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

is expected to be small, |Q 1 | ≤ 1 MeV. In our case,[ ( 3Q = T o5 u2/3 2 2/3 − 14 ) [ β′′+(29 18 + 2/3 − 5 ) ]β L′′ u 5/33[ ( η+18 + 2 1/3 − 4 ] (3.10))η L]u 1+ǫ3Thus, for S v = 30 MeV and S v ′ = 15 MeV, Q 1 ≃ −6.5 MeV. Therefore, in general, thenon-quadratic terms involving β L ′′ and η L lead to neutron matter energy and pressure thatare not well-behaved at higher densities in comparison to symmetric energies and pressures.Another possibility is to require that Q 1 ≈ 0, specificallyβ(2 ′′18 + 2/3 − 5 )β L ′′ + η (3 18 + 2 1/3 − 4 )η L = 0. (3.11)3Using Eqs.(3.8) and (3.11), we can evaluate α L = −4.0, β L ′′ = −1.4, and η L = 4.6 usingǫ = 1/3, S v = 30 MeV, and S v ′ = 15 MeV. Once again, the negative value of β′′ L rendersneutron matter unstable at high densities.3.2 Alternate ModificationMost Skyrme forces explicitly incorporate the η terms with a quadratic x dependence sothat they, unlike the α terms, vanish in Eq. (3.10), and the surviving β L ′′ term can be madesmall. It will be necessary for us to formulate a force with similar properties. One way todo this is to make the extra terms in C L,U functionals of ρ 1 only (or ρ 2 ). For example,C ηL = − h34 T oρ o( 34πP 3 oC ηU = − h34 T oρ o( 34πP 3 o) 2 ( ) ǫ ( ) 2 ρǫη 1 +ρ ǫ 2Lρ o 2) 2 ( ) ǫ ( ) (3.12)2 ρǫη 1 +ρ ǫ 2Uρ o 2Then,We find∆E = 1 ∫T o2ρ o= T oρ o∫d 3 r 1 d 3 r 2 f(r 12 /a)×[(ρ1) ǫ+ρ o( ) ǫ ρ ∑d 3 rρ o(ρ2∆E = T ( ) ǫo ρ ∑ρ o ρ otρ o) ǫ ] ∑tρ t[η L˜ρ t +η U ρ˜t ′tρ t1[η L ρ t2 +η U ρ t ′ 2].](3.13)]ρ t[η L˜ρ t +η U ρ˜t ′ , (3.14)42