Constraints on the nuclear equation of state from nuclear masses and radii in a Thomas-Fermi meta-modeling approach

The question of correlations among empirical equation of state (EoS) parameters constrained by nuclear observables is addressed in a Thomas-Fermi meta-modeling approach. A recently proposed meta-modeling for the nuclear EoS in nuclear matter is augmented with a single finite size term to produce a minimal unified EoS functional able to describe the smooth part of the nuclear ground state properties. This meta-model can reproduce the predictions of a large variety of models, and interpolate continuously between them. An analytical approximation to the full Thomas-Fermi integrals is further proposed giving a fully analytical meta-model for nuclear masses. The parameter space is sampled and filtered through the constraint of nuclear mass reproduction with Bayesian statistical tools. We show that this simple analytical meta-modeling has a predictive power on masses, radii, and skins comparable to full Hartree-Fock or extended Thomas-Fermi calculations with realistic energy functionals. The covariance analysis on the posterior distribution shows that no physical correlation is present between the different EoS parameters. Concerning nuclear observables, a strong correlation between the slope of the symmetry energy and the neutron skin is observed, in agreement with previous studies.

Figure 1

Difference between theoretical and experimental energy per particle of symmetric nuclei for the reference analytical EoS, in the case where a single surface parameter $C_{\mathrm{fin}}$ is added to the empirical set (filled circles), and in the case where a spin-orbit coupling $C_{so}$ is also added (open circles). The result for the full ETF using an optimized EoS (filled squares) and spherical Hartree-Fock results using the Sly4 [45] parameter set (open squares) are also given.

Figure 2

Difference between theoretical and experimental energy per particle vs asymmetry $I=(N-Z)/A$ for different $Z$ values: 20 (Ca), 28 (Ni), 50 (Sn), and 82 (Pb), for the reference empirical EoS (filled circles). The result for the full ETF mass model using an optimized EoS (filled squares) and spherical H-F results with the Sly4 parameter set (open squares) are also given.

Figure 3

Difference between theoretical and experimental binding energy per particle, for values of empirical parameters varied between $\pm \sigma$. (a) Isoscalar parameters for symmetric nuclei, as a function of $A$. (b) Isovector parameters for the Sn isotopic chain, as a function of $(N-Z)/A$.

Figure 4

Effect of cutoff in $\mathrm{\Delta }$ on the average (central purple line) and standard deviation (band width) for two different observables—the ${}^{208}\mathrm{Pb}$ binding energy per nucleon (left) and its neutron radius (center)—and the $n_{\mathrm{sat}}$ empirical parameter (right). In each panel, the inset shows the same for ${\mathrm{\Delta }}_{\mathrm{cutoff}}<0.5$.

Figure 5

Variance of the posterior distribution $p_{\mathrm{\Delta }}$ of the finite size $C_{\mathrm{fin}}$ parameter as a function of the cut-off (see text).

Figure 6

Posterior correlation matrix for empirical parameters and nuclear observables, after application of the mass filter.

Figure 7

rms charge radii vs $I$ for semimagic isotopic chains for different $Z$ values (a) 20, (b) 28, (c) 50, and (d) 82. Symbols with error bars: experimental data. Bands: predictions at $1\sigma$ (and $2\sigma$) for the empirical EoS filtered through the binding energy constraint. Lines with filled squares: prediction from the full variational ETF with the optimized empirical parameter set.

Figure 8

Neutron skin as a function of global asymmetry for various nuclei. Symbols with error bars: experimental data. Bands: predictions at $1\sigma$ (and $2\sigma$) for the empirical EoS filtered through binding the energy constraint. Lines with filled squares: prediction from the full variational ETF with the optimized empirical parameter set.