Isospin symmetry breaking at band termination

Terminating states or seniority isomers are fully-stretched p-h configurations with a maximum-spin $I_{max}$ that can be built within a given SM space of valence particles. Because of their simple SM character, terminating states provide a robust probe of SM and MF theories and corresponding effective interactions. In this context, of particular interest are the terminating states associated with the $[f_{7/2}^n]_{I_{max}}$ and $[d_{3/2}^{-1} f_{7/2}^{n+1}]_{I_{max}}$ configurations (in the following, $n$ denotes a number of valence particles outside the $^{40}$Ca core) in $20\leq Z\leq N \leq 24$ nuclei from the lower$-fp$ shell ($A\sim 44$), which were systematically measured during the last decade [60,61,62,63,64,65].

According to MF calculations, these specific states appear to have almost spherical shapes; hence, the correlations resulting from the angular-momentum restoration are practically negligible there [66]. Hence, they can be regarded as extreme cases of an almost undisturbed s.p. motion, thus offering an excellent playground to study, among others, time-odd densities and fields, spin-orbit force [67,68,69], tensor interactions [70], and the isospin dependence of cross-shell ($sd-fp$) p-h matrix elements [68,71].

While most of the terminating states are uniquely defined, the $[d_{3/2}^{-1} f_{7/2}^{n+1}]_{I_{max}}$ states in the $N$=$Z$ nuclei provide a notable exception. Indeed, within the MF approximation these particular states can be created by promoting either one proton, $[d_{3/2}^{-1} f_{7/2}^{n+1}]_{I_{max}}^\pi$, or one neutron, $[d_{3/2}^{-1} f_{7/2}^{n+1}]_{I_{max}}^\nu$, across the magic gap 20, Because the Coulomb energy difference between these configurations is small, the resulting energy levels are almost degenerate: $E([d_{3/2}^{-1} f_{7/2}^{n+1}]_{I_{max}}^\pi) \approx E([d_{3/2}^{-1} f_{7/2}^{n+1}]_{I_{max}}^\nu)$. Therefore, we encounter problems related to isospin symmetry similar to those discussed earlier in Sec. 2.

As in the case of $^{56}$Ni, we encounter the situation where the isospin symmetry is manifestly broken by the MF approximation and the predictions are at variance with empirical data. The difficulty with the isospin content of terminating states was recognized in Ref. [68], where a purely phenomenological method of isospin restoration was proposed. It has resulted in a good MF description of experimental data, at the level of the state-of-the-art SM calculations. Quantitatively, however, the estimated energy correction due to the isospin projection appeared to have surprisingly strong $A$-dependence, changing quite rapidly from $\delta E_T$$\approx$2MeV in $^{40}$Ca down to $\delta E_T$$\approx$1MeV in $^{46}$V, see Fig. 4 of Ref. [68]. This trend has been found to depend weakly on the EDF parameterization.

Figure 5: Energy differences $\Delta E$ (60) between the terminating states in Ca,Sc,Ti,and Cr $N$=$Z$ nuclei. Open and full triangles mark, respectively, standard and isospin-projected HF results for the $[d_{3/2}^{-1} f_{7/2}^{n+1}]_{I_{max}}^\pi$ (up triangles) and $[d_{3/2}^{-1} f_{7/2}^{n+1}]_{I_{max}}^\nu$ (down triangles) configurations. Calculations were performed by using the SIII$_L$ functional [59]. Experimental data (dots) and SM results of Ref. [68] (circles) are shown for comparison. Note the limited energy scale.

In the present work, we repeat calculations of Ref. [68], however, by using the mathematically rigorous isospin projection of Sec. 3. The energies of the isospin-projected states $[d_{3/2}^{-1} f_{7/2}^{n+1}]_{I_{max}}^{T=0}$ are shown in Fig. 5 relative to those of the $[f_{7/2}^n]_{I_{max}}$ configurations:

$$\Delta E = E([d_{3/2}^{-1} f_{7/2}^{n+1}]_{I_{max}})- E([f_{7/2}^n]_{I_{max}}) .$$ (60)

Full up and down triangles show results obtained by projecting the $T$=0 component of the Slater determinant corresponding to either one proton, $\vert[d_{3/2}^{-1} f_{7/2}^{n+1}]_{I_{max}}^\pi \rangle$, or one neutron, $\vert[d_{3/2}^{-1} f_{7/2}^{n+1}]_{I_{max}}^\nu \rangle$, p-h excitation through the magic gap 20, respectively. For comparison, the unprojected HF energies for the proton and neutron configurations are shown with the open up and down triangles, respectively. Here, all calculations were performed by using the SIII [41] Skyrme functional.

The isospin-projected results obtained from the $\vert[d_{3/2}^{-1} f_{7/2}^{n+1}]_{I_{max}}^\pi \rangle$ and $\vert[d_{3/2}^{-1} f_{7/2}^{n+1}]_{I_{max}}^\nu \rangle$ HF configurations are similar but not identical, reflecting small polarization differences due to proton and neutron p-h excitations. Note, that slightly better results are obtained by projecting from the proton configurations as the resulting levels appear slightly lower in energy. This is probably not surprising, because these states include directly the polarization of the Coulomb field by the proton p-h excitation and thus they have slightly richer isospin structure than their neutron counterparts.

The results of rigorous isospin projection closely follow those obtained in the phenomenological approach of Ref. [68]. As shown in Fig. 6, the energy corrections calculated in the isospin-projected HF,

$$\delta E_T = \tfrac{1}{2} \left[E([d_{3/2}^{-1} f_{7/2}^{n+1}]_{I_{max}}^{T=1}) -E([d_{3/2}^{-1} f_{7/2}^{n+1}]_{I_{max}}^{T=0}) \right],$$ (61)

also exhibit an appreciable $A$-dependence, in a close analogy to the previously obtained phenomenological trend. Consequently, the $A$-dependence of $\Delta E$ (60), calculated using the isospin projected EDF approach also shows a different pattern than experiment and SM. This result is weakly dependent on EDF parameterization, suggesting that there is some generic problem pertaining to the standard form of the Skyrme EDF.

Figure: Full symbols label energy correction $\delta E_T$ (61) calculated by projecting good isospin from the HF states corresponding to the $[d_{3/2}^{-1} f_{7/2}^{n+1}]_{I_{max}}^\pi$ configurations in $N$=$Z$ nuclei and subsequent Coulomb rediagonalization. Open symbols represent results of a phenomenological method proposed in Ref. [68]. Calculations were carried out by using the SIII$_L$ (dots) and SkP$_{LT}$ (triangles) Skyrme functionals [59]. See text for further details.

Recently, we have developed a new class of the Skyrme functionals with spin-orbit and tensor terms locally refitted, to reproduce the $f_{7/2}-f_{5/2}$ spin-orbit splitting in $^{40}$Ca, $^{48}$Ca, and $^{56}$Ni [59]. The spin-orbit and the tensor strengths obtained in this way turned out to be fairly independent of other coupling constants of the Skyrme functional. This result indicates that the strong dependence of the spin-orbit strength on EDF parameterization, in particular on the isoscalar effective mass [72], is likely to be an artifact of fitting protocols based predominantly on data pertaining to bulk nuclear properties. Indeed, as discussed in Ref. [73], the use of inaccurate models in the fitting procedure can lead to results that strongly depend on the fitting protocol itself; hence, can result in contradictory information on the key model parameters (compare, e.g., results of Refs. [72] and [74]). It seems that this is exactly the case for the current parameterizations of the Skyrme EDF. As shown recently in Ref. [75], parameterizations that correctly describe the spin-orbit properties in light nuclei do not fare well in heavier systems. This again points to limitations of the second-order Skyrme EDF [76] and to a danger of drawing conclusions on tensor interactions from global fits [72].

Applications of new functionals to the terminating states in $N\ne Z$, $A\sim 44$ nuclei [77] have revealed that removal of the artificial isoscalar effective mass scaling from the spin-orbit restores the effective mass scaling in the s.p. level density. As a consequence, only the forces having large isoscalar effective masses ( $\frac{M^*}{M}\geq 0.9$) such as SkP$_T$ and SkO$_T$ [70], are able to reproduce empirical data involving s.p. levels in light nuclei.

Figure 7: Similar to Fig. 5 except for the modified Skyrme functional SkP$_{LT}$ [59].

This conclusion is nicely corroborated by results presented in Fig. 7, which shows the predictions for terminating states using the new Skyrme parameterization SkP$_{LT}$. It is rewarding to see that the modified functional yields results that are very consistent with SM. This is particularly true for the isospin projection from the $\vert[d_{3/2}^{-1} f_{7/2}^{n+1}]_{I_{max}}^\pi \rangle$ configurations. The new values of $\Delta E$ reasonably agree with experiment, considering the energy range of the plot, While the detailed $A$-dependence is reproduced very well, theoretical $\Delta E$ curve slightly underestimates experiment. Identification of a specific source of this remaining discrepancy requires further studies.