Quantified Gamow shell model interaction for $psd$-shell nuclei

Background: The structure of weakly bound and unbound nuclei close to particle drip lines is one of the major science drivers of nuclear physics. A comprehensive understanding of these systems goes beyond the traditional configuration interaction approach formulated in the Hilbert space of localized states (nuclear shell model) and requires an open quantum system description. The complex-energy Gamow shell model (GSM) provides such a framework as it is capable of describing resonant and nonresonant many-body states on equal footing.

Purpose: To make reliable predictions, quality input is needed that allows for the full uncertainty quantification of theoretical results. In this study, we carry out the optimization of an effective GSM (one-body and two-body) interaction in the $psdf$-shell-model space. The resulting interaction is expected to describe nuclei with $5\le A\lesssim 12$ at the $p-sd$-shell interface.

Method: The one-body potential of the ${}^4\mathrm{He}$ core is modeled by a Woods-Saxon + spin-orbit + Coulomb potential, and the finite-range nucleon-nucleon interaction between the valence nucleons consists of central, spin-orbit, tensor, and Coulomb terms. The GSM is used to compute key fit observables. The ${\chi }^2$ optimization is performed using the Gauss-Newton algorithm augmented by the singular value decomposition technique. The resulting covariance matrix enables quantification of statistical errors within the linear regression approach.

Results: The optimized one-body potential reproduces nucleon-${}^4\mathrm{He}$ scattering phase shifts up to an excitation energy of 20 MeV. The two-body interaction built on top of the optimized one-body field is adjusted to the bound and unbound ground-state binding energies and selected excited states of the helium, lithium, and beryllium isotopes up to $A=9$. A very good agreement with experimental results was obtained for binding energies. First applications of the optimized interaction include predictions for two-nucleon correlation densities and excitation spectra of light nuclei with quantified uncertainties.

Conclusion: The new interaction will enable comprehensive and fully quantified studies of structure and reactions aspects of nuclei from the $psd$ region of the nuclear chart.

Figure 1

Optimized $s_{1/2}$ (top), $p_{3/2}$ (middle), and $p_{1/2}$ (bottom) neutron-${}^4\mathrm{He}$ nuclear phase shifts as functions of the energy of the neutron in the laboratory obtained using the Woods-Saxon parameters given in Table 2. The experimental values represented by crosses are taken from Ref. [64].

Figure 2

Similar as in Fig. 1 but for the proton-${}^4\mathrm{He}$ scattering. The experimental data from 0 to 3 MeV are taken from Ref. [65] and the data from 3 to 20 MeV from Ref. [66].

Figure 3

Energies of the helium, lithium, and beryllium isotopic chains with $A\le 9$ computed using the optimized GSM interaction with parameters given in Tables 2 and 6. The experimental values shown by stars are taken from Ref. [69]. The widths of unbound states are listed in Table 5.

Figure 4

Two-nucleon correlation densities (in ${\mathrm{fm}}^{-2}$) calculated for states in ${}^6\mathrm{He}$ and ${}^6\mathrm{Li}$ using the optimized GSM interaction. For the $2^{+}$ state in ${}^6\mathrm{He}$, the density was multiplied by a factor of two to maintain the same scale as in other panels.

Figure 5

Spectra of ${}^7\mathrm{He}, {}^7\mathrm{Be}$, and ${}^7\mathrm{B}$. The experimental values are taken from Ref. [69]. The g.s. energies of ${}^7\mathrm{He}$ and ${}^7\mathrm{Be}$ were included in the optimization and hence are shown without uncertainties. The uncertainties on the widths, not shown in the figure, are listed in Table 9.

Figure 6

Spectra of ${}^8\mathrm{He}$ and ${}^9\mathrm{He}$. The experimental values come from Refs. [69] (${}^8\mathrm{He}$) and [82] (${}^9\mathrm{He}$). The energy of the $2^{+}$ of ${}^8\mathrm{He}$ is unresolved experimentally and the two presently adopted values are shown. The ground state of ${}^8\mathrm{He}$ was included in the optimization and is shown without uncertainties. The uncertainties on the widths, not shown in the figure, are listed in Table 9.