Toward a global description of nuclear charge radii: Exploring the Fayans energy density functional

Background: Binding energies and charge radii are fundamental properties of atomic nuclei. When inspecting their particle-number dependence, both quantities exhibit pronounced odd-even staggering. While the odd-even effect in binding energy can be attributed to nucleonic pairing, the origin of staggering in charge radii is less straightforward to ascertain.

Purpose: In this work, we study the odd-even effect in binding energies and charge radii, and systematic behavior of differential radii, to identify the underlying components of the effective nuclear interaction.

Method: We apply nuclear density functional theory using a family of Fayans and Skyrme energy density functionals fitted to similar data sets but using different optimization protocols. We inspect various correlations between differential charge radii, odd-even staggering in energies and radii, and nuclear matter properties. The Fayans functional is assumed to be in the local ${\mathrm{FaNDF}}^0$ form. Detailed analysis is carried out for medium-mass and heavy semimagic nuclei with a particular focus on the Ca chain.

Results: By making the surface and pairing terms dependent on density gradients, the Fayans functional offers the superb simultaneous description of odd-even staggering effects in energies and charge radii. Conversely, when the data on differential radii are added to the pool of fit observables, the coupling constants determining the strengths of the gradient terms of Fayans functional are increased by orders of magnitude. The Skyrme functional optimized in this work with the generalized Fayans pairing term offers results of similar quality. We quantify these findings by performing correlation analysis based on the statistical linear regression technique. The nuclear matter parameters characterizing Fayans and Skyrme functionals optimized to similar data sets are fairly close.

Conclusion: The Fayans paring functional, with its generalized density dependence, significantly improves the description of charge radii in odd and even nuclei. Adding differential charge radii to the set of fit observables in the optimization protocol is helpful for both description of radii and for improving the pairing functional. In particular, the Fayans functional ${\mathrm{FaNDF}}^0$ constrained in this way is capable of explaining charge radii in the even-even Ca isotopes. However, in order to obtain good description of differential radii data in both medium-mass and heavy nuclei, an $A$-dependent scaling of the Fayans pairing functional is still needed. Various extensions of the current model are envisioned that carry out a promise for the global description.

Figure 1

Quality measure ${\chi }^2$ (top) and odd-even binding-energy staggering ${\mathrm{\Delta }}_E^{\left(3\right)}$ for ${}^{204}\mathrm{Pb}$ (bottom) and as functions of the $\gamma$ parameter of the Fayans pairing functional for Fy(std), $\mathrm{Fy}(\mathrm{\Delta }r)$, $\mathrm{Fy}(h_{\mathbf{\nabla }}^s)$, and Sk+PFy.

Figure 2

Summary of global performance of Fayans and Skyrme parametrizations used/optimized in this work; for naming convention see Sec. 2f. Top: average and r.m.s. deviations (21) for selected observables of the fit. Middle: Differential charge radii in Ca. Bottom: odd-even staggering of charge radii and binding energies in Ca isotopes.

Figure 3

Three-point binding energy difference ${\mathrm{\Delta }}_E^{\mathrm{ee}}$ (multiplied by the mass number to easily compare different systems) for the spherical open-shell nuclei ${}^{44}\mathrm{Ca},{}^{124}\mathrm{Sn},{}^{204}\mathrm{Pb}$, and ${}^{212}\mathrm{Pb}$ computed with selected Fayans and Skyrme functionals and compared to experiment.

Figure 4

Differential charge radii $\delta {\langle r^2\rangle }^{48,A}$ (15) in Ca predicted by Fayans and Skyrme parametrizations as indicated, and compared to experiment.

Figure 5

Predicted mean values and variances of three-point neutron differences ${\mathrm{\Delta }}_r^{\left(3\right)}$ (top) and ${\mathrm{\Delta }}_E^{\left(3\right)}$ (bottom) in ${}^{44}\mathrm{Ca},{}^{64}\mathrm{Ni},{}^{124}\mathrm{Sn}$, and ${}^{204}\mathrm{Pb}$, predicted by selected Fayans and Skyrme functionals, and compared to experiment.

Figure 6

Neutron pairing densities ${\breve{\rho }}_n$ in spherical configurations of ${}^{44}\mathrm{Ca},{}^{124}\mathrm{Sn},{}^{204}\mathrm{Pb}$, and ${}^{214}\mathrm{U}$, predicted by selected Fayans and Skyrme functionals.

Figure 7

Proton HF potential for ${}^{44}\mathrm{Ca}$ (top), ${}^{48}\mathrm{Ca}$ (middle) and ${}^{52}\mathrm{Ca}$ (bottom) obtained in Fy(std), $\mathrm{Fy}(\mathrm{\Delta }r)$, and ${\mathrm{FaNDF}}^0$.

Figure 8

Charge form factor characteristics, ${\sigma }_{\mathrm{ch}},R_{\mathrm{diff}}$, and $r_{\mathrm{ch}}$, along the chain of Ca isotopes for Fy(std), $\mathrm{Fy}(\mathrm{\Delta }r)$, and ${\mathrm{FaNDF}}^0$. Experimental data are taken from Ref. [38].

Figure 9

Matrices of coefficients of determination $r_{AB}^2$ for a selection of observables computed with the Fayans functionals Fy(nogap) (top triangle) and $\mathrm{Fy}(\mathrm{\Delta }r)$ (bottom triangle).

Figure 10

Similar as in Fig. 9 except for the Skyrme parametrizations SV-min (top triangle) and Sk-PFy (bottom triangle).