The properties of asymmetric nuclear matter

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.

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THE PROPERTIES OF ASYMMETRIC NUCLEAR MATTER

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

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.

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: hoalv@hnue.edu.vn

2 Content

2.1 The effective potential of one-loop approximation

Let us begin with the asymmetric nuclear matter given by the Lagrangian density

+ 1

2(∂

µ σ∂ µ σ − m2

σ σ2)−1

4F µν F

µν +1

2m

2

ω ω µ ω µ −

4

⃗

2m

2

ρ ⃗ µ ⃗ µ+1

2

(

∂ µ ⃗ δ∂ µ ⃗ δ − m2

δ ⃗ δ2) + ¯ψγ0µψ, (2.1)

in which

2 ; µ n = µ B − µ I

2 .

Where ψ, σ, ω µ , ⃗ δ, ⃗ ρ are the field operators of the nucleon, sigma, omega, rho and

delta mesons, respectively; M N = 939M eV, m σ = 500M eV, m ω = 783M eV, m δ =

983M eV, m ρ = 770M eV are the "base" mass of the nucleon, meson sigma, meson omega, meson delta and meson rho; g σ , g ω , g δ , g ρ are the coupling constants; ⃗ τ = ⃗ σ/2,

In the mean-field approximation, the σ, ω µ , ⃗ δ, and ⃗ ρ fields are replaced by the

ground-state expectation values

Inserting (2.2) into (2.1) we arrive at

L M F T = ¯ψ {iγ µ ∂ µ − M ∗

where

d

b

2 = µ p = µ B ± µ I

2 − g ω ω0∓ g ρ

b

2, (2.5)

2m

2

σ σ02+ 1

2m

2

δ d2−1

2m

2

ω ω02− 1

2m

2

ρ b2. (2.6) Starting with (2.3) we obtain the inverse propagator in the tree approximation

S −1 (k; σ0, ω0, b, d) =











(k0 +µ ∗ p)−M ∗

⃗

0 0 (k0+µ ∗ n)−M ∗

0 0 ⃗ σ.⃗k −(k0+µ ∗ n)−M ∗

n









(2.7),

and thus

det S −1 (k; σ0, ω0, b, d) = (k0+ E p+)(k0− E −

p )(k0 + E n+)(k0− E −

n ), (2.8)

in which

2 − g ω ω0− g ρ

b

2,

p = E k p − µ B − µ I

2 + g ω ω0+ g ρ

b

2,

2 − g ω ω0+ g ρ b

2,

n = E k n − µ B+ µ I

2 + g ω ω0− g ρ

b

2,

√

p , E k n=

√

Based on (2.6) and (2.8) the effective potential at finite temperature is derived:

= U (σ0, ω0, b, d) − T

∫ ∞ 0

[

ln(1 + e −E p+/T) +

+ ln(1 + e −E − p /T ) + ln(1 + e −E+n /T ) + ln(1 + e −E n − /T)

]

.(2.10)

The ground state of nuclear matter is determined by the minimum conditions:

∂Ω

∂Ω

∂Ω

∂Ω

or

σ π2

∫ ∞ 0

{

M p ∗

+

p + n − p) + M

∗ n

k (n+n + n − n)

}

σ (ρ s p + ρ s n ),

2m2

δ π2

∫ ∞ 0

{

M p ∗

+

p + n − p)− M n ∗

k (n+n + n − n)

}

2m2

δ (ρ s p − ρ s n ),

ω π2

∫ ∞ 0

(n − p − n+

p ) + (n − n − n+

n)}

ω (ρ B p + ρ B n ),

2m2

ρ π2

∫ ∞ 0

(n − p − n+

p)− (n −

n − n+

n)}

2m2

ρ (ρ B p − ρ B n ). (2.12) Here

e E ± p,n /T

+ 1]−1

,

∫ ∞ 0

∗ p

+

p + n − p ), ρ s n = 1

∫ ∞ 0

∗ n

k (n+n + n − n ),

∫ ∞ 0

p ), ρ B n = 1

∫ ∞ 0

n ). (2.13)

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

and introduce the isospin asymmetry α:

in which ρ B = ρ B n + ρ B p is the baryon density, and ρ B n , ρ B p are the neutron, proton densities, respectively

Combining equations (2.14), (2.4), and (2.10) together produces the following expression for the pressure

2f σ

(

2

)2

2f δ

(

p

)2 +f ω

2 ρ

2

B+f ρ

8α

2

ρ2B

+ T

∫ ∞ 0

[

ln(1+e −E p+/T

) + ln(1 + e −E − p /T

)

+ ln(1 + e −E n+/T

) + ln(1 + e −E − n /T

)

]

Here f i = g2i

m2

i

Based on (2.10) the entropy density is derived

1

∫ ∞ 0

k2dk(E p+n+p + E p − n − p + E n+n+n + E n − n − n)

+ 1

∫ ∞ 0

ln(1 + e −E p+/T ) + ln(1 + e −E − p /T)

+ ln(1 + e −E n+/T ) + ln(1 + e −E − n /T)]

The energy density is obtained by the Legendre transform ofP:

= 1

2f σ

(

2

)2 + 1

2f δ

(

p

)2 +f ω

2 ρ

2

B+f ρ

8α

2ρ2B

+ 1

∫ ∞ 0

(E k p (n+p + n − p ) + E k n (n+n + n − n)}

Eqs (2.16) and (2.18) constitute the equations of state governing all thermodynamical processes of nuclear matter

2.2.2 Numerical study

In order to understand the properties of nuclear matter one has to carry out the

numerical study We first fix the coupling constants f i = g i2

m2

i

end, Eq (2.4) is solved numerically for symmetric nuclear matter (G δ,ρ = 0) at T = 0.

Its solution is then substituted into the nuclear binding energyE bin = −M + E/ρ B with

14.49f m2and f ω = 10.97f m2 Figure 1 shows the graph of binding energy in relation to baryon density

ρB/ ρ0 -20

-10 0 10 20 30

Ebin

-15.8MeV

fω=10.97 fm2

Figure 1 Nuclear binding energy as a function of baryon density.

As to fixing f ρ let us follow the method developed in [5] where f δ is chosen as

f δ = 0 and f δ = 2.5f m2 Then, f ρis fitted to give

2

(

)

T =0, α=0, ρ B =ρ0

It is found that f ρ = 3.04(f m2) and f ρ = 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 14.49(f m2) 10.97(f m2) 0 3.04(f m2)

Set II 14.49(f m2) 10.97(f m2) 2.5(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

0 1 2 3

ρB/ ρ0

0.2 0.4 0.6 0.8 1

* p,n

T=0 T=5 T=10 T=15 T=20 T=30 T=50

α=0.2

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

-2

0

2

4

P

T = 0,

T = 5,

T = 10,

T = 20,

Figure 3a The equations of state for

several T steps at α = 0

-1 0 1 2 3 4

P

T =0,

T =5,

T =10,

T =20,

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

-1

0

1

2

3

P

T = 0,

T = 5,

T = 10,

T = 15,

T = 25,

Figure 4a The equations of state for

several T steps at α = 0.5

0 5

10

P

T = 0,

T = 5,

T = 10,

T = 15,

T = 25,

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

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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