No-core configuration-interaction model for the isospin- and angular-momentum-projected states

Background: Single-reference density functional theory is very successful in reproducing bulk nuclear properties like binding energies, radii, or quadrupole moments throughout the entire periodic table. Its extension to the multireference level allows for restoring symmetries and, in turn, for calculating transition rates.

Purpose: We propose a new variant of the no-core-configuration-interaction (NCCI) model treating properly isospin and rotational symmetries. The model is applicable to any nucleus irrespective of its mass and neutron- and proton-number parity. It properly includes polarization effects caused by an interplay between the long- and short-range forces acting in the atomic nucleus.

Methods: The method is based on solving the Hill-Wheeler-Griffin equation within a model space built of linearly dependent states having good angular momentum and properly treated isobaric spin. The states are generated by means of the isospin and angular-momentum projection applied to a set of low-lying (multi)particle-(multi)hole deformed Slater determinants calculated using the self-consistent Skyrme-Hartree-Fock approach.

Results: The theory is applied to calculate energy spectra in $N\approx Z$ nuclei that are relevant from the point of view of a study of superallowed Fermi $\beta$ decays. In particular, a new set of the isospin-symmetry-breaking corrections to these decays is given.

Conclusions: It is demonstrated that the NCCI model is capable of capturing main features of low-lying energy spectra in light and medium-mass nuclei using relatively small model space and without any local readjustment of its low-energy coupling constants. Its flexibility and a range of applicability makes it an interesting alternative to the conventional nuclear shell model.

Figure 1

Computational scheme of the NCCI model. See text for details.

Figure 2

(a) Eigenvalues of the norm matrix obtained in the NCCI calculations for the $I=0$ states of odd-odd nuclei. Depicted are typical results obtained for two representative examples of axial (${}^{46}\mathrm{V}$) and triaxial (${}^{50}\mathrm{Mn}$) nuclei. The boxes give values of eigenenergies obtained by including three, six or nine eigenvalues of the norm matrix. (b) Dependence of the ISB corrections to superallowed ${}^{46}\mathrm{V}\to {}^{46}\mathrm{Ti}$ and ${}^{50}\mathrm{Mn}\to {}^{50}\mathrm{Cr}$ decays on a number of collective states retained in the mixing calculations. Dimension of the collective space decreases from the left- to the right-hand side from $D=9\left(6\right)$ in ${}^{50}\mathrm{Mn}$(${}^{46}\mathrm{V}$), respectively.

Figure 3

Comparison between experimental and theoretical energy spectra of ${}^6\mathrm{Li}$ and ${}^8\mathrm{Li}$.

Figure 4

(a) Excitation energies of the $0_2^{+}$ states with respect to the $0_1^{+}$ states in the $A=38$ isobaric triplet nuclei: Ca, K, and Ar. Theoretical predictions and data [53] are shown with solid and dashed lines, respectively. Calculated values of ${\delta }_{\mathrm{C}}$ are also shown. Decays $0_2^{+}\to 0_1^{+}$ indicated in the figure are predicted to be strongly hindered. (b) Comparison between the total binding energies of the $0_1^{+}$ states ($N=10$ harmonic oscillator shells were used).

Figure 5

Excitation energies of the isovector (circles) and isoscalar (squares) multiplets in ${}^{42}\mathrm{Sc}$ with respect to the $0^{+}$ state. Theoretical and experimental results are marked with open and solid symbols, respectively.

Figure 6

Same as in Fig. 5, but for the isovector multiplets in ${}^{42}\mathrm{Sc}$ (circles) and ${}^{42}\mathrm{Ca}$ (diamonds).

Figure 7

Schematic illustration of the interplay between the primary physical ingredients contributing to the excitation energy of the intruder state within our model. See text for details.

Figure 8

The lowest $0^{+}$ states projected from ${\left(f_{7/2}\right)}^2$ (0p-0h), ${\left(f_{7/2}\right)}^4{\left(d_{3/2}\right)}^{-2}$ (2p-2h), ${\left(f_{7/2}\right)}^6{\left(d_{3/2}\right)}^{-4}$ (4p-4h), and ${\left(f_{7/2}\right)}^8{\left(d_{3/2}\right)}^{-6}$ (6p-6h) reference states. Open (solid) diamonds refer to calculations performed using the SV and ${\mathrm{SV}}_{\mathrm{SO}}$ functionals, respectively. These results do not include configuration mixing. The right part shows excitation energies of the intruder state obtained within the NCCI theory with SV and ${\mathrm{SV}}_{\mathrm{SO}}$ interactions.

Figure 9

The low-lying $0^{+}$ states in ${}^{62}\mathrm{Zn}$. The first two columns show old and new experimental data; see [60] for details. The next three columns collect the results of the shell-model calculations using interactions MSDI3 [61], GXPF1 [62], and GXPF1A [63], respectively. The last three columns show results obtained within the NCCI approach, angular-momentum projection, and pure HF method, respectively. From Ref. [18].

Figure 10

(a) The low-lying $0^{+}$ states in ${}^{62}\mathrm{Zn}$ in the function of the number of configurations included in the NCCI calculations. (b) Calculated ISB corrections versus the number of configurations taken into account in the daughter nucleus. Different curves correspond to different sets of configurations taken to calculate the $0^{+}$ state in ${}^{62}\mathrm{Ga}$. (c) The HF energies of configurations included in the calculation of ${}^{62}\mathrm{Ga}$; see Table 10. Further details are given in the text.