Theory of Nuclear Matter for Neutron Stars and ... - Graduate Physics

The energy exchange from the gradient has the formβ = D pp +2D np ξ +D nn ξ 2 , ξ = − ∂µ p/∂ρ n∂µ n /∂ρ n(4.35)where the coefficients of the gradient terms are given by D pp = D np = D nn = 132MeV·fm 5[18]. The k TF in the Coulomb interaction represents the inverse Thomas-Fermi screeninglength of the electrons. When we see the change in the sign of v, the uniform matter phaseis more stable than the periodic structure of the nuclei. The v has a minimum atwhen q 2 min = (4πe2 /β) 1/2 −k 2 TF .v min = v 0 +2(4πe 2 β) 1/2 −βk 2 TF , (4.36)Another way to see the phase transition is to use the thermodynamic instability. Thethermodynamic stability condition can be described using the inequalities [18, 19],( ) ∂P− >0,∂vµ( (4.37)∂µ− >0.∂q c)vwhere P = P b + P e is the total pressure from electrons and baryons and µ = µ n − µ p isthe difference between the neutron and proton chemical potentials, which is the electronchemical potential in beta-stable matter. q c is defined as q c = x p −ρ e /ρ. Mathematically,the inequalities in Eq. (4.37) show that the energy per baryon is convex. Eq. (4.37) can beverified to be [18, 19]( ) [ ∂P− = ρ 2 2ρ ∂E(ρ,x (p)+ρ 2∂2 E(ρ,x p ) ∂ 2 E(ρ,x p )− ρ∂vµ∂ρ ∂ρ 2 ∂ρ∂x p( ∂µ− =∂q c)v( ∂ 2 E(ρ,x p )∂x 2 p) 2/ ] ∂ 2 E(ρ,x p )> 0,∂x 2 p) −1+ µ2 eπ 2 3 ρ > 0. (4.38)The second of Eq. (4.38) always holds, so the first will determine the phase transition in theneutron star crust. Xu et al.[18], use a simple equation to determine the instability usingthe thermodynamic relation,2∂E∂ 2 Eρ ∂ρ ∂x 2 p+ ∂2 E∂ρ 2 ∂ 2 E∂x 2 p−( ) ∂ 2 2E= ∂µ ( ) 2n ∂µ p ∂µn− . (4.39)∂ρ∂x p ∂ρ n ∂ρ p ∂ρ pEq. (4.39) is equivalent to the volume part of the thermodynamic perturbation Eq. (4.34)method except that there is a ∂µ n /∂ρ n difference. Comparing the two methods (pertubationand thermodynamic instability) shows the effects of the gradient and Coulomb terms in theperturbation method on the transition densities. Fig. 4.7 shows transition densities using65