Nuclear energy density optimization: Shell structure

Background: Nuclear density functional theory is the only microscopical theory that can be applied throughout the entire nuclear landscape. Its key ingredient is the energy density functional.

Purpose: In this work, we propose a new parametrization unedf2 of the Skyrme energy density functional.

Methods: The functional optimization is carried out using the pounders optimization algorithm within the framework of the Skyrme Hartree-Fock-Bogoliubov theory. Compared to the previous parametrization unedf1, restrictions on the tensor term of the energy density have been lifted, yielding a very general form of the energy density functional up to second order in derivatives of the one-body density matrix. In order to impose constraints on all the parameters of the functional, selected data on single-particle splittings in spherical doubly-magic nuclei have been included into the experimental dataset.

Results: The agreement with both bulk and spectroscopic nuclear properties achieved by the resulting unedf2 parametrization is comparable with unedf1. While there is a small improvement on single-particle spectra and binding energies of closed shell nuclei, the reproduction of fission barriers and fission isomer excitation energies has degraded. As compared to previous unedf parametrizations, the parameter confidence interval for unedf2 is narrower. In particular, our results overlap well with those obtained in previous systematic studies of the spin-orbit and tensor terms.

Conclusions: unedf2 can be viewed as an all-around Skyrme EDF that performs reasonably well for both global nuclear properties and shell structure. However, after adding new data aiming to better constrain the nuclear functional, its quality has improved only marginally. These results suggest that the standard Skyrme energy density has reached its limits, and significant changes to the form of the functional are needed.

Figure 1

Sensitivity of the unedf2 parametrization to different data types: masses, charge radii, OES, fission isomer excitation energies (FI), and s.p. energies.

Figure 2

Overall change of the unedf2 parametrization when data point $d_{i,j}$ is changed (in isolation) by $0.1w_i$.

Figure 3

Position of the lowest critical densities for $S=0,M=0$. Solid line: isoscalar $T=0$ channel; dashed line: isovector $T=1$ channel. The horizontal dashed line shows the saturation density.

Figure 4

Position of the lowest critical densities for $S=0$ in pure neutron matter. The horizontal dashed line shows the saturation density.

Figure 5

Absolute values of correlations between various observables for (a) SV-min and (b) unedf2. The expectation values of individual observables and their uncertainties are listed in Table 7.

Figure 6

The E1 strength distribution for ${}^{208}$Pb computed with unedf2.

Figure 7

Neutron single-particle energies in ${}^{48}$Ca calculated with the unedf0 (UN0), unedf1 (UN1), and unedf2 (UN2) parametrizations of the Skyrme energy density. These are compared with the empirical values (Exp) of Ref. [53].

Figure 8

Same as Fig. 7 but for proton single-particle energies in ${}^{208}$Pb.

Figure 9

Same as Fig. 8 but for neutron single-particle energies in ${}^{208}$Pb.

Figure 10

The residuals of nuclear binding energies of even-even nuclei calculated with unedf2. Panel (a) shows isotopic chains, panel (b) the isotonic chains.

Figure 11

The residuals of (a) $S_{2n}$ and (b) $S_{2p}$ obtained in unedf2 for even-even nuclei.

Figure 12

The residuals of the inner fission barriers, $\Delta E_A$, panels (a)–(d); fission isomer excitation energies, $\Delta E_{\mathrm{II}}$, panels (e)–(h); and outer fission barriers, $\Delta E_B$, panels (i)–(l), for various actinide nuclei. Residuals are defined as the difference between the computed values with unedf2, unedf1, D1S, and FRLDM models and the empirical values [83, 84]. The shaded area represents an average experimental uncertainty for each quantity.

Figure 13

Neutron droplet energies predicted with unedf0, unedf1, and unedf2 compared to the ab initio AFDMC results and DFT calculations with Ly4 and adjusted SLy4 EDFs of Ref. [92].