Isospin-invariant Skyrme energy-density-functional approach with axial symmetry

Background: Density functional theory (DFT) is the microscopic tool of choice to describe properties of nuclei over the entire nuclear landscape, with a focus on medium-mass and heavy complex systems. Modern energy density functionals (EDFs) often offer a level of accuracy typical of phenomenological approaches based on parameters locally fitted to the data. It is clear, however, that in order to achieve high quality of predictions to guide spectroscopic studies, current functionals must be improved, especially in the isospin channel. In this respect, experimental studies of short-lived nuclei far from stability offer a unique test of isospin aspects of the many-body theory.

Purpose: We develop the isospin-invariant Skyrme-EDF method by considering local densities in all possible isospin channels and proton-neutron ($p$-$n$) mixing terms as mandated by the isospin symmetry. The EDF employed has the most general form that depends quadratically on the isoscalar and isovector densities. We test and benchmark the resulting $p$-$n$ EDF approach, and study the general properties of the new scheme by means of the cranking in the isospin space.

Methods: We extend the existing axial DFT solver hfbtho to the case of isospin-invariant EDF approach with all possible $p$-$n$ mixing terms. Explicit expressions have been derived for all the densities and potentials that appear in the isospin representation. In practical tests, we consider the Skyrme EDF SkM${}^{*}$ and, as a first application, concentrate on Hartree-Fock aspects of the problem, i.e., pairing has been disregarded.

Results: Calculations have been performed for the ($A=78,T\simeq 11$), ($A=40,T\simeq 8$), and ($A=48,T\simeq 4$) isobaric analog chains. Isospin structure of self-consistent $p$-$n$ mixed solutions has been investigated with and without the Coulomb interaction, which is the sole source of isospin symmetry breaking in our approach. The extended axial hfbtho solver has been benchmarked against the symmetry-unrestricted hfodd code for deformed and spherical states.

Conclusions: We developed and tested a general isospin-invariant Skyrme-EDF framework. The new approach permits spin-isospin densities that may give rise to hitherto unexplored modes in the excitation spectrum. The new formalism has been tested in the Hartree-Fock limit. A systematic comparison between hfodd and hfbtho results show a maximum deviation of about 10 keV on the total binding energy for deformed nuclei when the Coulomb term is included. Without this term, the results of both solvers agree down to a $\sim$10 eV level.

Figure 1

Total HF energy of the $A=78$, $T\approx 11$ IASs as a function of ${\theta }^{\prime }$ with (solid line) and without (dashed line) the isospin-symmetry-breaking Coulomb term.

Figure 2

Similar as in Fig. 1, but for the expectation values of $\langle {\widehat{T}}_x\rangle$ (a) and $\langle {\widehat{T}}_z\rangle$ (b).

Figure 3

Similar as in Fig. 1, but for $\langle {\widehat{T}}^2\rangle$.

Figure 4

Single-particle Routhians as functions of ${\theta }^{\prime }$ for $T\approx 11$ configurations in the $A=78$ systems. The Coulomb interaction is not included. Points are colored according to the s.p. expectation values of $\langle {\widehat{\tau }}_z\rangle$. At ${\theta }^{\prime }=0^{\circ }$, neutron and proton states are plotted up to 1$g_{7/2}$ and 2$p_{1/2}$, respectively.

Figure 5

Similar to Fig. 4, but with the Coulomb interaction included.

Figure 6

Similar to Fig. 4 but for the s.p. energies with the Coulomb interaction included. Only the occupied states are plotted; that is, at ${\theta }^{\prime }=0^{\circ }$, neutron and proton states are plotted up to 1$g_{9/2}$ and 1$f_{7/2}$ shells, respectively.

Figure 7

Evolution of s.p. energies of occupied HF states (with Coulomb) originating from proton (top) and neutron (bottom) shells at ${\theta }^{\prime }=180$ with respect to the energy of the 1$f_{7/2}$ shell, and relative to the corresponding values at ${\theta }^{\prime }=0^{\circ }$. That is, the energies plotted are $[{\varepsilon }_i\left({\theta }^{\prime }\right)-{\varepsilon }_{f_{7/2}}\left({\theta }^{\prime }\right)]-[{\varepsilon }_i\left(0^{\circ }\right)-{\varepsilon }_{f_{7/2}}\left(0^{\circ }\right)]$.

Figure 8

The total HF energy (with Coulomb) for $A=78$, $T\approx 11$ calculated with $N_{\mathrm{sh}}=10$, 12, and 14 HO shells relative to that with $N_{\mathrm{sh}}=16$ shells as a function of ${\theta }^{\prime }$.

Figure 9

Neutron and proton rms radii as functions of ${\theta }^{\prime }$.

Figure 10

Lines: proton effective HF potential (with Coulomb and centrifugal terms included), calculated for the $A=78$, $T\simeq |T_z|$ isobars for $\ell =4$ (a) and $\ell =1$ (b). Dots: rms radii and s.p. energies of the proton 1$g_{9/2}$ (a) and 1$p_{1/2}$ states.