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.