Superdeformed bands in $^{56}$Ni

Figure 4: Superdeformed bands in $^{56}$Ni. Experimental data for Band 1 (dots) and Band 2 (circles) of Ref. [20] are compared with HF+SLy4$_L$ predictions marked by full and open triangles, respectively. The left panel shows the standard HF results while the predicted $T$=0 SD bands obtained in the isospin-projected approach are displayed in the right panel.

We begin discussion from the case of SD bands in $^{56}$Ni. As discussed in Sec. 2, the standard cranked HF theory gives a reasonable reproduction of Band 1 while there are problems with theoretical interpretation of Band 2 built on p-h excitations to the [440]1/2 proton and neutron levels. The results are summarized in Fig. 4. The HF (left panel) and isospin-projected (right panel) calculations were performed with SLy4$_L$ functional. Since the HF configuration corresponding to Band 1 is predominantly isospin symmetric, it is only weakly affected by the isospin projection. The calculated isospin impurity within this band is small ($\sim 2$%). This value is similar to that obtained for the spherical g.s. configuration of $^{56}$Ni. On the other hand, the Band 2 based on the $\pi$[440]1/2 orbital represents, before rediagonalization, almost equal mixture of the $T$=0 and $T$=1 components. Therefore, its isospin impurity assumes an unrealistic value of about 50%.

Following the isospin projection, the low-spin part of the $T$=0 SD band originating from Band 2 is shifted down by about 0.95MeV with respect to the unprojected HF result. The isospin impurity in this band is in the range of 6% to 8%, and it slowly increases as a function of the angular momentum. In fact, these values may indicate a presence of an uncontrolled isospin mixing related, most likely, to the angular-momentum non-conservation. As shown schematically in Fig. 2, a $T$=1 SD partner structure is expected to lie higher in energy. We indeed calculate the $T$=1 band (not shown in Fig. 4) to lie about 2.5MeV above the $T$=0 band. As expected, the $T$=0 and $T$=1 bands projected from the $\nu$[440]1/2 MF configuration are almost identical to those projected from the $\pi$[440]1/2 MF configuration.

By comparing the results of isospin-projected calculations with experiment, we see a significant improvement as compared to standard HF. The $T$=0 Band 2 agrees well with experiment, both in terms of excitation energy and MoI. The predicted crossing between $T$=0 Bands 1 and 2 occurs around spin 14$\hbar$, i.e., $\sim 2\,\hbar$ too high as compared to the data. This discrepancy, however, rather reflects a deficiency of our model in describing the MoI of Band 1, which is slightly overestimated as our calculations neglect pairing correlations that are expected to be important in this configuration [20]. The large energy splitting between the $T$=0 and $T$=1 Band 2 doublet is consistent with observation of only one SD side band. In short, the isospin-projected MF theory provides a quantitative explanation of experimental data on collective band structures in $^{56}$Ni.