ЯДЕРНАЯ ФИЗИКА, 2014, том 77, № 3, с. 314–322

ЯДРА

THE PROPERTIES OF NUCLEAR MATTERAT ZERO AND FINITE TEMPERATURES

c© 2014 H. M. M. Mansour1), Kh. S. A. Hassaneen2), 3)

Received June 17, 2013

The properties of nuclear matter are studied in the frame of the Brueckner theory. The Brueckner–Hartree–Fock approximation plus two-body density-dependent Skyrme potential which is equivalent tothree-body interaction are used. Various modern nucleon–nucleon potentials are used in the frameworkof the Brueckner–Hartree–Fock approximation, e.g.: CD-Bonn potential, Nijm1 potential, and Reid93 potential. These modern nucleon–nucleon potentials fit the deuteron properties and are phase shiftsequivalent. The equation of state at T = 0, pressure at T = 0, 8, and 12 MeV, free energy at T = 8 and12 MeV, nuclear matter incompressibility, and the symmetry energy calculation are presented. The hotproperties of nuclear matter are calculated using T 2-approximation method at low temperatures. Goodagreement is obtained in comparison with previous theoretical estimates and experimental data, especiallyat low densities.

DOI: 10.7868/S0044002714020135

1. INTRODUCTION

The determination of thermodynamic propertiesof hot nuclear matter is extremely relevant for itsapplications in astrophysics and heavy-ion reactions.Pressure and entropy at finite temperature are crucialingredients in the modeling of core-collapse super-novae and protoneutron stars, which involve pro-cesses characterized by densities up to several timesthe nuclear saturation density and temperatures upto few tens of MeV [1]. The nuclear equation of state(EOS) plays an important role also in the interpreta-tion of nucleus–nucleus collisions, in which a hot anddense state of matter is formed. Of particular interestis the study of a possible phase transition occurringat subsaturation densities. Because of the van derWaals nature of nucleon–nucleon (NN ) interactions,at low densities and high temperatures nuclear matteris expected to undergo a first-order transition to a gasphase [2].

The calculation of the EOS of nuclear matterstarting from realistic models of the NN interac-tion is still one of the challenging open problemsin nuclear physics. In fact, the presence of strongshort–range and tensor components in the realisticNN interactions, which are required to fit the NN

1)Physics Department, Faculty of Science, Cairo University,Egypt.

2)Physics Department, Faculty of Science, Sohag University,Egypt.

3)Physics Department, Faculty of Science, Taif University,Saudi Arabia.

scattering data, are the origin of the correspondingcorrelations in the nuclear wave function. The studyof these correlations and their influence on differentobservables has recently made important progressnot only on the theoretical side but also on theexperimental point of view [3–5].

On a microscopic basis, the properties of hot sym-metric nuclear matter have been studied in the frame-work of the finite temperature Brueckner–Hartree–Fock (BHF) approach using Argonne v14 potential[6]. It is well known that three-body forces (3BF)are necessary for reproducing the empirical satura-tion properties of cold symmetric nuclear matter ina nonrelativistic microscopic approach [7–11]. Twokinds of 3BF have been chosen in the BHF formalism[7–10]. One is the semi-phenomenological 3BF [12]which has two or few adjustable parameters whichare determined by fitting the empirical saturationdensity and energy of the cold symmetric nuclearmatter in the BHF calculations [10]. The other is themicroscopic 3BF based on meson exchange coupledto the intermediate virtual excitations of nucleon–antinucleon pairs and nucleon resonances [7–9].

The results of BHF calculations depend on thechoice of the single-particle potential, U(k) [13]. Inthe “conventional” or “gap” choice one assumes thatthe auxiliary potential U(k) is zero above the Fermimomentum kF. In the “continuous choice” [14] thesame definition of the self-consistent single-particlepotential U(k) is extended to momenta k larger thankF, thus making U(k) a continuous function throughthe Fermi surface. This continuous choice leads to an

314

THE PROPERTIES OF NUCLEAR MATTER 315

enhancement of correlation effects in the medium andtends to predict larger binding energies for symmetricnuclear matter than the gap choice [14]. In the presentwork a “continuous” choice of the single-particlespectrum is used.

In the present work the equation of state, pressure,etc. are computed using different modern NN po-tentials like CD-Bonn [15], Nijm1, and Reid 93 [16]potentials in the framework of BHF approach (con-tinuous) choice plus a two-body density-dependentSkyrme potential with two and four terms with dif-ferent powers of the densities (Eq. (9) below). Theadditional density-dependent term is added after theBHF self-consistent procedure has been completed.The same results have been done with the conven-tional choice of the auxiliary potential U(k) in [17]using Skyrme force with two terms in the summationin Eq. (2.9). Nuclear matter is determined to saturateat density, ρ0 = 0.17 ± 0.02 fm−3, and energy pernucleon, E/A = −16 ± 1 MeV.

Also we computed the symmetry energy as. Thesymmetry energy at high densities is very importantin nuclear astrophysics for understanding, e.g., thecooling mechanisms in a neutron star [18]. In thenext section we give a brief description of the BHFapproach theory and the Bethe–Goldstone equation(BGE). Actually, Section 3 is for the properties ofnuclear matter at T = 0, and Section 4 is for theproperties of nuclear matter at finite T . Summary isgiven in Section 5.

2. THE MODEL

The G matrix, which describes in an effective waythe interaction between two nucleons in the presenceof a surrounding medium, is written schematically as

G(ω) = V + VQ

ω − H0 + iηG(ω). (1)

This is known as the BGE. In the above expression,V denotes the bare NN potential, η is infinitesimalsmall number, H0 is the unperturbed energy of theintermediate scattering states, Q is the Pauli operatorwhich allows only intermediate states compatiblewith the Pauli principle, and ω, the so-called startingenergy which is usually the sum of nonrelativisticsingle-particle energies of the states of the interactingnucleons,

ω = e(k) + e(k′). (2)

The single-particle energy e(k) is given by

e(k) =�

2k2

2m+ U(k), (3)

where U(k) is called the single-particle potential.Physically, U(k) describes the mean field felt by the

nucleon due to its interaction with the other nucleonsof the medium. In the BHF approach, U(k) is givenby

U(k) = (4)

= Re∑

k′

θ(k′)〈kk′|G(ω = e(k) + e(k′))|kk′〉A,

where |kk′〉 refers to antisymmetrized two-bodystates, and

θ(k) =

{1, if k � kF,

0, otherwise,(5)

is the step function that represents the occupationprobability of a free Fermi gas with a Fermi mo-mentum kF. In the present calculation, we adoptthe so-called continuous choice [14] for the single-particle potential. The continuous choice has beenshown to provide a much faster convergence of thehole-line expansion for the energy per nucleon inthe nuclear matter at T = 0 than the gap choice[13]. The use of the continuous choice potentialimplies that the G-matrix elements needed in the self-consistent calculation are complex. The prescriptionadvocated by Mahaux for the average binding energyper nucleon is obtained from the Brueckner G matrix,G(ω), according to the equation

E

A=

∑

k

�2k2

2m+ (6)

+12

∑

k,k′<kF

〈kk′|G(ω = e(k) + e(k′))|kk′〉 =

=3�

2k2F

10m+

32k3

F

kF∫

0

k2dkU(k).

The k momentum dependence of the single-particleenergy is usually approximated by a quadratic func-tion which consists of a constant term Δ, the meanfield, and a squared momentum-dependent term,related to the nucleon effective mass m∗, reading

e(k) =�

2k2

2m∗ + Δ. (7)

From Eqs. (3) and (7), one can calculate the effectivemass m∗ from the slope of U(k) at the Fermi momen-tum as [19]

m∗

m=

[1 +

m

�2kF

(dU(k)

dk

)

k=kF

]−1

. (8)

The parameters m∗ and the constant Δ can then bereadjusted in such a way that the parameterizationEq. (8) reproduces these two energies. This procedure

ЯДЕРНАЯ ФИЗИКА том 77 № 3 2014 2*

316 MANSOUR, HASSANEEN

0.8

0.7

0.6

0.5

m */

m

0.1 0.3 0.5 0.7

ρ

, fm

–3

CD-BonnNijm 1Reid 93

1.0

0.9

0.8

0.7

m

*/

m

0.1 0.3 0.5 0.7

Fig. 1. The effective mass (m∗/m) as a function of density (ρ) of symmetric nuclear matter (left panel) and pure neutron matter(right panel) using different potentials.

is then iterated until a self-consistent solution isobtained. Finally we want to stress that, despite thefact that the parabolic approximation for the single-particle energy is not accurate [13] and we use anot so large cutoff for the single-particle momentum,we believe that the differences in the results forvarious NN interactions, obtained within the sameapproximation scheme, are sensible and meaningful.

In this section we show the results for the nu-cleon effective mass over the nucleon mass m∗/mcalculations as a function of Fermi momentum kF,i.e., density ρ, where m = 938.9260 MeV. In Fig. 1we draw the dependence of m∗/m on the densityρ for different potentials. The left panel of Fig. 1 isfor symmetric nuclear matter, the right panel is forpure neutron matter. We observe in the symmetricnuclear matter that at low density ρ, the resultsfor all potentials are identical, while at high densitythere are differences between the results obtainedusing different potentials. For pure neutron matter,the right panel of Fig. 1, we note that the effectivemass decreases almost linearly with increasing thedensity ρ. At low density the values of the effective

Table 1. Parameters ti and xi defining a two-body density-dependent potential of Eq. (9) as obtained for the fit tothe saturation point ρ0 = 0.17 fm−3; E/A = −16 MeV fordifferent potentials

PotentialContinuous choice

t1 t2 x1 x2

CD-Bonn –656.5 1197.8 –0.2991 –0.7721

Nijm1 –673.2 1156 –0.5146 –0.9705

Reid 93 –782.4 1176.2 –0.4472 –1.0043

mass for pure neutron matter are larger than that ofthe symmetric nuclear matter.

Recently, Hassaneen et al. [13] have studied theproperties of nuclear matter in the framework of BHFapproach with an angle-average approximation to thePauli operator using several modern NN potentials.It is found that BHF theory is able to produce satura-tion due to short-range correlations but fails to repro-duce the empirical region of saturation quantitatively(ρ0 = 0.17 fm−3; E/A = −16 MeV). This is true forall realistic and high-precision NN potentials. Thismeans that to obtain the empirical saturation pointsone should add the three-body forces to the model[6–12] or go beyond the suggested model [20]. In [20]the authors discussed the properties of the nuclearmatter in different approaches. For the microscopicapproaches they suggested to supplement those witha density-dependent contact interaction to achievethe empirical saturation point (see Eq. (30) in thisreference).

In the present study, we will examine the propertiesof nuclear matter like the binding energy per nucleon,pressure, and incompressibility using BHF approachwith the addition of three-body forces.

In the present work one may introduce a Skyrmeeffective interaction density-dependent term [21] inaddition to the BHF potential,

V (r1, r2) =4∑

i=1

ti(1 + xiPσ)ραiδ(r1 − r2). (9)

This is a two-body density-dependent potentialwhich is equivalent to three-body interaction. Hereti and xi are interaction parameters, Pσ is the spin-exchange operator, ρ is the density, r1 and r2 arethe position vectors of particle (1) and particle (2),

ЯДЕРНАЯ ФИЗИКА том 77 № 3 2014

THE PROPERTIES OF NUCLEAR MATTER 317

20

10

0

–10

E

/

A

, MeV

0.08 0.16 0.24 0.32

ρ

, fm

–3

CD-BonnNijm 1Reid 93

0.40 0.48–20

BHF (CD-Bonn)BHF (Nijm 1)BHF (Reid 93)Exp. point

50

40

0

E

/

A

, MeV

0.04 0.12 0.16 0.20 0.24 0.280.08

10

20

30

Fig. 2. E/A for symmetric nuclear matter (left panel) and pure neutron matter (right panel) using different potentials at T = 0as a function of density using different potentials for continuous choice of the auxiliary potential.

respectively, and αi = 1/3, 2/3, 1/2, and 1. Theparameter α is added to this term to soften thedensity dependence which could otherwise lead to toohigh incompressibility for nuclear matter. In order toreproduce the empirical saturation point of symmetricnuclear matter, we have to fit the parameters ti.The parameters would be then useful just to give amore quantitative estimate of the needed correctionto the BHF results (which do not reproduce thecorrect position and value of the saturation pointof the nuclear matter EOS). The parameters xi aredetermined by fitting with the experimental symmetryenergy. In actual fact we attempted to take one, two,three, and four terms of the above equation to fit thedata and we found that a satisfactory fit is obtainedaround the empirical point of the EOS using only twoterms of the above summation with αi = 1/3 and 2/3only. The results for these fitting parameters are listedin Table 1.

Table 2. The saturation properties of nuclear matter thatare extracted from the equation of state at saturation point(these values are the Fermi momentum kF or saturationdensity ρ0, saturation energy E0 and the incompressibility.All results are calculated within BHF approach with a two-body density-dependent potential using the continuouschoice for the auxiliary potential)

PotentialContinuous choice

kF, fm−1 ρ0, fm−3 E0, MeV κ0, MeV

CD-Bonn 1.3413 0.163 –16.055 208.47

Nijm1 1.3468 0.165 –16.065 184.56

Reid 93 1.3495 0.166 –16.034 228.16

3. PROPERTIES OF NUCLEAR MATTERAT T = 0

3.1. EOS of the Symmetric and Pure Nuclear Matter

Figure 2 shows the energy per nucleon (E/A)for symmetric nuclear matter (left panel) and pureneutron matter (right panel) in MeV as a functionof density ρ in fm−3. The calculations have beenperformed using BHF approach with and withouta density-dependent interaction as 3BF with ad-justable parameters given in Table 1 using differentpotentials. The relevance of the three-body forcesis immediately seen from the shift of the saturationpoint to ρ0 ≈ 0.17 fm−3; E/A ≈−16 MeV, close tothe empirical one. The effect of these forces is alsovery small at low densities and becomes larger atincreasing densities, where a much stronger repulsionis apparent. In symmetric nuclear matter the isovec-tor components of the interaction do not contributeto the energy per nucleon and therefore the variousinteractions considered here are expected to givesimilar EOS up to moderate values of the density. Theresults for the calculated saturation points in Table 2are supplemented by the corresponding values of thenuclear incompressibility κ0 [22],

κ0 = 9ρ2 ∂2(E/A)(ρ)∂ρ2

∣∣∣∣ρ=ρ0

. (10)

This nuclear incompressibility κ0, which is calcu-lated at the saturation density ρ0, together with theincrease of the energy at large density displayed inFig. 2 characterizes the stiffness of the EOS of sym-metric nuclear matter. The experimental value of theincompressibility of nuclear matter at its saturationdensity ρ0 has been determined to be 210 ± 30 MeV[23].

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318 MANSOUR, HASSANEEN

For more enhancement of the EOS of symmetricnuclear matter, we have added more parameters asdefined in Eq. (9) choosing values of αi = 1/3, 2/3,1/2, and 1 and fitting the parameters ti and xi. Theresults for these fitting parameters are listed in Table 3and the corresponding energy versus density curvesare displayed in Fig. 3 using CD-Bonn interaction.The suggestion was very general to fit the propertiesof the nuclear and neutron matter. Therefore we takea number of points which is equal to the numberof parameters to proceed with the fitting procedure.We started with two terms of the potential, i.e., 4-parameters. In this case we took the saturation pointsfor E/A and the pressure besides two more pointsof the experimental data of the symmetry energy.Good fit was observed near the minimum of theEOS and the symmetry energy as well. We gavean example if one uses four terms of the potentialwith 8-parameters which shows a better fit for E/Aand the pressure over a wider density scale. In thiscase we took two more points on both the E/A andpressure data to obtain the rest of the parameters.Our observation was that a complete and good fit wasobtained for the nuclear matter data but satisfactoryfit for the neutron matter data. To obtain a better fitin this last case one has to add more terms to thepotential. Such work is currently in progress.

Our results are compared with the results obtainedby Friedman and Pandharipande (F and P) usingthe variational calculations [24]. One sees that agood agreement is obtained for a wide range of ρ.Also, the present calculations are compared withthe microscopic calculations with the BHF approachsupplemented by three-body forces using CD-Bonnpotential by Baldo and Shaban [25], self-consistentGreen’s function (SCGF) approach with an exacttreatment of Pauli operator [26]. As expected, thethree-body force or a simple two-body density-dependent term, like in Eq. (9), shifts the minimumof each curve towards the empirical saturation point.The softest EOS for symmetric matter among thoseapproaches which fit the empirical saturation point isprovided by the BHF approximation. Also, the steep-ness of the EOS at higher densities can depend, ofcourse, on the particular three-body forces introducedin the calculations, but the region around saturationis expected to be insensitive to the details of the forceused, since they are constrained to reproduce thisregion.

3.2. Pressure of Symmetric Nuclear Matter

The pressure for symmetric nuclear matter at T == 0 is defined in terms of the energy per particle as

p(ρ) = ρ2 ∂(E/A)(ρ)∂ρ

. (11)

75

30

0

–15

E

/

A

, MeV

0.08 0.24

ρ

, fm

–3

BHFBHF + 8 termsBHF + 4 terms

0.40 0.56–30

BHF + 3BFSCGFExp. pointF and P [24]

15

45

60

Fig. 3. E/A for symmetric nuclear matter at T = 0 as afunction of density using CD-Bonn potential for contin-uous choice in comparison with different approaches.

The results are shown in Fig. 4 using Table 2. Thevalues of the pressure are plotted against the densityρ for symmetric nuclear matter for continuous choiceusing the CD-Bonn potential (dotted line), the Nijm1potential (dashed line) and the Reid 93 potential(dash-dotted line) in comparison with Argonne v14

plus microscopic 3BF (short-dashed line) potentialby Baldo and Ferreira [6]. From Fig. 4 it is observedthat when the density of symmetric nuclear matterincreases, the pressure of nuclear matter increases.The agreement is good for small values of ρ. Also, allresults lie in the experimental region [27].

3.3. Symmetry Energy

Most of the microscopic calculations have beendevoted to study symmetric nuclear matter (SNM)and pure neutron matter (PNM) [13, 28, 29]. Thestudy of asymmetric nuclear matter is technicallymore involved and only a few BHF calculations areavailable [30–34]. In most of the cases, one assumesa quadratic dependence of the energy per particle,

E/A(ρ, α) = E/A(ρ, 0) + as(ρ)α2 + . . . (12)

Table 3. All parameters ti and xi defining a two-bodydensity-dependent potential of Eq. (9) as obtained forthe fit to the saturation point ρ0 = 0.17 fm−3; E/A == −16 MeV for CD-Bonn potential

t1 t2 t3 t4 x1 x2 x3 x4

–1326.3 –2 1852 –128 –0.878 466.075 –0.9394 –8.1141

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THE PROPERTIES OF NUCLEAR MATTER 319

10

1

Pressure, MeV fm

–3

0.2 0.3

ρ

, fm

–3

CD-BonnNijm 1Reid 93

0.5 0.7

Arg. v14 + 3BFExp. data

0.4 0.6 0.8

10

2

10

3

Fig. 4. The pressure of symmetric nuclear matter at T == 0 as a function of density using different potentials forcontinuous choice in comparison with Argonne v14 plusmicroscopic 3BF by Baldo and Ferreira [6]. All the resultsare compared with the experimental values of pressure,closed area [27].

in terms of the asymmetry parameter α = (N −Z)/Aand the symmetry energy as(ρ). In this way, calcu-lation of the energy of SNM and PNM allows oneto determine the symmetry energy as(ρ), and, usingthe previous equation, one can estimate the energyfor any asymmetry parameter α. That this quadraticexpression is a good approximation has been directlyconfirmed in BHF calculations of asymmetric nuclearmatter [33, 34]. This enables us to calculate thesymmetry energy as in terms of the difference betweenthe binding energy of pure neutron matter E/A(ρ, 1)and that of symmetric nuclear matter E/A(ρ, 0),

as = E/A(ρ, 1) − E/A(ρ, 0). (13)

But one would refrain from applying it at very high

density.

In Fig. 5 for continuous choice, symmetry energiesin MeV are plotted against the density ρ in fm−3

using Table 2, in comparison with the experimentaldata [35] represented by solid line, using the CD-Bonn potential, the Nijm1 potential, and the Reid93 potential with dot line. It is observed that whenthe density of nuclear matter increases, the symmetryenergy of the system increases in agreement withthe experimental results. For all potentials at thesaturation density (ρ0 = 0.16 fm−3) it is found thatthe nuclear symmetry energy is around the empiricalvalue, 32 MeV.

50

45

40

35

30

25

20

15

Symmetry energy, MeV

Exp.Reid 93

50

45

40

35

30

25

20

15

Exp.Nijm1

50

45

40

35

30

25

20

15

Exp.CD-Bonn

0.08 0.12 0.16 0.20 0.24 0.28 0.32

ρ

, fm

–3

Fig. 5. The symmetry energy as a function of densityin comparison with the experimental data [35] usingdifferent potentials for the continuous choice.

4. PROPERTIES OF NUCLEAR MATTERAT FINITE TEMPERATURES

4.1. Free Energy of the Symmetric Nuclear Matterat Different T

The many-body problem at finite temperatures hasbeen considered by several authors within differentapproaches, such as the finite temperature Green’sfunction method [36], the thermo-field method [37],or the Bloch–De Domicis (BD) diagrammatic expan-sion [38]. The latter, developed soon after the Brueck-ner theory, represents the “natural” extension to finite

ЯДЕРНАЯ ФИЗИКА том 77 № 3 2014

320 MANSOUR, HASSANEEN

temperature of the BBG expansion, to which it leadsin the zero temperature limit. Baldo and Ferreira [6]showed that the dominant terms in the BD expansionwere those that correspond to the zero temperature ofthe Brueckner–Bethe–Goldstone (BBG) diagrams,where the temperature is introduced only through theFermi–Dirac distribution

f(k, T ) =[1 + exp

(e(k, T ) − μ(T )

T

)]−1

. (14)

Therefore, at the BHF level, finite temperature effectscan be introduced in a very good approximation justreplacing in the BGE (1):

(i) the zero temperature Pauli operator Q = (1 −− θ1(k))(1 − θ2(k)) by the corresponding finite-temperature one Q(T ) = (1 − f1)(1 − f2),

(ii) the single-particle energies e(k) by thetemperature-dependent ones e(k, T ), obtained fromEqs. (3) and (4) when θ(k) is replaced by f(k, T ).

In the present work, two simplifications are usedto calculate the thermodynamic properties of nuclearmatter. Firstly, the G-matrix calculation is performedat T = 0 MeV and using the continuous choice forU(k). Secondly, the internal energy of the systemF → F/A is computed by using the entropy of thefree Fermi gas with effective mass m∗, where theinternal energy of nuclear mater is defined by

F = E − TST , (15)

where E → E/A is the total energy at T = 0, ST

is the entropy of the system at temperature T .In addition, thermal effects are treated in a low-temperature limit of the internal energy. Starting fromEq. (15), in the low-temperature limit the energy andentropy behave as E = ET=0 + aT 2 and ST = 2aT ,respectively, where a is the so-called level densityparameter. Therefore, for the internal energy we havethe following expression:

F = E + aT 2 − 2aT 2 = E − a(ρ)T 2 (16)

with

a(ρ) =14π2

(2m∗(kF)

�2k2F

)= (17)

=16

(2m∗(ρ)

�2

)(3π2

2

)1/3

ρ−2/3,

where the level-density parameter a is a function ofthe nucleon effective mass m∗ at T = 0 MeV withk = kF. By using equation (16) the internal energy[39] of the system at temperature T is defined by

F = ET=0 −T 2

6

(2m∗

�2

)(3π2

2

)1/3

ρ−2/3, (18)

0.16 0.32 0.48 0.64 0.80

ρ

, fm

–3

0.09

0.06

0.03

0

a

, MeV

CD-BonnNijm 1Reid 93

Fig. 6. Level-density parameter as a function of thedensity calculated for the BHF approach.

where m∗ is the effective mass of the nucleon atzero temperature with k = kF, defined in Eq. (8).It should be pointed out that the same expressionsare obtained for zero-range forces [40]. In fact, theyreflect a general property of the Landau theory ofnormal Fermi liquids. The level-density parameter isplotted in Fig. 6 as a function of the density. The level-density parameter increases quite rapidly when thedensity decreases due to the increment of the availablevolume per particle.

Many attempts were made to use the BHF cal-culations at finite temperature [6, 41–43]. In Fig. 7,the internal energy F of nuclear matter in MeV isplotted against the density ρ in fm−3 and the valuesobtained with the low-temperature expansion (18).The results are shown in Fig. 7, for symmetric nuclearmatter using different potentials at T = 8 (uppergraph), and 12 MeV (lower one), for continuouschoice. Fig. 7 gives the results obtained using theCD-Bonn potential (solid line), the Nijm1 potential(dashed line), and the Reid 93 potential (dash-dottedline) in comparison with a more elaborate calculationusing Argonne v14 plus microscopic 3BF [6] (dash-double-dotted line). From the plotted figures it isobserved that the internal energy first decreases withincreasing the density until it reaches a minimumthen it increases with increasing the density. Ourresults are comparable to those obtained in [6].

4.2. Pressure of the Symmetric Nuclear Matterat Different T

At T = 8 and 12 MeV, the following equation isused for calculating the pressure of symmetric nuclearmatter by using T 2 approximation [39]:

PT = pT=0 +T 2

9

(2m∗

�2

)(3π2

2

)1/3

ρ1/3. (19)

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THE PROPERTIES OF NUCLEAR MATTER 321

CD-BonnNijm 1Reid 93

20

10

0

–10

–20

–30

F

, MeV

Arg. v14 + 3BF

20

10

0

–10

–20

–300.16 0.24 0.32 0.40 0.48

ρ

, fm

–3

Fig. 7. The internal energy at T = 8 (upper panel) and12 MeV (lower panel) for symmetric nuclear matter as afunction of density using different potentials for continu-ous choice compared with Argonne v14 plus microscopic3BF by Baldo and Ferreira [6].

The results are shown in Fig. 8, the values ofpressure at T = 8 (upper part) and 12 MeV (lowerpart) are plotted against the density ρ for sym-metric nuclear matter for continuous choice, us-ing the CD-Bonn potential (solid line), the Nijm1potential (dashed line), and the Reid 93 potential(dash-dotted line) in comparison with Argonne v14

plus microscopic 3BF [6] (dash-double-dotted line).From Fig. 8 it is observed that when the density ofsymmetric nuclear matter increases, the pressure ofsymmetric nuclear matter first decreases and afterthat increases. Also, the present results of the pres-sure are comparable to those obtained in [6] at lowdensities.

5. SUMMARYThe bulk properties of symmetric nuclear matter

are computed, such as the equation of state, pressure,internal energy, and nuclear-matter symmetry energyand incompressibility as a function of density. Thecalculations of the above properties for symmetricnuclear matter are made by using BHF interaction ++ two-body density-dependent Skyrme interaction

CD-BonnNijm 1

Reid 93

60

10

0

–10

Pressure, MeV fm

–3

Arg. v14 + 3BF

50

40

30

20

60

10

0

–10

50

40

30

20

0.08 0.16 0.24 0.32 0.40 0.48

ρ

, fm

–3

Fig. 8. The pressure for symmetric nuclear matter at T == 8 (upper panel) and 12 MeV (lower panel) as a functionof density using different potentials for continuous choicecompared with Argonne v14 plus microscopic 3BF [6].

which is equivalent to three-body interaction. ModernNN interactions as the CD-Bonn potential, theNijm1 potential, and the Reid 93 potential are usedin order to analyze the dependence of the results onthe nuclear interaction. The EOS for both symmetricnuclear and pure neutron matter are calculated atT = 0. The free energy and the pressure of symmetricnuclear matter at T = 8 and 12 MeV have beencalculated by using T 2 approximation [39].

Good values are obtained for the incompressibilityshowing the stiffness of each potential with respectto the others. The symmetry energy also shows agood agreement with the experimental data. Oneconcludes that the calculations for the energy pernucleon are suitable at low densities and low temper-atures and are comparable at higher densities. By thismethod good agreement is obtained in comparisonwith previous elaborate theoretical calculation usingArgonne v14 plus microscopic 3BF [6], the BHF ++ 3BF by Baldo and Shaban [25], and the exper-imental data. We conclude that the BHF theory inaddition to our suggested contact interaction is able

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322 MANSOUR, HASSANEEN

to produce the experimental saturation point for theequation of state. Comparable results are obtainedfor finite temperatures. Among the different choicesof the sets of parameters ti and xi best results wereobtained for the set of parameters given in Table 1.Two terms only are used in our suggested potential.One can add other terms to calculate other physicalquantities. In fact, terms with other values of theparameter α (4/3, 5/3, 3/2, and 2) may be envisagedin order to get a fit with the inclusion of neutron-matter properties. This will be the subject of a futurework.

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ЯДЕРНАЯ ФИЗИКА том 77 № 3 2014