Physics Overview

The MetaModel (MM) allows you to compute Equations of State (EOS) in \((n_B, \delta)\) space, where \(n_B\) is the baryon number density and \(\delta\) is the isospin asymmetry. The model requires specifying the saturation (or bulk) properties of nuclear matter, the nucleon mass and the splitting at saturation, and low density corrections described below.

The energy expansion around saturation density and symmetric nuclear matter is given by:

\[E(n_B, \delta) = E_{sat}(n_B) + E_{sym}(n_B) \delta^2 + O(\delta^4)\]

with

\[E_{sat}(n_B) = BE + \frac{x^2}{2!} K_0 + \frac{x^3}{3!} Q_0 + \frac{x^4}{4!} Z_0\]

and

\[E_{sym}(n_B) = E_{sym} + L_{sym} x + \frac{x^2}{2!} K_{sym} + \frac{x^3}{3!} Q_{sym} + \frac{x^4}{4!} Z_{sym}\]

where \(x = (n_B - n_0)/(3 n_0)\), and \(n_0\) is the saturation density.

The MetaModel computes the energy per particle from a kinetic term and a potential term, as:

\[E(n_B, \delta) = t_{kin}(n_B, \delta) + v_{pot}(n_B, \delta)\]

with

\[v_{pot}(n_B, \delta) = \sum_{n=0}^{N} \frac{1}{n!} (v_{sat,i} + v_{sym,i} \delta^2) x^n u_N(x)\]

where \(u_N(x) = 1 - (-3x)^{N+1-n} \exp(-b \, n_B/n_0)\) is a low density correction to assure \(e(n_B\rightarrow 0) = 0\).

The kinetic term is given by:

\[t_{kin}(n_B, \delta) = \frac{t_{sat}^{FG}}{2} (\frac{n_B}{n_0})^{2/3} [ (1 + \kappa_{sat} \frac{n_B}{n_0}) f_1(\delta) + \kappa_{sym} \frac{n_B}{n_0} f_2(\delta)]\]

where \(t_{sat}^{FG} = 3\hbar^2/(10m) (3\pi^2/2)^{2/3} n_0^{2/3}\) is the kinetic energy of a free Fermi gas at saturation density, and \(\kappa_{sat}\) is the isoscalar effective mass splitting at saturation. The functions \(f_1(\delta) = (1+\delta)^{5/3} + (1-\delta)^{5/3}\) and \(f_2(\delta) = \delta [ (1+\delta)^{5/3} - (1-\delta)^{5/3}]\) account for the isospin asymmetry.

The parameters \(v_{sat,i}\) and \(v_{sym,i}\) are determined from the empirical parameters through analytical relations, by matching the metamodel expansion to the expansion in bulk parameteres, as described in [Margueron2018].