Journal of Modern Physics, 2013, 4, 37-41
doi:10.4236/jmp.2013.45B008 Published Online May 2013 (
Bulk Properties of Symmetric Nuclear and Pure
Neutron Matter
Khaled Hassaneen1,2, Hesham Mansour3
1Department of Physics, Faculty of Science, Sohag University, Sohag, Egypt
2Department of Physics, Faculty of Science, Taif University, Taif, Saudi Arabia
3Department of Physics, Faculty of Science, Cairo University, Gizza, Egypt
Received 2013
We study the equation of state (EOS) of symmetric nuclear and neutron matter within the framework of the Brueck-
ner-Hartree-Fock (BHF) approach which is extended by including a density-dependent contact interaction to achieve
the empirical saturation property of symmetric nuclear matter. This method is shown to affect significantly the nuclear
matter EOS and the density dependence of nuclear symmetry energy at high densities above the normal nuclear matter
density, and it is necessary for reproducing the empirical saturation property of symmetric nuclear matter in a nonrela-
tivistic microscopic framework. Realistic nucleon-nucleon interactions which reproduce the nucleon-nucleon phase
shifts are used in the present calculations.
Keywords: Symmetric Nuclear Matter; Equation of State; Three-Body Force; Symmetry Energy
1. Introduction
Many-body calculations which are based on the realistic
bare nucleon-nucleon potentials are able to reproduce
qualitatively but not quantitatively the saturation proper-
ties of symmetric nuclear matter. The saturation points
calculated with different approaches implemented with
various choices of the nucleon-nucleon (NN) potential lie
on the so called Coester line in the energy per parti-
cle-density plane, away from the experimentally allowed
values [1]. The theoretical predictions give a saturation
density sensibly higher than the experimental value ρ0
0.16 fm3 (usually in the range 1.5 ρ0 - 2 ρ0) and often
over bind the nuclear system (up to 25%), failing to get
close to the empirical binding energy E0 16 MeV.
These discrepancies and uncertainties get amplified when
calculating the equation of state (EOS) of pure neutron
matter, which is necessary for the estimates of key quan-
tities such as the symmetry energy and in general for the
description of neutron-rich matter in neutron stars.
Different ways have been developed to obtain predict-
tions for the properties of nuclear systems. One way is to
start from phenomenological models which successfully
describe the properties of stable nuclei. A very popular
approach along this line is the use of an effective density
dependent Skyrme-type interaction [2,3]. Modern Skyrme
parameterizations have been developed, which were con-
strained in their fitting procedures to obtain results for
neutron-rich nuclear matter which are compatible to
those of microscopic calculations. Also the relativistic
mean-field approximation has very successfully been
used to describe the properties of stable nuclei [4].
Therefore, there are some of the microscopic ap-
proaches, which start from models of the NN interaction,
which are adjusted to describe the experimental phase
shifts of NN scattering at energies below the pion thresh-
old. The traditional models of such realistic NN interac-
tions like, e.g., the charge-dependent Bonn (CD-Bonn)
potential [5] or the Reid 93 or Nijm1 potentials [6]. Such
nonperturbative approximations include the Brueckner
hole-line expansion with the Brueckner Hartree-Fock
(BHF) [1] approximation, the self-consistent evaluation
of Green’s function using the T-matrix or G-matrix ap-
proximation [7-12] (SCGF) and also variational ap-
proaches using correlated basis functions [13].
In the present work we implement the self consistent
G-matrix scheme with three different realistic NN poten-
tials (CD-Bonn, Nijm1 and Reid 93) plus a density-de-
pendent contact interaction to achieve the empirical
2. BHF for Symmetric Nuclear Matter
In the BHF approximation, the nuclear matter total energy
EA is obtained from the Brueckner G-matrix, G(ω), ac-
cording to the equation:
kk k,kk
2m 2
 
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with |k1k2a = |k2k1, i.e., the subscript a indicates anti-
symmetrization of the matrix elements. Here kF is the
Fermi momentum, the summation over the momenta ki
include spin and isospin variables. The single particle
energies ek, appearing in the entry energy of the G-matrix,
are given by:
 
ek Uk
where, the single particle potential U(k) is determined by
the self-consistent equation:
k' kF
Ukkk'|Gee |kk'
The self-consistency is coupled with the integral equa-
tion for the G-matrix, i.e., in the BHF approach G(ω) is
obtained by solving the Bethe-Goldstone equation:
123412 34
F3 F4
34 34
k k|G|k kk k||k k
kk||k kee
kk |G|kk
 
 
where, F (k) = 1 defining the step function for k < kF
and is zero otherwise and ω denotes the starting energy.
The product Q(k, k’) = (1-F (k))(1-F (k’)), appearing
in the kernel of Equation (4), enforces the scattered mo-
menta to lie outside the Fermi sphere and it is commonly
referred to as the “Pauli operator”. In the case of the an-
gle-aver-age of Pauli operator this energy is given as,
If one assumes that the potential U(k), or equivalently
the single particle energy e(k), has approximately a quad-
ratic form
ek e 2m*
. (6)
where, e0 is the zero point energy. Then one can calculate
the potential, at each iteration step, in few points only
and interpolate the obtained values with a parabola. The
approximation of Equations (6) is usually called the ef-
fective mass approximation, since then the spectrum has
the same shape as the free one but with an effective mass
m*. From Equations (2) and (6) the effective mass m*
can be evaluated from the slope of U(k) at the Fermi
momentum [15],
m*m dU
In the present work one may introduce a Skyrme ef-
fective interaction density dependent term [2] in addition
to the BHF potential.
12 12
V(r ,r)t(1xP)(rr).
This is a two-body density dependent potential which
is equivalent to three-body interaction. Where ti and xi
are interaction parameters, Pσ is the spin exchange op-
erator, ρ is the density, r1 and r2 are the position vectors
of the particle (1) and particle (2) respectively and αi =
(1/3, 2/3, 1/2 and 1). The parameter α is added to this
term to soften the density dependence which could oth-
erwise lead to too high incompressibility for nuclear
matter. In order to reproduce the empirical saturation
point of symmetric nuclear matter, we have to fit the pa-
rameters ti. The parameters would be then useful just to
give a more quantitative estimate of the needed correc-
tion to the BHF results, (which do not reproduce the cor-
rect position and value of the saturation point of the nu-
clear matter EOS). The parameters xi are determined by
fitting with the experimental symmetry energy. In actual
fact we attempted to take one, two, three and four terms
of the above equation to fit the data and we found that a
satisfactory fit is obtained around the empirical point of
the EOS using only two terms of the above summation
with αi = 1/3 and 2/3 only. The results for these fitting
parameters are listed in Table 1.
3. Results and Discussion
The energy per particle (E/A) as a function of density ρ
in fm-3 for nuclear matter is shown in Figure 1. Upper
panel represents the symmetric nuclear matter and pure
neutron matter is shown by lower panel. The calculations
have been performed using BHF approach with and
without a density dependent interaction as 3BF with
Table 1. Parameters ti and xi defining a two–body density
dependent potential of Eq. (9) as obtained for the fit to the
saturation point 0 = 0.17 fm-3 ; E/A = -16 MeV for different
Continuous choice
x1 t2 t1
-0.7721 -0.2991 1197.8 -656.5 CD-Bonn
-0.9705 -0.5146 1156 -673.2 Nijm1
-1.0043 -0.4472 1176.2 -782.4 Reid 93
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0.08 0.12 0.16 0.20 0.240.28 0.32 0.36 0.40
E/A (MeV)
(fm-3 )
Reid 93
BH F(CD-Bonn)
BH F(Reid93)
exp. point
0.04 0.08 0.12 0.160.20 0.24 0.28
(fm-3 )
E/A (MeV)
C D-Bonn
R eid93
BHF (C D-Bonn)
BHF ( N ijm1 )
BHF (R eid93)
Figure 1. E/A in M eV for symmetric nuclear matter (upper
panel) and pure neutron matter (lower panel) using differ-
ent potentials at (T = 0) as a function of density using dif-
ferent potentials for continuous choice of the auxiliary po-
Adjustable parameters given in Table 1 using different
potentials. The relevance of the three–body forces is im-
mediately seen from the shift of the saturation point to
0.17 fm-3 ; E/A -16 MeV, close to the empirical one.
The effect of these forces is also very small at low densities
and becomes larger at increasing densities, where a much
stronger repulsion is apparent. In symmetric nuclear
matter the is-ovector components of the interaction does
not contribute to the energy per nucleon and therefore,
the various in-teractions considered here are expected to
give similar EOS up to moderate values of the density.
For more enhancement of the EOS of symmetric nu-
clear matter, we have added more parameters as defined
in Eqution (8) choosing values of αi = (1/3, 2/3, 1/2 and 1)
and fitting the parameters ti and xi. The results for these
fitting parameters are listed in Table 2 and the corre-
sponding energy versus density curves are displayed in
Figure 2 using CD-Bonn interaction. The suggestion was
very general to fit the properties of the nuclear and neu-
tron matter. Therefore we take a number of points which
is equal to the number of parameters to proceed with the
fitting procedure. We started with two terms of the po-
tential i.e.4-parameters. In this case we took the satura-
tion points for E/A and the pressure besides two more
points of the experimental data of the symmetry energy.
Good fit was observed near the minimum of the EOS and
the symmetry energy as well. We gave an example if one
uses four terms of the potential with 8-parameters which
shows a better fit for E/A and the pressure over a wider
densities scale. In this case we took two more points on
both the E/A and pressure data to obtain the rest of the
parameters. Our observation was that a complete and
good fit was obtained for the nuclear matter data but sat-
isfactory fit for the neutron matter data.
Our results are compared with the results obtained by
Freidman and Pandharipande (F and P) using the varia-
tional calculations [16]. One sees that a good agreement
is obtained for a wide range of ρ. Also, the present cal-
culations are compared with the microscopic calculations
with the BHF approach supplemented by three–body
forces using CD-Bonn potential by Baldo and Shaban
[17], self-consistent Green’s function (SCGF) approach
with an exact treatment of Pauli operator [18]. As ex-
pected, the three-body force or a simple two-body den-
sity dependent term, like as Equation (8), shifts the
minimum of each curve towards the empirical saturation
point. The softest EOS for symmetric matter among those
approaches which fit the empirical saturation point is
provided by the BHF approximation. Also, the steepness
of the EOS at higher densities can depend of course on
the particular three–body forces introduced in the calcu-
lations, but the region around saturation is expected to be
insensitive to the details of the force used, since they are
constrained to reproduce this region.
Table 2. All parameters ti and xi defining a two-body den-
sity dependent potential of Equation (8) as obtained for the
fit to the saturation point 0 = 0.17 fm-3; E/A = -16 MeV for
CD- Bonn potential.
x 4 x 3 x 2 x1 t 4 t 3 tt1
-8.1141-0.9394466.075 -0.878 -1281852-2-1326.3
0.10 0.15 0.20 0.25 0.30 0.35 0.40
E/A (MeV)
BHF+8 terms
BHF+4 terms
exp. point
F and P
Figure 2. E/A in MeV for symmetric nuclear matter at (T =
0) as a function of density using CD-Bonn potential for con-
tinuous choice in comparison with differe nt appr oac hes.
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An important quantity in determining the equation of
state of isospin asymmetric nuclear matter is the symme-
try energy. The symmetry energy is defined from the
energy per nucleon E/A as follows
 
sym 2
Both ρn and ρp are the neutron and proton densities in
Asymmetric Nuclear Matter (ANM) and ρ = ρn + ρp is the
total density of asymmetric nuclear matter. It is well es-
tablished [19-20] that the binding energy per nucleon EA
fulfills the simple a2-law not only for a«1 as assumed in
the empirical nuclear mass formula [21], but also in the
whole asymmetry range. The a2-law of the EOS of ANM
at any isospin asymmetry leads to two important conse-
quences. This enables us to calculate the symmetry en-
ergy Esym in terms of the difference between the binding
energy of pure neutron matter EA (ρ,1) and that of sym-
metric nuclear matter EA (ρ,0), i.e.,:
sym AA
EE,1E, 0,
But one would refrain from applying it at very high
density. The results of our calculation for the symmetry
energy as a function of baryonic density in terms of the
Fermi momentum kF are depicted in Figure 3.
0.08 0.12 0.16 0.20 0.24 0.28 0.32
symmetry energy (MeV)
(fm -3 )
0.08 0.12 0.16 0.20 0.24 0.280.32
symmetry energy (MeV)
(fm -3 )
0.08 0.12 0.160.20 0.24 0.28 0.32
symmetry energy (MeV)
(fm -3 )
Reid 93
Figure 3. The symmetry energy in MeV as a function of den-
sity in fm-3 in comparison with the experimental data [22]
using different potentials for the continuous choice.
In Figure 3 for continuous choice, symmetry energies
incompressibility κ 0
characterizes the
MeV are plotted against the density ρ in fm-3 in com-
parison with the experimental data [22] represented by
solid line using the CD-Bonn potential, the Nijm1 poten-
tial, and the Reid 93 potential with dot lines from Figure
3. It is observed that when the density of nuclear matter
increases the symmetry energy of the system increases in
agreement with the experimental results. For all poten-
tials at the saturation density (ρ0 = 0.16 fm-3) it is found
that the nuclear symmetry energy is around the empirical
value, 32 MeV.
The nuclear
iffness of the EOS of symmetric nuclear matter. The
experimental value of the incompressibility of nuclear
matter at its saturation density ρ0 has been determined to
be 210 ± 30 MeV [23]. The incompressibility κ 0 at the
saturation point ρ0 is given by
(E/A)(k )(E/A)( )
The results for the incompressibility for the pr
4. Summary
rties of symmetric nuclear matter are
lculations at the saturation points are 208.5, 184.6 and
228.2 MeV for the CD-Bonn, Nijm1 and Reid93 poten-
The bulk prope
computed such as the equation of state, nuclear matter
symmetry energy and incompressibility as a function of
the density. The calculations of the above properties for
symmetric nuclear matter are made by using BHF inter-
action + two–body density dependent Skyrme interaction
which is equivalent to three–body interaction. Modern
NN interactions such as the CD-Bonn potential, the
Nijm1 potential, and the Reid 93 potential are used in
order to analyze the dependence of the results on the nu-
clear interaction. Good values are obtained for the in-
compressibility showing the stiffness of each potential
with respect to the others. The symmetry energy also
shows a good agreement with the experimental data. We
conclude that the BHF theory in addition to our sug-
gested contact interaction is able to produce the experi-
mental saturation point for the equation of state. Among
the different choices of the sets of parameters ti and xi
best results were obtained for the set of parameters given
here. Two terms are used only in our suggested potential.
One can add other terms to calculate other physical
quantities. In fact terms with other values of the parame-
ter α 4/3, 5/3, 3/2 and 2 may be envisaged in order to get
a fit with the inclusion of neutron matter properties.
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