Energy density functional approach to superfluid nuclei

A. Bulgac and Y. Yu

A steady transition is taking place during the last several years from the mean-field description of nuclear properties in terms of effective forces to an energy density functional approach (EDF). A significant role is played in this transition process by the fact that an EDF approach has a strong theoretical underpinning, the Hohenberg-Kohn theorem concerning the existence of an EDF and the Kohn-Sham Local Density Approximation. Only recently it became clear that a theoretically consistent local EDF formulation of the (nuclear) pairing properties is indeed possible [4,5]. Even though the crucial role of the pairing phenomena in nuclei has been established firmly, it is surprising to realize how poor the quality of our knowledge still is. Phenomenologically, one cannot unambiguously decide whether the pairing correlations in nuclei have a volume or/and a surface character. The isospin character of the nuclear pairing correlations requires further clarification as well. These questions become even sharper in the language of a local EDF.

We have shown [6,7] that within the framework of a simple local nuclear energy density functional (EDF), one can describe accurately the one- and two-nucleon separation energies of semi-magic nuclei. While for the normal part of the EDF we have used previously suggested parameterizations, for the superfluid part of the EDF we used the simplest possible local form compatible with known nuclear symmetries. We were able to infer that pairing properties of either kind of nucleons can be accounted for with a single constant $g=g_0+g_1$, where the superfluid EDF has the following structure

$${\cal{E}}_S({\mathbf{r}}) = g_0({\mathbf{r}})\vert\nu _p({... ...bf{r}})\vert\nu _p({\mathbf{r}})-\nu _n({\mathbf{r}})\vert^2 ,$$

and $\nu_{p,n}({\mathbf{r}})$ are the $S=0$ proton/neutron anomalous densities. This particular aspect has to be contrasted with the fact that in order to achieve a similar accuracy other authors have been forced to use up to five parameters and often even break nuclear symmetries in the pairing channel. It remains to be seen whether the other (non-perturbative) combination $g^\prime=g_0-g_1$ (never considered by other authors) could ever become relevant. In essentially all cases in which we have been able to perform a comparison between our results and those available in literature, 212 odd and even nuclei all together, our results were either qualitatively superior or, in a few separate cases, as good as the best available results.