1. Introduction
The recent investigation [1, 2, 3] shows that most of the ground state properties
of the large number of nuclei over the entire range of the periodical table can be very
well reproduced by relativistic mean field theory (RMF). A number of theoretical articles
have been published [4, 5], based on simplified models of strongly interacting nucleons
is of great interest for understanding nuclear matter under different conditions. In the
case of asymmetric matter, however, few articles have been published, since it is more
complex [6, 7]. There is an additional degree of freedom that needs to be taken into
account: the isospin. In this respect, this article considers phase transition of asymmetric
nuclear matter.

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JOURNAL OF SCIENCE OF HNUE
Mathematical and Physical Sci., 2014, Vol. 59, No. 7, pp. 52-57
This paper is available online at
PHASE TRANSITION IN ASYMMETRIC NUCLEAR MATTER
IN ONE-LOOP APPROXIMATION
Le Viet Hoa1, Nguyen Tuan Anh2 and Le Duc Anh1
1Faculty of Physics, Hanoi National University of Education
2Faculty of Energy Technology, Electric Power University
Abstract. The equations of state (EoS) of asymmetric nuclear matter (ANM)
starting from the effective potential in a one-loop approximation is investigated.
The numerical computation showed that chiral symmetry is restored asymptotically
at high nuclear density and liquid-gas phase transition in asymmetric nuclear matter
is first-order.
Keywords: Asymmetric, effective potential, isospin, first-order.
1. Introduction
The recent investigation [1, 2, 3] shows that most of the ground state properties
of the large number of nuclei over the entire range of the periodical table can be very
well reproduced by relativistic mean field theory (RMF). A number of theoretical articles
have been published [4, 5], based on simplified models of strongly interacting nucleons
is of great interest for understanding nuclear matter under different conditions. In the
case of asymmetric matter, however, few articles have been published, since it is more
complex [6, 7]. There is an additional degree of freedom that needs to be taken into
account: the isospin. In this respect, this article considers phase transition of asymmetric
nuclear matter.
2. Content
2.1. The effective potential in one-loop approximation
Let us begin with the asymmetry nuclear matter given by the Lagrangian density
$ = ψ¯(i∂ˆ −M)ψ + Gs
2
(ψ¯ψ)2 − Gv
2
(ψ¯γµψ)2 − Gr
2
(ψ¯τ⃗γµψ)2 + ψ¯γ0µψ. (2.1)
Received March 20, 2014. Accepted September 30, 2014.
Contact Le Viet Hoa, e-mail address: hoalv@hnue.edu.vn
52
Phase transition in asymmetric nuclear matter in one-loop approximation
where ψ is the field operator of the nucleon; µ = µp+µn;µp, µn are the chemical potential
of the proton and neutron respectively; M is the "base" mass of the nucleon; Gs, Gv, Gr
are the coupling constants; τ⃗ = σ⃗/2, σ⃗ = (σ1, σ2, σ3) are the Pauli matrices and γµ are
the Dirac matrices.
Bosonizing
σˇ = ψ¯ψ, ωˇµ = ψ¯γµψ,
⃗ˇbµ = ψ¯τ⃗γµψ,
leads to
$ = ψ¯(i∂ˆ −M)ψ +Gsψ¯σˇψ −Gvψ¯γµωˇµψ −Grψ¯γµτ⃗ .⃗bˇµψ
− Gs
2
σˇ2 +
Gv
2
ωˇµωˇµ +
Gr
2
⃗ˇbµ⃗bˇµ + ψ¯γ
0µψ. (2.2)
In the mean field approximation
⟨σˇ⟩ = σ, ⟨ωˇµ⟩ = ωδ0µ, ⟨bˇaµ⟩ = bδ3aδ0µ
we arrive at
$
MFT
= ψ¯{i∂ˆ −M∗ + γ0µ∗}ψ − U(σ, ω, b), (2.3)
in which
M∗ = M −Gsσ, (2.4)
µ∗ = µ−Gvω −Grτ 3b, (2.5)
U(σ, ω, b) =
1
2
[Gsσ
2 −Gvω2 −Grb2]. (2.6)
Starting with (2.3) we obtain the inverse propagator in the momentum space
S−1(k; σ, ω, b) =
(k0+µ
∗
p)−M∗ −σ⃗.⃗k 0 0
σ⃗.⃗k −(k0+µ∗p)−M∗ 0 0
0 0 (k0+µ
∗
n)−M∗ −σ⃗.⃗k
0 0 σ⃗.⃗k −(k0+µ∗n)−M∗
.
(2.7)
According to Refs. [7, 10] the thermodynamic potential at finite temperature reads.
Ω(σ, ω, b) = U(σ, ω, b)− T
π2
∫ ∞
0
k2dk
[
ln(1 + e−E
−
−/T )
+ ln(1 + e−E
−
+/T ) + ln(1 + e−E
+
−/T ) + ln(1 + e−E
+
+/T )
]
, (2.8)
in which
E±∓ = E
±
k ∓
(
µ
B
−Gωω
)
, E±k = Ek ± (
µ
I
2
− Gρ
2
b), Ek =
√
k⃗2 +M∗2
µB = µp + µn, µI = µp − µn. (2.9)
53
Le Viet Hoa, Nguyen Tuan Anh and Le Duc Anh
The ground state of nuclear matter is determined by the minimum condition
∂Ω
∂σ
= 0,
∂Ω
∂ω
= 0,
∂Ω
∂b
= 0 (2.10)
which yields
σ =
1
π2
∫ ∞
0
k2dk
M∗
Ek
{
(n−p + n
+
p ) + (n
−
n + n
+
n )
} ≡ ρs
ω =
1
π2
∫ ∞
0
k2dk
{
(n−p − n+p ) + (n−n − n+n )
} ≡ ρ
B
b =
1
2π2
∫ ∞
0
k2dk
{
(n−p − n+p )− (n−n − n+n )
} ≡ ρ
I
. (2.11)
where
n−− = n
−
p ; n
+
+ = n
+
p ; n
−
+ = n
+
n ; n
+
− = n
−
n ; n
±
∓ =
[
eE
±
∓/T + 1
]−1
.
Based on the (2.8) and (2.11) the pressure is derived
P = −Ω|at min = −Gs
2
ρ2s +
Gv
2
ρ2
B
+
Gr
2
ρ2I +
T
π2
∫ ∞
0
k2dk
[
ln(1 + e−E
−
−/T )
+ ln(1 + e−E
−
+/T ) + ln(1 + e−E
+
−/T ) + ln(1 + e−E
+
+/T )
]
. (2.12)
The energy density is obtained by the Legendre transform:
E(T, ρ
B
, y) = Ω(σ, ω, b) + Tς + µ
B
ρ
B
+ µ
I
ρ
I
=
Gs
2
ρ2s +
Gv
2
ρ2B +
Gr
2
ρ2I
+
1
π2
∫ ∞
0
k2dkEk(n
−
p +n
+
p + n
−
n + n
+
n ). (2.13)
Introducing the dimensionless parameter Y = ρp/ρB equations (2.12) and (2.16)
are rewritten as
E(T, ρ
B
, y) =
(M −M∗)2
2Gs
+
[
Gv
2
+
Gr(Y − 0.5)2
2
]
ρ2
B
+
+
1
π2
∫ ∞
0
k2dkEk(n
−
p + n
+
p + n
−
n + n
+
n ). (2.14)
P = −(M −M
∗)2
2Gs
+
[
Gv
2
+
Gr(Y − 0.5)2
2
]
ρ2
B
+
T
π2
∫ ∞
0
k2dk
[
ln(1 + e−
Ek−∗p
T ) +
+ ln(1 + e−
Ek+
∗
p
T ) + ln(1 + e−
Ek+
∗
n
T ) + ln(1 + e−
Ek−∗n
T )
]
. (2.15)
Eqs. (2.14) and (2.15) constitute the equations of state governing all
thermodynamical processes in the considered nuclear matter.
54
Phase transition in asymmetric nuclear matter in one-loop approximation
2.2. Phase transition in one-loop approximation
In order to get insight into the nature of phase transition one has to carry out a
numerical study. We follow the method developed by Walecka [8, 9] to determine the
three parameters Gs, Gv, and Gr for symmetric nuclear matter based on the saturation
condition: The saturation mechanism requires that at normal density ρB = ρ0 =
0.16fm−3 the binding energy
Ebin = −M + E/ρB (2.16)
attains its minimum value (Ebin)ρ0 = -15.8 MeV. It is found that Gs = 13.62 fm2,
Gv/Gs = 0.75. As to fixing Gr let us employ the expression of nuclear symmetry energy
at saturation density
Esym =
ρ2B
8
(
∂2Ebin
∂ρ2I
)
ρI=0; ρB=ρ0
= 32MeV.
Its value is Gr = 0.198Gs.
The next step is to solve numerically Eq.(2.4). Figure 1 shows the density
dependence of effective nucleon masses at temperature T = 10K and several values of
Y . It is clear that the chiral symmetry is restored asymptotically at high nuclear density.
Figure 1. The density dependence of effective nucleon masses
We also obtain in Figures 2a and 2b the evolution of the nucleon effective mass
versus µB at various values of temperature T and isospin asymmetry α =
ρn−ρp
ρB
= 1−2Y .
It is clear that depending on α for T ≥ Tc the nucleon effective mass is a
single-valued function of the baryon chemical potential µB and smoothly tends to zero.
For lower temperatures, 0 ≤ T < Tc, the nucleon effective mass turns out to be a
multi-valued function of µB, where a first-order liquid-gas phase transition emerges.
The EoS at several values of the isospin asymmetry α and some fixed temperatures T
55
Le Viet Hoa, Nguyen Tuan Anh and Le Duc Anh
is presented in Figures 3 and 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 also has new distinct features: the critical temperature for the phase transition
dependence on temperature T and the decreases with increasing neutron excess. These
results are in good agreement with that based on the Skyrme interaction and relativistic
mean-field theory.
Figure 2a. The density dependence of effective nucleon masses
Figure 2b. The density dependence of effective nucleon masses
Figure 3. The EoS for several α steps at some fixed temperatures
56
Phase transition in asymmetric nuclear matter in one-loop approximation
Figure 4. The EoS for several α steps at some fixed temperatures
3. Conclusion
Due to the important role of the isospin degree of freedom in ANM, we have
investigated the EoS of asymmetric nuclear matter. Our main results are summarized
as follows:
- Based on the effective potential in the one-loop approximation we reproduced the
equation of state of ANM.
- The numerical computation showed that chiral symmetry is restored
asymptotically at high nuclear density and the liquid-gas phase transition in asymmetric
nuclear matter is the first-order and strongly influenced by the isospin degree of freedom.
This is our major success.
Acknowledgment. The authors would like to thank the HNUE project under the code
SPHN-13-239 for financial support.
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