The properties of asymmetric nuclear matter

JOURNAL OF SCIENCE OF HNUE
Mathematical and Physical Sci., 2013, Vol. 58, No. 7, pp. 125-131
This paper is available online at

THE PROPERTIES OF ASYMMETRIC NUCLEAR MATTER

Le Viet Hoa1 , Le Duc Anh1 and Dang Thi Minh Hue2
1
2

Faculty of Physics, Hanoi National University of Education
Faculty of Mathematics, Water Resources University, Hanoi

Abstract. The equations of state of asymmetric nuclear matter (ANM) starting
from the effective potential in a one-loop approximation is investigated. It was
showen that chiral symmetry is restored at high nuclear density and the liquid-gas
phase transition are both strongly influenced by the isospin degree of freedom.
Keywords: Asymmetric nuclear matter, effective potential, chiral symmetry.

1. Introduction
One of the most important thrusts of modern nuclear physics is the use of
high-energy heavy-ion reactions to study the properties of excited nuclear matter and
find evidence of nuclear phase transition between different thermodynamic states at finite
temperature and density. Such ambitious objectives have attracted intense experimental
and theoretical investigation. A number of theoretical articles have been published [3,
4, 8, 10] among them, and research based on simplified models of strongly interacting
nucleons is of great interest to those who wish to understand nuclear matters under
different conditions. In the case of asymmetric matter, however, few articles have been
published because it is more complex [7, 9]. An additional degree of freedom needs to
be taken into account: the isospin. For asymmetric systems, the phenomenon of isospin
distillation demonstrates that the proton fraction is an order parameter. Such matter plays

an important role in astrophysics, where neutron-rich systems are involved in neutron
stars and supernova evolution [2, 5, 6]. In this respect, this article considers properties of
asymmetric nuclear matter.

Received July 22, 2013. Accepted September 24, 2013.
Contact Le Viet Hoa, e-mail address:

125


Le Viet Hoa, Le Duc Anh and Dang Thi Minh Hue

2. Content
2.1. The effective potential of one-loop approximation
Let us begin with the asymmetric nuclear matter given by the Lagrangian density
¯ µ ∂ µ − MN + gσ σ + gδ ⃗τ ⃗δ − gω γµ ω µ − gρ γµ⃗τ ρ⃗µ ]ψ +
£ = ψ[iγ
1 µ
1
1
+
(∂ σ∂µ σ − m2σ σ 2 ) − Fµν F µν + m2ω ωµ ω µ −
2
4
2
)
1 ( ⃗ µ⃗
1 ⃗ ⃗ µν 1 2
µ
¯ 0 µψ,

−
Gµν G + mρ ρ⃗µ ρ⃗ + ∂µ δ∂ δ − m2δ ⃗δ2 + ψγ
4
2
2

(2.1)

in which
⃗ µν = ∂µ ρ⃗ν − ∂ν ρ⃗µ
Fµν = ∂µ ων − ∂ν ωµ ; G
µI
µI
µ = diag(µp , µn ), µp = µB + ; µn = µB − .
2
2
Where ψ, σ, ωµ , ⃗δ, ρ⃗ are the field operators of the nucleon, sigma, omega, rho and
delta mesons, respectively; MN = 939M eV, mσ = 500M eV, mω = 783M eV, mδ =
983M eV, mρ = 770M eV are the "base" mass of the nucleon, meson sigma, meson
omega, meson delta and meson rho; gσ , gω , gδ , gρ are the coupling constants; ⃗τ = ⃗σ /2,
⃗σ = (σ 1 , σ 2 , σ 3 ) are the Pauli matrices and γµ are the Dirac matrices.
In the mean-field approximation, the σ, ωµ , ⃗δ, and ρ⃗ fields are replaced by the
ground-state expectation values
⟨σ⟩ = σ0 ,

⟨ωµ ⟩ = ω0 δ0µ ,

⟨ρaµ ⟩ = bδ3a δ0µ ,

⟨δi ⟩ = dδ3i .

(2.2)

Inserting (2.2) into (2.1) we arrive at
¯ µ ∂ µ − M ∗ + γ 0 µ∗ }ψ − U (σ0 , ω0 , b, d),
LM F T = ψ{iγ
p,n
p,n

(2.3)

where
d
∗
Mp,n
= MN − gσ σ0 ∓ gδ ,
2
b
µI
b
µ∗p,n = µp,n − gω ω0 ∓ gρ = µp = µB ±
− gω ω0 ∓ gρ ,
2
2
2
1 2 2 1 2 2 1 2 2 1 2 2
U (σ0 , ω0 , b, d) =
m σ + m d − mω ω 0 − mρ b .
2 σ 0 2 δ
2

2

(2.4)
(2.5)
(2.6)

Starting with (2.3) we obtain the inverse propagator in the tree approximation
S −1 (k; σ0 , ω0 , b, d) =

−⃗σ .⃗k
0
0
(k0 +µ∗p )−Mp∗

∗
∗
⃗
0
0
⃗σ .k
−(k0 +µp )−Mp


∗
∗

−⃗σ .⃗k
0
0
(k0 +µn )−Mn

0
0
⃗σ .⃗k
−(k0 +µ∗n )−Mn∗
126




,
(2.7)



The properties of asymmetric nuclear matter

and thus
det S −1 (k; σ0 , ω0 , b, d) = (k0 + Ep+ )(k0 − Ep− )(k0 + En+ )(k0 − En− ),

(2.8)

in which
µI
b
− gω ω0 − gρ ,
2
2
µ
b
I

Ekp − µ∗p = Ekp − µB −
+ gω ω0 + gρ ,
2
2
µI
b
Ekn + µ∗n = Ekn + µB −
− gω ω0 + gρ ,
2
2
µI
b
n
∗
n
Ek − µ n = Ek − µ B +
+ gω ω0 − gρ ,
2
√2
√
2
2
⃗k 2 + M ∗ , E n = ⃗k 2 + M ∗ .

Ep+ = Ekp + µ∗p = Ekp + µB +
Ep− =
En+ =
En− =
Ekp =

p

k

(2.9)

n

Based on (2.6) and (2.8) the effective potential at finite temperature is derived:
Ω(σ0 , ω0 , b, d, T ) = U (σ0 , ω0 , b, d) + i Tr ln S −1 (k; σ0 , ω0 , b, d) =
[
∫ ∞
+
T
2
= U (σ0 , ω0 , b, d) − 2
k dk ln(1 + e−Ep /T ) +
π 0
−Ep− /T

+ ln(1 + e

) + ln(1 + e

+
−En
/T

) + ln(1 + e

−
−En
/T

]
) .(2.10)

The ground state of nuclear matter is determined by the minimum conditions:
∂Ω
= 0,
∂σ0
or
σ0 =
d =
ω0 =
b =
Here

∂Ω
= 0,
∂d

∂Ω
= 0,
∂ω0

∂Ω
= 0,
∂b

(2.11)

}
{ ∗
∫ ∞
Mp +
gσ
Mn∗ +
gσ
−
−
2
(n
+
n
)
+
(n
+
n
)
≡
(ρs + ρsn ),
k
dk
p
p
p
n
n

m2σ π 2 0
Ek
Ekn
m2σ p
}
{ ∗
∫ ∞
Mp +
Mn∗ +
gδ
gδ
−
−
2
(nn + nn ) ≡
(ρs − ρsn ),
k dk
p (np + np ) −
2 2
n
2mδ π 0
Ek
Ek
2m2δ p
∫ ∞
{ −
}
gω
gω
2

+
−
+
k
dk
(n
−
n
)
+
(n
−
n
)
≡ 2 (ρBp + ρBn ),
p
p
n
n
2
2
mω π 0
mω
∫ ∞
{
}
gρ
gρ
+
−

+
k 2 dk (n−
(ρBp − ρBn ).
(2.12)
p − np ) − (nn − nn ) ≡
2
2
2mρ π 0
2m2ρ
[ E ± /T
]−1
e p,n + 1 ,
∫ ∞
∫ ∞
Mp∗ +
1
1
Mn∗ +
2
−
2
),
ρ
=
=
k
dk
+
n
k

dk
(n
(nn + n−
s
p
n
p
n ),
p
n
2
2
π 0
Ek
π 0
Ek
∫ ∞
∫ ∞
1
1
+
+
2
−
(2.13)
=
k dk(np − np ), ρBn = 2
k 2 dk(n−
n − nn ).
2

π 0
π 0

n±
p,n =
ρsp
ρBp

127


Le Viet Hoa, Le Duc Anh and Dang Thi Minh Hue

2.2. Physical properties
2.2.1. Equations of state
Let us now consider equations of state starting with the effective potential. To this
end, we begin with the pressure defined by
P = −Ω|at minimum ,

(2.14)

and introduce the isospin asymmetry α:
α=

ρBn − ρBp
,
ρB

(2.15)

in which ρB = ρBn + ρBp is the baryon density, and ρBn , ρBp are the neutron, proton
densities, respectively.
Combining equations (2.14), (2.4), and (2.10) together produces the following
expression for the pressure
(
)
(
)2
Mp∗ + Mn∗ 2 1
1
fω
fρ
∗
∗
P(ρB , α, T ) = −
MN −
−
Mn − Mp + ρ2B + α2 ρ2B
2fσ
2
2fδ
2
8
[
∫ ∞
+
−
T
+
k 2 dk ln(1+e−Ep /T ) + ln(1 + e−Ep /T )

2
π 0
]
+
−
−En
/T
−En
/T
+ ln(1 + e
) + ln(1 + e
) .
(2.16)
Here fi =

gi2
,
m2i

(i ≡ σ, ω, δ, ρ).

Based on (2.10) the entropy density is derived
∫ ∞
∂Ω
1
− −
+ +
− −
ς = −
=

k 2 dk(Ep+ n+
p + Ep np + En nn + En nn )
∂T
T π2 0
∫ ∞
[
+
−
1
+
k 2 dk ln(1 + e−Ep /T ) + ln(1 + e−Ep /T )
2
π 0
]
+
−
+ ln(1 + e−En /T ) + ln(1 + e−En /T ) .

(2.17)

The energy density is obtained by the Legendre transform of P:
E(ρB , α, T ) = Ω + T ς + µp ρBp + µn ρBn
)2
(
(
)
Mp∗ + Mn∗ 2
fω
fρ
1

1
∗
∗
MN −
+
Mn − Mp + ρ2B + α2 ρ2B
=
2fσ
2
2fδ
2
8
∫ ∞
}
{
1
−
+
n
−
(2.18)
+
k 2 dk (Ekp (n+
p + np ) + Ek (nn + nn ) .
π2 0
Eqs. (2.16) and (2.18) constitute the equations of state governing all thermodynamical
processes of nuclear matter.
128

The properties of asymmetric nuclear matter

2.2.2. Numerical study
In order to understand the properties of nuclear matter one has to carry out the
g2
numerical study. We first fix the coupling constants fi = mi2 , (i ≡ σ, ω, δ, ρ). To this
i
end, Eq. (2.4) is solved numerically for symmetric nuclear matter (Gδ,ρ = 0) at T = 0.
Its solution is then substituted into the nuclear binding energy Ebin = −M + E/ρB with
E given in (2.18). Two parameters fσ and fω are adjusted to yield the the binding energy
εbin |T =0 = −15.8M eV at normal density ρB = ρ0 = 0.16f m−3 . It is found that fσ =
14.49f m2 and fω = 10.97f m2 . Figure 1 shows the graph of binding energy in relation to
baryon density.
30
fω=10.97 fm

Ebin(MeV)

20

2

10
0
-10
-15.8MeV
-20
0

0.5

1
ρB/ρ0

1.5

2

Figure 1. Nuclear binding energy as a function of baryon density.
As to fixing fρ let us follow the method developed in [5] where fδ is chosen as
fδ = 0 and fδ = 2.5f m2 . Then, fρ is fitted to give
(
)
1 ∂ 2 Ebin
Esym =
= 32M eV.
(2.19)
2 ∂α2 T =0, α=0, ρB =ρ0
It is found that fρ = 3.04(f m2 ) and fρ = 5.02(f m2 ) respectively. Thus, all of
the model parameters are known as in Table 1, which are in good agreement with those
widely expected in the literature [10].

Set I
Set II

fσ
14.49(f m2 )
14.49(f m2 )

fω

10.97(f m2 )
10.97(f m2 )

fδ
0
2.5(f m2 )

fρ
3.04(f m2 )
5.02(f m2 )

Now we are ready to carry out the numerical computation. Figure 2 shows the
density dependence of effective nucleon masses at several values of temperature and
isospin asymmetry α = 0.2. It is clear that the chiral symmetry is restored at high nuclear
density.
129


Le Viet Hoa, Le Duc Anh and Dang Thi Minh Hue
1
T=0
T=5
T=10
T=15
T=20
T=30
T=40
T=50

α=0.2

0.6

*

M p,n/MN

0.8

0.4

0.2
0

1

ρB/ρ0

2

3

Figure 2. The density dependence of effective nucleon masses
The phenomena of liquid-gas phase transition are governed by the equations of
state (2.16) and (2.18). In Figures (3a - 4b), we obtain a set of isotherms at fixed isospin
asymmetry. These bear the typical structure of the van der Waals equations of state [1,
4]. As we can see from the these figures the liquid-gas phase transition in asymmetric
nuclear matter is not only more complex than in symmetric matter but it also has new
distinct features. This is because they are strongly influenced by the isospin degree of
freedom.

4

4

P

T =0,

T = 0,

P3

T = 5,

T =5,
T =10,

T = 10,

T =15,

T = 15,

2

T = 20,

T =20,

2

T =25,

T = 25,

1
0
0.5

1.0

1.5

0
0.5

1.0

1.5

-1
-2

Figure 3a. The equations of state for
several T steps at α = 0

Figure 3b. The equations of state for
several T steps at α = 0.25
10

3

P

T = 0,

P

T = 5,
T = 10,

T = 0,

T = 15,

2

T = 5,

T = 20,

T = 10,

T = 25,

T = 15,

5

1

T = 20,
T = 25,

0
0.5

1.0

-1

0
0.5

Figure 4a. The equations of state for
several T steps at α = 0.5
130

1.0

Figure 4b. The equations of state for
several T steps at α = 1


The properties of asymmetric nuclear matter

3. Conclusion
Due to the important role of the isospin degree of freedom in ANM, we have
investigated the isospin dependence of pressure on asymmetric nuclear matter. Our main
results are summarized as follows:

1-Based on the effective potential in one-loop approximation we reproduced the
expression for the pressure and energy density. They constitute the equations of state of
nuclear matter.
2-It was shown that chiral symmetry is restored at high nuclear density and
liquid-gas phase transition in asymmetric nuclear matter is strongly influenced by the
isospin degree of freedom. This is our major success.
In order to understand better the properties of asymmetric nuclear matter a more
detailed study phase structure should be carried out by means of numerical computation.
This is a promising task for future research.
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