in this mean-field frameworkand find that all these results compare extremely well with those obtained from the S -matrix method, partic-ularly in the low-density high-temperature regime. Thisgives one confidence in the applicability of this mean-fieldapproach in dealing with the EOS of warm dilute bary-onic matter and the possibility of extending this methodto higher densities. The price, however, is considerationof a larger number of fragment species and a numericallyinvolved calculation. Acknowledgments
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