Surface symmetry energy of nuclear energy density functionals

We study the bulk deformation properties of the Skyrme nuclear energy density functionals (EDFs). Following simple arguments based on the leptodermous expansion and liquid drop model, we apply the nuclear density functional theory to assess the role of the surface symmetry energy in nuclei. To this end, we validate the commonly used functional parametrizations against the data on excitation energies of superdeformed band heads in Hg and Pb isotopes and fission isomers in actinide nuclei. After subtracting shell effects, the results of our self-consistent calculations are consistent with macroscopic arguments and indicate that experimental data on strongly deformed configurations in neutron-rich nuclei are essential for optimizing future nuclear EDFs. The resulting survey provides a useful benchmark for further theoretical improvements. Unlike in nuclei close to the stability valley, whose macroscopic deformability hangs on the balance of surface and Coulomb terms, the deformability of neutron-rich nuclei strongly depends on the surface symmetry energy; hence, its proper determination is crucial for the stability of deformed phases of the neutron-rich matter and description of fission rates for r-process nucleosynthesis.

Figure 1

Correlation between the symmetry and surface symmetry coefficients extracted from Skyrme EDFs from Table (dots) and Skyrme EDFs of Ref. [28] (circles). The phenomenological LDM values of Table are also indicated (stars) as well as the hydrodynamical [38] and mass [43, 44] estimates: $r_{S/V}=-2$ and $-1.6$, respectively (dashed lines). The linear fit to the values of Ref. [28] is shown by a dotted line.

Figure 2

Contributions to the microscopic LDM energy per nucleon along the LDM valley of stability: volume symmetry term $a_{\mathrm{sym}}{I}^2$ (top), surface symmetry term $a_{\mathrm{ssym}}{I}^2{A}^{-1/3}$ (middle), and the total symmetry energy (bottom) for the microscopic LDM derived from Skyrme EDFs of Table .

Figure 3

Individual contributions to the total deformation energy per nucleon of the microscopic LDM for nine Skyrme EDFs for two $A=100$ isobars: ${}^{100}\mathrm{Sn}$ ($I=0$, top) and ${}^{100}\mathrm{Zn}$ ($I=0.4$, bottom). The assumed quadrupole deformation is ${\mathrm{\beta ̃}}_2=0.6$.

Figure 4

Potential energy curves for ${}^{194}\mathrm{Pb}$ (top) and ${}^{236}\mathrm{U}$ (bottom) versus quadrupole deformation $\beta$ calculated with SkI3, SkI4, SkM*, SkO, SLy4, and SLy6 Skyrme EDFs. All curves are normalized to the spherical point. Axial symmetry is assumed.

Figure 5

Residuals $\Delta E=E_{\mathrm{th}}^{*}-E_{\exp }^{*}$ (top) and rms deviations from experiment (bottom) for various Skyrme EDFs. Additional references for the Skyrme forces: SLy5-SLy7 [88], SkMP [126], SkX-SkXC [127], SIII [128], MSk1-MSk6 [129], BSk2 [87], SkO’ [90]. The average rms deviation for the set of EDFs considered (marked by a dashed line in the lower panel) is 1.26 MeV for the nine data points of Table and 1.34 MeV for the fission isomers nuclei.

Figure 6

Convergence of the HO basis expansion for the HFB$+$SkM* binding energy of the fission isomer and g.s. (top panel) and the excitation energy of the fission isomer (bottom panel) in ${}^{240}\mathrm{Pu}$ as a function of the HO basis size. Results are compared with the benchmark numbers obtained with the precise coordinate-space solver hfb-ax [130].

Figure 7

Proton smoothed single-particle density $\mathrm{g̃}(e)$ in the g.s. and SD configuration of ${}^{240}\mathrm{Pu}$ calculated with the SkM* EDF and $N_{\max }=16$ HO shells.

Figure 8

Convergence of the shell-correction contribution to the deformation energy, $\delta E_{\mathrm{shell}}^{\mathrm{SD}}-\delta E_{\mathrm{shell}}^{\mathrm{g}.\mathrm{s}.}$, for ${}^{240}\mathrm{Pu}$ as a function of $N_{\max }$. Calculations were performed with with the SkM* EDF.

Figure 9

Extraction of the LDM deformation energy from constrained HF$+$SLy4 calculations for ${}^{236}\mathrm{U}$ at several values of quadrupole deformation $\beta$. Shown are the total shell correction $\delta E_{\mathrm{shell}}$ (squares), the smooth HFB deformation energy ${\mathrm{Ẽ}}_{\mathrm{def}}^{\mathrm{HF}}(\beta )$ (dots), and the corresponding LDM deformation energy $E_{\mathrm{def}}^{\mathrm{LDM}}(\beta )$ (triangles). The inset shows the equivalent LDM deformations ${\mathrm{\beta ̃}}_l$ with $l=4$, 6, and 8.

Figure 10

Smooth excitation energy ${\mathrm{Ẽ}}^{*}$ of fission isomers in ${}^{236,238}\mathrm{U}$, ${}^{240}\mathrm{Pu}$, and ${}^{242}\mathrm{Cu}$ calculated in HFB and LDM for seven EDFs. See text for details.

Figure 11

Same as in Fig. 10 except for SD band heads in ${}^{192,194}\mathrm{Hg}$ and ${}^{192,194}\mathrm{Pb}$.

Figure 12

Surface and surface symmetry contributions to the LDM excitation energy of fission isomers in the actinides compared to the smooth HFB excitation energy for the same Skyrme EDFs as in Figs. 10 and 11. All curves are shown relative to SkM* results.

Figure 13

Quadrupole deformations $\beta$ in SkM* for g.s.s and fission isomers in a sequence of neutron-rich U isotopes (top) and the corresponding excitation energies (bottom).

Figure 14

Relative contributions of the Coulomb, surface symmetry, curvature, and surface terms to the equivalent LDM excitation energy of fission isomers for the same U isotopes as in Fig. 13. Calculations are based on SkM* and BSk6.

Figure 15

The upper part of the chart of the nuclides with the $x=1$ limit indicated for SkI3 and SkM* EDFs, the value of $\eta =1.7826$ used in Ref. [143] (LDM), and no isospin dependence ($\eta =0$). The region of known nuclides is marked by black squares.