Isospin-symmetry restoration within the nuclear density functional theory: Formalism and applications

Isospin symmetry of atomic nuclei is explicitly broken by the charge-dependent interactions, primarily the Coulomb force. Within the nuclear density functional theory, isospin is also broken spontaneously. We propose a projection scheme rooted in a Hartree-Fock theory that allows the consistent treatment of isospin breaking in both ground and exited nuclear states. We demonstrate that this scheme is free from spurious divergences plaguing particle-number and angular-momentum restoration approaches. Applications of the new technique include excited high-spin states in medium-mass $N=Z$ nuclei, such as superdeformed bands and many-particle–many-hole terminating states. Owing to the large spin polarization and/or high seniority of these states, pairing correlations have been ignored.

Figure 1

The upper panel schematically shows two possible g.s. configurations of an odd-odd $N=Z$ nucleus, as described by the conventional deformed MF theory. These degenerate configurations are called aligned (left) and anti-aligned (right), depending on what levels are occupied by the valence particles. The lower panel shows what happens if the isospin-symmetry is restored. The aligned configuration is isoscalar; hence, it is insensitive to the isospin projection. The anti-aligned configuration represents a mixture of the $T=0$ and $T=1$ states. The isospin projection removes the degeneracy by shifting the $T=0$ component down.

Figure 2

The upper panel schematically shows the s.p. configuration corresponding to the SD Band 1 in ${}^{56}\mathrm{Ni}$, which is based on a 4p4h excitation with respect to the doubly magic core. The middle panel depicts two possible configurations of Band 2 involving a single proton (left) or neutron (right) promoted to the $0g_{9/2}$ shell. To facilitate the discussion, we also show the relevant asymptotic Nilsson quantum numbers associated with deformed orbitals involved. The lower panel illustrates the isospin splitting induced by the isospin-symmetry restoration.

Figure 3

Excitation energies of the doorway $T=1$ states in the e-e $N=Z$ nuclei relative to the SHF g.s. energies $E_{\mathit{HF}}$. Diamonds, dots, and circles show the self-consistent results obtained in the isospin-projected DFT with SIII [45], SLy4 [60], and SkP [61] Skyrme functionals, respectively. Horizontal lines mark the mean DFT values. Thick dotted lines show estimates based on the perturbation theory, $E_{T=1}\approx 82/A^{1/3}$ MeV [58, 59], and on the hydrodynamical model, $E_{T=1}\approx 169/A^{1/3}$ MeV [59].

Figure 4

Superdeformed bands in ${}^{56}\mathrm{Ni}$. Experimental data for Band 1 (dots) and Band 2 (circles) of Ref. [22] are compared with HF$+$SLy$4_L$ 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.

Figure 5

Energy differences $\Delta E$ (66) 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 [64]. Experimental data (dots) and SM results of Ref. [73] (circles) are shown for comparison. Note the limited energy scale.

Figure 6

Full symbols label energy correction $\delta E_T$ (67) 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. [73]. Calculations were carried out by using the SIII${}_L$ (dots) and SkP${}_{\mathit{LT}}$ (triangles) Skyrme functionals [64]. See text for further details.

Figure 7

Similar to Fig. 5 except for the modified Skyrme functional SkP${}_{\mathit{LT}}$ [64].