Gamow-Teller response in the configuration space of a density-functional-theory–rooted no-core configuration-interaction model

Background: The atomic nucleus is a unique laboratory in which to study fundamental aspects of the electroweak interaction. This includes a question concerning in medium renormalization of the axial-vector current, which still lacks satisfactory explanation. Study of spin-isospin or Gamow-Teller (GT) response may provide valuable information on both the quenching of the axial-vector coupling constant as well as on nuclear structure and nuclear astrophysics.

Purpose: We have performed a seminal calculation of the GT response by using the no-core configuration-interaction approach rooted in multireference density functional theory (DFT-NCCI). The model treats properly isospin and rotational symmetries and can be applied to calculate both the nuclear spectra and transition rates in atomic nuclei, irrespectively of their mass and particle-number parity.

Methods: The DFT-NCCI calculation proceeds as follows: First, one builds a configuration space by computing relevant, for a given physical problem, (multi)particle-(multi)hole Slater determinants. Next, one applies the isospin and angular-momentum projections and performs the isospin and $K$ mixing in order to construct a model space composed of linearly dependent states of good angular momentum. Eventually, one mixes the projected states by solving the Hill-Wheeler-Griffin equation.

Results: The method is applied to compute the GT strength distribution in selected $N\approx Z$ nuclei including the $p$-shell ${}^8\mathrm{Li}$ and ${}^8\mathrm{Be}$ nuclei and the $sd$-shell well-deformed nucleus ${}^{24}\mathrm{Mg}$. In order to demonstrate a flexibility of the approach we present also a calculation of the superallowed GT $\beta$ decay in doubly-magic spherical ${}^{100}\mathrm{Sn}$ and the low-spin spectrum in ${}^{100}\mathrm{In}$.

Conclusions: It is demonstrated that the DFT-NCCI model is capable of capturing the GT response satisfactorily well by using a relatively small configuration space, exhausting simultaneously the GT sum rule. The model, due to its flexibility and broad range of applicability, may either serve as a complement or even as an alternative to other theoretical approaches, including the conventional nuclear shell model.

Figure 1

The leftmost column shows the HF energies of the g.s. and the lowest $\nu \mathrm{p}-\nu \mathrm{h}$ (red, dashed line) and $\pi \mathrm{p}-\pi \mathrm{h}$ (blue, solid line) excitations in ${}^{24}\mathrm{Mg}$. The second column illustrates $4^{+}$ states projected from these SR configurations without configuration mixing. The last two columns depict the DFT-NCCI results involving different configurations. The left (right) part shows the CI results involving the g.s. and $\nu \mathrm{p}-\nu \mathrm{h}$ ($\pi \mathrm{p}-\pi \mathrm{h}$) configurations, respectively. Numbers in the last two columns indicate the calculated GT matrix elements for $|{}^{24}\mathrm{Al};4_1^{+}\rangle \to |{}^{24}\mathrm{Mg};4_i^{+}\rangle$ decay.

Figure 2

Nilsson neutron (left) and proton (right) mean-field single-particle orbitals for the SR ground state of ${}^8\mathrm{Be}, {}^8\mathrm{Li}$, and ${}^8\mathrm{He}$. The orbitals are labeled with approximate Nilsson quantum numbers. Dots indicate occupied levels, and colors differentiate orbitals between parity blocks.

Figure 3

The GT strength distribution in $1^{+}$ states of ${}^8\mathrm{Li}$ in logarithmic scale smoothed with a Lorentzian function with half-width of $\mathrm{\Gamma }=0.5\mathrm{MeV}$. The dashed curve represents experimental data obtained by means of the $R$-matrix theory in Ref. [39, 40]. The dotted line marks the shell-model input to the $R$ matrix [39] calculated using the $p$-shell residual interaction proposed by Kumar in Ref. [41]. The continuous line labels the DFT-NCCI calculations. See text for discussion.

Figure 4

Decomposition of the wave functions of the first and fourth (GT resonance) $1^{+}$ states in ${}^8\mathrm{Li}$ in terms of the HF configurations, ${\phi }_n$, included in the space. An abscissa numbers the configurations in accordance with Table 1.

Figure 5

Low-spin states in ${}^8\mathrm{Be}$ below 30 MeV. The panels show, counting from the left, the groups of levels having spins $I^{\pi }=0^{+}, 1^{+}, 2^{+}, 3^{+}$, and $4^{+}$, respectively. Each panel shows experimental (left), DFT-NCCI (center), and ab initio NCCI (right) spectra, each normalized with respect to its g.s. energy.

Figure 6

GTSRs for the lowest $1_1^{+}, 2_1^{+}$, and $3_1^{+}$ initial state in ${}^8\mathrm{Li}$ against the number of configurations included in ${}^8\mathrm{Be}$. The configurations are listed in Table 1.

Figure 7

GTSRs for the first $1_1^{+}, 2_1^{+}$, and $3_1^{+}$ initial state in ${}^8\mathrm{Li}$ against the number of configurations included in ${}^8\mathrm{Li}$. The bottom panel shows the GTSR calculated without the cutoff. The upper panel represents the results obtained for $\varepsilon \approx 0.01$. See text for further details.

Figure 8

Eigenvalues of the norm matrix for $0^{+}, 1^{+}, 2^{+}, 3^{+}$, and $4^{+}$ states in ${}^8\mathrm{Be}$ (left) and in ${}^8\mathrm{Li}$ (right). The eigenvalues are plotted in ascending order.

Figure 9

Neutron mean-field s.p. levels in the g.s. of ${}^{24}\mathrm{Mg}$. The levels are labeled using the asymptotic Nilsson quantum numbers. Dots indicate occupied levels.

Figure 10

GTME stability analysis against configuration mixing in the parent nucleus. The figure shows GTMEs for $|{}^{24}\mathrm{Al};4_{\mathrm{g}}.\mathrm{s}{.}^{+}\rangle \to |{}^{24}\mathrm{Mg};4^{+}\rangle$ decay. The results were obtained by using 17 configurations in the daughter nucleus. The number of Slater determinants (SDs) used to correlate the parent nucleus changes from 1 (bottom) to 3 (top).

Figure 11

GTMEs between the $4^{+}$ g.s. of ${}^{24}\mathrm{Al}$ and the $4^{+}$ states in ${}^{24}\mathrm{Mg}$ for various dimensions of the used configuration space in ${}^{24}\mathrm{Mg}$. See text for details.

Figure 12

Complete GTSD in ${}^{24}\mathrm{Mg}$ following $\beta$ decay of the $4^{+}$ g.s. in ${}^{24}\mathrm{Al}$. The GTSD includes matrix elements for $3^{+}, 4^{+}$, and $5^{+}$ states in ${}^{24}\mathrm{Mg}$. The DFT-NCCI result is compared to experimental data and the shell-model calculations of Ref. [49]. Curves are smoothed with a Lorentzian function with a half-width of $\mathrm{\Gamma }=0.5$ MeV.

Figure 13

The content of the wave function of first $4^{+}$ Gamow-Teller resonance (red) and of the second GTR peak in terms of HF configurations listed in Table 3.

Figure 14

GTMEs connecting the $4^{+}$ g.s. in ${}^{24}\mathrm{Al}$ with the $|2025/2\rangle$ resonant peak (solid line) and the $|2111/2\rangle$ secondary peak (dotted line). The calculations were done using the ${\mathrm{SV}}_{\mathrm{T}}$ parametrization with spin-orbit strength increased by $10\%$ (a), $20\%$ (b), $30\%$ (c), and $40\%$ (d) with respect to the original value. See text for further details.

Figure 15

Low-lying states in ${}^{100}\mathrm{In}$ calculated by using the DFT-NCCI (middle) and LSSM (right). The left part shows the experimental (dotted line) and DFT-NCCI (solid line) binding energies in ${}^{100}\mathrm{Sn}$ relative to the ground state energy in ${}^{100}\mathrm{In}$. See text for details.