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

For the finite-range term, we use a Gaussian phenomenological model for the nuclear potential,E FR = ∑ ∫1[]d 3 rπ 3/2 r 3 1 d 3 r 2 e −r2 12 /r2 0V 1L ρ t (r 1 )ρ t (r 2 )+V 1U ρ t (r 1 )ρ t ′(r 2 )t 0+ ∑ ∫1[]d 3 rπ 3/2 r 3 1 d 3 r 2 e −r2 12 /r2 0V 2L ρ 1+ǫt (r 1 )ρ 1+ǫt (r 2 )+V 2U ρ 1+ǫt (r 1 )ρ 1+ǫt(r ′ 2 )t 0+ ∑ ∫(4.5)1d 3 rπ 3/2 r 3 1 d 3 r 2 e −r2 12 /r2 0 ×t 0[∫∫]d 3 p t1 d 3 p t2 f t1 f t2 V 3L p 2 12 + d 3 p t1 d 3 p t ′ 2f t1 f t ′ 2V 3U p 2 12 ,where p 12 = |p 1 −p 2 |, r 12 = |r 1 −r 2 |, r 0 is the length of interaction, and V 1L , V 1U , V 2L , V 2U ,V 3L , and V 3U are interaction parameters to be determined. The last term is added to explainthe effective mass of nucleons in dense matter.The zero-range term in the nuclear force can be regarded as the energy contribution fromthree-body nuclear forces. The three-body force is quite important if the baryon densityincreases beyond two or three times the saturation density. One possible form of the threebodyforce is [34]E ZR = 1 ∫4 t 3 d 3 rρ n (r)ρ p (r)ρ(r), (4.6)where t 3 istheinteractionstrengthforathree-bodyforce, ρ n (ρ p )isneutron(proton)density,and the ρ is total density.The energy functional for the Coulomb interaction has an exchange term which is absent inclassical physics,E C = E pp∫ ∫= e22C +Eex Cd 3 r 1 d 3 r 2ρ p (r 1 )ρ p (r 2 )r 12− 34π (3π2 ) 1/3 e 2 ∫d 3 rρ 4/3p (r).(4.7)In bulk nuclear matter, the spin-orbit and Coulomb interactions constitute a small portionof the total energy, so we neglect these two terms. Then the bulk density functional wouldbeE B = T kin +E FR +E ZR∫(4.8)= d 3 rE B (r),53