Variational study of asymmetric nuclear matter and a new term in the mass formula

Asymmetric nuclear matter at zero temperature is studied using a variational method which is an extension of the methods used by the present authors previously for simpler systems. An approximate expression for the energy per nucleon in asymmetric nuclear matter is derived through a combination of two procedures, one used for symmetric nuclear matter and the other for spin-polarized liquid ³He with spin polarization replaced by isospin polarization. The approximate expression for the energy is obtained as a functional of various spin-isospin-dependent radial distribution functions, tensor distribution functions, and spin-orbit distribution functions. The Euler-Lagrange equations are derived to minimize this approximate expression for the energy; they consist of 16 coupled integrodifferential equations for various distribution functions. These equations were solved numerically for several values of the nucleon number density ρ and for many degrees of asymmetry ζ[ζ = (ρ_n - ρ_p)/ρ, where ρ_n(ρ_p) is the neutron (proton) number density]. Unexpectedly, we find that the energies at a fixed density cannot be represented by a power series in ζ². A new energy term, ε₁ (ζ² + ζ₀²)^1/2i where ζ₀ is a small number and ε₁ is a positive coefficient, is proposed. It is shown that if the power series is supplemented with this new term, it reproduces the energies obtained by variational calculations very accurately. This new term is studied in relation to cluster formation in nuclear matter, and some mention is made of a possible similar term in the mass formula for finite nuclei.