Nuclear charge and neutron radii and nuclear matter: Trend analysis in Skyrme density-functional-theory approach

Background: Radii of charge and neutron distributions are fundamental nuclear properties. They depend on both nuclear interaction parameters related to the equation of state of infinite nuclear matter and on quantal shell effects, which are strongly impacted by the presence of nuclear surface.

Purpose: In this work, by studying the correlation of charge and neutron radii, and neutron skin, with nuclear matter parameters, we assess different mechanisms that drive nuclear sizes.

Method: We apply nuclear density functional theory using a family of Skyrme functionals obtained by means of optimization protocols, which do not include any radius information. By performing the Monte Carlo sampling of reasonable functionals around the optimal parametrization, we scan all correlations between nuclear matter properties and observables characterizing charge and neutron distributions of spherical closed-shell nuclei ${}^{48}\mathrm{Ca},{}^{208}\mathrm{Pb}$, and ${}^{298}\mathrm{Fl}$.

Results: By considering the influence of various nuclear matter properties on charge and neutron radii in a multidimensional parameter space of Skyrme functionals, we demonstrate the existence of two strong relationships: (i) between the nuclear charge radii and the saturation density of symmetric nuclear matter ${\rho }_0$, and (ii) between the neutron skins and the slope of the symmetry energy $L$. The impact of other nuclear matter properties on nuclear radii is weak or nonexistent. For functionals optimized to experimental binding energies only, proton and neutron radii are found to be weakly correlated due to canceling trends from different nuclear matter characteristics.

Conclusion: The existence of only two strong relations connecting nuclear radii with nuclear matter properties has important consequences. First, by requiring that the nuclear functional reproduces the empirical saturation point of symmetric nuclear matter practically fixes the charge (or proton) radii, and vice versa. This explains the recent results of ab initio calculations with two-nucleon and three-nucleon forces optimized simultaneously to binding energies and radii of selected nuclei. Second, since the neutron skin uncertainty primarily depends on the slope of the symmetry energy, imposing constraints on both ${\rho }_0$ and $L$ practically determines the nuclear size, modulo small variations due to shell effects.

Figure 1

Covariance matrices for a selection of observables and NMP computed with SV-min [36] (below the diagonal) and SV-E [31, 35] (above the diagonal). Nuclear observables (in ${}^{208}\mathrm{Pb})$ are charge surface thickness ${\sigma }_{\mathrm{ch}}$; root-mean-square (rms) charge radius $r_{\mathrm{ch}}$; rms neutron radius $r_{\mathrm{n}}$; neutron skin $r_{\mathrm{skin}}$; electric dipole polarizability ${\alpha }_{\mathrm{D}}$; and giant resonance energies $E_{\mathrm{GMR}},E_{\mathrm{GDR}}$, and $E_{\mathrm{GQR}}$. The NMP corresponding to the symmetric nuclear matter include incompressibility $K$, symmetry energy $J$, symmetry energy slope $L$, isoscalar effective mass $m^{*}/m$; TRK sum-rule enhancement ${\kappa }_{\mathrm{TRK}}$; and density ${\rho }_0$ and energy $E/A$ at the saturation point. The correlations with the charge form factor data ${\sigma }_{\mathrm{ch}}$ and $r_{\mathrm{ch}}$ are not shown for SV-min as these quantities were included in the correspondins set of fit observables.

Figure 2

Variance ellipsoids with SV-E in the $\left(r_{\mathrm{ch}},r_{\mathrm{n}}\right)$ plane for ${}^{298}\mathrm{Fl}$ (a) and ${}^{48}\mathrm{Ca}$ (b). The arrows and segments indicate the direction of changing radii when varying one NMP as indicated. Their lengths represent the magnitude of corresponding variations. The narrow ellipsoids mark the SV-min results. The correlation coefficient $c_{AB}$ between $r_{\mathrm{ch}}$ and $r_{\mathrm{n}}$ is indicated in both cases.

Figure 3

Proton (red) and neutron (green) rms radii in ${}^{208}\mathrm{Pb}$ with respect to the SV-E values from the ensemble of 2000 parametrizations in the vicinity of the optimal fit SV-E drawn vs different NMP: ${\rho }_0$ (a), $K$ (b), $m^{*}/m$ (c), and ${\kappa }_{\mathrm{TRK}}$ (d).

Figure 4

Similar as in Fig. 3 but vs ${\rho }_0$ (left) and $L$ (right) for ${}^{298}\mathrm{Fl}$ (top), ${}^{208}\mathrm{Pb}$ (middle), and ${}^{48}\mathrm{Ca}$ (bottom). To illustrate the trends, the right panels show also averages and variances of radii taken over bins in $L$.

Figure 5

Neutron skins from the ensemble of 2000 parametrizations in the vicinity of the optimal fit SV-E vs $J$ (left) and $L$ (right) for the three nuclei under consideration.

Figure 6

Uncertainties in the predictions of rms neutron and charge radii (bottom), neutron skins (middle), and surface thicknesses (top) for different EDF fits. The reference EDF is SV-E. Other EDF fits use the same pool of fit observables as SV-E but constrain one or two NMP, as indicated, at the SV-E values.