Elucidating the role of the surface energy in density functional theory

Abstract

The saturation of symmetric nuclear matter -- reflected in the nearly constant interior density of heavy nuclei -- is a defining property of nuclear matter. Modern relativistic energy density functionals (EDFs) calibrated exclusively to the properties of finite nuclei, make robust predictions with quantified uncertainties about the bulk properties of symmetric nuclear matter in the vicinity of the saturation density. Following the same fitting protocol, nonrelativistic Skyrme EDFs systematically predict higher saturation densities than their relativistic counterparts. To investigate this tension in the bulk limit, we study the ground-state properties of hypothetical symmetric macroscopic nuclei containing thousands of nucleons. Using both relativistic and non-relativistic EDF frameworks, we extract the corresponding liquid-drop parameters. We find a clear correlation between the volume and surface energy coefficients: Skyrme models, which saturate at higher densities, develop softer and more diffuse surfaces with lower surface energies, whereas relativistic EDFs, which saturate at lower densities, produce more defined and less diffuse surfaces with higher surface energies. This compensating behavior allows both classes of models to reproduce empirical nuclear radii despite their distinct saturation properties. Our analysis suggests that the apparent disparity in saturation densities arises from the intrinsic balance among saturation density, bulk binding energy, and surface tension, rather than from the fitting protocol.

Figures (5)

• Figure 1: (a) Binding energy per nucleon of symmetric nuclear matter as predicted by the five covariant energy density functionals used in this work. The small circle in the figure denotes the saturation point; see Table \ref{['Table3']}. (b) Predictions from the same five models for the density dependence of the symmetry energy. The small circle in the figure indicates that theoretical uncertainties in the value of the symmetry energy are minimized at a density $\rho\!\approx\!(2/3)\rho_{\raisebox{-1.0pt}{\tiny!0}}$. Panels (c) and (d) are the corresponding predictions from the five Skyrme EDFs.
• Figure 2: Energy per nucleon for a collection of symmetric nuclei (without Coulomb) fitted to a semi-empirical mass formula $\mathlarger{\varepsilon}=a_{\rm v}+a_{\rm s}A^{-1/3}$. Predictions are displayed with both Skyrme and FSUGarnet energy density functionals.
• Figure 3: Corner plot displaying the probability distribution and correlation coefficients for three empirical parameters of the liquid-drop formula as obtained via a Metropolis-Monte-Carlo method. The solid line and the associated labels represent the results obtained assuming a normal distribution. Predictions are displayed with the SLy4 (in blue) and FSUGarnet (in garnet) energy density functionals.
• Figure 4: The characteristic $A^{1/3}$ scaling of nuclear radii, illustrating a direct manifestation of nuclear saturation. Panels (a) and (b) display the results obtained with the SLy4 and FSUGarnet functionals, respectively, for a collection of symmetric nuclei spanning the $A\!=\!250$-$3500$ range.
• Figure 5: Baryon density as predicted by FSUGarnet and SLy4 for hypothetical symmetric nuclei with $A\!=\!252, 1500, 2504$.