Central depression in nucleonic densities: Trend analysis in the nuclear density functional theory approach

Background: The central depression of nucleonic density, i.e., a reduction of density in the nuclear interior, has been attributed to many factors. For instance, bubble structures in superheavy nuclei are believed to be due to the electrostatic repulsion. In light nuclei, the mechanism behind the density reduction in the interior has been discussed in terms of shell effects associated with occupations of $s$ orbits.

Purpose: The main objective of this work is to reveal mechanisms behind the formation of central depression in nucleonic densities in light and heavy nuclei. To this end, we introduce several measures of the internal nucleonic density. Through the statistical analysis, we study the information content of these measures with respect to nuclear matter properties.

Method: We apply nuclear density functional theory with Skyrme functionals. Using the statistical tools of linear least square regression, we inspect correlations between various measures of central depression and model parameters, including nuclear matter properties. We study bivariate correlations with selected quantities as well as multiple correlations with groups of parameters. Detailed correlation analysis is carried out for ${}^{34}\mathrm{Si}$ for which a bubble structure has been reported recently, ${}^{48}\mathrm{Ca}$, and $N=82$, 126, and 184 isotonic chains.

Results: We show that the central depression in medium-mass nuclei is very sensitive to shell effects, whereas for superheavy systems it is firmly driven by the electrostatic repulsion. An appreciable semibubble structure in proton density is predicted for ${}^{294}\mathrm{Og}$, which is currently the heaviest nucleus known experimentally.

Conclusion: Our correlation analysis reveals that the central density indicators in nuclei below ${}^{208}\mathrm{Pb}$ carry little information on parameters of nuclear matter; they are predominantly driven by shell structure. On the other hand, in the superheavy nuclei there exists a clear relationship between the central nucleonic density and symmetry energy.

Figure 1

Top: proton (left) and neutron (right) densities of ${}^{48}\mathrm{Ca}$ and ${}^{302}\mathrm{Og}$, normalized to ${\rho }_{\max }$, calculated with SV-min in the ($x,z$) plane at $y=0$. The densities are displayed in a $20\mathrm{fm}\times 20$ fm box. Bottom: proton (left) and neutron (right) densities of ${}^{34}\mathrm{Si}$, ${}^{48}\mathrm{Ca}$, ${}^{208}\mathrm{Pb}$, ${}^{302}\mathrm{Og}$, and ${}^{472}164$ obtained with SV-min as functions of $r$. The shaded areas mark the spread of results obtained with SV-min, SLy6, and UNEDF1.

Figure 2

Neutron (left) and proton (right) densities of ${}^{294}\mathrm{Og}$ (top) and ${}^{326}\mathrm{Og}$ (bottom) calculated with SV-min in the ($x,z$) plane at $y=0$. The densities, normalized to ${\rho }_{\max }$, are displayed in a $20\mathrm{fm}\times 20$ fm box.

Figure 3

Central proton depression ${\overline{w}}_p$ (a), central density ${\rho }_{t,\mathrm{c}}$ (b), and CoD between the Coulomb energy $E_{\mathrm{Coul}}$ and central proton density ${\rho }_{p,\mathrm{c}}$ (c) for ${}^{34}\mathrm{Si}$, ${}^{48}\mathrm{Ca}$, and $N=82$, 126, and 184 isotonic chains predicted by SV-min. The $\times$ symbol marks the values of $\overline{w}$ and ${\rho }_{t,\mathrm{c}}$ in ${}^{34}\mathrm{Si}$ obtained without pairing.

Figure 4

Matrices of coefficients of determination for SV-min parameters and selected observables characterizing central densities in ${}^{48}\mathrm{Ca}$ (upper triangle) and ${}^{302}\mathrm{Og}$ (lower triangle).

Figure 5

Multiple correlation coefficients with $E_{\mathrm{Coul}}$ (a) and ${\overline{\rho }}_{p,\mathrm{c}}$ (b) in heavy nuclei with four groups of SV-min parameters: LDM ($E/A$, ${\rho }_{\mathrm{eq}}$, $K$, $J$, $L$, $a_{\mathrm{surf}}$, $a_{\mathrm{surf},\mathrm{s}}$); bulk ($E/A$, ${\rho }_{\mathrm{eq}}$, $K$, $J$, $L$); sym ($J$, $L$); and ls+pair (spin-orbit parameters $C_0^{\rho \nabla J}$ and $C_1^{\rho \nabla J}$ and pairing parameters $V_{\mathrm{pair},n},V_{\mathrm{pair},p}$, and ${\rho }_{\mathrm{pair}}$). The surface constants $a_{\mathrm{surf}}$ and $a_{\mathrm{surf},\mathrm{s}}$ are defined as in Ref. [66]. For other parameters, see Refs. [52, 65].