Skyrme pseudopotentials at next-to-next-to-leading order: Construction of local densities and first symmetry-breaking calculations

There is an ongoing quest to improve on the spectroscopic quality of nuclear energy density functionals (EDFs) of the Skyrme type through extensions of its traditional form. One direction for such activities is the inclusion of terms of higher order in gradients in the EDF. We report on exploratory symmetry-breaking calculations performed for an extension of the Skyrme EDF that includes central terms with four gradients at next-to-next-to-leading order (N2LO) and for which the high-quality parametrization SN2LO1 has been constructed recently [Becker et al., Phys. Rev. C 96, 044330 (2017)]. Up to now, the investigation of such functionals with higher-order terms was limited to infinite matter and spherically symmetric configurations of singly and doubly magic nuclei. We address here nuclei and phenomena that require us to consider axial and nonaxial deformation, both for reflection-symmetric and also reflection-asymmetric shapes, as well as the breaking of time-reversal invariance. Achieving these calculations demanded a number of formal developments. These all resulted from the formulation of the N2LO EDF requiring the introduction of new local densities with additional gradients that are not present in the EDF at NLO. Their choice is not unique, but can differ in the way the gradients are coupled. While designing a numerical implementation of N2LO EDFs in Cartesian three-dimensional coordinate-space representation, we have developed a novel definition and a new unifying notation for normal and pair densities that contain gradients at arbitrary order. Besides having mnemonic advantages, the new notation allows for the easy identification of redundancies and reducibilities in a given set of local densities, and the new definition makes it straightforward to construct densities that automatically adopt the symmetries of the many-body state they are constructed from. The resulting scheme resolves several issues with some of the choices that have been made for local densities in the past, in particular when breaking time-reversal symmetry. Guided by general practical considerations, we propose an alternative form of the N2LO contribution to the Skyrme EDF that is built from a different set of densities. It has exactly the same physics content, but is much more efficient to handle in formal discussions and, compared to the original formulation, leads to a substantial reduction of computational cost and memory requirements in deformed codes. As representative examples for the performance of SN2LO1, we have chosen the ground states of even-even Kr and Nd isotopes, the fission barrier of ${}^{240}\mathrm{Pu}$, as well as the superdeformed rotational band of ${}^{194}\mathrm{Hg}$. Overall, for the nuclei and phenomena studied here, the SN2LO1 parametrization does not yet present a systematic improvement over standard NLO parametrizations. This finding calls for improved fit protocols that better discriminate between NLO and N2LO terms and better exploit the unique features of the additional degrees of freedom offered by the latter.

Figure 1

Deformation energy curve for the lighter krypton isotopes ${}^{72–82}\mathrm{Kr}$ as a function of the quadrupole moment ${\beta }_{20}$, normalized to the spherical configurations. Top: results for SN2LO1; bottom, results for SLy5*. Black diamonds indicate the global minimum for each isotope.

Figure 2

Same as Fig. 1 but for the heavier isotopes ${}^{84–94}\mathrm{Kr}$.

Figure 3

Isotopic shifts $\delta {\langle r^2\rangle }^{N,82}$ of mean-square charge radii for the Nd isotope chain calculated with SLy5* and SN2LO1. Experimental values were taken from Ref. [78]. The evolution of charge radii when assuming the global $A^{2/3}$ scaling of the spherical liquid-drop model is also given for comparison.

Figure 4

Deformation ${\beta }_{20,p}$ of the proton distribution in the Nd isotope chain calculated with SLy5* and SN2LO1, compared to values of the transition charge deformation deduced from experimental $B(E2,0_1^{+}\to 2_1^{+})$ values [79] under the assumption that the nuclei are rigid rotors.

Figure 5

Static fission path of ${}^{240}\mathrm{Pu}$ as a function of quadrupole deformation ${\beta }_{20}$, as calculated with the SLy5s1, SLy5*, and SN2LO1 parametrizations. The energy curves are normalized to their respective prolate minimum. Along the path, nuclear shapes explore nonaxial and/or reflection asymmetric deformations (see text). The horizontal grey bars 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 same sources as in Ref. [66]

Figure 6

Dynamical moment of inertia ${\mathcal{I}}^{\left(2\right)}$ of the superdeformed band in ${}^{194}\mathrm{Hg}$ as a function of the cranking frequency $\omega$ for the SLy5s1, SLy5*, and the SN2LO1 parametrizations. As before, results for NLO (N2LO) parametrizations are plotted with dashed (full) lines. The empty diamond symbols are experimental results for the yrast SD-1 band taken from Ref. [84].