Impact of choices for center-of-mass correction energy on the surface energy of Skyrme energy density functionals

Background: In the framework of nuclear energy density functional (EDF) methods, many nuclear phenomena can be related to the deformation of intrinsic states. Their accurate modeling relies on the correct description of the change of nuclear binding energy with deformation. The two most important contributions to the deformation energy have their origin in shell effects and the surface energy coefficient of nuclear matter.

Purpose: It has been pointed out before that the choices made for the center-of-mass (c.m.) correction energy and the effective mass during the parameter adjustment influence the deformation properties of nuclear EDFs. We study the impact of these two properties by means of a set of purpose-built parametrizations of the standard Skyrme EDF at next-to-leading (NLO) order in gradients.

Methods: In a first step, we build nine series of parametrizations with a systematically varied surface-energy coefficient $a_{\text{surf}}$ for three frequently used options for the c.m. correction (none, one-body term only, full one-body and two-body contributions) combined with three values for the isoscalar effective mass $m_0^{*}/m$ (0.7, 0.8, 0.85) and analyze how well each of these parametrizations can be adjusted to the properties of spherical nuclei and infinite nuclear matter. In a second step, we performed additional fits without the constraint on surface energy, adding one “best-fit” parametrization to each of the nine series. We then benchmark these parametrizations to the deformation properties of heavy nuclei by means of three-dimensional Hartree-Fock-Bogoliubov calculations that allow for nonaxial and/or nonreflection symmetric configurations.

Results: We perform a detailed correlation analysis between surface and volume properties of nuclear matter using the nine series of parametrizations. The best fits out of each series are then benchmarked on the fission barriers of ${}^{240}\mathrm{Pu}$ and ${}^{180}\mathrm{Hg}$, as well as on the properties of deformed states at normal and superdeformation for actinides and nuclei in the neutron-deficient Hg region.

Conclusions: The main conclusions are as follows: (i) Each combination of choices for c.m. correction and $m_0^{*}/m$ leads to a significantly different optimal value of $a_{\text{surf}}$, reason being that the effective interaction has to absorb the contribution of the c.m. correction to the total binding energy. (ii) Many properties of symmetric and asymmetric infinite nuclear matter of Skyrme NLO EDFs are strongly correlated to the value of $a_{\text{surf}}$. (iii) Omitting the c.m. correction results in values of $a_{\text{surf}}$ that are systematically too small. On the other hand, including the one-body term but neglecting the computationally expensive two-body term means $a_{\text{surf}}$ will be too large. Both choices result in unrealistic predictions for fission barriers and superdeformed states of heavy nuclei. Only by incorporating the complete c.m. correction does one obtain quite realistic surface properties from an adjustment protocol that only constrains properties of infinite nuclear matter and spherical nuclei. (iv) Lowering $a_{\text{surf}}$ increases the susceptibility of finite nuclei to take an exotic shape.

Figure 1

Differences between the surface energy coefficient as calculated in either (a) the MTF or (b) ETF approach and its HF value for the nine series of parametrizations as indicated as a function of their $a_{\text{surf}}^{\text{MTF}}$. Different colors indicate different schemes for c.m. correction, whereas different markers indicate different isoscalar effective mass $m_0^{*}/m$.

Figure 2

Penalty function ${\chi }^2$ for all parametrizations out of the nine series of fits as a function of their value of (a) $a_{\text{surf}}^{\text{MTF}}$ and (b) $a_{\text{surf}}^{\text{HF}}$. The dots indicate the parametrizations with systematically varied $a_{\text{surf}}^{\text{MTF}}$ whose properties are studied in Sec. 4c. For each of the nine series, the minimum of ${\chi }^2$ is indicated by a filled marker. The properties of the corresponding “best fit” parametrizations are studied beginning with Sec. 4d.

Figure 3

Surface symmetry energy coefficient $a_{\text{ssym}}^{\text{HF}}$ calculated in HF approximation for all parametrizations out of the nine series of fits, plotted as a function of their value for $a_{\text{surf}}^{\text{MTF}}$. Panels (a)–(c) compare parametrizations with same effective mass but different scheme for c.m. correction, whereas panels (d)–(f) compare parametrizations with same scheme for c.m. correction but different effective mass $m_0^{*}/m$.

Figure 4

Location of the saturation point: Energy per particle ${\varepsilon }_{\text{sat}}$ at the saturation density ${\rho }_{\text{sat}}$ of the parametrizations as indicated. The horizontal lines indicate the target value of ${\varepsilon }_{\text{sat}}$ and its tolerance in the penalty function.

Figure 5

Symmetry energy $J$ and its slope $L$ at saturation density ${\rho }_{\text{sat}}$ and at ${\rho }_0=0.1{\text{fm}}^{-3}$ (see text) for the parametrizations as indicated. The horizontal lines indicate the target values of $J$ and $L$ in the penalty function, as well as their respective tolerances. Colors and symbols are as in Fig. 4.

Figure 6

Higher-order derivatives $K_{\infty }, Q_{\infty }, K_{\text{sym}}$, and $Q_{\text{sym}}$ of the binding energy per particle at saturation density ${\rho }_{\text{sat}}$. Colors and symbols are as in Fig. 4.

Figure 7

Isoscalar effective mass $m_0^{*}/m$, the enhancement factor of the Thomas-Reiche-Kuhn sum rule ${\kappa }_v$, and the splitting $\mathrm{\Delta }{m}_{np}^{*}$ of the effective masses of neutrons and protons in neutron matter. Colors and symbols as in Fig. 4.

Figure 8

Value of the surface energy coefficient $a_{\text{surf}}$ calculated with the HF, ETF, and MTF methods for the nine “best fits.”

Figure 9

Size of the c.m. correction energy $E_{\text{c.m.}}$ (full markers) for the three parametrizations with $m_0^{*}/m=0.70$ as indicated for the seven doubly magic nuclei entering the penalty function of the new fits plotted as a function of their mass number. The lines are estimates for the size of the c.m. correction based on differences of the LDM coefficients obtained for the three fits (see text). Panel (a) compares with the LDM estimates assuming $I=0$ for all nuclei, i.e., considering only the volume and surface energy, whereas for the lines in panel (b) also the symmetry and surface symmetry contributions to the LDM energies are taken into account.

Figure 10

(a) Deformation-energy curve of ${}^{240}\mathrm{Pu}$ and (b) change of the center-of-mass correction as a function of the quadrupole deformation ${\beta }_{20}$ for the parametrizations as indicated. In both cases, the energies are normalized to the value at the respective ground-state deformation. To facilitate the comparison, both panels share the same energy scale. The horizontal gray bars in panel (a) indicate experimental values for the height of the inner and outer barriers as well as the excitation energy of the fission isomer, taken from the sources mentioned in the text. The inserts on top of the figure indicate the evolution of shapes along the fission path, the upper and lower halves representing isodensities at ${\rho }_0=0.08$ and $0.15{\mathrm{fm}}^{-3}$ in the $x$ and $y$ directions.

Figure 11

Characteristic energies of the fission barrier of ${}^{240}\mathrm{Pu}$ as a function of (a) $a_{\text{surf}}^{\text{MTF}}$, (b) $a_{\text{surf}}^{\text{HF}}$, and (c) $a_{\text{surf,eff}}^{\text{HF}}$(${}^{240}\mathrm{Pu}$). Colors indicate families of parameter sets with same scheme for c.m. correction as in Fig. 10. Markers, however, indicate here the excitation energy of the fission isomer and the heights of the inner and outer barrier, respectively. To guide the eye, lines connect results obtained with the three parameter sets with same scheme for c.m. correction, but different effective mass. Within each family of parameter sets, $a_{\text{surf}}$ decreases with increasing effective mass, see Table 4. As in Fig. 10, the experimental data are indicated by horizontal gray bars, where those for the inner and outer barrier overlap.

Figure 12

(a) Deformation energy curve of ${}^{180}\mathrm{Hg}$ and (b) change of the center-of-mass correction as a function of the quadrupole deformation ${\beta }_{20}$ drawn in the same way as in Fig. 10. The energy curves end at the deformation at which the calculation jumps to a solution with two separate fragments.

Figure 13

Height of the primary fission barrier of ${}^{180}\mathrm{Hg}$ as a function of (a) $a_{\text{surf}}^{\text{MTF}}$, (b) $a_{\text{surf}}^{\text{HF}}$, and (c) $a_{\text{surf,eff}}^{\text{HF}}$(${}^{180}\mathrm{Hg}$) plotted in the same way as in Fig. 11.

Figure 14

Dimensionless quadrupole and hexadecapole deformation of the charge-density distribution of the ground states of even-even U ($Z=92$) and Pu ($Z=94$) isotopes compared with experimental data where available, plotted in the same colors and line styles as in Fig. 10. The error bars of the experimental ${\beta }_{20}$ values are smaller than the markers used to plot them. For the two Pu isotopes for which we found reflection-asymmetric ground-state shapes, we also show the calculated octupole deformation.

Figure 15

Energy of the normal-deformed configuration of ${}^{240}\mathrm{Pu}$ as a function of dimensionless octupole deformation ${\beta }_{30}$, plotted in the same colors and line styles as in Fig. 10. The insets on top of the figure indicate the typical evolution of shapes along the energy curve.

Figure 16

Charge quadrupole (a), (b) deformation and (c), (d) excitation energy of the $0^{+}$ fission isomers of even-even (a), (c) U and (b), (d) Pu isotopes. Colors and line styles are the same as in the previous figures.

Figure 17

Charge quadrupole (a), (b) deformation and (c), (d) excitation energy of the hypothetical (see text) $0^{+}$ bandheads of the superdeformed rotational bands of even-even (a), (c) Hg and (b), (d) Pb isotopes. Colors and line styles are the same as in the previous figures.

Figure 18

Shape coexistence at normal deformation in even-even neutron-deficient Hg isotopes. (a) Calculated dimensionless quadrupole charge deformation ${\beta }_2$ of the three minima compared with experimental data where available (see text). (b)–(d) Energy of the weakly and strongly deformed prolate states relative to the oblate state shown separately for the 1F2F(X), 1T2F(X) and 1T2T(X) parametrizations. Colors and line styles are the same as in the previous figures.