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 1Faculty of Physics, Hanoi National University of Education 2Faculty 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 $ =  [i @  MN + g + g~~ g! ! g ~~] + + 1 2 (@@ m22) 1 4 FF  + 1 2 m2!!!  1 4 ~G ~G  + 1 2 m2~~  + 1 2 @~@ ~ m2~2
+  0 ; (2.1) in which F = @! @!; ~G = @~ @~  = diag(p; n); p = B + I 2 ; n = B I 2 : Where ; ; !; ~; ~ are the field operators of the nucleon, sigma, omega, rho and delta mesons, respectively;MN = 939MeV; m = 500MeV; m! = 783MeV; m = 983MeV; m = 770MeV are the "base" mass of the nucleon, meson sigma, meson omega, meson delta and meson rho; g; g!; g; g are the coupling constants; ~ = ~=2, ~ = (1; 2; 3) are the Pauli matrices and  are the Dirac matrices. In the mean-field approximation, the ; !; ~, and ~ fields are replaced by the ground-state expectation values hi = 0; h!i = !00; hai = b3a0; hii = d3i: (2.2) Inserting (2.2) into (2.1) we arrive at L MFT =  fi @ Mp;n + 0p;ng U(0; !0; b; d); (2.3) where Mp;n = MN g0  g d 2 ; (2.4) p;n = p;n g!!0  g b 2 = p = B  I 2 g!!0  g b 2 ; (2.5) U(0; !0; b; d) = 1 2 m2 2 0 + 1 2 m2d 2 1 2 m2!! 2 0 1 2 m2b 2: (2.6) Starting with (2.3) we obtain the inverse propagator in the tree approximation S1(k;0; !0; b; d) =0BBB@ (k0+  p)Mp ~:~k 0 0 ~:~k (k0+p)Mp 0 0 0 0 (k0+  n)Mn ~:~k 0 0 ~:~k (k0+n)Mn 1CCCA ;(2.7) 126 The properties of asymmetric nuclear matter and thus detS1(k;0; !0; b; d) = (k0 + E+p )(k0 Ep )(k0 + E+n )(k0 En ); (2.8) in which E+p = E p k +   p = E p k + B + I 2 g!!0 g b 2 ; Ep = E p k p = Epk B I 2 + g!!0 + g b 2 ; E+n = E n k +   n = E n k + B I 2 g!!0 + g b 2 ; En = E n k n = Enk B + I 2 + g!!0 g b 2 ; Epk = q ~k2 +M2p ; E n k = q ~k2 +M2n : (2.9) Based on (2.6) and (2.8) the effective potential at finite temperature is derived: (0; !0; b; d; T ) = U(0; !0; b; d) + i Tr lnS 1(k;0; !0; b; d) = = U(0; !0; b; d) T 2 Z 1 0 k2dk  ln(1 + eE + p =T ) + + ln(1 + eE p =T ) + ln(1 + eE + n =T ) + ln(1 + eE n =T )  :(2.10) The ground state of nuclear matter is determined by the minimum conditions: @ @0 = 0; @ @d = 0; @ @!0 = 0; @ @b = 0; (2.11) or 0 = g m2 2 Z 1 0 k2dk  Mp Epk (n+p + n p ) + Mn Enk (n+n + n n )   g m2 (sp + sn); d = g 2m2 2 Z 1 0 k2dk  Mp Epk (n+p + n p ) Mn Enk (n+n + n n )   g 2m2 (sp sn); !0 = g! m2! 2 Z 1 0 k2dk  (np n+p ) + (nn n+n )  g! m2! (Bp + Bn); b = g 2m2 2 Z 1 0 k2dk  (np n+p ) (nn n+n )  g 2m2 (Bp Bn): (2.12) Here np;n =  eE  p;n=T + 1 1 ; sp = 1 2 Z 1 0 k2dk Mp Epk (n+p + n p ); sn = 1 2 Z 1 0 k2dk Mn Enk (n+n + n n ); Bp = 1 2 Z 1 0 k2dk(np n+p ); Bn = 1 2 Z 1 0 k2dk(nn n+n ): (2.13) 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 = jat minimum; (2.14) and introduce the isospin asymmetry :  = Bn Bp B ; (2.15) in which B = Bn + Bp is the baryon density, and Bn ; Bp are the neutron, proton densities, respectively. Combining equations (2.14), (2.4), and (2.10) together produces the following expression for the pressure P(B; ; T ) = 1 2f  MN Mp +M  n 2 2 1 2f  Mn Mp 2 + f! 2 2B+ f 8 22B + T 2 Z 1 0 k2dk  ln(1+eE + p =T ) + ln(1 + eE p =T ) + ln(1 + eE + n =T ) + ln(1 + eE n =T )  : (2.16) Here fi = g2i m2i ; (i  ; !; ; ). Based on (2.10) the entropy density is derived & = @ @T = 1 T2 Z 1 0 k2dk(E+p n + p + E p n p + E + n n + n + E n n n ) + 1 2 Z 1 0 k2dk  ln(1 + eE + p =T ) + ln(1 + eE p =T ) + ln(1 + eE + n =T ) + ln(1 + eE n =T )  : (2.17) The energy density is obtained by the Legendre transform of P: E( B ; ; T ) = + T& + pBp + nBn = 1 2f  MN Mp +M  n 2 2 + 1 2f  Mn Mp 2 + f! 2 2B+ f 8 22B + 1 2 Z 1 0 k2dk  (Epk(n + p + n p ) + E n k (n + n + n n ) : (2.18) 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 numerical study. We first fix the coupling constants fi = g2i m2i ; (i  ; !; ; ). To this end, Eq. (2.4) is solved numerically for symmetric nuclear matter (G; = 0) at T = 0. Its solution is then substituted into the nuclear binding energy Ebin = M + E=B with E given in (2.18). Two parameters f and f! are adjusted to yield the the binding energy "binjT=0 = 15:8MeV at normal density B = 0 = 0:16fm3. It is found that f = 14:49fm2 and f! = 10:97fm2. Figure 1 shows the graph of binding energy in relation to baryon density. 0 0.5 1 1.5 2 ρB/ρ0 -20 -10 0 10 20 30 E b in (M eV ) -15.8MeV f ω =10.97 fm2 Figure 1. Nuclear binding energy as a function of baryon density. As to fixing f let us follow the method developed in [5] where f is chosen as f = 0 and f = 2:5fm2. Then, f is fitted to give Esym = 1 2  @2Ebin @2  T=0; =0; B=0 = 32MeV: (2.19) It is found that f = 3:04(fm2) and f = 5:02(fm2) 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]. f f! f f Set I 14:49(fm2) 10:97(fm2) 0 3:04(fm2) Set II 14:49(fm2) 10:97(fm2) 2:5(fm2) 5:02(fm2) 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  = 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 0 1 2 3 ρB/ρ0 0.2 0.4 0.6 0.8 1 M * p, n/M N T=0 T=5 T=10 T=15 T=20 T=30 T=40 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. 0.5 1.0 1.5 -2 0 2 4 P T = 0, T = 5, T = 10, T = 15, T = 20, T = 25, Figure 3a. The equations of state for several T steps at  = 0 0.5 1.0 1.5 -1 0 1 2 3 4 P T =0, T =5, T =10, T =15, T =20, T =25, Figure 3b. The equations of state for several T steps at  = 0:25 0.5 1.0 -1 0 1 2 3 P T = 0, T = 5, T = 10, T = 15, T = 20, T = 25, Figure 4a. The equations of state for several T steps at  = 0:5 0.5 1.0 0 5 10 P T = 0, T = 5, T = 10, T = 15, T = 20, T = 25, Figure 4b. The equations of state for several T steps at  = 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. 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