29

Chapter 3

Equation of state of Symmetric Nuclear matter and Neutron matter

3.1

In view of the failure of the two-body forces to predict the correct saturation properties of

nuclear matter it has become essential to use the three-body forces (TBF). We have used two

models of TBF. The Urbana VII (UVII) three nucleon potential [9,10], and a

Introduction

Microscopic optical potential calculated within first order Brueckner theory has been

extensively used to calculate saturation properties of zero temperature symmetric nuclear

matter (SNM), pure neutron matter (PNM) and analyze the nucleon scattering data [1-3]

using several realistic two nucleon interactions. In this chapter we confine ourselves to only

the saturation property of SNM and PNM. A detailed discussion concerning the calculation

of microscopic optical potential for scattering of nucleons from finite nuclei is given in

chapter 4. The only input required for these calculations is the realistic two-body inter-

nucleon potential. We have used soft core Urbana v14 (UV14) [4] and Hard core Hamada

Johnston (HJ) [5] inter-nucleon potential in the present work. The old hard-core potential has

been used only to compare results with soft-core potential. We have calculated the properties

of pure neutron matter at densities up to about five times the saturation density ρ=0.17fm-3

,

appropriate for neutron star studies. The method of calculation have been described in detail

in Chapter 2

No inter-nucleon potential [6] has been successful in obtaining the correct binding energy

and saturation density in the non relativistic BHF or variational approach. The predicted

binding energies lie in a famous coester band [7]. Estimates of higher order terms( third and

fourth order) show that the Brueckner Goldstone series converges rapidly with continuous

choice [8], and hence there is no hope that inclusion of higher order terms using only two

body force would be helpful in obtaining correct saturation property.

30

phenomenological density dependent three nucleon interaction(TNI) model of Lagris,

Friedman, and Pandharipande [11,12] in our effective interaction code to calculate EOS of

SNM and PNM.

3.2 Nucleon-Nucleon (NN) Potential Model

In this section we have briefly described the local hard-core Hamada-Johnston (HJ) and soft

core Urbana v14 (UV14) two-nucleon potential used in the present work. It is useful to

mention that most of the experimental elastic phase shifts are extracted from the pp and np

differential cross sections [13]. In these models, the phase shift data are fitted in the energy

range 0-350 MeV. For higher energies E >350 MeV the pion production and other relativistic

effects become important and the Schrödinger two-nucleon equation is therefore no longer

sufficient. Hamada-Johnston and Urbana v14 potentials satisfactorily reproduce all the two-

body scattering data as a function of energy over the energy range of several hundred MeV.

3.2.1

(3.1) where

Hamada - Johnston (HJ) potential

The general form of HJ potential, is as follows

, (3.2) and

, (3.3a)

,

(3.3b)

, (3.3c) , (3.3d)

where μ, x, and M are the pion mass in MeV, the inter-nucleon distance measured in units of

the pion Compton wavelength (1.415 fm-1) and the nucleon mass (is taken to be 6.73μ)

respectively and the functions

31

The values of the parameters ac, bc, at, bt, …….are determined from a detailed fit to the

scattering data [5]. The radial shapes of the potential are used outside the hard core radius

xc= 0.342. The predicted values of the binding energy, electric quadratic moment and D state

probability of deuteron are -2.226MeV, 0.285 fm2, and 6.97% respectively [5].

3.2.2

pijij

psij

pI

pij

pij Orvrvrvv ))()()((

14,1++= ∑

=π

Urbana v14Potential (UV14)

The UrbanaV14 potential is written as a sum of 14 operator components.

(3.4)

The parameters of the radial functions multiplying the first eight operators are obtained by

fitting the NN phase shifts up to 325 MeV in S, P, D, F waves, and deuteron properties,

where

The next six operators are:

(six "quadratic L" terms) are relatively weak, and chosen in order to make many-body

calculations with this operator simpler. From now on, instead of the functions associated with

the above 14 operators, we use c, σ , τ, στ , t, t τ, b, b τ, q, qσ , q τ, qστ, bb, bbτ respectively

to denote them. The three radial component of Eq. (3.4) includes: long range one-pion-

exchange )( ijP rvπ ,which is non-zero only for p =στ , tτ operators

(3.5)

32

intermediate range part )( ijPI rv that comes from two pion exchange processes whose shape

is represented by the square of the one- pion exchange tensor function T(r)

)()( 2 rTIrv PpI π= (3.6)

and the short-range part )( ijPS rv is attributed to ω -and ρ – exchange and taken to have a

Yukawa shape. However, since the believed size of nucleon is at least of the order of the

Compton wavelength of ω - and ρ -mesons, the Yukawa shape will be very much modified.

Hence, in the UV14 interaction model, )( ijPS rv is taken to be a sum of two Woods-Saxon

potentials.

It is possible to obtain reasonable fits to the scattering data with S′p = 0 for all values of p

except b and bτ. The spin-orbit potential in I=1 states is required to have a smaller range than

that of the central part to fit the scattering data. Hence, the W′(r) terms are needed for only

p=b and bτ. The model parameters and comprehensive description for this model are given in

Ref. [4]. The predicted values of the binding energy, electric quadratic moment and D state

probability of deuteron are -2.225MeV, 0.273 fm2, and 5.2% respectively [4]. The quality of

fit to phase-shift obtained by UV14 is similar to that of Paris potential [14].

3.3 Comparison of Nuclear Matter Potential from Hamada-Johnston (HJ) and Urbana

In this section we compare the BHF results of nuclear matter potential from the use of Hard-

core (HJ) and the soft core (UV14) inter-nucleon potential. Our BHF results for the

v14 (UV14)

33

calculated nuclear matter potential and its partial wave breakup at kF =1.4fm-1(approximately

corresponding to the density at the center of a nucleus) are given in Tables 1 and 2 at two

incident nucleon energies; 65.0 and 200 MeV. We have chosen these two energies to show

the changes in the calculated nuclear matter potential with energy. Further as shown in (Fig.

9 of chapter 4(section B)) the resulting real central potential from the soft-core inter-nucleon

potential is about 5-7 MeV deeper as compared to the nucleon optical potential from HJ

inter-nucleon potential. In the following we have investigated the source of this deeper

potential resulting from the use of the soft-core interaction in BHF.

At 65.0 MeV we see (Table 1) that the inter-nucleon 1S0 and 3S1 states contribute about 4

MeV to the greater depth of the optical potential calculated from UV14 as compared with

that derived from HJ potential. As discussed in Ref. [15] the differences between 1S0 and 3S1

phase-shifts resulting from the two potentials are quite small at low energies and these

differences are of opposite signs at medium energies. Therefore, it seems that the effect of

the Pauli operator in the Bethe-Goldstone Equation is to cause more attraction from a soft

core potential, UV14, than a hard core HJ potential.

Table 2 shows that the difference between the optical potentials from the inter-nucleon 3S1

states becomes smaller at 200 MeV. This is expected, as there would be a weakening of the

Pauli effect at these energies. However for 1S0 state we find no effect of the relaxation of the

Pauli operator and the greater attraction from a soft core potential is marginally more at 200

MeV than at low energies. In Ref. [15] this is shown that the free UV14 is slightly more

attractive than HJ at 200 MeV (UV14 phase-shift is about 50 more than HJ phase shift at 200

MeV as shown in Fig. (2.3) in Ref. [15] ). The situation is reversed for the case of 3S1 state as

far as free space phase shifts are concerned (as shown in Fig. (2.4 in Ref. [15] )). Further we

notice from Tables 1 and 2 that 3S1 state has tensor coupling to 3D1 state, which is suppressed

at low energies due to Pauli effect. At 200 MeV the total S-state contribution is only about 4

MeV to the greater depth from the UV14 potential. Also we note that of the four inter-

nucleon D-states, 1D2 and 3D2 contribute significantly to the greater depth of the optical

potential. We find that the net contribution of the inter-nucleon D-state to the greater depth of

34

the optical potential is about 4 MeV from UV14 potential. Of the four P-states the major

contribution to the optical potential comes from the greater repulsion from 3P2 and 3P1 state.

However the greater attraction from 3P2 almost cancels the greater repulsion from 3P1 state.

As a result of this cancellation the total P-state contribution for UV14 is about 2.0 MeV more

repulsive than for HJ potential. This helps in reducing the difference, though marginally,

between the calculated optical potential from the two inter-nucleon potentials considered

here.

The contribution of the higher partial waves to the greater depth for UV14 is quite small at

low energies. However, at 200 MeV, the G-states contribute about 2 MeV to the deeper

potential from UV14 as compared with HJ potential. This is about 20% of the difference

between the calculated optical potential from the two inter-nucleon potentials at 200 MeV.

Further Tables 1 and 2 shows that additional attraction for UV14 comes from 3S1 and 3D1

states at both the energies considered here.

In view of the above results we can conclude that the greater depth of the calculated real

central optical potential for the soft core potential as compared with HJ inter-nucleon

potential comes mainly from the inter-nucleon S- and D- states over the whole energy range.

3.4

We have been able to obtain a self-consistent microscopic nucleon nuclear matter optical

potential for a range of incident local momentum k: 0 ≤ k ≤ 8 fm-1(calculational details are

given in Chapter 2). Fig. 1(a) shows the real part of the calculated nuclear matter optical

potential (NMOP) as a function of the incident local momentum at various Fermi momenta,

ranging from 0.60 fm-1 to 1.80 fm-1, using Urbana v14 realistic interaction. The results

indicate the following. Firstly, at the low incident momentum (i.e. low incident energy) the

real NMOP remains attractive and its strength smoothly decreases with decreasing nuclear

BHF results for SNM (only two-body forces)

In this section we discuss our results concerning the calculation of nuclear matter optical

potential and binding energy of infinite nuclear matter using UV14 soft-core and Hamada–

Johnston (HJ) hard-core realistic interactions.

35

matter density. Secondly, at high energy, k around 3.80 fm-1, the real potential becomes

repulsive for a high nuclear matter density though remains attractive for small densities

(small kF) up to quite high values of k. This is reflected in the calculated nucleon optical

potential for finite nuclei and the shape of the real optical potential resembles wine bottle

bottom type at high energies. Further, these changes suggest that the radial shape of real

potential changes substantially with increasing energy.

Fig.1(b) shows the calculated imaginary part of NMOP as a function of incident local

momentum at various Fermi momenta from 0.60 fm-1 to 1.80 fm-1, using Urbana V14

realistic interaction. We note that the imaginary NMOP remains attractive at all incident

momenta and at low incident energies (small values of k) the calculated imaginary potential

is small, for high kF and large for low kF values. This is reflected as surface enhancement in

the imaginary potential for low incident nucleon energies.

Fig. 1(c) and 1(d) respectively show the calculated real and imaginary NMOP using Hamada-

Johnston hard – core interaction .The behaviour shown in Fig. 1(a) are qualitatively similar to

the ones shown in Fig. 1(c) except that the use of Hamada-Johnston interaction gives a real

potential which is less attractive as compared with the results using Urbana v14 realistic

interaction. The results for the calculated imaginary potential are also similar (compare Fig.

1(d) with Fig. 1(b)).

The results of our calculations for NMOP (Fig. 1(a), 1(b), 1(c), 1(d)) using Hamada-Johnston

and Urbana v14 realistic interaction agree with a recent calculation of Arellano et al. [14]

(see Fig. 1 and 2 of Ref. [16]).

The calculated average energy per nucleon of symmetric nuclear matter using UV14 (solid

line) and HJ (dash line) interaction is shown in Fig. 2(a). Empirical saturation point

(ρ=0.17±1fm-3, E/a=-16±1 MeV) of nuclear matter lies inside the rectangular box shown in

Fig. 2(a). We notice that the lowest order Brueckner theory using UV14 interaction gives rise

to a nuclear matter which saturates at ρ =0.256 fm-3 with E/A=-19.01 MeV. Thus it predicts a

large saturation density and an over binding of the infinite nuclear matter. The use of HJ

36

interaction gives rise to a nuclear matter which saturates at ρ = 0.148 fm-3 with E/A = -12.4

MeV. Though the saturation density is quite close to the empirical value, but the

predicted energy is small as compared with the empirical value. Thus the density is correct

but the system is under bound as expected from hard-core interaction [17].

For comparison we also report in Fig. 2(b) the results of binding energy per nucleon for SNM

in variational approach [18] using UV14 (dotted line) and BHF approach using Argonne v14

(AV14) [19] interaction (dash dot line). Variational calculation [18] gives saturation at a

higher density ρ =0.326 fm-3, E/A=-17.1 MeV and BHF approach [19] using AV14 gives

saturation at ρ =0.256 fm-3, E/A=-18.6 MeV. We notice that in BHF approach the binding

energy for SNM is larger for UV14 than for AV14. This is consistent with the result of (Fig.4

of Ref. [19]), where they have shown a comparison of results from these two models (UV14

and AV14) in variational approach. Results from Brueckner and variational approach using

UV14 interactions are qualitatively similar with differences in only small quantitative details.

Both interactions give rise to a large saturation density and an over binding of the nuclear

matter. In particulars, BHF predicts slightly larger binding energy (by about 2 MeV per

particle) at a comparatively lower saturation density compared to the results using variational

approach. These differences are due to different calculational procedure.

In Table 3 and 4 we have given the partial wave contribution to the energy per nucleon of

nuclear matter at several densities from UV14 and HJ interaction. We note that the

contribution of H-wave is less than 1% at kF=1.4fm-1. Hence it seems justified to neglect the

contribution of partial waves with l≥ 5 in our calculation. Further, the major contribution to

nuclear matter binding comes from S, P and D-waves.

3.5

Non relativistic calculations based on purely two-body interactions fail to reproduce the

correct saturation properties of symmetric nuclear matter [5]. Further two body potentials

under binds 3H and 3He. This well known deficiency is commonly corrected by introducing

three-body-forces (TBF). Unfortunately, it seems not possible to reproduce the experimental

Three-Body Forces (TBF)

37

binding energies of light nuclei along with the correct saturation property of SNM accurately

with one simple set of TBF. Presently, the most widely used model for TBF are the Urbana

VII model and the density dependent three nucleon interaction (TNI) model of Lagris,

Friedman, and Pandharipande [11,12]. These models are briefly described below.

3.5.1

Rijkijkijk VVV += π2

UrbanaVII (UVII) model

A realistic model for nuclear TBF has been introduced by Urbana Group [10].The Urbana

model [9,10] includes two terms :

(3.8)

The two pion exchange term V π2ijk is attractive and is a cyclic sum over the nucleon indices i,

j, k of products of commutator and anticommutator terms:

V {( }{ } ][ [ ])kjjijkijkjjijkcyc

ijijk XXXXA ττττττττπ .,.41.,., ,

2 += ∑ , (3.9)

Where ( ) ( ) ijijjiijij SrTrYX += σσ . (3.10)

is the one pion exchange term and Sij is the tensor operator. Y(r) and T(r) are the Yukawa and

Tensor functions associated with the one-pion exchange interaction. π2ijkV mainly contributes

at low densities.

The repulsive part is taken as:

22 )()( jkcyc

ijR

ijk rTrTUV ∑= . (3.11)

This repulsive part is dominant at high densities and hence is helpful in obtaining correct

saturation properties.

The strengths A (<0) and U(>0) are adjusted to fit the saturation properties of nuclear matter.

We have used the method of including this three-body force in the BHF formalism as

discussed in Ref. [20]. To avoid solving Bethe Fadeev equation the three body interaction is

38

reduced to an effective two-body interaction by averaging over the third particle [21]. This

averaging is done with the weighted probability of the relevant two body defect function:

[ ] [ ]∑∫ −−=jj

jkijijkjikeff rgrgVrdrv

τσ

ρ,

2233 )(1)(1)( , (3.12)

Where g(r) are the defect functions calculated self consistently from an earlier BHF

calculation using only two-body force and ρ is the nuclear matter density.

This procedure yields an effective two-nucleon potential of a simple structure [9]

])().)(([)(3 ijTjisjieff Srvrvrv += σσττ +VR(r) (3.13)

Using the triangle relation rjk=rik + rij, Vs , VT and VR are [9]

(3.14)

(3.15)

∫ ∫ −−= 22222 ))(1())(1)]((()()[(cos2)( jkijjkijjkjkikR rgrgrTrTddrrUrV θρπ (3.16)

39

where the z axis was taken along the vector rik, )(cosϑlP are the Legendre polynomials of

order l. This effective force is added to the nuclear Hamiltonian H and the calculation

proceeds along the same ordinary Brueckner scheme with only two-body force plus the

averaged three-body force (Eq.(3.13)).

3.5.2 Three Nucleon Interaction (TNI) As shown by lagaris and Pandharipande in Ref. [12], realistic two-nucleon interaction seem

to overbind nuclear matter very significantly at kF >1.5 fm-1, whereas at low kF <1.3 fm-1

nuclear matter is an underbound, this strongly suggest the need for more attraction at low

densities and higher repulsion at high densities. Lagris and pandharipande [12] have taken a

phenomenological point of view, and add contribution of TNI to the Urbana v14 model to get

the correct E (kF) around kF=1.33 fm-1.

Lagris and pandharipande [12] argued that a reasonable procedure for constructing a three

body potential is to make an expansion of the form:

where Ul are strength parameters, ul(r) are functions of interparticle distance, θi is the angle

between vectors rij and rik and Σcycl represents cyclic permutation of the indices i,j,k. At high

densities l = 0 term dominates and empirically it should be repulsive. The l ≠ 0 terms can be

attractive but these should saturate at high density.

The UV14 plus TNI model approximates the effect of Vijk by adding two density dependent

terms to the UV14 two-body potential: a three Nucleon repulsion (TNR) term designed to

represent l=0 part of equation (3.16) and a three nucleon attractive term for l ≠ 0. The TNR

term is taken as the product of an exponential of the density with the intermediate range part

of vij (Eq. 3.4), such that

(3.17)

3.16

40

with 31 15.0 −= fmγ . The primary effect if this term is the reduction of the intermediate range

attraction of the two nucleon potential with three-body interactions effectively contributing PIvργ 1− .

The attractive Vijk interaction is not treated microscopically by FP [11]. They assume that its

contribution to the nuclear matter has the form

(3.18)

where β = (N – Z) /A,

where N and Z are numbers of neutrons and protons.

We follow Ref. [12] and calculate E (kF, v14+TNR) with the interaction (Eq.3.17) using BHF

method, and add the TNA contribution (Eq.3.18) to obtain the nuclear matter energy.

The effect of the attractive Vijk on the wave function is also neglected by FP. The values

of 1γ , 2γ and 3γ used by FP [11] are 0.15 fm-3,-700 MeV fm6 and 13.6 fm3 respectively.

3.5.3

In Urbana VII model for three body force we follow the procedure of Ref. [9]. In Fig. 3 we

have shown BHF defect function g(r) for 1S0 and 3S1 waves as a function of inter particle

distance at several densities. It turns out that density dependence of the defect function in the

relevant region is very small. Fig. 4 shows the defect function at kF =1.4 fm-1at different

incident momenta and it can be seen that dependence of g(r) over incident momenta is also

quite weak.

BHF results for SNM (two plus three body force) In this section we discuss our results concerning the calculation of binding energy of infinite

nuclear matter using Urbana v14 two body nuclear force and the two models for three-body

force (TBF) namely the Urbana VII model for the three body force (UV14 plus UVII) [9,10]

as given in subsection (3.5.1) and the phenomenological density dependent three nucleon

interaction model (UV14 plus TNI) of Lagris, Friedman and Pandharipande [11,12], see

subsection (3.5.2).

41

In our calculation we construct g(r) for kF =1.33 fm-1, averaged over different incident

momenta.

In Fig. 5 we show different components Vs ,VT ,VR (Eqs. (3.14-3.16)) of the averaged BHF

three body force potential in symmetric matter at kF =1.4 fm-1. Our results are very close to

those in Ref. [3].

In Fig. 6 we have shown (solid line) calculated binding energy per nucleon E(ρ) for

symmetric nuclear matter as a function of density using UV14 plus the Urbana UVII model

for TBF. We notice that symmetric matter with three body force saturates at ρ =0.178 fm-3,

E/A = -14.62 MeV a value close to the empirical value [5]. For comparison we have also

shown (the same figure) results of a non relativistic BHF calculation using AV14 plus UVII

[19] (dashed line), and from the variational approach using UV14 plus UVII (dotted line)

[18]. In table 5 we have given results for the saturation property of nuclear matter from some

earlier calculations along with our results. We note that AV14 plus UVII [19] is more

repulsive than our results for UV14 plus UVII (see Fig. 4) for densities larger than 0.3 fm-3.

Further the variational approach using UV14 plus UVII [18] underestimates the binding

energy as compared to the BHF approach used in the present work. A similar situation exist

(see Fig. 2) when we use same two body force and compare BHF results with variational

approach. As a consequence the value of the strengths (A and U) of the attractive and

repulsive TBF needed (Eqs.9-11) to reproduce empirical saturation point are different. In

Ref. [19] where BHF approach is used with AV14 plus UVII the values of parameters A’= -

0.0038 and U’= 0.0036. While our values are A= -0.0058 and U=0.0016. As a result of

U<U’, our repulsive TBF is weaker. Consequently our EOS is softer at high density as

compared to Ref. [19], where the repulsive component of the TBF is dominant.

In Fig. 7 the solid line shows our results for the energy E (ρ) for symmetric nuclear matter

(SNM) using UV14 plus TNI (Eqs. 3.17-3.18). We note that the symmetric nuclear matter

with UV14 plus TNI saturates at ρ =0.157 fm-3, E/A = -16.6 MeV. We have also compared

our results with variational approach using UV14 plus TNI [18] (dashed line). In variational

42

approach the parameters are: 31 15.0 −= fmγ , γ2=-700MeVfm6 and γ3=13.6 fm3. In order to

reproduce the correct saturation of SNM we found 31 15.0 −= fmγ , γ2=-260 MeVfm6 and

γ3=11 fm3. Table 5 shows that our results are quite close to variational results and both BHF

and variational approaches [19] predict nearly the same saturation property in close

agreement with the empirical value.

3.6 Neutron Matter

Pure neutron matter is defined as an idealized infinite, homogenous system of neutrons. At a

given density the properties of such a system, treated as a gas of interacting fermions at T =

0°K, are determined by the neutron-neutron interaction.

To calculate the EOS of neutron matter we follow the procedure given in Ref. [22], and

remove all T = 0 interactions, and also T=1, T3=0 interaction. The Fermi momentum kF is

related to the density ρ of neutron matter

(3.19)

We have calculated energy per nucleon of neutron matter E(ρ)as a function of density in first

order Brueckner theory using UV14 and HJ interactions. Results are discussed in section

(3.6.1).

The energy density ε (ρ) and pressure P(ρ) are obtained the E(ρ), where E(ρ) is the energy

per nucleon , ρ is the number density[18]:

ε (ρ) = ρ (E(ρ) + MNC2), (3.20)

(3.21)

The cold equation of state P(ρ) is obtained by eliminating ρ from (3.20) and (3.21). Velocity

sound in neutron matter(in units of c) is given by

43

(3.22)

(3.6.1) BHF results for PNM

(a)

355.0 −= fmρ

(Two body force)

In this section we discuss our results concerning of binding energy of neutron Matter using

UV14 soft-core and Hamada–Johnston (HJ) hard-core realistic interaction in BHF.

In Fig. 8 we have shown the calculated binding energy per nucleon as a function of density

for PNM using UV14 and HJ potentials. Results for UV14 are shown by solid line and for HJ

interaction by dash-dot line. In Fig. 8(a) we show our results for densities up to

and in Fig. 8(b) the results are shown up to higher densities( 31.2 −= fmρ )

typically encountered in the core of neutron star.

We also compare our results with those in Refs. [19,18] where they obtain equation of state

for neutron matter in BHF approach using AV14 interaction (open circle) and from

variational approach using UV14 (open stars) interaction respectively. From Fig. 8 we notice

that all interactions (except HJ) are in reasonable agreement with each other up to density ρ =

0.2 fm-3. At density greater then 0.2 fm-3 we note that our results in BHF approach using

UV14 is more repulsive than Baldo’s results from AV14 [19]. Again BHF approach with

UV14 is more repulsive than Variational approach of UV14 [18].

(b) (Two plus three-body force)

In this section we discuss our results concerning the calculation of binding energy of neutron

matter using Urbana v14 two body nuclear force and two types of three-body forces: namely

the Urbana VII model for the three body force (UV14 plus UVII) [9,10] as given in

subsection (3.5.1) and the phenomenological density dependent three nucleon interaction

model (UV14 plus TNI) of Lagris, Friedman and Pandharipande [11,12], see subsection

(3.5.2).

44

In Fig. 9 we have shown the calculated energy per nucleon for PNM in BHF approach using

UV14 plus UVII (solid line) and UV14 plus TNI (dashed line) interaction. Fig. 9(a) shows

our results for densities up to=0.55fm-3 and in Fig. 9(b) we show the corresponding results up

to densities (ρ=2.1 fm-3) typically encountered in the core of neutron star. For comparison we

have also shown in the same figure BHF results for AV14 plus UVII [19] (open stars), the

variational results using UV14 plus UVII (solid triangles) [18] and the variational results for

UV14 plus TNI (open circles) [18]. We have already seen in case of nuclear matter that

addition of three body force in the Hamiltonian significantly reduces the saturation density in

nuclear matter, making the equation of state much stiffer. The results in neutron matter are

similar ie. the inclusion of three body force stiffens the equation of state. We notice that all

non relativistic calculations give similar results for densities up to 0.3fm-3. In BHF approach,

at densities greater than 0.3 fm-3 the results for AV14 plus UVII [19] is more repulsive than

our results for UV14 plus UVII because of weaker repulsive contribution of UVII as

discussed above. Further comparing our BHF results with variational approach we again find

more repulsion in the variational approach for UV14 plus UVII [18].

Our results in BHF approach for UV14 plus TNI are in close agreement with results using

variational approach for UV14 plus TNI [18]. Thus we have studied the behavior of E(p)

using two types of three body force in BHF for both SNM and PNM.

Results for the mass density, pressure and sound velocity in neutron matter using UV14

(solid line), UU14 plus UVII (dashed line) and UV14 plus TNI (dotted line) are shown in

Fig. 10. These results are in agreement with others [18].

018801 (2004).

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Schiavilla, V. R. Pandharipande, and R. B. Wiringa, Nucl. Phys. A 449, 219 (1986).

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Phys. Rev. D 8, 800 (1973).

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46

Table 1. Contribution of some inter-nucleon states to the calculated nucleon-nuclear matter optical potential using Hamada -Johnston and Urbana v14 potential at 65.0 MeV for kF =1.40 fm-1

State HJ

UV14

V(MeV) W(MeV) V(MeV) W(MeV) 1S0 -10.08 -1.731 -13.20 -2.592 3S1 -16.737 -4.438 -17.309 -5.102 3P2 -27.21 -1.805 -28.700 -2.198 3PO -2.858 -0.174 -3.265 -0.250 3P1 25.84 -2.346 28.02 -3.130 1P1 8.905 -0.890 +9.743 -1.170 3D1 5.297 -1.314 +4.553 -0.846 3D3 -0.435 -0.131 -1.512 -0.160 1D2 -11.58 -1.274 -11.65 -0.379 3D2 -13.99 -1.274 -14.95 -1.622

Total (S) -26.82 -6.169 -30.51 -7.694

Total (P) +4.680 -5.217 +5.807 -6.749

Total(D)

-20.713 -3.057 -23.564 -3.008

Total (F) +3.898 -0.273 +3.882 -0.075

Total(G)

-4.488 -0.235 -3.899 -0.340

Total(H)

+2.355 -0.037 +2.576 -0.038

Total -41.09 -14.98 -47.21 -18.12

47

Table 2. Contribution of some inter-nucleon states to the calculated nucleon-nuclear matter

optical potential using Hamada –Johnston(HJ) and Urbana v14(UV14) potential at 200.0

MeV for kF=1.40 fm-1

State HJ

UV14

V(MeV) W(MeV) V(MeV) W(MeV) 1S0 1.106 -0.821 -2.279 -0.805 3S1 -2.960 -1.908 -2.916 -1.486 3P2 -28.943 -4.094 -29.580 -4.451 3PO 2.0271 -0.380 2.142 -0.469 3P1 27.457 -6.379 29.807 -7.850 1P1 10.514 -2.966 +10.936 -3.288 3D1 7.589 -2.549 +5.086 -1.140 3D3 -1.290 -0.343 -2.158 -0.350 1D2 -15.162 -1.151 -15.014 -1.146 3D2 -13.586 -2.620 -14.12 -2.916

Total (S) -1.854 -2.729 -5.196 -2.291

Total (P) +11.051 -13.820 +13.306 -16.059

Total(D)

-22.450 -6.664 -26.213 -5.554

Total (F) +2.598 -0.883 +3.419 -0.753

Total(G)

-5.618 -0.813 -7.549 -0.914

Total(H)

+3.183 -0.124 +3.342 -0.140

Total -13.089 -25.036 -18.892 -25.714

48

Table 3. Partial wave contribution to the binding energy of nuclear matter at several

densities from UV14 interaction .

Channel

1S0 3S1 3P2 3P0 3P1 1P1 3D1 3D3 1D2 3D2 3F3 1F3 3F4 3F2 1G4 3G3 3G5 3G4 3H5 3H6 1H5 3H4

K.E P.E B.E

Fermi mom =1.1 fm-1

-10.845 -16.312 -3.177 -1.996 4.692 1.712 0.535 -0.029 -1.000 -1.667 0.532 0.297 -0.089 -0.190 -0.127 0.055 0.019 -0.205 0.073 -0.008 0.048 -0.019

15.054 -27.701 -12.647

Fermi mom =1.33 fm-1

-16.036 -21.420 -6.722 -4.468 9.463 3.341 1.174 -0.068 -2.283 -3.658 1.335 0.703 -0.278 -0.501 -0.363 0.166 0.061 -0.601 0.254 -0.032 0.161 -0.068

21.007 -38.840 -17.833

Fermi mom =1.40 fm-1

-17.458 -22.536 -8 .389 -4.947 11.544 4.041 1.455 -0.095 -2.908 -4.581 1.731 0.890 -0.392 -0.659 -0.489 0.227 0.086 -0.819 0.363 -0.048 0.225 -0.100

23.384 -41.859 -18.475

Fermi mom =1.60 fm-1

-22.103 -25.569 -13.920 -6.316 19.012 6.651 2.391 -0.155 -5.118 -7.681 3.155 1.536 -0.853 -1.232 -0.965 0.462 0.182 -1.648 0.790 -0.119 0.478 -0.230

30.848 -50.252 -19.404

Fermi mom =1.8 fm-1

-26.381 -26.907 -21.371 -6.356 29.644 10.597 3.589 -0.215 -9.408

-11.927 5.212 2.395 -1.653 -2.047 -1.706 0.828 0.334 -2.947 1.498 -0.255 0.882 -0.209

39.309 -55.403 -16.094

Fermi mom =2.0 fm-1

-29.710 -25.927 -30.936 -6.596 44.297 16.371 5.001 -0.222

-13.931 -17.303 8.005 3.467 -2.928 -3.083 -2.790 1.349 0.556 -4.832 2.569 -0.491 1.476 -0.825

48.764 -55.483 -6.719

49

Table 4 Partial wave contribution to the binding energy of nuclear matter at several densities from HJ interaction .

Channel

1S0 3S1 3P2 3P0 3P1 1P1 3D1 3D3 1D2 3D2 3F3 1F3 3F4 3F2 1G4 3G3 3G5 3G4 3H5 3H6 1H5 3H4

K.E P.E B.E

Fermi mom =1.1 fm-1

-10.05

-14.53 -2.79 -1.79 4.47 1.51 0.54

0.026 -1.05 -1.62 0.51 0.29 -0.06 -0.21 -0.12 0.053 0.021 -0.19 0.07 -0.01 0.045 -0.02

15.05 -24.90 -9.85

Fermi mom =1.33 fm-1

-15.19

-18.72 -7.08 -3.07 6.98 2.96 1.23

0.086 -2.41 -0.35 1.26 0.71 -0.19 -0.56 -0.36 0.16 0.08 -0.57 0.23 -0.02 0.16 -0.08

21.00 -34.76 -13.76

Fermi mom =1.60 fm-1

-17.34 -19.37 -15.13 -5.470 14.92 6.04 2.66 0.24 -6.54 -0.07 3.06 1.62 -0.66 -1.43 -0.11 0.46 0.24 -1.62 0.76 -0.14 0.52 -0.28

30.84

-37.64 -6.80

Fermi mom =1.8 fm-1

-18.50 -15.33 -20.01 -7.29 26.55 11.71 5.12 0.47 -7.85

-11.50 5.98

3.57 -1.32 -2.34 -1.67 0.81 0.44 -2.80 1.38 -0.14 0.94 -0.56

39.30 -32.34 6.96

Fermi mom =2.0 fm-1

-16.94

-4.51 -29.40 -5.30 40.74 15.28 6.05

0.84 -13.61 -16.65

7.70 3.94 -2.62 -3.58 -2.73 1.34 0.79 -4.56 2.37 -0.25 1.59 -1.05

48.76

-20.56 28.20

50

Table 5. Saturation property of Nuclear Matter obtained from different potentials

E/A(MeV) ρ (fm-3) K (MeV) HJ -12.4 0.148 155.7

UV14 -19.01 0.256 241.163 UV14 plus UVII -14.62 0.178 177.16 UV14 plus TNI

Our results (BHF) -16.6 0.157 258.0

AV14 -18.26 0.256

AV14 plus UVII Ref. [19] (BHF) -16.01 0.176 253.0

UV14 -17.1 0.326 243.0

UV14 plus UVII -11.5 0.175 202.0 UV14 plus TNI

Ref. [18] Variational -16.6 0.157 261.0

0 1 2 3 4-150

-100

-50

0

50

0 1 2 3 4 5-120

-80

-40

0

(a)UV14

A-- kF(fm-1)=0.60B---- 0.80C---- 0.95D---- 1.10E---- 1.25F---- 1.33G---- 1.40H---- 1.50I---- 1.70J---- 1.80

H

JI

GFED

CBA

Rea

l Nuc

lear

Mat

ter P

oten

tial (

MeV

)

Incident Momentum (fm-1)

(b)

A-- kF(fm -1)=0.60B---- =0.80C---- =0.95D---- =1.10E---- =1.25F---- =1.33G---- =1.40H---- =1.50I---- =1.70J---- =1.80

J

I

HGFE

DCBA

Imag

inar

y Nu

clea

r Mat

ter P

oten

tial (

MeV

)

Incident Momentum(fm-1)

Figure 1(a)-1(b). Real and imaginary part of nuclear matter potential as a function of

incident local momentum for various Fermi momenta between 0.60 fm-1 and 1.8 fm-1,

using Urbana v14 soft-core interaction.

51

0 1 2 3 4 5

-120

-100

-80

-60

-40

-20

0

0 1 2 3 4

-100

-50

0

50

(d)

HJ

Imag

inar

y Nu

clea

r Mat

ter P

oten

tial (

MeV

)

Incident Momentum(fm-1)

A-- kF(fm-1)=0.60B---- =0.80C---- =0.95D---- =1.10E---- =1.25F---- =1.33G---- =1.40H---- =1.50I---- =1.70J---- =1.80 J

I

H

GFED

CB

A

(c)

HJ

Rea

l Nuc

lear

Mat

ter P

oten

tial (

MeV

)

Incident Momentum(fm-1)

A-kF(fm-1) =0.60B---- =0.80C---- =0.95D---- =1.10E---- =1.25F---- =1.33G---- =1.40H---- =1.50I---- =1.70J---- =1.80

JI

HGFE

D

C

B

A

Figure 1(c)-1(d). Real and imaginary part of nuclear matter potential as a function of

incident local momentum for various Fermi momenta between 0.60 fm-1 and 1.8 fm-1,

using Hamada-Johnston hard-core interaction.

52

0.0 0.1 0.2 0.3 0.4 0.5 0.6

-20

-16

-12

-8

-4-20

-16

-12

-8

-4

(b)

density (fm-3)

E(ρ)

/ A

(MeV

/ A)

UV14 Ref. [18] Ref. [19]

(a)

UV14 HJ

Figure 2. (a) Energy per nucleon as a function of density for symmetric nuclear matter.

Solid line and Dashed line are our results using Urbana v14 soft-core and Hamada-

Johnston hard-core interactions respectively. (b) dotted line and dash-dote line

corresponds to the variational calculation [18] using UV14 and BHF calculation [19]

using AV14 respectively. Empirical saturation point of nuclear matter lies inside the

rectangular box shown in the figure.

53

0 1 2 3 4 5

-0.2

0.0

0.2

0.4

0.6

0.8

0 1 2 3 4 5-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

(a)

KF=1.4 fm-1

r(fm-1)

1S0

g(r)

KF=1.1 fm-1

KF=1.33 fm-1

KF=1.4 fm-1

KF=1.5 fm-1

KF=2.0 fm-1

(b)

3S1

g(r)

KF=1.4 fm-1

KF=1.5 fm-1

KF=1.8 fm-1

KF=1.33 fm-1

KF=1.18 fm-1

KF=2.00 fm-1

Figure 3. BHF defect functions in symmetric nuclear matter at different densities (a) in 1S0 and (b) in 3S1 state.

54

0 1 2 3 4 5

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0 1 2 3 4 5

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

(a)

1S0

KF=1.4 fm-1

g(r)

Kinc=0.1fm-1

Kinc=0.75 fm-1

Kinc=1.00 fm-1

Kinc=2.00 fm-1

Kinc=4.00 fm-1

Kinc=5.00 fm-1

Kinc=6.00 fm-1

Kinc=8.00 fm-1

(b)

r(fm-1)

3S1

Kinc=0.1 fm-1

Kinc=0.75fm-1

Kinc=1.0 fm-1

Kinc=2.00 fm-1

Kinc=4.00 fm-1

Kinc=5.00 fm-1

Kinc=6.00 fm-1

Kinc=7.00 fm-1

g(r)

Figure 4. BHF defect functions in symmetric nuclear matter at different incident

momenta (a) in 1S0 and (b) in 3S1 state.

55

0.0 0.5 1.0 1.5 2.0 2.5

-8

-6

-4

-2

0

2

4

6 kF=1.4 fm -1

A = -0.0058U = 0.0016

VR(r)

VT(r)

VS(r)/5

V (M

eV)

r(fm)

Figure 5. Components of effective two-body potential after taking the average over the

third nucleon in the Urbana Model [9,10] of three body force.

56

0.0 0.1 0.2 0.3 0.4 0.5 0.6-20

-15

-10

-5

0

5

10

15

20

25

30

UV14+UVII AV14+UVII Ref. [19] UV14+UVII Ref. [18]

density (fm-3)

E(ρ)

/ A

( MeV

/ A)

Figure 6. Energy per nucleon as a function of density for symmetric nuclear matter with

three-body force. Solid line shows our results using UV14 plus UVII, dotted line and

dashed line corresponds to the variational calculation [18] using UV14 plus UVII and

BHF calculation [19] using AV14 plus UVII respectively. Empirical saturation point of

nuclear matter lies inside the rectangular box shown in the figure.

57

0.0 0.1 0.2 0 .3 0 .4 0 .5

-20

-10

0

10

20

30

U V 14+T N I U V 14+T N I R e f. [18 ]

density(fm -3)

E(ρ)

/ A

(MeV

/ A)

Figure 7. Same as for Figure 6. Solid line show our results using UV14 plus TNI, dotted

line corresponds to the variational calculation [18] using UV14 plus TNI. Empirical

saturation point of nuclear matter lies inside the rectangular box shown in the figure.

58

0.0 0.5 1.0 1.5 2.0 2.5

0

200

400

600

800

1000

1200

0.0 0.1 0.2 0.3 0.4 0.50

10

20

30

40

50

60

70

80

(b)

density (fm-3)

E(ρ)

/ A

(MeV

/A)

(a)

UV14 HJ AV14 Ref. [19] UV14 Ref. [18]

Figure 8. (a) Energy per nucleon as a function of density for Pure Neutron Matter. Solid

and Dashed line are our calculations using UV14 and HJ interactions respectively. Open

stars and open circles correspond to the variational calculation [18] using UV14 and BHF

calculation [19] using AV14 respectively. (b) same as for fig 8(a) up to density ρ=2.1 fm-3

59

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.00

100

200

300

400

500

600

700

8000.1 0.2 0.3 0.4 0.5

0

20

40

60

80

density (fm-3)

E(ρ)

/ A

(MeV

/ A)

(b)

(a)

UV14+UVII AV14+UVII Ref.[19] UV14+UVII Ref.[18] UV14+TNI Ref.[18] UV14+TNI

Figure 9. Energy per nucleon as a function of density for Pure Neutron Matter with

three-body force. Solid and dashed line show our results using UV14 plus UVII and

UV14 plus TNI. solid triangle and open stars correspond to the variational calculation

[18] using UV14 plus UVII and BHF calculation [19] using AV14 plus UVII

respectively. Open circles correspond to the variational calculation [18] using UV14 plus

TNI. (b) same as fig 9 (a) up to density ρ = 2.1fm-3.

60

0.0 0.5 1.0 1.5 2.0

100

101

102

103

104

c

c/10

density(fm-3)

MeV

-fm3

sound velocity

pressure

mass density

UV14+TNI UV14+TBF UV14

Figure 10. Mass density, Pressure and Sound velocity for Neutron Matter using UV14

(solid line), UU14 plus UVII (dashed line) and UV14 plus TNI (dotted line).

61