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.
JOURNAL OF SCIENCE OF HNUE Mathematical and Physical Sci., 2014, Vol 59, No 7, pp 52-57 This paper is available online at http://stdb.hnue.edu.vn PHASE TRANSITION IN ASYMMETRIC NUCLEAR MATTER IN ONE-LOOP APPROXIMATION Le Viet Hoa1 , Nguyen Tuan Anh2 and Le Duc Anh1 Faculty of Physics, Hanoi National University of Education Faculty 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 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 Content 2.1 The effective potential in one-loop approximation Let us begin with the asymmetry nuclear matter given by the Lagrangian density ¯ ∂ˆ − M )ψ + Gs (ψψ) ¯ − Gv (ψγ ¯ µ ψ)2 − Gr (ψ⃗ ¯τ γ µ ψ)2 + ψγ ¯ µψ £ = ψ(i 2 Received March 20, 2014 Accepted September 30, 2014 Contact Le Viet Hoa, e-mail address: hoalv@hnue.edu.vn 52 (2.1) 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; ⃗τ = ⃗σ /2, ⃗σ = (σ , σ , σ ) are the Pauli matrices and γ µ are the Dirac matrices Bosonizing ¯ σ ˇ = ψψ, ¯ µ ψ, ⃗ˇbµ = ψ⃗ ¯τ γµ ψ, ω ˇ µ = ψγ leads to ¯ ∂ˆ − M )ψ + Gs ψˇ ¯σ ψ − Gv ψγ ¯ µω ¯ µ⃗τ ⃗ˇbµ ψ £ = ψ(i ˇ µ ψ − Gr ψγ Gs Gv µ Gr ¯ µψ − σ ˇ + ω ˇ ω ˇ µ + ⃗ˇbµ⃗ˇbµ + ψγ 2 (2.2) In the mean field approximation ⟨ˇ σ ⟩ = σ, ⟨ˇ ωµ ⟩ = ωδ0µ , ⟨ˇbaµ ⟩ = bδ3a δ0µ we arrive at ¯ ∂ˆ − M ∗ + γ µ∗ }ψ − U (σ, ω, b), £M F T = ψ{i (2.3) M ∗ = M − Gs σ, µ∗ = µ − Gv ω − Gr τ b, U (σ, ω, b) = [Gs σ − Gv ω − Gr b2 ] (2.4) (2.5) in which (2.6) Starting with (2.3) we obtain the inverse propagator in the momentum space (k0 +µ∗p )−M ∗ −⃗σ ⃗k 0 ⃗σ ⃗k −(k0 +µ∗p )−M ∗ 0 −1 S (k; σ, ω, b) = ∗ ∗ ⃗ 0 (k0 +µn )−M −⃗σ k ∗ ∗ ⃗ 0 ⃗σ k −(k0 +µn )−M (2.7) According to Refs [7, 10] the thermodynamic potential at finite temperature reads [ ∫ ∞ − T Ω(σ, ω, b) = U (σ, ω, b) − k dk ln(1 + e−E− /T ) π ] − + + −E+ /T −E− /T −E+ /T + ln(1 + e ) + ln(1 + e ) + ln(1 + e ) , (2.8) in which E∓± = Ek± ( ) ∓ µB − Gω ω , Ek± µB = µp + µn , µI = µp − µn µ Gρ = Ek ± ( I − b), Ek = 2 √ ⃗k + M ∗2 (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 = 0, ∂σ which yields ∂Ω = 0, ∂ω ∂Ω =0 ∂b (2.10) ∫ ∞ } M∗ { − + − + σ = k dk (n + n ) + (n + n ) ≡ ρs p p n n π2 Ek ∫ ∞ } { − + − + ) ≡ ρB − n ) + (n − n ω = k dk (n n n p p π2 ∫ ∞ } { − + − + b = ) ≡ ρI − n ) − (n − n k dk (n n n p p 2π where − n− − = np ; + n+ + = np ; [ E ± /T ]−1 ± − + + ∓ = e + ; n = n ; n = n n− ∓ n − n + Based on the (2.8) and (2.11) the pressure is derived P = −Ω|at (2.11) Gs Gv Gr T = − ρ2s + ρB + ρ + 2 I π2 ∫ ∞ [ − k dk ln(1 + e−E− /T ) ] − + + −E+ /T −E− /T −E+ /T + ln(1 + e ) + ln(1 + e ) + ln(1 + e ) (2.12) The energy density is obtained by the Legendre transform: E(T, ρB , y) = Ω(σ, ω, b) + T ς + µB ρB + µI ρI Gs Gv Gr = ρs + ρB + ρ 2∫ 2 I ∞ + − + + k dkEk (n− p +np + nn + nn ) π (2.13) Introducing the dimensionless parameter Y = ρp /ρB equations (2.12) and (2.16) are rewritten as [ ] Gv Gr (Y − 0.5)2 (M − M ∗ )2 E(T, ρB , y) = + + ρB + 2Gs 2 ∫ ∞ + − + k dkEk (n− (2.14) + p + np + nn + nn ) π [ ] ∫ ∞ Ek −µ∗ [ p (M − M ∗ )2 Gv Gr (Y − 0.5)2 T P = − + + k dk ln(1 + e− T ) + ρB + 2Gs 2 π ∗ Ek +µ∗ E +µ Ek −µ∗ p n n ] k (2.15) + ln(1 + e− T ) + ln(1 + e− T ) + ln(1 + e− T ) 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 = 0.16f m−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 f m2 , Gv /Gs = 0.75 As to fixing Gr let us employ the expression of nuclear symmetry energy at saturation density ( ) ρ2B ∂ Ebin Esym = = 32M eV ∂ρ2I ρI =0; ρB =ρ0 Its value is Gr = 0.198Gs The next step is to solve numerically Eq.(2.4) Figure 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 The density dependence of effective nucleon masses We also obtain in Figures 2a and 2b the evolution of the nucleon effective mass p versus µB at various values of temperature T and isospin asymmetry α = ρnρ−ρ = 1−2Y B 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, ≤ 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 and 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 The EoS for several α steps at some fixed temperatures 56 Phase transition in asymmetric nuclear matter in one-loop approximation Figure The EoS for several α steps at some fixed temperatures 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 REFERENCES [1] I K Gambhir, P.Ring and A Thimet, 1990 Ann Phys 198, 132 [2] H Muller and B D Serot, 1995 Phys Rev C 52, 2072 [3] P Wang, D B Leinweber, A W Thomas and A G Williams, 2004 Phys Rev C 70, 055204 [4] P Huovinen, 2005 Anisotropy of flow and the order of phase transition in relativistic heavy ion collisions Nucl Phys A 761, 296 [5] H R Jaqaman, A Z Mekjian and L Zamick, 1983 Nuclear condensation Phys Rev C 27, 2782 [6] B Liu, V Greco, V Baran, M Colonnal and M Di Toro, 2001 Asymmetric nuclear matter: the role of the isovector scalar channel arXiv: nucl-th/0112034V1 [7] Tran Huu Phat et al 2011 Communications in Physics, Vol 21, No , pp 117-123 [8] J D Walecka, 1974 Ann Phys 83, 491 [9] B D Serot and J D Walecka, 1997 Phys Lett B 87, 172 [10] Tran Huu Phat, Nguyen Tuan Anh, Nguyen Van Long and Le Viet Hoa, 2007 Phys Rev C 76, 045202 57 ... 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. .. 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. .. temperatures 56 Phase transition in asymmetric nuclear matter in one-loop approximation Figure The EoS for several α steps at some fixed temperatures Conclusion Due to the important role of the isospin degree