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

specific heat) of the surface of symmetric mater. This description is given below.What appears in the liquid droplet model, however, is the surface free energy per unitarea σ+µ n ν n +µ p ν p , where ν t represent the areal density of particles in the surface. It willbe found that these are determined byν t = − ∂σ∂µ t∣ ∣∣T, (6.18)as is expected from thermodynamics. Since our liquid droplet model takes the referenceinterface to be located at the proton surface, this model explicitly considers a skin madeonly of neutrons. There are no ‘surface protons’, and ν p = 0 . This is consistent withthe statement above that, at a given temperature, the quantity σ is a function of a singlequantity, µ n . Therefore, the surface free energy per area is simply σ(µ n ,T)+µ n ν n . In whatfollows, the neutron chemical potential for the surface is denoted by µ s , but it will be shownthat free energy minimization requires µ s = µ no .The surface free energy density for spherical nuclei is thusf surf = 4πr 2 Nρ n (σ(µ s ,T)+µ s ν n ) = 3ur N(σ(µ s ,T)+µ s ν n ). (6.19)Once again, following the discussion of Lattimer & Swesty [22], the consequences ofnuclear deformation or shape change are taken into account by the surface shape factor s(u),so thatf surf = 3s(u)r N(σ(µ s ,T)+µ s ν n ). (6.20)Obviously, s(u) = u for a sphere. As is the case for c(u), the precise form for s(u) will berequired.In practice, the results of plane-parallel surface energy calculations are most easily renderedin terms of the functions σ(x,T) and µ s (x,T), where x is the reference asymptoticproton fraction of the dense phase. Note that x is equivalent to x i for a plane-parallelsurface, but is not necessarily equivalent to x i for a finite liquid droplet nucleus. The quantityx is simply a convenient intermediate variable used to simplify the construction of σ(µ s ).As in section 3.4, we write the surface tension as a two-parameter function times atemperature dependence:( )2·2 α +qσ(x,T) = σ o h(x,T). (6.21)x −α +q +(1−x) −αThe quantity σ o is the zero-temperature, symmetric matter, surface tension. In case ofSkyrme force model, the numerical calculation of surface tension is not stable since it is adifferential equation rather than an integral equation. The numerical method is describedin Ravenhall et al. [29].92