The properties of asymmetric nuclear matter

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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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JOURNAL OF SCIENCE OF HNUE Mathematical and Physical Sci., 2013, Vol. 58, No. 7, pp. 125-131 This paper is available online at http://stdb.hnue.edu.vn THE PROPERTIES OF ASYMMETRIC NUCLEAR MATTER Le Viet Hoa1 , Le Duc Anh1 and Dang Thi Minh Hue2 1 Faculty of Physics, Hanoi National University of Education 2 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: hoalv@hnue.edu.vn 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 ⃗ ⃗ µν 1 2 1 ( ⃗ µ⃗ ) − Gµν G + mρ ρ⃗µ ρ⃗ + ∂µ δ∂ δ − m2δ ⃗δ2 + ψγ µ ¯ 0 µψ, (2.1) 4 2 2 in which Fµν = ∂µ ων − ∂ν ωµ ; G⃗ µν = ∂µ ρ⃗ν − ∂ν ρ⃗µ 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 (2.3) p,n p,n where ∗ d Mp,n = MN − gσ σ0 ∓ gδ , (2.4) 2 b µI b µ∗p,n = µp,n − gω ω0 ∓ gρ = µp = µB ± − gω ω0 ∓ gρ , (2.5) 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 . (2.6) 2 σ 0 2 δ 2 2 Starting with (2.3) we obtain the inverse propagator in the tree approximation S −1 (k; σ0 , ω0 , b, d) =   (k0 +µ∗p )−Mp∗ −⃗σ .⃗k 0 0  ⃗ ∗ ∗   ⃗σ .k −(k0 +µp )−Mp 0 0   (2.7) ,  0 0 ∗ (k0 +µn )−Mn ∗ −⃗σ .⃗k  0 0 ⃗σ .⃗k −(k0 +µ∗n )−Mn∗ 126
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 Ep+ = Ekp + µ∗p = Ekp + µB + − gω ω0 − gρ , 2 2 µ b Ep− = Ekp − µ∗p = Ekp − µB − I + gω ω0 + gρ , 2 2 µI b En+ = Ekn + µ∗n = Ekn + µB − − gω ω0 + gρ , 2 2 µI b En− = n ∗ Ek − µ n = Ek − µ B + n + gω ω0 − gρ , √ √2 2 Ekp = ⃗k 2 + M ∗ , E n = ⃗k 2 + M ∗ . 2 2 (2.9) p k 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 + = U (σ0 , ω0 , b, d) − 2 k dk ln(1 + e−Ep /T ) + 2 π 0 ] −Ep− /T + −En /T − −En /T + ln(1 + e ) + ln(1 + e ) + ln(1 + e ) .(2.10) The ground state of nuclear matter is determined by the minimum conditions: ∂Ω ∂Ω ∂Ω ∂Ω = 0, = 0, = 0, = 0, (2.11) ∂σ0 ∂d ∂ω0 ∂b or ∫ ∞ { ∗ } gσ Mp + − Mn∗ + − gσ σ0 = k 2 dk p (n p + n p ) + (n n + n n ) ≡ (ρs + ρsn ), m2σ π 2 0 Ek Ekn m2σ p ∫ ∞ { ∗ } gδ Mp + − Mn∗ + − gδ p (np + np ) − (nn + nn ) ≡ (ρs − ρsn ), 2 d = 2 2 k dk n 2mδ π 0 Ek Ek 2m2δ p ∫ ∞ gω { − − } gω ω0 = 2 2 k 2 dk (np − n + p ) + (nn − n + n ) ≡ 2 (ρBp + ρBn ), mω π 0 mω ∫ ∞ gρ { } gρ b = 2 2 k 2 dk (n− − p − np ) − (nn − nn ) ≡ + + (ρBp − ρBn ). (2.12) 2mρ π 0 2m2ρ Here [ E ± /T ]−1 n± p,n = e p,n + 1 , ∫ ∞ ∫ ∞ 1 Mp∗ + − 1 Mn∗ + ρsp = 2 k 2 dk p (n p + n p ), ρ s n = 2 k 2 dk n (nn + n− n ), π 0 Ek π 0 Ek ∫ ∞ ∫ ∞ 1 − 1 ρBp = 2 k dk(np − np ), ρBn = 2 2 + k 2 dk(n− n − nn ). + (2.13) π 0 π 0 127
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: ρBn − ρBp α= , (2.15) ρB 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 1 Mp∗ + Mn∗ 2 1 ∗ ∗ fω fρ P(ρB , α, T ) = − MN − − Mn − Mp + ρ2B + α2 ρ2B 2fσ 2 2fδ 2 8 ∫ ∞ [ T + − + 2 k 2 dk ln(1+e−Ep /T ) + ln(1 + e−Ep /T ) π 0 ] + − −En /T −En /T + ln(1 + e ) + ln(1 + e ) . (2.16) gi2 Here fi = 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 [ + − + 2 k 2 dk ln(1 + e−Ep /T ) + ln(1 + e−Ep /T ) 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 1 Mp∗ + Mn∗ 2 1 ∗ ∗ fω fρ = MN − + Mn − Mp + ρ2B + α2 ρ2B 2fσ 2 2fδ 2 8 ∫ ∞ 1 { − − } + k 2 dk (Ekp (n+ n + p + np ) + Ek (nn + nn ) . (2.18) 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 2 20 fω=10.97 fm Ebin(MeV) 10 0 -10 -15.8MeV -20 0 0.5 1 1.5 2 ρB/ρ0 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]. fσ fω fδ fρ 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. 129
Le Viet Hoa, Le Duc Anh and Dang Thi Minh Hue 1 T=0 α=0.2 T=5 0.8 T=10 T=15 T=20 M p,n/MN T=30 0.6 T=40 T=50 * 0.4 0.2 0 1 2 3 ρB/ρ0 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 T = 0, T =0, P T = 5, P3 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 Figure 3b. The equations of state for several T steps at α = 0 several T steps at α = 0.25 10 T = 0, 3 T = 5, P P T = 10, T = 0, T = 15, T = 5, 2 T = 20, T = 10, T = 25, T = 15, 5 T = 20, 1 T = 25, 0 0.5 1.0 0 -1 0.5 1.0 Figure 4a. The equations of state for Figure 4b. The equations of state for several T steps at α = 0.5 several T steps at α = 1 130
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. REFERENCES [1] L. P. CSernai et al., 1986. Entropy and cluster production in nuclear collisions. Phys. Rep. 131, 223. [2] N. K. Glendenning, 2001. Phase transitions and crystalline structures in neutron star cores. Phys. Rep. 342, 393. [3] P. Huovinen, 2005. Anisotropy of flow and the order of phase transition in relativistic heavy ion collisions. Nucl. Phys. A761, 296. [4] H. R. Jaqaman, A. Z. Mekjian and L. Zamick, 1983. Nuclear condensation. Phys. Rev. C27, 2782. [5] S. Kubis, M. Kutschera and S. Stachniewicz, 1998. Neutron stars in relativistic mean field theory with isovector scalar meson. arXiv:astro-ph/9802303V1. [6] J. M. Lattimer and M. Prakash, 2000. Nuclear matter and its role in supernovae, neutron stars and compact object binary mergers. Phys. Rep. 333, 121. [7] B. Liu, V. Greco, V. Baran, M. Colonna1 and M. Di Toro, 2001. Asymmetric nuclear matter: the role of the isovector scalar channel. arXiv: nucl-th/0112034V1. [8] H. Muller and B. D. Serot, 1995. Phase transitions in warm, asymmetric nuclear matter. Phys. Rev. C52, 2072. [9] Tran Huu Phat, Le Viet Hoa, Nguyen Tuan Anh, Le Duc Anh and Dinh Thanh Tam, 2012. Phase Structure in an Asymmetric Model of Nuclear Matter. Nuclear Science and Technology, 1, pp. 1-25. [10] J. D. Walecka, 1974. Theoretical nuclear and subnuclear physics, second edition. Ann. Phys. 83, 491. 131